what is z a n u p f
the ruling party in Zimbabwe
2253141
ZANU–PF The Zimbabwe African National Union – Patriotic Front (ZANU–PF) has been the ruling party in Zimbabwe since independence in 1980. The party was led for many years under Robert Mugabe, first as Prime Minister with the Zimbabwe African National Union (ZANU) and then as President from 1987 after the merger with the Zimbabwe African People's Union (ZAPU) and retaining the name ZANU–PF, until 2017 when he was removed as leader. At the 2008 parliamentary election, the ZANU–PF lost sole control of parliament for the first time in party history and brokered a difficult power-sharing deal with the Movement for
ZANU–PF
1843895
and "Y" is locally compact and Hausdorff and "p" : "X" → "Y" is a proper local homeomorphism, then "p" is a covering map. It is shown in complex analysis that a complex analytic function "f" : "U" → C (where "U" is an open subset of the complex plane C) is a local homeomorphism precisely when the derivative "f" ′("z") is non-zero for all "z" ϵ "U". The function "f"("z") = "z" on an open disk around 0 is not a local homeomorphism at 0 when "n" is at least 2. In that case 0 is a point of
Local homeomorphism
5888451
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
Udi language
16075990
read in Zo ccaw( Zo literature) today. Historical records are needed to confirm the identity of these people groups. A AW B C CC (cc sounds like 'th' in English) D E F G H I J K L M N O P R S T U V Y Z a aw b c cc d e f g h I j k l m n o p r s t u v y z Vowels in Zo Ccaw A AW E I O U Y Zotung (Chin) people are heavily affected by natural disaster that hits the country in
Zotung
5817080
1930: A a, B в, C c, D d, Ə ə, F f, G g, Ƣ ƣ, I i, J j, K k, Q q, M m, N n, N̡ n̡, O o, Ө ө, P p, R r, S s, T t, U u, V v, Ş ş, Z z, Ƶ ƶ, L l, Ь ь, Y y, Į į. The order of the letters was later changed to correspond with alphabets for other languages in the Soviet Union, the letter Ә ә was replaced with E e, and the letter Į į was dropped. In 1938 the Latin
Shor language
12229746
B • C • D • E • F • G • H • I • J • K • L • M • N • O • P • Q • R • S • T • U • V • W • X • Y • Z • see also "Jump to:" 0-9 • A • B • C • D • E • F • G • H • I • J • K • L • M • N • O • P • Q • R • S • T • U • V •
Engineering drawing abbreviations and symbols
17237477
as well as position coordinates x = ("x", "y", "z") and time "t" in frame F are transformed to "E"′ = "m"′"c", p′ = ("p"′, "p"′, "p"′), x′ = ("x"′, "y"′, "z"′), and "t"′ in F′ according to the Lorentz transformations The Lorentz factor here applies to the velocity v, the relative velocity between the frames. This is not necessarily the same as the velocity u of an object. For the orbital 3-angular momentum L as a pseudovector, we have In the second terms of "L"′ and "L"′, the "y" and "z" components of the cross product v×N can be
Relativistic angular momentum
11489407
K, L, M, N, NG, O, Ô, P, R, S, T, U, V, Z Mara Alphabet (small letters):a, aw, y, b, ch, d, e, f, h, i, k, l, m, n, ng, o, ô, p, r, s, t, u, v, z Mara diphthongs:ao, yu, ai, ei, ia, ie, ua The plural form of a noun is formed by affixing one of the following terms to the end of the noun: Words inside bracket were how a foreign author N.E. Parry (1937) wrote according to his understanding of the sound. But now the Maras have their own alphabet and the correct
Mara language
9377041
i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z (which also have capital forms: A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z). The spelling system, or orthography, of English is multi-layered, with elements of French, Latin, and Greek spelling on top of the native Germanic system. Further complications have arisen through sound changes with which the orthography has not kept pace. Compared to European languages for which official organisations have promoted spelling reforms,
English language
2705108
K L M N Ꞑ O Ɵ P R S Ş T U V X Y Z Ƶ Ь a ʙ c d e f g ƣ i j k l m n ꞑ o ө p r s ş t u v x y z ƶ ь Examples By September 1943, this Latin-based alphabet was replaced by a Cyrillic-based one, which is still in use to the present day. In the post-Soviet era, Tuvan and other scholars have taken a renewed interest in the history of Tuvan letters. There is no official transliteration standard for transforming the Cyrillic-based Tuvan
Tuvan language
1066450
matrices with the commutator serving as the Lie bracket. Unlike the real case, is connected. This follows, in part, since the multiplicative group of complex numbers C is connected. The group manifold is not compact; rather its maximal compact subgroup is the unitary group U("n"). As for U("n"), the group manifold is not simply connected but has a fundamental group isomorphic to Z. If "F" is a finite field with "q" elements, then we sometimes write instead of . When "p" is prime, is the outer automorphism group of the group Z, and also the automorphism group, because Z is
General linear group
525166
there is a "u" in U such that "ua"="c" and "ub"="d". This relation is in fact an equivalence relation. The points of the projective line over D are equivalence classes in B under this relation: P(D) = B/ ~. Consider the embedding D → P(D) by "z" → U("z",1) where U("z",1) is the equivalence class of ("z",1). Then points U(1,"n"), "n" = 0, are in P(D) but are not the image of any point under the embedding. P(D) is projected onto a cylinder by projection: Take a cylinder tangent to the double number plane on the line {"y" ε: "y"
Dual number
326365
but this scheme has the advantage of using 12 consecutive letters of the alphabet. It is used by convention in discussing Conway's Game of Life, where, for example, one speaks of the R-pentomino instead of the F-pentomino. The F, L, N, P, Y, and Z pentominoes are chiral; adding their reflections (F′, L′, N′, Q, Y′, Z′) brings the number of "one-sided" pentominoes to 18. If rotations are also considered distinct, then the pentominoes from the first category count eightfold, the ones from the next three categories (T, U, V, W, Z) count fourfold, I counts twice, and X counts
Pentomino
13599012
set is "U" ∪ "V" where "U" ∩ "V" = ∅. Let "N"("v") denote the neighborhood of a vertex "v" ∈ "V". The graph "G" is convex over "U" if and only if there exists a bijective mapping, "f": "U" → {1, …, |"U"|}, such that for all "v" ∈ "V", for any two vertices "x","y" ∈ "N"("v") ⊆ "U" there does not exist a "z" ∉ "N"("v") such that "f"("x") < "f"("z") < "f"("y"). Convex bipartite graph In the mathematical field of graph theory, a convex bipartite graph is a bipartite graph with specific properties. A bipartite graph, ("U"
Convex bipartite graph
3550184
"U" in Sp("N") is conjugate to a block diagonal matrix with entries where |"z"| = 1. Thus the eigenvalues of "U" are ("z"). The irreducible representations of Sp("N") are labelled by signatures where the "f" are integers. The character of the corresponding irreducible representation "σ" is given by The branching rule from Sp("N") to Sp("N" – 1) states that Here "f" = 0 and the multiplicity "m"(f, g) is given by where is the non-increasing rearrangement of the 2"N" non-negative integers ("f"), ("g") and 0. Example. The branching from U(2"N") to Sp("N") relies on two identities of Littlewood: where Π
Restricted representation
2732647
is said to be "ramified" at a point "P" in "S"′ if there exist analytic coordinates near "P" and π("P") such that π takes the form π("z") = "z", and "n" > 1. An equivalent way of thinking about this is that there exists a small neighborhood "U" of "P" such that π("P") has exactly one preimage in "U", but the image of any other point in "U" has exactly "n" preimages in "U". The number "n" is called the "ramification index at P" and also denoted by "e". In calculating the Euler characteristic of "S"′ we notice the loss
Riemann–Hurwitz formula
30651
c'4 c' g' g' | a' a' g'2 | f'4 f' e' e' | d' d' c'2 | g'4 g' f' f' | e' e' d'2 | g'4 \times 2/3 { f'8 f' f' } e'4 d' | c' r r2 | \bar "|." } \addlyrics { A B C D E F G H I J K L M N O P Q R S T U V dub- a- U X Y "Z(ed)" }</score> Another alternate Zed version: Note that the third line is lengthened and the fourth line is shortened, to compensate for the Dutch pronunciations. A
Alphabet song
3083715
without the ascenders/descenders] Y, y = j [vertically mirrored Г] = ġ З, з [a clockwise-turned U] = ħ I, i = i J, j = j K, k = k [I with a small c superimposed on it] L, l = l M, m = m N, n = n O, o = o P, p = p R, r = r S, s = s Ɯ, ɯ = x V, v = v U, u = u W, w = w Z, z = z Ʒ, ʒ = ż Æ, æ = final e Five grave accented vowels
Maltese alphabet
2877249
elements of the form "f"("z")= "z" + "bz" + ... Composing this with translations yields "T" ∘ "f" ∘ "T"("z") = "cz" + "dz" + ... with "c", "d" ≠ 0. Unless "n" = 2, this contradicts the form of subgroup "A"; so "n" = 2. The Schwarzian derivative is related to the pseudogroup for the complex Möbius group. In fact if "f" is a biholomorphism defined on "V" then ("f") = "S"("f") is a quadratic differential on "V". If "g" is a bihomolorphism defined on "U" and "g"("V") ⊆ "U", "S"("f" ∘ "g") and "S"("g") are quadratic differentials on
Schwarzian derivative
15003454
K, Mungan Z. Definition of C282Y mutation in a hereditary hemochromatosis family from Turkey. Turk J Gastroenterol. 18:53-57 (2007) 33. Deglincerti A, De Giorgio R, Cefle K, Devoto M, Pippucci T, Castegnaro G, Panza E, Barbara G, Cogliandro RF, Mungan Z, Palanduz S, Corinaldesi R, Romeo G, Seri M, Stanghellini V. A novel locus for syndromic chronic idiopathic intestinal pseudo-obstruction maps to chromosome 8q23-q24. Eur J Hum Genet. 15:889-97 (2007) 34. Demir K, Akyuz F, Ozdil S, Aksoy N, Kaymakoglu S, Poturoglu S, Akyüz U, Besisik F, Boztas G, Mungan Z, Cevikbas U, Cakaloglu Y, Okten A. What is the
Zeynel Mungan
627392
the number of different generators is "φ"("n"), where "φ" is the Euler totient function, the function that counts the number of numbers modulo "n" that are relatively prime to "n". Every finite cyclic group is isomorphic to a group Z/("n"), where "n" is the order of the group. The integer and modular addition operations, used to define the cyclic groups, are both the addition operations of commutative rings, also denoted Z and Z/("n"). If "p" is a prime, then Z/("p") is a finite field, and is usually instead written as F or GF("p"). Every field with "p" elements is isomorphic
Cyclic group
1894014
to include more than just profit, such as overhead, it can be included as such: or P = (1+μ) W. Where μ is the markup over costs. This is the pricing equation. W = F(u,z) Pe . This is the wage setting relation. u is unemployment which negatively affects wages and z the catch all variable positively affects wages. P = Pe(1+μ) F(u,z). This is the aggregate supply curve. Where the price is determined by expected price, unemployment and z the catch all variable. Markup (business) Markup is the ratio between the cost of a good or service and its
Markup (business)
2318241
series formula_1 such that not all formula_2 are in the maximal ideal formula_3 of "A", there is a unique unit "u" in formula_4 and a polynomial "F" of the form formula_5 with formula_6 (a so-called distinguished polynomial) such that Since formula_4 is again a complete local ring, the result can be iterated and therefore gives similar factorization results for formal power series in several variables. For example, this applies to the ring of integers in a p-adic field. In this case the theorem says that a power series "f"("z") can always be uniquely factored as π·"u"("z")·"p"("z"), where "u"("z") is a
Weierstrass preparation theorem
9917892
each person writes in his or her own dialect, sociolect, and idiolect. The descriptive texts concerning the ALUPEC claim that it is “a system composed by 23 letters and four digraphs”. What those texts do not specify is that the ALUPEC also includes the letter "Y" and the digraph "RR". Older documents, such as the 1994 "Proposed Criteria of the Unified Alphabet for the Cape Verdean Writing System," showed the following order: A B S D E F G H I J DJ L LH M N NH N̈ O P K R T U V X TX Z Later
ALUPEC
3654872
this theorem. Recently, Yabuta (2001) gave a simple proof. Given two coprime integers "P" and "Q", such that formula_1 and , let be the Lucas sequence of the first kind defined by Then, for "n" ≠ 1, 2, 6, "U"("P","Q") has at least one prime divisor that does not divide any "U"("P","Q") with "m" < "n", except "U"(1, -1)=F(12)=144, "U"(-1, -1)=-F(12)=-144. Such a prime "p" is called a "characteristic factor" or a "primitive prime divisor" of "U"("P","Q"). Indeed, Carmichael showed a slightly stronger theorem: For "n" ≠ 1, 2, 6, "U"("P","Q") has at least one primitive prime divisor not dividing
Carmichael's theorem
4273626
"f" : "M" → "N" such that the pushforward of linear frames "f" : "FM" → "FN" restricts to give a mapping of "P" into "Q". (Note that it is sufficient that "Q" be contained within the image of "f".) The "G"-structures "P" and "Q" are locally isomorphic if "M" admits a covering by open sets "U" and a family of diffeomorphisms "f" : "U" → "f"("U") ⊂ "N" such that "f" induces an isomorphism of "P"| → "Q"|. An automorphism of a "G"-structure is an isomorphism of a "G"-structure "P" with itself. Automorphisms arise frequently in the study of
G-structure on a manifold
2104460
function "F" ∈ "H" can be reconstructed from the real distribution Re("f") on the circle, because the Taylor coefficients "c" of "F" can be computed from the Fourier coefficients of Re("f"). Distributions on the circle are general enough for handling Hardy spaces when "p" < 1. Distributions that are not functions do occur, as is seen with functions "F"("z") = (1−"z") (for |"z"| < 1), that belong to "H" when 0 < "N" "p" < 1 (and "N" an integer ≥ 1). A real distribution on the circle belongs to real-"H"(T) iff it is the boundary value of the real
Hardy space
10276541
[0, +∞]) for each "i" = 1, ..., "m". Then The geometric Brascamp–Lieb inequality follows from the Brascamp–Lieb inequality as stated above by taking "n" = 1 and "B"("x") = "x" · "u". Then, for "z" ∈ R, It follows that "D" = 1 in this case. As another special case, take "n" = "n", "B" = id, the identity map on formula_1, replacing "f" by "f", and let "c" = 1 / "p" for 1 ≤ "i" ≤ "m". Then and the log-concavity of the determinant of a positive definite matrix implies that "D" = 1. This yields Hölder's
Brascamp–Lieb inequality
16877408
correct for chance similarity. The ASJP uses the following 40-word list. It is similar to the Swadesh–Yakhontov list, but has some differences. ASJP version from 2016 uses the following symbols to encode phonemes: p b f v m w 8 t d s z c n r l S Z C j T 5 y k g x N q X h 7 L 4 G ! i e E 3 a u o They represent 7 vowels and 34 consonants, all found on the standard QWERTY keyboard. Automated Similarity Judgment Program The Automated Similarity Judgment Program (ASJP) is a
Automated Similarity Judgment Program
12749307
E, F, Q, and X are not used except in loanwords): A B D G H I K L M N O P R S T U V W Y Z These characters together are called "Pimato". Semivowels /w j/ and rhotic /r/ only occur in most of the other dialects of the Dusun/Kadazan languages. Forschner (1978) and Antonissen (1958) list two fricatives /v z/ for the Rungus and Penampang Kadazan dialects. [x] is also listed as an allophone of /k/ in word-medial position. The vowels are divided into: Simple vowels: /i u a o/ Diphthongs: "aa ai" (sometimes pronounced
Dusun language
16001017
Vowels (Vawella̱ Sha̱kuup) a a̱ e e̱ i i̱ o u Long Vowels aa ee ii oo uu (Khwikhwir Vawel) ai au a̱u ou CONSONANTS b c d dh dz f g gy gb gh ghy h hw hwy hyw j k ky kh khy kp l m n nh ng ny p r s sh t th thn ts v w y yw z zh Hayab (2016:66) pinpoints, stating that "available data in Hyam by Koelle (1854: appendix, 2-188) and Meek (1931:120) reveals that the Ham contest in virtually a dissimilar way from what is obtained today." He added
Hyam language
3697134
š, t, u (ú), ū (ū́), ʏ (ʏ́), v, w (u̯), x, x°, y, z, ž, ع TITUS transcribes the alphabet as such: a (á), b, č, d, e (é), ĕ (ĕ́), ẹ (ẹ́), ẹ̆ (ẹ̆́), ə (ə́), f, g, ɣ, h, x̣, i (í), ĭ (ĭ́), ī (ī́), ǰ, k, q, l, m (m̃), n (ñ), o (ó), ọ (ọ́), p, r, s, š, t, u (ú), ŭ (ŭ́), ı̥ (í̥), v, u̯, x, x°, y, z, ž, ع In recent times, Sayfiddīn Mīrzozoda, from the Tajik Academy of Sciences, uses a modified Tajik alphabet for writing Yaghnobi. The
Yaghnobi language
9569517
I: Intermediate J: Intermediate Elite S: Standard R: Standard Elite F: Fullsize G: Fullsize Elite P: Premium U: Premium Elite L: Luxury W: Luxury Elite O: Oversize X: Special Type B: 2-3 Door C: 2/4 Door D: 4-5 Door W: Wagon/Estate V: Passenger Van L: Limousine S: Sport T: Convertible F: SUV J: Open Air All Terrain X: Special P: Pick up Regular Cab Q: Pick up Extended Cab Z: Special Offer Car E: Coupe M: Monospace R: Recreational Vehicle H: Motor Home Y: 2 Wheel Vehicle N: Roadster G: Crossover K: Commercial Van/Truck M: Manual (drive unspecified) N: Manual
Association of Car Rental Industry Systems Standards
3958667
is represented by aa. All long vowels are written this way. All alveolars () are dental-alveolars. They are represented in the orthography by t, d, n and nd. The sound is represented by y. The complete orthography of Mbula is as follows: a e i o u b d g k l m mb n nd ŋ ŋg p r s t w z y A E I O U B D G K L M Mb N Nd ¿ ¿g P R S T W Z Y In an ideal grammar each classificatory word type would belong only to
Mbula language
10360325
interest in the subject. A function "f"("z") is represented by a formal power series: where "c" ≠ 0, by convention. The ("m", "n")th entry "R" in the Padé table for "f"("z") is then given by where "P"("z") and "Q"("z") are polynomials of degrees not more than "m" and "n", respectively. The coefficients {"a"} and {"b"} can always be found by considering the expression and equating coefficients of like powers of "z" up through "m" + "n". For the coefficients of powers "m" + 1 to "m" + "n", the right hand side is 0 and the resulting system of linear
Padé table
1291052
non-zero element "a" of "F" can be written as "a" = ϖ"u" with "u" a unit, and "n" a unique integer. The normalized valuation of "F" is the surjective function "v" : "F" → Z ∪ {∞} defined by sending a non-zero "a" to the unique integer "n" such that "a" = ϖ"u" with "u" a unit, and by sending 0 to ∞. If "q" is the cardinality of the residue field, the absolute value on "F" induced by its structure as a local field is given by An equivalent and very important definition of a non-Archimedean local field is
Local field
5742546
one of the 19 letters used (A, Z, Y, X, U, T, S, R, P, M, L, K, J, H, G, E, D, C, B), with "A" corresponding to a remainder of 0, "Z" corresponding to 1, "Y" corresponding to 2 and so on. In the case of SBS 3229, the final letter should be a P; for E 23, the final letter should be a H. Note that letters F, I, N, O, Q, V and W are not used as checksum letters. Online Checksum Online calculator for the checksum and Singapore Online Vehicle Checksum Checksum suffix letters are
Vehicle registration plates of Singapore
619220
theorem for real quadratic polynomials. In other words, algebraic proofs of the fundamental theorem actually show that if "R" is any real-closed field, then its extension "C" = "R"() is algebraically closed. As mentioned above, it suffices to check the statement "every non-constant polynomial "p"("z") with real coefficients has a complex root". This statement can be proved by induction on the greatest non-negative integer "k" such that 2 divides the degree "n" of "p"("z"). Let "a" be the coefficient of "z" in "p"("z") and let "F" be a splitting field of "p"("z") over "C"; in other words, the field "F"
Fundamental theorem of algebra
4048584
P R S T U V Z X Y W. After that, it was 00000A-99999A, 00000D-99999D etc. Possible letters were, in this order, A D E F G H L M N P R S T V W X Y Z; then, the letter was moved to the second position, and then to third (same range as in second position). Schematic representation: Front Plate 1927-1932 rear plate The front plate was kept intact as in the 1927-1976 period. The rear plate, instead, began tobbe manufactured in two pices. One, sized 10,7 × 33 cm, had black background with white digits,
Vehicle registration plates of Italy
661211
Conversely, any harmonic function "u" on an open subset Ω of R is "locally" the real part of a holomorphic function. This is immediately seen observing that, writing "z" = "x" + "iy", the complex function "g"("z") := "u" − i "u" is holomorphic in Ω because it satisfies the Cauchy–Riemann equations. Therefore, "g" has locally a primitive "f", and "u" is the real part of "f" up to a constant, as "u" is the real part of formula_13 . Although the above correspondence with holomorphic functions only holds for functions of two real variables, harmonic functions in "n" variables
Harmonic function
15670156
to Intal or to unify. Intal uses the Latin script without special signs. The Intal alphabet has 23 letters, 5 vowels (a, e, i, o, u) and 18 consonants (b, c, d, f, g, h, j, k, l, m, n, p, r, s, t, v, y, z). The consonants q, w and x are only used in proper names and foreign words. "c" is pronounced /ʃ/ (like English "sh"), "g" always hard (as in German), "j" as in French, and "y" and "z" as in English. The stress of a word is on the vowel before the last consonant, unless
Intal language
1680504
was added to the alphabet in the 19th century by Otto Wilhelm Masing and stands for the vowel . In addition, the alphabet also differs from the Latin alphabet by the addition of the letters Š and Ž (S and Z with caron/háček), and by the position of Z in the alphabet: it has been moved from the end to between S and T (or Š and Ž). The official Estonian alphabet has 27 letters: A, B, D, E, F, G, H, I, J, K, L, M, N, O, P, R, S, Š, Z, Ž, T, U, V, Õ, Ä,
Estonian orthography
5332614
ĉ, d, e, ê, f, g, ĝ, h, ĥ, i, ĭ, ĭ:, ĭ́, î, î:, î́, j, ĵ, k, k̆, l, m, m̆, n, n̆, o, ô, p, pʷ (actually, an overstruck ʷ), s, ŝ, t, t̂, ť, u, û, ŭ, v, z, ẑ, z̆, q, q́, q̂, q̂́, w, ẃ, ŵ, ŵ́, x, x́, y, 2, 2́. The sound values of these are not well explained. It appears that the vowels a, e, i, o, u are as in Esperanto. A circumflex vowel is nasal. Ⱥ is as in English 'ash', and ⱥ̂ is nasal ⱥ. Ĭ is as
Poliespo
2877229
order ordinary differential equation There are "n"−3 linearly independent accessory parameters, which can be difficult to determine in practise. For a triangle, when "n" = 3, there are no accessory parameters. The ordinary differential equation is equivalent to the hypergeometric differential equation and "f"("z") can be written in terms of hypergeometric functions. For a quadrilateral the accessory parameters depend on one independent variable λ. Writing "U"("z") = "q"("z")"u"("z") for a suitable choice of "q"("z"), the ordinary differential equation takes the form Thus formula_35 are eigenfunctions of a Sturm-Liouville equation on the interval formula_36. By the Sturm separation theorem, the non-vanishing
Schwarzian derivative
8522215
of transform kernels if, for any suitable function "f"("z") or any suitable function "g"("z"), the following two relationships hold simultaneously: The pair of kernels is said to be symmetric if "k"("z","y") = "h"("z","y"). showed that the functions: are an asymmetric pair of transform kernels, where "γ" > 0, "n" − "p" = "m" − "q" > 0, and: along with further convergence conditions. In particular, if "p" = "q", "m" = "n", "a" + "b" = 0 for "j" = 1, 2, ..., "p" and "c" + "d" = 0 for "j" = 1, 2, ..., "m", then the pair
Meijer G-function
9608124
N' "K". In symbols, if then Ran("F' ") = [N' "K"]. That [N' "K"] ⊂ Ran("F' ") follows from the definition of commutant. On the other hand, [N' "K"] is invariant under every unitary "U" in N' . Therefore the projection onto [N' "K"] lies in (N')' = N. Minimality of "F' " then yields Ran("F' ") ⊂ [N' "K"]. Now if "E" is a projection in M, applying the above to the von Neumann algebra "Z"(M) gives One can deduce some simple consequences from the above description. Suppose "E" and "F" are projections in a von Neumann algebra M.
Central carrier
107834
to "N". They are "C" diffeomorphic if there is an "r" times continuously differentiable bijective map between them whose inverse is also "r" times continuously differentiable. Given a subset "X" of a manifold "M" and a subset "Y" of a manifold "N", a function "f" : "X" → "Y" is said to be smooth if for all "p" in "X" there is a neighborhood "U" ⊂ "M" of "p" and a smooth function "g" : "U" → "N" such that the restrictions agree formula_1 (note that "g" is an extension of "f"). "f" is said to be a diffeomorphism if
Diffeomorphism
6343184
points on "C" over "F", at least when "q" = "p". Because of Hasse's theorem on elliptic curves, knowing "N" modulo "p" determines "N" for "p" ≥ 5. This connection with local zeta-functions has been investigated in depth. For a plane curve defined by a cubic "f"("X","Y","Z") = 0, the Hasse invariant is zero if and only if the coefficient of ("XYZ") in "f" is zero. Hasse–Witt matrix In mathematics, the Hasse–Witt matrix "H" of a non-singular algebraic curve "C" over a finite field "F" is the matrix of the Frobenius mapping ("p"-th power mapping where "F" has "q" elements,
Hasse–Witt matrix
8647805
singular points on "C" (see Plücker formula). The "p-th" polar of a "C" for a natural number "p" is defined as Δ"f"("x, y, z") = 0. This is a curve of degree "n"−"p". When "p" is "n"−1 the "p"-th polar is a line called the "polar line" of "C" with respect to "Q". Similarly, when "p" is "n"−2 the curve is called the "polar conic" of "C". Using Taylor series in several variables and exploiting homogeneity, "f"(λ"a"+μ"p", λ"b"+μ"q", λ"c"+μ"r") can be expanded in two ways as and Comparing coefficients of λμ shows that In particular, the "p"-th polar of "C"
Polar curve
6144347
addition, U("n") acts freely on EU("n"). The spaces "F"(C) and "G"(C) are CW-complexes. One can find a decomposition of these spaces into CW-complexes such that the decomposition of "F"(C), resp. "G"(C), is induced by restriction of the one for "F"(C), resp. "G"(C). Thus EU("n") (and also "G"(C)) is a CW-complex. By Whitehead Theorem and the above Lemma, EU("n") is contractible. Proposition: The cohomology of the classifying space "H*"(BU("n")) is a ring of polynomials in "n" variables "c", ..., "c" where "c" is of degree 2"p". Proof: Let us first consider the case "n" = 1. In this case, U(1) is
Classifying space for U(n)
333147
not isomorphic. The Sylow subgroups of general linear groups are another fundamental family of examples. Let "V" be a vector space of dimension "n" with basis { "e", "e", …, "e" } and define "V" to be the vector space generated by { "e", "e", …, "e" } for 1 ≤ "i" ≤ "n", and define "V" = 0 when "i" > "n". For each 1 ≤ "m" ≤ "n", the set of invertible linear transformations of "V" which take each "V" to "V" form a subgroup of Aut("V") denoted "U". If "V" is a vector space over Z/"p"Z, then
P-group
7653444
"U" such that "e" = "f"("z") for all "z" in "U". Such a function "g" is necessarily holomorphic with "g′"("z") = "f′"("z")/"f"("z") for all "z" in "U". If "U" is a simply connected open subset of formula_16, and "f" is a nowhere-vanishing holomorphic function on "U", then a branch of log "f" defined on "U" can be constructed by choosing a starting point "a" in "U", choosing a logarithm "b" of "f"("a"), and defining for each "z" in "U". Complex logarithm In complex analysis, a complex logarithm of the non-zero complex number , denoted by , is defined to be
Complex logarithm
3204815
for official acceptance) is as follows: Zamanälif contains 35 letters. There are 10 vowels and 25 consonants. There are 10 extra letters: Çç, Ğğ, Şş, Ññ, Ää, Öö, Üü, Iı, İi and Íí. The other letters are the same in both alphabets, but they are pronounced differently. A, Ä, B, C, Ç, D, E, F, G, Ğ, H, I, İ, Í, J, K, L, M, N, Ñ, O, Ö, P, Q, R, S, Ş, T, U, Ü, V, W, X, Y, Z. Tatar vowels are: a/ä, o/ö, u/ü, í/i, ı/e. The symbol is used for the glottal stop (known as
Tatar alphabet
14559782
" n u l l " ] , " n a m 0000240 e " : " f a v o r i t e _ n u 0000260 m b e r " } , { " t y p e " : 0000300 [ " s t r i n g " , " n u l 0000320 l " ] , " n a m e " : " f a 0000360 024 a v r o . c o d e c \b n u l l 0000400 \0 211 266 / 030 334 ˪ **
Apache Avro
1623652
final, stretched product. The letters are not made in order of appearance in the name (B, L, A, C, K, P, O, O, L) but by their shape; "square" letters, (B, E, F, H, J, K, L, M, N, P, R, T, W, X, Y and Z), are made first, as they will not lose their shape, while "triangle" (A and V) and "round" (C, D, G, O, Q, S, U) letters are made last to prevent them from losing their shape, as the toffee is still reasonably soft at this point. For example, the letters that make up "BLACKPOOL
Rock (confectionery)
9923480
"F". Caveat: this notation clashes somewhat with the "n" of Dynkin diagrams, which is the rank. The general linear group GL("R") is the group of all "R"-linear automorphisms of "R". There is a subgroup: the special linear group SL("R"), and their quotients: the projective general linear group PGL("R") = GL("R")/Z(GL("R")) and the projective special linear group PSL("R") = SL("R")/Z(SL("R")). The projective special linear group PSL("F") over a field "F" is simple for "n" ≥ 2, except for the two cases when "n" = 2 and the field has order 2 or 3. The unitary group U("R") is a group preserving
Classical group
14081280
prime ideal "P" of "R" such that formula_7 is minimal over formula_8) and the "incomparability" property (If "P" and "Q" are prime ideals of "R" and formula_9 then formula_10) are satisfied. If "F" := Z is the free algebra in "N" variables and "R" is a PI-ring satisfying the polynomial "P" in "N" variables, then "P" is in the [[Kernel (algebra)|kernel]] of any homomorphism An ideal "I" of "F" is called T-ideal if formula_13 for every [[endomorphism]] "f" of "F". Given a PI-ring, "R", the set of all polynomial identities it satisfies is an [[Ideal (ring theory)|ideal]] but even more
Polynomial identity ring
9943508
"W" = "W"/"p""W" of Witt vectors of length "n" over a perfect field "k" of characteristic "p">0. For example, "k" could be the finite field of order "p", and "W" is then the ring Z/"p"Z. (More generally one can work over a base scheme "S" which has a fixed sheaf of ideals "I" with a divided power structure.) If "X" is a scheme over "k", then the crystalline site of "X" relative to "W", denoted Cris("X"/"W"), has as its objects pairs "U"→"T" consisting of a closed immersion of a Zariski open subset "U" of "X" into some "W"-scheme "T" defined
Crystalline cohomology
10440147
ïlwb Avi wnÂv kvK gv‡S gv‡S †`Lv †g‡j ‡bD‡j ,KvwQg, ¸uBmvc| wfb‡`‡ki H cvnvo †hb AšÍwiÿ †gvov †gNvj‡qi wPivcyÄx †`q Zv cvnviv| el©v G‡jB cvnvo †hb RxqbKvwVi e‡j AšÍwiZ cvnvo †e‡q SYv©aviv bv‡g| mybvgM‡Äi ag©cvkvq Rj`vb‡ei †Zvo Zvi Zjv‡ZB e‡q P‡j Uv½yqvi nvIo| `yB cv‡k Zvi QqPwjøk MuvI, gv‡S gv‡S RjUz½xi Ni ‡mBLv‡b‡Z kvOv, wS‡q _v‡K Rxebfi| mvwi mvwi e„ÿ †hb e„ÿ‡`‡ei gvqv wb_i R‡j Aciƒc KiP Mv‡Qi Qvqv| Mjv R‡j `uvov‡q eiæY, mv‡_ j‡q wbPzj MÖx‡®§i Zvc`‡n, †`Lvq Zv‡`i mg~j| weR‡b †duv‡U cy¯ú, wnR‡ji kv‡L cÖfv‡Z S‡i c‡o, ‡f‡m hvq †¯ªv‡Z| cv‡o duy‡U ey‡bv †Mvjvc, SY©v, Puv`gvjv ‡cŠl dz‡ji
Tanguar haor
3550187
representations. In fact if an orthogonal matrix "U" has eigenvalues "z" for 1 ≤ "i" ≤ "n", then the character of the corresponding irreducible representation is given by for "N" = 2"n" and by for "N" = 2"n"+1. The branching rules from SO("N") to SO("N" – 1) state that for "N" = 2"n" + 1 and for "N" = 2"n", where the differences "f" − "g" must be integers. Since the branching rules from "U"("N") to U("N" – 1) or SO("N") to SO("N" – 1) have multiplicity one, the irreducible summands corresponding to smaller and smaller "N" will eventually terminate
Restricted representation
7200188
identified by complete mt genome have been identified in southern, northern and western Australia. P haplogroup: P4b (now called P11), further diversified by into several distinct but culturally connected family lineages (van Holst Pellekaan 2011) and 2013. References: F, C, W, M, D, N, K, U, T, A, B, C, Z, U many number variants to each section Human mitochondrial DNA haplogroup In human genetics, a human mitochondrial DNA haplogroup is a haplogroup defined by differences in human mitochondrial DNA. Haplogroups are used to represent the major branch points on the mitochondrial phylogenetic tree. Understanding the evolutionary path of the
Human mitochondrial DNA haplogroup
12054889
position, and while the long-short distinction described above is maintained, the "r" has its own effect on the vowel, as in Elara (a long but closed syllable ending in "r"). Anglo-Latin (hereafter A-L) includes all of the letters of the English alphabet except "w", viz.: "a b c d e f g h i j k l m n o p q r s t u v x y z". It differs from Classical Latin in distinguishing "i" from "j" and "u" from "v". In addition to these letters the digraphs "æ" and "œ" may also be used (as in
Traditional English pronunciation of Latin
8647803
Define the operator Then Δ"f" is a homogeneous polynomial of degree "n"−1 and Δ"f"("x, y, z") = 0 defines a curve of degree "n"−1 called the "first polar" of "C" with respect of "Q". If "P"=("p", "q", "r") is a non-singular point on the curve "C" then the equation of the tangent at "P" is In particular, "P" is on the intersection of "C" and its first polar with respect to "Q" if and only if "Q" is on the tangent to "C" at "P". Note also that for a double point of "C", the partial derivatives of "f" are
Polar curve
4262585
not an image of an element under the canonical embedding. The whole of P(ℤ/nZ) is filled out by elements formula_7 the units of ℤ/nℤ. For examples, when "n" =6 there is U(2,3); when "n" =10 there is U(2,5). In ℤ/12ℤ, U = {1, 5, 7, 11} providing more extra points than in ℤ/6ℤ. The extra points can be associated with ℚ ⊂ ℝ ⊂ ℂ, the rationals in the extended complex upper-half plane. The group of homographies on P(ℤ/nℤ) is called a modular subgroup as it is a subgroup of the modular group, the homography group for P(Z). The projective
Projective line over a ring
15003453
in achalasia: Case report. Turk J Gastroenterol. 17:46-49, (2006) 30. Yonal O, Hatırnaz O, Akyuz F, Ozbek U, Demir K, Kaymakoglu S, Otken A, Mungan Z. HFE gene mutation, chronic liver disease, and iron overload In Turkey. Dig Dis Sci. 52:3298-302 (2007). 31. Akyuz F, Demir K, Ozdil S, Aksoy N, Poturoglu S, Ibrisim D, Kaymakoglu S, Besisik F, Boztas G, Cakaloglu Y, Mungan Z, Cevikbas U. Otken A. The effects of rosiglitazone, metformin, and diet with exercise in nonalcoholic fatty liver disease. Dig Dis Sci. 52:2359-67, (2007) 32. Yonal O, Hatirnaz O, Akyuz F, Koroglu G, Ozbek U, Cefle
Zeynel Mungan
3654874
divide any earlier Pell number. The smallest primitive prime divisor of "n"th Pell number are Carmichael's theorem In number theory, Carmichael's theorem, named after the American mathematician R.D. Carmichael, states that, for any nondegenerate Lucas sequence of the first kind "U"("P","Q") with relatively prime parameters "P, Q" and positive discriminant, an element "U" with "n" ≠ 1, 2, 6 has at least one prime divisor that does not divide any earlier one except the 12th Fibonacci number F(12)="U"(1, -1)=144 and its equivalent "U"(-1, -1)=-144. In particular, for "n" greater than 12, the "n"th Fibonacci number F("n") has at least one
Carmichael's theorem
1594943
product of classes "u" ∈ "H"("X","R") and "v" ∈ "H"("Y","R") is: Another interpretation of Poincaré duality is that the cohomology ring of a closed oriented manifold is self-dual in a strong sense. Namely, let "X" be a closed connected oriented manifold of dimension "n", and let "F" be a field. Then "H"("X","F") is isomorphic to "F", and the product is a perfect pairing for each integer "i". In particular, the vector spaces "H"("X","F") and "H"("X","F") have the same (finite) dimension. Likewise, the product on integral cohomology modulo torsion with values in "H"("X",Z) ≅ Z is a perfect pairing over Z.
Cohomology
11955398
any "w" in "D". This is because "h"("z") = "g"("z") + ("w" - "w"), and for "z" on the boundary of "B", |"g"("z")| ≥ "e" > |"w" - "w"|. Thus, for every "w" in "D", there exists at least one "z" in "B" such that "f"("z") = "w". This means that the disk "D" is contained in "f"("B"). The image of the ball "B", "f"("B") is a subset of the image of "U", "f"("U"). Thus "w" is an interior point of "f"("U"). Since "w" was arbitrary in "f"("U") we know that "f"("U") is open. Since "U" was arbitrary, the function
Open mapping theorem (complex analysis)
14269830
written with one or more capital letters (e.g. +DA, +lAr, +GA, etc.), these capital letters indicate that these sounds are harmonic, that is, variable: D= d/t, G= gh/q/g/k; K= k/q; A= a/e; I= i/u/ü or ø/i/u/ü. There are 32 basic sounds in Modern Uyghur. Uyghur has 24 consonants (listed here according to the Arabic-script alphabet): b, p, t, j, ch, x, d, r, z, zh, s, sh, gh, f, q, k, g, ng, l, m, n, x, h, w, y (and 25 consonants if the glottal stop ‘ is counted). Most are not pronounced much differently than their English counterparts
Uyghur grammar
3654871
Carmichael's theorem In number theory, Carmichael's theorem, named after the American mathematician R.D. Carmichael, states that, for any nondegenerate Lucas sequence of the first kind "U"("P","Q") with relatively prime parameters "P, Q" and positive discriminant, an element "U" with "n" ≠ 1, 2, 6 has at least one prime divisor that does not divide any earlier one except the 12th Fibonacci number F(12)="U"(1, -1)=144 and its equivalent "U"(-1, -1)=-144. In particular, for "n" greater than 12, the "n"th Fibonacci number F("n") has at least one prime divisor that does not divide any earlier Fibonacci number. Carmichael (1913, Theorem 21) proved
Carmichael's theorem
4857803
bundle of differential forms of top degree on "U". Equivalently, the sheaf "O"("K") on "X" is the direct image sheaf "j"Ω, where "n" is the dimension of "X". Example: Let "X" = P be the projective "n"-space with the homogeneous coordinates "x", ..., "x". Let "U" = { "x" ≠ 0 }. Then "U" is isomorphic to the affine "n"-space with the coordinates "y" = "x"/"x". Let Then ω is a rational differential form on "U"; thus, it is a rational section of Ω which has simple poles along "Z" = { "x" = 0 }, "i" = 1, ...,
Divisor (algebraic geometry)
13651095
− 1)/4 from ("F" − 1)"H" to ("F" + 1)"H". Thus "u" → Ind ("F" + 1) u ("F" − 1)/4 gives an additive mapping from "K"("A") to Z. When the spectral triple is finitely summable, one may write the above indexes using the (super) trace, and a product of "F", "e" (resp. "u") and commutator of "F" with "e" (resp. "u"). This can be encoded as a ("p" + 1)-functional on "A" satisfying some algebraic conditions and give Hochschild / cyclic cohomology cocycles, which describe the above maps from K-theory to the integers. Spectral triple In noncommutative geometry and
Spectral triple
5344139
"p" + "q" and the "w" from the right member is the number of variations of a Sturm chain (while "w" refers to a generalized Sturm chain in the present theorem). We can easily determine a stability criterion using this theorem as it is trivial that "f"("z") is Hurwitz-stable iff "p" − "q" = "n". We thus obtain conditions on the coefficients of "f"("z") by imposing "w"(+∞) = "n" and "w"(−∞) = 0. Routh–Hurwitz theorem In mathematics, the Routh–Hurwitz theorem gives a test to determine whether all roots of a given polynomial lie in the left half-plane. Polynomials with this
Routh–Hurwitz theorem
2312797
the Eritrean Ministry of Education, and the Ethiopian Afar Language Studies and Enrichment Center have since worked with Afar linguists, authors and community representatives to select a standard orthography for Afar from among the various existing writing systems used to transcribe the language. A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z a, ba, ca, da, e, fa, ga, ha, i, ja, ka, la, ma, na, o, pa, qa, ra, sa, ta, u, va, wa, ya, za Afar language The Afar language ()
Afar language
1855337
"Parlett" and "Betza" contain the notation describing how each piece moves. The notation systems are explained in the last sections of this article. A – B – C – D – E – F – G – H – I – J – K – L – M – N – O – P – Q – R – S – T – U – V – W – X, Y, Z While a large amount of information can be found concerning the relative value of variant chess pieces, there are few resources where it is in a concise format
Fairy chess piece
711048
May 28–30, 2000). Alphabetical order: a á b c d ď e f g ĝ h i í j k l m n o p q r ř s š t ť u ú v w x y z ž ay aw (33 letters and 2 digraphs) Dictionaries and Lexicographical Works Orthography Courses and Study Guides Etymological and Historical Studies -Elfenbein, J. 1985. Balochi from Khotan. In: Studia Iranica. Vol. XIV (2): 223-238. -Gladstone, C. E. 1874. Biluchi Handbook. Lahore. -Hashmi, S. Z. S. 1986. Balúcí Zabán va Adab kí Táríx [The History of Balochi language and Literature: A Survey].
Balochi language
8140462
place names: A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P | Q | R | S | T | U | V | W | Y | Z Bulgarian toponyms in Antarctica Bulgarian toponyms in Antarctica are approved by the Antarctic Place-names Commission in compliance with its "", and formally given by the President of the Republic according to the Bulgarian Constitution and the established international and Bulgarian practice. Place naming is confined to nameless geographic features
Bulgarian toponyms in Antarctica
15003451
T, Turgut S, Mungan Z. Heterotopic gastric mucosa in the cervical esophagus (inlet patch): Endoscopic prevalence, histological and clinical characteristics. Journal of Gastroenterology and Hepatology, 19:891-6 (2004) 25. Boztaş G, Mungan Z, Özdil S, Akyüz F, Karaca C, Demir K, Kaymakoglu S, Besısık F, Çakaaloglu Y, Ökten A. Pneumatic balloon dilatation in primary achalasia: The long term following results . Hepatogastroenterology. 52:475-80,(2005) 26. Poturoğlu Ş, Mungan Z, Boztaş G, Kaymakoğlu S, Ozdil S, Akyüz F, Aksoy N, Kamaci S, Pınarbaşı B, Çevikbaş U, Ökten A. Peritoneal amyloidosis caused by familial mediterranean fever. J Gastroenterol Hepatol, 20:325-6, (2005) 27. Akbayir N,
Zeynel Mungan
2104459
0 < "p" < 1, a function "F" in "H" cannot be reconstructed from the real part of its boundary limit "function" on the circle, because of the lack of convexity of "L" in this case. Convexity fails but a kind of ""complex convexity"" remains, namely the fact that "z" → |"z"| is subharmonic for every "q" > 0. As a consequence, if is in "H", it can be shown that "c" = O("n"). It follows that the Fourier series converges in the sense of distributions to a distribution "f" on the unit circle, and "F"("re") =("f" ∗ "P")(θ). The
Hardy space
6876532
Carathéodory–Jacobi–Lie theorem The Carathéodory–Jacobi–Lie theorem is a theorem in symplectic geometry which generalizes Darboux's theorem. Let "M" be a 2"n"-dimensional symplectic manifold with symplectic form ω. For "p" ∈ "M" and "r" ≤ "n", let "f", "f", ..., "f" be smooth functions defined on an open neighborhood "V" of "p" whose differentials are linearly independent at each point, or equivalently where {f, f} = 0. (In other words, they are pairwise in involution.) Here {–,–} is the Poisson bracket. Then there are functions "f", ..., "f", "g", "g", ..., "g" defined on an open neighborhood "U" ⊂ "V" of "p"
Carathéodory–Jacobi–Lie theorem
6403491
way as before but taking the "fundamental class" in Z homology. In this case deg("f") is an element of Z (the field with two elements), the manifolds need not be orientable and if "n" is the number of preimages of "p" as before then deg("f") is "n" modulo 2. Integration of differential forms gives a pairing between (C-)singular homology and de Rham cohomology: formula_17, where formula_18 is a homology class represented by a cycle formula_18 and formula_20 a closed form representing a de Rham cohomology class. For a smooth map "f" : "X"→"Y" between orientable "m"-manifolds, one has where "f"
Degree of a continuous mapping
11955396
in "f"("U"). Consider an arbitrary "w" in "f"("U"). Then there exists a point "z" in "U" such that "w" = "f"("z"). Since "U" is open, we can find "d" > 0 such that the closed disk "B" around "z" with radius "d" is fully contained in "U". Consider the function "g"("z") = "f"("z")−"w". Note that "z" is a root of the function. We know that "g"("z") is not constant and holomorphic. The roots of "g" are isolated by the identity theorem, and by further decreasing the radius of the image disk "d", we can assure that "g"("z") has only a
Open mapping theorem (complex analysis)
20156020
writing. Because it was largely unavailable to the masses, McConnell determined to create a system everyone could use. This original writing system contained 33 symbols: Oral Vowels: a, è, é, i, o, ò, u Nasal Vowels: â, ê, ô Semi-Vowels: i/y, w, u Consonants: b, d, f, g, h, j, k, l, m, n, p, r, s, t, v, z Digraphs: sh, gn Frank Laubach was not a speaker of Creole, though he was an expert in American literacy. In 1943, he helped McConnell to revise his orthographic system, which resulted in what's known as the McConnell–Laubach Orthography. This contained
McConnell–Laubach orthography
4087909
always been fairly small, even before the Europeans invaded, because of the limited resources that each atoll had, which limited the number of people that could be supported on each. Tokelauan is written in the Latin script, albeit only using 15 letters: A, E, I, O, U, F, G, K, L, M, N, P, H, T, and V. Its alphabet consists of 5 vowels: a (pronounced: /a/), e (pronounced: /e/), i (pronounced: /i/), o (pronounced: /o/) and u (pronounced: /u/); and 10 consonants: /p t k f v h m n ŋ l/, /ŋ/ spelled g Long vowels are marked
Tokelauan language
1752216
are the second in these pairs, together with the corresponding kerning values. Here are a few examples of glyph classes in the Minion Pro font for the first character in a kerning pair: (d i l u), (h m n), (j q), (b o p), (v w y), (D O Q), (H I), (V W); and for the second character in a pair: (f i m n r), (h k l), (j p t u), (c d e o q), (v w y), (C G O Q), (B D E F H I K L N P R). A type
Kerning
9175314
subset of "k", and let "f", ..., "f" be a collection of analytic functions on "U". Denote by "Z" the common vanishing locus of "f", ..., "f", that is, let "Z" = { "x" | "f"("x") = ... = "f"("x") = 0 }. "Z" is an analytic variety. Suppose that the structure sheaf of "U" is formula_1. Then "Z" has a structure sheaf formula_2, where formula_3 is the ideal generated by "f", ..., "f". In other words, the structure sheaf of "Z" consists of all functions on "U" modulo the possible ways they can differ outside of "Z". An analytic
Analytic space
12881929
Dalmatian grammar This article outlines the grammar of the Dalmatian language. A B Č D E F G H I J K L M N O P Q R S T U V W X Y Z A Dalmatian noun has a gender (masculine or feminine) and is inflected for number (singular or plural). The plural is formed with the ending -i for masculine and -e for feminine nouns. The indefinite article is "yoin" (one), whose feminine form is "yoina". Examples: The definite article for masculine nouns is "el" in singular and "i" in plural. The definite article for feminine
Dalmatian grammar
12881932
the verb "to be" and the infinitive: The verb "to be": Infinitive: Saite Singular Plural Adverbs of place and direction: Adverbs of time: Dalmatian grammar This article outlines the grammar of the Dalmatian language. A B Č D E F G H I J K L M N O P Q R S T U V W X Y Z A Dalmatian noun has a gender (masculine or feminine) and is inflected for number (singular or plural). The plural is formed with the ending -i for masculine and -e for feminine nouns. The indefinite article is "yoin" (one), whose feminine
Dalmatian grammar
6118994
ė, ę, ę̇, ı, i, o, ȯ, ø, ø̇, u, u̇, y, ẏ Samhljóðendr (Consonants): b, ʙ, c, ᴋ, d, ᴅ, f, ꜰ, g, ɢ, ǥ, h, l, ʟ, m, ᴍ, n, ɴ, p, ᴘ, r, ʀ, ſ, ꜱ, t, ᴛ, þ Samsettar (Letters for composite sounds): x, z Other: ⁊, ˜ The author is unknown, and is usually referred to as "First Grammarian". Scholars have hypothesized various identities for the First Grammarian. One probable candidate is Hallr Teitsson (born ca. 1085, died 1150). Þóroddr Gamlason has also been suggested. First Grammatical Treatise The First Grammatical Treatise () is a
First Grammatical Treatise
3095467
except in linguistic or highly formalized texts. Also, the letters "ñ" and "ll" are sometimes replaced by "ny" and "ly" in informal texts. The Chavacano alphabet has 30 letters, including <ch>, <ll>, <ñ> and <rr>: a, b, c, ch, d, e, f, g, h, i, j, k, l, ll, m, n, ñ, o, p, q, r, rr, s, t, u, v, w, x, y, z Other letter combinations include "rr" (erre), which is pronounced or , and "ng," which is pronounced . Another combination was "ñg," which was pronounced but is now obsolete and is only written as "ng". Some
Chavacano
7585112
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
Minakshisundaram–Pleijel zeta function
17082333
of eighteen graphemes: a, b, d, e, g, h, i, k, l, m, n, p, r, s, t, u, w, y. The graphemes c, f, j, o, q, v, x, z are used in recently borrowed words and the names of people and places. Punctuation standards follow those of the Philippine national language. The glottal stop is represented by a hyphen when it occurs word medially, but not where it occurs intervocalically. For example, the word [manʔʌʔ] 'again' is written as "man-e", while the word [tiʔaŋ] 'carry on the shoulder' is written as "tiang". Matigsalug has 4 vowels: /i, ʌ,
Matigsalug language
5954571
a prime-to-"m" ideal ("n") is simply "n" (mod "m") in (Z/mZ). Let "p" and ℓ be distinct odd primes. For convenience, let ℓ* = (−1)ℓ (which is always 1 (mod 4)). Then, quadratic reciprocity states that The relation between the quadratic and Artin reciprocity laws is given by studying the quadratic field formula_32 and the cyclotomic field formula_33 as follows. First, "F" is a subfield of "L", so if "H" = Gal("L"/"F") and "G" = Gal("L"/Q), then Gal("F"/Q) = "G"/"H". Since the latter has order 2, the subgroup "H" must be the group of squares in (Z/ℓZ). A basic property
Artin reciprocity law
7637431
groups are assumed finite. Suppose "V" formula_2 (Z/"pZ) is an elementary abelian group. Since Z/"pZ formula_2 "F", the finite field of "p" elements, we have "V" = (Z/"pZ) formula_2 "F", hence "V" can be considered as an "n"-dimensional vector space over the field "F". Note that an elementary abelian group does not in general have a distinguished basis: choice of isomorphism "V" formula_5 (Z/"pZ) corresponds to a choice of basis. To the observant reader, it may appear that "F" has more structure than the group "V", in particular that it has scalar multiplication in addition to (vector/group) addition. However, "V"
Elementary abelian group
187532
"df" in a neighborhood "U" of "z" is equal to "f"′("z") "dz" for some continuous function "f"′. It follows from that "df"′ is also proportional to "dz", implying that the derivative "f"′ is itself holomorphic and thus that "f" is infinitely differentiable. Similarly, the fact that implies that any function "f" that is holomorphic on the simply connected region "U" is also integrable on "U". (For a path γ from "z" to "z" lying entirely in "U", define in light of the Jordan curve theorem and the generalized Stokes' theorem, "F"("z") is independent of the particular choice of path γ,
Holomorphic function
15003448
Turk J Gastroenterol. 13:226-8, (2002) 16. Kaymakoglu S, Kahraman T, Kudat H, Demir K, Cakaloglu Y, Adalet I, Dincer D, Besısık F, Boztas G, Sozen AB, Mungan Z, Okten A. Hepatopulmonary syndrome in noncirrhotic portal hypertensive patients. Dig Dis Sci, 48:556-60 (2003) 17. Mungan Z, Akyuz F, Bugra Z, Yonal O, Ozturk S, Acar A, Cevikbas U. A family of visceral myopathy; with pseudo-obstruction, megaduodenum, Barrett’s esophagus and cardiac abnormalities. Am J Gastroenterol, 98:2556-60 (2003) 18. Ozdil S, Demir K, Boztas G, Danalioglu A, Karaca C, Akyuz F, Aksoy N, Kaymakoglu S, Mungan Z, Besisik F, Cakaloglu Y, Okten A.
Zeynel Mungan
2748010
but it is usually now called Zarma, the name that its speakers use in their own language. The Zarma alphabet uses the following letters: a, b, c, d, e, f, g, h, i, j, k, l, m, n, ɲ (or ny), ŋ, o, p, r, s, t, u, w, y, z. Also, v may be used in a few words of foreign origin, but many Zarma cannot pronounce it. Most of the letters are pronounced with the same values as the International Phonetic Alphabet (IPA), the exceptions being (approximately English "j" but more palatalized), , (a flap). The letter is
Zarma language
4851011
represent the sounds [c], [ɟ], and [r]. Double vowel characters indicate that the vowels are elongated. An apostrophe (ʼ) is used as a glottal stop. It uses the five vowel system denoting vowel sounds and their lengths. In Nigeria ʼy substitutes ƴ, and in Senegal Ñ/ñ is used instead of ɲ. a, aa, b, mb, ɓ, c, d, nd, ɗ, e, ee, f, g, ng, h, i, ii, j, nj, k, l, m, n, ŋ, ɲ (ny or ñ), o, oo, p, r, s, t, u, uu, w, y, ƴ, ' The letters q, v, x, z are used in
Fula language
4231465
– but "rz" is pronounced exactly the same as in Polish. Kashubian alphabet The Kashubian or Cassubian alphabet ("kaszëbsczi alfabét", "kaszëbsczé abecadło") is the script of the Kashubian language, based on the Latin alphabet. The Kashubian alphabet consists of 34 letters: A, Ą, Ã, B, C, D, E, É, Ë, F, G, H, I, J, K, L, Ł, M, N, Ń, O, Ò, Ó, Ô, P, R, S, T, U, Ù, W, Y, Z, Ż The Kashubian language also use some digraphs: ch, cz, dz, dż, rz and sz. The digraphs "cz", "dż", "sz", "ż" are pronounced in a different
Kashubian alphabet
2613191
is noted by the "Guinness Book of World Records" as being "the most successful pianist in the world". A B C D E F G H I J L M N O P Q R S T U V W Z Numbers Richard Clayderman Richard Clayderman (; born Philippe Pagès , 28 December 1953 in Paris) is a French pianist who has released numerous albums including the compositions of Paul de Senneville and Olivier Toussaint, instrumental renditions of popular music, rearrangements of movie soundtracks, ethnic music, and easy-listening arrangements of popular works of classical music. Clayderman learned piano from his
Richard Clayderman
16606756
World Orthography The World Orthography (WO) is an alphabet and transcription system based on the Africa Alphabet and the International Phonetic Alphabet. In Daniel Jones’s 1948 "Difference between Spoken and Written Language", an adaptation of WO for English is given with the letters a b c d ð e ə f g h i j k l m n ŋ o p r s ʃ t θ u v w x y z ʒ. The capitals of ð, ə, ŋ, ʃ, θ, and ʒ are: Ð (shaped like Ƌ), Ə, the Ŋ (shaped like large ŋ), Ʃ (shaped like sigma
World Orthography