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Ledgers.
A "ledger" is a record of accounts. The ledger is a permanent summary of all amounts entered in supporting Journals which list individual transactions by date. These accounts are recorded separately, showing their beginning/ending balance. A journal lists financial transactions in chronological order, without showing their balance but showing how much is going to be entered in each account. A ledger takes each financial transaction from the journal and records it into the corresponding accounts. The ledger also determines the balance of every account, which is transferred into the balance sheet or the income statement. There are three different kinds of ledgers that deal with book-keeping:
Chart of accounts.
A chart of accounts is a list of the accounts codes that can be identified with numeric, alphabetical, or alphanumeric codes allowing the account to be located in the general ledger. The equity section of the chart of accounts is based on the fact that the legal structure of the entity is of a particular legal type. Possibilities include "sole trader", "partnership", "trust", and "company".
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Computerized bookkeeping.
Computerized bookkeeping removes many of the paper "books" that are used to record the financial transactions of a business entity; instead, relational databases are used today, but typically, these still enforce the norms of bookkeeping including the single-entry and double-entry bookkeeping systems. Certified Public Accountants (CPAs) supervise the internal controls for computerized bookkeeping systems, which serve to minimize errors in documenting the numerous activities a business entity may initiate or complete over an accounting period. |
Bézier curve
A Bézier curve ( , ) is a parametric curve used in computer graphics and related fields. A set of discrete "control points" defines a smooth, continuous curve by means of a formula. Usually the curve is intended to approximate a real-world shape that otherwise has no mathematical representation or whose representation is unknown or too complicated. The Bézier curve is named after French engineer Pierre Bézier (1910–1999), who used it in the 1960s for designing curves for the bodywork of Renault cars. Other uses include the design of computer fonts and animation. Bézier curves can be combined to form a Bézier spline, or generalized to higher dimensions to form Bézier surfaces. The Bézier triangle is a special case of the latter.
In vector graphics, Bézier curves are used to model smooth curves that can be scaled indefinitely. "Paths", as they are commonly referred to in image manipulation programs, are combinations of linked Bézier curves. Paths are not bound by the limits of rasterized images and are intuitive to modify.
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Bézier curves are also used in the time domain, particularly in animation, user interface design and smoothing cursor trajectory in eye gaze controlled interfaces. For example, a Bézier curve can be used to specify the velocity over time of an object such as an icon moving from A to B, rather than simply moving at a fixed number of pixels per step. When animators or interface designers talk about the "physics" or "feel" of an operation, they may be referring to the particular Bézier curve used to control the velocity over time of the move in question.
This also applies to robotics where the motion of a welding arm, for example, should be smooth to avoid unnecessary wear.
Invention.
The mathematical basis for Bézier curves—the Bernstein polynomials—was established in 1912, but the polynomials were not applied to graphics until some 50 years later when mathematician Paul de Casteljau in 1959 developed de Casteljau's algorithm, a numerically stable method for evaluating the curves, and became the first to apply them to computer-aided design at French automaker Citroën. De Casteljau's method was patented in France but not published until the 1980s while the Bézier polynomials were widely publicised in the 1960s by the French engineer Pierre Bézier, who discovered them independently and used them to design automobile bodies at Renault.
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Specific cases.
A Bézier curve is defined by a set of "control points" P0 through P"n", where "n" is called the order of the curve ("n" = 1 for linear, 2 for quadratic, 3 for cubic, etc.). The first and last control points are always the endpoints of the curve; however, the intermediate control points generally do not lie on the curve. The sums in the following sections are to be understood as affine combinations – that is, the coefficients sum to 1.
Linear Bézier curves.
Given distinct points P0 and P1, a linear Bézier curve is simply a line between those two points. The curve is given by
This is the simplest and is equivalent to linear interpolation. The quantity formula_2 represents the displacement vector from the start point to the end point.
Quadratic Bézier curves.
A quadratic Bézier curve is the path traced by the function B("t"), given points P0, P1, and P2,
which can be interpreted as the linear interpolant of corresponding points on the linear Bézier curves from P0 to P1 and from P1 to P2 respectively. Rearranging the preceding equation yields:
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This can be written in a way that highlights the symmetry with respect to P1:
Which immediately gives the derivative of the Bézier curve with respect to "t":
from which it can be concluded that the tangents to the curve at P0 and P2 intersect at P1. As "t" increases from 0 to 1, the curve departs from P0 in the direction of P1, then bends to arrive at P2 from the direction of P1.
The second derivative of the Bézier curve with respect to "t" is
Cubic Bézier curves.
Four points P0, P1, P2 and P3 in the plane or in higher-dimensional space define a cubic Bézier curve.
The curve starts at P0 going toward P1 and arrives at P3 coming from the direction of P2. Usually, it will not pass through P1 or P2; these points are only there to provide directional information. The distance between P1 and P2 determines "how far" and "how fast" the curve moves towards P1 before turning towards P2.
Writing BP"i",P"j",P"k"("t") for the quadratic Bézier curve defined by points P"i", P"j", and P"k", the cubic Bézier curve can be defined as an affine combination of two quadratic Bézier curves:
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The explicit form of the curve is:
For some choices of P1 and P2 the curve may intersect itself, or contain a cusp.
Any series of 4 distinct points can be converted to a cubic Bézier curve that goes through all 4 points in order.
Given the starting and ending point of some cubic Bézier curve, and the points along the curve corresponding to "t" = 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
General definition.
Bézier curves can be defined for any degree "n".
Recursive definition.
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_12 denote the Bézier curve determined by any selection of points P0, P1, ..., P"k". Then to start,
This recursion is elucidated in the animations below.
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Explicit definition.
The formula can be expressed explicitly as follows (where t0 and (1-t)0 are extended continuously to be 1 throughout [0,1]):
where formula_16 are the binomial coefficients.
For example, when "n" = 5:
Terminology.
Some terminology is associated with these parametric curves. We have
where the polynomials
are known as Bernstein basis polynomials of degree "n".
"t"0 = 1, (1 − "t")0 = 1, and the binomial coefficient, formula_16, is:
The points P"i" are called "control points" for the Bézier curve. The polygon formed by connecting the Bézier points with lines, starting with P0 and finishing with P"n", is called the "Bézier polygon" (or "control polygon"). The convex hull of the Bézier polygon contains the Bézier curve.
Polynomial form.
Sometimes it is desirable to express the Bézier curve as a polynomial instead of a sum of less straightforward Bernstein polynomials. Application of the binomial theorem to the definition of the curve followed by some rearrangement will yield
where
This could be practical if formula_24 can be computed prior to many evaluations of formula_25; however one should use caution as high order curves may lack numeric stability (de Casteljau's algorithm should be used if this occurs). Note that the empty product is 1.
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Second-order curve is a parabolic segment.
A quadratic Bézier curve is also a segment of a parabola. As a parabola is a conic section, some sources refer to quadratic Béziers as "conic arcs". With reference to the figure on the right, the important features of the parabola can be derived as follows:
Derivative.
The derivative for a curve of order "n" is
Constructing Bézier curves.
Linear curves.
Let "t" denote the fraction of progress (from 0 to 1) the point B("t") has made along its traversal from P0 to P1. For example, when "t"=0.25, B("t") is one quarter of the way from point P0 to P1. As "t" varies from 0 to 1, B("t") draws a line from P0 to P1.
Quadratic curves.
For quadratic Bézier curves one can construct intermediate points Q0 and Q1 such that as "t" varies from 0 to 1:
Higher-order curves.
For higher-order curves one needs correspondingly more intermediate points. For cubic curves one can construct intermediate points Q0, Q1, and Q2 that describe linear Bézier curves, and points R0 and R1 that describe quadratic Bézier curves:
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For fourth-order curves one can construct intermediate points Q0, Q1, Q2 and Q3 that describe linear Bézier curves, points R0, R1 and R2 that describe quadratic Bézier curves, and points S0 and S1 that describe cubic Bézier curves:
For fifth-order curves, one can construct similar intermediate points.
These representations rest on the process used in De Casteljau's algorithm to calculate Bézier curves.
Offsets (or stroking) of Bézier curves.
The curve at a fixed offset from a given Bézier curve, called an offset or parallel curve in mathematics (lying "parallel" to the original curve, like the offset between rails in a railroad track), cannot be exactly formed by a Bézier curve (except in some trivial cases). In general, the two-sided offset curve of a cubic Bézier is a 10th-order algebraic curve and more generally for a Bézier of degree "n" the two-sided offset curve is an algebraic curve of degree 4"n" − 2. However, there are heuristic methods that usually give an adequate approximation for practical purposes.
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In the field of vector graphics, painting two symmetrically distanced offset curves is called "stroking" (the Bézier curve or in general a path of several Bézier segments). The conversion from offset curves to filled Bézier contours is of practical importance in converting fonts defined in Metafont, which require stroking of Bézier curves, to the more widely used PostScript type 1 fonts, which only require (for efficiency purposes) the mathematically simpler operation of filling a contour defined by (non-self-intersecting) Bézier curves.
Degree elevation.
A Bézier curve of degree "n" can be converted into a Bézier curve of degree "n" + 1 "with the same shape". This is useful if software supports Bézier curves only of specific degree. For example, systems that can only work with cubic Bézier curves can implicitly work with quadratic curves by using their equivalent cubic representation.
To do degree elevation, we use the equality formula_40 Each component formula_41 is multiplied by (1 − "t") and "t", thus increasing a degree by one, without changing the value. Here is the example of increasing degree from 2 to 3.
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In other words, the original start and end points are unchanged. The new control points are formula_43 and formula_44.
For arbitrary "n" we use equalities
Therefore:
introducing arbitrary formula_47 and formula_48.
Therefore, new control points are
Repeated degree elevation.
The concept of degree elevation can be repeated on a control polygon R to get a sequence of control polygons R, R1, R2, and so on. After "r" degree elevations, the polygon R"r" has the vertices P0,"r", P1,"r", P2,"r", ..., P"n"+"r","r" given by
It can also be shown that for the underlying Bézier curve "B",
Degree reduction.
Degree reduction can only be done exactly when the curve in question is originally elevated from a lower degree. A number of approximation algorithms have been proposed and used in practice.
Rational Bézier curves.
The rational Bézier curve adds adjustable weights to provide closer approximations to arbitrary shapes. The numerator is a weighted Bernstein-form Bézier curve and the denominator is a weighted sum of Bernstein polynomials. Rational Bézier curves can, among other uses, be used to represent segments of conic sections exactly, including circular arcs.
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Given "n" + 1 control points P0, ..., P"n", the rational Bézier curve can be described by
or simply
The expression can be extended by using number systems besides reals for the weights. In the complex plane the points {1}, {-1}, and {1} with weights {formula_54}, {1}, and {formula_55} generate a full circle with radius one. For curves with points and weights on a circle, the weights can be scaled without changing the curve's shape. Scaling the central weight of the above curve by 1.35508 gives a more uniform parameterization.
Applications.
Computer graphics.
Bézier curves are widely used in computer graphics to model smooth curves. As the curve is completely contained in the convex hull of its control points, the points can be graphically displayed and used to manipulate the curve intuitively. Affine transformations such as translation and rotation can be applied on the curve by applying the respective transform on the control points of the curve.
Quadratic and cubic Bézier curves are most common. Higher degree curves are more computationally expensive to evaluate. When more complex shapes are needed, low order Bézier curves are patched together, producing a composite Bézier curve. A composite Bézier curve is commonly referred to as a "path" in vector graphics languages (like PostScript), vector graphics standards (like SVG) and vector graphics programs (like Artline, Timeworks Publisher, Adobe Illustrator, CorelDraw, Inkscape, and Allegro). In order to join Bézier curves into a composite Bézier curve without kinks, a property called "G1 continuity" suffices to force the control point at which two constituent Bézier curves meet to lie on the line defined by the two control points on either side.
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The simplest method for scan converting (rasterizing) a Bézier curve is to evaluate it at many closely spaced points and scan convert the approximating sequence of line segments. However, this does not guarantee that the rasterized output looks sufficiently smooth, because the points may be spaced too far apart. Conversely it may generate too many points in areas where the curve is close to linear. A common adaptive method is recursive subdivision, in which a curve's control points are checked to see if the curve approximates a line to within a small tolerance. If not, the curve is subdivided parametrically into two segments, 0 ≤ "t" ≤ 0.5 and 0.5 ≤ "t" ≤ 1, and the same procedure is applied recursively to each half. There are also forward differencing methods, but great care must be taken to analyse error propagation.
Analytical methods where a Bézier is intersected with each scan line involve finding roots of cubic polynomials (for cubic Béziers) and dealing with multiple roots, so they are not often used in practice.
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The rasterisation algorithm used in Metafont is based on discretising the curve, so that it is approximated by a sequence of "rook moves" that are purely vertical or purely horizontal, along the pixel boundaries. To that end, the plane is first split into eight 45° sectors (by the coordinate axes and the two lines formula_56), then the curve is decomposed into smaller segments such that the "direction" of a curve segment stays within one sector; since the curve velocity is a second degree polynomial, finding the formula_57 values where it is parallel to one of these lines can be done by solving quadratic equations. Within each segment, either horizontal or vertical movement dominates, and the total number of steps in either direction can be read off from the endpoint coordinates; in for example the 0–45° sector horizontal movement to the right dominates, so it only remains to decide between which steps to the right the curve should make a step up.
There is also a modified curve form of Bresenham's line drawing algorithm by Zingl that performs this rasterization by subdividing the curve into rational pieces and calculating the error at each pixel location such that it either travels at a 45° angle or straight depending on compounding error as it iterates through the curve. This reduces the next step calculation to a series of integer additions and subtractions.
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Animation.
In animation applications, such as Adobe Flash and Synfig, Bézier curves are used to outline, for example, movement. Users outline the wanted path in Bézier curves, and the application creates the needed frames for the object to move along the path.
In 3D animation, Bézier curves are often used to define 3D paths as well as 2D curves for keyframe interpolation. Bézier curves are now very frequently used to control the animation easing in CSS, JavaScript, JavaFx and Flutter SDK.
Fonts.
TrueType fonts use composite Bézier curves composed of quadratic Bézier curves. Other languages and imaging tools (such as PostScript, Asymptote, Metafont, and SVG) use composite Béziers composed of cubic Bézier curves for drawing curved shapes. OpenType fonts can use either kind of curve, depending on which font technology underlies the OpenType wrapper.
Font engines, like FreeType, draw the font's curves (and lines) on a pixellated surface using a process known as font rasterization. Typically font engines and vector graphics engines render Bézier curves by splitting them recursively up to the point where the curve is flat enough to be drawn as a series of linear or circular segments. The exact splitting algorithm is implementation dependent, only the flatness criteria must be respected to reach the necessary precision and to avoid non-monotonic local changes of curvature. The "smooth curve" feature of charts in Microsoft Excel also uses this algorithm.
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Because arcs of circles and ellipses cannot be exactly represented by Bézier curves, they are first approximated by Bézier curves, which are in turn approximated by arcs of circles. This is inefficient as there exists also approximations of all Bézier curves using arcs of circles or ellipses, which can be rendered incrementally with arbitrary precision. Another approach, used by modern hardware graphics adapters with accelerated geometry, can convert exactly all Bézier and conic curves (or surfaces) into NURBS, that can be rendered incrementally without first splitting the curve recursively to reach the necessary flatness condition. This approach also preserves the curve definition under all linear or perspective 2D and 3D transforms and projections.
Robotics.
Because the control polygon allows to tell whether or not the path collides with any obstacles, Bézier curves are used in producing trajectories of the end effectors. Furthermore, joint space trajectories can be accurately differentiated using Bézier curves. Consequently, the derivatives of joint space trajectories are used in the calculation of the dynamics and control effort (torque profiles) of the robotic manipulator. |
Banach algebra
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra formula_1 over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach space, that is, a normed space that is complete in the metric induced by the norm. The norm is required to satisfy
formula_2
This ensures that the multiplication operation is continuous with respect to the metric topology.
A Banach algebra is called "unital" if it has an identity element for the multiplication whose norm is formula_3 and "commutative" if its multiplication is commutative.
Any Banach algebra formula_1 (whether it is unital or not) can be embedded isometrically into a unital Banach algebra formula_5 so as to form a closed ideal of formula_5. Often one assumes "a priori" that the algebra under consideration is unital because one can develop much of the theory by considering formula_5 and then applying the outcome in the original algebra. However, this is not the case all the time. For example, one cannot define all the trigonometric functions in a Banach algebra without identity.
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The theory of real Banach algebras can be very different from the theory of complex Banach algebras. For example, the spectrum of an element of a nontrivial complex Banach algebra can never be empty, whereas in a real Banach algebra it could be empty for some elements.
Banach algebras can also be defined over fields of formula_8-adic numbers. This is part of formula_8-adic analysis.
Examples.
The prototypical example of a Banach algebra is formula_10, the space of (complex-valued) continuous functions, defined on a locally compact Hausdorff space formula_11, that vanish at infinity. formula_10 is unital if and only if formula_11 is compact. The complex conjugation being an involution, formula_10 is in fact a C*-algebra. More generally, every C*-algebra is a Banach algebra by definition.
Properties.
Several elementary functions that are defined via power series may be defined in any unital Banach algebra; examples include the exponential function and the trigonometric functions, and more generally any entire function. (In particular, the exponential map can be used to define abstract index groups.) The formula for the geometric series remains valid in general unital Banach algebras. The binomial theorem also holds for two commuting elements of a Banach algebra.
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The set of invertible elements in any unital Banach algebra is an open set, and the inversion operation on this set is continuous (and hence is a homeomorphism), so that it forms a topological group under multiplication.
If a Banach algebra has unit formula_35 then formula_36 cannot be a commutator; that is, formula_37 for any formula_38 This is because formula_39 and formula_40 have the same spectrum except possibly formula_41
The various algebras of functions given in the examples above have very different properties from standard examples of algebras such as the reals. For example:
Spectral theory.
Unital Banach algebras over the complex field provide a general setting to develop spectral theory. The "spectrum" of an element formula_49 denoted by formula_50, consists of all those complex scalars formula_51 such that formula_52 is not invertible in formula_48 The spectrum of any element formula_54 is a closed subset of the closed disc in formula_55 with radius formula_56 and center formula_57 and thus is compact. Moreover, the spectrum formula_50 of an element formula_54 is non-empty and satisfies the spectral radius formula:
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formula_60
Given formula_49 the holomorphic functional calculus allows to define formula_62 for any function formula_63 holomorphic in a neighborhood of formula_64 Furthermore, the spectral mapping theorem holds:
formula_65
When the Banach algebra formula_1 is the algebra formula_67 of bounded linear operators on a complex Banach space formula_11 (for example, the algebra of square matrices), the notion of the spectrum in formula_1 coincides with the usual one in operator theory. For formula_70 (with a compact Hausdorff space formula_11), one sees that:
formula_72
The norm of a normal element formula_54 of a C*-algebra coincides with its spectral radius. This generalizes an analogous fact for normal operators.
Let formula_1 be a complex unital Banach algebra in which every non-zero element formula_54 is invertible (a division algebra). For every formula_76 there is formula_77 such that
formula_78 is not invertible (because the spectrum of formula_79 is not empty) hence formula_80 this algebra formula_1 is naturally isomorphic to formula_55 (the complex case of the Gelfand–Mazur theorem).
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Ideals and characters.
Let formula_1 be a unital "commutative" Banach algebra over formula_84 Since formula_1 is then a commutative ring with unit, every non-invertible element of formula_1 belongs to some maximal ideal of formula_48 Since a maximal ideal formula_88 in formula_1 is closed, formula_90 is a Banach algebra that is a field, and it follows from the Gelfand–Mazur theorem that there is a bijection between the set of all maximal ideals of formula_1 and the set formula_92 of all nonzero homomorphisms from formula_1 to formula_84 The set formula_92 is called the "structure space" or "character space" of formula_96 and its members "characters".
A character formula_97 is a linear functional on formula_1 that is at the same time multiplicative, formula_99 and satisfies formula_100 Every character is automatically continuous from formula_1 to formula_102 since the kernel of a character is a maximal ideal, which is closed. Moreover, the norm (that is, operator norm) of a character is one. Equipped with the topology of pointwise convergence on formula_1 (that is, the topology induced by the weak-* topology of formula_104), the character space, formula_105 is a Hausdorff compact space.
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For any formula_49
formula_107
where formula_108 is the Gelfand representation of formula_54 defined as follows: formula_108 is the continuous function from formula_92 to formula_55 given by formula_113 The spectrum of formula_114 in the formula above, is the spectrum as element of the algebra formula_115 of complex continuous functions on the compact space formula_116 Explicitly,
formula_117
As an algebra, a unital commutative Banach algebra is semisimple (that is, its Jacobson radical is zero) if and only if its Gelfand representation has trivial kernel. An important example of such an algebra is a commutative C*-algebra. In fact, when formula_1 is a commutative unital C*-algebra, the Gelfand representation is then an isometric *-isomorphism between formula_1 and formula_120
Banach *-algebras.
A Banach *-algebra formula_1 is a Banach algebra over the field of complex numbers, together with a map formula_122 that has the following properties:
In other words, a Banach *-algebra is a Banach algebra over formula_55 that is also a *-algebra.
In most natural examples, one also has that the involution is isometric, that is,
formula_135
Some authors include this isometric property in the definition of a Banach *-algebra.
A Banach *-algebra satisfying formula_136 is a C*-algebra. |
Boris Pasternak
Boris Leonidovich Pasternak (; ; 30 May 1960) was a Russian and Soviet poet, novelist, composer, and literary translator.
Composed in 1917, Pasternak's first book of poems, "My Sister, Life", was published in Berlin in 1922 and soon became an important collection in the Russian language. Pasternak's translations of stage plays by Goethe, Schiller, Calderón de la Barca and Shakespeare remain very popular with Russian audiences.
Pasternak was the author of "Doctor Zhivago" (1957), a novel that takes place between the Russian Revolution of 1905 and the First World War. "Doctor Zhivago" was rejected for publication in the USSR, but the manuscript was smuggled to Italy and was first published there in 1957.
Pasternak was awarded the Nobel Prize in Literature in 1958, an event that enraged the Communist Party of the Soviet Union, which forced him to decline the prize. In 1989, Pasternak's son Yevgeny finally accepted the award on his father's behalf. "Doctor Zhivago" has been part of the main Russian school curriculum since 2003.
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Early life.
Pasternak was born in Moscow on into a wealthy, assimilated Jewish family. His father was the post-Impressionist painter Leonid Pasternak, who taught as a professor at the Moscow School of Painting, Sculpture, and Architecture. His mother was Rosa Kaufman, a concert pianist and the daughter of Odessa industrialist Isadore Kaufman and his wife. Pasternak had a younger brother, Alex, and two sisters, Lydia and Josephine. The family claimed descent on the paternal line from Isaac Abarbanel, the famous 15th-century Sephardic Jewish philosopher, Bible commentator, and treasurer of Portugal.
Early education.
From 1904 to 1907, Boris Pasternak was the cloister-mate of Peter Minchakievich (1890–1963) in Holy Dormition Pochayev Lavra (now in Ukraine). Minchakievich came from an Orthodox Ukrainian family and Pasternak came from a Jewish family. Some confusion has arisen as to Pasternak attending a military academy in his boyhood years. The uniforms of their monastery Cadet Corp were only similar to those of The Czar Alexander the Third Military Academy, as Pasternak and Minchakievich never attended any military academy. Most schools used a distinctive military-looking uniform particular to them as was the custom of the time in Eastern Europe and Russia. Boyhood friends, they parted in 1908, friendly but with different politics, never to see each other again. Pasternak went to the Moscow Conservatory to study music (and later to Germany to study philosophy), and Minchakievich went to Lvov University to study history and philosophy. The good dimension of the character Strelnikov in "Dr. Zhivago" is based upon Peter Minchakievich. Several of Pasternak's characters are composites. After World War One and the Revolution, fighting for the Provisional or Republican government under Kerensky, and then escaping a Communist jail and execution, Minchakievich trekked across Siberia in 1917 and became an American citizen. Pasternak stayed in Russia.
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In a 1959 letter to Jacqueline de Proyart, Pasternak recalled:
Shortly after his birth, Pasternak's parents had joined the Tolstoyan Movement. Novelist Leo Tolstoy was a close family friend, as Pasternak recalled,
In a 1956 essay, Pasternak recalled his father's feverish work creating illustrations for Tolstoy's novel "Resurrection". The novel was serialized in the journal "Niva" by the publisher Fyodor Marx, based in St Petersburg. The sketches were drawn from observations in such places as courtrooms, prisons and on trains, in a spirit of realism. To ensure that the sketches met the journal deadline, train conductors were enlisted to personally collect the illustrations. Pasternak wrote,
According to Max Hayward,
Regular visitors to the Pasternaks' home also included Sergei Rachmaninoff, Alexander Scriabin, Lev Shestov, and Rainer Maria Rilke. Pasternak aspired first to be a musician. Inspired by Scriabin, Pasternak briefly was a student at the Moscow Conservatory. In 1910, he abruptly left for the University of Marburg in Germany, where he studied under neo-Kantian philosophers Hermann Cohen, Nicolai Hartmann, and Paul Natorp.
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Life and career.
Olga Freidenberg.
In 1910 Pasternak was reunited with his cousin Olga Freidenberg (1890–1955). They had shared the same nursery but had been separated when the Freidenberg family moved to Saint Petersburg. They fell in love immediately but were never lovers. The romance, however, is made clear from their letters, Pasternak writing:
The cousins' initial passion developed into a lifelong close friendship. From 1910 Pasternak and Freidenberg exchanged frequent letters, and their correspondence lasted over 40 years until 1954. The cousins last met in 1936.
Ida Wissotzkaya.
Pasternak fell in love with Ida Wissotzkaya, a girl from a notable Moscow Jewish , whose company Wissotzky Tea was the largest tea company in the world. Pasternak had tutored her in the final class of high school. He helped her prepare for finals. They met in Marburg during the summer of 1912 when Boris' father, Leonid Pasternak, painted her portrait.
Although Professor Cohen encouraged him to remain in Germany and to pursue a Philosophy doctorate, Pasternak decided against it. He returned to Moscow around the time of the outbreak of the First World War. In the aftermath of events, Pasternak proposed marriage to Ida. However, the Wissotzky family was disturbed by Pasternak's poor prospects and persuaded Ida to refuse him. She turned him down and he told of his love and rejection in the poem "Marburg" (1917):
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<poem>I quivered. I flared up, and then was extinguished.
I shook. I had made a proposal—but late,
Too late. I was scared, and she had refused me.
I pity her tears, am more blessed than a saint.</poem>
Around this time, when he was back in Russia, he joined the Russian Futurist group Centrifuge (Tsentrifuga) as a pianist; poetry was still only a hobby for him at that time. It was in their group journal, "Lirika", where some of his earliest poems were published. His involvement with the Futurist movement as a whole reached its peak when, in 1914, he published a satirical article in "Rukonog", which attacked the jealous leader of the "Mezzanine of Poetry", Vadim Shershenevich, who was criticizing "Lirika" and the Ego-Futurists because Shershenevich himself was barred from collaborating with Centrifuge, the reason being that he was such a talentless poet. The action eventually caused a verbal battle amongst several members of the groups, fighting for recognition as the first, truest Russian Futurists; these included the Cubo-Futurists, who were by that time already notorious for their scandalous behaviour. Pasternak's first and second books of poetry were published shortly after these events.
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Another failed love affair in 1917 inspired the poems in his third and first major book, "My Sister, Life". His early verse cleverly dissimulates his preoccupation with Immanuel Kant's philosophy. Its fabric includes striking alliterations, wild rhythmic combinations, day-to-day vocabulary, and hidden allusions to his favourite poets such as Rilke, Lermontov, Pushkin and German-language Romantic poets.
During World War I, Pasternak taught and worked at a chemical factory in Vsevolodo-Vilva near Perm, which undoubtedly provided him with material for "Dr. Zhivago" many years later. Unlike the rest of his family and many of his closest friends, Pasternak chose not to leave Russia after the October Revolution of 1917. According to Max Hayward,
When it finally was published in 1922, Pasternak's "My Sister, Life" revolutionised Russian poetry. It made Pasternak the model for younger poets, and decisively changed the poetry of Osip Mandelshtam, Marina Tsvetayeva and others.
Following "My Sister, Life", Pasternak produced some hermetic pieces of uneven quality, including his masterpiece, the lyric cycle "Rupture" (1921). Both Pro-Soviet writers and their White émigré equivalents applauded Pasternak's poetry as pure, unbridled inspiration.
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In the late 1920s, he also participated in the much celebrated tripartite correspondence with Rilke and Tsvetayeva. As the 1920s wore on, however, Pasternak increasingly felt that his colourful style was at odds with a less educated readership. He attempted to make his poetry more comprehensible by reworking his earlier pieces and starting two lengthy poems on the Russian Revolution of 1905. He also turned to prose and wrote several autobiographical stories, notably "The Childhood of Luvers" and "Safe Conduct". (The collection "Zhenia's Childhood and Other Stories" would be published in 1982.)
In 1922 Pasternak married Evgeniya Lurye (Евгения Лурье), a student at the Art Institute. The following year their son Yevgeny was born.
Evidence of Pasternak's support of still-revolutionary members of the leadership of the Communist Party as late as 1926 is indicated by his poem "In Memory of Reissner" presumably written upon the premature death from typhus of Bolshevik leader Larissa Reissner aged 30 in February of that year.
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By 1927, Pasternak's close friends Vladimir Mayakovsky and Nikolai Aseyev were advocating the complete subordination of the arts to the needs of the Communist Party of the Soviet Union. In a letter to his sister Josephine, Pasternak wrote of his intentions to "break off relations" with both of them. Although he expressed that it would be deeply painful, Pasternak explained that it could not be prevented. He explained:
By 1932, Pasternak had strikingly reshaped his style to make it more understandable to the general public and printed the new collection of poems, aptly titled "The Second Birth". Although its Caucasian pieces were as brilliant as the earlier efforts, the book alienated the core of Pasternak's refined audience abroad, which was largely composed of anti-communist émigrés.
In 1932, Pasternak fell in love with Zinaida Neuhaus, the wife of the Russian pianist Heinrich Neuhaus. They both got divorces and married two years later.
Pasternak continued to change his poetry, simplifying his style and language through the years, as expressed in his next book, "Early Trains" (1943).
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Stalin Epigram.
In April 1934 Osip Mandelstam recited his "Stalin Epigram" to Pasternak. After listening, Pasternak told Mandelstam:
On the night of 14 May 1934, Mandelstam was arrested at his home based on a warrant signed by NKVD boss Genrikh Yagoda. Devastated, Pasternak went immediately to the offices of "Izvestia" and begged Nikolai Bukharin to intercede on Mandelstam's behalf.
Soon after his meeting with Bukharin, the telephone rang in Pasternak's Moscow apartment. A voice from the Kremlin said, According to Ivinskaya, Pasternak was struck dumb. Flustered, Pasternak denied that there was any discussion or that there were any literary circles left in Soviet Russia. Stalin went on to ask him for his own opinion of Mandelstam. In an "eager fumbling manner" Pasternak explained that he and Mandelstam each had a completely different philosophy about poetry. Stalin finally said, in a mocking tone of voice: , and put down the receiver.
Great Purge.
According to Pasternak, during the 1937 trial of General Iona Yakir and Marshal Mikhail Tukhachevsky, the Union of Soviet Writers requested all members to add their names to a statement supporting the death penalty for the defendants. Pasternak refused to sign, even after leadership of the Union visited him.
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Soon after, Pasternak appealed directly to Stalin, describing his family's strong Tolstoyan convictions and putting his own life at Stalin's disposal; he said that he could not stand as a self-appointed judge of life and death. Pasternak was certain that he would be arrested, but instead Stalin is said to have crossed Pasternak's name off an execution list, reportedly declaring, (or, in another version, ).
Pasternak's close friend Titsian Tabidze did fall victim to the Great Purge. In an autobiographical essay published in the 1950s, Pasternak described the execution of Tabidze and the suicides of Marina Tsvetaeva and Paolo Iashvili.
Ivinskaya wrote,
World War II.
When the Luftwaffe began bombing Moscow, Pasternak immediately began to serve as a fire warden on the roof of the writer's building on Lavrushinski Street. According to Ivinskaya, he repeatedly helped to dispose of German bombs which fell on it.
In 1943, Pasternak was finally granted permission to visit the soldiers at the front. He bore it well, considering the hardships of the journey (he had a weak leg from an old injury), and he wanted to go to the most dangerous places. He read his poetry and talked extensively with the active and injured troops.
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Pasternak later said,
Olga Ivinskaya.
In October 1946, the twice-married Pasternak met Olga Ivinskaya, a 34 year old single mother employed by "Novy Mir". Deeply moved by her resemblance to his first love Ida Vysotskaya, Pasternak gave Ivinskaya several volumes of his poetry and literary translations. Although Pasternak never left his wife Zinaida, he started an extramarital relationship with Ivinskaya that would last for the remainder of Pasternak's life. Ivinskaya later recalled,
She gave him the phone number of her neighbour Olga Volkova who resided below. In the evenings, Pasternak would phone and Volkova would signal by Olga banging on the water pipe which connected their apartments.
When they first met, Pasternak was translating the verse of the Hungarian national poet, Sándor Petőfi. Pasternak gave his lover a book of Petőfi with the inscription,
Pasternak later noted on a photograph of himself: Ivinskaya would later describe the Petőfi translations as "a first declaration of love".
According to Ivinskaya, Zinaida Pasternak was infuriated by her husband's infidelity. Once, when his younger son Leonid fell seriously ill, Zinaida extracted a promise from her husband, as they stood by the boy's sickbed, that he would end his affair with Ivinskaya. Pasternak asked Luisa Popova, a mutual friend, to tell Ivinskaya about his promise. Popova told him that he must do it himself. Soon after, Ivinskaya happened to be ill at Popova's apartment, when suddenly Zinaida Pasternak arrived and confronted her.
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Ivinskaya later recalled,
In 1948, Pasternak advised Ivinskaya to resign her job at "Novy Mir", which was becoming extremely difficult due to their relationship. In the aftermath, Pasternak began to instruct her in translating poetry. In time, they began to refer to her apartment on Potapov Street as, "Our Shop".
On the evening of 6 October 1949, Ivinskaya was arrested at her apartment by the KGB. Ivinskaya relates in her memoirs that, when the agents burst into her apartment, she was at her typewriter working on translations of the Korean poet Won Tu-Son. Her apartment was ransacked and all items connected with Pasternak were piled up in her presence. Ivinskaya was taken to the Lubyanka Prison and repeatedly interrogated, where she refused to say anything incriminating about Pasternak. At the time, she was pregnant with Pasternak's child and had a miscarriage early in her ten-year sentence in the GULAG.
Upon learning of his mistress' arrest, Pasternak telephoned Luisa Popova and asked her to come at once to Gogol Boulevard. She found him sitting on a bench near the Palace of Soviets Metro Station. Weeping, Pasternak told her,
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According to Ivinskaya,
In a 1958 letter to a friend in West Germany, Pasternak wrote,
Translating Goethe.
Pasternak's translation of the of "Faust" led him to be attacked in the August 1950 edition of "Novy Mir". The critic accused Pasternak of distorting Goethe's "progressive" meanings to support , as well as introducing aesthetic and individualist values. In a subsequent letter to the daughter of Marina Tsvetaeva, Pasternak explained that the attack was motivated by the fact that the supernatural elements of the play, which "Novy Mir" considered, "irrational", had been translated as Goethe had written them. Pasternak further declared that, despite the attacks on his translation, his contract for the had not been revoked.
Khrushchev thaw.
When Stalin died of a stroke on 5 March 1953, Ivinskaya was still imprisoned in the Gulag, and Pasternak was in Moscow. Across the nation, there were waves of panic, confusion, and public displays of grief. Pasternak wrote,
After her release, Pasternak's relationship with Ivinskaya picked up where it had left off. Soon after he confided in her, During this period, Pasternak delighted in reading a clandestine copy of George Orwell's "Animal Farm" in English. In conversation with Ivinskaya, Pasternak explained that the pig dictator Napoleon, in the novel, "vividly reminded" him of Soviet Premier Nikita Khrushchev.
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"Doctor Zhivago".
Although it contains passages written in the 1910s and 1920s, "Doctor Zhivago" was not completed until 1955. Pasternak submitted the novel to "Novy Mir" in 1956, which refused publication due to its rejection of socialist realism. The author, like his protagonist Yuri Zhivago, showed more concern for the welfare of individual characters than for the "progress" of society. Censors also regarded some passages as anti-Soviet, especially the novel's criticisms of Stalinism, Collectivisation, the Great Purge, and the Gulag.
Pasternak's fortunes were soon to change, however. In March 1956, the Italian Communist Party sent a journalist, Sergio D'Angelo, to work in the Soviet Union, and his status as a journalist as well as his membership in the Italian Communist Party allowed him to have access to various aspects of the cultural life in Moscow at the time. A Milan publisher, the communist Giangiacomo Feltrinelli, had also given him a commission to find new works of Soviet literature that would be appealing to Western audiences, and upon learning of "Doctor Zhivago"s existence, D'Angelo travelled immediately to Peredelkino and offered to submit Pasternak's novel to Feltrinelli's company for publication. At first Pasternak was stunned. Then he brought the manuscript from his study and told D'Angelo with a laugh,
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According to Lazar Fleishman, Pasternak was aware that he was taking a huge risk. No Soviet author had attempted to deal with Western publishers since the 1920s, when such behavior led the Soviet State to declare war on Boris Pilnyak and Evgeny Zamyatin. Pasternak, however, believed that Feltrinelli's Communist affiliation would not only guarantee publication, but might even force the Soviet State to publish the novel in Russia.
In a rare moment of agreement, both Olga Ivinskaya and Zinaida Pasternak were horrified by the submission of "Doctor Zhivago" to a Western publishing house. Pasternak, however, refused to change his mind and informed an emissary from Feltrinelli that he was prepared to undergo any sacrifice in order to see "Doctor Zhivago" published.
In 1957, Feltrinelli announced that the novel would be published by his company. Despite repeated demands from visiting Soviet emissaries, Feltrinelli refused to cancel or delay publication. According to Ivinskaya, The Soviet government forced Pasternak to cable the publisher to withdraw the manuscript, but he sent separate, secret letters advising Feltrinelli to ignore the telegrams.
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Helped considerably by the Soviet campaign against the novel (as well as by the U.S. Central Intelligence Agency's secret purchase of hundreds of copies of the book as it came off the presses around the world – see "Nobel Prize" section below), "Doctor Zhivago" became an instant sensation throughout the non-Communist world upon its release in November 1957. In the State of Israel, however, Pasternak's novel was sharply criticized for its assimilationist views towards the Jewish people. When informed of this, Pasternak responded, According to Lazar Fleishman, Pasternak had written the disputed passages prior to Israeli independence. At the time, Pasternak had also been regularly attending Russian Orthodox Divine Liturgy. Therefore, he believed that Soviet Jews converting to Christianity was preferable to assimilating into atheism and Stalinism.
The first English translation of "Doctor Zhivago" was hastily produced by Max Hayward and Manya Harari in order to coincide with overwhelming public demand. It was released in August 1958, and remained the only edition available for more than fifty years. Between 1958 and 1959, the English language edition spent 26 weeks at the top of "The New York Times"' bestseller list.
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Ivinskaya's daughter Irina circulated typed copies of the novel in Samizdat. Although no Soviet critics had read the banned novel, "Doctor Zhivago" was pilloried in the State-owned press. Similar attacks led to a humorous Russian saying, "I haven't read Pasternak, but I condemn him".
During the aftermath of the Second World War, Pasternak had composed a series of poems on Gospel themes. According to Ivinskaya, Pasternak had regarded Stalin as a Therefore, Pasternak's decision to write Christian poetry was .
On 9 September 1958, the "Literary Gazette" critic Viktor Pertsov retaliated by denouncing Furthermore, the author received much hate mail from Communists both at home and abroad. According to Ivinskaya, Pasternak continued to receive such letters for the remainder of his life.
In a letter written to his sister Josephine, however, Pasternak recalled the words of his friend Ekaterina Krashennikova upon reading "Doctor Zhivago". She had said,
Nobel Prize.
According to Yevgeni Borisovich Pasternak, According to the former Nobel Committee head Lars Gyllensten, his nomination was discussed every year from 1946 to 1950, then again in 1957 (it was finally awarded in 1958). Pasternak guessed at this from the growing waves of criticism in USSR. Sometimes he had to justify his European fame: 'According to the Union of Soviet Writers, some literature circles of the West see unusual importance in my work, not matching its modesty and low productivity...'
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Meanwhile, Pasternak wrote to Renate Schweitzer and his sister, Lydia Pasternak Slater. In both letters, the author expressed hope that he would be passed over by the Nobel Committee in favour of Alberto Moravia. Pasternak wrote that he was wracked with torments and anxieties at the thought of placing his loved ones in danger.
On 23 October 1958, Boris Pasternak was announced as the winner of the Nobel Prize. The citation credited Pasternak's contribution to Russian lyric poetry and for his role in On 25 October, Pasternak sent a telegram to the Swedish Academy: That same day, the Literary Institute in Moscow demanded that all its students sign a petition denouncing Pasternak and his novel. They were further ordered to join a "spontaneous" demonstration demanding Pasternak's exile from the Soviet Union. Also on that day, the "Literary Gazette" published a letter which was sent to B. Pasternak in September 1956 by the editors of the Soviet literary journal Novy Mir to justify their rejection of Doctor Zhivago. In publishing this letter the Soviet authorities wished to justify the measures they had taken against the author and his work. On 26 October, the "Literary Gazette" ran an article by David Zaslavski entitled, "Reactionary Propaganda Uproar over a Literary Weed".
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According to Solomon Volkov:
Furthermore, Pasternak was informed that, if he traveled to Stockholm to collect his Nobel Medal, he would be refused re-entry to the Soviet Union. As a result, on 29 October Pasternak sent a second telegram to the Nobel Committee: The Swedish Academy announced:
According to Yevgenii Pasternak,
Deportation plans.
Despite his decision to decline the award, the Union of Soviet Writers continued to demonise Pasternak in the State-owned press. Furthermore, he was threatened at the very least with formal exile to the West. In response, Pasternak wrote directly to Soviet Premier Nikita Khrushchev,
In "The Oak and the Calf", Alexander Solzhenitsyn sharply criticized Pasternak, both for declining the Nobel Prize and for sending such a letter to Khrushchev. In her own memoirs, Olga Ivinskaya blames herself for pressuring her lover into making both decisions.
According to Yevgenii Pasternak,
On 31 October 1958, the Union of Soviet Writers held a trial behind closed doors. According to the meeting minutes, Pasternak was denounced as an internal émigré and a Fascist fifth columnist. Afterwards, the attendees announced that Pasternak had been expelled from the Union. They further signed a petition to the Politburo, demanding that Pasternak be stripped of his Soviet citizenship and exiled to According to Yevgenii Pasternak, however, author Konstantin Paustovsky refused to attend the meeting. Yevgeny Yevtushenko did attend, but walked out in disgust.
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According to Yevgenii Pasternak, his father would have been exiled had it not been for Indian Prime Minister Jawaharlal Nehru, who telephoned Khrushchev and threatened to organize a Committee for Pasternak's protection.
It is possible that the 1958 Nobel Prize prevented Pasternak's imprisonment due to the Soviet State's fear of international protests. Yevgenii Pasternak believes, however, that the resulting persecution fatally weakened his father's health.
Meanwhile, Bill Mauldin produced that won the 1959 Pulitzer Prize for Editorial Cartooning. The cartoon depicts Pasternak as a GULAG inmate splitting trees in the snow, saying to another inmate:
Last years.
Pasternak's post-"Zhivago" poetry probes the universal questions of love, immortality, and reconciliation with God. Boris Pasternak wrote his last complete book, "When the Weather Clears", in 1959.
According to Ivinskaya, Pasternak continued to stick to his daily writing schedule even during the controversy over "Doctor Zhivago". He also continued translating the writings of Juliusz Słowacki and Pedro Calderón de la Barca. In his work on Calderon, Pasternak received the discreet support of Nikolai Mikhailovich Liubimov, a senior figure in the Party's literary apparatus. Ivinskaya describes Liubimov as, "a shrewd and enlightened person who understood very well that all the mudslinging and commotion over the novel would be forgotten, but that there would always be a Pasternak." In a letter to his sisters in Oxford, England, Pasternak claimed to have finished translating one of Calderon's plays in less than a week.
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During the summer of 1959, Pasternak began writing "The Blind Beauty", a trilogy of stage plays set before and after Alexander II's abolition of serfdom in Russia. In an interview with Olga Carlisle from "The Paris Review", Pasternak enthusiastically described the play's plot and characters. He informed Olga Carlisle that, at the end of "The Blind Beauty", he wished to depict "the birth of an enlightened and affluent middle class, open to occidental influences, progressive, intelligent, artistic". However, Pasternak fell ill with terminal lung cancer before he could complete the first play of the trilogy.
Death.
Boris Pasternak died of lung cancer in his dacha in Peredelkino on the evening of 30 May 1960. He first summoned his sons, and in their presence said, Pasternak's last words were,
Funeral demonstration.
Despite only a small notice appearing in the "Literary Gazette", handwritten notices carrying the date and time of the funeral were posted throughout the Moscow subway system. As a result, thousands of admirers braved Militia and KGB surveillance to attend Pasternak's funeral in Peredelkino.
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Before Pasternak's civil funeral, Ivinskaya had a conversation with Konstantin Paustovsky. According to her,
Then, in the presence of a large number of foreign journalists, the body of Pasternak was removed to the cemetery. According to Ivinskaya,
To the horror of the assembled Party officials, however, someone with "a young and deeply anguished voice" began reciting Pasternak's banned poem "Hamlet".
According to Ivinskaya,
The final speaker at the graveside service said,
As the spectators cheered, the bells of Peredelkino's Church of the Transfiguration began to toll. Written prayers for the dead were then placed upon Pasternak's forehead and the coffin was closed and buried. Pasternak's gravesite would go on to become a major shrine for members of the Soviet dissident movement.
Legacy.
After Pasternak's death, Ivinskaya was arrested for the second time, with her daughter, Irina Emelyanova. Both were accused of being Pasternak's link with Western publishers and of dealing in hard currency for "Doctor Zhivago". All of Pasternak's letters to Ivinskaya, as well as many other manuscripts and documents, were seized by the KGB. The KGB quietly released them, Irina after one year, in 1962, and Olga in 1964. By this time, Ivinskaya had served four years of an eight-year sentence, in retaliation for her role in "Doctor Zhivago"'s publication. In 1978, her memoirs were smuggled abroad and published in Paris. An English translation by Max Hayward was published the same year under the title "A Captive of Time: My Years with Pasternak".
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Ivinskaya was rehabilitated only in 1988. After the dissolution of the Soviet Union, Ivinskaya sued for the return of the letters and documents seized by the KGB in 1961. The Russian Supreme Court ultimately ruled against her, stating that "there was no proof of ownership" and that the "papers should remain in the state archive". Ivinskaya died of cancer on 8 September 1995. A reporter on NTV compared her role to that of other famous muses for Russian poets: "As Pushkin would not be complete without Anna Kern, and Yesenin would be nothing without Isadora, so Pasternak would not be Pasternak without Olga Ivinskaya, who was his inspiration for "Doctor Zhivago".".
Meanwhile, Boris Pasternak continued to be pilloried by the Soviet State until Mikhail Gorbachev proclaimed Perestroika during the 1980s.
In 1980, an asteroid was named 3508 Pasternak after Boris Pasternak.
In 1988, after decades of circulating in Samizdat, "Doctor Zhivago" was serialized in the literary journal "Novy Mir".
In December 1989, Yevgenii Borisovich Pasternak was permitted to travel to Stockholm in order to collect his father's Nobel Medal. At the ceremony, acclaimed cellist and Soviet dissident Mstislav Rostropovich performed a Bach serenade in honor of his deceased countryman.
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The Pasternak family papers are stored at the Hoover Institution Archives, Stanford University. They contain correspondence, drafts of "Doctor Zhivago" and other writings, photographs, and other material, of Boris Pasternak and other family members.
Since 2003, during the first presidency of Vladimir Putin, the novel "Doctor Zhivago" has entered the Russian school curriculum, where it is read in the 11th grade of secondary school.
Commemoration.
In October 1984 by decision of a court, Pasternak's dacha in Peredelkino was taken from the writer's relatives and transferred to state ownership. Two years later, in 1986, the House-Museum of Boris Pasternak was founded (the first house-museum in the USSR).
In 1990, the year of the poet's 100th anniversary, the Pasternak Museum opened its doors in Chistopol, in the house where the poet evacuated to during the Great Patriotic War (1941–1943), and in Peredelkino, where he lived for many years until his death. The head of the poet's house-museum is Natalia Pasternak, his daughter-in-law (widow of the youngest son Leonid).
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In 2008 a museum was opened in Vsevolodo-Vilva in the house where the budding poet lived from January to June 1916.
In 2009 on the City Day in Perm the first Russian monument to Pasternak was erected in the square near the Opera Theater (sculptor: Elena Munc).
A memorial plaque was installed on the house where Pasternak was born.
In memory of the poet's three-time stay in Tula, on 27 May 2005 a marble memorial plaque to Pasternak was installed on the Wörmann hotel's wall, as Pasternak was a Nobel laureate and dedicated several of his works to Tula.
On 20 February 2008, in Kyiv, a memorial plaque was put up on the house No.9 on Lipinsky Street, but seven years later it was stolen by vandals.
In 2012 a monument to Boris Pasternak was erected in the district center of Muchkapsky by Z. Tsereteli.
In 1990, as part of the series "Nobel Prize Winners", the USSR and Sweden ("Nobel Prize Winners – Literature") issued stamps depicting Boris Pasternak.
In 2015, as part of the series "125th Annive. of the Birth of Boris Pasternak, 1890–1960", Mozambique issued a miniature sheet depicting Boris Pasternak. Although this issue was acknowledged by the postal administration of Mozambique, the issue was not placed on sale in Mozambique, and was only distributed to the new issue trade by Mozambique's philatelic agent.
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In 2015, as part of the series "125th Birth Anniversary of Boris Pasternak", Maldives issued a miniature sheet depicting Boris Pasternak. The issue was acknowledged by the Maldive postal authorities, but only distributed by the Maldive philatelic agent for collecting purposes.
On the occasion of the 50th anniversary of B. Pasternak's Nobel Prize, the Principality of Monaco issued a postage stamp in his memory.
On 27 January 2015, in honor of the poet's 125th birthday, the Russian Post issued an envelope with the original stamp.
On 1 October 2015, a monument to Pasternak was erected in Chistopol.
On 10 February 2020, a celebration of the 130th birthday anniversary was held at Exhibition of Achievements of National Economy in Moscow.
On 10 February 2021, Google celebrated his 131st birthday with a Google Doodle. The Doodle was displayed in Russia, Sweden, some Middle Eastern countries and some Mediterranean countries.
Adaptations.
The first screen adaptation of "Doctor Zhivago", adapted by Robert Bolt and directed by David Lean, appeared in 1965. The film, which toured in the roadshow tradition, starred Omar Sharif, Geraldine Chaplin, and Julie Christie. Concentrating on the love triangle aspects of the novel, the film became a worldwide blockbuster, but was unavailable in Russia until perestroika.
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In 2002, the novel was adapted as a television miniseries. Directed by Giacomo Campiotti, the serial starred Hans Matheson, Alexandra Maria Lara, Keira Knightley, and Sam Neill.
The Russian TV version of 2006, directed by Aleksandr Proshkin and starring Oleg Menshikov as Zhivago, is considered more faithful to Pasternak's novel than David Lean's 1965 film.
Work.
Poetry.
Thoughts on poetry.
According to Olga Ivinskaya:
For this reason, Pasternak avoided literary cafes where young poets regularly invited them to read their verse. According to Ivinskaya,
Also according to Ivinskaya,
Translation.
Reluctant to conform to socialist realism, Pasternak turned to translation in order to provide for his family. He soon produced acclaimed translations of Sándor Petőfi, Johann Wolfgang von Goethe, Rainer Maria Rilke, Paul Verlaine, Taras Shevchenko, and Nikoloz Baratashvili. Osip Mandelstam, however, privately warned him,
In a 1942 letter, Pasternak declared,
According to Ivinskaya, Pasternak believed in not being too literal in his translations, which he felt could confuse the meaning of the text. He instead advocated observing each poem from afar to plumb its true depths.
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Pasternak's translations of William Shakespeare ("Romeo and Juliet", "Antony and Cleopatra", "Othello", "King Henry IV" (Part I) and (Part II), "Hamlet", "Macbeth", "King Lear") remain deeply popular with Russian audiences because of their colloquial, modernised dialogues. Pasternak's critics, however, accused him of "pasternakizing" Shakespeare. In a 1956 essay, Pasternak wrote:
According to Ivinskaya:
While they were both collaborating on translating Rabindranath Tagore from Bengali into Russian, Pasternak advised Ivinskaya: "1) Bring out the theme of the poem, its subject matter, as clearly as possible; 2) tighten up the fluid, non-European form by rhyming internally, not at the end of the lines; 3) use loose, irregular meters, mostly ternary ones. You may allow yourself to use assonances."
Later, while she was collaborating with him on a translation of Vítězslav Nezval, Pasternak told Ivinskaya:
According to Ivinskaya, however, translation was not a genuine vocation for Pasternak. She later recalled:
Music.
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Boris Pasternak was also a composer, and had a promising musical career as a musician ahead of him, had he chosen to pursue it. He came from a musical family: his mother was a concert pianist and a student of Anton Rubinstein and Theodor Leschetizky, and Pasternak's early impressions were of hearing piano trios in the home. The family had a dacha (country house) close to one occupied by Alexander Scriabin. Sergei Rachmaninoff, Rainer Maria Rilke and Leo Tolstoy were all visitors to the family home. His father Leonid was a painter who produced one of the most important portraits of Scriabin, and Pasternak wrote many years later of witnessing with great excitement the creation of Scriabin's Symphony No. 3 ("The Divine Poem"), in 1903.
Pasternak began to compose at the age of 13. The high achievements of his mother discouraged him from becoming a pianist, but – inspired by Scriabin – he entered the Moscow Conservatory, but left abruptly in 1910 at the age of twenty, to study philosophy in Marburg University. Four years later he returned to Moscow, having finally decided on a career in literature, publishing his first book of poems, influenced by Aleksandr Blok and the Russian Futurists, the same year.
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Pasternak's early compositions show the clear influence of Scriabin. His single-movement Piano Sonata of 1909 shows a more mature and individual voice. Nominally in B minor, it moves freely from key to key with frequent changes of key-signature and a chromatic dissonant style that defies easy analysis. Although composed during his time at the Conservatory, the Sonata was composed at Rayki, some 40km north-east of Moscow, where Leonid Pasternak had his painting studio and taught his students. |
Binomial coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written formula_1 It is the coefficient of the term in the polynomial expansion of the binomial power ; this coefficient can be computed by the multiplicative formula
which using factorial notation can be compactly expressed as
For example, the fourth power of is
and the binomial coefficient formula_5 is the coefficient of the term.
Arranging the numbers formula_6 in successive rows for gives a triangular array called Pascal's triangle, satisfying the recurrence relation
The binomial coefficients occur in many areas of mathematics, and especially in combinatorics. In combinatorics the symbol formula_8 is usually read as " choose " because there are formula_8 ways to choose an (unordered) subset of elements from a fixed set of elements. For example, there are formula_10 ways to choose elements from , namely , , , , and .
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The first form of the binomial coefficients can be generalized to formula_11 for any complex number and integer , and many of their properties continue to hold in this more general form.
History and notation.
Andreas von Ettingshausen introduced the notation formula_12 in 1826, although the numbers were known centuries earlier (see Pascal's triangle). In about 1150, the Indian mathematician Bhaskaracharya gave an exposition of binomial coefficients in his book "Līlāvatī".
Alternative notations include , , , , , and , in all of which the stands for "combinations" or "choices"; the notation means the number of ways to choose "k" out of "n" objects. Many calculators use variants of the because they can represent it on a single-line display. In this form the binomial coefficients are easily compared to the numbers of -permutations of, written as , etc.
Definition and interpretations.
For natural numbers (taken to include 0) and , the binomial coefficient formula_12 can be defined as the coefficient of the monomial in the expansion of . The same coefficient also occurs (if ) in the binomial formula
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(valid for any elements , of a commutative ring),
which explains the name "binomial coefficient".
Another occurrence of this number is in combinatorics, where it gives the number of ways, disregarding order, that objects can be chosen from among objects; more formally, the number of -element subsets (or -combinations) of an -element set. This number can be seen as equal to the one of the first definition, independently of any of the formulas below to compute it: if in each of the factors of the power one temporarily labels the term with an index (running from to ), then each subset of indices gives after expansion a contribution , and the coefficient of that monomial in the result will be the number of such subsets. This shows in particular that formula_12 is a natural number for any natural numbers and . There are many other combinatorial interpretations of binomial coefficients (counting problems for which the answer is given by a binomial coefficient expression), for instance the number of words formed of bits (digits 0 or 1) whose sum is is given by formula_12, while the number of ways to write formula_16 where every is a nonnegative integer is given by . Most of these interpretations can be shown to be equivalent to counting -combinations.
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Computing the value of binomial coefficients.
Several methods exist to compute the value of formula_8 without actually expanding a binomial power or counting -combinations.
Recursive formula.
One method uses the recursive, purely additive formula
formula_18 for all integers formula_19 such that formula_20
with boundary values
formula_21
for all integers .
The formula follows from considering the set and counting separately (a) the -element groupings that include a particular set element, say ", in every group (since " is already chosen to fill one spot in every group, we need only choose from the remaining ) and (b) all the "k"-groupings that don't include ""; this enumerates all the possible -combinations of elements. It also follows from tracing the contributions to "X""k" in . As there is zero or in , one might extend the definition beyond the above boundaries to include formula_22 when either or . This recursive formula then allows the construction of Pascal's triangle, surrounded by white spaces where the zeros, or the trivial coefficients, would be.
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Multiplicative formula.
A more efficient method to compute individual binomial coefficients is given by the formula
formula_23
where the numerator of the first fraction, formula_24, is a falling factorial.
This formula is easiest to understand for the combinatorial interpretation of binomial coefficients.
The numerator gives the number of ways to select a sequence of distinct objects, retaining the order of selection, from a set of objects. The denominator counts the number of distinct sequences that define the same -combination when order is disregarded. This formula can also be stated in a recursive form. Using the "C" notation from above, formula_25, where formula_26. It is readily derived by evaluating formula_27 and can intuitively be understood as starting at the leftmost coefficient of the formula_28-th row of Pascal's triangle, whose value is always formula_29, and recursively computing the next coefficient to its right until the formula_30-th one is reached.
Due to the symmetry of the binomial coefficients with regard to and , calculation of the above product, as well as the recursive relation, may be optimised by setting its upper limit to the smaller of and .
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Factorial formula.
Finally, though computationally unsuitable, there is the compact form, often used in proofs and derivations, which makes repeated use of the familiar factorial function:
formula_31
where denotes the factorial of . This formula follows from the multiplicative formula above by multiplying numerator and denominator by ; as a consequence it involves many factors common to numerator and denominator. It is less practical for explicit computation (in the case that is small and is large) unless common factors are first cancelled (in particular since factorial values grow very rapidly). The formula does exhibit a symmetry that is less evident from the multiplicative formula (though it is from the definitions)
which leads to a more efficient multiplicative computational routine. Using the falling factorial notation,
formula_32
Generalization and connection to the binomial series.
The multiplicative formula allows the definition of binomial coefficients to be extended by replacing "n" by an arbitrary number "α" (negative, real, complex) or even an element of any commutative ring in which all positive integers are invertible:
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formula_33
With this definition one has a generalization of the binomial formula (with one of the variables set to 1), which justifies still calling the formula_34 binomial coefficients:
This formula is valid for all complex numbers "α" and "X" with |"X"| < 1. It can also be interpreted as an identity of formal power series in "X", where it actually can serve as definition of arbitrary powers of power series with constant coefficient equal to 1; the point is that with this definition all identities hold that one expects for exponentiation, notably
formula_35
If "α" is a nonnegative integer "n", then all terms with are zero, and the infinite series becomes a finite sum, thereby recovering the binomial formula. However, for other values of "α", including negative integers and rational numbers, the series is really infinite.
Pascal's triangle.
Pascal's rule is the important recurrence relation
which can be used to prove by mathematical induction that formula_36 is a natural number for all integer "n" ≥ 0 and all integer "k", a fact that is not immediately obvious from formula (1). To the left and right of Pascal's triangle, the entries (shown as blanks) are all zero.
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Pascal's rule also gives rise to Pascal's triangle:
Row number contains the numbers formula_8 for . It is constructed by first placing 1s in the outermost positions, and then filling each inner position with the sum of the two numbers directly above. This method allows the quick calculation of binomial coefficients without the need for fractions or multiplications. For instance, by looking at row number 5 of the triangle, one can quickly read off that
Combinatorics and statistics.
Binomial coefficients are of importance in combinatorics because they provide ready formulas for certain frequent counting problems:
Binomial coefficients as polynomials.
For any nonnegative integer "k", the expression formula_45 can be written as a polynomial with denominator :
this presents a polynomial in "t" with rational coefficients.
As such, it can be evaluated at any real or complex number "t" to define binomial coefficients with such first arguments. These "generalized binomial coefficients" appear in Newton's generalized binomial theorem.
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For each "k", the polynomial formula_47 can be characterized as the unique degree "k" polynomial satisfying and .
Its coefficients are expressible in terms of Stirling numbers of the first kind:
The derivative of formula_47 can be calculated by logarithmic differentiation:
This can cause a problem when evaluated at integers from formula_51 to formula_52, but using identities below we can compute the derivative as:
Binomial coefficients as a basis for the space of polynomials.
Over any field of characteristic 0 (that is, any field that contains the rational numbers), each polynomial "p"("t") of degree at most "d" is uniquely expressible as a linear combination formula_54 of binomial coefficients, because the binomial coefficients consist of one polynomial of each degree. The coefficient "a""k" is the "k"th difference of the sequence "p"(0), "p"(1), ..., "p"("k"). Explicitly,
Integer-valued polynomials.
Each polynomial formula_47 is integer-valued: it has an integer value at all integer inputs formula_56. (One way to prove this is by induction on "k" using Pascal's identity.) Therefore, any integer linear combination of binomial coefficient polynomials is integer-valued too. Conversely, () shows that any integer-valued polynomial is an integer linear combination of these binomial coefficient polynomials. More generally, for any subring "R" of a characteristic 0 field "K", a polynomial in "K"["t"] takes values in "R" at all integers if and only if it is an "R"-linear combination of binomial coefficient polynomials.
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Example.
The integer-valued polynomial can be rewritten as
Identities involving binomial coefficients.
The factorial formula facilitates relating nearby binomial coefficients. For instance, if "k" is a positive integer and "n" is arbitrary, then
and, with a little more work,
We can also get
Moreover, the following may be useful:
For constant "n", we have the following recurrence:
To sum up, we have
Sums of the binomial coefficients.
The formula
says that the elements in the th row of Pascal's triangle always add up to 2 raised to the th power. This is obtained from the binomial theorem () by setting and . The formula also has a natural combinatorial interpretation: the left side sums the number of subsets of {1, ..., "n"} of sizes "k" = 0, 1, ..., "n", giving the total number of subsets. (That is, the left side counts the power set of {1, ..., "n"}.) However, these subsets can also be generated by successively choosing or excluding each element 1, ..., "n"; the "n" independent binary choices (bit-strings) allow a total of formula_64 choices. The left and right sides are two ways to count the same collection of subsets, so they are equal.
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The formulas
and
follow from the binomial theorem after differentiating with respect to (twice for the latter) and then substituting .
The Chu–Vandermonde identity, which holds for any complex values "m" and "n" and any non-negative integer "k", is
and can be found by examination of the coefficient of formula_66 in the expansion of using equation (). When , equation () reduces to equation (). In the special case , using (), the expansion () becomes (as seen in Pascal's triangle at right)
where the term on the right side is a central binomial coefficient.
Another form of the Chu–Vandermonde identity, which applies for any integers "j", "k", and "n" satisfying , is
The proof is similar, but uses the binomial series expansion () with negative integer exponents.
When , equation () gives the hockey-stick identity
and its relative
Let "F"("n") denote the "n"-th Fibonacci number.
Then
This can be proved by induction using () or by Zeckendorf's representation. A combinatorial proof is given below.
Multisections of sums.
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For integers "s" and "t" such that formula_70 series multisection gives the following identity for the sum of binomial coefficients:
For small , these series have particularly nice forms; for example,
Partial sums.
Although there is no closed formula for partial sums
of binomial coefficients, one can again use () and induction to show that for ,
with special case
for . This latter result is also a special case of the result from the theory of finite differences that for any polynomial "P"("x") of degree less than "n",
Differentiating () "k" times and setting "x" = −1 yields this for
formula_83,
when 0 ≤ "k" < "n",
and the general case follows by taking linear combinations of these.
When "P"("x") is of degree less than or equal to "n",
where formula_84 is the coefficient of degree "n" in "P"("x").
More generally for (),
where "m" and "d" are complex numbers. This follows immediately applying () to the polynomial instead of , and observing that still has degree less than or equal to "n", and that its coefficient of degree "n" is "dnan".
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The series formula_86 is convergent for "k" ≥ 2. This formula is used in the analysis of the German tank problem. It follows from formula_87 which is proved by induction on "M".
Identities with combinatorial proofs.
Many identities involving binomial coefficients can be proved by combinatorial means. For example, for nonnegative integers formula_88, the identity
(which reduces to () when "q" = 1) can be given a double counting proof, as follows. The left side counts the number of ways of selecting a subset of ["n"] = {1, 2, ..., "n"} with at least "q" elements, and marking "q" elements among those selected. The right side counts the same thing, because there are formula_90 ways of choosing a set of "q" elements to mark, and formula_91 to choose which of the remaining elements of ["n"] also belong to the subset.
In Pascal's identity
both sides count the number of "k"-element subsets of ["n"]: the two terms on the right side group them into those that contain element "n" and those that do not.
The identity () also has a combinatorial proof. The identity reads
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Suppose you have formula_94 empty squares arranged in a row and you want to mark (select) "n" of them. There are formula_95 ways to do this. On the other hand, you may select your "n" squares by selecting "k" squares from among the first "n" and formula_96 squares from the remaining "n" squares; any "k" from 0 to "n" will work. This gives
Now apply () to get the result.
If one denotes by the sequence of Fibonacci numbers, indexed so that , then the identity
formula_98
has the following combinatorial proof. One may show by induction that counts the number of ways that a strip of squares may be covered by and tiles. On the other hand, if such a tiling uses exactly of the tiles, then it uses of the tiles, and so uses tiles total. There are formula_99 ways to order these tiles, and so summing this coefficient over all possible values of gives the identity.
Sum of coefficients row.
The number of "k"-combinations for all "k", formula_100, is the sum of the "n"th row (counting from 0) of the binomial coefficients. These combinations are enumerated by the 1 digits of the set of base 2 numbers counting from 0 to formula_101, where each digit position is an item from the set of "n".
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Dixon's identity.
Dixon's identity is
or, more generally,
where "a", "b", and "c" are non-negative integers.
Continuous identities.
Certain trigonometric integrals have values expressible in terms of binomial coefficients: For any formula_104
These can be proved by using Euler's formula to convert trigonometric functions to complex exponentials, expanding using the binomial theorem, and integrating term by term.
Congruences.
If "n" is prime, then formula_108 for every "k" with formula_109
More generally, this remains true if "n" is any number and "k" is such that all the numbers between 1 and "k" are coprime to "n".
Indeed, we have
Generating functions.
Ordinary generating functions.
For a fixed , the ordinary generating function of the sequence formula_111 is
For a fixed , the ordinary generating function of the sequence formula_113 is
The bivariate generating function of the binomial coefficients is
A symmetric bivariate generating function of the binomial coefficients is
which is the same as the previous generating function after the substitution formula_117.
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Exponential generating function.
A symmetric exponential bivariate generating function of the binomial coefficients is:
Divisibility properties.
In 1852, Kummer proved that if "m" and "n" are nonnegative integers and "p" is a prime number, then the largest power of "p" dividing formula_119 equals "p""c", where "c" is the number of carries when "m" and "n" are added in base "p".
Equivalently, the exponent of a prime "p" in formula_39
equals the number of nonnegative integers "j" such that the fractional part of "k"/"p""j" is greater than the fractional part of "n"/"p""j". It can be deduced from this that formula_39 is divisible by "n"/gcd("n","k"). In particular therefore it follows that "p" divides formula_122 for all positive integers "r" and "s" such that . However this is not true of higher powers of "p": for example 9 does not divide formula_123.
A somewhat surprising result by David Singmaster (1974) is that any integer divides almost all binomial coefficients. More precisely, fix an integer "d" and let "f"("N") denote the number of binomial coefficients formula_39 with "n" < "N" such that "d" divides formula_39. Then
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Since the number of binomial coefficients formula_39 with "n" < "N" is "N"("N" + 1) / 2, this implies that the density of binomial coefficients divisible by "d" goes to 1.
Binomial coefficients have divisibility properties related to least common multiples of consecutive integers. For example:
Another fact:
An integer is prime if and only if
all the intermediate binomial coefficients
are divisible by "n".
Proof:
When "p" is prime, "p" divides
because formula_134 is a natural number and "p" divides the numerator but not the denominator.
When "n" is composite, let "p" be the smallest prime factor of "n" and let . Then and
otherwise the numerator has to be divisible by , this can only be the case when is divisible by "p". But "n" is divisible by "p", so "p" does not divide and because "p" is prime, we know that "p" does not divide and so the numerator cannot be divisible by "n".
Bounds and asymptotic formulas.
The following bounds for formula_39 hold for all values of "n" and "k" such that :
formula_137
The first inequality follows from the fact that
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formula_138
and each of these formula_139 terms in this product is formula_140. A similar argument can be made to show the second inequality. The final strict inequality is equivalent to formula_141, that is clear since the RHS is a term of the exponential series formula_142.
From the divisibility properties we can infer that
formula_143
where both equalities can be achieved.
The following bounds are useful in information theory:
formula_144
where formula_145 is the binary entropy function. It can be further tightened to
formula_146
for all formula_147.
Both "n" and "k" large.
Stirling's approximation yields the following approximation, valid when formula_148 both tend to infinity:
formula_149
Because the inequality forms of Stirling's formula also bound the factorials, slight variants on the above asymptotic approximation give exact bounds.
In particular, when formula_28 is sufficiently large, one has
formula_151 and formula_152. More generally, for and (again, by applying Stirling's formula to the factorials in the binomial coefficient),
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formula_153
If "n" is large and "k" is linear in "n", various precise asymptotic estimates exist for the binomial coefficient formula_154. For example, if formula_155 then
formula_156
where "d" = "n" − 2"k".
much larger than.
If is large and is (that is, if ), then
formula_157
where again is the little o notation.
Sums of binomial coefficients.
A simple and rough upper bound for the sum of binomial coefficients can be obtained using the binomial theorem:
formula_158
More precise bounds are given by
formula_159
valid for all integers formula_160 with formula_161.
Generalized binomial coefficients.
The infinite product formula for the gamma function also gives an expression for binomial coefficients
formula_162
which yields the asymptotic formulas
formula_163
as formula_164.
This asymptotic behaviour is contained in the approximation
formula_165
as well. (Here formula_166 is the "k"-th harmonic number and formula_167 is the Euler–Mascheroni constant.)
Further, the asymptotic formula
formula_168
hold true, whenever formula_169 and formula_170 for some complex number formula_171.
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Generalizations.
Generalization to multinomials.
Binomial coefficients can be generalized to multinomial coefficients defined to be the number:
where
While the binomial coefficients represent the coefficients of , the multinomial coefficients
represent the coefficients of the polynomial
The case "r" = 2 gives binomial coefficients:
The combinatorial interpretation of multinomial coefficients is distribution of "n" distinguishable elements over "r" (distinguishable) containers, each containing exactly "ki" elements, where "i" is the index of the container.
Multinomial coefficients have many properties similar to those of binomial coefficients, for example the recurrence relation:
and symmetry:
where formula_178 is a permutation of (1, 2, ..., "r").
Taylor series.
Using Stirling numbers of the first kind the series expansion around any arbitrarily chosen point formula_179 is
Binomial coefficient with.
The definition of the binomial coefficients can be extended to the case where formula_28 is real and formula_30 is integer.
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In particular, the following identity holds for any non-negative integer formula_30:
This shows up when expanding formula_185 into a power series using the Newton binomial series :
Products of binomial coefficients.
One can express the product of two binomial coefficients as a linear combination of binomial coefficients:
where the connection coefficients are multinomial coefficients. In terms of labelled combinatorial objects, the connection coefficients represent the number of ways to assign labels to a pair of labelled combinatorial objects—of weight "m" and "n" respectively—that have had their first "k" labels identified, or glued together to get a new labelled combinatorial object of weight . (That is, to separate the labels into three portions to apply to the glued part, the unglued part of the first object, and the unglued part of the second object.) In this regard, binomial coefficients are to exponential generating series what falling factorials are to ordinary generating series.
The product of all binomial coefficients in the "n"th row of the Pascal triangle is given by the formula:
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Partial fraction decomposition.
The partial fraction decomposition of the reciprocal is given by
Newton's binomial series.
Newton's binomial series, named after Sir Isaac Newton, is a generalization of the binomial theorem to infinite series:
The identity can be obtained by showing that both sides satisfy the differential equation .
The radius of convergence of this series is 1. An alternative expression is
where the identity
is applied.
Multiset (rising) binomial coefficient.
Binomial coefficients count subsets of prescribed size from a given set. A related combinatorial problem is to count multisets of prescribed size with elements drawn from a given set, that is, to count the number of ways to select a certain number of elements from a given set with the possibility of selecting the same element repeatedly. The resulting numbers are called "multiset coefficients"; the number of ways to "multichoose" (i.e., choose with replacement) "k" items from an "n" element set is denoted formula_193.
To avoid ambiguity and confusion with "n"'s main denotation in this article, let and .
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Multiset coefficients may be expressed in terms of binomial coefficients by the rule
formula_194
One possible alternative characterization of this identity is as follows:
We may define the falling factorial as
formula_195
and the corresponding rising factorial as
formula_196
so, for example,
formula_197
Then the binomial coefficients may be written as
formula_198
while the corresponding multiset coefficient is defined by replacing the falling with the rising factorial:
formula_199
Generalization to negative integers "n".
For any "n",
In particular, binomial coefficients evaluated at negative integers "n" are given by signed multiset coefficients. In the special case formula_201, this reduces to formula_202
For example, if "n" = −4 and "k" = 7, then "r" = 4 and "f" = 10:
Two real or complex valued arguments.
The binomial coefficient is generalized to two real or complex valued arguments using the gamma function or beta function via
This definition inherits these following additional properties from formula_205:
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moreover,
The resulting function has been little-studied, apparently first being graphed in . Notably, many binomial identities fail: formula_208 but formula_209 for "n" positive (so formula_210 negative). The behavior is quite complex, and markedly different in various octants (that is, with respect to the "x" and "y" axes and the line formula_211), with the behavior for negative "x" having singularities at negative integer values and a checkerboard of positive and negative regions:
Generalization to "q"-series.
The binomial coefficient has a q-analog generalization known as the Gaussian binomial coefficient.
Generalization to infinite cardinals.
The definition of the binomial coefficient can be generalized to infinite cardinals by defining:
where is some set with cardinality formula_220. One can show that the generalized binomial coefficient is well-defined, in the sense that no matter what set we choose to represent the cardinal number formula_220, formula_222 will remain the same. For finite cardinals, this definition coincides with the standard definition of the binomial coefficient.
Assuming the Axiom of Choice, one can show that formula_223 for any infinite cardinal formula_220. |
Bill Holbrook
Bill Holbrook (born 1958) is an American cartoonist and webcomic writer and artist, best known for his syndicated comic strip "On the Fastrack".
Born in Los Angeles, Holbrook grew up in Huntsville, Alabama, and began drawing at an early age. While majoring in illustration and visual design at Auburn University, Holbrook served as art director of the student newspaper, doing editorial cartoons and a weekly comic strip. At the same time, his work was being published in the "Huntsville Times" and the "Monroe Journal". After graduation in 1980, he joined the "Atlanta Constitution" as an editorial staff artist.
During a 1982 visit to relatives on the West Coast, Holbrook met "Peanuts" creator, Charles Schulz. Following his advice and encouragement, Holbrook created a strip in the fall of that year about a college graduate working in a rundown diner. It did not stir syndicate interest, but what he learned on the strip helped him when he created "On the Fastrack".
Eleven days before "On the Fastrack" made its syndicated debut (March 19, 1984), Holbrook met Teri Peitso on a blind date. They were married on Pearl Harbor Day, 1985. They have two daughters, Chandler and Haviland. Peitso-Holbrook's novels have been nominated for both Edgar Awards and Agatha Awards. She is currently an assistant professor in literacy education at Georgia State University. The family lives in the Atlanta area.
On October 3, 1988, Holbrook began his second strip, "Safe Havens", and his third strip, "Kevin and Kell" was launched in September 1995.
Comic strips.
Every week Holbrook writes the story line for the next three weeks for one of his strips and draws the next three weeks' worth of strips for another. In 2010, characters from "On the Fastrack" and "Safe Havens" began appearing in both strips. |
Bruce Campbell
Bruce Lorne Campbell (born June 22, 1958) is an American actor and filmmaker. He is known best for his role as Ash Williams in Sam Raimi's "Evil Dead" horror series, beginning with the short movie "Within the Woods" (1978). He has also featured in many low-budget cult movies, such as "Crimewave" (1985), "Maniac Cop" (1988), "" (1989), and "Bubba Ho-Tep" (2002).
Campbell had the main roles of the television series "The Adventures of Brisco County, Jr." (1993–1994) and "Jack of All Trades" (2000), and a recurring role as in ' (1995–1999) and ' (1995–1999). He played Sam Axe on the USA Network series "Burn Notice" (2007–2013) and reprised his role as Ash for the Starz series "Ash vs. Evil Dead" (2015–2018). He also appeared in "The Escort" (2015).
Campbell directed, produced, and featured in the documentaries "Fanalysis" (2002) and "A Community Speaks" (2004); co-wrote, directed, produced, and featured in the movie "Man with the Screaming Brain" (2005); and directed, produced, and featured in a parody of his career "My Name Is Bruce" (2007).
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Campbell is known for frequent collaborations with the aforementioned Raimi, his brother Ted, Josh Becker, and Scott Spiegel.
Early life.
Bruce Lorne Campbell was born in Royal Oak, Michigan, on June 22, 1958, the son of advertising executive and college professor Charles Newton Campbell and homemaker Joanne Louise (née Pickens). He is of English and Scottish ancestry, and has an older brother named Don and an older half-brother named Michael. His father was also an actor and director for local theater. Campbell began acting and making short Super 8 movies with friends as a teenager. After meeting future moviemaker Sam Raimi while the two attended Wylie E. Groves High School, they became good friends and collaborators. Campbell attended Western Michigan University and continued to pursue an acting career.
Career.
Film.
Campbell and Raimi collaborated with a 30-minute Super 8 version of the first "Evil Dead" movie, titled "Within the Woods" (1978), which was initially used to attract investors. He and Raimi got together with family and friends to begin working on "The Evil Dead" (1981). While featuring as the protagonist, Campbell also had participation with the production of the movie, receiving a co-executive producer credit. Raimi wrote, directed, and edited the movie, while Rob Tapert produced. After an endorsement by horror author Stephen King, the movie slowly began to receive attention and offers for distribution. Four years after its original release, it became the most popular movie in the UK. It was then distributed in the United States, resulting in the sequels "Evil Dead II" (1987) and "Army of Darkness" (1992).
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Campbell was also drawn in the Marvel Zombie comics as his character, Ash Williams. He is featured in five comics, all in the series "Marvel Zombies vs. Army of Darkness". In them, he fights alongside the Marvel heroes against the heroes and people who have become zombies (deadites) while in search of the Necronomicon (Book of the Names of the Dead). Campbell also played as Coach Boomer in the movie “Sky High”.
He has appeared in several of Raimi's movies other than the "Evil Dead" series, notably having cameo appearances in the director's "Spider-Man" film series. Campbell also joined the cast of Raimi's movie "Darkman" and "The Quick and the Dead", though having no actual screen time in the latter movie's theatrical version. In March 2022, Campbell was announced to have a cameo in Raimi's Marvel Cinematic Universe film "Doctor Strange in the Multiverse of Madness".
Campbell often performs quirky roles, such as Elvis Presley for the movie "Bubba Ho-Tep". Along with "Bubba Ho-Tep", he played a supporting role in "Maniac Cop" and "Maniac Cop 2", and spoofed his career in the self-directed "My Name is Bruce."
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Other mainstream movies for Campbell include supporting or featured roles in the Coen Brothers movie "The Hudsucker Proxy", the Michael Crichton adaptation "Congo", the movie version of "McHale's Navy", "Escape from L.A." (the sequel to John Carpenter's "Escape from New York"), the Jim Carrey drama "The Majestic" and the 2005 Disney movie "Sky High".
Campbell had a major voice role for the 2009 animated adaptation of the children's book "Cloudy with a Chance of Meatballs", and a supporting voice role for Pixar's "Cars 2".
Campbell produced the 2013 remake of "The Evil Dead", along with Raimi and Rob Tapert, appearing in the movie's post-credits scene in a cameo role with the expectation he would reprise that role in "Army of Darkness 2". The next year, the comedy metal band Psychostick released a song titled "Bruce Campbell" on their album "" that pays a comedic tribute to his past roles.
Campbell worked as an executive producer for the 2023 movie "Evil Dead Rise".
Television.
Other than cinema, Campbell has appeared in a number of television series. He featured in "The Adventures of Brisco County, Jr." a boisterous science fiction comedy western created by Jeffrey Boam and Carlton Cuse that played for one season. He played a lawyer turned bounty hunter who was trying to hunt down John Bly, the man who killed his father. He featured in the television series "Jack of All Trades", set on a fictional island occupied by the French in 1801. Campbell was also credited as co-executive producer, among others. The show was directed by Eric Gruendemann, and was produced by various people, including Sam Raimi. The show was broadcast for two seasons, from 2000 to 2001. He had a recurring role as "Bill Church Jr." based upon the character of Morgan Edge from the Superman comics on "".
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From 1996 to 1997, Campbell was a recurring guest actor of the television series "Ellen" as Ed Billik, who becomes Ellen's boss when she sells her bookstore in season four.
He is also known for his supporting role as the recurring character Autolycus ("King of Thieves") on both ' and ', which reunited him with producer Rob Tapert. Campbell played "Hercules"/"Xena" series producer Tapert in two episodes of "Hercules" set in the present. He directed a number of episodes of "Hercules" and "Xena", including the "Hercules" series finale.
Campbell also obtained the main role of race car driver Hank Cooper for the Disney made-for-television remake of "The Love Bug".
Campbell had a critically acclaimed dramatic guest role as a grief-stricken detective seeking revenge for his father's murder in a two-part episode of the fourth season of "". Campbell later played the part of a bigamous demon in "The X-Files" episode "Terms of Endearment". He also featured as Agent Jackman in the episode "Witch Way Now?" of the WB series "Charmed", as well as playing a state police officer in an episode of the short-lived series "American Gothic" titled "Meet The Beetles".
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Campbell co-featured in the television series "Burn Notice", which was broadcast from 2007 to 2013 by USA Network. He portrayed Sam Axe, a beer-chugging, former Navy SEAL now working as an unlicensed private investigator and occasional mercenary with his old friend Michael Westen, the show's main character. When working undercover, his character frequently used the alias Chuck Finley, which Bruce later revealed was the name of one of his father's old co-workers. Campbell was the star of a 2011 "Burn Notice" made-for-television prequel focusing on Sam's Navy SEAL career, titled "".
In 2014, Campbell played Santa Claus for an episode of "The Librarians". Campbell played Ronald Reagan in season 2 of the FX original series "Fargo". More recently Campbell reprised his role as Ashley "Ash" Williams in "Ash vs Evil Dead", a series based upon the "Evil Dead" series that began his career. "Ash vs Evil Dead" began airing on Starz on October 31, 2015, and was renewed by the cable channel for second and third seasons, before being cancelled.
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In January 2019, Travel Channel announced a new version of the "Ripley's Believe It or Not!" reality series, with Campbell serving as host and executive producer. The 10-episode season debuted on June 9, 2019.
Voice acting.
Campbell is featured as a voice actor for several video games. He provides the voice of Ash in the four games based on the "Evil Dead" movies series: ', ', ' and '. He has expressed his intent to return to the role in an upcoming "Evil Dead" animated series. He also provided the voice of Ash in "Dead by Daylight". Despite the inclusion of his character "Ash Williams" in Telltale Games' "Poker Night 2", Danny Webber voices the character in the game, instead of Bruce Campbell.
Campbell also provided voice talent in other titles such as "", "Spider-Man", "Spider-Man 2", "Spider-Man 3", and "The Amazing Spider-Man".
He provided the voice of main character Jake Logan for the PC game, ', the voice of main character Jake Burton for the PlayStation game "Broken Helix" and the voice of Magnanimous for "Megas XLR". Campbell voiced the pulp adventurer Lobster Johnson in ' and has done voice-over work for the Codemaster's game "Hei$t", a game which was announced on January 28, 2010, to have been "terminated". He also provided the voice of The Mayor for the 2009 movie "Cloudy With a Chance of Meatballs", the voice of Rod "Torque" Redline in "Cars 2", the voice of Himcules in the 2003 Nickelodeon TV series "My Life as a Teenage Robot", and the voice of Fugax in the 2006 movie "The Ant Bully".
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He has a voice in the online MOBA game, "Tome: Immortal Arena" in 2014. Campbell also provided voice-over and motion capture for Sgt. Lennox in the Exo Zombies mode of "".
Writing.
In addition to acting and occasionally directing, Campbell has become a writer, starting with an autobiography, "If Chins Could Kill: Confessions of a B Movie Actor", published in June 2001. The autobiography was a successful "New York Times" Best Seller. It describes Campbell's career to date as an actor in low-budget movies and television, providing his insight into "Blue-Collar Hollywood". The paperback version of the book adds details about the reactions of fans during book signings: "Whenever I do mainstream stuff, I think they're pseudo-interested, but they're still interested in seeing weirdo, offbeat stuff, and that's what I'm attracted to".
Campbell's next book "Make Love! The Bruce Campbell Way" was published on May 26, 2005. The book's plot involves him (depicted in a comical way) as the main character struggling to make it into the world of A-list movies. He later recorded an audio play adaptation of "Make Love" with fellow Michigan actors, including longtime collaborator Ted Raimi. This radio drama was released by the independent label Rykodisc and spans six discs with a six-hour running time.
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In addition to his books, Campbell also wrote a column for "X-Ray Magazine" in 2001, an issue of the popular comic series "The Hire", and comic book adaptations of his "Man with the Screaming Brain". Most recently he wrote the introduction to Josh Becker's "The Complete Guide to Low-Budget Feature Filmmaking".
In late 2016, Campbell announced that he would be releasing a third book, "Hail to the Chin: Further Confessions of a B Movie Actor," which will detail his life from where "If Chins Could Kill" ended. "Hail to the Chin" was released in August 2017, and accompanied by a book tour across the United States and Europe.
Campbell maintained a weblog on his official website, where he posted mainly about politics and the movie industry. However, the website has since been deleted.
Bruce Campbell Horror Film Festival.
Since 2014, the Bruce Campbell Horror Film Festival, narrated and organized by Campbell, was held in the Muvico Theater in Rosemont, Illinois. The first festival was originally from August 21 to 25, 2014, presented by Wizard World, as part of the Chicago Comicon. The second festival was from August 20 to 23, 2015, with guests Tom Holland and Eli Roth. The third festival took place over four days in August 2016. Guests of the event were Sam Raimi, Robert Tapert and Doug Benson.
Personal life.
Campbell married Christine Deveau in 1983; they had two children before divorcing in 1989. He met costume designer Ida Gearon while working on "Mindwarp." They married in 1991. They reside in Jacksonville, Oregon.
Campbell is also an ordained minister, and has performed marriage ceremonies.
Campbell is a Kentucky Colonel.
Filmography.
Video games.
2024- Spooky Pinball’s-Evil Dead Callouts |
Baron Aberdare
Baron Aberdare, of Duffryn in the County of Glamorgan, is a title in the Peerage of the United Kingdom. It was created on 23 August 1873 for the Liberal politician Henry Bruce. He served as Home Secretary from 1868 to 1873.
His grandson, the third Baron, was a soldier, cricketer and tennis player and a member of the International Olympic Committee. His son, the fourth Baron, held office in the Conservative administration of Edward Heath and was later a Deputy Speaker of the House of Lords. Lord Aberdare was one of the ninety-two elected hereditary peers that were allowed to remain in the House of Lords after the passing of the House of Lords Act 1999.
, the title is held by his son, the fifth Baron, who succeeded in 2005 and was elected to the House of Lords in 2009.
Lineage.
Henry Bruce, 1st Baron Aberdare, was the son of John Bruce-Pryce (born John Bruce Knight in Barnstaple, Devon), the eldest son of John Knight (died 1799) and Margaret Bruce (died 1809), daughter and heir of William Bruce of Llanblethian, Glamorgan. He descends from the Welsh Bruces, a branch of Bruce of Kennet of Clackmannan that dates to the 14th century.
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In 1805, Lord Aberdare's father changed his surname from Knight to Bruce when he reached the age of majority and inherited the Bruce estates in Llanblethian, as per his grandfather's will. His younger brothers were Rev. William Bruce Knight (1785–1845) and Sir James Knight-Bruce (1791–1866).
In 1837, he changed his name again to Bruce-Pryce when he inherited Duffryn and the Monknash estates from a distant cousin, Frances Anne Grey ("née" Pryce), the daughter and heiress of Thomas Pryce, who had married Hon. William Booth Grey (1773–1852; the second son of the 5th Earl of Stamford).
Lord Aberdare's great-grandmother, Jane Lewis, was the daughter of MP Gabriel Lewis, of Llanishen House. The Lewis family of Van Castle, Glamorgan, had been prominent in politics since the 16th century.
Coat of arms.
The heraldic blazon for the coat of arms of the family is: "Or, a saltire gules, on a chief of the last a martlet of the field".
Baron Aberdare (1873).
The heir apparent is the present holder's son, the Hon. Hector Morys Napier Bruce (born 1974). |
Boy band
A boy band is a vocal group consisting of young male singers, usually in their teenage years or in their twenties at the time of formation. Generally, boy bands perform love songs marketed towards girls and young women. Many boy bands dance as well as sing, usually giving highly choreographed performances. South Korean boy bands usually also have designated rappers. Most boy band members do not play musical instruments, either in recording sessions or on stage. They are similar in concept to their counterparts known as girl groups.
Some boy bands are formed on their own, but most are created by talent managers or record producers who hold auditions. The popularity of boy bands has peaked three times: first in the 1960s to '70s, with e.g. the Jackson 5 and the Osmonds; the second time during the late 1980s, the 1990s and the 2000s, when acts such as New Kids on the Block, Take That, Backstreet Boys, Boyzone, NSYNC, Five, Westlife and Blue dominated global pop charts; and the latest time in the 2010s up to the present, with the emergence of groups such as Big Time Rush, the Wanted, One Direction, and K-pop acts such as BTS and Seventeen.
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History.
The term was not established until the late 1980s when Lou Pearlman decided to form a record company to promote a new singing and dancing group after becoming fascinated with the success of New Kids on the Block.
Although generally described as a rock band, the highest-selling band in history – the Beatles – have been described by several journalists as "the first" or "the original" boy band, "before anyone had thought of the term", exclusively due to the enthusiastic response they received from their young female audience.
Other critics, however, have pointed out that this assessment of the Beatles as a "boy band" could be applied to all other bands of the 1960s, saying, "if they were a (boy band), so was everyone else" and is countered by others, including Ringo Starr, who point out that, from the beginning, the Beatles wrote and exercised creative control over their own music, played their own instruments, were not manufactured by a record label, and did not feature the choreographed dance moves that later came to be associated with boy bands. The Beatles did, however, inspire the production of the 1966 television series "The Monkees", which featured a music group of the same name, created for the show, that consisted of the four starring actors. The Monkees had a career as a rock and pop band after their songs from the TV series were released as successful records.
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Late 1960s and 1970s: The Jackson 5 and the Osmonds.
Although the term "boy band" was not commonly used then, the earliest predecessors of this format were groups such as the Jackson 5 and the Osmonds which helped form the template for boy bands. The Jackson 5 were a sibling group that established many musical conventions that boy bands follow. For instance, their music featured close harmonies from soul music and catchy pop hooks influenced as much as they were by Motown and acts like the Supremes. The group also incorporated choreographed dance moves to their performances. All members of the band sang, which is a common convention of a boy band, as opposed to having a front man and the rest on instruments; thus, no one person dominated the stage. Also a sibling group, The Osmonds first started singing barbershop music for local audiences, before being hired to perform at Disneyland early in their career. Their appearance in a televised Disney special earned them additional TV spots, such as "The Andy Williams Show" and "The Jerry Lewis Show".
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Late 1970s and 1980s: Menudo, New Edition, and New Kids on the Block.
The Puerto Rican boy band Menudo, appealing to young Latina audiences, was founded in 1977. Menudo had a convention unique among boy bands: when a member turned 16, became too tall, or their voice changed, they were replaced. The members of Menudo were generally aged 12–16. Menudo had a large impact in Latin America and in Asia; Menudo fever there was compared to Beatlemania and it was nicknamed "Menudomania".
Boston group New Edition was formed in 1978 and reached their height of popularity in the 1980s, meaning they are often credited for starting the boy-band trend, even though the term "boy band" did not exist until the 1990s. Maurice Starr was influenced by New Edition and popularized it with his protégé New Kids on the Block (NKOTB), the first commercially successful modern boy band, which formed in 1984 and found international success in 1988. Starr's idea was to take the traditional template from the R&B genre (in this case his teenage band New Edition) and apply it to a pop genre.
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Bros (abbreviation of the word "brothers") were a British boy band active in the late 1980s and early 1990s, consisting of twin brothers Matt and Luke Goss along with Craig Logan. Formed in 1986, they scored multiple top 10 hits between 1987 and 1989 and in 1988 became the first modern era–style boy band to have a multiple platinum-selling album in the UK, with "Push", still one of the most successful boy-band albums in the UK. Other big boy bands in Britain during the late 1980s were Big Fun and Brother Beyond.
1990s: Boyz II Men, Take That, Backstreet Boys, NSYNC, Westlife, Seo Taiji and Boys and the birth of modern K-pop.
The ongoing international success of New Kids on the Block inspired music managers in Europe to create their own acts, beginning with Nigel Martin-Smith's Take That in the UK (formed in 1990) and followed by Tom Watkins, who had success with Bros in the late 1980s and formed East 17 in 1991. East 17 were marketed and pitted against Take That as "rivals" with a rougher or harsher attitude, style and sound. Take That reformed in 2006 after a decade-long hiatus and became one of the most successful groups in British music chart history, with renewed chart success internationally, especially in Europe. Irish music manager Louis Walsh, who had witnessed the impact of these British boy bands, put out an advert for an "Irish Take That", thereby creating Boyzone in 1993. MN8 (formed in 1992), Let Loose (formed in 1993), and Damage and 911 (formed in 1995) were also successful boy bands in Britain; however, by the late 1990s all these bands had split up.
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All these artists were very successful on both the singles and albums charts domestically and internationally; however, with the emergence of Britpop and the commercial co-option of indie rock, many boy bands were ridiculed by the British music press as having no artistic credibility, although some, such as East 17 and Take That, did write most of their own material. The media attention was then placed on the "Battle of Britpop", and the bands Oasis and Blur replaced the importance and rivalry of Take That and East 17 as the two new biggest bands in Britain. However, boy bands continued to find success in the late 1990s, such as Five, Another Level, Point Break and Westlife. In 1995 successful German music manager Frank Farian, who had been manager of Boney M and Milli Vanilli, put together Latin American band No Mercy who scored a few worldwide hits during the mid-90s.
Although being American and the sons of Tito Jackson, a member of the Jackson 5, 3T had several hits singles across Europe in the mid-1990s, despite limited success in the US, and finished the second biggest selling act of 1996 in Europe behind the Spice Girls. With the success of North American boy bands like New Kids on the Block in East Asia, Japanese entertainment company Johnny & Associates formed SMAP in 1992. The group enjoyed tremendous success, selling over 35 million records.
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In 1992, after the disbandment of the heavy metal band Sinawe, in which he had a brief stint, Seo Taiji formed the boy band Seo Taiji and Boys (Korean: 서태지와 아이들) together with dancers Lee Juno and Yang Hyun-suk, which went on to become highly successful and created a craze at the time. Seo Taiji and Boys is credited with changing the South Korean music industry by pioneering the incorporation of rap and breakdance as well as the fusion of Korean music and various popular Western music genres in Korean popular music, and in turn creating the prototype for the modern hybrid K-pop genre or "rap-dance", as it was called at the time, and K-pop groups. They also left a lasting impact by explicitly putting social criticism at the forefront of their music, as well as paving the way for artistic freedom in South Korea by challenging censorship laws and the television networks hegemony over the music market. In 1995 the Korean Broadband is not 6 ft to a particular location casting Ethics Committee demanded that Seo Taiji and Boys change the lyrics for "Regret of the Times". As a result, Seo decided to release the song as a purely instrumental track. |
In 1995 the Korean Broadband is not 6 ft to a particular location casting Ethics Committee demanded that Seo Taiji and Boys change the lyrics for "Regret of the Times". As a result, Seo decided to release the song as a purely instrumental track. This incited protests and resulted in the abolishment of music pre-censorship in Korea. Seo Taiji also did not have to rely on television networks due to the fact that he owned his own studio. This autonomy allowed Seo to bring subcultures in Korea, such as heavy metal, to the forefront of popular culture and challenge pervasive social norms. The band's independent success diminished the power of the television networks to dictate which artists appeared on shows, and gave rise to the influence of record labels and talent agencies. In 1996, Seo Taiji and Boys disbanded. In April 1996, Billboard reported that the band's first three albums had each sold over 1.6 million copies, with the fourth nearing two million, making all four some of the best-selling albums of all time in South Korea to this day. |
In April 1996, Billboard reported that the band's first three albums had each sold over 1.6 million copies, with the fourth nearing two million, making all four some of the best-selling albums of all time in South Korea to this day. Lee Juno became a record producer, and Yang Hyun-suk was successful in founding YG Entertainment, one of the three biggest record companies in the country. Seo Taiji returned to music two years later with a successful solo career as a rock artist; he rose to become one of the most prominent and influential cultural icons in South Korea and was dubbed "the President of culture". In 2017, Seo Taiji released a 25th Anniversary album with his greatest hits and remakes by prominent Korean artists, including the group BTS. He also held a joint celebratory concert with the latter, in which he acknowledged them as his spiritual successors in K-pop due to the socially conscious thematic similarities in their music as well their shared hip hop leanings, and metaphorically passed the torch, saying "This is your generation now".
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In the early 1990s in North America, with New Kids on the Block's continued success and Color Me Badd also having success, boy bands became a continued staple of the Billboard charts. Continuing this success in the mid-1990s, most prominent boy bands were African American and had R&B and gospel elements, such as the groups All-4-One (formed in 1993) and Boyz II Men (formed in 1988). Boyz II Men are also the most successful boy band act on the U.S. Hot 100 as well as the Australian Singles Chart. Although they had success on the Billboard charts, they were not marketed towards youth but more towards adults. It was not until 1997 and the change to pop-oriented groups such as Backstreet Boys, 98 Degrees, NSYNC, the Moffatts, and Hanson that boy bands exploded commercially and dominated the market in the United States. This late 1990s marked the height of boy band popularity in North America, which has not been seen since.
Arguably the most successful boy band manager from the U.S. was Lou Pearlman, who founded commercially successful acts such as the Backstreet Boys in 1993, NSYNC and LFO in 1995, O-Town in 2000, and US5 in 2005. Backstreet Boys and NSYNC became the two biggest boy bands in the late 1990s and the early 2000s, and Backstreet Boys went on to become the best-selling boy band in history with over 100 million records sold.
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In the late 1990s in the UK, producer Simon Cowell (noted in the U.S. for the "American Idol/The X Factor" franchise) is also known for having managed British boyband Five (formed in 1997) and Irish boyband Westlife (formed in 1998). Westlife was created by Irishman Louis Walsh as a replacement for Boyzone and was initially managed by a former member of the band Ronan Keating. Westlife would eventually overtake Take That in number one's tally in the UK although Take That's overall UK sales are still higher. In 2012, the Official Charts Company revealed the biggest selling singles artists in British music chart history with Take That placed 15th overall and the highest selling boyband act (9.3 million), followed by Boyzone at 29 (7.1 million) and Westlife at 34 (6.8 million). Even though Cowell is known to have managed several successful boy bands, he is also infamous for passing on signing two of the biggest boybands to emerge from the 1990s and 2000s, Take That and Busted.
2000s: Backstreet Boys, NSYNC, Westlife and Jonas Brothers.
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