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R

Definition: R

R

Noun

1. A unit of radiation exposure; the dose of ionizing radiation that will produce 1 electrostatic unit of electricity in 1 cc of dry air.

2. (physics) the universal constant in the gas equation: pressure times volume = R times temperature; equal to 8.3143 joules per kelvin per mole.

3. The 18th letter of the Roman alphabet.

4. The length of a line segment between the center and circumference of a circle or sphere.

Source: WordNet 1.7.1 Copyright © 2001 by Princeton University. All rights reserved.
 

Date "R" was first used in popular English literature: sometime before 1350. (references)

 

Specialty Definition: R

DomainDefinition

Literature

R in prescriptions. The ornamental part of this letter is the symbol of Jupiter , under whose special protection all medicines were placed. The letter itself (Recipe, take) and its flourish may be thus paraphrased: "Under the good auspices of Jove, the patron of medicines, take the following drugs in the proportions set down." It has been suggested that the symbol is for Responsum Raphaelis, from the assertion of Dr. Napier and other physicians of the seventeenth century, that the angel Raphael imparted them.
R is called the dog-letter, because a dog in snarling utters the letter r-r-r-r, r-r, r-r-r-r-r, etc.- sometimes preceded by a g.
"Irritata canis quod RR quam plurima dicat."
Lucillus.
"[R] that's the dog's name. R is for the dog."
- Shakespeare: Romeo and Juliet, ii. 4.
The three R's. Sir William Curtis being asked to give a toast, said, "I will give you the three R's- writing, reading, and arithmetic."
"The House is aware that no payment is made except on the `three R's.' "- Mr. Cory. M.P.: Address to the House of Commons, February 28th, 1867. Source: Brewer's Dictionary.

Source: compiled by the editor from various references; see credits.

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Specialty Definition: Aozora Bunko: R

(From Wikipedia, the free Encyclopedia)

See Aozora Bunko

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Cantor's first uncountability proof

(From Wikipedia, the free Encyclopedia)

The misconception

Contrary to what most mathematicians believe, Georg Cantor's first proof that the set of all real numbers is uncountable was not his famous diagonal argument, and did not mention decimal expansions or any other numeral system. The theorem and proof below were found by Cantor in December 1873, and published in 1874 in Crelle's Journal, more formally known as Journal für die Reine und Angewandte Mathematik (German for Journal for Pure and Applied Mathematics). Cantor discovered the diagonal argument in 1877.

The theorem

Suppose a set R is

Then R is not countable.

The proof

The proof begins by assuming some sequence x1, x2, x3, ... has all of R as its range. Define two other sequences as follows:

a1 = x1.

b1 = xi, where i is the smallest index such that xi is not equal to a1.

an+1 = xi, where i is the smallest index greater than the one considered in the previous step such that xi is between an and bn.

bn+1 = xi, where i is the smallest index greater than the one considered in the previous step such that xi is between an+1 and bn.

The two monotone sequences a and b move toward each other. By the "gaplessness" of R, some point c must lie between them. The claim is that c cannot be in the range of the sequence x, and that is the contradiction. If c were in the range, then we would have c = xi for some index i. But then, when that index was reached in the process of defining a and b, then c would have been added as the next member of one or the other of those two sequences, contrary to the assumption that it lies between their ranges.

Real algebraic numbers and real transcendental numbers

In the same paper, published in 1874, Cantor showed that the set of all real algebraic numbers is countable, and inferred the existence of transcendental numbers as a corollary. That corollary had earlier been proved by quite different methods by Joseph Liouville.

Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Cantor's first uncountability proof."

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Casualties of the September 11, 2001 Terrorist Attacks

(From Wikipedia, the free Encyclopedia)

Any tributes to the individuals lost in this tragedy are welcome and encouraged at our memorial site. Some articles originally posted to wikipedia have been moved there - if you are looking for such an article, please check there.

See also Missing Persons, Foreign casualties, and Survivors.

Casualties

Planes - World Trade Center - Pentagon
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

As of October 29, 2003, 2,995 people were presumed dead as a result of all four September 11 attacks. This includes the casualties at the World Trade Center, the Pentagon, on the airplanes and the hijackers.

Planes

265 people killed on four planes; 232 passengers, 25 flight attendants, 8 pilots. (Note that this total includes the 19 hijackers, who reportedly boarded the planes as passengers.)

See also: Memorial wiki tributes to the occupants of each plane

World Trade Center

By October 29, 2003, 2605 people were listed as confirmed dead and 1058 bodies had been identified. (Note: this total does not include the 127 passengers and 20 crew on the two aircraft or the 10 hijackers).

The listing and memorial.

See also:

Missing Persons

The number of missing people grew to estimates as high as over 6000 in the months following the attack, but steadily declined as stories were checked and duplicate entries removed. (See Timeline of WTC missing).

As of August 2002, there were approximately 90 people who were officially missing; that is, their remains had not been identified and no family members had requested a death certificate.

Detailed listing.

Survivors

The great majority of the over 40,000 people working at the World Trade Center at the time of the attack evacuated safely, including 18 who escaped from above the impact zone in the second tower hit. By 9/20/2001 6291 people, including rescue and recovery workers, had been treated for injuries.

Detailed listing.

Pentagon

The Pentagon reports 125 staffers killed or missing, with 121 remains recovered and identified, as of Sept. 11, 2002. At least one person died later as a result of wounds incurred.

The listing and memorial.

Missing Persons

The Pentagon reports 4 staffers missing. One passenger on the airliner which hit the Pentagon was also never identified.

Detailed listing.

Survivors

88 treated at hospital.

Detailed entry.

Victim legends

Due to the very large number of World Trade Center casualties and missing persons, victim legends were a common form of September 11, Terrorist Attack urban legends. These were tales of victims who did not exist, spread by word-of-mouth and the Internet. Official sites, such as http://www.september11victims.com, contain accurate entries and are trusted content. Because Wikipedia, and many other websites allowed freely adding victims, there were no doubt many obvious fake entries. Fake victims added to these lists were often simply missing at the time of the attacks, or actually survivors of the attacks.

See also

September 11, 2001 Terrorist Attack - Donations - Assistance - Memorials and Services

Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Casualties of the September 11, 2001 Terrorist Attacks."

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Colombeau algebra

(From Wikipedia, the free Encyclopedia)

The algebra of moderate functions over Rn,

The ideal (subalgebra) of negligible functions:

The Colombeau algebra is the quotient algebra .

Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Colombeau algebra."

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Coordinates

(From Wikipedia, the free Encyclopedia)

coordinates in mathematics

A point P in the Euclidean space Rn is given by an n-tuple P=(r1,...,rn) of real numbers r1,...,rn.

These numbers r1,...,rn are called the coordinates of the point P.

If a subset S of an Euclidean space is mapped continuously onto another topological space, this defines coordinates in the image of S.

See also

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Covariant

(From Wikipedia, the free Encyclopedia)

In category theory, see covariant functor.

In tensor analysis, a covariant coordinate system is reciprocal to a corresponding contravariant coordinate system.

Roughly speaking, a covariant tensor is a vector field that defines the topology of a space; it is the base which one measures against.

A contravariant vector is thus a measurement or a displacement on this space.

Thus, their relationship can be represented simply as:

Another way of defining covariant vectors is to say that "covariant vectors" are actually one-forms, that is to say, real-valued linear functions on "contravariant" vectors. These one-forms can then be said to form a dual space to the vector space they take their arguments from.

If e1, e2, e3 are contravariant basis vectors of R3 (not necessarily orthogonal nor of unit norm) then the covariant basis vectors of their reciprocal system are:

Then the contravariant coordinates of any vector v can be obtained by the dot product of v with the contravariant basis vectors:

Likewise, the covariant components of v can be obtained from the dot product of v with covariant basis vectors, viz.

Then v can be expressed in two (reciprocal) ways, viz.

.

The indices of covariant coordinates, vectors, and tensors are subscripts. If the contravariant basis vectors are orthonormal then they are equivalent to the covariant basis vectors, so there is no need to distinguish between the covariant and contravariant coordinates, and all indices are subscripts.

Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Covariant."

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Earth radius

(From Wikipedia, the free Encyclopedia)

Earth radii are sometimes used to measure distance. The radius of Earth is approximately 6,378 km. This distance is usually denoted by RE.

See also: Effective Earth radius

Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Earth radius."

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Elementary group theory

(From Wikipedia, the free Encyclopedia)

First Theorems about Groups

A group (G,*) is usually defined as:

G is a set and * is an associative binary operation on G, obeying the following rules (or axioms):

A1. (G,*) has closure. That is, if a and b are in G, then a*b is in G
A2. The operation * is associative, that is, if a, b, and c are in G, then (a*b)*c=a*(b*c).
A3. G contains an identity element, often denoted e, such that for all a in G, e*a=a*e=a.
A4. Every element in (G,*) has an inverse; if a is in G, then there exists an element b in G such that a*b=b*a=e.

Axioms A1 and A2 follow from the definition of "associative binary operation", and are sometimes omitted, particularly A1.

Where no danger of confusion is possible, the group (G,*) will simply be referred to as "the group G"; but it is important to remember that the operation "*" is fundamental to the description of the group. For example, in the real numbers, we can speak of both the group (R,+), which is the additive group of reals with identity 0; and the group (R#, *), which is the multiplicative group of the reals (excluding 0), which has identity 1.

We can state simpler versions of A3 and A4:

A3'. G contains an identity element, often denoted e, such that for all a in G, a*e=a.
A4'. Every element in (G,*) has an inverse; for all a in G, there exists an element in G, denoted a -1, such that a*a -1 = e.

In the sequel, we will assume the axioms A1, A2, A3' and A4'. We first show in Theorem 1.1 and 1.2 that these assumptions actually imply A3 and A4. We will then go on to prove several other theorems, the most important of which are that every group has a unique identity, and every element in a group has a unique inverse.

Theorem 1.1: For all a in G, a -1*a = e.

Thus every right inverse in a group is necessarily a left inverse.

Theorem 1.2: For all a in G, e*a = a.

Thus, the identity in a group is both a left and right identity. (We will justify the use of the term "the identity" momentarily).

The following theorem demonstrates a fundamental property enjoyed by groups, which other more general structures (such as semigroups) lack:

Theorem 1.3: For all a,b in G, there exists a unique x in G such that a*x = b.

Theorem 1.4: The identity element of a group (G,*) is unique. As a result, we can speak of the identity element of (G,*) rather than an identity element. Where different groups are being discussed and compared, often eG will be used to identify the identity in (G,*). By analogy to the group of reals, the identity is also sometimes denoted as 1 (or 1G ) in groups that are written multiplicatively, and as 0 (or 0G ) in groups that are written additively.

Theorem 1.4: The inverse of each element in (G,*) is unique; equivalently, for all a in G, a*x = e if and only if x=a -1.

As a result, we can speak of the inverse of an element x, rather than an inverse. When the group operation is written multiplicatively (as it is here), we denote the inverse of x as x -1. When the group is written additively (i.e., in (G,+)), the inverse of x is written as -x.

Theorem 1.5: For all a belonging to a group (G,*), (a -1) -1=a.

Theorem 1.6: For all a,b belonging to a group (G,*), (a*b) -1=b -1*a -1. The results of the following theorem are often called the cancellation rules for a group:

Theorem 1.7: For all a,x,y, belonging to a group (G,*), if a*x=a*y, then x=y; and if x*a=y*a, then x=y.

Given a group (G, *), if the total number of elements in G is finite, then the group is called a finite group. The order of a group (G,*) is the number of elements in G (for a finite group), or the cardinality of the group if G is not finite. The order of a group G is written as |G| or (less frequently) o(G).

A subset H of G is called a subgroup of a group (G,*) if H satisfies the axioms of a group, using the same operator "*", and restricted to the subset H. Thus if H is a subgroup of (G,*), then (H,*) is also a group, and obeys the above theorems, restricted to H. The order of subgroup H is the number of elements in H.

A proper subgroup of a group G is a subgroup which is not identical to G. A non-trivial subgroup of G is (usually) any subgroup of G which contains an element other than e.

Theorem 2.1: If H is a subgroup of (G,*), then the identity eH in H is identical to the identity e in (G,*).

Theorem 2.2: If H is a subgroup of G, and h is an element of H, then the inverse of h in H is identical to the inverse of h in G. Given a subset S of G, we often want to determine whether or not S is also a subgroup of G. One handy theorem that covers the case for both both finite and infinite groups is:

Theorem 2.3: If S is a non-empty subset of G, then S is a subgroup of G if and only if for all a,b in S, a*b -1 is in S.

The intersection of two or more subgroups is again a subgroup.

Theorem 2.4: The intersection of any non-empty set of subgroups of a group G is a subgroup.

In a group (G,*), define x0 = e. We write x*x as x² ; and in general, x*x*x*...*x (n times) as xn. Similarly, we write x -n for (x -1)n.

Theorem: Let a be an element of a group (G,*). Then the set {an: n is an integer} is a subgroup of G.

A subgroup of this type is called a cyclic subgroup; the subgroup of the powers of a is often written as <a>, and we say that a generates <a>.

If there is a positive integer n such that an=e, then we say the element a has order n in G. Sometimes this is written as "o(a)=n.

If S and T are subsets of G, and a is an element of G, we write "a*S" to refer to the subset of G made up of all elements of the form a*s, where s is an element of S; similarly, we write "S*a" to indicate the set of elements of the form s*a. We write S*T for the subset of G made up of elements of the form s*t, where s is an element of S and t is an element of T.

If H is a subgroup of G, then a left coset of H is a set of the form a*H, for some a in G. A right coset is a subset of the form H*a.

Some useful theorems about cosets, stated without proof:

Theorem: If H is a subgroup of G, and x and y are elements of G, then either x*H = y*H, or x*H and y*H have empty intersection.

Theorem: If H is a subgroup of G, every left (right) coset of H in G contains the same number of elements.

\'Theorem': If H is a subgroup of G, then G is the disjoint union of the left (right) cosets of H.

Theorem: If H is a subgroup of G, then the number of distinct left cosets of H is the same as the number of distinct right cosets of H.

Define the index of a subgroup H of a group G (written "[G:H]" ) to be the number of distinct left cosets of H in G.

From these theorems, we can deduce the important Lagrange's Theorem relating the order of a subgroup to the order of a group:

Lagrange's Theorem: If H is a subgroup of G, then |G| = |H|*[G:H].

For finite groups, this also allows us to state:

Lagrange's Theorem: If H is a subgroup of a finite group G, then the order of H divides the order of G.

References

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List of airports: R

(From Wikipedia, the free Encyclopedia)

List of airports: 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

R

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List of Biblical names starting with R

(From Wikipedia, the free Encyclopedia)

List of Biblical names
A - B - C - D - E - F - G - H - I - J - K - L - M - N - O - P - Q - R - S - T - U - V - Y - Z

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List of books by title: R

(From Wikipedia, the free Encyclopedia)

List of books in alphabetical order by title:

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

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List of cities in Germany starting with R

(From Wikipedia, the free Encyclopedia)

List of cities in Germany: 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

TownPopulationDistrictBundesland
Rainau3,203OstalbkreisBaden-Württemberg
Rathenow29,000HavellandBrandenburg
Ratzeburg12,300LauenburgSchleswig-Holstein
Regensburg 125,100--Bavaria
Remscheid120,600--North Rhine-Westphalia
Remshalden13,510Rems-MurrBaden-Württemberg
Rendsburg29,400Rendsburg-EckernfördeSchleswig-Holstein
Rheine76,000SteinfurtNorth Rhine-Westphalia
Riesbürg2,384OstalbkreisBaden-Württemberg
Ribnitz-Damgarten17,600NordvorpommernMecklenburg-Western Pomerania
Rinteln28,500SchaumburgLower Saxony
Rosenberg2,639OstalbkreisBaden-Württemberg
Rosenheim58,800--Bavaria
Rostock212,700--Mecklenburg-Western Pomerania
Rotenburg21,500RotenburgLower Saxony

A "--" in the district column means, that the town is a district-free town, i.e. it is by itself a district.

Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "List of cities in Germany starting with R."

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List of colleges and universities starting with R

(From Wikipedia, the free Encyclopedia)

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
  1. Radford University
  2. Rajamangala Institute of Technology
  3. Ramrao Adik Institute of Engineering
  4. Rand Afrikaans University
  5. Randolph-Macon College
  6. Randolph-Macon Woman's College
  7. Rapperswil School of Engineering
  8. Rashtreeya Vidyalaya College of Engineering
  9. Rayapati Venkata Rangarao and Jagarlamudi Chandramouli College of Engineering
  10. Reading University
  11. Red River Community College
  12. Red de Interconexion Universitaria (RIU)
  13. Reed College (Portland, Oregon)
  14. Regent College
  15. Regent University
  16. Regent University College of Communication
  17. Regional Engineering College, Calicut
  18. Regional Engineering College, Durgapur
  19. Regional Engineering College, Rourkela
  20. Regional Engineering College, Suratkal
  21. Regional Engineering College, Tiruchirappalli, India
  22. Regional Engineering College, Trichy
  23. Regional Engineering College, Warangal
  24. Regional Institute of Technology, Jamshedpur
  25. Regional Technical College Cork
  26. Regional Technical College Galway
  27. Regional Technical College Letterkenny
  28. Regional Technical College Limerick
  29. Regional Technical College Sligo
  30. Regional Technical College Tallaght
  31. Regional Technical College, Carlow
  32. Regional Technical College, Dundalk
  33. Regis College
  34. Reitaku University
  35. Rensselaer Polytechnic Institute
  36. Rheinisch-Westfälische Technische Hochschule Aachen (RWTH)
  37. Rhode Island College
  38. Rhode Island School of Design
  39. Rhodes College
  40. Rhodes University
  41. Rice University (Houston, Texas)
  42. Richard Huish College
  43. Richard Stockton University
  44. Richland College
  45. Richland Community College
  46. Ricks College
  47. Rider University
  48. Riga Technical University
  49. Rio Salado Community College
  50. Ripon College
  51. Ritsumeikan University
  52. Riverside Community College
  53. Roanoke Bible College
  54. Roanoke College
  55. Robert Gordon University
  56. Rochester Institute of Technology
  57. Rockefeller University
  58. Rockford College
  59. Rockhurst College
  60. Rocky Mountain College
  61. Rollins College
  62. Rollins School of Public Health
  63. Roosevelt University
  64. Rose-Hulman Institute of Technology (Terre Haute, Indiana)
  65. Roskilde University
  66. Ross University School of Veterinary Medicine
  67. Ross University Schoole of Medicine
  68. Rostov State University
  69. Rowan College of New Jersey
  70. Royal College of Surgeons
  71. Royal Danish School of Educational Studies
  72. Royal Danish School of Pharmacy
  73. Royal Holloway, University of London
  74. Royal Institute of Technology (Stockholm, Sweden)
  75. Royal Melbourne Institute of Technology
  76. Royal Military Academy of Belgium
  77. Royal Military College of Canada
  78. Royal Postgraduate Medical School
  79. Royal Roads University
  80. Royal Veterinary College, London
  81. Royal Veterinary and Agricultural University
  82. Rudolf Steiner College
  83. Ruhr-Universitat Bochum
  84. Russell Sage College
  85. Russian Academy of Sciences
  86. Rutgers University
  87. Rutgers University - Campus at Newark
  88. Rutgers University, Camden
  89. Ryerson Polytechnic University

See also : Colleges and universities

Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "List of colleges and universities starting with R."

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List of people by name: R

(From Wikipedia, the free Encyclopedia)

List of people by name: 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 Ra - Rb - Rc - Rd - Re - Rf - Rg - Rh - Ri - Rj - Rk - Rl - Rm - Rn - Ro - Rp - Rq - Rr - Rs - Rt - Ru - Rv - Rw - Rx - Ry - Rz

Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "List of people by name: R."

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List of people by name: Re

(From Wikipedia, the free Encyclopedia)

List of people by name: 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 Ra - Rb - Rc - Rd - Re - Rf - Rg - Rh - Ri - Rj - Rk - Rl - Rm - Rn - Ro - Rp - Rq - Rr - Rs - Rt - Ru - Rv - Rw - Rx - Ry - Rz

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List of people by name: Rh

(From Wikipedia, the free Encyclopedia)

List of people by name: 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 Ra - Rb - Rc - Rd - Re - Rf - Rg - Rh - Ri - Rj - Rk - Rl - Rm - Rn - Ro - Rp - Rq - Rr - Rs - Rt - Ru - Rv - Rw - Rx - Ry - Rz

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List of people by name: Ri

(From Wikipedia, the free Encyclopedia)

List of people by name: 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 Ra - Rb - Rc - Rd - Re - Rf - Rg - Rh - Ri - Rj - Rk - Rl - Rm - Rn - Ro - Rp - Rq - Rr - Rs - Rt - Ru - Rv - Rw - Rx - Ry - Rz

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List of people by name: Ro

(From Wikipedia, the free Encyclopedia)

List of people by name: 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

Ra - Rb - Rc - Rd - Re - Rf - Rg - Rh - Ri - Rj - Rk - Rl - Rm - Rn - Ro - Rp - Rq - Rr - Rs - Rt - Ru - Rv - Rw - Rx - Ry - Rz

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List of people by name: Ru

(From Wikipedia, the free Encyclopedia)

List of people by name: 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 Ra - Rb - Rc - Rd - Re - Rf - Rg - Rh - Ri - Rj - Rk - Rl - Rm - Rn - Ro - Rp - Rq - Rr - Rs - Rt - Ru - Rv - Rw - Rx - Ry - Rz

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List of people by name: Ry

(From Wikipedia, the free Encyclopedia)

List of people by name: 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 Ra - Rb - Rc - Rd - Re - Rf - Rg - Rh - Ri - Rj - Rk - Rl - Rm - Rn - Ro - Rp - Rq - Rr - Rs - Rt - Ru - Rv - Rw - Rx - Ry - Rz

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List of rare diseases starting with R

(From Wikipedia, the free Encyclopedia)

This list of rare diseases was originally taken from the NIH public domain resource at http://ord.aspensys.com/asp/diseases/diseases.asp .

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