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Poisson bracket

In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. The Poisson bracket also distinguishes a certain class of coordinate transformations, called canonical transformations, which map canonical coordinate systems into canonical coordinate systems. A "canonical coordinate system" consists of canonical position and momentum variables (below symbolized by $$q_i$$ and $$p_i$$, respectively) that satisfy canonical Poisson bracket relations. The set of possible canonical transformations is always very rich. For instance, it is often possible to choose the Hamiltonian itself $$H =H(q, p, t)$$ as one of the new canonical momentum coordinates.

In a more general sense, the Poisson bracket is used to define a Poisson algebra, of which the algebra of functions on a Poisson manifold is a special case. There are other general examples, as well: it occurs in the theory of Lie algebras, where the tensor algebra of a Lie algebra forms a Poisson algebra; a detailed construction of how this comes about is given in the universal enveloping algebra article. Quantum deformations of the universal enveloping algebra lead to the notion of quantum groups.

All of these objects are named in honor of Siméon Denis Poisson. He introduced the Poisson bracket in his 1809 treatise on mechanics.

Properties
Given two functions $f$ and $g$ that depend on phase space and time, their Poisson bracket $$\{f, g\}$$ is another function that depends on phase space and time. The following rules hold for any three functions $$f,\, g,\, h$$ of phase space and time:
 * Anticommutativity: $$\{f, g\} = -\{g, f\}$$
 * Bilinearity: $$\{af + bg, h\} = a\{f, h\} + b\{g, h\}, \quad \{h, af + bg\} = a\{h, f\} + b\{h, g\}, \quad a, b \in \mathbb R$$
 * Leibniz's rule: $$\{fg, h\} = \{f, h\}g + f\{g, h\}$$
 * Jacobi identity: $$\{f, \{g, h\}\} + \{g, \{h, f\}\} + \{h, \{f, g\}\} = 0$$

Also, if a function $$k$$ is constant over phase space (but may depend on time), then $$\{f,\, k\} = 0$$ for any $$f$$.

Definition in canonical coordinates
In canonical coordinates (also known as Darboux coordinates) $$ (q_i,\, p_i)$$ on the phase space, given two functions $$ f(p_i,\, q_i, t)$$ and $$ g(p_i,\, q_i, t)$$, the Poisson bracket takes the form $$\{f, g\} = \sum_{i=1}^{N} \left( \frac{\partial f}{\partial q_{i}} \frac{\partial g}{\partial p_{i}} - \frac{\partial f}{\partial p_i} \frac{\partial g}{\partial q_i}\right).$$

The Poisson brackets of the canonical coordinates are $$\begin{align} \{q_i,q_j\} &= 0 \\ \{p_i,p_j\} &= 0 \\ \{q_i,p_j\} &= \delta_{ij} \end{align}$$ where $$\delta_{ij}$$ is the Kronecker delta.

Hamilton's equations of motion
Hamilton's equations of motion have an equivalent expression in terms of the Poisson bracket. This may be most directly demonstrated in an explicit coordinate frame. Suppose that $$f(p, q, t)$$ is a function on the solution's trajectory-manifold. Then from the multivariable chain rule, $$\frac{d}{dt} f(p, q, t) = \frac{\partial f}{\partial q} \frac{dq}{dt} + \frac {\partial f}{\partial p} \frac{dp}{dt} + \frac{\partial f}{\partial t}.$$

Further, one may take $$p = p(t)$$ and $$q = q(t)$$ to be solutions to Hamilton's equations; that is, $$\begin{cases} \dot{q} = \frac{\partial H}{\partial p} = \{q, H\}; \\ \dot{p} = -\frac{\partial H}{\partial q} = \{p, H\}. \end{cases}$$

Then $$\begin{align} \frac {d}{dt} f(p, q, t) &= \frac{\partial f}{\partial q} \frac{\partial H}{\partial p} - \frac{\partial f}{\partial p} \frac{\partial H}{\partial q} + \frac{\partial f}{\partial t} \\ &= \{f, H\} + \frac{\partial f}{\partial t} ~. \end{align}$$

Thus, the time evolution of a function $$f$$ on a symplectic manifold can be given as a one-parameter family of symplectomorphisms (i.e., canonical transformations, area-preserving diffeomorphisms), with the time $$t$$ being the parameter: Hamiltonian motion is a canonical transformation generated by the Hamiltonian. That is, Poisson brackets are preserved in it, so that any time $$t$$ in the solution to Hamilton's equations, $$ q(t) = \exp (-t \{ H, \cdot \} ) q(0), \quad p(t) = \exp (-t \{ H, \cdot \}) p(0), $$ can serve as the bracket coordinates. Poisson brackets are canonical invariants.

Dropping the coordinates, $$\frac{d}{dt} f = \left(\frac{\partial}{\partial t} - \{H, \cdot\}\right)f.$$

The operator in the convective part of the derivative, $$i\hat{L} = -\{H, \cdot\}$$, is sometimes referred to as the Liouvillian (see Liouville's theorem (Hamiltonian)).

Poisson matrix in canonical transformations
The concept of Poisson brackets can be expanded to that of matrices by defining the Poisson matrix.

Consider the following canonical transformation:$$\eta = \begin{bmatrix} q_1\\ \vdots \\ q_N\\ p_1\\ \vdots\\ p_N\\ \end{bmatrix} \quad \rightarrow \quad \varepsilon = \begin{bmatrix} Q_1\\ \vdots \\ Q_N\\ P_1\\ \vdots\\ P_N\\ \end{bmatrix} $$Defining $M := \frac{\partial (\mathbf{Q}, \mathbf{P})}{\partial (\mathbf{q}, \mathbf{p})}$, the Poisson matrix is defined as $\mathcal P(\varepsilon) = MJM^T $, where $$J$$ is the symplectic matrix under the same conventions used to order the set of coordinates. It follows from the definition that:$$\mathcal P_{ij}(\varepsilon) = [MJM^T]_{ij}=\sum_{k=1}^{N} \left( \frac{\partial \varepsilon_i}{\partial \eta_{k}} \frac{\partial \varepsilon_j}{\partial \eta_{N+k}} - \frac{\partial \varepsilon_i}{\partial \eta_{N+k}} \frac{\partial \varepsilon_j}{\partial \eta_k}\right)=\sum_{k=1}^{N} \left( \frac{\partial \varepsilon_i}{\partial q_{k}} \frac{\partial \varepsilon_j}{\partial p_k} - \frac{\partial \varepsilon_i}{\partial p_k} \frac{\partial \varepsilon_j}{\partial q_k}\right)=\{ \varepsilon_i,\varepsilon_j\}_\eta. $$

The Poisson matrix satisfies the following known properties:$$\begin{align} \mathcal P^T &= - \mathcal P \\ \mathcal P^{-1}(\varepsilon)&= -(M^{-1})^T J M^{-1} = - \mathcal L (\varepsilon)\\ \end{align} $$
 * \mathcal P| &= \frac{1}{|M|^2}\\

where the $\mathcal L(\varepsilon) $ is known as a Lagrange matrix and whose elements correspond to Lagrange brackets. The last identity can also be stated as the following:$$\sum_{k=1}^{2N} \{\eta_i,\eta_k\}[\eta_k,\eta_j] = -\delta_{ij} $$Note that the summation here involves generalized coordinates as well as generalized momentum.

The invariance of Poisson bracket can be expressed as: $\{ \varepsilon_i,\varepsilon_j\}_\eta=\{ \varepsilon_i,\varepsilon_j\}_\varepsilon = J_{ij} $, which directly leads to the symplectic condition: $MJM^T = J   $.

Constants of motion
An integrable system will have constants of motion in addition to the energy. Such constants of motion will commute with the Hamiltonian under the Poisson bracket. Suppose some function $$f(p, q)$$ is a constant of motion. This implies that if $$p(t), q(t)$$ is a trajectory or solution to Hamilton's equations of motion, then $$0 = \frac{df}{dt}$$ along that trajectory. Then $$0 = \frac{d}{dt} f(p,q) = \{f, H\}$$ where, as above, the intermediate step follows by applying the equations of motion and we assume that $$f$$ does not explicitly depend on time. This equation is known as the Liouville equation. The content of Liouville's theorem is that the time evolution of a measure given by a distribution function $$f$$ is given by the above equation.

If the Poisson bracket of $$f$$ and $$g$$ vanishes ($$\{f,g\} = 0$$), then $$f$$ and $$g$$ are said to be in involution. In order for a Hamiltonian system to be completely integrable, $$n$$ independent constants of motion must be in mutual involution, where $$n$$ is the number of degrees of freedom.

Furthermore, according to Poisson's Theorem, if two quantities $$A$$ and $$B$$ are explicitly time independent ($$A(p, q), B(p, q)$$) constants of motion, so is their Poisson bracket $$\{A,\, B\}$$. This does not always supply a useful result, however, since the number of possible constants of motion is limited ($$2n - 1$$ for a system with $$n$$ degrees of freedom), and so the result may be trivial (a constant, or a function of $$A$$ and $$B$$.)

The Poisson bracket in coordinate-free language
Let $$M$$ be a symplectic manifold, that is, a manifold equipped with a symplectic form: a 2-form $$\omega$$ which is both closed (i.e., its exterior derivative $$d \omega$$ vanishes) and non-degenerate. For example, in the treatment above, take $$M$$ to be $$\mathbb{R}^{2n}$$ and take $$\omega = \sum_{i=1}^{n} d q_i \wedge d p_i.$$

If $$ \iota_v \omega$$ is the interior product or contraction operation defined by $$ (\iota_v \omega)(u) = \omega(v,\, u)$$, then non-degeneracy is equivalent to saying that for every one-form $$\alpha$$ there is a unique vector field $$\Omega_\alpha$$ such that $$ \iota_{\Omega_\alpha} \omega =  \alpha$$. Alternatively, $$ \Omega_{d H} = \omega^{-1}(d H)$$. Then if $$H$$ is a smooth function on $$M$$, the Hamiltonian vector field $$X_H$$ can be defined to be $$ \Omega_{d H}$$. It is easy to see that $$\begin{align} X_{p_i} &= \frac{\partial}{\partial q_i} \\ X_{q_i} &= -\frac{\partial}{\partial p_i}. \end{align}$$

The Poisson bracket $$\ \{\cdot,\, \cdot\} $$ on $(M, ω)$ is a bilinear operation on differentiable functions, defined by $$ \{f,\, g\} \;=\; \omega(X_f,\, X_g) $$; the Poisson bracket of two functions on $M$ is itself a function on $M$. The Poisson bracket is antisymmetric because: $$\{f, g\} = \omega(X_f, X_g) = -\omega(X_g, X_f) = -\{g, f\} .$$

Furthermore,

Here $X_{g}f$ denotes the vector field $X_{g}$ applied to the function $f$ as a directional derivative, and $$\mathcal{L}_{X_g} f$$ denotes the (entirely equivalent) Lie derivative of the function $f$.

If $α$ is an arbitrary one-form on $M$, the vector field $Ω_{α}$ generates (at least locally) a flow $$ \phi_x(t)$$ satisfying the boundary condition $$ \phi_x(0) = x$$ and the first-order differential equation $$\frac{d\phi_x}{dt} = \left. \Omega_\alpha \right|_{\phi_x(t)}.$$

The $$ \phi_x(t)$$ will be symplectomorphisms (canonical transformations) for every $t$ as a function of $x$ if and only if $$ \mathcal{L}_{\Omega_\alpha}\omega \;=\; 0$$; when this is true, $Ω_{α}$ is called a symplectic vector field. Recalling Cartan's identity $$ \mathcal{L}_X\omega \;=\; d (\iota_X \omega) \,+\, \iota_X d\omega$$ and $dω = 0$, it follows that $$ \mathcal{L}_{\Omega_\alpha}\omega \;=\; d\left(\iota_{\Omega_\alpha} \omega\right) \;=\; d\alpha$$. Therefore, $Ω_{α}$ is a symplectic vector field if and only if α is a closed form. Since $$ d(df) \;=\; d^2f \;=\; 0$$, it follows that every Hamiltonian vector field $X_{f}$ is a symplectic vector field, and that the Hamiltonian flow consists of canonical transformations. From $$ above, under the Hamiltonian flow $X_{H}$, $$\frac{d}{dt}f(\phi_x(t)) = X_Hf = \{f,H\}.$$

This is a fundamental result in Hamiltonian mechanics, governing the time evolution of functions defined on phase space. As noted above, when ${f,H} = 0$, $f$ is a constant of motion of the system. In addition, in canonical coordinates (with $$ \{p_i,\, p_j\} \;=\; \{q_i,q_j\} \;=\; 0$$ and $$\{q_i,\, p_j\} \;=\; \delta_{ij}$$), Hamilton's equations for the time evolution of the system follow immediately from this formula.

It also follows from $$ that the Poisson bracket is a derivation; that is, it satisfies a non-commutative version of Leibniz's product rule:

The Poisson bracket is intimately connected to the Lie bracket of the Hamiltonian vector fields. Because the Lie derivative is a derivation, $$\mathcal L_v\iota_w\omega = \iota_{\mathcal L_vw}\omega + \iota_w\mathcal L_v\omega = \iota_{[v,w]}\omega + \iota_w\mathcal L_v\omega.$$

Thus if $v$ and $w$ are symplectic, using $$ \mathcal{L}_v\omega \;=\; 0$$, Cartan's identity, and the fact that $$\iota_w\omega$$ is a closed form, $$\iota_{[v,w]}\omega = \mathcal L_v\iota_w\omega = d(\iota_v\iota_w\omega) + \iota_vd(\iota_w\omega) = d(\iota_v\iota_w\omega) = d(\omega(w,v)).$$

It follows that $$[v,w] = X_{\omega(w,v)}$$, so that

Thus, the Poisson bracket on functions corresponds to the Lie bracket of the associated Hamiltonian vector fields. We have also shown that the Lie bracket of two symplectic vector fields is a Hamiltonian vector field and hence is also symplectic. In the language of abstract algebra, the symplectic vector fields form a subalgebra of the Lie algebra of smooth vector fields on $M$, and the Hamiltonian vector fields form an ideal of this subalgebra. The symplectic vector fields are the Lie algebra of the (infinite-dimensional) Lie group of symplectomorphisms of $M$.

It is widely asserted that the Jacobi identity for the Poisson bracket, $$\{f,\{g,h\}\} + \{g,\{h,f\}\} + \{h,\{f,g\}\} = 0$$ follows from the corresponding identity for the Lie bracket of vector fields, but this is true only up to a locally constant function. However, to prove the Jacobi identity for the Poisson bracket, it is sufficient to show that: $$\operatorname{ad}_{\{g,f\}}=\operatorname{ad}_{-\{f,g\}}=[\operatorname{ad}_f,\operatorname{ad}_g]$$ where the operator $$\operatorname{ad}_g$$ on smooth functions on $M$ is defined by $$\operatorname{ad}_g(\cdot) \;=\; \{\cdot,\, g\}$$ and the bracket on the right-hand side is the commutator of operators, $$ [\operatorname A,\, \operatorname B] \;=\; \operatorname A\operatorname B - \operatorname B\operatorname A$$. By $$, the operator $$\operatorname{ad}_g$$ is equal to the operator $X_{g}$. The proof of the Jacobi identity follows from $$ because, up to the factor of -1, the Lie bracket of vector fields is just their commutator as differential operators.

The algebra of smooth functions on M, together with the Poisson bracket forms a Poisson algebra, because it is a Lie algebra under the Poisson bracket, which additionally satisfies Leibniz's rule $$. We have shown that every symplectic manifold is a Poisson manifold, that is a manifold with a "curly-bracket" operator on smooth functions such that the smooth functions form a Poisson algebra. However, not every Poisson manifold arises in this way, because Poisson manifolds allow for degeneracy which cannot arise in the symplectic case.

A result on conjugate momenta
Given a smooth vector field $$X$$ on the configuration space, let $$P_X$$ be its conjugate momentum. The conjugate momentum mapping is a Lie algebra anti-homomorphism from the Lie bracket to the Poisson bracket: $$\{P_X, P_Y\} = -P_{[X, Y]}.$$

This important result is worth a short proof. Write a vector field $$X$$ at point $$q$$ in the configuration space as $$X_q = \sum_i X^i(q) \frac{\partial}{\partial q^i}$$ where $ \frac{\partial}{\partial q^i}$ is the local coordinate frame. The conjugate momentum to $$X$$ has the expression $$P_X(q, p) = \sum_i X^i(q) \;p_i$$ where the $$p_i$$ are the momentum functions conjugate to the coordinates. One then has, for a point $$(q,p)$$ in the phase space, $$\begin{align} \{P_X,P_Y\}(q,p) &= \sum_i \sum_j \left\{ X^i(q) \;p_i, Y^j(q)\; p_j \right\} \\ &= \sum_{ij} p_i Y^j(q) \frac{\partial X^i}{\partial q^j} -  p_j X^i(q) \frac{\partial Y^j}{\partial q^i} \\ &= -\sum_i p_i \; [X, Y]^i(q) \\ &= - P_{[X, Y]}(q, p). \end{align}$$

The above holds for all $$(q, p)$$, giving the desired result.

Quantization
Poisson brackets deform to Moyal brackets upon quantization, that is, they generalize to a different Lie algebra, the Moyal algebra, or, equivalently in Hilbert space, quantum commutators. The Wigner-İnönü group contraction of these (the classical limit, $ħ → 0$) yields the above Lie algebra.

To state this more explicitly and precisely, the universal enveloping algebra of the Heisenberg algebra is the Weyl algebra (modulo the relation that the center be the unit). The Moyal product is then a special case of the star product on the algebra of symbols. An explicit definition of the algebra of symbols, and the star product is given in the article on the universal enveloping algebra.

title: Poisson bracket


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Danielle Bunten Berry

Danielle Bunten Berry (February 19, 1949 – July 3, 1998), was an American game designer and programmer, known for the 1983 game M.U.L.E., one of the first influential multiplayer video games, and 1984's The Seven Cities of Gold.

In 1998, she was awarded the Lifetime Achievement Award by the Computer Game Developers Association. In 2007, the Academy of Interactive Arts & Sciences chose Berry as the 10th inductee into its Hall of Fame. In 2009, she was chosen by IGN as one of the top 100 game creators of all time.

Early life
Berry was born in St. Louis, Missouri and moved to Little Rock, Arkansas as a junior in high school. She was the oldest of six siblings. While growing up in Little Rock, Berry's family didn't always have enough money to make ends meet, so Berry took a job at a pharmacy. She also held a leadership role with the Boy Scouts.

According to Berry, one of her fondest childhood memories involved playing games with her family. She was quoted saying, “When I was a kid, the only times my family spent together that weren't totally dysfunctional were when we were playing games. Consequently, I believe games are a wonderful way to socialize.”

While attending the University of Arkansas, she opened up her own bike shop called Highroller Cyclerie. Berry acquired a degree in industrial engineering in 1974 and started programming text-based video games as a hobby. After she graduated from college, she was employed by the National Science Foundation, where she created urban models before starting a job at a video game company.

Career
In 1978, Berry sold a real-time auction game for the Apple II titled Wheeler Dealers to a Canadian software company, Speakeasy Software. This early multiplayer game required a custom controller, raising its price to USD$35 in an era of $15 games sold in plastic bags. It sold only 50 copies.

After producing three titles for Strategic Simulations, Berry, who by then had founded a software company called Ozark Softscape, caught the attention of Electronic Arts founder Trip Hawkins. M.U.L.E. was Berry's first game for EA, originally published for the Atari 8-bit computers because the Atari 800 had four controller ports. Berry later ported it to the Commodore 64. While its sales of 30,000 units were not high, the game developed a cult following and was widely pirated. The game setting was inspired by the novel Time Enough for Love by Robert A. Heinlein.

Along with the success of M.U.L.E., Berry also had close ties with the games Robot Rascals, Heart of Africa, and Cartels & Cutthroat$. Throughout her career, she was involved in the creation of 12 games, 10 of which revolved around multiplayer compatibility. The only two which didn't have a multiplayer focus were The Seven Cities of Gold and Heart of Africa.

Berry wanted to follow up M.U.L.E. with a game that would have been similar to the later game Sid Meier's Civilization, but after fellow Ozark Softscape partners balked at the idea, Berry followed with The Seven Cities of Gold, which proved popular because of its simplicity. By the time the continent data were stored in the computer's memory, there was little left for fancy graphics or complex gameplay - the game had only five resources. It was a hit, selling more than 150,000 copies.

The follow-up game, Heart of Africa, appeared in 1985 and was followed by Robot Rascals, a combination computer/card game that had no single-player mode and sold only 9,000 copies, and 1988's Modem Wars, one of the early games played by two players over a dial-up modem. Modem Wars was ahead of its time, as few people in the late 1980s had modems in their homes.

Berry departed EA for MicroProse. Allegedly, Trip Hawkins, CEO of EA, did not feel that pushing production of games onto a cartridge based system was a good idea. The shift was important to Berry, as computer games had previously been distributed on floppy discs, and a changeover to a cartridge system would allow games to be played on Nintendo systems. This was a significant factor in her decision to leave. She then developed a computer version of the board game Axis and Allies, which became 1990's Command HQ, a modem/network grand strategy wargame. Berry's second and last game for MicroProse was 1992's Global Conquest, a 4-player network/modem war game. It was the first 4-player network game from a major publisher. Berry was a strong advocate of multi-player online games, observing that, "No one ever said on their deathbed, 'Gee, I wish I had spent more time alone with my computer.'"

A port of M.U.L.E. to the Mega Drive/Genesis was cancelled after Berry refused to put guns and bombs in the game, feeling it would alter the game too much from its original concept. In 1997, Berry shifted focus to multiplayer games over the Internet with Warsport, a remake of Modem Wars that debuted on the MPlayer.com game network.

Less than a year after the release of Warsport, Berry was diagnosed with lung cancer presumably related to years of heavy smoking. She died on July 3, 1998. At the time, she was working on the design of an Internet version of M.U.L.E.

M.U.L.E
The game's primary premise consisted of players playing with and against one another to establish total control over a planet. The name of the game stands for Multiple Use Labor Element. The game was originally made for the Atari 400 and 800, then later ported to the Commodore 64, NES, and IBM PCjr. The game has a maximum of four players. Players are given different options and choices, and are allowed to create their colony the way they see fit. This can be done by changing races and giving respective colonies different advantages that will impact the way the game is played and determined later on down the line.

Ultimately there are two ways in which players can win the game. The first way is by having the most money out of all four players, and the second way is by being able to survive the colony itself. The game focuses heavily on going out and retrieving resources that can be used to benefit their character. Items such as food, energy, and crystite are some of a number of in-game items that players are able to retrieve and use to better themselves. In order for a player to be able to access these items, they will first have to have access to a M.U.L.E. The acquisition of these items has a direct reflection on what the player will be allowed to do. For example, if a player doesn't have enough food, they will have less time during their turn.

The Seven Cities of Gold
The Seven Cities of Gold was originally intended to be another multiplayer game. It was originally a single player format, focused heavily on having the players travel around the map and collect items to help them strengthen their colony. Once they felt as though they had a solid colony, the players could battle each other to see who could overtake who. After much consideration, Ozark Software came to the conclusion that this would not be doable. Instead, they went with a formula that had the game focus solely on developing a colony.

Ozark Softscape
Ozark Softscape was a computer game development team consisting initially of Berry, her brother Bill Bunten, Jim Rushing, and Alan Watson. Ozark was run out of Berry's basement. The company was based out of Little Rock, Arkansas and had profound success with a few of their early titles. Ozark Softscape had a publishing deal with Electronic Arts for several of its groundbreaking games. In the early 1990s, Ozark Softscape left its partnership with Electronic Arts over a dispute to port some games to cartridge format for the Nintendo Entertainment System. It began a partnership with MicroProse to produce two more titles: Command HQ and Global Conquest. A dispute occurred over creating a follow-up to M.U.L.E. with Sega in 1993, and the company dissolved. The employees of Ozark Softscape moved to different areas of the software industry.

Personal life
Berry was married three times. Berry had three children, one daughter and two sons.

After a third divorce, Berry, who had until then been living as male, transitioned to living as a woman. Berry underwent sex reassignment surgery in November 1992 and afterward kept a lower profile in the games industry. Berry later regretted having surgery, finding that for her, the drawbacks of surgical transition outweighed the benefits, and wishing she had considered alternative approaches. It caused difficulties in family relationships, and led to her being shunned by the video game industry. She joked that the surgery was to improve the industry's male/female ratio and aesthetics, but advised others considering a sex change not to proceed unless there was no alternative and warned them of the cost, saying "Being my 'real self' could have included having a penis and including more femininity in whatever forms made sense. I didn't know that until too late and now I have to make the best of the life I've stumbled into. I just wish I would have tried more options before I jumped off the precipice."

After her transition in fall 1992, Berry stayed out of the video game spotlight, mostly keeping to herself. She felt as though that after transitioning she was not as good at video game development as she had previously been, stating "So, I'm a little more than three years into my new life role as Ms. Danielle Berry, and her career looks to be somewhat different from old Mr. Dan Bunten's. For one thing, I'm not as good a programmer as he was."

On July 3, 1998, Berry died of lung cancer.

Recognition
Although many of Berry's titles were not commercially successful, they were widely recognized by the industry as being ahead of their time. On May 7, 1998, less than two months before her death, Berry was awarded the Lifetime Achievement Award by the Computer Game Developers Association.

In 2000, Will Wright dedicated his blockbuster hit The Sims to Berry's memory. In 2007, the Academy of Interactive Arts & Sciences chose Berry to be inducted into its Hall of Fame. Sid Meier, the mastermind behind the video game series Civilization, inducted her at the Hard Rock Hotel in Las Vegas.

Berry was known as someone who was very easy to talk to. If someone recognized her in public, she would be more than delighted to have a conversation with them.

In 2012, Arkansas Times wrote that Berry "is still considered something of a rock star among game designers and those interested in the history of games", crediting her emphasis on social interaction in gaming.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Danielle Bunten Berry


text:
William Paterson (judge)

William Paterson (December 24, 1745 – September 9, 1806) was an American statesman, lawyer, jurist, and signer of the United States Constitution. He was an Associate Justice of the United States Supreme Court, the second governor of New Jersey, and a Founding Father of the United States.

Born in County Antrim, Ireland, Paterson moved to the North American British colonies at a young age. After graduating from the College of New Jersey (now Princeton University) and studying law under Richard Stockton, he was admitted to the bar in 1768. He helped write the 1776 Constitution of New Jersey and served as the New Jersey Attorney General from 1776 to 1783. He represented New Jersey at the 1787 Philadelphia Convention, where he proposed the New Jersey Plan, which would have provided for equal representation among the states in Congress.

After the ratification of the Constitution, Paterson served in the United States Senate from 1789 to 1790, helping to draft the Judiciary Act of 1789. He resigned from the Senate to take office as governor of New Jersey. In 1793, he accepted appointment by President George Washington to serve as an associate justice of the Supreme Court. He served on the court until his death in 1806.

Early life
William Paterson was born December 24, 1745, in County Antrim, Ireland, to Richard Paterson, an Ulster Protestant. Paterson immigrated with his parents to New Castle, Pennsylvania, in 1747. At 14, he began college at Princeton. After graduating, he read law with the prominent lawyer Richard Stockton and was admitted to the bar in 1768. He also stayed connected to his alma mater and helped found the Cliosophic Society with Aaron Burr.

Early career
Paterson was selected as the Somerset County delegate for the first three provincial congresses of New Jersey, where, as secretary, he recorded the 1776 New Jersey State Constitution. Paterson was appointed as the first attorney general of New Jersey, serving from 1776 to 1783, establishing himself as one of the state's most prominent lawyers. He was sent to the 1787 Philadelphia Convention, where he proposed the New Jersey Plan for a unicameral legislative body with equal representation from each state. The Constitution of the United States was ultimately signed with the Connecticut Compromise that created a bicameral Congress with a Senate that equally represented each state and a House of Representatives with population-based representation.

United States Senator
Paterson, who was a strong nationalist who supported the Federalist Party, went on to become one of New Jersey's first U.S. senators (1789–90). As a member of the Senate Judiciary Committee, he played an important role in drafting the Judiciary Act of 1789 that established the federal court system. The first nine sections of this very important law are in his handwriting.

Governor of New Jersey
In 1790, he became the first person to resign from the U.S. Senate, when he did so in order to succeed fellow signer William Livingston as governor of New Jersey. As governor, Paterson pursued his interest in legal matters by codifying the English statutes that had been in force in New Jersey before the Revolution in Laws of the State of New Jersey. He also published a revision of the rules of the chancery and common law courts in Paterson, later adopted by the New Jersey Legislature.

United States Supreme Court
President George Washington nominated Paterson for the Supreme Court of the United States on February 27, 1793, to the seat vacated by Thomas Johnson. Washington withdrew the nomination the following day, having realized that since the Judiciary Act of 1789 (the law creating the Supreme Court) had been passed during Paterson's current term as a Senator, the nomination was a violation of the Ineligibility Clause (Article I, Section 6) of the Constitution. Washington re-nominated Paterson to the court on March 4, 1793, after his term as Senator had expired; Paterson was immediately confirmed by the Senate and received his commission.

He resigned from the governorship to become an associate justice of the U.S. Supreme Court. On the circuit, he presided over the trials of individuals indicted for treason in the Whiskey Rebellion, a revolt by farmers in western Pennsylvania over the federal excise tax on whiskey, the principal product of their cash crop. Militia sent out by President Washington successfully quelled the uprising, and for the first time, the courts had to interpret the provisions of the Constitution concerning the use of troops in civil disturbances. Here, and, throughout his long career, Paterson extolled the primacy of law over governments, a principle embodied in the Constitution he helped write. He declined an appointment as Secretary of State in 1795. Paterson was elected to the American Philosophical Society in 1789. He was elected a Fellow of the American Academy of Arts and Sciences in 1801. Paterson served on the Supreme Court until he died in 1806.

Personal life
In 1779, Paterson married Cornelia Bell (1755–1783), daughter of John Bell, a wealthy Somerset County landowner. Together, they had three children, but she died in 1783 shortly after giving birth to their only son. Their children were:

 * Cornelia Bell Paterson (1780–1844), who married Stephen Van Rensselaer (1764–1839) after the death of his first wife, Margaret "Peggy" Schuyler (1758–1801)
 * Frances Van Paterson (1781–1783), who died young
 * William Bell Paterson (1783–1832), who married Jane Eliza Neilson

In 1785, he married Euphemia White (1746–1832), sister of Anthony Walton White (1750–1803), daughter of Anthony White (1717–1787), a New Jersey landholder and judge of the Somerset court, and the granddaughter of Lewis Morris (1671–1746), chief justice of New York from 1715 to 1733 and governor of New Jersey from 1738 to 1746.

Death and interment
On September 9, 1806, Paterson, aged 60, died from the lingering effects of a coach accident suffered in 1803 while on circuit court duty in New Jersey. He was on his way to the spa at Ballston Springs, New York, to "take the waters", when he died at the Van Rensselaer Manor home of his daughter, Cornelia, and son-in-law, Stephen Van Rensselaer, in Albany, New York. He was laid to rest in the Van Renssalaer family vault. When the city acquired the property, Paterson's remains were relocated to Albany Rural Cemetery Menands in Albany County, New York. Also buried there are Associate Justice Rufus W. Peckham and President Chester A. Arthur.

Descendants
Through his eldest daughter, his grandchildren include Cortlandt Van Rensselaer (1808–1860), a noted Presbyterian clergyman, and Henry Bell Van Rensselaer (1810–1864), a politician and general in the Union Army during the American Civil War, who married Elizabeth Ray King, a granddaughter of U.S. Senator Rufus King.

Through his son, his grandchildren included twin brothers, William Paterson (1817–1899), who married Salvadora Meade, a Spanish-born woman living in Philadelphia, and Stephen Van Rensselaer Paterson (1817–1872), who married Emily Sophia King (1823–1853), daughter of Charles King (1789–1867), the president of Columbia University, and the second son Rufus King. Both grandsons were members of the Princeton University class of 1835 and William was admitted to the bar in 1838. He later served as a member of the New Jersey Assembly from 1842 to 1843, Secretary of the New Jersey Constitutional Convention of 1844, a lay judge of the Court of Errors and Appeals, and mayor of Perth Amboy for ten years in between 1846 and 1878.

Honors
Both the city of Paterson, and the college, William Paterson University, are named after him.

title: William Paterson (judge)


text:
Bilinear

Bilinear may refer to:

 * Bilinear sampling (also called "bilinear filtering"), a method in computer graphics for choosing the color of a texture
 * Bilinear form, a type of mathematical function from a vector space to the underlying field
 * Bilinear interpolation, an extension of linear interpolation for interpolating functions of two variables on a rectilinear 2D grid
 * Bilinear map, a type of mathematical function between vector spaces
 * Bilinear transform, a method of transforming from the S to Z domain in control theory and signal processing
 * Bilinear transformation (disambiguation)

title: Bilinear


text:
1839 in art

Events from the year 1839 in art.

Events

 * January 9 – The French Academy of Sciences announces the Daguerreotype photography process.
 * January 25 – H. Fox Talbot shows his "photogenic drawings" at the Royal Institution in London.
 * c. October – Robert Cornelius takes a daguerreotype self-portrait, the earliest known existing photographic portrait of a human in America.
 * Honoré de Balzac's novel Pierre Grassou concerns an artist who lives off forgeries.

Works

 * John Bell – Babes in the Wood (sculpture)
 * Sara Anne Bright – The Leaf (photogenic drawing)
 * Francesco Hayez – Reclining Odalisque
 * George Hayter – The Coronation of Queen Victoria
 * Edwin Landseer – Dignity and Impudence
 * Carl Spitzweg – The Poor Poet
 * J. M. W. Turner
 * Ancient Rome – Agrippina Landing with the Ashes of Germanicus
 * Modern Rome – Campo Vaccino
 * Cicero at his Villa
 * The Fighting Temeraire tugged to her last Berth to be broken up, 1838
 * Peter von Hess – The entry of King Otto of Greece into Athens and his reception in front of the Thiseion temple in 1833
 * Sir David Wilkie – General Sir David Baird Discovering the Body of Sultan Tippoo Sahib after having Captured Seringapatam, on the 4th May, 1799
 * William Wyon – Una and the Lion (design for five pounds British gold coin)

Births

 * January 19 – Paul Cézanne, French painter (died 1906)
 * March 16 – John Butler Yeats, Irish painter (died 1922)
 * May 24 – Arthur Quartley, painter (died 1886)
 * June 9 – Joseph Paelinck, Belgian painter (born 1781)
 * September 1 – Charles Edward Perugini, English painter (died 1918)
 * October 2 – Hans Thoma, German painter (died 1924)
 * October 29 – Kate Dickens Perugini, English painter (died 1929)
 * October 30 – Alfred Sisley, French impressionist painter (died 1899)
 * November 16 – William De Morgan, English ceramic artist (died 1917)
 * November 20 – Christian Wilberg, painter (died 1882)
 * date unknown – François Sallé, French realist painter (died 1899)

Deaths

 * January 12 – Joseph Anton Koch, Austrian painter of the German Romantic movement (born 1768)
 * January 14 – John Wesley Jarvis, American painter (born 1781)
 * January 28 – Sir William Beechey, English portrait painter (born 1753)
 * February 6 – François Louis Thomas Francia, French shorescape watercolorist (born 1772)
 * February 8 – Giovanni Cendramini, Italian painter and engraver (born 1760)
 * February 21 – John Charles Felix Rossi, English sculptor (born 1769)
 * June 2 – Wijnand Nuijen, Dutch land- and seascape painter (born 1813)
 * June 4 – Cornelia Scheffer, Dutch painter and portrait miniaturist (born 1769)
 * July 8 – John Laporte, English landscape painter and etcher (born 1761)
 * September 4 – Hendrik Voogd, Dutch painter and printmaker active in Italy (born 1768)
 * September 10 – Pieter Fontijn, Dutch painter and drawer (born 1773)
 * September 30 – John Thirtle, English watercolour painter and frame-maker (born 1777)
 * November 15 – Giocondo Albertolli, Swiss-born architect, painter and sculptor active in Italy (born 1743)
 * December – William Sadler, Irish landscape painter (born 1782)
 * December 22 – Cornelis van Spaendonck, Dutch painter (born 1756)
 * date unknown – Paolo Vincenzo Bonomini, Italian portrait and caricature painter (born 1757)

title: 1839 in art


text:
1906 in art

Events from the year 1906 in art.

Events

 * September–October – First group exhibition by Die Brücke, in Dresden.
 * Gwen John begins modelling for Auguste Rodin.
 * Hilma af Klint begins painting abstract art, including the first of her Primordial Chaos series.
 * Paula Modersohn-Becker begins a series of nude portraits of herself and of other women and children in Paris.
 * Juan Gris, Amedeo Modigliani and Gino Severini all arrive in Paris.
 * Walter Sickert paints music hall scenes in London and Paris.
 * Ferdinand Preiss opens his workshop in Berlin.
 * Museum of Fine Arts (Budapest) completed.
 * Dritte Deutsche Kunstgewerbeausstellung Dresden (third German arts and crafts exhibition) took place in Dresden from May 12 to October 31

Works

 * Umberto Boccioni – Self-portrait
 * Antoine Bourdelle – The Fruit (sculpture)
 * Olga Boznańska – Portrait of a Lady in a White Hat
 * Paul Cézanne – The Gardener Vallier
 * Robert Delaunay – L'homme à la tulipe (Portrait de Jean Metzinger)
 * André Derain
 * Charing Cross Bridge, London
 * The Pool of London
 * Daniel Chester French and Edward Clark Potter
 * Equestrian statue of Charles Devens
 * Progress of the State (quadriga)
 * J. W. Godward
 * Drusilla
 * Nerissa
 * The Tambourine Girl (two versions)
 * Wilhelm Hammershoi – Interior
 * William Le Baron Jenney and Charles Mulligan – Illinois Memorial
 * Robert Henri - El Matador (Felix Asiego)
 * Ishibashi Kazunori – Lady Reading Poetry
 * Theo Alice Ruggles Kitson – The Hiker (statue)
 * Peder Severin Krøyer – Midsummer Eve Bonfire on Skagen Beach
 * Albert Marquet
 * The Beach at Fécamp
 * Fécamp (The Beach at Sainte-Adresse)
 * Henri Matisse
 * Le bonheur de vivre
 * Self-Portrait in a Striped T-shirt
 * The Young Sailor I
 * The Young Sailor II
 * Willard Metcalf – May Night
 * Jean Metzinger
 * Coucher de soleil no. 1
 * Femme au Chapeau
 * La danse (Bacchante)
 * Portrait de Robert Delaunay
 * Paula Modersohn-Becker – Self-portrait
 * Claude Monet – Water Lilies
 * Edvard Munch
 * Portrait of Friedrich Nietzsche
 * Self-portrait with a bottle
 * Mikhail Nesterov
 * Portrait of E. P. Nesterov
 * Portrait of Olga Nesterova (Woman in a Riding Habit)
 * Maxfield Parrish – Old King Cole and his Fiddlers Three (mural at St. Regis Hotel, New York City)
 * Pablo Picasso
 * Boy Leading a Horse
 * Portrait of Gertrude Stein
 * Self-Portrait with Palette
 * Niko Pirosmani – Feast with Organ-Grinder Datiko Zemel
 * Medardo Rosso – Ecce Puer (sculpted bust)
 * John Singer Sargent – Self-portrait
 * Zinaida Serebriakova – Country Girl
 * Walter Sickert – La Hollandaise
 * Franz von Stuck – Salome
 * Albert Chevallier Tayler – Kent vs Lancashire at Canterbury
 * Douglas Tilden – California Volunteers (sculpture, San Francisco)
 * Ville Vallgren – Havis Amanda (bronze)

January to June

 * January 2 – Hans Mertens, German painter (died 1944)
 * January 3 – Óscar Domínguez, Spanish surrealist painter (died 1957)
 * January 9 – Janko Brašić, Serbian painter (died 1994)
 * January 13 – Burgoyne Diller, American abstract painter (died 1965)
 * February 28 – Percy Shakespeare, English painter (died 1943)
 * March 2 – Jessie Oonark, Canadian Inuit artist (died 1985)
 * March 8 – Victor Hasselblad, Swedish inventor and photographer (died 1978)
 * March 9 – David Smith, American abstract expressionist sculptor (died 1965)
 * March 26 – Henri Cadiou, French realist painter and lithographer (died 1989)
 * March 27 – Bernard Lefebvre, French photographer (died 1992)
 * April 11 – Dale Messick, first woman syndicated comic strip artist in the United States (died 2005)
 * April 14 – Elmyr de Hory, Hungarian-born painter and art forger (d. 1976)
 * April 21 – Lillian Browse, English art dealer and historian (died 2005)
 * May 16 – Alfred Pellan, Canadian artist (died 1988)
 * May 2 – Philippe Halsman, Latvian-born American portrait photographer (died 1979)
 * May 20 – Leon Polk Smith, American painter (died 1996)
 * June 9 – James Boswell, New Zealand painter and draughtsman (died 1971)

July to December

 * July 8 – Philip Johnson, American architect, art collector, curator (died 2005)
 * August 12 – Tedd Pierce, American animated cartoon writer, animator and artist (died 1972)
 * August 14 – Horst P. Horst, German American photographer (died 1999)
 * August 20 – Heinz Henghes, German sculptor (died 1975)
 * August 21 – Friz Freleng, American animator, cartoonist, director and producer (died 1995)
 * September 5 – Ralston Crawford, American painter, lithographer and photographer (died 1978)
 * October 18 – James Brooks, American painter and muralist (died 1992)
 * October 24 – Marie-Louise von Motesiczky, Austrian painter (died 1996)
 * October 27 – Peter Blume, American painter (died 1992)
 * November 26 – Krystyna Dąbrowska, Polish sculptor and painter (died 1944)
 * December 10 – Padraig Marrinan, Irish painter (died 1975)
 * December 14 – Maxwell Bates, Canadian architect and expressionist painter (died 1980)
 * December 27 – Andreas Feininger, French-born American photographer (died 1999)

Deaths

 * January 20 – Philippe Solari, French sculptor (born 1840)
 * February – Charles-Auguste Lebourg, French sculptor (born 1829)
 * February 14 – Carl Eneas Sjöstrand, Swedish sculptor (born 1828)
 * March 5 – Hugh Ramsay, Australian painter (born 1877)
 * March 17 – Thomas Dalziel, English engraver (born 1823)
 * March 27 – Eugène Carrière, French Symbolist painter (born 1849)
 * March 19 – Étienne Carjat, French portraitist (born 1828)
 * March 29 – Slava Raskaj, Croatian watercolour painter (born 1877)
 * April 5 – Wyke Bayliss, English painter (born 1835)
 * April 14 – Walter Williams, English landscape painter (born 1834)
 * April 19 – Daniel Huntington, American painter (born 1816)
 * May – John Mulvany, Irish-born American painter (b. circa 1839)
 * June 20 – John Clayton Adams, English landscape artist (born 1840)
 * August 21 – Eugen Felix, Austrian painter (born 1836)
 * August 29 – Alfred Stevens, Belgian painter (born 1828)
 * October 2 – Raja Ravi Varma, Indian painter (born 1848)
 * October 22 – Paul Cézanne, French painter (born 1839)
 * October 28 – Jean Benner, French artist (born 1836)
 * November 5 – Frits Thaulow, Norwegian Impressionist painter (born 1847)
 * December 11 – Axel Lindahl, Swedish landscape photographer (born 1841)
 * November 20 – Charles Eastlake, English architect and furniture designer (born 1836)
 * December 31 – Julia Goodman, English portrait painter (born 1812)
 * date unknown
 * Antonio Beato, Italian-born photographer (born c.1825)
 * Charles Edwin Fripp, English-born war artist (born 1854)
 * Maeda Genzō, Japanese photographer (born 1831)
 * William Linnell, British painter (born 1826)

title: 1906 in art


text:
Beloved

Beloved may refer to:

Books

 * Beloved (novel), a 1987 novel by Toni Morrison
 * The Beloved (Faulkner novel), a 2012 novel by Australian author Annah Faulkner
 * Beloved, a 1993 historical romance about Zenobia, by Bertrice Small

Film

 * Beloved (1934 film), an American drama film directed by Victor Schertzinger
 * Beloved (1965 film), a Soviet romance film directed by Richard Viktorov
 * Beloved (1998 film), based on the Toni Morrison novel
 * Beloved (2011 film), a French film written and directed by Christophe Honoré
 * The Beloved (1940 film), a Soviet film directed by Ivan Pyryev
 * The Beloved, also known as Sin, a 1971 British film written and directed by George P. Cosmatos
 * The Beloved (1991 film), a Georgian film
 * The Beloved (2015 film), a Chinese film directed by Cao Dawei

Bands

 * Beloved (band), an American post-hardcore band
 * The Beloved (band), a British electronic music group

Albums

 * Beloved (Dave East and Styles P album), a 2018 mixtape by Dave East with Styles P
 * Beloved (Glay album), 1996
 * Beloved (Jordan Feliz album), 2015
 * Beloved (Jewelry album), 2003
 * Beloved (I Killed the Prom Queen album), 2014
 * Beloved (Mesut Kurtis album), 2009
 * Beloved, a 2018 album by Snatam Kaur

Songs

 * "Beloved" (Jordan Feliz song), a song from the eponymous 2015 album
 * "Beloved" (Mumford & Sons song), a song from the 2018 album Delta
 * "Beloved" (Wendy Matthews song), a 1998 single from Wendy Matthews
 * "Beloved" (VNV Nation song), a song by VNV Nation from the 2002 album Futureperfect
 * "Beloved", a song by Ayumi Hamasaki's from the 2011 EP Five
 * "Beloved", a song by Leeland from the 2016 album Invisible
 * "Beloved", an art song by Rhea Silberta (1900–1959)
 * "Beloved", a song by Luscious Jackson from the 1999 album Electric Honey

People and biblical figures

 * John the Apostle or Saint John the Beloved, one of the Twelve Apostles of Jesus
 * Charles VI of France (1368–1422), King of France known as "the Beloved"
 * Louis XV (1710–1774), King of France known as "the Beloved"

Other uses

 * The Beloved (Rossetti painting), an 1865 oil painting by Dante Gabriel Rossetti
 * Beloved, the main antagonist of The Unicorn Chronicles book series

title: Beloved


text:
Chern class

In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since become fundamental concepts in many branches of mathematics and physics, such as string theory, Chern–Simons theory, knot theory, Gromov–Witten invariants. Chern classes were introduced by.

Basic idea and motivation
Chern classes are characteristic classes. They are topological invariants associated with vector bundles on a smooth manifold. The question of whether two ostensibly different vector bundles are the same can be quite hard to answer. The Chern classes provide a simple test: if the Chern classes of a pair of vector bundles do not agree, then the vector bundles are different. The converse, however, is not true.

In topology, differential geometry, and algebraic geometry, it is often important to count how many linearly independent sections a vector bundle has. The Chern classes offer some information about this through, for instance, the Riemann–Roch theorem and the Atiyah–Singer index theorem.

Chern classes are also feasible to calculate in practice. In differential geometry (and some types of algebraic geometry), the Chern classes can be expressed as polynomials in the coefficients of the curvature form.

Construction
There are various ways of approaching the subject, each of which focuses on a slightly different flavor of Chern class.

The original approach to Chern classes was via algebraic topology: the Chern classes arise via homotopy theory which provides a mapping associated with a vector bundle to a classifying space (an infinite Grassmannian in this case). For any complex vector bundle V over a manifold M, there exists a map f from M to the classifying space such that the bundle V is equal to the pullback, by f, of a universal bundle over the classifying space, and the Chern classes of V can therefore be defined as the pullback of the Chern classes of the universal bundle. In turn, these universal Chern classes can be explicitly written down in terms of Schubert cycles.

It can be shown that for any two maps f, g from M to the classifying space whose pullbacks are the same bundle V, the maps must be homotopic. Therefore, the pullback by either f or g of any universal Chern class to a cohomology class of M must be the same class. This shows that the Chern classes of V are well-defined.

Chern's approach used differential geometry, via the curvature approach described predominantly in this article. He showed that the earlier definition was in fact equivalent to his. The resulting theory is known as the Chern–Weil theory.

There is also an approach of Alexander Grothendieck showing that axiomatically one need only define the line bundle case.

Chern classes arise naturally in algebraic geometry. The generalized Chern classes in algebraic geometry can be defined for vector bundles (or more precisely, locally free sheaves) over any nonsingular variety. Algebro-geometric Chern classes do not require the underlying field to have any special properties. In particular, the vector bundles need not necessarily be complex.

Regardless of the particular paradigm, the intuitive meaning of the Chern class concerns 'required zeroes' of a section of a vector bundle: for example the theorem saying one can't comb a hairy ball flat (hairy ball theorem). Although that is strictly speaking a question about a real vector bundle (the "hairs" on a ball are actually copies of the real line), there are generalizations in which the hairs are complex (see the example of the complex hairy ball theorem below), or for 1-dimensional projective spaces over many other fields.

See Chern–Simons theory for more discussion.

The Chern class of line bundles
(Let X be a topological space having the homotopy type of a CW complex.)

An important special case occurs when V is a line bundle. Then the only nontrivial Chern class is the first Chern class, which is an element of the second cohomology group of X. As it is the top Chern class, it equals the Euler class of the bundle.

The first Chern class turns out to be a complete invariant with which to classify complex line bundles, topologically speaking. That is, there is a bijection between the isomorphism classes of line bundles over X and the elements of $$H^2(X;\Z)$$, which associates to a line bundle its first Chern class. Moreover, this bijection is a group homomorphism (thus an isomorphism): $$c_1(L \otimes L') = c_1(L) + c_1(L');$$ the tensor product of complex line bundles corresponds to the addition in the second cohomology group.

In algebraic geometry, this classification of (isomorphism classes of) complex line bundles by the first Chern class is a crude approximation to the classification of (isomorphism classes of) holomorphic line bundles by linear equivalence classes of divisors.

For complex vector bundles of dimension greater than one, the Chern classes are not a complete invariant.

Via the Chern–Weil theory
Given a complex hermitian vector bundle V of complex rank n over a smooth manifold M, representatives of each Chern class (also called a Chern form) $$c_k(V)$$ of V are given as the coefficients of the characteristic polynomial of the curvature form $$\Omega$$ of V.

$$\det \left(\frac {it\Omega}{2\pi} +I\right) = \sum_k c_k(V) t^k$$

The determinant is over the ring of $$n \times n$$ matrices whose entries are polynomials in t with coefficients in the commutative algebra of even complex differential forms on M. The curvature form $$\Omega$$ of V is defined as $$\Omega = d\omega+\frac{1}{2}[\omega,\omega]$$ with ω the connection form and d the exterior derivative, or via the same expression in which ω is a gauge field for the gauge group of V. The scalar t is used here only as an indeterminate to generate the sum from the determinant, and I denotes the n × n identity matrix.

To say that the expression given is a representative of the Chern class indicates that 'class' here means up to addition of an exact differential form. That is, Chern classes are cohomology classes in the sense of de Rham cohomology. It can be shown that the cohomology classes of the Chern forms do not depend on the choice of connection in V.

If follows from the matrix identity $$\mathrm{tr}(\ln(X))=\ln(\det(X))$$ that $$ \det(X) =\exp(\mathrm{tr}(\ln(X)))$$. Now applying the Maclaurin series for $$\ln(X+I)$$, we get the following expression for the Chern forms:

$$\sum_k c_k(V) t^k = \left[ I       + i \frac{\mathrm{tr}(\Omega)}{2\pi} t        +   \frac{\mathrm{tr}(\Omega^2)-\mathrm{tr}(\Omega)^2}{8\pi^2} t^2 + i \frac{-2\mathrm{tr}(\Omega^3)+3\mathrm{tr}(\Omega^2)\mathrm{tr}(\Omega)-\mathrm{tr}(\Omega)^3}{48\pi^3} t^3 + \cdots \right].$$

Via an Euler class
One can define a Chern class in terms of an Euler class. This is the approach in the book by Milnor and Stasheff, and emphasizes the role of an orientation of a vector bundle.

The basic observation is that a complex vector bundle comes with a canonical orientation, ultimately because $$\operatorname{GL}_n(\Complex)$$ is connected. Hence, one simply defines the top Chern class of the bundle to be its Euler class (the Euler class of the underlying real vector bundle) and handles lower Chern classes in an inductive fashion.

The precise construction is as follows. The idea is to do base change to get a bundle of one-less rank. Let $$\pi\colon E \to B$$ be a complex vector bundle over a paracompact space B. Thinking of B as being embedded in E as the zero section, let $$B' = E \setminus B$$ and define the new vector bundle: $$E' \to B'$$ such that each fiber is the quotient of a fiber F of E by the line spanned by a nonzero vector v in F (a point of B′ is specified by a fiber F of E and a nonzero vector on F.) Then $$E'$$ has rank one less than that of E. From the Gysin sequence for the fiber bundle $$\pi|_{B'}\colon B' \to B$$: $$\cdots \to \operatorname{H}^k(B; \Z) \overset{\pi|_{B'}^*} \to \operatorname{H}^k(B'; \Z) \to \cdots,$$ we see that $$\pi|_{B'}^*$$ is an isomorphism for $$k < 2n-1$$. Let $$c_k(E) = \begin{cases} {\pi|_{B'}^*}^{-1} c_k(E') & k < n\\ e(E_{\R}) & k = n \\ 0 & k > n \end{cases}$$

It then takes some work to check the axioms of Chern classes are satisfied for this definition.

See also: The Thom isomorphism.

The complex tangent bundle of the Riemann sphere
Let $$\mathbb{CP}^1$$ be the Riemann sphere: 1-dimensional complex projective space. Suppose that z is a holomorphic local coordinate for the Riemann sphere. Let $$V=T\mathbb{CP}^1$$ be the bundle of complex tangent vectors having the form $$a \partial/\partial z$$ at each point, where a is a complex number. We prove the complex version of the hairy ball theorem: V has no section which is everywhere nonzero.

For this, we need the following fact: the first Chern class of a trivial bundle is zero, i.e., $$c_1(\mathbb{CP}^1\times \Complex)=0.$$

This is evinced by the fact that a trivial bundle always admits a flat connection. So, we shall show that $$c_1(V) \not= 0.$$

Consider the Kähler metric $$h = \frac{dz d\bar{z}}{(1+|z|^2)^2}.$$

One readily shows that the curvature 2-form is given by $$\Omega=\frac{2dz\wedge d\bar{z}}{(1+|z|^2)^2}.$$

Furthermore, by the definition of the first Chern class $$c_1= \left[\frac{i}{2\pi} \operatorname{tr} \Omega\right] .$$

We must show that this cohomology class is non-zero. It suffices to compute its integral over the Riemann sphere: $$\int c_1 =\frac{i}{\pi}\int \frac{dz\wedge d\bar{z}}{(1+|z|^2)^2}=2$$ after switching to polar coordinates. By Stokes' theorem, an exact form would integrate to 0, so the cohomology class is nonzero.

This proves that $$T\mathbb{CP}^1$$ is not a trivial vector bundle.

Complex projective space
There is an exact sequence of sheaves/bundles: $$0 \to \mathcal{O}_{\mathbb{CP}^n} \to \mathcal{O}_{\mathbb{CP}^n}(1)^{\oplus (n+1)} \to T\mathbb{CP}^n \to 0$$ where $$\mathcal{O}_{\mathbb{CP}^n} $$ is the structure sheaf (i.e., the trivial line bundle), $$\mathcal{O}_{\mathbb{CP}^n}(1)$$ is Serre's twisting sheaf (i.e., the hyperplane bundle) and the last nonzero term is the tangent sheaf/bundle.

There are two ways to get the above sequence:

1. Let $z_0, \ldots, z_n$ be the coordinates of $\Complex^{n+1},$ let $\pi\colon \Complex^{n+1} \setminus \{0\} \to \Complex\mathbb{P}^n$ be the canonical projection, and let $U = \mathbb{CP}^n \setminus \{ z_0 = 0\}$. Then we have: $\pi^* d(z_i / z_0) = {z_0 dz_i - z_i d z_0 \over z_0^2}, \, i \ge 1.$ In other words, the cotangent sheaf $\Omega_{\Complex\mathbb{P}^n}

2. _U$, which is a free $\mathcal{O}_U$-module with basis $d(z_i / z_0)$, fits into the exact sequence $ 0 \to \Omega_{\Complex\mathbb{P}^n}

3. _U \overset{dz_i \mapsto e_i}\to \oplus_1^{n+1} \mathcal{O}(-1)

4. _U \overset{e_i \mapsto z_i}\to \mathcal{O}_U \to 0, \, i \ge 0,$ where $e_i$ are the basis of the middle term. The same sequence is clearly then exact on the whole projective space and the dual of it is the aforementioned sequence.

5. Let L be a line in $\Complex^{n+1}$ that passes through the origin. It is an elementary geometry to see that the complex tangent space to $\Complex\mathbb{P}^n$ at the point L is naturally the set of linear maps from L to its complement. Thus, the tangent bundle $T\Complex\mathbb{P}^n$ can be identified with the hom bundle $\operatorname{Hom}(\mathcal{O}(-1), \eta)$ where η is the vector bundle such that $\mathcal{O}(-1) \oplus \eta = \mathcal{O}^{\oplus (n+1)}$. It follows: $T\Complex \mathbb{P}^n \oplus \mathcal{O} = \operatorname{Hom}(\mathcal{O}(-1), \eta) \oplus \operatorname{Hom}(\mathcal{O}(-1), \mathcal{O}(-1)) = \mathcal{O}(1)^{\oplus(n+1)}.$

By the additivity of total Chern class $$c = 1 + c_1 + c_2 + \cdots$$ (i.e., the Whitney sum formula), $$c(\Complex\mathbb{P}^n) \overset{\mathrm{def}}= c(T\mathbb{CP}^n) = c(\mathcal{O}_{\Complex\mathbb{P}^n}(1))^{n+1} = (1+a)^{n+1},$$ where a is the canonical generator of the cohomology group $$H^2(\Complex\mathbb{P}^n, \Z )$$; i.e., the negative of the first Chern class of the tautological line bundle $$\mathcal{O}_{\Complex\mathbb{P}^n}(-1)$$ (note: $$c_1(E^*) = -c_1(E)$$ when $$E^*$$ is the dual of E.)

In particular, for any $$k\ge 0$$, $$c_k(\Complex\mathbb{P}^n) = \binom{n+1}{k} a^k.$$

Chern polynomial
A Chern polynomial is a convenient way to handle Chern classes and related notions systematically. By definition, for a complex vector bundle E, the Chern polynomial ct of E is given by: $$c_t(E) =1 + c_1(E) t + \cdots + c_n(E) t^n.$$

This is not a new invariant: the formal variable t simply keeps track of the degree of ck(E). In particular, $$c_t(E)$$ is completely determined by the total Chern class of E: $$c(E) =1 + c_1(E) + \cdots + c_n(E)$$ and conversely.

The Whitney sum formula, one of the axioms of Chern classes (see below), says that ct is additive in the sense: $$c_t(E \oplus E') = c_t(E) c_t(E').$$ Now, if $$E = L_1 \oplus \cdots \oplus L_n$$ is a direct sum of (complex) line bundles, then it follows from the sum formula that: $$c_t(E) = (1+a_1(E) t) \cdots (1+a_n(E) t)$$ where $$a_i(E) = c_1(L_i)$$ are the first Chern classes. The roots $$a_i(E)$$, called the Chern roots of E, determine the coefficients of the polynomial: i.e., $$c_k(E) = \sigma_k(a_1(E), \ldots, a_n(E))$$ where σk are elementary symmetric polynomials. In other words, thinking of ai as formal variables, ck "are" σk. A basic fact on symmetric polynomials is that any symmetric polynomial in, say, ti's is a polynomial in elementary symmetric polynomials in ti's. Either by splitting principle or by ring theory, any Chern polynomial $$c_t(E)$$ factorizes into linear factors after enlarging the cohomology ring; E need not be a direct sum of line bundles in the preceding discussion. The conclusion is

Example: We have polynomials sk $$t_1^k + \cdots + t_n^k = s_k(\sigma_1(t_1, \ldots, t_n), \ldots, \sigma_k(t_1, \ldots, t_n))$$ with $$s_1 = \sigma_1, s_2 = \sigma_1^2 - 2 \sigma_2$$ and so on (cf. Newton's identities). The sum $$\operatorname{ch}(E) = e^{a_1(E)} + \cdots + e^{a_n(E)} = \sum s_k(c_1(E), \ldots, c_n(E)) / k!$$ is called the Chern character of E, whose first few terms are: (we drop E from writing.) $$\operatorname{ch}(E) = \operatorname{rk} + c_1 + \frac{1}{2}(c_1^2 - 2c_2) + \frac{1}{6} (c_1^3 - 3c_1c_2 + 3c_3) + \cdots.$$

Example: The Todd class of E is given by: $$\operatorname{td}(E) = \prod_1^n {a_i \over 1 - e^{-a_i}} = 1 + {1 \over 2} c_1 + {1 \over 12} (c_1^2 + c_2) + \cdots.$$

Remark: The observation that a Chern class is essentially an elementary symmetric polynomial can be used to "define" Chern classes. Let Gn be the infinite Grassmannian of n-dimensional complex vector spaces. It is a classifying space in the sense that, given a complex vector bundle E of rank n over X, there is a continuous map $$f_E: X \to G_n$$ unique up to homotopy. Borel's theorem says the cohomology ring of Gn is exactly the ring of symmetric polynomials, which are polynomials in elementary symmetric polynomials σk; so, the pullback of fE reads: $$f_E^*: \Z [\sigma_1, \ldots, \sigma_n] \to H^*(X, \Z ).$$ One then puts: $$c_k(E) = f_E^*(\sigma_k).$$

Remark: Any characteristic class is a polynomial in Chern classes, for the reason as follows. Let $$\operatorname{Vect}_n^{\Complex}$$ be the contravariant functor that, to a CW complex X, assigns the set of isomorphism classes of complex vector bundles of rank n over X and, to a map, its pullback. By definition, a characteristic class is a natural transformation from $$\operatorname{Vect}_n^{\Complex } = [-, G_n]$$ to the cohomology functor $$H^*(-, \Z ).$$ Characteristic classes form a ring because of the ring structure of cohomology ring. Yoneda's lemma says this ring of characteristic classes is exactly the cohomology ring of Gn: $$\operatorname{Nat}([-, G_n], H^*(-, \Z )) = H^*(G_n, \Z ) = \Z [\sigma_1, \ldots, \sigma_n].$$

Computation formulae
Let E be a vector bundle of rank r and $$c_t(E) = \sum_{i = 0}^r c_i(E)t^i$$ the Chern polynomial of it. c_t(\operatorname{Sym}^p E) &= \prod_{i_1 \le \cdots \le i_p} (1 + (\alpha_{i_1} + \cdots + \alpha_{i_p})t), \\ c_t(\wedge^p E) &= \prod_{i_1 < \cdots < i_p} (1 + (\alpha_{i_1} + \cdots + \alpha_{i_p})t). \end{align}$$ In particular, $$c_1(\wedge^r E) = c_1(E).$$
 * For the dual bundle $$E^*$$ of $$E$$, $$c_i(E^*) = (-1)^i c_i(E)$$.
 * If L is a line bundle, then $$c_t(E \otimes L) = \sum_{i=0}^r c_i(E) c_t(L)^{r-i} t^i$$ and so $$c_i(E \otimes L), i = 1, 2, \dots, r$$ are $$c_1(E) + r c_1(L), \dots, \sum_{j=0}^i \binom{r-i+j}{j} c_{i-j}(E) c_1(L)^j, \dots, \sum_{j=0}^r c_{r-j}(E) c_1(L)^j.$$
 * For the Chern roots $$\alpha_1, \dots, \alpha_r$$ of $$E$$, $$\begin{align}
 * For example, for $$c_i = c_i(E)$$,
 * when $$r = 2$$, $$c(\operatorname{Sym}^2 E) = 1 + 3c_1 + 2 c_1^2 + 4 c_2 + 4 c_1 c_2,$$
 * when $$r = 3$$, $$c(\operatorname{Sym}^2 E) = 1 + 4c_1 + 5 c_1^2 + 5 c_2 + 2 c_1^3 + 11 c_1 c_2 + 7 c_3.$$
 * (cf. Segre class.)

Applications of formulae
We can use these abstract properties to compute the rest of the chern classes of line bundles on $$\mathbb{CP}^1$$. Recall that $$\mathcal{O}(-1)^* \cong \mathcal{O}(1)$$ showing $$c_1(\mathcal{O}(1)) = 1 \in H^2(\mathbb{CP}^1;\mathbb{Z})$$. Then using tensor powers, we can relate them to the chern classes of $$c_1(\mathcal{O}(n)) = n$$ for any integer.

Properties
Given a complex vector bundle E over a topological space X, the Chern classes of E are a sequence of elements of the cohomology of X. The k-th Chern class of E, which is usually denoted ck(E), is an element of $$H^{2k}(X;\Z),$$ the cohomology of X with integer coefficients. One can also define the total Chern class $$c(E) = c_0(E) + c_1(E) + c_2(E) + \cdots .$$

Since the values are in integral cohomology groups, rather than cohomology with real coefficients, these Chern classes are slightly more refined than those in the Riemannian example.

Classical axiomatic definition
The Chern classes satisfy the following four axioms:
 * 1) $$c_0(E) = 1$$ for all E.
 * 2) Naturality: If $$f : Y \to X$$ is continuous and f*E is the vector bundle pullback of E, then $$c_k(f^* E) = f^* c_k(E)$$.
 * 3) Whitney sum formula: If $$F \to X$$ is another complex vector bundle, then the Chern classes of the direct sum $$E \oplus F$$ are given by $$c(E \oplus F) = c(E) \smile c(F);$$ that is, $$c_k(E \oplus F) = \sum_{i = 0}^k c_i(E) \smile c_{k - i}(F).$$
 * 4) Normalization: The total Chern class of the tautological line bundle over $$\mathbb{CP}^k$$ is 1−H, where H is Poincaré dual to the hyperplane $$\mathbb{CP}^{k - 1} \subseteq \mathbb{CP}^k$$.

Grothendieck axiomatic approach
Alternatively, replaced these with a slightly smaller set of axioms:

 * Naturality: (Same as above)
 * Additivity: If $$ 0\to E'\to E\to E\to 0$$ is an exact sequence of vector bundles, then $$c(E)=c(E')\smile c(E)$$.
 * Normalization: If E is a line bundle, then $$c(E)=1+e(E_{\R})$$ where $$e(E_{\R})$$ is the Euler class of the underlying real vector bundle.

He shows using the Leray–Hirsch theorem that the total Chern class of an arbitrary finite rank complex vector bundle can be defined in terms of the first Chern class of a tautologically-defined line bundle.

Namely, introducing the projectivization $$\mathbb{P}(E)$$ of the rank n complex vector bundle E → B as the fiber bundle on B whose fiber at any point $$b\in B$$ is the projective space of the fiber Eb. The total space of this bundle $$\mathbb{P}(E)$$ is equipped with its tautological complex line bundle, that we denote $$\tau$$, and the first Chern class $$c_1(\tau)=: -a$$ restricts on each fiber $$\mathbb{P}(E_b)$$ to minus the (Poincaré-dual) class of the hyperplane, that spans the cohomology of the fiber, in view of the cohomology of complex projective spaces.

The classes $$1, a, a^2, \ldots, a^{n-1}\in H^*(\mathbb{P}(E))$$ therefore form a family of ambient cohomology classes restricting to a basis of the cohomology of the fiber. The Leray–Hirsch theorem then states that any class in $$H^*(\mathbb{P}(E))$$ can be written uniquely as a linear combination of the 1, a, a2, ..., an−1 with classes on the base as coefficients.

In particular, one may define the Chern classes of E in the sense of Grothendieck, denoted $$c_1(E), \ldots c_n(E)$$ by expanding this way the class $$-a^n$$, with the relation: $$ - a^n = c_1(E)\cdot a^{n-1}+ \cdots + c_{n-1}(E) \cdot a + c_n(E) .$$

One then may check that this alternative definition coincides with whatever other definition one may favor, or use the previous axiomatic characterization.

The top Chern class
In fact, these properties uniquely characterize the Chern classes. They imply, among other things:

 * If n is the complex rank of V, then $$c_k(V) = 0$$ for all k > n. Thus the total Chern class terminates.
 * The top Chern class of V (meaning $$c_n(V)$$, where n is the rank of V) is always equal to the Euler class of the underlying real vector bundle.

Axiomatic description
There is another construction of Chern classes which take values in the algebrogeometric analogue of the cohomology ring, the Chow ring. It can be shown that there is a unique theory of Chern classes such that if you are given an algebraic vector bundle $$E \to X$$ over a quasi-projective variety there are a sequence of classes $$c_i(E) \in A^i(X)$$ such that = [D]$$
 * 1) $$c_0(E) = 1$$
 * 2) For an invertible sheaf $$\mathcal{O}_X(D)$$ (so that $$D$$ is a Cartier divisor), $$c_1(\mathcal{O}_X(D))
 * 1) Given an exact sequence of vector bundles $$ 0 \to E' \to E \to E \to 0 $$ the Whitney sum formula holds: $$c(E) = c(E')c(E)$$
 * 2) $$c_i(E) = 0$$ for $$i > \text{rank}(E)$$
 * 3) The map $$E \mapsto c(E)$$ extends to a ring morphism $$c:K_0(X) \to A^\bullet(X)$$

Normal sequence
Computing the characteristic classes for projective space forms the basis for many characteristic class computations since for any smooth projective subvariety $$X \subset \mathbb{P}^n$$ there is the short exact sequence $$0 \to \mathcal{T}_X \to \mathcal{T}_{\mathbb{P}^n}|_X \to \mathcal{N}_{X/\mathbb{P}^n} \to 0$$

Quintic threefold
For example, consider the nonsingular quintic threefold in $$\mathbb{P}^4$$. Then the normal bundle is given by $$\mathcal{O}_X(5)$$ and we have the short exact sequence $$0 \to \mathcal{T}_X \to \mathcal{T}_{\mathbb{P}^4}|_X \to \mathcal{O}_X(5) \to 0$$

Let $$h$$ denote the hyperplane class in $$A^\bullet(X)$$. Then the Whitney sum formula gives us that $$c(\mathcal{T}_X)c(\mathcal{O}_X(5)) = (1+h)^5 = 1 + 5h + 10h^2 + 10h^3 $$

Since the Chow ring of a hypersurface is difficult to compute, we will consider this sequence as a sequence of coherent sheaves in $$\mathbb{P}^4$$. This gives us that $$\begin{align} c(\mathcal{T}_X) &= \frac{1 + 5h + 10h^2 + 10h^3}{1 + 5h} \\ &= \left(1 + 5h + 10h^2 + 10h^3\right)\left(1 - 5h + 25h^2 - 125h^3\right) \\ &= 1 + 10h^2 - 40h^3 \end{align}$$

Using the Gauss-Bonnet theorem we can integrate the class $$c_3(\mathcal{T}_X)$$ to compute the Euler characteristic. Traditionally this is called the Euler class. This is $$\int_{[X]} c_3(\mathcal{T}_X) = \int_{[X]} -40h^3 = -200$$ since the class of $$h^3$$ can be represented by five points (by Bézout's theorem). The Euler characteristic can then be used to compute the Betti numbers for the cohomology of $$X$$ by using the definition of the Euler characteristic and using the Lefschetz hyperplane theorem.

Degree d hypersurfaces
If $$X \subset \mathbb{P}^3$$ is a degree $$d$$ smooth hypersurface, we have the short exact sequence $$0 \to \mathcal{T}_X \to \mathcal{T}_{\mathbb{P}^3}|_X \to \mathcal{O}_X(d) \to 0$$ giving the relation $$c(\mathcal{T}_X) = \frac{c(\mathcal{T}_{\mathbb{P}^3|_X})}{c(\mathcal{O}_X(d))}$$ we can then calculate this as $$\begin{align} c(\mathcal{T}_X) &= \frac{(1+[H])^4}{(1 + d[H])} \\ &= (1 + 4[H] + 6[H]^2)(1-d[H]+d^2[H]^2) \\ &= 1 + (4-d)[H] + (6-4d+d^2)[H]^2 \end{align}$$ Giving the total chern class. In particular, we can find $$X$$ is a spin 4-manifold if $$4-d $$ is even, so every smooth hypersurface of degree $$2k$$ is a spin manifold.

The Chern character
Chern classes can be used to construct a homomorphism of rings from the topological K-theory of a space to (the completion of) its rational cohomology. For a line bundle L, the Chern character ch is defined by

$$\operatorname{ch}(L) = \exp(c_1(L)) := \sum_{m=0}^\infty \frac{c_1(L)^m}{m!}.$$

More generally, if $$V = L_1 \oplus \cdots \oplus L_n$$ is a direct sum of line bundles, with first Chern classes $$x_i = c_1(L_i),$$ the Chern character is defined additively $$ \operatorname{ch}(V) = e^{x_1} + \cdots + e^{x_n} :=\sum_{m=0}^\infty \frac{1}{m!}(x_1^m + \cdots + x_n^m). $$

This can be rewritten as:

$$ \operatorname{ch}(V) = \operatorname{rk}(V) + c_1(V) + \frac{1}{2}(c_1(V)^2 - 2c_2(V)) + \frac{1}{6} (c_1(V)^3 - 3c_1(V)c_2(V) + 3c_3(V)) + \cdots.$$

This last expression, justified by invoking the splitting principle, is taken as the definition ch(V) for arbitrary vector bundles V.

If a connection is used to define the Chern classes when the base is a manifold (i.e., the Chern–Weil theory), then the explicit form of the Chern character is $$\operatorname{ch}(V)=\left[\operatorname{tr}\left(\exp\left(\frac{i\Omega}{2\pi}\right)\right)\right]$$ where $Ω$ is the curvature of the connection.

The Chern character is useful in part because it facilitates the computation of the Chern class of a tensor product. Specifically, it obeys the following identities:

$$\operatorname{ch}(V \oplus W) = \operatorname{ch}(V) + \operatorname{ch}(W)$$ $$\operatorname{ch}(V \otimes W) = \operatorname{ch}(V) \operatorname{ch}(W).$$

As stated above, using the Grothendieck additivity axiom for Chern classes, the first of these identities can be generalized to state that ch is a homomorphism of abelian groups from the K-theory K(X) into the rational cohomology of X. The second identity establishes the fact that this homomorphism also respects products in K(X), and so ch is a homomorphism of rings.

The Chern character is used in the Hirzebruch–Riemann–Roch theorem.

Chern numbers
If we work on an oriented manifold of dimension $$2n$$, then any product of Chern classes of total degree $$2n$$ (i.e., the sum of indices of the Chern classes in the product should be $$n$$) can be paired with the orientation homology class (or "integrated over the manifold") to give an integer, a Chern number of the vector bundle. For example, if the manifold has dimension 6, there are three linearly independent Chern numbers, given by $$c_1^3$$, $$c_1 c_2$$, and $$c_3$$. In general, if the manifold has dimension $$2n$$, the number of possible independent Chern numbers is the number of partitions of $$n$$.

The Chern numbers of the tangent bundle of a complex (or almost complex) manifold are called the Chern numbers of the manifold, and are important invariants.

Generalized cohomology theories
There is a generalization of the theory of Chern classes, where ordinary cohomology is replaced with a generalized cohomology theory. The theories for which such generalization is possible are called complex orientable. The formal properties of the Chern classes remain the same, with one crucial difference: the rule which computes the first Chern class of a tensor product of line bundles in terms of first Chern classes of the factors is not (ordinary) addition, but rather a formal group law.

Algebraic geometry
In algebraic geometry there is a similar theory of Chern classes of vector bundles. There are several variations depending on what groups the Chern classes lie in:

 * For complex varieties the Chern classes can take values in ordinary cohomology, as above.
 * For varieties over general fields, the Chern classes can take values in cohomology theories such as etale cohomology or l-adic cohomology.
 * For varieties V over general fields the Chern classes can also take values in homomorphisms of Chow groups CH(V): for example, the first Chern class of a line bundle over a variety V is a homomorphism from CH(V) to CH(V) reducing degrees by 1. This corresponds to the fact that the Chow groups are a sort of analog of homology groups, and elements of cohomology groups can be thought of as homomorphisms of homology groups using the cap product.

Manifolds with structure
The theory of Chern classes gives rise to cobordism invariants for almost complex manifolds.

If M is an almost complex manifold, then its tangent bundle is a complex vector bundle. The Chern classes of M are thus defined to be the Chern classes of its tangent bundle. If M is also compact and of dimension 2d, then each monomial of total degree 2d in the Chern classes can be paired with the fundamental class of M, giving an integer, a Chern number of M. If M′ is another almost complex manifold of the same dimension, then it is cobordant to M if and only if the Chern numbers of M′ coincide with those of M.

The theory also extends to real symplectic vector bundles, by the intermediation of compatible almost complex structures. In particular, symplectic manifolds have a well-defined Chern class.

Arithmetic schemes and Diophantine equations
(See Arakelov geometry)

title: Chern class


text:
User:Geoffrey~enwiki/unmarked lists

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title: User:Geoffrey~enwiki/unmarked lists


text:
1992 in art

Events from the year 1992 in art.

Events

 * 16 March - British fashion designer Alexander McQueen shows his first collection, partly inspired by The Silence of the Lambs (film).
 * 12 October – Thyssen-Bornemisza Museum in Madrid is opened to the public as a gallery for the private art collection of the Thyssen-Bornemisza family.
 * 31 October – Kunsthal in Rotterdam, designed by Rem Koolhaas, is opened as a gallery for modern art.

Awards

 * Archibald Prize: – Bryan Westwood – The Prime Minister (Paul Keating)
 * Turner Prize: – Grenville Davey

Works

 * Magdalena Abakanowicz – bronzes
 * Becalmed Beings
 * Puellae
 * Banksy – First graffiti art (in Bristol)
 * Bust of Bernardo O'Higgins (Houston) (sculpture, Texas)
 * Muriel Castanis – Ideals (sculpture, Portland, Oregon)
 * Grenville Davey – Hal
 * Anya Gallaccio – Red on Green
 * Gibson/Ashbaugh – Agrippa (a book of the dead)
 * Damien Hirst – Pharmacy (installation)
 * Soraida Martinez – Verdadism
 * Simon Patterson – The Great Bear (lithograph)
 * George Rickey – Cluster of Four Cubes (sculpture, Washington, D.C.)
 * James Rosenquist – Time Dust
 * Brad Rude – A Donkey, 3 Rocks, and a Bird. (sculpture, Portland, Oregon)
 * George Segal – Street Crossing (sculpture)
 * Jack Vettriano – The Singing Butler
 * Christopher Wool - "If You"

Exhibitions

 * "Circa 1492: Art in the Age of Exploration" at the National Gallery of Art in Washington D.C.
 * Edward Delaney retrospective – Royal Hibernian Academy.
 * Richard Hamilton retrospective – Tate Gallery.
 * Sol LeWitt Drawings 1958–1992 – Gemeentemuseum Den Haag.
 * Young British Artists – Saatchi Gallery, London (featuring Damien Hirst's The Physical Impossibility of Death in the Mind of Someone Living).

Gifts (Bequests)

 * Herbert and Dorothy Vogel collection given to National Gallery of Art in Washington, D.C.

Births

 * 29 January – George Pocheptsov, American painter

January to June

 * 6 January – Richard Mortensen, Danish painter and educator (b. 1910)
 * 27 January – Isabel Rawsthorne, English painter and model (b. 1912)
 * 19 February – Lena Gurr, American painter and lithographer (b. 1897).
 * 4 March – Art Babbitt, American animator (b. 1907).
 * 6 March – Maria Helena Vieira da Silva, Portuguese-French abstract painter (b. 1908).
 * 11 April – Alejandro Obregón, Colombian painter, muralist, sculptor and engraver (b. 1920).
 * 28 April – Francis Bacon, Irish-born British figurative painter (b. 1909).
 * 13 May – F. E. McWilliam, Irish sculptor (b. 1909).
 * 6 June – Richard Eurich, English marine painter (b. 1903).
 * 15 June – Brett Whiteley, Australian avant-garde artist (b. 1939).
 * 18 June – Mordecai Ardon, Israeli painter (b. 1896).
 * 28 June – John Piper, English landscape painter and designer (b. 1903).
 * 30 June – André Hébuterne, French painter (b. 1894).
 * 6 July – Richard Eurich, English sea- and landscape painter (b. 1903).

July to December

 * 7 September – EQ Nicholson, English textile designer and painter (b.1908).
 * 25 September – César Manrique, Spanish artist and architect (b. 1919)
 * 30 October – Joan Mitchell, American Abstract Expressionist painter (b. 1925).
 * 11 November – Giulio Carlo Argan, Italian art historian and politician (b. 1909).
 * 27 November – Ivan Generalić, Croatian naïve art painter (b. 1914)
 * 30 November - Bernard Lefebvre, French photographer (b. 1906)
 * 23 December – Vincent Fourcade, French interior designer (b. 1934).
 * 24 December – Peyo, Belgian comics artist (b. 1928).

title: 1992 in art


text:
1949 Governor General's Awards

The 1949 Governor General's Awards for Literary Merit were the 14th rendition of the Governor General's Awards, Canada's annual national awards program which then comprised literary awards alone. The awards recognized Canadian writers for new English-language works published in Canada during 1949 and were presented early in 1950. There were no cash prizes.

The Governor General's Award for juvenile literature was introduced, and would be conferred annually through the 1958 cycle, after which there were several changes for the 1959 Governor General's Awards under the new administrator Canada Council. As every year from 1949 to 1958, there were two awards for non-fiction, and five awards in four categories that recognized English-language works only.

Winners

 * Fiction: Philip Child, Mr. Ames Against Time
 * Poetry or drama: James Reaney, The Red Heart
 * Non-fiction: Hugh MacLennan, Cross-country
 * Non-fiction: R. MacGregor Dawson, Democratic Government in Canada
 * Juvenile: Richard S. Lambert, Franklin of the Arctic: a life of adventure

title: 1949 Governor General's Awards


text:
State Anthem of the Soviet Union

The "State Anthem of the Union of Soviet Socialist Republics" was the national anthem of the Soviet Union and the regional anthem of the Russian Soviet Federative Socialist Republic from 1944 to 1991, replacing "The Internationale". Its original lyrics were written by Sergey Mikhalkov (1913–2009) in collaboration with Gabriyel’ Arkadyevich Ureklyan (1899–1945), and its music was composed by Alexander Alexandrov (1883–1946). For a two-decade interval following de-Stalinization, the anthem was performed without lyrics. The second set of lyrics, also written by Mikhalkov and in which Stalin's name was omitted, was adopted in 1977.

A decade after the dissolution of the Soviet Union, the same melody was used for the Soviet Union's successor state, as the State Anthem of the Russian Federation.

Origins
The anthem's music was originally composed by Alexander Alexandrov in 1938 for the Hymn of the Bolshevik Party. Its opening bars were borrowed from one of Alexandrov's previous pieces, "Life Has Become Better", which was based on a quote by former Soviet Union General Secretary Joseph Stalin at the First All-Union Meeting of the Stakhanovites on 17 November 1935. The song alludes to Stalin's cult of personality.

Other than "Life Has Become Better", the music of the anthem has several possible outside influences. Alexandrov himself has described it as the combination of a march with Russian traditional music, particularly that of bylina epic songs. The anthem shares several chord progressions with Vasily Kalinnikov's overture Bylina, Epic Poem (which, as its name indicates, is also inspired by the bylina tradition). There also exist similarities between Alexandrov's anthem and Robert Schumann's Frühlingsfahrt.

When the Communist International (Comintern) was dissolved in 1943 for the Soviet Union to maintain its alliance with the other Allies of World War II, a new composition was needed to replace "The Internationale" as the national anthem. A contest was held in mid to late 1943 for a new anthem, and more than 200 entries were submitted. Alexandrov's music was personally chosen by Stalin, who both praised and criticized it. The anthem's lyrics then had to be written. Stalin thought the song should be short, and that it should invoke the Red Army's impending victory over the forces of Germany on the Eastern Front. The poets Sergey Mikhalkov and Gabriyel’ Arkadyevich Ureklyan were chosen by Stalin's staffers, called to Moscow, and given the task of writing lyrics that referenced not only the Great Patriotic War, but also "a Country of Soviets". The first draft was completed overnight.

The anthem was first published on 7 November 1943. It was played for the first time on Soviet radio at midnight on 1 January 1944, and officially adopted on 15 March the same year. The new lyrics had three refrains following three different stanzas. In each refrain, the second line was modified to refer to friendship, then happiness, then glory. Joseph Stalin and the Soviet Union's war against Germany were originally invoked in the second and third verses, respectively. Reportedly, Stalin was opposed to including his name in the lyrics but relented after some Politburo members insisted.

Post-Stalin era
With the process of de-Stalinization after Stalin's death, the lyrics referring to him were considered unacceptable, and from 1956 to 1977 the anthem was performed without lyrics. A notable exception took place at the 1976 Canada Cup ice hockey tournament, where singer Roger Doucet insisted on performing the anthem with lyrics after consultations with Russian studies scholars from Université de Montréal and Soviet team officials. In 1977, to coincide with the 60th anniversary of the October Revolution, revised lyrics, written in 1970 by original author Sergey Mikhalkov, were adopted. The varying refrains were replaced by a uniform refrain after all stanzas, and the line praising Stalin was dropped, as were the lines referring to the Great Patriotic War. Another notable change was the replacement of a line referring to the Soviet national flag with one citing the Communist Party of the Soviet Union in the form of "Partiya Lenina" (The party of Lenin). These lyrics were also present in the original party anthem at the same place in the melody, but followed by the lyrics "Partiya Stalina" (The party of Stalin).

Use in the Russian Federation
After the dissolution of the Soviet Union in 1991, the Russian Federation adopted a new anthem, the Patriotic Song. It was previously the regional anthem of the Russian Soviet Federative Socialist Republic from 1990 until 1991 (until 1990 it used the State Anthem of the Soviet Union). Unlike most national anthems, it had no official lyrics (although unofficial ones written for it were proposed, they were not adopted).

The Patriotic Song was replaced soon after Vladimir Putin first took office on 7 May 2000. The federal legislature established and approved the music of the National Anthem of the Soviet Union, with newly written lyrics, in December 2000. Boris Yeltsin criticized Putin for supporting the semi-reintroduction of the Soviet-era national anthem, although some opinion polls showed that many Russians favored this decision. In late 2000, the current national anthem of Russia was introduced, which uses the music of the Soviet national anthem with new lyrics by Sergey Mikhalkov, who also wrote both the original Soviet anthem and its remake in 1977.

State Union of Russia and Belarus
The same music was used for a proposed anthem for the State Union of Russia and Belarus, entitled Derzhavny Soyuz Narodov ("Sovereign Union of Nations"). Its lyrics were not tied to any specific nationality, and there were official versions in the languages of every Soviet republic and several other Soviet languages; thus it could have been adopted by a broader union. However, it was never officially adopted, and there appear to be no plans to utilize it in any official role.

Decommunization
In some post-Soviet states which adopted decommunization laws banning Communist symbolism, publicly performing the Soviet anthem is illegal. For example, since 2015, offenders in Ukraine face up to five years in prison with the exception of Russian-occupied areas in Ukraine. Similar laws were adopted in Latvia and Lithuania.

Communist symbols are also banned in Indonesia.

Similarly, some communist symbols (particularly Soviet symbols) are banned in Georgia, Germany and Poland.

Official translations
The anthem has been officially translated into several languages:

Lyrics
This table shows the anthem's various lyrics. As there were two official lyric versions (in 1944 and 1977), both are included within their sections.

title: State Anthem of the Soviet Union


text:
Anticommutative property

In mathematics, anticommutativity is a specific property of some non-commutative mathematical operations. Swapping the position of two arguments of an antisymmetric operation yields a result which is the inverse of the result with unswapped arguments. The notion inverse refers to a group structure on the operation's codomain, possibly with another operation. Subtraction is an anticommutative operation because commuting the operands of a − b gives b − a = −(a − b); for example, 2 − 10 = −(10 − 2) = −8. Another prominent example of an anticommutative operation is the Lie bracket.

In mathematical physics, where symmetry is of central importance, these operations are mostly called antisymmetric operations, and are extended in an associative setting to cover more than two arguments.

Definition
If $$A, B$$ are two abelian groups, a bilinear map $$f\colon A^2 \to B$$ is anticommutative if for all $$x, y \in A$$ we have

 * $$f(x, y) = - f(y, x).$$

More generally, a multilinear map $$g : A^n \to B$$ is anticommutative if for all $$x_1, \dots x_n \in A$$ we have
 * $$g(x_1,x_2, \dots x_n) = \text{sgn}(\sigma) g(x_{\sigma(1)},x_{\sigma(2)},\dots x_{\sigma(n)})$$

where $$\text{sgn}(\sigma)$$ is the sign of the permutation $$\sigma$$.

Properties
If the abelian group $$B$$ has no 2-torsion, implying that if $$x = -x$$ then $$x = 0$$, then any anticommutative bilinear map $$f\colon A^2 \to B$$ satisfies

 * $$f(x, x) = 0.$$

More generally, by transposing two elements, any anticommutative multilinear map $$g\colon A^n \to B$$ satisfies

 * $$g(x_1, x_2, \dots x_n) = 0$$

if any of the $$x_i$$ are equal; such a map is said to be alternating. Conversely, using multilinearity, any alternating map is anticommutative. In the binary case this works as follows: if $$f\colon A^2 \to B$$ is alternating then by bilinearity we have

 * $$f(x+y, x+y) = f(x, x) + f(x, y) + f(y, x) + f(y, y) = f(x, y) + f(y, x) = 0$$

and the proof in the multilinear case is the same but in only two of the inputs.

Examples
Examples of anticommutative binary operations include:

 * Cross product
 * Lie bracket of a Lie algebra
 * Lie bracket of a Lie ring
 * Subtraction

title: Anticommutative property


text:
1956 in art

Events from the year 1956 in art.

Events

 * March 1 – Replica statue of the Discus Thrower dedicated in Washington, D.C., as a gift from the Italian government to commemorate the return of looted art objects after World War II.
 * March – 56 Group founded, to promote modernist art in Wales. Subsequently renamed 56 Group Wales.
 * September 17 – Release in the United States of the biographical film Lust for Life with Kirk Douglas portraying Vincent van Gogh and Anthony Quinn as Paul Gauguin.
 * Le mystère Picasso, a French documentary film, shows Pablo Picasso in the act of creating paintings for the camera (which he subsequently destroys so that they will exist only on film).
 * William Klein publishes his photo essay New York, 1954–55.
 * Shanghai Art Museum, the predecessor of the China Art Museum, opens.
 * English curator Jim Ede settles at Kettle's Yard, Cambridge, England.
 * English painter Edward Seago joins a tour of the Antarctic.
 * Two attacks are made on Leonardo da Vinci's Mona Lisa in the Louvre, Paris.

Awards

 * Archibald Prize: William Dargie – Mr Albert Namatjira
 * Kate Greenaway Medal – Edward Ardizzone for Tim All Alone

Films

 * Lust for Life

Works

 * Laurence Bradshaw – Monument to Karl Marx at Highgate Cemetery, London (including bronze bust)
 * Alexander Calder – Red Mobile
 * Frank Cadogan Cowper – The Golden Bowl
 * Salvador Dalí – Living Still Life
 * M. C. Escher – Print Gallery (lithograph)
 * Max Ernst – L’oiseau Rose
 * Helen Frankenthaler – Eden
 * Richard Hamilton – Just What Is It that Makes Today's Homes So Different, So Appealing? (collage)
 * Rudolf Hausner – The Ark of Odysseus
 * Eduardo Kingman – La Lavendera
 * Roy Lichtenstein – Ten Dollar Bill (lithograph)
 * O. Winston Link - Hotshot Eastbound (black-and-white photograph)
 * L. S. Lowry – The Floating Bridge
 * Joan Mitchell
 * Café
 * Hemlock
 * Candido Portinari – Self-portrait
 * Norman Rockwell – The Scoutmaster
 * Kay Sage – Le Passage
 * Alexander Nikolayevich Samokhvalov – Cafe Gurzuf
 * Charles Sheeler – On a Shaker Theme
 * David Wynne – Sir Thomas Beecham (bronzes)

Exhibitions

 * August 9–September 9 – This Is Tomorrow, Whitechapel Art Gallery, London, featuring principally the interdisciplinary ICA Independent Group, including early examples of Pop Art.

Births

 * January 2 – Lynda Barry, American cartoonist
 * January 19 – Junpei Satoh, Japanese Western-style painter
 * February 24 – Fiona Graham-Mackay, née Bain, British portrait painter
 * May 25 – Andrea Pazienza, Italian comics artist (d. 1988)
 * date unknown
 * Lee Bae, South Korean visual artist
 * Emma Biggs, English mosaicist
 * Cornelia Parker, English sculptor and installation artist
 * Veronica Ryan, Montserrat-born British sculptor
 * Mackenzie Thorpe, English painter and sculptor

Deaths

 * January 13 – Lyonel Feininger, German American painter and cartoonist (b. 1871)
 * April 23 – Cecile Walton, Scottish painter, illustrator and sculptor (b. 1891)
 * May 3 – Peter Watson, English arts benefactor (murdered) (b. 1908)
 * June 8 – Marie Laurencin, French painter and engraver (b. 1883)
 * June 11 - Frank Brangwyn, Welsh painter (b. 1867)
 * July 26 – Louis Raemaekers, Dutch painter and cartoonist (b. 1869)
 * August 7 – LeMoine FitzGerald, Canadian painter (b. 1890)
 * August 11 – Jackson Pollock, American painter (b. 1912)
 * August 16 – Theodor Pallady, Romanian painter (b. 1871)
 * November 3 – Jean Metzinger, French painter (b. 1883)
 * December 16 – Nina Hamnett, British painter, model and designer (b. 1890)
 * Mohamed Nagy, Egyptian painter (b. 1888)

title: 1956 in art


text:
User:Mkfairdpm

Michael K. Fairbanks, DPM La Mesa, CA

title: User:Mkfairdpm