text
stringlengths 100
500k
| subset
stringclasses 4
values |
|---|---|
Geometric mean theorem
In Euclidean geometry, the right triangle altitude theorem or geometric mean theorem is a relation between the altitude on the hypotenuse in a right triangle and the two line segments it creates on the hypotenuse. It states that the geometric mean of the two segments equals the altitude.
Theorem and applications
If h denotes the altitude in a right triangle and p and q the segments on the hypotenuse then the theorem can be stated as:[1]
$h={\sqrt {pq}}$
or in term of areas:
$h^{2}=pq.$
The latter version yields a method to square a rectangle with ruler and compass, that is to construct a square of equal area to a given rectangle. For such a rectangle with sides p and q we denote its top left vertex with D. Now we extend the segment q to its left by p (using arc AE centered on D) and draw a half circle with endpoints A and B with the new segment p + q as its diameter. Then we erect a perpendicular line to the diameter in D that intersects the half circle in C. Due to Thales' theorem C and the diameter form a right triangle with the line segment DC as its altitude, hence DC is the side of a square with the area of the rectangle. The method also allows for the construction of square roots (see constructible number), since starting with a rectangle that has a width of 1 the constructed square will have a side length that equals the square root of the rectangle's length.[1]
Another application of provides a geometrical proof of the AM–GM inequality in the case of two numbers. For the numbers p and q one constructs a half circle with diameter p + q. Now the altitude represents the geometric mean and the radius the arithmetic mean of the two numbers. Since the altitude is always smaller or equal to the radius, this yields the inequality.[2]
The theorem can also be thought of as a special case of the intersecting chords theorem for a circle, since the converse of Thales' theorem ensures that the hypotenuse of the right angled triangle is the diameter of its circumcircle.[1]
The converse statement is true as well. Any triangle, in which the altitude equals the geometric mean of the two line segments created by it, is a right triangle.
History
The theorem is usually attributed to Euclid (ca. 360–280 BC), who stated it as a corollary to proposition 8 in book VI of his Elements. In proposition 14 of book II Euclid gives a method for squaring a rectangle, which essentially matches the method given here. Euclid however provides a different slightly more complicated proof for the correctness of the construction rather than relying on the geometric mean theorem.[1][3]
Proof
Based on similarity
Proof of theorem:
The triangles △ADC , △ BCD are similar, since:
• consider triangles △ABC, △ACD ; here we have
$\angle ACB=\angle ADC=90^{\circ },\quad \angle BAC=\angle CAD;$
therefore by the AA postulate
$\triangle ABC\sim \triangle ACD.$
• further, consider triangles △ABC, △BCD ; here we have
$\angle ACB=\angle BDC=90^{\circ },\quad \angle ABC=\angle CBD;$
therefore by the AA postulate
$\triangle ABC\sim \triangle BCD.$
Therefore, both triangles △ACD, △BCD are similar to △ABC and themselves, i.e.
$\triangle ACD\sim \triangle ABC\sim \triangle BCD.$
Because of the similarity we get the following equality of ratios and its algebraic rearrangement yields the theorem:[1]
${\frac {h}{p}}={\frac {q}{h}}\,\Leftrightarrow \,h^{2}=pq\,\Leftrightarrow \,h={\sqrt {pq}}\qquad (h,p,q>0)$
Proof of converse:
For the converse we have a triangle △ABC in which $h^{2}=pq$ holds and need to show that the angle at C is a right angle. Now because of $h^{2}=pq$ we also have ${\tfrac {h}{p}}={\tfrac {q}{h}}.$ Together with $\angle ADC=\angle CDB$ the triangles △ADC, △BDC have an angle of equal size and have corresponding pairs of legs with the same ratio. This means the triangles are similar, which yields:
${\begin{aligned}\angle ACB&=\angle ACD+\angle DCB\\&=\angle ACD+(90^{\circ }-\angle DBC)\\&=\angle ACD+(90^{\circ }-\angle ACD)\\&=90^{\circ }\end{aligned}}$
Based on the Pythagorean theorem
In the setting of the geometric mean theorem there are three right triangles △ABC, △ADC and △DBC in which the Pythagorean theorem yields:
${\begin{aligned}h^{2}&=a^{2}-q^{2}\\h^{2}&=b^{2}-p^{2}\\c^{2}&=a^{2}+b^{2}\end{aligned}}$
Adding the first 2 two equations and then using the third then leads to:
${\begin{aligned}2h^{2}&=a^{2}+b^{2}-p^{2}-q^{2}\\&=c^{2}-p^{2}-q^{2}\\&=(p+q)^{2}-p^{2}-q^{2}\\&=2pq\\\therefore \ h^{2}&=pq.\end{aligned}}$
which finally yields the formula of the geometric mean theorem.[4]
Based on dissection and rearrangement
Dissecting the right triangle along its altitude h yields two similar triangles, which can be augmented and arranged in two alternative ways into a larger right triangle with perpendicular sides of lengths p + h and q + h. One such arrangement requires a square of area h2 to complete it, the other a rectangle of area pq. Since both arrangements yield the same triangle, the areas of the square and the rectangle must be identical.
Based on shear mappings
The square of the altitude can be transformed into an rectangle of equal area with sides p and q with the help of three shear mappings (shear mappings preserve the area):
References
• Hartmut Wellstein, Peter Kirsche: Elementargeometrie. Springer, 2009, ISBN 9783834808561, pp. 76-77 (German, online copy, p. 76, at Google Books)
1. Claudi Alsina, Roger B. Nelsen: Icons of Mathematics: An Exploration of Twenty Key Images. MAA 2011, ISBN 9780883853528, pp. 31–32 (online copy, p. 31, at Google Books)
2. Euclid: Elements, book II – prop. 14, book VI – pro6767800hshockedmake ,me uoppppp. 8, (online copy)
3. Ilka Agricola, Thomas Friedrich: Elementary Geometry. AMS 2008, ISBN 9780821843475, p. 25 (online copy, p. 25, at Google Books)
External links
• Geometric Mean at Cut-the-Knot
Ancient Greek mathematics
Mathematicians
(timeline)
• Anaxagoras
• Anthemius
• Archytas
• Aristaeus the Elder
• Aristarchus
• Aristotle
• Apollonius
• Archimedes
• Autolycus
• Bion
• Bryson
• Callippus
• Carpus
• Chrysippus
• Cleomedes
• Conon
• Ctesibius
• Democritus
• Dicaearchus
• Diocles
• Diophantus
• Dinostratus
• Dionysodorus
• Domninus
• Eratosthenes
• Eudemus
• Euclid
• Eudoxus
• Eutocius
• Geminus
• Heliodorus
• Heron
• Hipparchus
• Hippasus
• Hippias
• Hippocrates
• Hypatia
• Hypsicles
• Isidore of Miletus
• Leon
• Marinus
• Menaechmus
• Menelaus
• Metrodorus
• Nicomachus
• Nicomedes
• Nicoteles
• Oenopides
• Pappus
• Perseus
• Philolaus
• Philon
• Philonides
• Plato
• Porphyry
• Posidonius
• Proclus
• Ptolemy
• Pythagoras
• Serenus
• Simplicius
• Sosigenes
• Sporus
• Thales
• Theaetetus
• Theano
• Theodorus
• Theodosius
• Theon of Alexandria
• Theon of Smyrna
• Thymaridas
• Xenocrates
• Zeno of Elea
• Zeno of Sidon
• Zenodorus
Treatises
• Almagest
• Archimedes Palimpsest
• Arithmetica
• Conics (Apollonius)
• Catoptrics
• Data (Euclid)
• Elements (Euclid)
• Measurement of a Circle
• On Conoids and Spheroids
• On the Sizes and Distances (Aristarchus)
• On Sizes and Distances (Hipparchus)
• On the Moving Sphere (Autolycus)
• Optics (Euclid)
• On Spirals
• On the Sphere and Cylinder
• Ostomachion
• Planisphaerium
• Sphaerics
• The Quadrature of the Parabola
• The Sand Reckoner
Problems
• Constructible numbers
• Angle trisection
• Doubling the cube
• Squaring the circle
• Problem of Apollonius
Concepts
and definitions
• Angle
• Central
• Inscribed
• Axiomatic system
• Axiom
• Chord
• Circles of Apollonius
• Apollonian circles
• Apollonian gasket
• Circumscribed circle
• Commensurability
• Diophantine equation
• Doctrine of proportionality
• Euclidean geometry
• Golden ratio
• Greek numerals
• Incircle and excircles of a triangle
• Method of exhaustion
• Parallel postulate
• Platonic solid
• Lune of Hippocrates
• Quadratrix of Hippias
• Regular polygon
• Straightedge and compass construction
• Triangle center
Results
In Elements
• Angle bisector theorem
• Exterior angle theorem
• Euclidean algorithm
• Euclid's theorem
• Geometric mean theorem
• Greek geometric algebra
• Hinge theorem
• Inscribed angle theorem
• Intercept theorem
• Intersecting chords theorem
• Intersecting secants theorem
• Law of cosines
• Pons asinorum
• Pythagorean theorem
• Tangent-secant theorem
• Thales's theorem
• Theorem of the gnomon
Apollonius
• Apollonius's theorem
Other
• Aristarchus's inequality
• Crossbar theorem
• Heron's formula
• Irrational numbers
• Law of sines
• Menelaus's theorem
• Pappus's area theorem
• Problem II.8 of Arithmetica
• Ptolemy's inequality
• Ptolemy's table of chords
• Ptolemy's theorem
• Spiral of Theodorus
Centers
• Cyrene
• Mouseion of Alexandria
• Platonic Academy
Related
• Ancient Greek astronomy
• Attic numerals
• Greek numerals
• Latin translations of the 12th century
• Non-Euclidean geometry
• Philosophy of mathematics
• Neusis construction
History of
• A History of Greek Mathematics
• by Thomas Heath
• algebra
• timeline
• arithmetic
• timeline
• calculus
• timeline
• geometry
• timeline
• logic
• timeline
• mathematics
• timeline
• numbers
• prehistoric counting
• numeral systems
• list
Other cultures
• Arabian/Islamic
• Babylonian
• Chinese
• Egyptian
• Incan
• Indian
• Japanese
Ancient Greece portal • Mathematics portal
Wikimedia Commons has media related to Geometric mean theorem.
|
Wikipedia
|
Zero divisor
In abstract algebra, an element a of a ring R is called a left zero divisor if there exists a nonzero x in R such that ax = 0,[1] or equivalently if the map from R to R that sends x to ax is not injective.[lower-alpha 1] Similarly, an element a of a ring is called a right zero divisor if there exists a nonzero y in R such that ya = 0. This is a partial case of divisibility in rings. An element that is a left or a right zero divisor is simply called a zero divisor.[2] An element a that is both a left and a right zero divisor is called a two-sided zero divisor (the nonzero x such that ax = 0 may be different from the nonzero y such that ya = 0). If the ring is commutative, then the left and right zero divisors are the same.
An element of a ring that is not a left zero divisor (respectively, not a right zero divisor) is called left regular or left cancellable (respectively, right regular or right cancellable). An element of a ring that is left and right cancellable, and is hence not a zero divisor, is called regular or cancellable,[3] or a non-zero-divisor. A zero divisor that is nonzero is called a nonzero zero divisor or a nontrivial zero divisor. A non-zero ring with no nontrivial zero divisors is called a domain.
Examples
• In the ring $\mathbb {Z} /4\mathbb {Z} $, the residue class ${\overline {2}}$ is a zero divisor since ${\overline {2}}\times {\overline {2}}={\overline {4}}={\overline {0}}$.
• The only zero divisor of the ring $\mathbb {Z} $ of integers is $0$.
• A nilpotent element of a nonzero ring is always a two-sided zero divisor.
• An idempotent element $e\neq 1$ of a ring is always a two-sided zero divisor, since $e(1-e)=0=(1-e)e$.
• The ring of n × n matrices over a field has nonzero zero divisors if n ≥ 2. Examples of zero divisors in the ring of 2 × 2 matrices (over any nonzero ring) are shown here:
${\begin{pmatrix}1&1\\2&2\end{pmatrix}}{\begin{pmatrix}1&1\\-1&-1\end{pmatrix}}={\begin{pmatrix}-2&1\\-2&1\end{pmatrix}}{\begin{pmatrix}1&1\\2&2\end{pmatrix}}={\begin{pmatrix}0&0\\0&0\end{pmatrix}},$
${\begin{pmatrix}1&0\\0&0\end{pmatrix}}{\begin{pmatrix}0&0\\0&1\end{pmatrix}}={\begin{pmatrix}0&0\\0&1\end{pmatrix}}{\begin{pmatrix}1&0\\0&0\end{pmatrix}}={\begin{pmatrix}0&0\\0&0\end{pmatrix}}.$
• A direct product of two or more nonzero rings always has nonzero zero divisors. For example, in $R_{1}\times R_{2}$ with each $R_{i}$ nonzero, $(1,0)(0,1)=(0,0)$, so $(1,0)$ is a zero divisor.
• Let $K$ be a field and $G$ be a group. Suppose that $G$ has an element $g$ of finite order $n>1$. Then in the group ring $K[G]$ one has $(1-g)(1+g+\cdots +g^{n-1})=1-g^{n}=0$, with neither factor being zero, so $1-g$ is a nonzero zero divisor in $K[G]$.
One-sided zero-divisor
• Consider the ring of (formal) matrices ${\begin{pmatrix}x&y\\0&z\end{pmatrix}}$ with $x,z\in \mathbb {Z} $ and $y\in \mathbb {Z} /2\mathbb {Z} $. Then ${\begin{pmatrix}x&y\\0&z\end{pmatrix}}{\begin{pmatrix}a&b\\0&c\end{pmatrix}}={\begin{pmatrix}xa&xb+yc\\0&zc\end{pmatrix}}$ and ${\begin{pmatrix}a&b\\0&c\end{pmatrix}}{\begin{pmatrix}x&y\\0&z\end{pmatrix}}={\begin{pmatrix}xa&ya+zb\\0&zc\end{pmatrix}}$. If $x\neq 0\neq z$, then ${\begin{pmatrix}x&y\\0&z\end{pmatrix}}$ is a left zero divisor if and only if $x$ is even, since ${\begin{pmatrix}x&y\\0&z\end{pmatrix}}{\begin{pmatrix}0&1\\0&0\end{pmatrix}}={\begin{pmatrix}0&x\\0&0\end{pmatrix}}$, and it is a right zero divisor if and only if $z$ is even for similar reasons. If either of $x,z$ is $0$, then it is a two-sided zero-divisor.
• Here is another example of a ring with an element that is a zero divisor on one side only. Let $S$ be the set of all sequences of integers $(a_{1},a_{2},a_{3},...)$. Take for the ring all additive maps from $S$ to $S$, with pointwise addition and composition as the ring operations. (That is, our ring is $\mathrm {End} (S)$, the endomorphism ring of the additive group $S$.) Three examples of elements of this ring are the right shift $R(a_{1},a_{2},a_{3},...)=(0,a_{1},a_{2},...)$, the left shift $L(a_{1},a_{2},a_{3},...)=(a_{2},a_{3},a_{4},...)$, and the projection map onto the first factor $P(a_{1},a_{2},a_{3},...)=(a_{1},0,0,...)$. All three of these additive maps are not zero, and the composites $LP$ and $PR$ are both zero, so $L$ is a left zero divisor and $R$ is a right zero divisor in the ring of additive maps from $S$ to $S$. However, $L$ is not a right zero divisor and $R$ is not a left zero divisor: the composite $LR$ is the identity. $RL$ is a two-sided zero-divisor since $RLP=0=PRL$, while $LR=1$ is not in any direction.
Non-examples
• The ring of integers modulo a prime number has no nonzero zero divisors. Since every nonzero element is a unit, this ring is a finite field.
• More generally, a division ring has no nonzero zero divisors.
• A non-zero commutative ring whose only zero divisor is 0 is called an integral domain.
Properties
• In the ring of n × n matrices over a field, the left and right zero divisors coincide; they are precisely the singular matrices. In the ring of n × n matrices over an integral domain, the zero divisors are precisely the matrices with determinant zero.
• Left or right zero divisors can never be units, because if a is invertible and ax = 0 for some nonzero x, then 0 = a−10 = a−1ax = x, a contradiction.
• An element is cancellable on the side on which it is regular. That is, if a is a left regular, ax = ay implies that x = y, and similarly for right regular.
Zero as a zero divisor
There is no need for a separate convention for the case a = 0, because the definition applies also in this case:
• If R is a ring other than the zero ring, then 0 is a (two-sided) zero divisor, because any nonzero element x satisfies 0x = 0 = x 0.
• If R is the zero ring, in which 0 = 1, then 0 is not a zero divisor, because there is no nonzero element that when multiplied by 0 yields 0.
Some references include or exclude 0 as a zero divisor in all rings by convention, but they then suffer from having to introduce exceptions in statements such as the following:
• In a commutative ring R, the set of non-zero-divisors is a multiplicative set in R. (This, in turn, is important for the definition of the total quotient ring.) The same is true of the set of non-left-zero-divisors and the set of non-right-zero-divisors in an arbitrary ring, commutative or not.
• In a commutative noetherian ring R, the set of zero divisors is the union of the associated prime ideals of R.
Zero divisor on a module
Let R be a commutative ring, let M be an R-module, and let a be an element of R. One says that a is M-regular if the "multiplication by a" map $M\,{\stackrel {a}{\to }}\,M$ is injective, and that a is a zero divisor on M otherwise.[4] The set of M-regular elements is a multiplicative set in R.[4]
Specializing the definitions of "M-regular" and "zero divisor on M" to the case M = R recovers the definitions of "regular" and "zero divisor" given earlier in this article.
See also
• Zero-product property
• Glossary of commutative algebra (Exact zero divisor)
• Zero-divisor graph
Notes
1. Since the map is not injective, we have ax = ay, in which x differs from y, and thus a(x − y) = 0.
References
1. N. Bourbaki (1989), Algebra I, Chapters 1–3, Springer-Verlag, p. 98
2. Charles Lanski (2005), Concepts in Abstract Algebra, American Mathematical Soc., p. 342
3. Nicolas Bourbaki (1998). Algebra I. Springer Science+Business Media. p. 15.
4. Hideyuki Matsumura (1980), Commutative algebra, 2nd edition, The Benjamin/Cummings Publishing Company, Inc., p. 12
Further reading
• "Zero divisor", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Michiel Hazewinkel; Nadiya Gubareni; Nadezhda Mikhaĭlovna Gubareni; Vladimir V. Kirichenko. (2004), Algebras, rings and modules, vol. 1, Springer, ISBN 1-4020-2690-0
• Weisstein, Eric W. "Zero Divisor". MathWorld.
|
Wikipedia
|
Rigid cohomology
In mathematics, rigid cohomology is a p-adic cohomology theory introduced by Berthelot (1986). It extends crystalline cohomology to schemes that need not be proper or smooth, and extends Monsky–Washnitzer cohomology to non-affine varieties. For a scheme X of finite type over a perfect field k, there are rigid cohomology groups Hi
rig
(X/K) which are finite dimensional vector spaces over the field K of fractions of the ring of Witt vectors of k. More generally one can define rigid cohomology with compact supports, or with support on a closed subscheme, or with coefficients in an overconvergent isocrystal. If X is smooth and proper over k the rigid cohomology groups are the same as the crystalline cohomology groups.
The name "rigid cohomology" comes from its relation to rigid analytic spaces.
Kedlaya (2006) used rigid cohomology to give a new proof of the Weil conjectures.
References
• Berthelot, Pierre (1986), "Géométrie rigide et cohomologie des variétés algébriques de caractéristique p", Mémoires de la Société Mathématique de France, Nouvelle Série (23): 7–32, ISSN 0037-9484, MR 0865810
• Kedlaya, Kiran S. (2009), "p-adic cohomology", in Abramovich, Dan; Bertram, A.; Katzarkov, L.; Pandharipande, Rahul; Thaddeus., M. (eds.), Algebraic geometry---Seattle 2005. Part 2, Proc. Sympos. Pure Math., vol. 80, Providence, R.I.: Amer. Math. Soc., pp. 667–684, arXiv:math/0601507, Bibcode:2006math......1507K, ISBN 978-0-8218-4703-9, MR 2483951
• Kedlaya, Kiran S. (2006), "Fourier transforms and p-adic 'Weil II'", Compositio Mathematica, 142 (6): 1426–1450, arXiv:math/0210149, doi:10.1112/S0010437X06002338, ISSN 0010-437X, MR 2278753, S2CID 5233570
• Le Stum, Bernard (2007), Rigid cohomology, Cambridge Tracts in Mathematics, vol. 172, Cambridge University Press, ISBN 978-0-521-87524-0, MR 2358812
• Tsuzuki, Nobuo (2009), "Rigid cohomology", Mathematical Society of Japan. Sugaku (Mathematics), 61 (1): 64–82, ISSN 0039-470X, MR 2560145
External links
• Kedlaya, Kiran S., Rigid cohomology and its coefficients
• Le Stum, Bernard (2012), An introduction to rigid cohomology (PDF), Special week – Strasbourg
|
Wikipedia
|
Structural rigidity
In discrete geometry and mechanics, structural rigidity is a combinatorial theory for predicting the flexibility of ensembles formed by rigid bodies connected by flexible linkages or hinges.
Definitions
"Rigid graph" redirects here. For the meaning "has no nontrivial automorphisms", see Asymmetric graph.
Rigidity is the property of a structure that it does not bend or flex under an applied force. The opposite of rigidity is flexibility. In structural rigidity theory, structures are formed by collections of objects that are themselves rigid bodies, often assumed to take simple geometric forms such as straight rods (line segments), with pairs of objects connected by flexible hinges. A structure is rigid if it cannot flex; that is, if there is no continuous motion of the structure that preserves the shape of its rigid components and the pattern of their connections at the hinges.
There are two essentially different kinds of rigidity. Finite or macroscopic rigidity means that the structure will not flex, fold, or bend by a positive amount. Infinitesimal rigidity means that the structure will not flex by even an amount that is too small to be detected even in theory. (Technically, that means certain differential equations have no nonzero solutions.) The importance of finite rigidity is obvious, but infinitesimal rigidity is also crucial because infinitesimal flexibility in theory corresponds to real-world minuscule flexing, and consequent deterioration of the structure.
A rigid graph is an embedding of a graph in a Euclidean space which is structurally rigid.[1] That is, a graph is rigid if the structure formed by replacing the edges by rigid rods and the vertices by flexible hinges is rigid. A graph that is not rigid is called flexible. More formally, a graph embedding is flexible if the vertices can be moved continuously, preserving the distances between adjacent vertices, with the result that the distances between some nonadjacent vertices are altered.[2] The latter condition rules out Euclidean congruences such as simple translation and rotation.
It is also possible to consider rigidity problems for graphs in which some edges represent compression elements (able to stretch to a longer length, but not to shrink to a shorter length) while other edges represent tension elements (able to shrink but not stretch). A rigid graph with edges of these types forms a mathematical model of a tensegrity structure.
Mathematics of rigidity
The fundamental problem is how to predict the rigidity of a structure by theoretical analysis, without having to build it. Key results in this area include the following:
• In any dimension, the rigidity of rod-and-hinge linkages is described by a matroid. The bases of the two-dimensional rigidity matroid (the minimally rigid graphs in the plane) are the Laman graphs.
• Cauchy's theorem states that a three-dimensional convex polyhedron constructed with rigid plates for its faces, connected by hinges along its edges, forms a rigid structure.
• Flexible polyhedra, non-convex polyhedra that are not rigid, were constructed by Raoul Bricard, Robert Connelly, and others. The bellows conjecture, now proven, states that every continuous motion of a flexible polyhedron preserves its volume.
• In the grid bracing problem, where the framework to be made rigid is a square grid with added diagonals as cross bracing, the rigidity of the structure can be analyzed by translating it into a problem on the connectivity of an underlying bipartite graph.[3][4]
However, in many other simple situations it is not yet always known how to analyze the rigidity of a structure mathematically despite the existence of considerable mathematical theory.
History
One of the founders of the mathematical theory of structural rigidity was the great physicist James Clerk Maxwell. The late twentieth century saw an efflorescence of the mathematical theory of rigidity, which continues in the twenty-first century.
"[A] theory of the equilibrium and deflections of frameworks subjected to the action of forces is acting on the hardnes of quality... in cases in which the framework ... is strengthened by additional connecting pieces ... in cases of three dimensions, by the regular method of equations of forces, every point would have three equations to determine its equilibrium, so as to give 3s equations between e unknown quantities, if s be the number of points and e the number of connexions[sic]. There are, however, six equations of equilibrium of the system which must be fulfilled necessarily by the forces, on account of the equality of action and reaction in each piece. Hence if e = 3s − 6, the effect of any eternal force will be definite in producing tensions or pressures in the different pieces; but if e > 3s − 6, these forces will be indeterminate...."[5]
See also
• Chebychev–Grübler–Kutzbach criterion
• Counting on Frameworks
• Kempe's universality theorem
Notes
1. Weisstein, Eric W. "Rigid Graph". MathWorld.
2. Weisstein, Eric W. "Flexible Graph". MathWorld.
3. Baglivo, Jenny A.; Graver, Jack E. (1983), "3.10 Bracing structures", Incidence and Symmetry in Design and Architecture, Cambridge Urban and Architectural Studies, Cambridge, UK: Cambridge University Press, pp. 76–87, ISBN 9780521297844
4. Graver, Jack E. (2001), Counting on Frameworks: Mathematics to Aid the Design of Rigid Structures, The Dolciani Mathematical Expositions, vol. 25, Washington, DC: Mathematical Association of America, ISBN 0-88385-331-0, MR 1843781. See in particular sections 1.2 ("The grid bracing problem", pp. 4–12), 1.5 ("More about the grid problem", pp. 19–22), 2.6 ("The solution to the grid problem", pp. 50–55), and 4.4 ("Tensegrity: tension bracings", particularly pp. 158–161).
5. Maxwell, James Cleark (1864), "On reciprocal figures and diagrams of forces", Philosophical Magazine, 4th Series, vol. 27, pp. 250–261, doi:10.1080/14786446408643663
References
• Alfakih, Abdo Y. (2007), "On dimensional rigidity of bar-and-joint frameworks", Discrete Applied Mathematics, 155 (10): 1244–1253, doi:10.1016/j.dam.2006.11.011, MR 2332317.
• Connelly, Robert (1980), "The rigidity of certain cabled frameworks and the second-order rigidity of arbitrarily triangulated convex surfaces", Advances in Mathematics, 37 (3): 272–299, doi:10.1016/0001-8708(80)90037-7, MR 0591730.
• Crapo, Henry (1979), "Structural rigidity", Structural Topology (1): 26–45, 73, hdl:2099/521, MR 0621627.
• Maxwell, J. C. (1864), "On reciprocal figures and diagrams of forces", Philosophical Magazine, 4th Series, 27 (182): 250–261, doi:10.1080/14786446408643663.
• Rybnikov, Konstantin; Zaslavsky, Thomas (2005), "Criteria for balance in abelian gain graphs, with applications to piecewise-linear geometry", Discrete and Computational Geometry, 34 (2): 251–268, arXiv:math/0210052, doi:10.1007/s00454-005-1170-6, MR 2155721, S2CID 14391276.
• Whiteley, Walter (1988), "The union of matroids and the rigidity of frameworks", SIAM Journal on Discrete Mathematics, 1 (2): 237–255, doi:10.1137/0401025, MR 0941354
|
Wikipedia
|
Rigid category
In category theory, a branch of mathematics, a rigid category is a monoidal category where every object is rigid, that is, has a dual X* (the internal Hom [X, 1]) and a morphism 1 → X ⊗ X* satisfying natural conditions. The category is called right rigid or left rigid according to whether it has right duals or left duals. They were first defined (following Alexander Grothendieck) by Neantro Saavedra Rivano in his thesis on Tannakian categories.[1]
Definition
There are at least two equivalent definitions of a rigidity.
• An object X of a monoidal category is called left rigid if there is an object Y and morphisms $\eta _{X}:\mathbf {1} \to X\otimes Y$ and $\epsilon _{X}:Y\otimes X\to \mathbf {1} $ such that both compositions
$X~{\xrightarrow {\eta _{X}\otimes \mathrm {id} _{X}}}~(X\otimes Y)\otimes X~{\xrightarrow {\alpha _{X,Y,X}^{-1}}}~X\otimes (Y\otimes X)~{\xrightarrow {\mathrm {id} _{X}\otimes \epsilon _{X}}}~X$
$Y~{\xrightarrow {\mathrm {id} _{Y}\otimes \eta _{X}}}~Y\otimes (X\otimes Y)~{\xrightarrow {~\alpha _{X,Y,X}~}}~(Y\otimes X)\otimes Y~{\xrightarrow {\epsilon _{X}\otimes \mathrm {id} _{Y}}}~Y$
are identities. A right rigid object is defined similarly.
An inverse is an object X−1 such that both X ⊗ X−1 and X−1 ⊗ X are isomorphic to 1, the identity object of the monoidal category. If an object X has a left (respectively right) inverse X−1 with respect to the tensor product then it is left (respectively right) rigid, and X* = X−1.
The operation of taking duals gives a contravariant functor on a rigid category.
Uses
One important application of rigidity is in the definition of the trace of an endomorphism of a rigid object. The trace can be defined for any pivotal category, i. e. a rigid category such that ( )**, the functor of taking the dual twice repeated, is isomorphic to the identity functor. Then for any right rigid object X, and any other object Y, we may define the isomorphism
$\phi _{X,Y}:\left\{{\begin{array}{rcl}\mathrm {Hom} (\mathbf {1} ,X^{*}\otimes Y)&\longrightarrow &\mathrm {Hom} (X,Y)\\f&\longmapsto &(\epsilon _{X}\otimes id_{Y})\circ (id_{X}\otimes f)\end{array}}\right.$
and its reciprocal isomorphism
$\psi _{X,Y}:\left\{{\begin{array}{rcl}\mathrm {Hom} (X,Y)&\longrightarrow &\mathrm {Hom} (\mathbf {1} ,X^{*}\otimes Y)\\g&\longmapsto &(id_{X^{*}}\otimes g)\circ \eta _{X}\end{array}}\right.$.
Then for any endomorphism $f:X\to X$, the trace is of f is defined as the composition:
$\mathop {\mathrm {tr} } f:\mathbf {1} {\xrightarrow {\psi _{X,X}(f)}}X^{*}\otimes X{\xrightarrow {\gamma _{X,X}}}X\otimes X^{*}{\xrightarrow {\epsilon _{X}}}\mathbf {1} ,$
We may continue further and define the dimension of a rigid object to be:
$\dim X:=\mathop {\mathrm {tr} } \ \mathrm {id} _{X}$.
Rigidity is also important because of its relation to internal Hom's. If X is a left rigid object, then every internal Hom of the form [X, Z] exists and is isomorphic to Z ⊗ Y. In particular, in a rigid category, all internal Hom's exist.
Alternative terminology
A monoidal category where every object has a left (respectively right) dual is also sometimes called a left (respectively right) autonomous category. A monoidal category where every object has both a left and a right dual is sometimes called an autonomous category. An autonomous category that is also symmetric is called a compact closed category.
Discussion
A monoidal category is a category with a tensor product, precisely the sort of category for which rigidity makes sense.
The category of pure motives is formed by rigidifying the category of effective pure motives.
Notes
1. Rivano, N. Saavedra (1972). Catégories Tannakiennes. Lecture Notes in Mathematics (in French). Vol. 265. Springer. doi:10.1007/BFb0059108. ISBN 978-3-540-37477-0.
References
• Davydov, A. A. (1998). "Monoidal categories and functors". Journal of Mathematical Sciences. 88 (4): 458–472. doi:10.1007/BF02365309.
• Rigid monoidal category at the nLab
|
Wikipedia
|
Rigid origami
Rigid origami is a branch of origami which is concerned with folding structures using flat rigid sheets joined by hinges. That is, unlike in traditional origami, the panels of the paper cannot be bent during the folding process; they must remain flat at all times, and the paper only folded along its hinges. A rigid origami model would still be foldable if it was made from glass sheets with hinges in place of its crease lines.
However, there is no requirement that the structure start as a single flat sheet – for instance shopping bags with flat bottoms are studied as part of rigid origami.
Rigid origami is a part of the study of the mathematics of paper folding, and rigid origami structures can be considered as a type of mechanical linkage. Rigid origami has great practical utility.
Mathematics
The number of standard origami bases that can be folded using rigid origami is restricted by its rules.[1] Rigid origami does not have to follow the Huzita–Hatori axioms, the fold lines can be calculated rather than having to be constructed from existing lines and points. When folding rigid origami flat, Kawasaki's theorem and Maekawa's theorem restrict the folding patterns that are possible, just as they do in conventional origami, but they no longer form an exact characterization: some patterns that can be folded flat in conventional origami cannot be folded flat rigidly.[2]
The Bellows theorem says that a flexible polyhedron has constant volume when flexed rigidly.[3]
The napkin folding problem asks whether it is possible to fold a square so the perimeter of the resulting flat figure is increased. That this can be solved within rigid origami was proved by A.S. Tarasov in 2004.[4]
Blooming is a rigid origami motion of a net of a polyhedron from its flat unfolded state to the folded polyhedron, or vice versa. Although every convex polyhedron has a net with a blooming, it is not known whether there exists a blooming that does not cut across faces of the polyhedron, or whether all nets of convex polyhedra have bloomings.[5]
Complexity theory
Determining whether all creases of a crease pattern can be folded simultaneously as a piece of rigid origami, or whether a subset of the creases can be folded, are both NP-hard. This is true even for determining the existence of a folding motion that keeps the paper arbitrarily close to its flat state, so (unlike for other results in the hardness of folding origami crease patterns) this result does not rely on the impossibility of self-intersections of the folded paper.[6]
Applications
The Miura fold is a rigid fold that has been used to pack large solar panel arrays for space satellites, which have to be folded before deployment.
Robert J. Lang has applied rigid origami to the problem of folding a space telescope.[7]
Although paper shopping bags are commonly folded flat and then unfolded open, the standard folding pattern for doing so is not rigid; the sides of the bag bend slightly when it is folded and unfolded. The tension in the paper from this bending causes it to snap into its two flat states, the flat-folded and opened bag.[8]
Recreational uses
Martin Gardner has popularised flexagons which are a form of rigid origami and the flexatube.[9]
Kaleidocycles are toys, usually made of paper, which give an effect similar to a kaleidoscope when convoluted.
References
1. Demaine, E. D. (2001). Folding and Unfolding. Doctoral Thesis (PDF). University of Waterloo, Canada.{{cite book}}: CS1 maint: location missing publisher (link)
2. Abel, Zachary; Cantarella, Jason; Demaine, Erik D.; Eppstein, David; Hull, Thomas C.; Ku, Jason S.; Lang, Robert J.; Tachi, Tomohiro (2016). "Rigid origami vertices: conditions and forcing sets". Journal of Computational Geometry. 7 (1): 171–184. doi:10.20382/jocg.v7i1a9. MR 3491092. S2CID 7181079.
3. Connelly, R.; Sabitov, I.; Walz, A. (1997). "The bellows conjecture". Beiträge zur Algebra und Geometrie. 38 (1): 1–10. MR 1447981.
4. Tarasov, A. S. (2004). "Solution of Arnold's "folded ruble" problem". Chebyshevskii Sbornik (in Russian). 5 (1): 174–187. Archived from the original on 2007-08-25.
5. Miller, Ezra; Pak, Igor (2008). "Metric combinatorics of convex polyhedra: Cut loci and nonoverlapping unfoldings". Discrete & Computational Geometry. 39 (1–3): 339–388. doi:10.1007/s00454-008-9052-3. MR 2383765. S2CID 10227925.. Announced in 2003.
6. Akitaya, Hugo; Demaine, Erik; Horiyama, Takashi; Hull, Thomas; Ku, Jason; Tachi, Tomohiro (2020). "Rigid foldability is NP-hard". Journal of Computational Geometry. 11 (1). arXiv:1812.01160.
7. "The Eyeglass Space Telescope" (PDF).
8. Devin. J. Balkcom, Erik D. Demaine, Martin L. Demaine (November 2004). "Folding Paper Shopping Bags". Abstracts from the 14th Annual Fall Workshop on Computational Geometry. Cambridge, Massachusetts: 14–15.{{cite journal}}: CS1 maint: multiple names: authors list (link)
9. Weisstein, Eric W. "Flexatube". Wolfram MathWorld.
External links
• Hull, Tom. "Rigid Origami".
Mathematics of paper folding
Flat folding
• Big-little-big lemma
• Crease pattern
• Huzita–Hatori axioms
• Kawasaki's theorem
• Maekawa's theorem
• Map folding
• Napkin folding problem
• Pureland origami
• Yoshizawa–Randlett system
Strip folding
• Dragon curve
• Flexagon
• Möbius strip
• Regular paperfolding sequence
3d structures
• Miura fold
• Modular origami
• Paper bag problem
• Rigid origami
• Schwarz lantern
• Sonobe
• Yoshimura buckling
Polyhedra
• Alexandrov's uniqueness theorem
• Blooming
• Flexible polyhedron (Bricard octahedron, Steffen's polyhedron)
• Net
• Source unfolding
• Star unfolding
Miscellaneous
• Fold-and-cut theorem
• Lill's method
Publications
• Geometric Exercises in Paper Folding
• Geometric Folding Algorithms
• Geometric Origami
• A History of Folding in Mathematics
• Origami Polyhedra Design
• Origamics
People
• Roger C. Alperin
• Margherita Piazzola Beloch
• Robert Connelly
• Erik Demaine
• Martin Demaine
• Rona Gurkewitz
• David A. Huffman
• Tom Hull
• Kôdi Husimi
• Humiaki Huzita
• Toshikazu Kawasaki
• Robert J. Lang
• Anna Lubiw
• Jun Maekawa
• Kōryō Miura
• Joseph O'Rourke
• Tomohiro Tachi
• Eve Torrence
|
Wikipedia
|
Level structure (algebraic geometry)
In algebraic geometry, a level structure on a space X is an extra structure attached to X that shrinks or eliminates the automorphism group of X, by demanding automorphisms to preserve the level structure; attaching a level structure is often phrased as rigidifying the geometry of X.[1][2]
In applications, a level structure is used in the construction of moduli spaces; a moduli space is often constructed as a quotient. The presence of automorphisms poses a difficulty to forming a quotient; thus introducing level structures helps overcome this difficulty.
There is no single definition of a level structure; rather, depending on the space X, one introduces the notion of a level structure. The classic one is that on an elliptic curve (see #Example: an abelian scheme). There is a level structure attached to a formal group called a Drinfeld level structure, introduced in (Drinfeld 1974).[3]
Level structures on elliptic curves
Classically, level structures on elliptic curves $E=\mathbb {C} /\Lambda $ are given by a lattice containing the defining lattice of the variety. From the moduli theory of elliptic curves, all such lattices can be described as the lattice $\mathbb {Z} \oplus \mathbb {Z} \cdot \tau $ for $\tau \in {\mathfrak {h}}$ in the upper-half plane. Then, the lattice generated by $1/n,\tau /n$ gives a lattice which contains all $n$-torsion points on the elliptic curve denoted $E[n]$. In fact, given such a lattice is invariant under the $\Gamma (n)\subset {\text{SL}}_{2}(\mathbb {Z} )$ action on ${\mathfrak {h}}$, where
${\begin{aligned}\Gamma (n)&={\text{ker}}({\text{SL}}_{2}(\mathbb {Z} )\to {\text{SL}}_{2}(\mathbb {Z} /n))\\&=\left\{M\in {\text{SL}}_{2}(\mathbb {Z} ):M\equiv {\begin{pmatrix}1&0\\0&1\end{pmatrix}}{\text{ (mod n)}}\right\}\end{aligned}}$
hence it gives a point in $\Gamma (n)\backslash {\mathfrak {h}}$[4] called the moduli space of level N structures of elliptic curves $Y(n)$, which is a modular curve. In fact, this moduli space contains slightly more information: the Weil pairing
$e_{n}\left({\frac {1}{n}},{\frac {\tau }{n}}\right)=e^{2\pi i/n}$
gives a point in the $n$-th roots of unity, hence in $\mathbb {Z} /n$.
Example: an abelian scheme
Let $X\to S$ be an abelian scheme whose geometric fibers have dimension g.
Let n be a positive integer that is prime to the residue field of each s in S. For n ≥ 2, a level n-structure is a set of sections $\sigma _{1},\dots ,\sigma _{2g}$ such that[5]
1. for each geometric point $s:S\to X$, $\sigma _{i}(s)$ form a basis for the group of points of order n in ${\overline {X}}_{s}$,
2. $m_{n}\circ \sigma _{i}$ is the identity section, where $m_{n}$ is the multiplication by n.
See also: modular curve#Examples, moduli stack of elliptic curves.
See also
• Siegel modular form
• Rigidity (mathematics)
• Local rigidity
Notes
1. Mumford, Fogarty & Kirwan 1994, Ch. 7.
2. Katz & Mazur 1985, Introduction
3. Deligne, P.; Husemöller, D. (1987). "Survey of Drinfeld's modules" (PDF). Contemp. Math. 67 (1): 25–91. doi:10.1090/conm/067/902591.
4. Silverman, Joseph H., 1955- (2009). The arithmetic of elliptic curves (2nd ed.). New York: Springer-Verlag. pp. 439–445. ISBN 978-0-387-09494-6. OCLC 405546184.{{cite book}}: CS1 maint: multiple names: authors list (link)
5. Mumford, Fogarty & Kirwan 1994, Definition 7.1.
References
• Drinfeld, V. (1974). "Elliptic modules". Math USSR Sbornik. 23 (4): 561–592. Bibcode:1974SbMat..23..561D. doi:10.1070/sm1974v023n04abeh001731.
• Katz, Nicholas M.; Mazur, Barry (1985). Arithmetic Moduli of Elliptic Curves. Princeton University Press. ISBN 0-691-08352-5.
• Harris, Michael; Taylor, Richard (2001). The Geometry and Cohomology of Some Simple Shimura Varieties. Annals of Mathematics Studies. Vol. 151. Princeton University Press. ISBN 978-1-4008-3720-5.
• Mumford, David; Fogarty, J.; Kirwan, F. (1994). Geometric invariant theory. Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)]. Vol. 34 (3rd ed.). Springer-Verlag. ISBN 978-3-540-56963-3. MR 1304906.
Further reading
• Notes on principal bundles
• J. Lurie, Level Structures on Elliptic Curves.
|
Wikipedia
|
Rigidity matroid
In the mathematics of structural rigidity, a rigidity matroid is a matroid that describes the number of degrees of freedom of an undirected graph with rigid edges of fixed lengths, embedded into Euclidean space. In a rigidity matroid for a graph with n vertices in d-dimensional space, a set of edges that defines a subgraph with k degrees of freedom has matroid rank dn − k. A set of edges is independent if and only if, for every edge in the set, removing the edge would increase the number of degrees of freedom of the remaining subgraph.[1][2][3]
Definition
A framework is an undirected graph, embedded into d-dimensional Euclidean space by providing a d-tuple of Cartesian coordinates for each vertex of the graph. From a framework with n vertices and m edges, one can define a matrix with m rows and nd columns, an expanded version of the incidence matrix of the graph called the rigidity matrix. In this matrix, the entry in row e and column (v,i) is zero if v is not an endpoint of edge e. If, on the other hand, edge e has vertices u and v as endpoints, then the value of the entry is the difference between the ith coordinates of v and u.[1][3]
The rigidity matroid of the given framework is a linear matroid that has as its elements the edges of the graph. A set of edges is independent, in the matroid, if it corresponds to a set of rows of the rigidity matrix that is linearly independent. A framework is called generic if the coordinates of its vertices are algebraically independent real numbers. Any two generic frameworks on the same graph G determine the same rigidity matroid, regardless of their specific coordinates. This is the (d-dimensional) rigidity matroid of G.[1][3]
Statics
A load on a framework is a system of forces on the vertices (represented as vectors). A stress is a special case of a load, in which equal and opposite forces are applied to the two endpoints of each edge (which may be imagined as a spring) and the forces formed in this way are added at each vertex. Every stress is an equilibrium load, a load that does not impose any translational force on the whole system (the sum of its force vectors is zero) nor any rotational force. A linear dependence among the rows of the rigidity matrix may be represented as a self-stress, an assignment of equal and opposite forces to the endpoints of each edge that is not identically zero but that adds to zero at every vertex. Thus, a set of edges forms an independent set in the rigidity matroid if and only if it has no self-stress.[3]
The vector space of all possible loads, on a system of n vertices, has dimension dn, among which the equilibrium loads form a subspace of dimension $dn-{\binom {d+1}{2}}$. An independent set in the rigidity matroid has a system of equilibrium loads whose dimension equals the cardinality of the set, so the maximum rank that any set in the matroid can have is $dn-{\binom {d+1}{2}}$. If a set has this rank, it follows that its set of stresses is the same as the space of equilibrium loads. Alternatively and equivalently, in this case every equilibrium load on the framework may be resolved by a stress that generates an equal and opposite set of forces, and the framework is said to be statically rigid.[3]
Kinematics
If the vertices of a framework are in a motion, then that motion may be described over small scales of distance by its gradient, a vector for each vertex specifying its speed and direction. The gradient describes a linearized approximation to the actual motion of the points, in which each point moves at constant velocity in a straight line. The gradient may be described as a row vector that has one real number coordinate for each pair $(v,i)$ where $v$ is a vertex of the framework and $i$ is the index of one of the Cartesian coordinates of $d$-dimensional space; that is, the dimension of the gradient is the same as the width of the rigidity matrix.[1][3]
If the edges of the framework are assumed to be rigid bars that can neither expand nor contract (but can freely rotate) then any motion respecting this rigidity must preserve the lengths of the edges: the derivative of length, as a function of the time over which the motion occurs, must remain zero. This condition may be expressed in linear algebra as a constraint that the gradient vector of the motion of the vertices must have zero inner product with the row of the rigidity matrix that represents the given edge. Thus, the family of gradients of (infinitesimally) rigid motions is given by the nullspace of the rigidity matrix.[1][3] For frameworks that are not in generic position, it is possible that some infinitesimally rigid motions (vectors in the nullspace of the rigidity matrix) are not the gradients of any continuous motion, but this cannot happen for generic frameworks.[2]
A rigid motion of the framework is a motion such that, at each point in time, the framework is congruent to its original configuration. Rigid motions include translations and rotations of Euclidean space; the gradients of rigid motions form a linear space having the translations and rotations as bases, of dimension ${\binom {d+1}{2}}$, which must always be a subspace of the nullspace of the rigidity matrix. Because the nullspace always has at least this dimension, the rigidity matroid can have rank at most $dn-{\binom {d+1}{2}}$, and when it does have this rank the only motions that preserve the lengths of the edges of the framework are the rigid motions. In this case the framework is said to be first-order (or infinitesimally) rigid.[1][3] More generally, an edge $e$ belongs to the matroid closure operation of a set $S$ if and only if there does not exist a continuous motion of the framework that changes the length of $e$ but leaves the lengths of the edges in $S$ unchanged.[1]
Although defined in different terms (column vectors versus row vectors, or forces versus motions) static rigidity and first-order rigidity reduce to the same properties of the underlying matrix and therefore coincide with each other. In two dimensions, the generic rigidity matroid also describes the number of degrees of freedom of a different kind of motion, in which each edge is constrained to stay parallel to its original position rather than being constrained to maintain the same length; however, the equivalence between rigidity and parallel motion breaks down in higher dimensions.[3]
Unique realization
A framework has a unique realization in d-dimensional space if every placement of the same graph with the same edge lengths is congruent to it. Such a framework must necessarily be rigid, because otherwise there exists a continuous motion bringing it to a non-congruent placement with the same edge lengths, but unique realizability is stronger than rigidity. For instance, the diamond graph (two triangles sharing an edge) is rigid in two dimensions, but it is not uniquely realizable because it has two different realizations, one in which the triangles are on opposite sides of the shared edge and one in which they are both on the same side. Uniquely realizable graphs are important in applications that involve reconstruction of shapes from distances, such as triangulation in land surveying,[4] the determination of the positions of the nodes in a wireless sensor network,[5] and the reconstruction of conformations of molecules via nuclear magnetic resonance spectroscopy.[4]
Bruce Hendrickson defined a graph to be redundantly rigid if it remains rigid after removing any one of its edges. In matroidal terms, this means that the rigidity matroid has the full rank $dn-{\binom {d+1}{2}}$ and that the matroid does not have any coloops. Hendrickson proved that every uniquely realizable framework (with generic edge lengths) is either a complete graph or a $(d+1)$-vertex-connected, redundantly rigid graph, and he conjectured that this is an exact characterization of the uniquely realizable frameworks.[6] The conjecture is true for one and two dimensions; in the one-dimensional case, for instance, a graph is uniquely realizable if and only if it is connected and bridgeless.[7] However, Henrickson's conjecture is false for three or more dimensions.[8] For frameworks that are not generic, it is NP-hard to determine whether a given framework is uniquely realizable.[9]
Relation to sparsity
Streinu & Theran (2009) define a graph as being $(k,l)$-sparse if every nonempty subgraph with $n$ vertices has at most $kn-l$ edges, and $(k,l)$-tight if it is $(k,l)$-sparse and has exactly $kn-l$ edges.[10] From the consideration of loads and stresses it can be seen that a set of edges that is independent in the rigidity matroid forms a $(d,{\binom {d+1}{2}})$-sparse graph, for if not there would exist a subgraph whose number of edges would exceed the dimension of its space of equilibrium loads, from which it follows that it would have a self-stress. By similar reasoning, a set of edges that is both independent and rigid forms a $(d,{\binom {d+1}{2}})$-tight graph. For instance, in one dimension, the independent sets form the edge sets of forests, (1,1)-sparse graphs, and the independent rigid sets form the edge sets of trees, (1,1)-tight graphs. In this case the rigidity matroid of a framework is the same as the graphic matroid of the corresponding graph.[2]
In two dimensions, Laman (1970) showed that the same characterization is true: the independent sets form the edge sets of (2,3)-sparse graphs and the independent rigid sets form the edge sets of (2,3)-tight graphs.[11] Based on this work the (2,3)-tight graphs (the graphs of minimally rigid generic frameworks in two dimensions) have come to be known as Laman graphs. The family of Laman graphs on a fixed set of $n$ vertices forms the set of bases of the rigidity matroid of a complete graph, and more generally for every graph $G$ that forms a rigid framework in two dimensions, the spanning Laman subgraphs of $G$ are the bases of the rigidity matroid of $G$.
However, in higher dimensions not every $(d,{\binom {d+1}{2}})$-tight graph is minimally rigid, and characterizing the minimally rigid graphs (the bases of the rigidity matroid of the complete graph) is an important open problem.[12]
References
1. Graver, Jack E. (1991), "Rigidity matroids", SIAM Journal on Discrete Mathematics, 4 (3): 355–368, doi:10.1137/0404032, MR 1105942.
2. Whiteley, Walter (1992), "Matroids and rigid structures", Matroid Applications, Encyclopedia of Mathematics and its Applications, vol. 40, Cambridge: Cambridge Univ. Press, pp. 1–53, doi:10.1017/CBO9780511662041.002, MR 1165538.
3. Whiteley, Walter (1996), "Some matroids from discrete applied geometry", Matroid theory (Seattle, WA, 1995), Contemporary Mathematics, vol. 197, Providence, RI: American Mathematical Society, pp. 171–311, doi:10.1090/conm/197/02540, MR 1411692.
4. Hendrickson, Bruce (1995), "The molecule problem: exploiting structure in global optimization", SIAM Journal on Optimization, 5 (4): 835–857, CiteSeerX 10.1.1.55.2335, doi:10.1137/0805040, MR 1358807.
5. Eren, T.; Goldenberg, O.K.; Whiteley, W.; Yang, Y.R.; Morse, A.S.; Anderson, B.D.O.; Belhumeur, P.N. (2004), "Rigidity, computation, and randomization in network localization", Proc. Twenty-third Annual Joint Conference of the IEEE Computer and Communications Societies (INFOCOM 2004), vol. 4, pp. 2673–2684, doi:10.1109/INFCOM.2004.1354686.
6. Hendrickson, Bruce (1992), "Conditions for unique graph realizations", SIAM Journal on Computing, 21 (1): 65–84, doi:10.1137/0221008, MR 1148818.
7. Jackson, Bill; Jordán, Tibor (2005), "Connected rigidity matroids and unique realizations of graphs", Journal of Combinatorial Theory, Series B, 94 (1): 1–29, doi:10.1016/j.jctb.2004.11.002, MR 2130278.
8. Connelly, Robert (1991), "On generic global rigidity", Applied Geometry and Discrete Mathematics, DIMACS Series on Discrete Mathematics and Theoretical Computer Science, vol. 4, Providence, RI: American Mathematical Society, pp. 147–155, MR 1116345.
9. Saxe, J. B. (1979), Embeddability of weighted graphs in k-space is strongly NP-hard, Technical Report, Pittsburgh, PA: Computer Science Department, Carnegie-Mellon University. As cited by Jackson & Jordán (2005).
10. Streinu, I.; Theran, L. (2009), "Sparse hypergraphs and pebble game algorithms", European Journal of Combinatorics, 30 (8): 1944–1964, arXiv:math/0703921, doi:10.1016/j.ejc.2008.12.018.
11. Laman, G. (1970), "On graphs and the rigidity of plane skeletal structures", J. Engineering Mathematics, 4 (4): 331–340, Bibcode:1970JEnMa...4..331L, doi:10.1007/BF01534980, MR 0269535.
12. Jackson, Bill; Jordán, Tibor (2006), "On the rank function of the 3-dimensional rigidity matroid" (PDF), International Journal of Computational Geometry & Applications, 16 (5–6): 415–429, doi:10.1142/S0218195906002117, MR 2269396.
|
Wikipedia
|
Rijndael MixColumns
The MixColumns operation performed by the Rijndael cipher, along with the ShiftRows step, is the primary source of diffusion in Rijndael. Each column is treated as a four-term polynomial $b(x)=b_{3}x^{3}+b_{2}x^{2}+b_{1}x+b_{0}$ which are elements within the field $\operatorname {GF} (2^{8})$. The coefficients of the polynomials are elements within the prime sub-field $\operatorname {GF} (2)$.
Each column is multiplied with a fixed polynomial $a(x)=3x^{3}+x^{2}+x+2$ modulo $x^{4}+1$; the inverse of this polynomial is $a^{-1}(x)=11x^{3}+13x^{2}+9x+14$.
Demonstration
The polynomial $a(x)=3x^{3}+x^{2}+x+2$ will be expressed as $a(x)=a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}$.
Polynomial multiplication
${\begin{aligned}a(x)\bullet b(x)=c(x)&=\left(a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}\right)\bullet \left(b_{3}x^{3}+b_{2}x^{2}+b_{1}x+b_{0}\right)\\&=c_{6}x^{6}+c_{5}x^{5}+c_{4}x^{4}+c_{3}x^{3}+c_{2}x^{2}+c_{1}x+c_{0}\end{aligned}}$
where:
${\begin{aligned}c_{0}&=a_{0}\bullet b_{0}\\c_{1}&=a_{1}\bullet b_{0}\oplus a_{0}\bullet b_{1}\\c_{2}&=a_{2}\bullet b_{0}\oplus a_{1}\bullet b_{1}\oplus a_{0}\bullet b_{2}\\c_{3}&=a_{3}\bullet b_{0}\oplus a_{2}\bullet b_{1}\oplus a_{1}\bullet b_{2}\oplus a_{0}\bullet b_{3}\\c_{4}&=a_{3}\bullet b_{1}\oplus a_{2}\bullet b_{2}\oplus a_{1}\bullet b_{3}\\c_{5}&=a_{3}\bullet b_{2}\oplus a_{2}\bullet b_{3}\\c_{6}&=a_{3}\bullet b_{3}\end{aligned}}$
Modular reduction
The result $c(x)$ is a seven-term polynomial, which must be reduced to a four-byte word, which is done by doing the multiplication modulo $x^{4}+1$.
If we do some basic polynomial modular operations we can see that:
${\begin{aligned}x^{6}{\bmod {\left(x^{4}+1\right)}}&=-x^{2}=x^{2}{\text{ over }}\operatorname {GF} \left(2^{8}\right)\\x^{5}{\bmod {\left(x^{4}+1\right)}}&=-x=x{\text{ over }}\operatorname {GF} \left(2^{8}\right)\\x^{4}{\bmod {\left(x^{4}+1\right)}}&=-1=1{\text{ over }}\operatorname {GF} \left(2^{8}\right)\end{aligned}}$
In general, we can say that $x^{i}{\bmod {\left(x^{4}+1\right)}}=x^{i{\bmod {4}}}.$
So
${\begin{aligned}a(x)\otimes b(x)&=c(x){\bmod {\left(x^{4}+1\right)}}\\&=\left(c_{6}x^{6}+c_{5}x^{5}+c_{4}x^{4}+c_{3}x^{3}+c_{2}x^{2}+c_{1}x+c_{0}\right){\bmod {\left(x^{4}+1\right)}}\\&=c_{6}x^{6{\bmod {4}}}+c_{5}x^{5{\bmod {4}}}+c_{4}x^{4{\bmod {4}}}+c_{3}x^{3{\bmod {4}}}+c_{2}x^{2{\bmod {4}}}+c_{1}x^{1{\bmod {4}}}+c_{0}x^{0{\bmod {4}}}\\&=c_{6}x^{2}+c_{5}x+c_{4}+c_{3}x^{3}+c_{2}x^{2}+c_{1}x+c_{0}\\&=c_{3}x^{3}+\left(c_{2}\oplus c_{6}\right)x^{2}+\left(c_{1}\oplus c_{5}\right)x+c_{0}\oplus c_{4}\\&=d_{3}x^{3}+d_{2}x^{2}+d_{1}x+d_{0}\end{aligned}}$
where
$d_{0}=c_{0}\oplus c_{4}$
$d_{1}=c_{1}\oplus c_{5}$
$d_{2}=c_{2}\oplus c_{6}$
$d_{3}=c_{3}$
Matrix representation
The coefficient $d_{3}$, $d_{2}$, $d_{1}$ and $d_{0}$ can also be expressed as follows:
$d_{0}=a_{0}\bullet b_{0}\oplus a_{3}\bullet b_{1}\oplus a_{2}\bullet b_{2}\oplus a_{1}\bullet b_{3}$
$d_{1}=a_{1}\bullet b_{0}\oplus a_{0}\bullet b_{1}\oplus a_{3}\bullet b_{2}\oplus a_{2}\bullet b_{3}$
$d_{2}=a_{2}\bullet b_{0}\oplus a_{1}\bullet b_{1}\oplus a_{0}\bullet b_{2}\oplus a_{3}\bullet b_{3}$
$d_{3}=a_{3}\bullet b_{0}\oplus a_{2}\bullet b_{1}\oplus a_{1}\bullet b_{2}\oplus a_{0}\bullet b_{3}$
And when we replace the coefficients of $a(x)$ with the constants ${\begin{bmatrix}3&1&1&2\end{bmatrix}}$ used in the cipher we obtain the following:
$d_{0}=2\bullet b_{0}\oplus 3\bullet b_{1}\oplus 1\bullet b_{2}\oplus 1\bullet b_{3}$
$d_{1}=1\bullet b_{0}\oplus 2\bullet b_{1}\oplus 3\bullet b_{2}\oplus 1\bullet b_{3}$
$d_{2}=1\bullet b_{0}\oplus 1\bullet b_{1}\oplus 2\bullet b_{2}\oplus 3\bullet b_{3}$
$d_{3}=3\bullet b_{0}\oplus 1\bullet b_{1}\oplus 1\bullet b_{2}\oplus 2\bullet b_{3}$
This demonstrates that the operation itself is similar to a Hill cipher. It can be performed by multiplying a coordinate vector of four numbers in Rijndael's Galois field by the following circulant MDS matrix:
${\begin{bmatrix}d_{0}\\d_{1}\\d_{2}\\d_{3}\end{bmatrix}}={\begin{bmatrix}2&3&1&1\\1&2&3&1\\1&1&2&3\\3&1&1&2\end{bmatrix}}{\begin{bmatrix}b_{0}\\b_{1}\\b_{2}\\b_{3}\end{bmatrix}}$
Implementation example
This can be simplified somewhat in actual implementation by replacing the multiply by 2 with a single shift and conditional exclusive or, and replacing a multiply by 3 with a multiply by 2 combined with an exclusive or. A C example of such an implementation follows:
void gmix_column(unsigned char *r) {
unsigned char a[4];
unsigned char b[4];
unsigned char c;
unsigned char h;
/* The array 'a' is simply a copy of the input array 'r'
* The array 'b' is each element of the array 'a' multiplied by 2
* in Rijndael's Galois field
* a[n] ^ b[n] is element n multiplied by 3 in Rijndael's Galois field */
for (c = 0; c < 4; c++) {
a[c] = r[c];
/* h is 0xff if the high bit of r[c] is set, 0 otherwise */
h = (r[c] >> 7) & 1; /* arithmetic right shift, thus shifting in either zeros or ones */
b[c] = r[c] << 1; /* implicitly removes high bit because b[c] is an 8-bit char, so we xor by 0x1b and not 0x11b in the next line */
b[c] ^= h * 0x1B; /* Rijndael's Galois field */
}
r[0] = b[0] ^ a[3] ^ a[2] ^ b[1] ^ a[1]; /* 2 * a0 + a3 + a2 + 3 * a1 */
r[1] = b[1] ^ a[0] ^ a[3] ^ b[2] ^ a[2]; /* 2 * a1 + a0 + a3 + 3 * a2 */
r[2] = b[2] ^ a[1] ^ a[0] ^ b[3] ^ a[3]; /* 2 * a2 + a1 + a0 + 3 * a3 */
r[3] = b[3] ^ a[2] ^ a[1] ^ b[0] ^ a[0]; /* 2 * a3 + a2 + a1 + 3 * a0 */
}
A C# example
private byte GMul(byte a, byte b) { // Galois Field (256) Multiplication of two Bytes
byte p = 0;
for (int counter = 0; counter < 8; counter++) {
if ((b & 1) != 0) {
p ^= a;
}
bool hi_bit_set = (a & 0x80) != 0;
a <<= 1;
if (hi_bit_set) {
a ^= 0x1B; /* x^8 + x^4 + x^3 + x + 1 */
}
b >>= 1;
}
return p;
}
private void MixColumns() { // 's' is the main State matrix, 'ss' is a temp matrix of the same dimensions as 's'.
Array.Clear(ss, 0, ss.Length);
for (int c = 0; c < 4; c++) {
ss[0, c] = (byte)(GMul(0x02, s[0, c]) ^ GMul(0x03, s[1, c]) ^ s[2, c] ^ s[3, c]);
ss[1, c] = (byte)(s[0, c] ^ GMul(0x02, s[1, c]) ^ GMul(0x03, s[2, c]) ^ s[3,c]);
ss[2, c] = (byte)(s[0, c] ^ s[1, c] ^ GMul(0x02, s[2, c]) ^ GMul(0x03, s[3, c]));
ss[3, c] = (byte)(GMul(0x03, s[0,c]) ^ s[1, c] ^ s[2, c] ^ GMul(0x02, s[3, c]));
}
ss.CopyTo(s, 0);
}
Test vectors for MixColumn()
Hexadecimal Decimal
Before After Before After
db 13 53 45 8e 4d a1 bc 219 19 83 69 142 77 161 188
f2 0a 22 5c 9f dc 58 9d 242 10 34 92 159 220 88 157
01 01 01 01 01 01 01 01 1 1 1 1 1 1 1 1
c6 c6 c6 c6 c6 c6 c6 c6 198 198 198 198 198 198 198 198
d4 d4 d4 d5 d5 d5 d7 d6 212 212 212 213 213 213 215 214
2d 26 31 4c 4d 7e bd f8 45 38 49 76 77 126 189 248
InverseMixColumns
The MixColumns operation has the following inverse (numbers are decimal):
${\begin{bmatrix}b_{0}\\b_{1}\\b_{2}\\b_{3}\end{bmatrix}}={\begin{bmatrix}14&11&13&9\\9&14&11&13\\13&9&14&11\\11&13&9&14\end{bmatrix}}{\begin{bmatrix}d_{0}\\d_{1}\\d_{2}\\d_{3}\end{bmatrix}}$
Or:
${\begin{aligned}b_{0}&=14\bullet d_{0}\oplus 11\bullet d_{1}\oplus 13\bullet d_{2}\oplus 9\bullet d_{3}\\b_{1}&=9\bullet d_{0}\oplus 14\bullet d_{1}\oplus 11\bullet d_{2}\oplus 13\bullet d_{3}\\b_{2}&=13\bullet d_{0}\oplus 9\bullet d_{1}\oplus 14\bullet d_{2}\oplus 11\bullet d_{3}\\b_{3}&=11\bullet d_{0}\oplus 13\bullet d_{1}\oplus 9\bullet d_{2}\oplus 14\bullet d_{3}\end{aligned}}$
Galois Multiplication lookup tables
Commonly, rather than implementing Galois multiplication, Rijndael implementations simply use pre-calculated lookup tables to perform the byte multiplication by 2, 3, 9, 11, 13, and 14.
For instance, in C# these tables can be stored in Byte[256] arrays. In order to compute
p * 3
The result is obtained this way:
result = table_3[(int)p]
Some of the most common instances of these lookup tables are as follows:
Multiply by 2:
0x00,0x02,0x04,0x06,0x08,0x0a,0x0c,0x0e,0x10,0x12,0x14,0x16,0x18,0x1a,0x1c,0x1e,
0x20,0x22,0x24,0x26,0x28,0x2a,0x2c,0x2e,0x30,0x32,0x34,0x36,0x38,0x3a,0x3c,0x3e,
0x40,0x42,0x44,0x46,0x48,0x4a,0x4c,0x4e,0x50,0x52,0x54,0x56,0x58,0x5a,0x5c,0x5e,
0x60,0x62,0x64,0x66,0x68,0x6a,0x6c,0x6e,0x70,0x72,0x74,0x76,0x78,0x7a,0x7c,0x7e,
0x80,0x82,0x84,0x86,0x88,0x8a,0x8c,0x8e,0x90,0x92,0x94,0x96,0x98,0x9a,0x9c,0x9e,
0xa0,0xa2,0xa4,0xa6,0xa8,0xaa,0xac,0xae,0xb0,0xb2,0xb4,0xb6,0xb8,0xba,0xbc,0xbe,
0xc0,0xc2,0xc4,0xc6,0xc8,0xca,0xcc,0xce,0xd0,0xd2,0xd4,0xd6,0xd8,0xda,0xdc,0xde,
0xe0,0xe2,0xe4,0xe6,0xe8,0xea,0xec,0xee,0xf0,0xf2,0xf4,0xf6,0xf8,0xfa,0xfc,0xfe,
0x1b,0x19,0x1f,0x1d,0x13,0x11,0x17,0x15,0x0b,0x09,0x0f,0x0d,0x03,0x01,0x07,0x05,
0x3b,0x39,0x3f,0x3d,0x33,0x31,0x37,0x35,0x2b,0x29,0x2f,0x2d,0x23,0x21,0x27,0x25,
0x5b,0x59,0x5f,0x5d,0x53,0x51,0x57,0x55,0x4b,0x49,0x4f,0x4d,0x43,0x41,0x47,0x45,
0x7b,0x79,0x7f,0x7d,0x73,0x71,0x77,0x75,0x6b,0x69,0x6f,0x6d,0x63,0x61,0x67,0x65,
0x9b,0x99,0x9f,0x9d,0x93,0x91,0x97,0x95,0x8b,0x89,0x8f,0x8d,0x83,0x81,0x87,0x85,
0xbb,0xb9,0xbf,0xbd,0xb3,0xb1,0xb7,0xb5,0xab,0xa9,0xaf,0xad,0xa3,0xa1,0xa7,0xa5,
0xdb,0xd9,0xdf,0xdd,0xd3,0xd1,0xd7,0xd5,0xcb,0xc9,0xcf,0xcd,0xc3,0xc1,0xc7,0xc5,
0xfb,0xf9,0xff,0xfd,0xf3,0xf1,0xf7,0xf5,0xeb,0xe9,0xef,0xed,0xe3,0xe1,0xe7,0xe5
Multiply by 3:
0x00,0x03,0x06,0x05,0x0c,0x0f,0x0a,0x09,0x18,0x1b,0x1e,0x1d,0x14,0x17,0x12,0x11,
0x30,0x33,0x36,0x35,0x3c,0x3f,0x3a,0x39,0x28,0x2b,0x2e,0x2d,0x24,0x27,0x22,0x21,
0x60,0x63,0x66,0x65,0x6c,0x6f,0x6a,0x69,0x78,0x7b,0x7e,0x7d,0x74,0x77,0x72,0x71,
0x50,0x53,0x56,0x55,0x5c,0x5f,0x5a,0x59,0x48,0x4b,0x4e,0x4d,0x44,0x47,0x42,0x41,
0xc0,0xc3,0xc6,0xc5,0xcc,0xcf,0xca,0xc9,0xd8,0xdb,0xde,0xdd,0xd4,0xd7,0xd2,0xd1,
0xf0,0xf3,0xf6,0xf5,0xfc,0xff,0xfa,0xf9,0xe8,0xeb,0xee,0xed,0xe4,0xe7,0xe2,0xe1,
0xa0,0xa3,0xa6,0xa5,0xac,0xaf,0xaa,0xa9,0xb8,0xbb,0xbe,0xbd,0xb4,0xb7,0xb2,0xb1,
0x90,0x93,0x96,0x95,0x9c,0x9f,0x9a,0x99,0x88,0x8b,0x8e,0x8d,0x84,0x87,0x82,0x81,
0x9b,0x98,0x9d,0x9e,0x97,0x94,0x91,0x92,0x83,0x80,0x85,0x86,0x8f,0x8c,0x89,0x8a,
0xab,0xa8,0xad,0xae,0xa7,0xa4,0xa1,0xa2,0xb3,0xb0,0xb5,0xb6,0xbf,0xbc,0xb9,0xba,
0xfb,0xf8,0xfd,0xfe,0xf7,0xf4,0xf1,0xf2,0xe3,0xe0,0xe5,0xe6,0xef,0xec,0xe9,0xea,
0xcb,0xc8,0xcd,0xce,0xc7,0xc4,0xc1,0xc2,0xd3,0xd0,0xd5,0xd6,0xdf,0xdc,0xd9,0xda,
0x5b,0x58,0x5d,0x5e,0x57,0x54,0x51,0x52,0x43,0x40,0x45,0x46,0x4f,0x4c,0x49,0x4a,
0x6b,0x68,0x6d,0x6e,0x67,0x64,0x61,0x62,0x73,0x70,0x75,0x76,0x7f,0x7c,0x79,0x7a,
0x3b,0x38,0x3d,0x3e,0x37,0x34,0x31,0x32,0x23,0x20,0x25,0x26,0x2f,0x2c,0x29,0x2a,
0x0b,0x08,0x0d,0x0e,0x07,0x04,0x01,0x02,0x13,0x10,0x15,0x16,0x1f,0x1c,0x19,0x1a
Multiply by 9:
0x00,0x09,0x12,0x1b,0x24,0x2d,0x36,0x3f,0x48,0x41,0x5a,0x53,0x6c,0x65,0x7e,0x77,
0x90,0x99,0x82,0x8b,0xb4,0xbd,0xa6,0xaf,0xd8,0xd1,0xca,0xc3,0xfc,0xf5,0xee,0xe7,
0x3b,0x32,0x29,0x20,0x1f,0x16,0x0d,0x04,0x73,0x7a,0x61,0x68,0x57,0x5e,0x45,0x4c,
0xab,0xa2,0xb9,0xb0,0x8f,0x86,0x9d,0x94,0xe3,0xea,0xf1,0xf8,0xc7,0xce,0xd5,0xdc,
0x76,0x7f,0x64,0x6d,0x52,0x5b,0x40,0x49,0x3e,0x37,0x2c,0x25,0x1a,0x13,0x08,0x01,
0xe6,0xef,0xf4,0xfd,0xc2,0xcb,0xd0,0xd9,0xae,0xa7,0xbc,0xb5,0x8a,0x83,0x98,0x91,
0x4d,0x44,0x5f,0x56,0x69,0x60,0x7b,0x72,0x05,0x0c,0x17,0x1e,0x21,0x28,0x33,0x3a,
0xdd,0xd4,0xcf,0xc6,0xf9,0xf0,0xeb,0xe2,0x95,0x9c,0x87,0x8e,0xb1,0xb8,0xa3,0xaa,
0xec,0xe5,0xfe,0xf7,0xc8,0xc1,0xda,0xd3,0xa4,0xad,0xb6,0xbf,0x80,0x89,0x92,0x9b,
0x7c,0x75,0x6e,0x67,0x58,0x51,0x4a,0x43,0x34,0x3d,0x26,0x2f,0x10,0x19,0x02,0x0b,
0xd7,0xde,0xc5,0xcc,0xf3,0xfa,0xe1,0xe8,0x9f,0x96,0x8d,0x84,0xbb,0xb2,0xa9,0xa0,
0x47,0x4e,0x55,0x5c,0x63,0x6a,0x71,0x78,0x0f,0x06,0x1d,0x14,0x2b,0x22,0x39,0x30,
0x9a,0x93,0x88,0x81,0xbe,0xb7,0xac,0xa5,0xd2,0xdb,0xc0,0xc9,0xf6,0xff,0xe4,0xed,
0x0a,0x03,0x18,0x11,0x2e,0x27,0x3c,0x35,0x42,0x4b,0x50,0x59,0x66,0x6f,0x74,0x7d,
0xa1,0xa8,0xb3,0xba,0x85,0x8c,0x97,0x9e,0xe9,0xe0,0xfb,0xf2,0xcd,0xc4,0xdf,0xd6,
0x31,0x38,0x23,0x2a,0x15,0x1c,0x07,0x0e,0x79,0x70,0x6b,0x62,0x5d,0x54,0x4f,0x46
Multiply by 11 (0xB):
0x00,0x0b,0x16,0x1d,0x2c,0x27,0x3a,0x31,0x58,0x53,0x4e,0x45,0x74,0x7f,0x62,0x69,
0xb0,0xbb,0xa6,0xad,0x9c,0x97,0x8a,0x81,0xe8,0xe3,0xfe,0xf5,0xc4,0xcf,0xd2,0xd9,
0x7b,0x70,0x6d,0x66,0x57,0x5c,0x41,0x4a,0x23,0x28,0x35,0x3e,0x0f,0x04,0x19,0x12,
0xcb,0xc0,0xdd,0xd6,0xe7,0xec,0xf1,0xfa,0x93,0x98,0x85,0x8e,0xbf,0xb4,0xa9,0xa2,
0xf6,0xfd,0xe0,0xeb,0xda,0xd1,0xcc,0xc7,0xae,0xa5,0xb8,0xb3,0x82,0x89,0x94,0x9f,
0x46,0x4d,0x50,0x5b,0x6a,0x61,0x7c,0x77,0x1e,0x15,0x08,0x03,0x32,0x39,0x24,0x2f,
0x8d,0x86,0x9b,0x90,0xa1,0xaa,0xb7,0xbc,0xd5,0xde,0xc3,0xc8,0xf9,0xf2,0xef,0xe4,
0x3d,0x36,0x2b,0x20,0x11,0x1a,0x07,0x0c,0x65,0x6e,0x73,0x78,0x49,0x42,0x5f,0x54,
0xf7,0xfc,0xe1,0xea,0xdb,0xd0,0xcd,0xc6,0xaf,0xa4,0xb9,0xb2,0x83,0x88,0x95,0x9e,
0x47,0x4c,0x51,0x5a,0x6b,0x60,0x7d,0x76,0x1f,0x14,0x09,0x02,0x33,0x38,0x25,0x2e,
0x8c,0x87,0x9a,0x91,0xa0,0xab,0xb6,0xbd,0xd4,0xdf,0xc2,0xc9,0xf8,0xf3,0xee,0xe5,
0x3c,0x37,0x2a,0x21,0x10,0x1b,0x06,0x0d,0x64,0x6f,0x72,0x79,0x48,0x43,0x5e,0x55,
0x01,0x0a,0x17,0x1c,0x2d,0x26,0x3b,0x30,0x59,0x52,0x4f,0x44,0x75,0x7e,0x63,0x68,
0xb1,0xba,0xa7,0xac,0x9d,0x96,0x8b,0x80,0xe9,0xe2,0xff,0xf4,0xc5,0xce,0xd3,0xd8,
0x7a,0x71,0x6c,0x67,0x56,0x5d,0x40,0x4b,0x22,0x29,0x34,0x3f,0x0e,0x05,0x18,0x13,
0xca,0xc1,0xdc,0xd7,0xe6,0xed,0xf0,0xfb,0x92,0x99,0x84,0x8f,0xbe,0xb5,0xa8,0xa3
Multiply by 13 (0xD):
0x00,0x0d,0x1a,0x17,0x34,0x39,0x2e,0x23,0x68,0x65,0x72,0x7f,0x5c,0x51,0x46,0x4b,
0xd0,0xdd,0xca,0xc7,0xe4,0xe9,0xfe,0xf3,0xb8,0xb5,0xa2,0xaf,0x8c,0x81,0x96,0x9b,
0xbb,0xb6,0xa1,0xac,0x8f,0x82,0x95,0x98,0xd3,0xde,0xc9,0xc4,0xe7,0xea,0xfd,0xf0,
0x6b,0x66,0x71,0x7c,0x5f,0x52,0x45,0x48,0x03,0x0e,0x19,0x14,0x37,0x3a,0x2d,0x20,
0x6d,0x60,0x77,0x7a,0x59,0x54,0x43,0x4e,0x05,0x08,0x1f,0x12,0x31,0x3c,0x2b,0x26,
0xbd,0xb0,0xa7,0xaa,0x89,0x84,0x93,0x9e,0xd5,0xd8,0xcf,0xc2,0xe1,0xec,0xfb,0xf6,
0xd6,0xdb,0xcc,0xc1,0xe2,0xef,0xf8,0xf5,0xbe,0xb3,0xa4,0xa9,0x8a,0x87,0x90,0x9d,
0x06,0x0b,0x1c,0x11,0x32,0x3f,0x28,0x25,0x6e,0x63,0x74,0x79,0x5a,0x57,0x40,0x4d,
0xda,0xd7,0xc0,0xcd,0xee,0xe3,0xf4,0xf9,0xb2,0xbf,0xa8,0xa5,0x86,0x8b,0x9c,0x91,
0x0a,0x07,0x10,0x1d,0x3e,0x33,0x24,0x29,0x62,0x6f,0x78,0x75,0x56,0x5b,0x4c,0x41,
0x61,0x6c,0x7b,0x76,0x55,0x58,0x4f,0x42,0x09,0x04,0x13,0x1e,0x3d,0x30,0x27,0x2a,
0xb1,0xbc,0xab,0xa6,0x85,0x88,0x9f,0x92,0xd9,0xd4,0xc3,0xce,0xed,0xe0,0xf7,0xfa,
0xb7,0xba,0xad,0xa0,0x83,0x8e,0x99,0x94,0xdf,0xd2,0xc5,0xc8,0xeb,0xe6,0xf1,0xfc,
0x67,0x6a,0x7d,0x70,0x53,0x5e,0x49,0x44,0x0f,0x02,0x15,0x18,0x3b,0x36,0x21,0x2c,
0x0c,0x01,0x16,0x1b,0x38,0x35,0x22,0x2f,0x64,0x69,0x7e,0x73,0x50,0x5d,0x4a,0x47,
0xdc,0xd1,0xc6,0xcb,0xe8,0xe5,0xf2,0xff,0xb4,0xb9,0xae,0xa3,0x80,0x8d,0x9a,0x97
Multiply by 14 (0xE):
0x00,0x0e,0x1c,0x12,0x38,0x36,0x24,0x2a,0x70,0x7e,0x6c,0x62,0x48,0x46,0x54,0x5a,
0xe0,0xee,0xfc,0xf2,0xd8,0xd6,0xc4,0xca,0x90,0x9e,0x8c,0x82,0xa8,0xa6,0xb4,0xba,
0xdb,0xd5,0xc7,0xc9,0xe3,0xed,0xff,0xf1,0xab,0xa5,0xb7,0xb9,0x93,0x9d,0x8f,0x81,
0x3b,0x35,0x27,0x29,0x03,0x0d,0x1f,0x11,0x4b,0x45,0x57,0x59,0x73,0x7d,0x6f,0x61,
0xad,0xa3,0xb1,0xbf,0x95,0x9b,0x89,0x87,0xdd,0xd3,0xc1,0xcf,0xe5,0xeb,0xf9,0xf7,
0x4d,0x43,0x51,0x5f,0x75,0x7b,0x69,0x67,0x3d,0x33,0x21,0x2f,0x05,0x0b,0x19,0x17,
0x76,0x78,0x6a,0x64,0x4e,0x40,0x52,0x5c,0x06,0x08,0x1a,0x14,0x3e,0x30,0x22,0x2c,
0x96,0x98,0x8a,0x84,0xae,0xa0,0xb2,0xbc,0xe6,0xe8,0xfa,0xf4,0xde,0xd0,0xc2,0xcc,
0x41,0x4f,0x5d,0x53,0x79,0x77,0x65,0x6b,0x31,0x3f,0x2d,0x23,0x09,0x07,0x15,0x1b,
0xa1,0xaf,0xbd,0xb3,0x99,0x97,0x85,0x8b,0xd1,0xdf,0xcd,0xc3,0xe9,0xe7,0xf5,0xfb,
0x9a,0x94,0x86,0x88,0xa2,0xac,0xbe,0xb0,0xea,0xe4,0xf6,0xf8,0xd2,0xdc,0xce,0xc0,
0x7a,0x74,0x66,0x68,0x42,0x4c,0x5e,0x50,0x0a,0x04,0x16,0x18,0x32,0x3c,0x2e,0x20,
0xec,0xe2,0xf0,0xfe,0xd4,0xda,0xc8,0xc6,0x9c,0x92,0x80,0x8e,0xa4,0xaa,0xb8,0xb6,
0x0c,0x02,0x10,0x1e,0x34,0x3a,0x28,0x26,0x7c,0x72,0x60,0x6e,0x44,0x4a,0x58,0x56,
0x37,0x39,0x2b,0x25,0x0f,0x01,0x13,0x1d,0x47,0x49,0x5b,0x55,0x7f,0x71,0x63,0x6d,
0xd7,0xd9,0xcb,0xc5,0xef,0xe1,0xf3,0xfd,0xa7,0xa9,0xbb,0xb5,0x9f,0x91,0x83,0x8d
References
• FIPS PUB 197: the official AES standard (PDF file)
See also
• Advanced Encryption Standard
|
Wikipedia
|
Riley slice
In the mathematical theory of Kleinian groups, the Riley slice of Schottky space is a family of Kleinian groups generated by two parabolic elements. It was studied in detail by Keen & Series (1994) and named after Robert Riley by them. Some subtle errors in their paper were corrected by Komori & Series (1998).
Definition
The Riley slice consists of the complex numbers ρ such that the two matrices
${\begin{pmatrix}1&1\\0&1\\\end{pmatrix}},{\begin{pmatrix}1&0\\\rho &1\\\end{pmatrix}}$
generate Kleinian group G with regular set Ω such that Ω/G is a 4-times punctured sphere.
The Riley slice is the quotient of the Teichmuller space of a 4-times punctured sphere by a group generated by Dehn twists around a curve, and so is topologically an annulus.
See also
• Bers slice
References
• Keen, Linda; Series, Caroline (1994), "The Riley slice of Schottky space", Proceedings of the London Mathematical Society, Third Series, 69 (1): 72–90, doi:10.1112/plms/s3-69.1.72, ISSN 0024-6115, MR 1272421
• Komori, Yohei; Series, Caroline (1998), "The Riley slice revisited", The Epstein birthday schrift, Geom. Topol. Monogr., vol. 1, Geom. Topol. Publ., Coventry, pp. 303–316, arXiv:math/9810194, doi:10.2140/gtm.1998.1.303, MR 1668296
|
Wikipedia
|
Rimhak Ree
Rimhak Ree (alternative spelling: Im-hak Ree, December 18, 1922 – January 9, 2005) was a Korean Canadian mathematician. He contributed in the field of group theory, most notably with the concept of the Ree group in (Ree 1960, 1961).
Rimhak Ree
이임학 (李林學)
Born(1922-12-18)18 December 1922
Hamhung, South Hamgyong, Korea
Died9 January 2005(2005-01-09) (aged 82)
Vancouver, Canada
NationalityKorean, de facto stateless
CitizenshipCanadian
Alma materKeijō Imperial University
University of British Columbia
Known forRee group
SpouseRhoda Ree
Awards
• Korea Science & Technology Hall of Fame (2007)
Scientific career
FieldsMathematics
Group Theory
Institutions
• University of British Columbia
ThesisWitt Algebras (1955)
Doctoral advisorStephen Arthur Jennings
Korean name
Hangul
이임학
Hanja
李林學
Revised RomanizationI Imhak
McCune–ReischauerYi Imhak
Early life
Ree received his early education in Hamhung, South Hamgyong, in what is now North Korea; he attended the Hamhung #1 Public Ordinary School (함흥 제 1공립보통학교), and in 1934 entered the Hamhung Public High School (함흥공립고등보통학교).[1] He went onto Keijō Imperial University, where he studied physics, which was an unusual choice for Koreans at the time. Ree graduated in 1944 with a physics degree; he then went to Fengtian, Manchukuo (today Shenyang, Liaoning in the People's Republic of China) to work for an aircraft company.
Career
After the surrender of Japan in 1945 and the end of Japanese rule in Korea, Ree returned to his home country and in 1947 took up a teaching position in the mathematics department at Seoul National University as an assistant professor. Later that year, in Namdaemun Market, Ree found an issue of the Bulletin of the American Mathematical Society, which proposedly was left by an American soldier. On the Bulletin was the paper 'Note on power series', in which Max Zorn solved a problem about the convergence of certain power series with complex coefficients. In the paper, Zorn posed a question of whether the same result held for power series with real coefficients.[2] Ree solved the problem and sent the solution to Max Zorn. When Zorn received Ree's solution, it was sent to the Bulletin of the American Mathematical Society to be published in 1949 with the title 'On a problem of Max Zorn' and become the first mathematical paper published by a Korean in an international journal.[2]
During the Korean War, he fled south to Busan, and in 1953 he was awarded a Canadian scholarship to allow him to study for a Ph.D. degree at the University of British Columbia in Vancouver, Canada.[3] He completed his dissertation on Witt algebras in 1955. His thesis advisor was Stephen Arthur Jennings. Following the award of his doctorate, Ree was appointed as a lecturer at Montana State University, despite facing several problems regarding his labour permission and nationality.
In mid-1955, Ree received a grant from the National Research Council of Canada and he worked with Jennings on Lie algebras. In 1958, he published a solution to a problem of Paul Erdős regarding a certain class of irrational numbers.[4] Ree's two most renowned papers were written from 1960 to 1961, in which he suggested a Lie type group over a finite field now named after him. In 1962 after being promoted to an assistant professor in mathematics at University of British Columbia, he was granted an academic year which he spent in Yale. He was elected a member of Royal Society of Canada in 1964.
Personal life
Family
Ree had two daughters Erran and Hiran from his first marriage.[5] He later married Rhoda Mah, a doctor and the daughter of John Ming Mah, who owned Northwest Food Products, a manufacturers of noodles. She would go on to work as staff physician for Canadian Pacific Airlines. Rimhak and Rhoda's first son Ronald was followed by another son Robert in December 1971.[6] They also had a third son Richard.[5]
Ree died on January 9, 2005, in Vancouver, Canada.[7]
Statelessness
Around the time Ree received his doctorate, his passport was approaching its expiration date, so he approached the South Korean consulate in San Francisco to extend it, but instead the consular officer confiscated his passport and ordered him to return to South Korea.[8][9] Ree refused the order, which caused him considerable difficulty, but in the end the Canadian government treated him as a de facto stateless person and granted him permanent residency in Canada. Afterwards, he continued to work at the University of British Columbia.[8][2]
Though Ree secured his immigration status in Canada, he continued to encounter difficulties with the South Korean government. Ree's family was divided by the Korean War, with his father, older sister, and other relatives having stayed in their hometown of Hamhung. Hamhung was the site of a munitions factory built during Japanese rule, making the city a frequent target for bombing by the United States Air Force during the Korean War, and Lee did not know if any of his relatives there had survived the war. He visited North Korea using his Canadian passport various times for academic exchanges, but he was not able to travel freely in North Korea and thus had no success in making contact with his relatives; furthermore, his visits to North Korea led South Korea's Park Chung-hee military government to place an entry ban on him. Ree requested help from Erdős, who as an internationally-famous Hungarian citizen faced fewer restrictions on travel or communication in either capitalist or communist countries. Finally, in the 1980s, Erdős was able to make contact with several relatives of Ree's with the help of Hungary's Ministry of Foreign Affairs and the country's embassy in Pyongyang, and sent Ree an envelope containing their letters, photographs, and addresses. Ree was so excited by the news that he forwarded the envelope to his mother and younger sister in South Korea, which reportedly resulted in them being investigated by South Korea's intelligence services. Ree remained banned from South Korea until 1996, when the ban was cancelled as he was invited to the 50th anniversary ceremony of the Korean Mathematical Society.[10][11] According to his colleagues, Rimhak Ree identified his nationality as "Joseon", which is a former name of Korea as well as a current autonym of North Korea.
Publications
• Ree, Rimhak (1960), "A family of simple groups associated with the simple Lie algebra of type (G2)", Bulletin of the American Mathematical Society, 66: 508–510, doi:10.1090/S0002-9904-1960-10523-X, ISSN 0002-9904, MR 0125155
• Ree, Rimhak (1961), "A family of simple groups associated with the simple Lie algebra of type (F4)", Bulletin of the American Mathematical Society, 67: 115–116, doi:10.1090/S0002-9904-1961-10527-2, ISSN 0002-9904, MR 0125155
Notes
1. Ju 2007, p. 2
2. "Rimhak Ree". MacTutor History of Mathematics. Retrieved 3 April 2018.
3. Ju 2007, p. 3
4. Ree, Rimhak. "4773: A class of irrational numbers. Proposed by Paul Erdős". The American Mathematical Monthly. 65 (10): 782. doi:10.2307/2310698. JSTOR 2310698. The following year he also co-authored a paper with Michael Marcus, who had an Erdős number of two via Ralph P. Boas Jr. Marcus, M.; Ree, R (1959). "Diagonals of doubly stochastic matrices". The Quarterly Journal of Mathematics. 10 (1): 296–302. doi:10.1093/qmath/10.1.296.
5. "Rimhak Ree, December 18, 1922 – January 9, 2005". Vancouver Sun. 12 January 2005. Retrieved 27 June 2022.
6. "Season's Greetings". Chinatown News. Vol. 18, no. 2. 1971. p. 56.
7. 주진구 [Ju Jin-gu] (2005). "선구적 수학자 이임학(李林學)선생을 추모하며" [In memory of pioneering mathematician Ree Rimhak]. 과학과 기술 [Science & Technology]. Korean Federation of Science and Technology Societies. 38 (2): 21. ISSN 1599-7340.
8. 주진구 [Ju Jin-gu]; 권경환 [Gwon Gyeong-hwan]; 고영소 [Go Yeong-so]; 이정림 [Yi Jeong-rim] (1996). "원로와의 대담: 세계적 수학권위자 李林學 박사" [Conversations with doyens: world-renowned mathematics authority Dr. Ree Rimhak] (PDF). 과학과 기술 [Science & Technology]. Korean Federation of Science and Technology Societies. 29 (12): 80–82. ISSN 1599-7340.
9. 신동호 [Sin Dong-ho] (11 June 1997). "두종류의'단순군'찾아내 최고 수학자 세계적 영예". The Hankyoreh. p. 13. Retrieved 27 June 2022 – via Naver News Library.
10. 박은하 [Bak Eun-ha] (30 October 2015). "세계적 천재 수학자 이임학을 기억하는 국가의 방식". Kyunghyang Shinmun. Retrieved 30 October 2015.
11. 박근태 [Bak Geun-tae] (21 May 2016). "국가가 버린 세계적 수학자 이임학, '리군이론'으로 수학사에 족적". Korea Economic Daily. p. A19. Retrieved 27 June 2022.
References
• Carol Tretkoff; Marvin Tretkoff (January 1979), "On a theorem of Rimhak Ree about permutations", Journal of Combinatorial Theory, Series A, 26 (1): 84–86, doi:10.1016/0097-3165(79)90056-6
• 주진순 [Ju Jin-sun] (March 2007), "세계적인 수학자 이임학 형을 그리워하며" (PDF), Newsletter of the Korean Mathematical Society, vol. 112, no. 1, pp. 2–4, retrieved 2010-10-08
• "News from the Departments", Canadian Mathematical Bulletin, 6 (3): 465, 1963, retrieved 2018-05-30
External links
• Rimhak Ree at the Mathematics Genealogy Project
Authority control
International
• VIAF
National
• Germany
Academics
• MathSciNet
• Mathematics Genealogy Project
• zbMATH
People
• Deutsche Biographie
|
Wikipedia
|
Computability logic
Computability logic (CoL) is a research program and mathematical framework for redeveloping logic as a systematic formal theory of computability, as opposed to classical logic which is a formal theory of truth. It was introduced and so named by Giorgi Japaridze in 2003.[1]
In classical logic, formulas represent true/false statements. In CoL, formulas represent computational problems. In classical logic, the validity of a formula depends only on its form, not on its meaning. In CoL, validity means being always computable. More generally, classical logic tells us when the truth of a given statement always follows from the truth of a given set of other statements. Similarly, CoL tells us when the computability of a given problem A always follows from the computability of other given problems B1,...,Bn. Moreover, it provides a uniform way to actually construct a solution (algorithm) for such an A from any known solutions of B1,...,Bn.
CoL formulates computational problems in their most general – interactive sense. CoL defines a computational problem as a game played by a machine against its environment. Such a problem is computable if there is a machine that wins the game against every possible behavior of the environment. Such a game-playing machine generalizes the Church-Turing thesis to the interactive level. The classical concept of truth turns out to be a special, zero-interactivity-degree case of computability. This makes classical logic a special fragment of CoL. Thus CoL is a conservative extension of classical logic. Computability logic is more expressive, constructive and computationally meaningful than classical logic. Besides classical logic, independence-friendly (IF) logic and certain proper extensions of linear logic and intuitionistic logic also turn out to be natural fragments of CoL.[2][3] Hence meaningful concepts of "intuitionistic truth", "linear-logic truth" and "IF-logic truth" can be derived from the semantics of CoL.
CoL systematically answers the fundamental question of what can be computed and how; thus CoL has many applications, such as constructive applied theories, knowledge base systems, systems for planning and action. Out of these, only applications in constructive applied theories have been extensively explored so far: a series of CoL-based number theories, termed "clarithmetics", have been constructed[4][5] as computationally and complexity-theoretically meaningful alternatives to the classical-logic-based Peano arithmetic and its variations such as systems of bounded arithmetic.
Traditional proof systems such as natural deduction and sequent calculus are insufficient for axiomatizing nontrivial fragments of CoL. This has necessitated developing alternative, more general and flexible methods of proof, such as cirquent calculus.[6][7]
Language
The full language of CoL extends the language of classical first-order logic. Its logical vocabulary has several sorts of conjunctions, disjunctions, quantifiers, implications, negations and so called recurrence operators. This collection includes all connectives and quantifiers of classical logic. The language also has two sorts of nonlogical atoms: elementary and general. Elementary atoms, which are nothing but the atoms of classical logic, represent elementary problems, i.e., games with no moves that are automatically won by the machine when true and lost when false. General atoms, on the other hand, can be interpreted as any games, elementary or non-elementary. Both semantically and syntactically, classical logic is nothing but the fragment of CoL obtained by forbidding general atoms in its language, and forbidding all operators other than ¬, ∧, ∨, →, ∀, ∃.
Japaridze has repeatedly pointed out that the language of CoL is open-ended, and may undergo further extensions. Due to the expressiveness of this language, advances in CoL, such as constructing axiomatizations or building CoL-based applied theories, have usually been limited to one or another proper fragment of the language.
Semantics
The games underlying the semantics of CoL are called static games. Such games have no turn order; a player can always move while the other players are "thinking". However, static games never punishes a player for "thinking" too long (delaying its own moves), so such games never become contests of speed. All elementary games are automatically static, and so are the games allowed to be interpretations of general atoms.
There are two players in static games: the machine and the environment. The machine can only follow algorithmic strategies, while there are no restrictions on the behavior of the environment. Each run (play) is won by one of these players and lost by the other.
The logical operators of CoL are understood as operations on games. Here we informally survey some of those operations. For simplicity we assume that the domain of discourse is always the set of all natural numbers: {0,1,2,...}.
The operation ¬ of negation ("not") switches the roles of the two players, turning moves and wins by the machine into those by the environment, and vice versa. For instance, if Chess is the game of chess (but with ties ruled out) from the white player's perspective, then ¬Chess is the same game from the black player's perspective.
The parallel conjunction ∧ ("pand") and parallel disjunction ∨ ("por") combine games in a parallel fashion. A run of A∧B or A∨B is a simultaneous play in the two conjuncts. The machine wins A∧B if it wins both of them. The machine wins A∨B if it wins at least one of them. For example, Chess∨¬Chess is a game on two boards, one played white and one black, and where the task of the machine is to win on at least one board. Such a game can be easily won regardless who the adversary is, by copying his moves from one board to the other.
The parallel implication operator → ("pimplication") is defined by A→B = ¬A∨B. The intuitive meaning of this operation is reducing B to A, i.e., solving A as long as the adversary solves B.
The parallel quantifiers ∧ ("pall") and ∨ ("pexists") can be defined by ∧xA(x) = A(0)∧A(1)∧A(2)∧... and ∨xA(x) = A(0)∨A(1)∨A(2)∨.... These are thus simultaneous plays of A(0),A(1),A(2),..., each on a separate board. The machine wins ∧xA(x) if it wins all of these games, and ∨xA(x) if it wins some.
The blind quantifiers ∀ ("blall") and ∃ ("blexists"), on the other hand, generate single-board games. A run of ∀xA(x) or ∃xA(x) is a single run of A. The machine wins ∀xA(x) (resp. ∃xA(x)) if such a run is a won run of A(x) for all (resp. at least one) possible values of x, and wins ∃xA(x) if this is true for at least one.
All of the operators characterized so far behave exactly like their classical counterparts when they are applied to elementary (moveless) games, and validate the same principles. This is why CoL uses the same symbols for those operators as classical logic does. When such operators are applied to non-elementary games, however, their behavior is no longer classical. So, for instance, if p is an elementary atom and P a general atom, p→p∧p is valid while P→P∧P is not. The principle of the excluded middle P∨¬P, however, remains valid. The same principle is invalid with all three other sorts (choice, sequential and toggling) of disjunction.
The choice disjunction ⊔ ("chor") of games A and B, written A⊔B, is a game where, in order to win, the machine has to choose one of the two disjuncts and then win in the chosen component. The sequential disjunction ("sor") AᐁB starts as A; it also ends as A unless the machine makes a "switch" move, in which case A is abandoned and the game restarts and continues as B. In the toggling disjunction ("tor") A⩛B, the machine may switch between A and B any finite number of times. Each disjunction operator has its dual conjunction, obtained by interchanging the roles of the two players. The corresponding quantifiers can further be defined as infinite conjunctions or disjunctions in the same way as in the case of the parallel quantifiers. Each sort disjunction also induces a corresponding implication operation the same way as this was the case with the parallel implication →. For instance, the choice implication ("chimplication") A⊐B is defined as ¬A⊔B.
The parallel recurrence ("precurrence") of A can be defined as the infinite parallel conjunction A∧A∧A∧... The sequential ("srecurrence") and toggling ("trecurrence") sorts of recurrences can be defined similarly.
The corecurrence operators can be defined as infinite disjunctions. Branching recurrence ("brecurrence") ⫰, which is the strongest sort of recurrence, does not have a corresponding conjunction. ⫰A is a game that starts and proceeds as A. At any time, however, the environment is allowed to make a "replicative" move, which creates two copies of the then-current position of A, thus splitting the play into two parallel threads with a common past but possibly different future developments. In the same fashion, the environment can further replicate any of positions of any thread, thus creating more and more threads of A. Those threads are played in parallel, and the machine needs to win A in all threads to be the winner in ⫰A. Branching corecurrence ("cobrecurrence") ⫯ is defined symmetrically by interchanging "machine" and "environment".
Each sort of recurrence further induces a corresponding weak version of implication and weak version of negation. The former is said to be a rimplication, and the latter a refutation. The branching rimplication ("brimplication") A⟜B is nothing but ⫰A→B, and the branching refutation ("brefutation") of A is A⟜⊥, where ⊥ is the always-lost elementary game. Similarly for all other sorts of rimplication and refutation.
As a problem specification tool
The language of CoL offers a systematic way to specify an infinite variety of computational problems, with or without names established in the literature. Below are some examples.
Let f be a unary function. The problem of computing f will be written as ⊓x⊔y(y=f(x)). According to the semantics of CoL, this is a game where the first move ("input") is by the environment, which should choose a value m for x. Intuitively, this amounts to asking the machine to tell the value of f(m). The game continues as ⊔y(y=f(m)). Now the machine is expected to make a move ("output"), which should be choosing a value n for y. This amounts to saying that n is the value of f(m). The game is now brought down to the elementary n=f(m), which is won by the machine if and only if n is indeed the value of f(m).
Let p be a unary predicate. Then ⊓x(p(x)⊔¬p(x)) expresses the problem of deciding p, ⊓x(p(x)&ᐁ¬p(x)) expresses the problem of semideciding p, and ⊓x(p(x)⩛¬p(x)) the problem of recursively approximating p.
Let p and q be two unary predicates. Then ⊓x(p(x)⊔¬p(x))⟜⊓x(q(x)⊔¬q(x)) expresses the problem of Turing-reducing q to p (in the sense that q is Turing reducible to p if and only if the interactive problem ⊓x(p(x)⊔¬p(x))⟜⊓x(q(x)⊔¬q(x)) is computable). ⊓x(p(x)⊔¬p(x))→⊓x(q(x)⊔¬q(x)) does the same but for the stronger version of Turing reduction where the oracle for p can be queried only once. ⊓x⊔y(q(x)↔p(y)) does the same for the problem of many-one reducing q to p. With more complex expressions one can capture all kinds of nameless yet potentially meaningful relations and operations on computational problems, such as, for instance, "Turing-reducing the problem of semideciding r to the problem of many-one reducing q to p". Imposing time or space restrictions on the work of the machine, one further gets complexity-theoretic counterparts of such relations and operations.
As a problem solving tool
The known deductive systems for various fragments of CoL share the property that a solution (algorithm) can be automatically extracted from a proof of a problem in the system. This property is further inherited by all applied theories based on those systems. So, in order to find a solution for a given problem, it is sufficient to express it in the language of CoL and then find a proof of that expression. Another way to look at this phenomenon is to think of a formula G of CoL as program specification (goal). Then a proof of G is – more precisely, translates into – a program meeting that specification. There is no need to verify that the specification is met, because the proof itself is, in fact, such a verification.
Examples of CoL-based applied theories are the so-called clarithmetics. These are number theories based on CoL in the same sense as Peano arithmetic PA is based on classical logic. Such a system is usually a conservative extension of PA. It typically includes all Peano axioms, and adds to them one or two extra-Peano axioms such as ⊓x⊔y(y=x') expressing the computability of the successor function. Typically it also has one or two non-logical rules of inference, such as constructive versions of induction or comprehension. Through routine variations in such rules one can obtain sound and complete systems characterizing one or another interactive computational complexity class C. This is in the sense that a problem belongs to C if and only if it has a proof in the theory. So, such a theory can be used for finding not merely algorithmic solutions, but also efficient ones on demand, such as solutions that run in polynomial time or logarithmic space. It should be pointed out that all clarithmetical theories share the same logical postulates, and only their non-logical postulates vary depending on the target complexity class. Their notable distinguishing feature from other approaches with similar aspirations (such as bounded arithmetic) is that they extend rather than weaken PA, preserving the full deductive power and convenience of the latter.
See also
• Game semantics
• Interactive computation
• Logic
• Logics for computability
References
1. G. Japaridze, Introduction to computability logic. Annals of Pure and Applied Logic 123 (2003), pages 1–99. doi:10.1016/S0168-0072(03)00023-X
2. G. Japaridze, In the beginning was game semantics?. Games: Unifying Logic, Language and Philosophy. O. Majer, A.-V. Pietarinen and T. Tulenheimo, eds. Springer 2009, pp. 249–350. doi:10.1007/978-1-4020-9374-6_11 Prepublication
3. G. Japaridze, The intuitionistic fragment of computability logic at the propositional level. Annals of Pure and Applied Logic 147 (2007), pages 187–227. doi:10.1016/j.apal.2007.05.001
4. G. Japaridze, Introduction to clarithmetic I. Information and Computation 209 (2011), pp. 1312–1354. doi:10.1016/j.ic.2011.07.002 Prepublication
5. G. Japaridze, Build your own clarithmetic I: Setup and completeness. Logical Methods is Computer Science 12 (2016), Issue 3, paper 8, pp. 1–59.
6. G. Japaridze, Introduction to cirquent calculus and abstract resource semantics. Journal of Logic and Computation 16 (2006), pages 489–532. doi:10.1093/logcom/exl005 Prepublication
7. G. Japaridze, The taming of recurrences in computability logic through cirquent calculus, Part I. Archive for Mathematical Logic 52 (2013), pp. 173–212. doi:10.1007/s00153-012-0313-8 Prepublication
External links
• Computability Logic Homepage Comprehensive survey of the subject.
• Giorgi Japaridze
• Game Semantics or Linear Logic?
• Lecture Course on Computability Logic
• On abstract resource semantics and computabilty logic Video lecture by N.Vereshchagin.
• A Survey of Computability Logic (PDF) Downloadable equivalent of the above homepage.
Logic
• Outline
• History
Major fields
• Computer science
• Formal semantics (natural language)
• Inference
• Philosophy of logic
• Proof
• Semantics of logic
• Syntax
Logics
• Classical
• Informal
• Critical thinking
• Reason
• Mathematical
• Non-classical
• Philosophical
Theories
• Argumentation
• Metalogic
• Metamathematics
• Set
Foundations
• Abduction
• Analytic and synthetic propositions
• Contradiction
• Paradox
• Antinomy
• Deduction
• Deductive closure
• Definition
• Description
• Entailment
• Linguistic
• Form
• Induction
• Logical truth
• Name
• Necessity and sufficiency
• Premise
• Probability
• Reference
• Statement
• Substitution
• Truth
• Validity
Lists
topics
• Mathematical logic
• Boolean algebra
• Set theory
other
• Logicians
• Rules of inference
• Paradoxes
• Fallacies
• Logic symbols
• Philosophy portal
• Category
• WikiProject (talk)
• changes
|
Wikipedia
|
Rinat Kedem
Rinat Kedem (born 5 December 1965) is an American mathematician and mathematical physicist.
Kedem graduated in 1988 with BA in physics from Macalester College.[1] She received her PhD in physics in 1993 from Stony Brook University (the State University of New York at Stony Brook) with thesis advisor Barry M. McCoy.[2] She was a postdoc from 1993 to 1995 at Kyoto University's Research Institute for Mathematical Sciences (RIMS), from 1995 to 1996 at the University of Melbourne, and from 1996 to 1997 at the University of California, Berkeley. At the University of Massachusetts Amherst, she was an assistant professor of mathematics from 1997 to 2001. In the mathematics department of the University of Illinois at Urbana-Champaign, she was from 2001 to 2006 an assistant professor and from 2006 to 2012 an associate professor and is since 2012 a full professor.[1]
Kedem's research deals with mathematical physics, Lie algebras, integrable models, and cluster algebras.[1] In 2014 she was an invited speaker with talk Fermionic spectra in integrable systems at the International Congress of Mathematicians in Seoul.[3] She was a plenary speaker in 2012 at the 24th International Conference on Formal Power Series and Algebraic Combinatorics in Nagoya and in 2019 at the 11th International Symposium on Quantum Theory and Symmetries (QTS2019) in Montreal.[1] For the academic year 2019–2020 she was awarded a Simons Fellowship.[4]
Selected publications
• Kedem, R.; Klassen, T.R.; McCoy, B.M.; Melzer, E. (1993). "Fermionic quasi-particle representations for characters of (($G$(1))1$\times $($G$(1))1)/($G$(1))2". Physics Letters B. 304 (3–4): 263–270. arXiv:hep-th/9211102. doi:10.1016/0370-2693(93)90292-P. S2CID 2243256.
• Kedem, R.; Klassen, T.R.; McCoy, B.M.; Melzer, E. (1993). "Fermionic sum representations for conformal field theory characters". Physics Letters B. 307 (1–2): 68–76. arXiv:hep-th/9301046. Bibcode:1993PhLB..307...68K. doi:10.1016/0370-2693(93)90194-M. S2CID 99529.
• Jimbo, Michio; Kedem, Rinat; Kojima, Takeo; Konno, Hitoshi; Miwa, Tetsuji (1995). "XXZ chain with a boundary". Nuclear Physics B. 441 (3): 437–470. arXiv:hep-th/9411112. Bibcode:1995NuPhB.441..437J. doi:10.1016/0550-3213(95)00062-W. S2CID 118907908.
• Jimbo, Michio; Kedem, Rinat; Konno, Hitoshi; Miwa, Tetsuji; Weston, Robert (1995). "Difference equations in spin chains with a boundary". Nuclear Physics B. 448 (3): 429–456. arXiv:hep-th/9502060. Bibcode:1995NuPhB.448..429J. doi:10.1016/0550-3213(95)00218-H. S2CID 2947742.
• Kedem, Rinat (2008). "Q-systems as cluster algebras". Journal of Physics A: Mathematical and Theoretical. 41 (19): 194011. arXiv:0712.2695. Bibcode:2008JPhA...41s4011K. doi:10.1088/1751-8113/41/19/194011. S2CID 115158426.
• Di Francesco, Philippe; Kedem, Rinat (2009). "Q-systems as Cluster Algebras II: Cartan Matrix of Finite Type and the Polynomial Property". Letters in Mathematical Physics. 89 (3): 183–216. arXiv:0803.0362. Bibcode:2009LMaPh..89..183D. doi:10.1007/s11005-009-0354-z. S2CID 16365405.
• Di Francesco, Philippe; Kedem, Rinat (2010). "Q-Systems, Heaps, Paths and Cluster Positivity". Communications in Mathematical Physics. 293 (3): 727–802. arXiv:0811.3027. Bibcode:2010CMaPh.293..727F. doi:10.1007/s00220-009-0947-5. S2CID 115154922.
References
1. "Curriculum Vitae - Rinat Kedem". Department of Mathematics, University of Illinois.
2. Rinat Kedem at the Mathematics Genealogy Project
3. Kedem, Rinat (2014). "Fermionic spectra in integrable systems". arXiv:1405.5585 [math-ph].
4. "2019 Simons Fellows in Mathematics and Theoretical Physics Announced". Simons Foundation. 15 March 2019.
External links
• "Rinat Kedem, Professor". Mathematics Department, UIUC.
• "ICM2014 VideoSeries IL11.8: Rinat Kedem of Aug19Tue". YouTube. Seoul ICM VOD. 20 August 2014.
• "Rinat Kedem: From Q-systems to quantum affine algebras and beyond". YouTube. Centre International de Rencontres Mathématiques. 4 April 2018.
Authority control: Academics
• Google Scholar
• MathSciNet
• Mathematics Genealogy Project
• ORCID
• zbMATH
|
Wikipedia
|
Ring class field
In mathematics, a ring class field is the abelian extension of an algebraic number field K associated by class field theory to the ring class group of some order O of the ring of integers of K.[1]
Properties
Let K be an algebraic number field.
• The ring class field for the maximal order O = OK is the Hilbert class field H.
Let L be the ring class field for the order Z[√−n] in the number field K = Q(√−n).
• If p is an odd prime not dividing n, then p splits completely in L if and only if p splits completely in K.
• L = K(a) for a an algebraic integer with minimal polynomial over Q of degree h(−4n), the class number of an order with discriminant −4n.
• If O is an order and a is a proper fractional O-ideal (i.e. {x ϵ K * : xa ⊂ a} = O), write j(a) for the j-invariant of the associated elliptic curve. Then K(j(a)) is the ring class field of O and j(a) is an algebraic integer.
References
1. Frey, Gerhard; Lange, Tanja (2006), "Varieties over special fields", Handbook of elliptic and hyperelliptic curve cryptography, Discrete Math. Appl. (Boca Raton), Chapman & Hall/CRC, Boca Raton, Florida, pp. 87–113, MR 2162721. See in particular p. 99.
External links
• Ring class fields. Archived 27 September 2018 at the Wayback Machine
|
Wikipedia
|
Ring learning with errors
In post-quantum cryptography, ring learning with errors (RLWE) is a computational problem which serves as the foundation of new cryptographic algorithms, such as NewHope, designed to protect against cryptanalysis by quantum computers and also to provide the basis for homomorphic encryption. Public-key cryptography relies on construction of mathematical problems that are believed to be hard to solve if no further information is available, but are easy to solve if some information used in the problem construction is known. Some problems of this sort that are currently used in cryptography are at risk of attack if sufficiently large quantum computers can ever be built, so resistant problems are sought. Homomorphic encryption is a form of encryption that allows computation on ciphertext, such as arithmetic on numeric values stored in an encrypted database.
RLWE is more properly called learning with errors over rings and is simply the larger learning with errors (LWE) problem specialized to polynomial rings over finite fields.[1] Because of the presumed difficulty of solving the RLWE problem even on a quantum computer, RLWE based cryptography may form the fundamental base for public-key cryptography in the future just as the integer factorization and discrete logarithm problem have served as the base for public key cryptography since the early 1980s.[2] An important feature of basing cryptography on the ring learning with errors problem is the fact that the solution to the RLWE problem can be used to solve a version of the shortest vector problem (SVP) in a lattice (a polynomial-time reduction from this SVP problem to the RLWE problem has been presented[1]).
Background
The security of modern cryptography, in particular public-key cryptography, is based on the assumed intractability of solving certain computational problems if the size of the problem is large enough and the instance of the problem to be solved is chosen randomly. The classic example that has been used since the 1970s is the integer factorization problem. It is believed that it is computationally intractable to factor the product of two prime numbers if those prime numbers are large enough and chosen at random.[3] As of 2015 research has led to the factorization of the product of two 384-bit primes but not the product of two 512-bit primes. Integer factorization forms the basis of the widely used RSA cryptographic algorithm.
The ring learning with errors (RLWE) problem is built on the arithmetic of polynomials with coefficients from a finite field.[1] A typical polynomial $ a(x)$ is expressed as:
$a(x)=a_{0}+a_{1}x+a_{2}x^{2}+\ldots +a_{n-2}x^{n-2}+a_{n-1}x^{n-1}$
Polynomials can be added and multiplied in the usual fashion. In the RLWE context the coefficients of the polynomials and all operations involving those coefficients will be done in a finite field, typically the field $ \mathbf {Z} /q\mathbf {Z} =\mathbf {F} _{q}$ for a prime integer $ q$. The set of polynomials over a finite field with the operations of addition and multiplication forms an infinite polynomial ring ($ \mathbf {F} _{q}[x]$). The RLWE context works with a finite quotient ring of this infinite ring. The quotient ring is typically the finite quotient (factor) ring formed by reducing all of the polynomials in $ \mathbf {F} _{q}[x]$ modulo an irreducible polynomial $ \Phi (x)$. This finite quotient ring can be written as $\mathbf {F} _{q}[x]/\Phi (x)$ though many authors write $\mathbf {Z} _{q}[x]/\Phi (x)$ .[1]
If the degree of the polynomial $\Phi (x)$ is $ n$, the quotient ring becomes the ring of polynomials of degree less than $n$ modulo $\Phi (x)$ with coefficients from $F_{q}$. The values $ n$, $ q$, together with the polynomial $\Phi (x)$ partially define the mathematical context for the RLWE problem.
Another concept necessary for the RLWE problem is the idea of "small" polynomials with respect to some norm. The typical norm used in the RLWE problem is known as the infinity norm (also called the uniform norm). The infinity norm of a polynomial is simply the largest coefficient of the polynomial when these coefficients are viewed as integers. Hence, $||a(x)||_{\infty }=b$ states that the infinity norm of the polynomial $a(x)$ is $b$. Thus $b$ is the largest coefficient of $a(x)$.
The final concept necessary to understand the RLWE problem is the generation of random polynomials in $\mathbf {F} _{q}[x]/\Phi (x)$ and the generation of "small" polynomials . A random polynomial is easily generated by simply randomly sampling the $n$ coefficients of the polynomial from $\mathbf {F} _{q}$, where $\mathbf {F} _{q}$ is typically represented as the set $\{-(q-1)/2,...,-1,0,1,...,(q-1)/2\}$.
Randomly generating a "small" polynomial is done by generating the coefficients of the polynomial from $\mathbf {F} _{q}$ in a way that either guarantees or makes very likely small coefficients. When $q$ is a prime integer, there are two common ways to do this:
1. Using Uniform Sampling – The coefficients of the small polynomial are uniformly sampled from a set of small coefficients. Let $ b$ be an integer that is much less than $ q$. If we randomly choose coefficients from the set: $ \{-b,-b+1,-b+2,\ldots ,-2,-1,0,1,2,\ldots ,b-2,b-1,b\}$ the polynomial will be small with respect to the bound ($ b$).
2. Using discrete Gaussian sampling – For an odd value for $ q$, the coefficients of the polynomial are randomly chosen by sampling from the set $ \{-(q-1)/2,\ldots ,(q-1)/2\}$ according to a discrete Gaussian distribution with mean $0$ and distribution parameter $ \sigma $. The references describe in full detail how this can be accomplished. It is more complicated than uniform sampling but it allows for a proof of security of the algorithm. The paper "Sampling from Discrete Gaussians for Lattice-Based Cryptography on a Constrained Device" by Dwarakanath and Galbraith provide an overview of this problem.[4]
The RLWE Problem
The RLWE problem can be stated in two different ways: a "search" version and a "decision" version. Both begin with the same construction. Let
• $a_{i}(x)$ be a set of random but known polynomials from $\mathbf {F} _{q}[x]/\Phi (x)$ with coefficients from all of $\mathbf {F} _{q}$.
• $e_{i}(x)$ be a set of small random and unknown polynomials relative to a bound $b$ in the ring $\mathbf {F} _{q}[x]/\Phi (x)$.
• $s(x)$ be a small unknown polynomial relative to a bound $b$ in the ring $\mathbf {F} _{q}[x]/\Phi (x)$.
• $b_{i}(x)=(a_{i}(x)\cdot s(x))+e_{i}(x)$.
The Search version entails finding the unknown polynomial $s(x)$ given the list of polynomial pairs $(a_{i}(x),b_{i}(x))$.
The Decision version of the problem can be stated as follows. Given a list of polynomial pairs $(a_{i}(x),b_{i}(x))$, determine whether the $b_{i}(x)$ polynomials were constructed as $b_{i}(x)=(a_{i}(x)\cdot s(x))+e_{i}(x)$ or were generated randomly from $\mathbf {F} _{q}[x]/\Phi (x)$ with coefficients from all of $\mathbf {F} _{q}$.
The difficulty of this problem is parameterized by the choice of the quotient polynomial ($\Phi (x)$), its degree ($n$), the field ($\mathbf {F} _{q}$), and the smallness bound ($b$). In many RLWE based public key algorithms the private key will be a pair of small polynomials $s(x)$ and $e(x)$. The corresponding public key will be a pair of polynomials $a(x)$, selected randomly from $\mathbf {F} _{q}[x]/\Phi (x)$, and the polynomial $t(x)=(a(x)\cdot s(x))+e(x)$. Given $a(x)$ and $t(x)$, it should be computationally infeasible to recover the polynomial $s(x)$.
Security Reduction
In cases where the polynomial $\Phi (x)$ is a cyclotomic polynomial, the difficulty of solving the search version of RLWE problem is equivalent to finding a short vector (but not necessarily the shortest vector) in an ideal lattice formed from elements of $\mathbf {Z} [x]/\Phi (x)$ represented as integer vectors.[1] This problem is commonly known as the Approximate Shortest Vector Problem (α-SVP) and it is the problem of finding a vector shorter than α times the shortest vector. The authors of the proof for this equivalence write:
"... we give a quantum reduction from approximate SVP (in the worst case) on ideal lattices in $\mathbf {R} $ to the search version of ring-LWE, where the goal is to recover the secret $s\in \mathbf {R} _{q}$ (with high probability, for any $s$) from arbitrarily many noisy products."[1]
In that quote, The ring $\mathbf {R} $ is $\mathbf {Z} [x]/\Phi (x)$ and the ring $\mathbf {R} _{q}$ is $\mathbf {F} _{q}[x]/\Phi (x)$.
The α-SVP in regular lattices is known to be NP-hard due to work by Daniele Micciancio in 2001, although not for values of α required for a reduction to general learning with errors problem.[5] However, there is not yet a proof to show that the difficulty of the α-SVP for ideal lattices is equivalent to the average α-SVP. Rather we have a proof that if there are any α-SVP instances that are hard to solve in ideal lattices then the RLWE Problem will be hard in random instances.[1]
Regarding the difficulty of Shortest Vector Problems in Ideal Lattices, researcher Michael Schneider writes, "So far there is no SVP algorithm making use of the special structure of ideal lattices. It is widely believed that solving SVP (and all other lattice problems) in ideal lattices is as hard as in regular lattices."[6] The difficulty of these problems on regular lattices is provably NP-hard.[5] There are, however, a minority of researchers who do not believe that ideal lattices share the same security properties as regular lattices.[7]
Peikert believes that these security equivalences make the RLWE problem a good basis for future cryptography. He writes: "There is a mathematical proof that the only way to break the cryptosystem (within some formal attack model) on its random instances is by being able to solve the underlying lattice problem in the worst case" (emphasis in the original).[8]
RLWE Cryptography
A major advantage that RLWE based cryptography has over the original learning with errors (LWE) based cryptography is found in the size of the public and private keys. RLWE keys are roughly the square root of keys in LWE.[1] For 128 bits of security an RLWE cryptographic algorithm would use public keys around 7000 bits in length.[9] The corresponding LWE scheme would require public keys of 49 million bits for the same level of security.[1] On the other hand, RLWE keys are larger than the keys sizes for currently used public key algorithms like RSA and Elliptic Curve Diffie-Hellman which require public key sizes of 3072 bits and 256 bits, respectively, to achieve a 128-bit level of security. From a computational standpoint, however, RLWE algorithms have been shown to be the equal of or better than existing public key systems.[10]
Three groups of RLWE cryptographic algorithms exist:
Ring learning with errors key exchanges (RLWE-KEX)
The fundamental idea of using LWE and Ring LWE for key exchange was proposed and filed at the University of Cincinnati in 2011 by Jintai Ding. The basic idea comes from the associativity of matrix multiplications, and the errors are used to provide the security. The paper[11] appeared in 2012 after a provisional patent application was filed in 2012.
In 2014, Peikert[12] presented a key transport scheme following the same basic idea of Ding's, where the new idea of sending additional 1 bit signal for rounding in Ding's construction is also utilized. An RLWE version of the classic MQV variant of a Diffie-Hellman key exchange was later published by Zhang et al.[13] The security of both key exchanges is directly related to the problem of finding approximate short vectors in an ideal lattice.
Ring learning with errors signature (RLWE-SIG)
A RLWE version of the classic Feige–Fiat–Shamir Identification protocol was created and converted to a digital signature in 2011 by Lyubashevsky.[14] The details of this signature were extended in 2012 by Gunesyu, Lyubashevsky, and Popplemann in 2012 and published in their paper "Practical Lattice Based Cryptography – A Signature Scheme for Embedded Systems."[15] These papers laid the groundwork for a variety of recent signature algorithms some based directly on the ring learning with errors problem and some which are not tied to the same hard RLWE problems.[16]
Ring learning with errors homomorphic encryption (RLWE-HOM)
Main article: Homomorphic encryption
The purpose of homomorphic encryption is to allow the computations on sensitive data to occur on computing devices that should not be trusted with the data. These computing devices are allowed to process the ciphertext which is output from a homomorphic encryption. In 2011, Brakersky and Vaikuntanathan, published "Fully Homomorphic Encryption from Ring-LWE and Security for Key Dependent Messages" which builds a homomorphic encryption scheme directly on the RLWE problem.[17]
References
1. Lyubashevsky, Vadim; Peikert, Chris; Regev, Oded (2012). "On Ideal Lattices and Learning with Errors Over Rings". Cryptology ePrint Archive.
2. Peikert, Chris (2014). "Lattice Cryptography for the Internet". In Mosca, Michele (ed.). Post-Quantum Cryptography. Lecture Notes in Computer Science. Vol. 8772. Springer International Publishing. pp. 197–219. CiteSeerX 10.1.1.800.4743. doi:10.1007/978-3-319-11659-4_12. ISBN 978-3-319-11658-7. S2CID 8123895.
3. Shor, Peter (20 November 1994). Algorithms for quantum computation: discrete logarithms and factoring. 35th Annual Symposium on Foundations of Computer Science. Santa Fe: IEEE. doi:10.1109/SFCS.1994.365700. ISBN 0-8186-6580-7. This paper gives Las Vegas algorithms for finding discrete logarithms and factoring integers on a quantum computer that take a number of steps which is polynomial in the input size, e.g., the number of digits of the integer to be factored. These two problems are generally considered hard on a classical computer and have been used as the basis of several proposed cryptosystems.
4. Dwarakanath, Nagarjun C.; Galbraith, Steven D. (2014-03-18). "Sampling from discrete Gaussians for lattice-based cryptography on a constrained device". Applicable Algebra in Engineering, Communication and Computing. 25 (3): 159–180. doi:10.1007/s00200-014-0218-3. ISSN 0938-1279. S2CID 13718364.
5. Micciancio, D. (January 1, 2001). "The Shortest Vector in a Lattice is Hard to Approximate to within Some Constant". SIAM Journal on Computing. 30 (6): 2008–2035. CiteSeerX 10.1.1.93.6646. doi:10.1137/S0097539700373039. ISSN 0097-5397.
6. Schneider, Michael (2011). "Sieving for Shortest Vectors in Ideal Lattices". Cryptology ePrint Archive.
7. "cr.yp.to: 2014.02.13: A subfield-logarithm attack against ideal lattices". blog.cr.yp.to. Retrieved 2015-07-03.
8. "What does GCHQ's "cautionary tale" mean for lattice cryptography?". www.eecs.umich.edu. Archived from the original on 2016-03-17. Retrieved 2016-01-05.
9. Singh, Vikram (2015). "A Practical Key Exchange for the Internet using Lattice Cryptography". Cryptology ePrint Archive.
10. Verbauwhede, Ruan de Clercq, Sujoy Sinha Roy, Frederik Vercauteren, Ingrid (2014). "Efficient Software Implementation of Ring-LWE Encryption". Cryptology ePrint Archive.{{cite journal}}: CS1 maint: multiple names: authors list (link)
11. Ding, Jintai; Xie, Xiang; Lin, Xiaodong (2012-01-01). "A Simple Provably Secure Key Exchange Scheme Based on the Learning with Errors Problem". Cryptology ePrint Archive.
12. Peikert, Chris (2014-01-01). "Lattice Cryptography for the Internet". Cryptology ePrint Archive.
13. Zhang, Jiang; Zhang, Zhenfeng; Ding, Jintai; Snook, Michael; Dagdelen, Özgür (2014). "Authenticated Key Exchange from Ideal Lattices". Cryptology ePrint Archive.
14. Lyubashevsky, Vadim (2011). "Lattice Signatures Without Trapdoors". Cryptology ePrint Archive.
15. Güneysu, Tim; Lyubashevsky, Vadim; Pöppelmann, Thomas (2012). Prouff, Emmanuel; Schaumont, Patrick (eds.). Practical Lattice-Based Cryptography: A Signature Scheme for Embedded Systems. Lecture Notes in Computer Science. Springer Berlin Heidelberg. pp. 530–547. doi:10.1007/978-3-642-33027-8_31. ISBN 978-3-642-33026-1.
16. "BLISS Signature Scheme". bliss.di.ens.fr. Retrieved 2015-07-04.
17. Brakerski, Zvika; Vaikuntanathan, Vinod (2011). Rogaway, Phillip (ed.). Fully Homomorphic Encryption from Ring-LWE and Security for Key Dependent Messages. Lecture Notes in Computer Science. Springer Berlin Heidelberg. pp. 505–524. doi:10.1007/978-3-642-22792-9_29. ISBN 978-3-642-22791-2.
Computational hardness assumptions
Number theoretic
• Integer factorization
• Phi-hiding
• RSA problem
• Strong RSA
• Quadratic residuosity
• Decisional composite residuosity
• Higher residuosity
Group theoretic
• Discrete logarithm
• Diffie-Hellman
• Decisional Diffie–Hellman
• Computational Diffie–Hellman
Pairings
• External Diffie–Hellman
• Sub-group hiding
• Decision linear
Lattices
• Shortest vector problem (gap)
• Closest vector problem (gap)
• Learning with errors
• Ring learning with errors
• Short integer solution
Non-cryptographic
• Exponential time hypothesis
• Unique games conjecture
• Planted clique conjecture
|
Wikipedia
|
Ring lemma
In the geometry of circle packings in the Euclidean plane, the ring lemma gives a lower bound on the sizes of adjacent circles in a circle packing.[1]
Statement
The lemma states: Let $n$ be any integer greater than or equal to three. Suppose that the unit circle is surrounded by a ring of $n$ interior-disjoint circles, all tangent to it, with consecutive circles in the ring tangent to each other. Then the minimum radius of any circle in the ring is at least the unit fraction
${\frac {1}{F_{2n-3}-1}}$
where $F_{i}$ is the $i$th Fibonacci number.[1][2]
The sequence of minimum radii, from $n=3$, begins
$ \displaystyle 1,{\frac {1}{4}},{\frac {1}{12}},{\frac {1}{33}},{\frac {1}{88}},{\frac {1}{232}},\dots $ (sequence A027941 in the OEIS)
Generalizations to three-dimensional space are also known.[3]
Construction
An infinite sequence of circles can be constructed, containing rings for each $n$ that exactly meet the bound of the ring lemma, showing that it is tight. The construction allows halfplanes to be considered as degenerate circles with infinite radius, and includes additional tangencies between the circles beyond those required in the statement of the lemma. It begins by sandwiching the unit circle between two parallel halfplanes; in the geometry of circles, these are considered to be tangent to each other at the point at infinity. Each successive circle after these first two is tangent to the central unit circle and to the two most recently added circles; see the illustration for the first six circles (including the two halfplanes) constructed in this way. The first $n$ circles of this construction form a ring, whose minimum radius can be calculated by Descartes' theorem to be the same as the radius specified in the ring lemma. This construction can be perturbed to a ring of $n$ finite circles, without additional tangencies, whose minimum radius is arbitrarily close to this bound.[4]
History
A version of the ring lemma with a weaker bound was first proven by Burton Rodin and Dennis Sullivan as part of their proof of William Thurston's conjecture that circle packings can be used to approximate conformal maps.[5] Lowell Hansen gave a recurrence relation for the tightest possible lower bound,[6] and Dov Aharonov found a closed-form expression for the same bound.[2]
Applications
Beyond its original application to conformal mapping,[5] the circle packing theorem and the ring lemma play key roles in a proof by Keszegh, Pach, and Pálvölgyi that planar graphs of bounded degree can be drawn with bounded slope number.[7]
References
1. Stephenson, Kenneth (2005), Introduction to Circle Packing: The Theory of Discrete Analytic Functions, Cambridge University Press, ISBN 978-0-521-82356-2, MR 2131318; see especially Lemma 8.2 (Ring Lemma), pp. 73–74, and Appendix B, The Ring Lemma, pp. 318–321.
2. Aharonov, Dov (1997), "The sharp constant in the ring lemma", Complex Variables, 33 (1–4): 27–31, doi:10.1080/17476939708815009, MR 1624890
3. Vasilis, Jonatan (2011), "The ring lemma in three dimensions", Geometriae Dedicata, 152: 51–62, doi:10.1007/s10711-010-9545-0, MR 2795235, S2CID 120113578
4. Aharonov, D.; Stephenson, K. (1997), "Geometric sequences of discs in the Apollonian packing", Algebra i Analiz, 9 (3): 104–140, MR 1466797
5. Rodin, Burt; Sullivan, Dennis (1987), "The convergence of circle packings to the Riemann mapping", Journal of Differential Geometry, 26 (2): 349–360, doi:10.4310/jdg/1214441375, MR 0906396
6. Hansen, Lowell J. (1988), "On the Rodin and Sullivan ring lemma", Complex Variables, 10 (1): 23–30, doi:10.1080/17476938808814284, MR 0946096
7. Keszegh, Balázs; Pach, János; Pálvölgyi, Dömötör (2011), "Drawing planar graphs of bounded degree with few slopes", in Brandes, Ulrik; Cornelsen, Sabine (eds.), Graph Drawing: 18th International Symposium, GD 2010, Konstanz, Germany, September 21-24, 2010, Revised Selected Papers, Lecture Notes in Computer Science, vol. 6502, Heidelberg: Springer, pp. 293–304, arXiv:1009.1315, doi:10.1007/978-3-642-18469-7_27, MR 2781274
|
Wikipedia
|
Ring homomorphism
In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if R and S are rings, then a ring homomorphism is a function f : R → S such that f is:[1][2][3][4][5][6][7][lower-alpha 1]
addition preserving:
$f(a+b)=f(a)+f(b)$ for all a and b in R,
multiplication preserving:
$f(ab)=f(a)f(b)$ for all a and b in R,
and unit (multiplicative identity) preserving:
$f(1_{R})=1_{S}$.
Algebraic structure → Ring theory
Ring theory
Basic concepts
Rings
• Subrings
• Ideal
• Quotient ring
• Fractional ideal
• Total ring of fractions
• Product of rings
• Free product of associative algebras
• Tensor product of algebras
Ring homomorphisms
• Kernel
• Inner automorphism
• Frobenius endomorphism
Algebraic structures
• Module
• Associative algebra
• Graded ring
• Involutive ring
• Category of rings
• Initial ring $\mathbb {Z} $
• Terminal ring $0=\mathbb {Z} _{1}$
Related structures
• Field
• Finite field
• Non-associative ring
• Lie ring
• Jordan ring
• Semiring
• Semifield
Commutative algebra
Commutative rings
• Integral domain
• Integrally closed domain
• GCD domain
• Unique factorization domain
• Principal ideal domain
• Euclidean domain
• Field
• Finite field
• Composition ring
• Polynomial ring
• Formal power series ring
Algebraic number theory
• Algebraic number field
• Ring of integers
• Algebraic independence
• Transcendental number theory
• Transcendence degree
p-adic number theory and decimals
• Direct limit/Inverse limit
• Zero ring $\mathbb {Z} _{1}$
• Integers modulo pn $\mathbb {Z} /p^{n}\mathbb {Z} $
• Prüfer p-ring $\mathbb {Z} (p^{\infty })$
• Base-p circle ring $\mathbb {T} $
• Base-p integers $\mathbb {Z} $
• p-adic rationals $\mathbb {Z} [1/p]$
• Base-p real numbers $\mathbb {R} $
• p-adic integers $\mathbb {Z} _{p}$
• p-adic numbers $\mathbb {Q} _{p}$
• p-adic solenoid $\mathbb {T} _{p}$
Algebraic geometry
• Affine variety
Noncommutative algebra
Noncommutative rings
• Division ring
• Semiprimitive ring
• Simple ring
• Commutator
Noncommutative algebraic geometry
Free algebra
Clifford algebra
• Geometric algebra
Operator algebra
Additive inverses and the additive identity are part of the structure too, but it is not necessary to require explicitly that they too are respected, because these conditions are consequences of the three conditions above.
If in addition f is a bijection, then its inverse f−1 is also a ring homomorphism. In this case, f is called a ring isomorphism, and the rings R and S are called isomorphic. From the standpoint of ring theory, isomorphic rings cannot be distinguished.
If R and S are rngs, then the corresponding notion is that of a rng homomorphism,[lower-alpha 2] defined as above except without the third condition f(1R) = 1S. A rng homomorphism between (unital) rings need not be a ring homomorphism.
The composition of two ring homomorphisms is a ring homomorphism. It follows that the class of all rings forms a category with ring homomorphisms as the morphisms (cf. the category of rings). In particular, one obtains the notions of ring endomorphism, ring isomorphism, and ring automorphism.
Properties
Let $f\colon R\rightarrow S$ be a ring homomorphism. Then, directly from these definitions, one can deduce:
• f(0R) = 0S.
• f(−a) = −f(a) for all a in R.
• For any unit element a in R, f(a) is a unit element such that f(a−1) = f(a)−1. In particular, f induces a group homomorphism from the (multiplicative) group of units of R to the (multiplicative) group of units of S (or of im(f)).
• The image of f, denoted im(f), is a subring of S.
• The kernel of f, defined as ker(f) = {a in R : f(a) = 0S}, is an ideal in R. Every ideal in a ring R arises from some ring homomorphism in this way.
• The homomorphism f is injective if and only if ker(f) = {0R}.
• If there exists a ring homomorphism f : R → S then the characteristic of S divides the characteristic of R. This can sometimes be used to show that between certain rings R and S, no ring homomorphisms R → S exists.
• If Rp is the smallest subring contained in R and Sp is the smallest subring contained in S, then every ring homomorphism f : R → S induces a ring homomorphism fp : Rp → Sp.
• If R is a field (or more generally a skew-field) and S is not the zero ring, then f is injective.
• If both R and S are fields, then im(f) is a subfield of S, so S can be viewed as a field extension of R.
• If I is an ideal of S then f−1(I) is an ideal of R.
• If R and S are commutative and P is a prime ideal of S then f−1(P) is a prime ideal of R.
• If R and S are commutative, M is a maximal ideal of S, and f is surjective, then f−1(M) is a maximal ideal of R.
• If R and S are commutative and S is an integral domain, then ker(f) is a prime ideal of R.
• If R and S are commutative, S is a field, and f is surjective, then ker(f) is a maximal ideal of R.
• If f is surjective, P is prime (maximal) ideal in R and ker(f) ⊆ P, then f(P) is prime (maximal) ideal in S.
Moreover,
• The composition of ring homomorphisms is a ring homomorphism.
• For each ring R, the identity map R → R is a ring homomorphism.
• Therefore, the class of all rings together with ring homomorphisms forms a category, the category of rings.
• The zero map R → S sending every element of R to 0 is a ring homomorphism only if S is the zero ring (the ring whose only element is zero).
• For every ring R, there is a unique ring homomorphism Z → R. This says that the ring of integers is an initial object in the category of rings.
• For every ring R, there is a unique ring homomorphism from R to the zero ring. This says that the zero ring is a terminal object in the category of rings.
Examples
• The function f : Z → Z/nZ, defined by f(a) = [a]n = a mod n is a surjective ring homomorphism with kernel nZ (see modular arithmetic).
• The complex conjugation C → C is a ring homomorphism (this is an example of a ring automorphism).
• For a ring R of prime characteristic p, R → R, x → xp is a ring endomorphism called the Frobenius endomorphism.
• If R and S are rings, the zero function from R to S is a ring homomorphism if and only if S is the zero ring. (Otherwise it fails to map 1R to 1S.) On the other hand, the zero function is always a rng homomorphism.
• If R[X] denotes the ring of all polynomials in the variable X with coefficients in the real numbers R, and C denotes the complex numbers, then the function f : R[X] → C defined by f(p) = p(i) (substitute the imaginary unit i for the variable X in the polynomial p) is a surjective ring homomorphism. The kernel of f consists of all polynomials in R[X] that are divisible by X2 + 1.
• If f : R → S is a ring homomorphism between the rings R and S, then f induces a ring homomorphism between the matrix rings Mn(R) → Mn(S).
• Let V be a vector space over a field k. Then the map $\rho :k\to \operatorname {End} (V)$ given by $\rho (a)v=av$ is a ring homomorphism. More generally, given an abelian group M, a module structure on M over a ring R is equivalent to giving a ring homomorphism $R\to \operatorname {End} (M)$.
• A unital algebra homomorphism between unital associative algebras over a commutative ring R is a ring homomorphism that is also R-linear.
Non-examples
• The function f : Z/6Z → Z/6Z defined by f([a]6) = [4a]6 is a rng homomorphism (and rng endomorphism), with kernel 3Z/6Z and image 2Z/6Z (which is isomorphic to Z/3Z).
• There is no ring homomorphism Z/nZ → Z for any n ≥ 1.
• If R and S are rings, the inclusion $R\to R\times S$ sending each r to (r,0) is a rng homomorphism, but not a ring homomorphism (if S is not the zero ring), since it does not map the multiplicative identity 1 of R to the multiplicative identity (1,1) of $R\times S$.
The category of rings
Endomorphisms, isomorphisms, and automorphisms
• A ring endomorphism is a ring homomorphism from a ring to itself.
• A ring isomorphism is a ring homomorphism having a 2-sided inverse that is also a ring homomorphism. One can prove that a ring homomorphism is an isomorphism if and only if it is bijective as a function on the underlying sets. If there exists a ring isomorphism between two rings R and S, then R and S are called isomorphic. Isomorphic rings differ only by a relabeling of elements. Example: Up to isomorphism, there are four rings of order 4. (This means that there are four pairwise non-isomorphic rings of order 4 such that every other ring of order 4 is isomorphic to one of them.) On the other hand, up to isomorphism, there are eleven rngs of order 4.
• A ring automorphism is a ring isomorphism from a ring to itself.
Monomorphisms and epimorphisms
Injective ring homomorphisms are identical to monomorphisms in the category of rings: If f : R → S is a monomorphism that is not injective, then it sends some r1 and r2 to the same element of S. Consider the two maps g1 and g2 from Z[x] to R that map x to r1 and r2, respectively; f ∘ g1 and f ∘ g2 are identical, but since f is a monomorphism this is impossible.
However, surjective ring homomorphisms are vastly different from epimorphisms in the category of rings. For example, the inclusion Z ⊆ Q is a ring epimorphism, but not a surjection. However, they are exactly the same as the strong epimorphisms.
See also
• Change of rings
Citations
1. Artin 1991, p. 353.
2. Atiyah & Macdonald 1969, p. 2.
3. Bourbaki 1998, p. 102.
4. Eisenbud 1995, p. 12.
5. Jacobson 1985, p. 103.
6. Lang 2002, p. 88.
7. Hazewinkel 2004, p. 3.
Notes
1. Hazewinkel initially defines "ring" without the requirement of a 1, but very soon states that from now on, all rings will have a 1.
2. Some authors do not require a ring to contain a multiplicative identity; instead of "rng", "ring", and "rng homomorphism", they use the terms "ring", "ring with identity", and "ring homomorphism", respectively. Because of this, some other authors, to avoid ambiguity, explicitly specify that rings are unital and that homomorphisms preserve the identity.
References
• Artin, Michael (1991). Algebra. Englewood Cliffs, N.J.: Prentice Hall.
• Atiyah, Michael F.; Macdonald, Ian G. (1969), Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., MR 0242802
• Bourbaki, N. (1998). Algebra I, Chapters 1–3. Springer.
• Eisenbud, David (1995). Commutative algebra with a view toward algebraic geometry. Graduate Texts in Mathematics. Vol. 150. New York: Springer-Verlag. xvi+785. ISBN 0-387-94268-8. MR 1322960.
• Hazewinkel, Michiel (2004). Algebras, rings and modules. Springer-Verlag. ISBN 1-4020-2690-0.
• Jacobson, Nathan (1985). Basic algebra I (2nd ed.). ISBN 9780486471891.
• Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, vol. 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556
|
Wikipedia
|
Adele ring
In mathematics, the adele ring of a global field (also adelic ring, ring of adeles or ring of adèles[1]) is a central object of class field theory, a branch of algebraic number theory. It is the restricted product of all the completions of the global field and is an example of a self-dual topological ring.
An adele derives from a particular kind of idele. "Idele" derives from the French "idèle" and was coined by the French mathematician Claude Chevalley. The word stands for 'ideal element' (abbreviated: id.el.). Adele (French: "adèle") stands for 'additive idele' (that is, additive ideal element).
The ring of adeles allows one to describe the Artin reciprocity law, which is a generalisation of quadratic reciprocity, and other reciprocity laws over finite fields. In addition, it is a classical theorem from Weil that $G$-bundles on an algebraic curve over a finite field can be described in terms of adeles for a reductive group $G$. Adeles are also connected with the adelic algebraic groups and adelic curves.
The study of geometry of numbers over the ring of adeles of a number field is called adelic geometry.
Definition
Let $K$ be a global field (a finite extension of $\mathbf {Q} $ or the function field of a curve $X/\mathbf {F_{\mathit {q}}} $ over a finite field). The adele ring of $K$ is the subring
$\mathbf {A} _{K}\ =\ \prod (K_{\nu },{\mathcal {O}}_{\nu })\ \subseteq \ \prod K_{\nu }$
consisting of the tuples $(a_{\nu })$ where $a_{\nu }$ lies in the subring ${\mathcal {O}}_{\nu }\subset K_{\nu }$ for all but finitely many places $\nu $. Here the index $\nu $ ranges over all valuations of the global field $K$, $K_{\nu }$ is the completion at that valuation and ${\mathcal {O}}_{\nu }$ the corresponding valuation ring.[2]
Motivation
The ring of adeles solves the technical problem of "doing analysis on the rational numbers $\mathbf {Q} $." The classical solution was to pass to the standard metric completion $\mathbf {R} $ and use analytic techniques there. But, as was learned later on, there are many more absolute values other than the Euclidean distance, one for each prime number $p\in \mathbf {Z} $, as was classified by Ostrowski. The Euclidean absolute value, denoted $|\cdot |_{\infty }$, is only one among many others, $|\cdot |_{p}$, but the ring of adeles makes it possible to compromise and use all of the valuations at once. This has the advantage of enabling analytic techniques while also retaining information about the primes, since their structure is embedded by the restricted infinite product.
The purpose of the adele ring is to look at all completions of $K$ at once. The adele ring is defined with the restricted product, rather than the Cartesian product. There are two reasons for this:
• For each element of $K$ the valuations are zero for almost all places, i.e., for all places except a finite number. So, the global field can be embedded in the restricted product.
• The restricted product is a locally compact space, while the Cartesian product is not. Therefore, there cannot be any application of harmonic analysis to the Cartesian product. This is because local compactness ensures the existence (and uniqueness) of Haar measure, a crucial tool in analysis on groups in general.
Why the restricted product?
The restricted infinite product is a required technical condition for giving the number field $\mathbf {Q} $ a lattice structure inside of $\mathbf {A} _{\mathbf {Q} }$, making it possible to build a theory of Fourier analysis (cf. Harmonic analysis) in the adelic setting. This is analogous to the situation in algebraic number theory where the ring of integers of an algebraic number field embeds
${\mathcal {O}}_{K}\hookrightarrow K$
as a lattice. With the power of a new theory of Fourier analysis, Tate was able to prove a special class of L-functions and the Dedekind zeta functions were meromorphic on the complex plane. Another natural reason for why this technical condition holds can be seen by constructing the ring of adeles as a tensor product of rings. If defining the ring of integral adeles $\mathbf {A} _{\mathbf {Z} }$ as the ring
$\mathbf {A} _{\mathbf {Z} }=\mathbf {R} \times {\hat {\mathbf {Z} }}=\mathbf {R} \times \prod _{p}\mathbf {Z} _{p},$
then the ring of adeles can be equivalently defined as
${\begin{aligned}\mathbf {A} _{\mathbf {Q} }&=\mathbf {Q} \otimes _{\mathbf {Z} }\mathbf {A} _{\mathbf {Z} }\\&=\mathbf {Q} \otimes _{\mathbf {Z} }\left(\mathbf {R} \times \prod _{p}\mathbf {Z} _{p}\right).\end{aligned}}$
The restricted product structure becomes transparent after looking at explicit elements in this ring. The image of an element $b/c\otimes (r,(a_{p}))\in \mathbf {A} _{\mathbf {Q} }$ inside of the unrestricted product $ \mathbf {R} \times \prod _{p}\mathbf {Q} _{p}$ is the element
$\left({\frac {br}{c}},\left({\frac {ba_{p}}{c}}\right)\right).$
The factor $ba_{p}/c$ lies in $\mathbf {Z} _{p}$ whenever $p$ is not a prime factor of $c$, which is the case for all but finitely many primes $p$.[3]
Origin of the name
The term "idele" (French: idèle) is an invention of the French mathematician Claude Chevalley (1909–1984) and stands for "ideal element" (abbreviated: id.el.). The term "adele" (French: adèle) stands for additive idele. Thus, an adele is an additive ideal element.
Examples
Ring of adeles for the rational numbers
The rationals $K={\mathbf {Q}}$ have a valuation for every prime number $p$, with $(K_{\nu },{\mathcal {O}}_{\nu })=(\mathbf {Q} _{p},\mathbf {Z} _{p})$, and one infinite valuation ∞ with $\mathbf {Q} _{\infty }=\mathbf {R} $. Thus an element of
$\mathbf {A} _{\mathbf {Q} }\ =\ \mathbf {R} \times \prod _{p}(\mathbf {Q} _{p},\mathbf {Z} _{p})$
is a real number along with a p-adic rational for each $p$ of which all but finitely many are p-adic integers.
Ring of adeles for the function field of the projective line
Secondly, take the function field $K=\mathbf {F} _{q}(\mathbf {P} ^{1})=\mathbf {F} _{q}(t)$ of the projective line over a finite field. Its valuations correspond to points $x$ of $X=\mathbf {P} ^{1}$, i.e. maps over ${\text{Spec}}\mathbf {F} _{q}$
$x\ :\ {\text{Spec}}\mathbf {F} _{q^{n}}\ \longrightarrow \ \mathbf {P} ^{1}.$ :\ {\text{Spec}}\mathbf {F} _{q^{n}}\ \longrightarrow \ \mathbf {P} ^{1}.}
For instance, there are $q+1$ points of the form ${\text{Spec}}\mathbf {F} _{q}\ \longrightarrow \ \mathbf {P} ^{1}$. In this case ${\mathcal {O}}_{\nu }={\widehat {\mathcal {O}}}_{X,x}$ is the completed stalk of the structure sheaf at $x$ (i.e. functions on a formal neighbourhood of $x$) and $K_{\nu }=K_{X,x}$ is its fraction field. Thus
$\mathbf {A} _{\mathbf {F} _{q}(\mathbf {P} ^{1})}\ =\ \prod _{x\in X}({\mathcal {K}}_{X,x},{\widehat {\mathcal {O}}}_{X,x}).$
The same holds for any smooth proper curve $X/\mathbf {F_{\mathit {q}}} $ over a finite field, the restricted product being over all points of $x\in X$.
Related notions
The group of units in the adele ring is called the idele group
$I_{K}=\mathbf {A} _{K^{\times }}$.
The quotient of the ideles by the subgroup $K^{\times }\subseteq I_{K}$ is called the idele class group
$C_{K}\ =\ I_{K}/K^{\times }.$
The integral adeles are the subring
$\mathbf {O} _{K}\ =\ \prod O_{\nu }\ \subseteq \ \mathbf {A} _{K}.$
Applications
Stating Artin reciprocity
The Artin reciprocity law says that for a global field $K$,
${\widehat {C_{K}}}={\widehat {\mathbf {A} _{K}^{\times }/K^{\times }}}\ \simeq \ {\text{Gal}}(K^{\text{ab}}/K)$
where $K^{ab}$ is the maximal abelian algebraic extension of $K$ and ${\widehat {(\dots )}}$ means the profinite completion of the group.
Giving adelic formulation of Picard group of a curve
If $X/\mathbf {F_{\mathit {q}}} $ is a smooth proper curve then its Picard group is[4]
${\text{Pic}}(X)\ =\ K^{\times }\backslash \mathbf {A} _{X}^{\times }/\mathbf {O} _{X}^{\times }$
and its divisor group is ${\text{Div}}(X)=\mathbf {A} _{X}^{\times }/\mathbf {O} _{X}^{\times }$. Similarly, if $G$ is a semisimple algebraic group (e.g. $ SL_{n}$, it also holds for $GL_{n}$) then Weil uniformisation says that[5]
${\text{Bun}}_{G}(X)\ =\ G(K)\backslash G(\mathbf {A} _{X})/G(\mathbf {O} _{X}).$
Applying this to $G=\mathbf {G} _{m}$ gives the result on the Picard group.
Tate's thesis
There is a topology on $\mathbf {A} _{K}$ for which the quotient $\mathbf {A} _{K}/K$ is compact, allowing one to do harmonic analysis on it. John Tate in his thesis "Fourier analysis in number fields and Hecke Zeta functions"[6] proved results about Dirichlet L-functions using Fourier analysis on the adele ring and the idele group. Therefore, the adele ring and the idele group have been applied to study the Riemann zeta function and more general zeta functions and the L-functions.
Proving Serre duality on a smooth curve
If $X$ is a smooth proper curve over the complex numbers, one can define the adeles of its function field $\mathbf {C} (X)$ exactly as the finite fields case. John Tate proved[7] that Serre duality on $X$
$H^{1}(X,{\mathcal {L}})\ \simeq \ H^{0}(X,\Omega _{X}\otimes {\mathcal {L}}^{-1})^{*}$
can be deduced by working with this adele ring $\mathbf {A} _{\mathbf {C} (X)}$. Here L is a line bundle on $X$.
Notation and basic definitions
Global fields
Throughout this article, $K$ is a global field, meaning it is either a number field (a finite extension of $\mathbb {Q} $) or a global function field (a finite extension of $\mathbb {F} _{p^{r}}(t)$ for $p$ prime and $r\in \mathbb {N} $). By definition a finite extension of a global field is itself a global field.
Valuations
For a valuation $v$ of $K$ it can be written $K_{v}$ for the completion of $K$ with respect to $v.$ If $v$ is discrete it can be written $O_{v}$ for the valuation ring of $K_{v}$ and ${\mathfrak {m}}_{v}$ for the maximal ideal of $O_{v}.$ If this is a principal ideal denoting the uniformising element by $\pi _{v}.$ A non-Archimedean valuation is written as $v<\infty $ or $v\nmid \infty $ and an Archimedean valuation as $v|\infty .$ Then assume all valuations to be non-trivial.
There is a one-to-one identification of valuations and absolute values. Fix a constant $C>1,$ the valuation $v$ is assigned the absolute value $|\cdot |_{v},$ defined as:
$\forall x\in K:\quad |x|_{v}:={\begin{cases}C^{-v(x)}&x\neq 0\\0&x=0\end{cases}}$
Conversely, the absolute value $|\cdot |$ is assigned the valuation $v_{|\cdot |},$ defined as:
$\forall x\in K^{\times }:\quad v_{|\cdot |}(x):=-\log _{C}(|x|).$
A place of $K$ is a representative of an equivalence class of valuations (or absolute values) of $K.$ Places corresponding to non-Archimedean valuations are called finite, whereas places corresponding to Archimedean valuations are called infinite. Infinite places of a global field form a finite set, which is denoted by $P_{\infty }.$
Define $\textstyle {\widehat {O}}:=\prod _{v<\infty }O_{v}$ and let ${\widehat {O}}^{\times }$ be its group of units. Then $\textstyle {\widehat {O}}^{\times }=\prod _{v<\infty }O_{v}^{\times }.$
Finite extensions
Let $L/K$ be a finite extension of the global field $K.$ Let $w$ be a place of $L$ and $v$ a place of $K.$ If the absolute value $|\cdot |_{w}$ restricted to $K$ is in the equivalence class of $v$, then $w$ lies above $v,$ which is denoted by $w|v,$ and defined as:
${\begin{aligned}L_{v}&:=\prod _{w|v}L_{w},\\{\widetilde {O_{v}}}&:=\prod _{w|v}O_{w}.\end{aligned}}$
(Note that both products are finite.)
If $w|v$, $K_{v}$ can be embedded in $L_{w}.$ Therefore, $K_{v}$ is embedded diagonally in $L_{v}.$ With this embedding $L_{v}$ is a commutative algebra over $K_{v}$ with degree
$\sum _{w|v}[L_{w}:K_{v}]=[L:K].$
The adele ring
The set of finite adeles of a global field $K,$ denoted $\mathbb {A} _{K,{\text{fin}}},$ is defined as the restricted product of $K_{v}$ with respect to the $O_{v}:$
$\mathbb {A} _{K,{\text{fin}}}:={\prod _{v<\infty }}^{'}K_{v}=\left\{\left.(x_{v})_{v}\in \prod _{v<\infty }K_{v}\right|x_{v}\in O_{v}{\text{ for almost all }}v\right\}.$
It is equipped with the restricted product topology, the topology generated by restricted open rectangles, which have the following form:
$U=\prod _{v\in E}U_{v}\times \prod _{v\notin E}O_{v}\subset {\prod _{v<\infty }}^{'}K_{v},$
where $E$ is a finite set of (finite) places and $U_{v}\subset K_{v}$ are open. With component-wise addition and multiplication $\mathbb {A} _{K,{\text{fin}}}$ is also a ring.
The adele ring of a global field $K$ is defined as the product of $\mathbb {A} _{K,{\text{fin}}}$ with the product of the completions of $K$ at its infinite places. The number of infinite places is finite and the completions are either $\mathbb {R} $ or $\mathbb {C} .$ In short:
$\mathbb {A} _{K}:=\mathbb {A} _{K,{\text{fin}}}\times \prod _{v|\infty }K_{v}={\prod _{v<\infty }}^{'}K_{v}\times \prod _{v|\infty }K_{v}.$
With addition and multiplication defined as component-wise the adele ring is a ring. The elements of the adele ring are called adeles of $K.$ In the following, it is written as
$\mathbb {A} _{K}={\prod _{v}}^{'}K_{v},$
although this is generally not a restricted product.
Remark. Global function fields do not have any infinite places and therefore the finite adele ring equals the adele ring.
Lemma. There is a natural embedding of $K$ into $\mathbb {A} _{K}$ given by the diagonal map: $a\mapsto (a,a,\ldots ).$
Proof. If $a\in K,$ then $a\in O_{v}^{\times }$ for almost all $v.$ This shows the map is well-defined. It is also injective because the embedding of $K$ in $K_{v}$ is injective for all $v.$
Remark. By identifying $K$ with its image under the diagonal map it is regarded as a subring of $\mathbb {A} _{K}.$ The elements of $K$ are called the principal adeles of $\mathbb {A} _{K}.$
Definition. Let $S$ be a set of places of $K.$ Define the set of the $S$-adeles of $K$ as
$\mathbb {A} _{K,S}:={\prod _{v\in S}}^{'}K_{v}.$
Furthermore, if
$\mathbb {A} _{K}^{S}:={\prod _{v\notin S}}^{'}K_{v}$
the result is: $\mathbb {A} _{K}=\mathbb {A} _{K,S}\times \mathbb {A} _{K}^{S}.$
The adele ring of rationals
By Ostrowski's theorem the places of $\mathbb {Q} $ are $\{p\in \mathbb {N} :p{\text{ prime}}\}\cup \{\infty \},$ it is possible to identify a prime $p$ with the equivalence class of the $p$-adic absolute value and $\infty $ with the equivalence class of the absolute value $|\cdot |_{\infty }$ defined as:
$\forall x\in \mathbb {Q} :\quad |x|_{\infty }:={\begin{cases}x&x\geq 0\\-x&x<0\end{cases}}$ :\quad |x|_{\infty }:={\begin{cases}x&x\geq 0\\-x&x<0\end{cases}}}
The completion of $\mathbb {Q} $ with respect to the place $p$ is $\mathbb {Q} _{p}$ with valuation ring $\mathbb {Z} _{p}.$ For the place $\infty $ the completion is $\mathbb {R} .$ Thus:
${\begin{aligned}\mathbb {A} _{\mathbb {Q} ,{\text{fin}}}&={\prod _{p<\infty }}^{'}\mathbb {Q} _{p}\\\mathbb {A} _{\mathbb {Q} }&=\left({\prod _{p<\infty }}^{'}\mathbb {Q} _{p}\right)\times \mathbb {R} \end{aligned}}$
Or for short
$\mathbb {A} _{\mathbb {Q} }={\prod _{p\leq \infty }}^{'}\mathbb {Q} _{p},\qquad \mathbb {Q} _{\infty }:=\mathbb {R} .$
the difference between restricted and unrestricted product topology can be illustrated using a sequence in $\mathbb {A} _{\mathbb {Q} }$:
Lemma. Consider the following sequence in $\mathbb {A} _{\mathbb {Q} }$:
${\begin{aligned}x_{1}&=\left({\frac {1}{2}},1,1,\ldots \right)\\x_{2}&=\left(1,{\frac {1}{3}},1,\ldots \right)\\x_{3}&=\left(1,1,{\frac {1}{5}},1,\ldots \right)\\x_{4}&=\left(1,1,1,{\frac {1}{7}},1,\ldots \right)\\&\vdots \end{aligned}}$
In the product topology this converges to $(1,1,\ldots )$, but it does not converge at all in the restricted product topology.
Proof. In product topology convergence corresponds to the convergence in each coordinate, which is trivial because the sequences become stationary. The sequence doesn't converge in restricted product topology. For each adele $a=(a_{p})_{p}\in \mathbb {A} _{\mathbb {Q} }$ and for each restricted open rectangle $\textstyle U=\prod _{p\in E}U_{p}\times \prod _{p\notin E}\mathbb {Z} _{p},$ it has: ${\tfrac {1}{p}}-a_{p}\notin \mathbb {Z} _{p}$ for $a_{p}\in \mathbb {Z} _{p}$ and therefore ${\tfrac {1}{p}}-a_{p}\notin \mathbb {Z} _{p}$ for all $p\notin F.$ As a result $x_{n}-a\notin U$ for almost all $n\in \mathbb {N} .$ In this consideration, $E$ and $F$ are finite subsets of the set of all places.
Alternative definition for number fields
Definition (profinite integers). The profinite integers are defined as the profinite completion of the rings $\mathbb {Z} /n\mathbb {Z} $ with the partial order $n\geq m\Leftrightarrow m|n,$ i.e.,
${\widehat {\mathbb {Z} }}:=\varprojlim _{n}\mathbb {Z} /n\mathbb {Z} ,$
Lemma. $\textstyle {\widehat {\mathbb {Z} }}\cong \prod _{p}\mathbb {Z} _{p}.$
Proof. This follows from the Chinese Remainder Theorem.
Lemma. $\mathbb {A} _{\mathbb {Q} ,{\text{fin}}}={\widehat {\mathbb {Z} }}\otimes _{\mathbb {Z} }\mathbb {Q} .$
Proof. Use the universal property of the tensor product. Define a $\mathbb {Z} $-bilinear function
${\begin{cases}\Psi :{\widehat {\mathbb {Z} }}\times \mathbb {Q} \to \mathbb {A} _{\mathbb {Q} ,{\text{fin}}}\\\left((a_{p})_{p},q\right)\mapsto (a_{p}q)_{p}\end{cases}}$ :{\widehat {\mathbb {Z} }}\times \mathbb {Q} \to \mathbb {A} _{\mathbb {Q} ,{\text{fin}}}\\\left((a_{p})_{p},q\right)\mapsto (a_{p}q)_{p}\end{cases}}}
This is well-defined because for a given $q={\tfrac {m}{n}}\in \mathbb {Q} $ with $m,n$ co-prime there are only finitely many primes dividing $n.$ Let $M$ be another $\mathbb {Z} $-module with a $\mathbb {Z} $-bilinear map $\Phi :{\widehat {\mathbb {Z} }}\times \mathbb {Q} \to M.$ :{\widehat {\mathbb {Z} }}\times \mathbb {Q} \to M.} It must be the case that $\Phi $ factors through $\Psi $ uniquely, i.e., there exists a unique $\mathbb {Z} $-linear map ${\tilde {\Phi }}:\mathbb {A} _{\mathbb {Q} ,{\text{fin}}}\to M$ such that $\Phi ={\tilde {\Phi }}\circ \Psi .$ ${\tilde {\Phi }}$ can be defined as follows: for a given $(u_{p})_{p}$ there exist $u\in \mathbb {N} $ and $(v_{p})_{p}\in {\widehat {\mathbb {Z} }}$ such that $u_{p}={\tfrac {1}{u}}\cdot v_{p}$ for all $p.$ Define ${\tilde {\Phi }}((u_{p})_{p}):=\Phi ((v_{p})_{p},{\tfrac {1}{u}}).$ One can show ${\tilde {\Phi }}$ is well-defined, $\mathbb {Z} $-linear, satisfies $\Phi ={\tilde {\Phi }}\circ \Psi $ and is unique with these properties.
Corollary. Define $\mathbb {A} _{\mathbb {Z} }:={\widehat {\mathbb {Z} }}\times \mathbb {R} .$ This results in an algebraic isomorphism $\mathbb {A} _{\mathbb {Q} }\cong \mathbb {A} _{\mathbb {Z} }\otimes _{\mathbb {Z} }\mathbb {Q} .$
Proof. $\mathbb {A} _{\mathbb {Z} }\otimes _{\mathbb {Z} }\mathbb {Q} =\left({\widehat {\mathbb {Z} }}\times \mathbb {R} \right)\otimes _{\mathbb {Z} }\mathbb {Q} \cong \left({\widehat {\mathbb {Z} }}\otimes _{\mathbb {Z} }\mathbb {Q} \right)\times (\mathbb {R} \otimes _{\mathbb {Z} }\mathbb {Q} )\cong \left({\widehat {\mathbb {Z} }}\otimes _{\mathbb {Z} }\mathbb {Q} \right)\times \mathbb {R} =\mathbb {A} _{\mathbb {Q} ,{\text{fin}}}\times \mathbb {R} =\mathbb {A} _{\mathbb {Q} }.$
Lemma. For a number field $K,\mathbb {A} _{K}=\mathbb {A} _{\mathbb {Q} }\otimes _{\mathbb {Q} }K.$
Remark. Using $\mathbb {A} _{\mathbb {Q} }\otimes _{\mathbb {Q} }K\cong \mathbb {A} _{\mathbb {Q} }\oplus \dots \oplus \mathbb {A} _{\mathbb {Q} },$ where there are $[K:\mathbb {Q} ]$ summands, give the right side receives the product topology and transport this topology via the isomorphism onto $\mathbb {A} _{\mathbb {Q} }\otimes _{\mathbb {Q} }K.$
The adele ring of a finite extension
If $L/K$ be a finite extension, then $L$ is a global field. Thus $\mathbb {A} _{L}$ is defined, and $\textstyle \mathbb {A} _{L}={\prod _{v}}^{'}L_{v}.$ $\mathbb {A} _{K}$ can be identified with a subgroup of $\mathbb {A} _{L}.$ Map $a=(a_{v})_{v}\in \mathbb {A} _{K}$ to $a'=(a'_{w})_{w}\in \mathbb {A} _{L}$ where $a'_{w}=a_{v}\in K_{v}\subset L_{w}$ for $w|v.$ Then $a=(a_{w})_{w}\in \mathbb {A} _{L}$ is in the subgroup $\mathbb {A} _{K},$ if $a_{w}\in K_{v}$ for $w|v$ and $a_{w}=a_{w'}$ for all $w,w'$ lying above the same place $v$ of $K.$
Lemma. If $L/K$ is a finite extension, then $\mathbb {A} _{L}\cong \mathbb {A} _{K}\otimes _{K}L$ both algebraically and topologically.
With the help of this isomorphism, the inclusion $\mathbb {A} _{K}\subset \mathbb {A} _{L}$ is given by
${\begin{cases}\mathbb {A} _{K}\to \mathbb {A} _{L}\\\alpha \mapsto \alpha \otimes _{K}1\end{cases}}$
Furthermore, the principal adeles in $\mathbb {A} _{K}$ can be identified with a subgroup of principal adeles in $\mathbb {A} _{L}$ via the map
${\begin{cases}K\to (K\otimes _{K}L)\cong L\\\alpha \mapsto 1\otimes _{K}\alpha \end{cases}}$
Proof.[8] Let $\omega _{1},\ldots ,\omega _{n}$ be a basis of $L$ over $K.$ Then for almost all $v,$
${\widetilde {O_{v}}}\cong O_{v}\omega _{1}\oplus \cdots \oplus O_{v}\omega _{n}.$
Furthermore, there are the following isomorphisms:
$K_{v}\omega _{1}\oplus \cdots \oplus K_{v}\omega _{n}\cong K_{v}\otimes _{K}L\cong L_{v}=\prod \nolimits _{w|v}L_{w}$
For the second use the map:
${\begin{cases}K_{v}\otimes _{K}L\to L_{v}\\\alpha _{v}\otimes a\mapsto (\alpha _{v}\cdot (\tau _{w}(a)))_{w}\end{cases}}$
in which $\tau _{w}:L\to L_{w}$ is the canonical embedding and $w|v.$ The restricted product is taken on both sides with respect to ${\widetilde {O_{v}}}:$
${\begin{aligned}\mathbb {A} _{K}\otimes _{K}L&=\left({\prod _{v}}^{'}K_{v}\right)\otimes _{K}L\\&\cong {\prod _{v}}^{'}(K_{v}\omega _{1}\oplus \cdots \oplus K_{v}\omega _{n})\\&\cong {\prod _{v}}^{'}(K_{v}\otimes _{K}L)\\&\cong {\prod _{v}}^{'}L_{v}\\&=\mathbb {A} _{L}\end{aligned}}$
Corollary. As additive groups $\mathbb {A} _{L}\cong \mathbb {A} _{K}\oplus \cdots \oplus \mathbb {A} _{K},$ where the right side has $[L:K]$ summands.
The set of principal adeles in $\mathbb {A} _{L}$ is identified with the set $K\oplus \cdots \oplus K,$ where the left side has $[L:K]$ summands and $K$ is considered as a subset of $\mathbb {A} _{K}.$
The adele ring of vector-spaces and algebras
Lemma. Suppose $P\supset P_{\infty }$ is a finite set of places of $K$ and define
$\mathbb {A} _{K}(P):=\prod _{v\in P}K_{v}\times \prod _{v\notin P}O_{v}.$
Equip $\mathbb {A} _{K}(P)$ with the product topology and define addition and multiplication component-wise. Then $\mathbb {A} _{K}(P)$ is a locally compact topological ring.
Remark. If $P'$ is another finite set of places of $K$ containing $P$ then $\mathbb {A} _{K}(P)$ is an open subring of $\mathbb {A} _{K}(P').$
Now, an alternative characterisation of the adele ring can be presented. The adele ring is the union of all sets $\mathbb {A} _{K}(P)$:
$\mathbb {A} _{K}=\bigcup _{P\supset P_{\infty },|P|<\infty }\mathbb {A} _{K}(P).$
Equivalently $\mathbb {A} _{K}$ is the set of all $x=(x_{v})_{v}$ so that $|x_{v}|_{v}\leq 1$ for almost all $v<\infty .$ The topology of $\mathbb {A} _{K}$ is induced by the requirement that all $\mathbb {A} _{K}(P)$ be open subrings of $\mathbb {A} _{K}.$ Thus, $\mathbb {A} _{K}$ is a locally compact topological ring.
Fix a place $v$ of $K.$ Let $P$ be a finite set of places of $K,$ containing $v$ and $P_{\infty }.$ Define
$\mathbb {A} _{K}'(P,v):=\prod _{w\in P\setminus \{v\}}K_{w}\times \prod _{w\notin P}O_{w}.$
Then:
$\mathbb {A} _{K}(P)\cong K_{v}\times \mathbb {A} _{K}'(P,v).$
Furthermore, define
$\mathbb {A} _{K}'(v):=\bigcup _{P\supset P_{\infty }\cup \{v\}}\mathbb {A} _{K}'(P,v),$
where $P$ runs through all finite sets containing $P_{\infty }\cup \{v\}.$ Then:
$\mathbb {A} _{K}\cong K_{v}\times \mathbb {A} _{K}'(v),$
via the map $(a_{w})_{w}\mapsto (a_{v},(a_{w})_{w\neq v}).$ The entire procedure above holds with a finite subset ${\widetilde {P}}$ instead of $\{v\}.$
By construction of $\mathbb {A} _{K}'(v),$ there is a natural embedding: $K_{v}\hookrightarrow \mathbb {A} _{K}.$ Furthermore, there exists a natural projection $\mathbb {A} _{K}\twoheadrightarrow K_{v}.$
The adele ring of a vector-space
Let $E$ be a finite dimensional vector-space over $K$ and $\{\omega _{1},\ldots ,\omega _{n}\}$ a basis for $E$ over $K.$ For each place $v$ of $K$:
${\begin{aligned}E_{v}&:=E\otimes _{K}K_{v}\cong K_{v}\omega _{1}\oplus \cdots \oplus K_{v}\omega _{n}\\{\widetilde {O_{v}}}&:=O_{v}\omega _{1}\oplus \cdots \oplus O_{v}\omega _{n}\end{aligned}}$
The adele ring of $E$ is defined as
$\mathbb {A} _{E}:={\prod _{v}}^{'}E_{v}.$
This definition is based on the alternative description of the adele ring as a tensor product equipped with the same topology that was defined when giving an alternate definition of adele ring for number fields. Next, $\mathbb {A} _{E}$ is equipped with the restricted product topology. Then $\mathbb {A} _{E}=E\otimes _{K}\mathbb {A} _{K}$ and $E$ is embedded in $\mathbb {A} _{E}$ naturally via the map $e\mapsto e\otimes 1.$
An alternative definition of the topology on $\mathbb {A} _{E}$ can be provided. Consider all linear maps: $E\to K.$ Using the natural embeddings $E\to \mathbb {A} _{E}$ and $K\to \mathbb {A} _{K},$ extend these linear maps to: $\mathbb {A} _{E}\to \mathbb {A} _{K}.$ The topology on $\mathbb {A} _{E}$ is the coarsest topology for which all these extensions are continuous.
The topology can be defined in a different way. Fixing a basis for $E$ over $K$ results in an isomorphism $E\cong K^{n}.$ Therefore fixing a basis induces an isomorphism $(\mathbb {A} _{K})^{n}\cong \mathbb {A} _{E}.$ The left-hand side is supplied with the product topology and transport this topology with the isomorphism onto the right-hand side. The topology doesn't depend on the choice of the basis, because another basis defines a second isomorphism. By composing both isomorphisms, a linear homeomorphism which transfers the two topologies into each other is obtained. More formally
${\begin{aligned}\mathbb {A} _{E}&=E\otimes _{K}\mathbb {A} _{K}\\&\cong (K\otimes _{K}\mathbb {A} _{K})\oplus \cdots \oplus (K\otimes _{K}\mathbb {A} _{K})\\&\cong \mathbb {A} _{K}\oplus \cdots \oplus \mathbb {A} _{K}\end{aligned}}$
where the sums have $n$ summands. In case of $E=L,$ the definition above is consistent with the results about the adele ring of a finite extension $L/K.$
[9]
The adele ring of an algebra
Let $A$ be a finite-dimensional algebra over $K.$ In particular, $A$ is a finite-dimensional vector-space over $K.$ As a consequence, $\mathbb {A} _{A}$ is defined and $\mathbb {A} _{A}\cong \mathbb {A} _{K}\otimes _{K}A.$ Since there is multiplication on $\mathbb {A} _{K}$ and $A,$ a multiplication on $\mathbb {A} _{A}$ can be defined via:
$\forall \alpha ,\beta \in \mathbb {A} _{K}{\text{ and }}\forall a,b\in A:\qquad (\alpha \otimes _{K}a)\cdot (\beta \otimes _{K}b):=(\alpha \beta )\otimes _{K}(ab).$
As a consequence, $\mathbb {A} _{A}$ is an algebra with a unit over $\mathbb {A} _{K}.$ Let ${\mathcal {B}}$ be a finite subset of $A,$ containing a basis for $A$ over $K.$ For any finite place $v$ , $M_{v}$ is defined as the $O_{v}$-module generated by ${\mathcal {B}}$ in $A_{v}.$ For each finite set of places, $P\supset P_{\infty },$ define
$\mathbb {A} _{A}(P,\alpha )=\prod _{v\in P}A_{v}\times \prod _{v\notin P}M_{v}.$
One can show there is a finite set $P_{0},$ so that $\mathbb {A} _{A}(P,\alpha )$ is an open subring of $\mathbb {A} _{A},$ if $P\supset P_{0}.$ Furthermore $\mathbb {A} _{A}$ is the union of all these subrings and for $A=K,$ the definition above is consistent with the definition of the adele ring.
Trace and norm on the adele ring
Let $L/K$ be a finite extension. Since $\mathbb {A} _{K}=\mathbb {A} _{K}\otimes _{K}K$ and $\mathbb {A} _{L}=\mathbb {A} _{K}\otimes _{K}L$ from the Lemma above, $\mathbb {A} _{K}$ can be interpreted as a closed subring of $\mathbb {A} _{L}.$ For this embedding, write $\operatorname {con} _{L/K}$. Explicitly for all places $w$ of $L$ above $v$ and for any $\alpha \in \mathbb {A} _{K},(\operatorname {con} _{L/K}(\alpha ))_{w}=\alpha _{v}\in K_{v}.$
Let $M/L/K$ be a tower of global fields. Then:
$\operatorname {con} _{M/K}(\alpha )=\operatorname {con} _{M/L}(\operatorname {con} _{L/K}(\alpha ))\qquad \forall \alpha \in \mathbb {A} _{K}.$
Furthermore, restricted to the principal adeles $\operatorname {con} $ is the natural injection $K\to L.$
Let $\{\omega _{1},\ldots ,\omega _{n}\}$ be a basis of the field extension $L/K.$ Then each $\alpha \in \mathbb {A} _{L}$ can be written as $\textstyle \sum _{j=1}^{n}\alpha _{j}\omega _{j},$ where $\alpha _{j}\in \mathbb {A} _{K}$ are unique. The map $\alpha \mapsto \alpha _{j}$ is continuous. Define $\alpha _{ij}$ depending on $\alpha $ via the equations:
${\begin{aligned}\alpha \omega _{1}&=\sum _{j=1}^{n}\alpha _{1j}\omega _{j}\\&\vdots \\\alpha \omega _{n}&=\sum _{j=1}^{n}\alpha _{nj}\omega _{j}\end{aligned}}$
Now, define the trace and norm of $\alpha $ as:
${\begin{aligned}\operatorname {Tr} _{L/K}(\alpha )&:=\operatorname {Tr} ((\alpha _{ij})_{i,j})=\sum _{i=1}^{n}\alpha _{ii}\\N_{L/K}(\alpha )&:=N((\alpha _{ij})_{i,j})=\det((\alpha _{ij})_{i,j})\end{aligned}}$
These are the trace and the determinant of the linear map
${\begin{cases}\mathbb {A} _{L}\to \mathbb {A} _{L}\\x\mapsto \alpha x\end{cases}}$
They are continuous maps on the adele ring, and they fulfil the usual equations:
${\begin{aligned}\operatorname {Tr} _{L/K}(\alpha +\beta )&=\operatorname {Tr} _{L/K}(\alpha )+\operatorname {Tr} _{L/K}(\beta )&&\forall \alpha ,\beta \in \mathbb {A} _{L}\\\operatorname {Tr} _{L/K}(\operatorname {con} (\alpha ))&=n\alpha &&\forall \alpha \in \mathbb {A} _{K}\\N_{L/K}(\alpha \beta )&=N_{L/K}(\alpha )N_{L/K}(\beta )&&\forall \alpha ,\beta \in \mathbb {A} _{L}\\N_{L/K}(\operatorname {con} (\alpha ))&=\alpha ^{n}&&\forall \alpha \in \mathbb {A} _{K}\end{aligned}}$
Furthermore, for $\alpha \in L,$$\operatorname {Tr} _{L/K}(\alpha )$ and $N_{L/K}(\alpha )$ are identical to the trace and norm of the field extension $L/K.$ For a tower of fields $M/L/K,$ the result is:
${\begin{aligned}\operatorname {Tr} _{L/K}(\operatorname {Tr} _{M/L}(\alpha ))&=\operatorname {Tr} _{M/K}(\alpha )&&\forall \alpha \in \mathbb {A} _{M}\\N_{L/K}(N_{M/L}(\alpha ))&=N_{M/K}(\alpha )&&\forall \alpha \in \mathbb {A} _{M}\end{aligned}}$
Moreover, it can be proven that:[10]
${\begin{aligned}\operatorname {Tr} _{L/K}(\alpha )&=\left(\sum _{w|v}\operatorname {Tr} _{L_{w}/K_{v}}(\alpha _{w})\right)_{v}&&\forall \alpha \in \mathbb {A} _{L}\\N_{L/K}(\alpha )&=\left(\prod _{w|v}N_{L_{w}/K_{v}}(\alpha _{w})\right)_{v}&&\forall \alpha \in \mathbb {A} _{L}\end{aligned}}$
Properties of the adele ring
Theorem.[11] For every set of places $S,\mathbb {A} _{K,S}$ is a locally compact topological ring.
Remark. The result above also holds for the adele ring of vector-spaces and algebras over $K.$
Theorem.[12] $K$ is discrete and cocompact in $\mathbb {A} _{K}.$ In particular, $K$ is closed in $\mathbb {A} _{K}.$
Proof. Prove the case $K=\mathbb {Q} .$ To show $\mathbb {Q} \subset \mathbb {A} _{\mathbb {Q} }$ is discrete it is sufficient to show the existence of a neighbourhood of $0$ which contains no other rational number. The general case follows via translation. Define
$U:=\left\{(\alpha _{p})_{p}\left|\forall p<\infty :|\alpha _{p}|_{p}\leq 1\quad {\text{and}}\quad |\alpha _{\infty }|_{\infty }<1\right.\right\}={\widehat {\mathbb {Z} }}\times (-1,1).$ :|\alpha _{p}|_{p}\leq 1\quad {\text{and}}\quad |\alpha _{\infty }|_{\infty }<1\right.\right\}={\widehat {\mathbb {Z} }}\times (-1,1).}
$U$ is an open neighbourhood of $0\in \mathbb {A} _{\mathbb {Q} }.$ It is claimed that $U\cap \mathbb {Q} =\{0\}.$ Let $\beta \in U\cap \mathbb {Q} ,$ then $\beta \in \mathbb {Q} $ and $|\beta |_{p}\leq 1$ for all $p$ and therefore $\beta \in \mathbb {Z} .$ Additionally, $\beta \in (-1,1)$ and therefore $\beta =0.$ Next, to show compactness, define:
$W:=\left\{(\alpha _{p})_{p}\left|\forall p<\infty :|\alpha _{p}|_{p}\leq 1\quad {\text{and}}\quad |\alpha _{\infty }|_{\infty }\leq {\frac {1}{2}}\right.\right\}={\widehat {\mathbb {Z} }}\times \left[-{\frac {1}{2}},{\frac {1}{2}}\right].$ :|\alpha _{p}|_{p}\leq 1\quad {\text{and}}\quad |\alpha _{\infty }|_{\infty }\leq {\frac {1}{2}}\right.\right\}={\widehat {\mathbb {Z} }}\times \left[-{\frac {1}{2}},{\frac {1}{2}}\right].}
Each element in $\mathbb {A} _{\mathbb {Q} }/\mathbb {Q} $ has a representative in $W,$ that is for each $\alpha \in \mathbb {A} _{\mathbb {Q} },$ there exists $\beta \in \mathbb {Q} $ such that $\alpha -\beta \in W.$ Let $\alpha =(\alpha _{p})_{p}\in \mathbb {A} _{\mathbb {Q} },$ be arbitrary and $p$ be a prime for which $|\alpha _{p}|>1.$ Then there exists $r_{p}=z_{p}/p^{x_{p}}$ with $z_{p}\in \mathbb {Z} ,x_{p}\in \mathbb {N} $ and $|\alpha _{p}-r_{p}|\leq 1.$ Replace $\alpha $ with $\alpha -r_{p}$ and let $q\neq p$ be another prime. Then:
$\left|\alpha _{q}-r_{p}\right|_{q}\leq \max \left\{|a_{q}|_{q},|r_{p}|_{q}\right\}\leq \max \left\{|a_{q}|_{q},1\right\}\leq 1.$
Next, it can be claimed that:
$|\alpha _{q}-r_{p}|_{q}\leq 1\Longleftrightarrow |\alpha _{q}|_{q}\leq 1.$
The reverse implication is trivially true. The implication is true, because the two terms of the strong triangle inequality are equal if the absolute values of both integers are different. As a consequence, the (finite) set of primes for which the components of $\alpha $ are not in $\mathbb {Z} _{p}$ is reduced by 1. With iteration, it can be deduced that there exists $r\in \mathbb {Q} $ such that $\alpha -r\in {\widehat {\mathbb {Z} }}\times \mathbb {R} .$ Now select $s\in \mathbb {Z} $ such that $\alpha _{\infty }-r-s\in \left[-{\tfrac {1}{2}},{\tfrac {1}{2}}\right].$ Then $\alpha -(r+s)\in W.$ The continuous projection $\pi :W\to \mathbb {A} _{\mathbb {Q} }/\mathbb {Q} $ is surjective, therefore $\mathbb {A} _{\mathbb {Q} }/\mathbb {Q} ,$ as the continuous image of a compact set, is compact.
Corollary. Let $E$ be a finite-dimensional vector-space over $K.$ Then $E$ is discrete and cocompact in $\mathbb {A} _{E}.$
Theorem. The following are assumed:
• $\mathbb {A} _{\mathbb {Q} }=\mathbb {Q} +\mathbb {A} _{\mathbb {Z} }.$
• $\mathbb {Z} =\mathbb {Q} \cap \mathbb {A} _{\mathbb {Z} }.$
• $\mathbb {A} _{\mathbb {Q} }/\mathbb {Z} $ is a divisible group.[13]
• $\mathbb {Q} \subset \mathbb {A} _{\mathbb {Q} ,{\text{fin}}}$ is dense.
Proof. The first two equations can be proved in an elementary way.
By definition $\mathbb {A} _{\mathbb {Q} }/\mathbb {Z} $ is divisible if for any $n\in \mathbb {N} $ and $y\in \mathbb {A} _{\mathbb {Q} }/\mathbb {Z} $ the equation $nx=y$ has a solution $x\in \mathbb {A} _{\mathbb {Q} }/\mathbb {Z} .$ It is sufficient to show $\mathbb {A} _{\mathbb {Q} }$ is divisible but this is true since $\mathbb {A} _{\mathbb {Q} }$ is a field with positive characteristic in each coordinate.
For the last statement note that $\mathbb {A} _{\mathbb {Q} ,{\text{fin}}}=\mathbb {Q} {\widehat {\mathbb {Z} }},$ because the finite number of denominators in the coordinates of the elements of $\mathbb {A} _{\mathbb {Q} ,{\text{fin}}}$ can be reached through an element $q\in \mathbb {Q} .$ As a consequence, it is sufficient to show $\mathbb {Z} \subset {\widehat {\mathbb {Z} }}$ is dense, that is each open subset $V\subset {\widehat {\mathbb {Z} }}$ contains an element of $\mathbb {Z} .$ Without loss of generality, it can be assumed that
$V=\prod _{p\in E}\left(a_{p}+p^{l_{p}}\mathbb {Z} _{p}\right)\times \prod _{p\notin E}\mathbb {Z} _{p},$
because $(p^{m}\mathbb {Z} _{p})_{m\in \mathbb {N} }$ is a neighbourhood system of $0$ in $\mathbb {Z} _{p}.$ By Chinese Remainder Theorem there exists $l\in \mathbb {Z} $ such that $l\equiv a_{p}{\bmod {p}}^{l_{p}}.$ Since powers of distinct primes are coprime, $l\in V$ follows.
Remark. $\mathbb {A} _{\mathbb {Q} }/\mathbb {Z} $ is not uniquely divisible. Let $y=(0,0,\ldots )+\mathbb {Z} \in \mathbb {A} _{\mathbb {Q} }/\mathbb {Z} $ and $n\geq 2$ be given. Then
${\begin{aligned}x_{1}&=(0,0,\ldots )+\mathbb {Z} \\x_{2}&=\left({\tfrac {1}{n}},{\tfrac {1}{n}},\ldots \right)+\mathbb {Z} \end{aligned}}$
both satisfy the equation $nx=y$ and clearly $x_{1}\neq x_{2}$ ($x_{2}$ is well-defined, because only finitely many primes divide $n$). In this case, being uniquely divisible is equivalent to being torsion-free, which is not true for $\mathbb {A} _{\mathbb {Q} }/\mathbb {Z} $ since $nx_{2}=0,$ but $x_{2}\neq 0$ and $n\neq 0.$
Remark. The fourth statement is a special case of the strong approximation theorem.
Haar measure on the adele ring
Definition. A function $f:\mathbb {A} _{K}\to \mathbb {C} $ is called simple if $\textstyle f=\prod _{v}f_{v},$ where $f_{v}:K_{v}\to \mathbb {C} $ are measurable and $f_{v}=\mathbf {1} _{O_{v}}$ for almost all $v.$
Theorem.[14] Since $\mathbb {A} _{K}$ is a locally compact group with addition, there is an additive Haar measure $dx$ on $\mathbb {A} _{K}.$ This measure can be normalised such that every integrable simple function $\textstyle f=\prod _{v}f_{v}$ satisfies:
$\int _{\mathbb {A} _{K}}f\,dx=\prod _{v}\int _{K_{v}}f_{v}\,dx_{v},$
where for $v<\infty ,dx_{v}$ is the measure on $K_{v}$ such that $O_{v}$ has unit measure and $dx_{\infty }$ is the Lebesgue measure. The product is finite, i.e., almost all factors are equal to one.
The idele group
Definition. Define the idele group of $K$ as the group of units of the adele ring of $K,$ that is $I_{K}:=\mathbb {A} _{K}^{\times }.$ The elements of the idele group are called the ideles of $K.$
Remark. $I_{K}$ is equipped with a topology so that it becomes a topological group. The subset topology inherited from $\mathbb {A} _{K}$ is not a suitable candidate since the group of units of a topological ring equipped with subset topology may not be a topological group. For example, the inverse map in $\mathbb {A} _{\mathbb {Q} }$ is not continuous. The sequence
${\begin{aligned}x_{1}&=(2,1,\ldots )\\x_{2}&=(1,3,1,\ldots )\\x_{3}&=(1,1,5,1,\ldots )\\&\vdots \end{aligned}}$
converges to $1\in \mathbb {A} _{\mathbb {Q} }.$ To see this let $U$ be neighbourhood of $0,$ without loss of generality it can be assumed:
$U=\prod _{p\leq N}U_{p}\times \prod _{p>N}\mathbb {Z} _{p}$
Since $(x_{n})_{p}-1\in \mathbb {Z} _{p}$ for all $p,$ $x_{n}-1\in U$ for $n$ large enough. However, as was seen above the inverse of this sequence does not converge in $\mathbb {A} _{\mathbb {Q} }.$
Lemma. Let $R$ be a topological ring. Define:
${\begin{cases}\iota :R^{\times }\to R\times R\\x\mapsto (x,x^{-1})\end{cases}}$
Equipped with the topology induced from the product on topology on $R\times R$ and $\iota ,R^{\times }$ is a topological group and the inclusion map $R^{\times }\subset R$ is continuous. It is the coarsest topology, emerging from the topology on $R,$ that makes $R^{\times }$ a topological group.
Proof. Since $R$ is a topological ring, it is sufficient to show that the inverse map is continuous. Let $U\subset R^{\times }$ be open, then $U\times U^{-1}\subset R\times R$ is open. It is necessary to show $U^{-1}\subset R^{\times }$ is open or equivalently, that $U^{-1}\times (U^{-1})^{-1}=U^{-1}\times U\subset R\times R$ is open. But this is the condition above.
The idele group is equipped with the topology defined in the Lemma making it a topological group.
Definition. For $S$ a subset of places of $K$ set: $I_{K,S}:=\mathbb {A} _{K,S}^{\times },I_{K}^{S}:=(\mathbb {A} _{K}^{S})^{\times }.$
Lemma. The following identities of topological groups hold:
${\begin{aligned}I_{K,S}&={\prod _{v\in S}}^{'}K_{v}^{\times }\\I_{K}^{S}&={\prod _{v\notin S}}^{'}K_{v}^{\times }\\I_{K}&={\prod _{v}}^{'}K_{v}^{\times }\end{aligned}}$
where the restricted product has the restricted product topology, which is generated by restricted open rectangles of the form
$\prod _{v\in E}U_{v}\times \prod _{v\notin E}O_{v}^{\times },$
where $E$ is a finite subset of the set of all places and $U_{v}\subset K_{v}^{\times }$ are open sets.
Proof. Prove the identity for $I_{K}$; the other two follow similarly. First show the two sets are equal:
${\begin{aligned}I_{K}&=\{x=(x_{v})_{v}\in \mathbb {A} _{K}:\exists y=(y_{v})_{v}\in \mathbb {A} _{K}:xy=1\}\\&=\{x=(x_{v})_{v}\in \mathbb {A} _{K}:\exists y=(y_{v})_{v}\in \mathbb {A} _{K}:x_{v}\cdot y_{v}=1\quad \forall v\}\\&=\{x=(x_{v})_{v}:x_{v}\in K_{v}^{\times }\forall v{\text{ and }}x_{v}\in O_{v}^{\times }{\text{ for almost all }}v\}\\&={\prod _{v}}^{'}K_{v}^{\times }\end{aligned}}$
In going from line 2 to 3, $x$ as well as $x^{-1}=y$ have to be in $\mathbb {A} _{K},$ meaning $x_{v}\in O_{v}$ for almost all $v$ and $x_{v}^{-1}\in O_{v}$ for almost all $v.$ Therefore, $x_{v}\in O_{v}^{\times }$ for almost all $v.$
Now, it is possible to show the topology on the left-hand side equals the topology on the right-hand side. Obviously, every open restricted rectangle is open in the topology of the idele group. On the other hand, for a given $U\subset I_{K},$ which is open in the topology of the idele group, meaning $U\times U^{-1}\subset \mathbb {A} _{K}\times \mathbb {A} _{K}$ is open, so for each $u\in U$ there exists an open restricted rectangle, which is a subset of $U$ and contains $u.$ Therefore, $U$ is the union of all these restricted open rectangles and therefore is open in the restricted product topology.
Lemma. For each set of places, $S,I_{K,S}$ is a locally compact topological group.
Proof. The local compactness follows from the description of $I_{K,S}$ as a restricted product. It being a topological group follows from the above discussion on the group of units of a topological ring.
A neighbourhood system of $1\in \mathbb {A} _{K}(P_{\infty })^{\times }$ is a neighbourhood system of $1\in I_{K}.$ Alternatively, take all sets of the form:
$\prod _{v}U_{v},$
where $U_{v}$ is a neighbourhood of $1\in K_{v}^{\times }$ and $U_{v}=O_{v}^{\times }$ for almost all $v.$
Since the idele group is a locally compact, there exists a Haar measure $d^{\times }x$ on it. This can be normalised, so that
$\int _{I_{K,{\text{fin}}}}\mathbf {1} _{\widehat {O}}\,d^{\times }x=1.$
This is the normalisation used for the finite places. In this equation, $I_{K,{\text{fin}}}$ is the finite idele group, meaning the group of units of the finite adele ring. For the infinite places, use the multiplicative lebesgue measure ${\tfrac {dx}{|x|}}.$
The idele group of a finite extension
Lemma. Let $L/K$ be a finite extension. Then:
$I_{L}={\prod _{v}}^{'}L_{v}^{\times }.$
where the restricted product is with respect to ${\widetilde {O_{v}}}^{\times }.$
Lemma. There is a canonical embedding of $I_{K}$ in $I_{L}.$
Proof. Map $a=(a_{v})_{v}\in I_{K}$ to $a'=(a'_{w})_{w}\in I_{L}$ with the property $a'_{w}=a_{v}\in K_{v}^{\times }\subset L_{w}^{\times }$ for $w|v.$ Therefore, $I_{K}$ can be seen as a subgroup of $I_{L}.$ An element $a=(a_{w})_{w}\in I_{L}$ is in this subgroup if and only if his components satisfy the following properties: $a_{w}\in K_{v}^{\times }$ for $w|v$ and $a_{w}=a_{w'}$ for $w|v$ and $w'|v$ for the same place $v$ of $K.$
The case of vector spaces and algebras
[15]
The idele group of an algebra
Let $A$ be a finite-dimensional algebra over $K.$ Since $\mathbb {A} _{A}^{\times }$ is not a topological group with the subset-topology in general, equip $\mathbb {A} _{A}^{\times }$ with the topology similar to $I_{K}$ above and call $\mathbb {A} _{A}^{\times }$ the idele group. The elements of the idele group are called idele of $A.$
Proposition. Let $\alpha $ be a finite subset of $A,$ containing a basis of $A$ over $K.$ For each finite place $v$ of $K,$ let $\alpha _{v}$ be the $O_{v}$-module generated by $\alpha $ in $A_{v}.$ There exists a finite set of places $P_{0}$ containing $P_{\infty }$ such that for all $v\notin P_{0},$$\alpha _{v}$ is a compact subring of $A_{v}.$ Furthermore, $\alpha _{v}$ contains $A_{v}^{\times }.$ For each $v,A_{v}^{\times }$ is an open subset of $A_{v}$ and the map $x\mapsto x^{-1}$ is continuous on $A_{v}^{\times }.$ As a consequence $x\mapsto (x,x^{-1})$ maps $A_{v}^{\times }$ homeomorphically on its image in $A_{v}\times A_{v}.$ For each $v\notin P_{0},$ the $\alpha _{v}^{\times }$ are the elements of $A_{v}^{\times },$ mapping in $\alpha _{v}\times \alpha _{v}$ with the function above. Therefore, $\alpha _{v}^{\times }$ is an open and compact subgroup of $A_{v}^{\times }.$[16]
Alternative characterisation of the idele group
Proposition. Let $P\supset P_{\infty }$ be a finite set of places. Then
$\mathbb {A} _{A}(P,\alpha )^{\times }:=\prod _{v\in P}A_{v}^{\times }\times \prod _{v\notin P}\alpha _{v}^{\times }$
is an open subgroup of $\mathbb {A} _{A}^{\times },$ where $\mathbb {A} _{A}^{\times }$ is the union of all $\mathbb {A} _{A}(P,\alpha )^{\times }.$[17]
Corollary. In the special case of $A=K$ for each finite set of places $P\supset P_{\infty },$
$\mathbb {A} _{K}(P)^{\times }=\prod _{v\in P}K_{v}^{\times }\times \prod _{v\notin P}O_{v}^{\times }$
is an open subgroup of $\mathbb {A} _{K}^{\times }=I_{K}.$ Furthermore, $I_{K}$ is the union of all $\mathbb {A} _{K}(P)^{\times }.$
Norm on the idele group
The trace and the norm should be transfer from the adele ring to the idele group. It turns out the trace can't be transferred so easily. However, it is possible to transfer the norm from the adele ring to the idele group. Let $\alpha \in I_{K}.$ Then $\operatorname {con} _{L/K}(\alpha )\in I_{L}$ and therefore, it can be said that in injective group homomorphism
$\operatorname {con} _{L/K}:I_{K}\hookrightarrow I_{L}.$
Since $\alpha \in I_{L},$ it is invertible, $N_{L/K}(\alpha )$ is invertible too, because $(N_{L/K}(\alpha ))^{-1}=N_{L/K}(\alpha ^{-1}).$ Therefore $N_{L/K}(\alpha )\in I_{K}.$ As a consequence, the restriction of the norm-function introduces a continuous function:
$N_{L/K}:I_{L}\to I_{K}.$
The Idele class group
Lemma. There is natural embedding of $K^{\times }$ into $I_{K,S}$ given by the diagonal map: $a\mapsto (a,a,a,\ldots ).$
Proof. Since $K^{\times }$ is a subset of $K_{v}^{\times }$ for all $v,$ the embedding is well-defined and injective.
Corollary. $A^{\times }$ is a discrete subgroup of $\mathbb {A} _{A}^{\times }.$
Defenition. In analogy to the ideal class group, the elements of $K^{\times }$ in $I_{K}$ are called principal ideles of $I_{K}.$ The quotient group $C_{K}:=I_{K}/K^{\times }$ is called idele class group of $K.$ This group is related to the ideal class group and is a central object in class field theory.
Remark. $K^{\times }$ is closed in $I_{K},$ therefore $C_{K}$ is a locally compact topological group and a Hausdorff space.
Lemma.[18] Let $L/K$ be a finite extension. The embedding $I_{K}\to I_{L}$ induces an injective map:
${\begin{cases}C_{K}\to C_{L}\\\alpha K^{\times }\mapsto \alpha L^{\times }\end{cases}}$
Absolute value on the idele group of K and 1-idele
Definition. For $\alpha =(\alpha _{v})_{v}\in I_{K}$ define: $\textstyle |\alpha |:=\prod _{v}|\alpha _{v}|_{v}.$ Since $\alpha $ is an idele this product is finite and therefore well-defined.
Remark. The definition can be extended to $\mathbb {A} _{K}$ by allowing infinite products. However, these infinite products vanish and so $|\cdot |$ vanishes on $\mathbb {A} _{K}\setminus I_{K}.$ $|\cdot |$ will be used to denote both the function on $I_{K}$ and $\mathbb {A} _{K}.$
Theorem. $|\cdot |:I_{K}\to \mathbb {R} _{+}$ is a continuous group homomorphism.
Proof. Let $\alpha ,\beta \in I_{K}.$
${\begin{aligned}|\alpha \cdot \beta |&=\prod _{v}|(\alpha \cdot \beta )_{v}|_{v}\\&=\prod _{v}|\alpha _{v}\cdot \beta _{v}|_{v}\\&=\prod _{v}(|\alpha _{v}|_{v}\cdot |\beta _{v}|_{v})\\&=\left(\prod _{v}|\alpha _{v}|_{v}\right)\cdot \left(\prod _{v}|\beta _{v}|_{v}\right)\\&=|\alpha |\cdot |\beta |\end{aligned}}$
where it is used that all products are finite. The map is continuous which can be seen using an argument dealing with sequences. This reduces the problem to whether $|\cdot |$ is continuous on $K_{v}.$ However, this is clear, because of the reverse triangle inequality.
Definition. The set of $1$-idele can be defined as:
$I_{K}^{1}:=\{x\in I_{K}:|x|=1\}=\ker(|\cdot |).$
$I_{K}^{1}$ is a subgroup of $I_{K}.$ Since $I_{K}^{1}=|\cdot |^{-1}(\{1\}),$ it is a closed subset of $\mathbb {A} _{K}.$ Finally the $\mathbb {A} _{K}$-topology on $I_{K}^{1}$ equals the subset-topology of $I_{K}$ on $I_{K}^{1}.$[19][20]
Artin's Product Formula. $|k|=1$ for all $k\in K^{\times }.$
Proof.[21] Proof of the formula for number fields, the case of global function fields can be proved similarly. Let $K$ be a number field and $a\in K^{\times }.$ It has to be shown that:
$\prod _{v}|a|_{v}=1.$
For finite place $v$ for which the corresponding prime ideal ${\mathfrak {p}}_{v}$ does not divide $(a)$, $v(a)=0$ and therefore $|a|_{v}=1.$ This is valid for almost all ${\mathfrak {p}}_{v}.$ There is:
${\begin{aligned}\prod _{v}|a|_{v}&=\prod _{p\leq \infty }\prod _{v|p}|a|_{v}\\&=\prod _{p\leq \infty }\prod _{v|p}|N_{K_{v}/\mathbb {Q} _{p}}(a)|_{p}\\&=\prod _{p\leq \infty }|N_{K/\mathbb {Q} }(a)|_{p}\end{aligned}}$
In going from line 1 to line 2, the identity $|a|_{w}=|N_{L_{w}/K_{v}}(a)|_{v},$ was used where $v$ is a place of $K$ and $w$ is a place of $L,$ lying above $v.$ Going from line 2 to line 3, a property of the norm is used. The norm is in $\mathbb {Q} $ so without loss of generality it can be assumed that $a\in \mathbb {Q} .$ Then $a$ possesses a unique integer factorisation:
$a=\pm \prod _{p<\infty }p^{v_{p}},$
where $v_{p}\in \mathbb {Z} $ is $0$ for almost all $p.$ By Ostrowski's theorem all absolute values on $\mathbb {Q} $ are equivalent to the real absolute value $|\cdot |_{\infty }$ or a $p$-adic absolute value. Therefore:
${\begin{aligned}|a|&=\left(\prod _{p<\infty }|a|_{p}\right)\cdot |a|_{\infty }\\&=\left(\prod _{p<\infty }p^{-v_{p}}\right)\cdot \left(\prod _{p<\infty }p^{v_{p}}\right)\\&=1\end{aligned}}$
Lemma.[22] There exists a constant $C,$ depending only on $K,$ such that for every $\alpha =(\alpha _{v})_{v}\in \mathbb {A} _{K}$ satisfying $\textstyle \prod _{v}|\alpha _{v}|_{v}>C,$ there exists $\beta \in K^{\times }$ such that $|\beta _{v}|_{v}\leq |\alpha _{v}|_{v}$ for all $v.$
Corollary. Let $v_{0}$ be a place of $K$ and let $\delta _{v}>0$ be given for all $v\neq v_{0}$ with the property $\delta _{v}=1$ for almost all $v.$ Then there exists $\beta \in K^{\times },$ so that $|\beta |\leq \delta _{v}$ for all $v\neq v_{0}.$
Proof. Let $C$ be the constant from the lemma. Let $\pi _{v}$ be a uniformising element of $O_{v}.$ Define the adele $\alpha =(\alpha _{v})_{v}$ via $\alpha _{v}:=\pi _{v}^{k_{v}}$ with $k_{v}\in \mathbb {Z} $ minimal, so that $|\alpha _{v}|_{v}\leq \delta _{v}$ for all $v\neq v_{0}.$ Then $k_{v}=0$ for almost all $v.$ Define $\alpha _{v_{0}}:=\pi _{v_{0}}^{k_{v_{0}}}$ with $k_{v_{0}}\in \mathbb {Z} ,$ so that $\textstyle \prod _{v}|\alpha _{v}|_{v}>C.$ This works, because $k_{v}=0$ for almost all $v.$ By the Lemma there exists $\beta \in K^{\times },$ so that $|\beta |_{v}\leq |\alpha _{v}|_{v}\leq \delta _{v}$ for all $v\neq v_{0}.$
Theorem. $K^{\times }$ is discrete and cocompact in $I_{K}^{1}.$
Proof.[23] Since $K^{\times }$ is discrete in $I_{K}$ it is also discrete in $I_{K}^{1}.$ To prove the compactness of $I_{K}^{1}/K^{\times }$ let $C$ is the constant of the Lemma and suppose $\alpha \in \mathbb {A} _{K}$ satisfying $\textstyle \prod _{v}|\alpha _{v}|_{v}>C$ is given. Define:
$W_{\alpha }:=\left\{\xi =(\xi _{v})_{v}\in \mathbb {A} _{K}||\xi _{v}|_{v}\leq |\alpha _{v}|_{v}{\text{ for all }}v\right\}.$
Clearly $W_{\alpha }$ is compact. It can be claimed that the natural projection $W_{\alpha }\cap I_{K}^{1}\to I_{K}^{1}/K^{\times }$ is surjective. Let $\beta =(\beta _{v})_{v}\in I_{K}^{1}$ be arbitrary, then:
$|\beta |=\prod _{v}|\beta _{v}|_{v}=1,$
and therefore
$\prod _{v}|\beta _{v}^{-1}|_{v}=1.$
It follows that
$\prod _{v}|\beta _{v}^{-1}\alpha _{v}|_{v}=\prod _{v}|\alpha _{v}|_{v}>C.$
By the Lemma there exists $\eta \in K^{\times }$ such that $|\eta |_{v}\leq |\beta _{v}^{-1}\alpha _{v}|_{v}$ for all $v,$ and therefore $\eta \beta \in W_{\alpha }$ proving the surjectivity of the natural projection. Since it is also continuous the compactness follows.
Theorem.[24] There is a canonical isomorphism $I_{\mathbb {Q} }^{1}/\mathbb {Q} ^{\times }\cong {\widehat {\mathbb {Z} }}^{\times }.$ Furthermore, ${\widehat {\mathbb {Z} }}^{\times }\times \{1\}\subset I_{\mathbb {Q} }^{1}$ is a set of representatives for $I_{\mathbb {Q} }^{1}/\mathbb {Q} ^{\times }$ and ${\widehat {\mathbb {Z} }}^{\times }\times (0,\infty )\subset I_{\mathbb {Q} }$ is a set of representatives for $I_{\mathbb {Q} }/\mathbb {Q} ^{\times }.$
Proof. Consider the map
${\begin{cases}\phi :{\widehat {\mathbb {Z} }}^{\times }\to I_{\mathbb {Q} }^{1}/\mathbb {Q} ^{\times }\\(a_{p})_{p}\mapsto ((a_{p})_{p},1)\mathbb {Q} ^{\times }\end{cases}}$ :{\widehat {\mathbb {Z} }}^{\times }\to I_{\mathbb {Q} }^{1}/\mathbb {Q} ^{\times }\\(a_{p})_{p}\mapsto ((a_{p})_{p},1)\mathbb {Q} ^{\times }\end{cases}}}
This map is well-defined, since $|a_{p}|_{p}=1$ for all $p$ and therefore $\textstyle \left(\prod _{p<\infty }|a_{p}|_{p}\right)\cdot 1=1.$ Obviously $\phi $ is a continuous group homomorphism. Now suppose $((a_{p})_{p},1)\mathbb {Q} ^{\times }=((b_{p})_{p},1)\mathbb {Q} ^{\times }.$ Then there exists $q\in \mathbb {Q} ^{\times }$ such that $((a_{p})_{p},1)q=((b_{p})_{p},1).$ By considering the infinite place it can be seen that $q=1$ proves injectivity. To show surjectivity let $((\beta _{p})_{p},\beta _{\infty })\mathbb {Q} ^{\times }\in I_{\mathbb {Q} }^{1}/\mathbb {Q} ^{\times }.$ The absolute value of this element is $1$ and therefore
$|\beta _{\infty }|_{\infty }={\frac {1}{\prod _{p}|\beta _{p}|_{p}}}\in \mathbb {Q} .$
Hence $\beta _{\infty }\in \mathbb {Q} $ and there is:
$((\beta _{p})_{p},\beta _{\infty })\mathbb {Q} ^{\times }=\left(\left({\frac {\beta _{p}}{\beta _{\infty }}}\right)_{p},1\right)\mathbb {Q} ^{\times }.$
Since
$\forall p:\qquad \left|{\frac {\beta _{p}}{\beta _{\infty }}}\right|_{p}=1,$
It has been concluded that $\phi $ is surjective.
Theorem.[24] The absolute value function induces the following isomorphisms of topological groups:
${\begin{aligned}I_{\mathbb {Q} }&\cong I_{\mathbb {Q} }^{1}\times (0,\infty )\\I_{\mathbb {Q} }^{1}&\cong I_{\mathbb {Q} ,{\text{fin}}}\times \{\pm 1\}.\end{aligned}}$
Proof. The isomorphisms are given by:
${\begin{cases}\psi :I_{\mathbb {Q} }\to I_{\mathbb {Q} }^{1}\times (0,\infty )\\a=(a_{\text{fin}},a_{\infty })\mapsto \left(a_{\text{fin}},{\frac {a_{\infty }}{|a|}},|a|\right)\end{cases}}\qquad {\text{and}}\qquad {\begin{cases}{\widetilde {\psi }}:I_{\mathbb {Q} ,{\text{fin}}}\times \{\pm 1\}\to I_{\mathbb {Q} }^{1}\\(a_{\text{fin}},\varepsilon )\mapsto \left(a_{\text{fin}},{\frac {\varepsilon }{|a_{\text{fin}}|}}\right)\end{cases}}$
Relation between ideal class group and idele class group
Theorem. Let $K$ be a number field with ring of integers $O,$ group of fractional ideals $J_{K},$ and ideal class group $\operatorname {Cl} _{K}=J_{K}/K^{\times }.$ Here's the following isomorphisms
${\begin{aligned}J_{K}&\cong I_{K,{\text{fin}}}/{\widehat {O}}^{\times }\\\operatorname {Cl} _{K}&\cong C_{K,{\text{fin}}}/{\widehat {O}}^{\times }K^{\times }\\\operatorname {Cl} _{K}&\cong C_{K}/\left({\widehat {O}}^{\times }\times \prod _{v|\infty }K_{v}^{\times }\right)K^{\times }\end{aligned}}$
where $C_{K,{\text{fin}}}:=I_{K,{\text{fin}}}/K^{\times }$ has been defined.
Proof. Let $v$ be a finite place of $K$ and let $|\cdot |_{v}$ be a representative of the equivalence class $v.$ Define
${\mathfrak {p}}_{v}:=\{x\in O:|x|_{v}<1\}.$
Then ${\mathfrak {p}}_{v}$ is a prime ideal in $O.$ The map $v\mapsto {\mathfrak {p}}_{v}$ is a bijection between finite places of $K$ and non-zero prime ideals of $O.$ The inverse is given as follows: a prime ideal ${\mathfrak {p}}$ is mapped to the valuation $v_{\mathfrak {p}},$ given by
${\begin{aligned}v_{\mathfrak {p}}(x)&:=\max\{k\in \mathbb {N} _{0}:x\in {\mathfrak {p}}^{k}\}\quad \forall x\in O^{\times }\\v_{\mathfrak {p}}\left({\frac {x}{y}}\right)&:=v_{\mathfrak {p}}(x)-v_{\mathfrak {p}}(y)\quad \forall x,y\in O^{\times }\end{aligned}}$
The following map is well-defined:
${\begin{cases}(\cdot ):I_{K,{\text{fin}}}\to J_{K}\\\alpha =(\alpha _{v})_{v}\mapsto \prod _{v<\infty }{\mathfrak {p}}_{v}^{v(\alpha _{v})},\end{cases}}$
The map $(\cdot )$ is obviously a surjective homomorphism and $\ker((\cdot ))={\widehat {O}}^{\times }.$ The first isomorphism follows from fundamental theorem on homomorphism. Now, both sides are divided by $K^{\times }.$ This is possible, because
${\begin{aligned}(\alpha )&=((\alpha ,\alpha ,\dotsc ))\\&=\prod _{v<\infty }{\mathfrak {p}}_{v}^{v(\alpha )}\\&=(\alpha )&&{\text{ for all }}\alpha \in K^{\times }.\end{aligned}}$
Please, note the abuse of notation: On the left hand side in line 1 of this chain of equations, $(\cdot )$ stands for the map defined above. Later, the embedding of $K^{\times }$ into $I_{K,{\text{fin}}}$ is used. In line 2, the definition of the map is used. Finally, use that $O$ is a Dedekind domain and therefore each ideal can be written as a product of prime ideals. In other words, the map $(\cdot )$ is a $K^{\times }$-equivariant group homomorphism. As a consequence, the map above induces a surjective homomorphism
${\begin{cases}\phi :C_{K,{\text{fin}}}\to \operatorname {Cl} _{K}\\\alpha K^{\times }\mapsto (\alpha )K^{\times }\end{cases}}$
To prove the second isomorphism, it has to be shown that $\ker(\phi )={\widehat {O}}^{\times }K^{\times }.$ Consider $\xi =(\xi _{v})_{v}\in {\widehat {O}}^{\times }.$ Then $\textstyle \phi (\xi K^{\times })=\prod _{v}{\mathfrak {p}}_{v}^{v(\xi _{v})}K^{\times }=K^{\times },$ because $v(\xi _{v})=0$ for all $v.$ On the other hand, consider $\xi K^{\times }\in C_{K,{\text{fin}}}$ with $\phi (\xi K^{\times })=OK^{\times },$ which allows to write $\textstyle \prod _{v}{\mathfrak {p}}_{v}^{v(\xi _{v})}K^{\times }=OK^{\times }.$ As a consequence, there exists a representative, such that: $\textstyle \prod _{v}{\mathfrak {p}}_{v}^{v(\xi '_{v})}=O.$ Consequently, $\xi '\in {\widehat {O}}^{\times }$ and therefore $\xi K^{\times }=\xi 'K^{\times }\in {\widehat {O}}^{\times }K^{\times }.$ The second isomorphism of the theorem has been proven.
For the last isomorphism note that $\phi $ induces a surjective group homomorphism ${\widetilde {\phi }}:C_{K}\to \operatorname {Cl} _{K}$ with
$\ker({\widetilde {\phi }})=\left({\widehat {O}}^{\times }\times \prod _{v|\infty }K_{v}^{\times }\right)K^{\times }.$
Remark. Consider $I_{K,{\text{fin}}}$ with the idele topology and equip $J_{K},$ with the discrete topology. Since $(\{{\mathfrak {a}}\})^{-1}$ is open for each ${\mathfrak {a}}\in J_{K},(\cdot )$ is continuous. It stands, that $(\{{\mathfrak {a}}\})^{-1}=\alpha {\widehat {O}}^{\times }$ is open, where $\alpha =(\alpha _{v})_{v}\in \mathbb {A} _{K,{\text{fin}}},$ so that $\textstyle {\mathfrak {a}}=\prod _{v}{\mathfrak {p}}_{v}^{v(\alpha _{v})}.$
Decomposition of the idele group and idele class group of K
Theorem.
${\begin{aligned}I_{K}&\cong I_{K}^{1}\times M:\quad {\begin{cases}M\subset I_{K}{\text{ discrete and }}M\cong \mathbb {Z} &\operatorname {char} (K)>0\\M\subset I_{K}{\text{ closed and }}M\cong \mathbb {R} _{+}&\operatorname {char} (K)=0\end{cases}}\\C_{K}&\cong I_{K}^{1}/K^{\times }\times N:\quad {\begin{cases}N=\mathbb {Z} &\operatorname {char} (K)>0\\N=\mathbb {R} _{+}&\operatorname {char} (K)=0\end{cases}}\end{aligned}}$
Proof. $\operatorname {char} (K)=p>0.$ For each place $v$ of $K,\operatorname {char} (K_{v})=p,$ so that for all $x\in K_{v}^{\times },$ $|x|_{v}$ belongs to the subgroup of $\mathbb {R} _{+},$ generated by $p.$ Therefore for each $z\in I_{K},$ $|z|$ is in the subgroup of $\mathbb {R} _{+},$ generated by $p.$ Therefore the image of the homomorphism $z\mapsto |z|$ is a discrete subgroup of $\mathbb {R} _{+},$ generated by $p.$ Since this group is non-trivial, it is generated by $Q=p^{m}$ for some $m\in \mathbb {N} .$ Choose $z_{1}\in I_{K},$ so that $|z_{1}|=Q,$ then $I_{K}$ is the direct product of $I_{K}^{1}$ and the subgroup generated by $z_{1}.$ This subgroup is discrete and isomorphic to $\mathbb {Z} .$
$\operatorname {char} (K)=0.$ For $\lambda \in \mathbb {R} _{+}$ define:
$z(\lambda )=(z_{v})_{v},\quad z_{v}={\begin{cases}1&v\notin P_{\infty }\\\lambda &v\in P_{\infty }\end{cases}}$
The map $\lambda \mapsto z(\lambda )$ is an isomorphism of $\mathbb {R} _{+}$ in a closed subgroup $M$ of $I_{K}$ and $I_{K}\cong M\times I_{K}^{1}.$ The isomorphism is given by multiplication:
${\begin{cases}\phi :M\times I_{K}^{1}\to I_{K},\\((\alpha _{v})_{v},(\beta _{v})_{v})\mapsto (\alpha _{v}\beta _{v})_{v}\end{cases}}$
Obviously, $\phi $ is a homomorphism. To show it is injective, let $(\alpha _{v}\beta _{v})_{v}=1.$ Since $\alpha _{v}=1$ for $v<\infty ,$ it stands that $\beta _{v}=1$ for $v<\infty .$ Moreover, it exists a $\lambda \in \mathbb {R} _{+},$ so that $\alpha _{v}=\lambda $ for $v|\infty .$ Therefore, $\beta _{v}=\lambda ^{-1}$ for $v|\infty .$ Moreover $\textstyle \prod _{v}|\beta _{v}|_{v}=1,$ implies $\lambda ^{n}=1,$ where $n$ is the number of infinite places of $K.$ As a consequence $\lambda =1$ and therefore $\phi $ is injective. To show surjectivity, let $\gamma =(\gamma _{v})_{v}\in I_{K}.$ It is defined that $\lambda :=|\gamma |^{\frac {1}{n}}$ :=|\gamma |^{\frac {1}{n}}} and furthermore, $\alpha _{v}=1$ for $v<\infty $ and $\alpha _{v}=\lambda $ for $v|\infty .$ Define $\textstyle \beta ={\frac {\gamma }{\alpha }}.$ It stands, that $\textstyle |\beta |={\frac {|\gamma |}{|\alpha |}}={\frac {\lambda ^{n}}{\lambda ^{n}}}=1.$ Therefore, $\phi $ is surjective.
The other equations follow similarly.
Characterisation of the idele group
Theorem.[25] Let $K$ be a number field. There exists a finite set of places $S,$ such that:
$I_{K}=\left(I_{K,S}\times \prod _{v\notin S}O_{v}^{\times }\right)K^{\times }=\left(\prod _{v\in S}K_{v}^{\times }\times \prod _{v\notin S}O_{v}^{\times }\right)K^{\times }.$
Proof. The class number of a number field is finite so let ${\mathfrak {a}}_{1},\ldots ,{\mathfrak {a}}_{h}$ be the ideals, representing the classes in $\operatorname {Cl} _{K}.$ These ideals are generated by a finite number of prime ideals ${\mathfrak {p}}_{1},\ldots ,{\mathfrak {p}}_{n}.$ Let $S$ be a finite set of places containing $P_{\infty }$ and the finite places corresponding to ${\mathfrak {p}}_{1},\ldots ,{\mathfrak {p}}_{n}.$ Consider the isomorphism:
$I_{K}/\left(\prod _{v<\infty }O_{v}^{\times }\times \prod _{v|\infty }K_{v}^{\times }\right)\cong J_{K},$
induced by
$(\alpha _{v})_{v}\mapsto \prod _{v<\infty }{\mathfrak {p}}_{v}^{v(\alpha _{v})}.$
At infinite places the statement is immediate, so the statement has been proved for finite places. The inclusion ″$\supset $″ is obvious. Let $\alpha \in I_{K,{\text{fin}}}.$ The corresponding ideal $\textstyle (\alpha )=\prod _{v<\infty }{\mathfrak {p}}_{v}^{v(\alpha _{v})}$ belongs to a class ${\mathfrak {a}}_{i}K^{\times },$ meaning $(\alpha )={\mathfrak {a}}_{i}(a)$ for a principal ideal $(a).$ The idele $\alpha '=\alpha a^{-1}$ maps to the ideal ${\mathfrak {a}}_{i}$ under the map $I_{K,{\text{fin}}}\to J_{K}.$ That means $\textstyle {\mathfrak {a}}_{i}=\prod _{v<\infty }{\mathfrak {p}}_{v}^{v(\alpha '_{v})}.$ Since the prime ideals in ${\mathfrak {a}}_{i}$ are in $S,$ it follows $v(\alpha '_{v})=0$ for all $v\notin S,$ that means $\alpha '_{v}\in O_{v}^{\times }$ for all $v\notin S.$ It follows, that $\alpha '=\alpha a^{-1}\in I_{K,S},$ therefore $\alpha \in I_{K,S}K^{\times }.$
Applications
Finiteness of the class number of a number field
In the previous section the fact that the class number of a number field is finite had been used. Here this statement can be proved:
Theorem (finiteness of the class number of a number field). Let $K$ be a number field. Then $|\operatorname {Cl} _{K}|<\infty .$
Proof. The map
${\begin{cases}I_{K}^{1}\to J_{K}\\\left((\alpha _{v})_{v<\infty },(\alpha _{v})_{v|\infty }\right)\mapsto \prod _{v<\infty }{\mathfrak {p}}_{v}^{v(\alpha _{v})}\end{cases}}$
is surjective and therefore $\operatorname {Cl} _{K}$ is the continuous image of the compact set $I_{K}^{1}/K^{\times }.$ Thus, $\operatorname {Cl} _{K}$ is compact. In addition, it is discrete and so finite.
Remark. There is a similar result for the case of a global function field. In this case, the so-called divisor group is defined. It can be shown that the quotient of the set of all divisors of degree $0$ by the set of the principal divisors is a finite group.[26]
Group of units and Dirichlet's unit theorem
Let $P\supset P_{\infty }$ be a finite set of places. Define
${\begin{aligned}\Omega (P)&:=\prod _{v\in P}K_{v}^{\times }\times \prod _{v\notin P}O_{v}^{\times }=(\mathbb {A} _{K}(P))^{\times }\\E(P)&:=K^{\times }\cap \Omega (P)\end{aligned}}$
Then $E(P)$ is a subgroup of $K^{\times },$ containing all elements $\xi \in K^{\times }$ satisfying $v(\xi )=0$ for all $v\notin P.$ Since $K^{\times }$ is discrete in $I_{K},$ $E(P)$ is a discrete subgroup of $\Omega (P)$ and with the same argument, $E(P)$ is discrete in $\Omega _{1}(P):=\Omega (P)\cap I_{K}^{1}.$
An alternative definition is: $E(P)=K(P)^{\times },$ where $K(P)$ is a subring of $K$ defined by
$K(P):=K\cap \left(\prod _{v\in P}K_{v}\times \prod _{v\notin P}O_{v}\right).$
As a consequence, $K(P)$ contains all elements $\xi \in K,$ which fulfil $v(\xi )\geq 0$ for all $v\notin P.$
Lemma 1. Let $0<c\leq C<\infty .$ The following set is finite:
$\left\{\eta \in E(P):\left.{\begin{cases}|\eta _{v}|_{v}=1&\forall v\notin P\\c\leq |\eta _{v}|_{v}\leq C&\forall v\in P\end{cases}}\right\}\right\}.$
Proof. Define
$W:=\left\{(\alpha _{v})_{v}:\left.{\begin{cases}|\alpha _{v}|_{v}=1&\forall v\notin P\\c\leq |\alpha _{v}|_{v}\leq C&\forall v\in P\end{cases}}\right\}\right\}.$
$W$ is compact and the set described above is the intersection of $W$ with the discrete subgroup $K^{\times }$ in $I_{K}$ and therefore finite.
Lemma 2. Let $E$ be set of all $\xi \in K$ such that $|\xi |_{v}=1$ for all $v.$ Then $E=\mu (K),$ the group of all roots of unity of $K.$ In particular it is finite and cyclic.
Proof. All roots of unity of $K$ have absolute value $1$ so $\mu (K)\subset E.$ For converse note that Lemma 1 with $c=C=1$ and any $P$ implies $E$ is finite. Moreover $E\subset E(P)$ for each finite set of places $P\supset P_{\infty }.$ Finally suppose there exists $\xi \in E,$ which is not a root of unity of $K.$ Then $\xi ^{n}\neq 1$ for all $n\in \mathbb {N} $ contradicting the finiteness of $E.$
Unit Theorem. $E(P)$ is the direct product of $E$ and a group isomorphic to $\mathbb {Z} ^{s},$ where $s=0,$ if $P=\emptyset $ and $s=|P|-1,$ if $P\neq \emptyset .$[27]
Dirichlet's Unit Theorem. Let $K$ be a number field. Then $O^{\times }\cong \mu (K)\times \mathbb {Z} ^{r+s-1},$ where $\mu (K)$ is the finite cyclic group of all roots of unity of $K,r$ is the number of real embeddings of $K$ and $s$ is the number of conjugate pairs of complex embeddings of $K.$ It stands, that $[K:\mathbb {Q} ]=r+2s.$
Remark. The Unit Theorem generalises Dirichlet's Unit Theorem. To see this, let $K$ be a number field. It is already known that $E=\mu (K),$ set $P=P_{\infty }$ and note $|P_{\infty }|=r+s.$
Then there is:
${\begin{aligned}E\times \mathbb {Z} ^{r+s-1}=E(P_{\infty })&=K^{\times }\cap \left(\prod _{v|\infty }K_{v}^{\times }\times \prod _{v<\infty }O_{v}^{\times }\right)\\&\cong K^{\times }\cap \left(\prod _{v<\infty }O_{v}^{\times }\right)\\&\cong O^{\times }\end{aligned}}$
Approximation theorems
Weak Approximation Theorem.[28] Let $|\cdot |_{1},\ldots ,|\cdot |_{N},$ be inequivalent valuations of $K.$ Let $K_{n}$ be the completion of $K$ with respect to $|\cdot |_{n}.$ Embed $K$ diagonally in $K_{1}\times \cdots \times K_{N}.$ Then $K$ is everywhere dense in $K_{1}\times \cdots \times K_{N}.$ In other words, for each $\varepsilon >0$ and for each $(\alpha _{1},\ldots ,\alpha _{N})\in K_{1}\times \cdots \times K_{N},$ there exists $\xi \in K,$ such that:
$\forall n\in \{1,\ldots ,N\}:\quad |\alpha _{n}-\xi |_{n}<\varepsilon .$
Strong Approximation Theorem.[29] Let $v_{0}$ be a place of $K.$ Define
$V:={\prod _{v\neq v_{0}}}^{'}K_{v}.$
Then $K$ is dense in $V.$
Remark. The global field is discrete in its adele ring. The strong approximation theorem tells us that, if one place (or more) is omitted, the property of discreteness of $K$ is turned into a denseness of $K.$
Hasse principle
Hasse-Minkowski Theorem. A quadratic form on $K$ is zero, if and only if, the quadratic form is zero in each completion $K_{v}.$
Remark. This is the Hasse principle for quadratic forms. For polynomials of degree larger than 2 the Hasse principle isn't valid in general. The idea of the Hasse principle (also known as local–global principle) is to solve a given problem of a number field $K$ by doing so in its completions $K_{v}$ and then concluding on a solution in $K.$
Characters on the adele ring
Definition. Let $G$ be a locally compact abelian group. The character group of $G$ is the set of all characters of $G$ and is denoted by ${\widehat {G}}.$ Equivalently ${\widehat {G}}$ is the set of all continuous group homomorphisms from $G$ to $\mathbb {T} :=\{z\in \mathbb {C} :|z|=1\}.$ :=\{z\in \mathbb {C} :|z|=1\}.} Equip ${\widehat {G}}$ with the topology of uniform convergence on compact subsets of $G.$ One can show that ${\widehat {G}}$ is also a locally compact abelian group.
Theorem. The adele ring is self-dual: $\mathbb {A} _{K}\cong {\widehat {\mathbb {A} _{K}}}.$
Proof. By reduction to local coordinates, it is sufficient to show each $K_{v}$ is self-dual. This can be done by using a fixed character of $K_{v}.$ The idea has been illustrated by showing $\mathbb {R} $ is self-dual. Define:
${\begin{cases}e_{\infty }:\mathbb {R} \to \mathbb {T} \\e_{\infty }(t):=\exp(2\pi it)\end{cases}}$
Then the following map is an isomorphism which respects topologies:
${\begin{cases}\phi :\mathbb {R} \to {\widehat {\mathbb {R} }}\\s\mapsto {\begin{cases}\phi _{s}:\mathbb {R} \to \mathbb {T} \\\phi _{s}(t):=e_{\infty }(ts)\end{cases}}\end{cases}}$ :\mathbb {R} \to {\widehat {\mathbb {R} }}\\s\mapsto {\begin{cases}\phi _{s}:\mathbb {R} \to \mathbb {T} \\\phi _{s}(t):=e_{\infty }(ts)\end{cases}}\end{cases}}}
Theorem (algebraic and continuous duals of the adele ring).[30] Let $\chi $ be a non-trivial character of $\mathbb {A} _{K},$ which is trivial on $K.$ Let $E$ be a finite-dimensional vector-space over $K.$ Let $E^{\star }$ and $\mathbb {A} _{E}^{\star }$ be the algebraic duals of $E$ and $\mathbb {A} _{E}.$ Denote the topological dual of $\mathbb {A} _{E}$ by $\mathbb {A} _{E}'$ and use $\langle \cdot ,\cdot \rangle $ and $[{\cdot },{\cdot }]$ to indicate the natural bilinear pairings on $\mathbb {A} _{E}\times \mathbb {A} _{E}'$ and $\mathbb {A} _{E}\times \mathbb {A} _{E}^{\star }.$ Then the formula $\langle e,e'\rangle =\chi ([e,e^{\star }])$ for all $e\in \mathbb {A} _{E}$ determines an isomorphism $e^{\star }\mapsto e'$ of $\mathbb {A} _{E}^{\star }$ onto $\mathbb {A} _{E}',$ where $e'\in \mathbb {A} _{E}'$ and $e^{\star }\in \mathbb {A} _{E}^{\star }.$ Moreover, if $e^{\star }\in \mathbb {A} _{E}^{\star }$ fulfils $\chi ([e,e^{\star }])=1$ for all $e\in E,$ then $e^{\star }\in E^{\star }.$
Tate's thesis
With the help of the characters of $\mathbb {A} _{K},$ Fourier analysis can be done on the adele ring.[31] John Tate in his thesis "Fourier analysis in Number Fields and Hecke Zeta Functions"[6] proved results about Dirichlet L-functions using Fourier analysis on the adele ring and the idele group. Therefore, the adele ring and the idele group have been applied to study the Riemann zeta function and more general zeta functions and the L-functions. Adelic forms of these functions can be defined and represented as integrals over the adele ring or the idele group, with respect to corresponding Haar measures. Functional equations and meromorphic continuations of these functions can be shown. For example, for all $s\in \mathbb {C} $ with $\Re (s)>1,$
$\int _{\widehat {\mathbb {Z} }}|x|^{s}d^{\times }x=\zeta (s),$
where $d^{\times }x$ is the unique Haar measure on $I_{\mathbb {Q} ,{\text{fin}}}$ normalised such that ${\widehat {\mathbb {Z} }}^{\times }$ has volume one and is extended by zero to the finite adele ring. As a result, the Riemann zeta function can be written as an integral over (a subset of) the adele ring.[32]
Automorphic forms
The theory of automorphic forms is a generalisation of Tate's thesis by replacing the idele group with analogous higher dimensional groups. To see this note:
${\begin{aligned}I_{\mathbb {Q} }&=\operatorname {GL} (1,\mathbb {A} _{\mathbb {Q} })\\I_{\mathbb {Q} }^{1}&=(\operatorname {GL} (1,\mathbb {A} _{\mathbb {Q} }))^{1}:=\{x\in \operatorname {GL} (1,\mathbb {A} _{\mathbb {Q} }):|x|=1\}\\\mathbb {Q} ^{\times }&=\operatorname {GL} (1,\mathbb {Q} )\end{aligned}}$
Based on these identification a natural generalisation would be to replace the idele group and the 1-idele with:
${\begin{aligned}I_{\mathbb {Q} }&\leftrightsquigarrow \operatorname {GL} (2,\mathbb {A} _{\mathbb {Q} })\\I_{\mathbb {Q} }^{1}&\leftrightsquigarrow (\operatorname {GL} (2,\mathbb {A} _{\mathbb {Q} }))^{1}:=\{x\in \operatorname {GL} (2,\mathbb {A} _{\mathbb {Q} }):|\det(x)|=1\}\\\mathbb {Q} &\leftrightsquigarrow \operatorname {GL} (2,\mathbb {Q} )\end{aligned}}$
And finally
$\mathbb {Q} ^{\times }\backslash I_{\mathbb {Q} }^{1}\cong \mathbb {Q} ^{\times }\backslash I_{\mathbb {Q} }\leftrightsquigarrow (\operatorname {GL} (2,\mathbb {Q} )\backslash (\operatorname {GL} (2,\mathbb {A} _{\mathbb {Q} }))^{1}\cong (\operatorname {GL} (2,\mathbb {Q} )Z_{\mathbb {R} })\backslash \operatorname {GL} (2,\mathbb {A} _{\mathbb {Q} }),$
where $Z_{\mathbb {R} }$ is the centre of $\operatorname {GL} (2,\mathbb {R} ).$ Then it define an automorphic form as an element of $L^{2}((\operatorname {GL} (2,\mathbb {Q} )Z_{\mathbb {R} })\backslash \operatorname {GL} (2,\mathbb {A} _{\mathbb {Q} })).$ In other words an automorphic form is a function on $\operatorname {GL} (2,\mathbb {A} _{\mathbb {Q} })$ satisfying certain algebraic and analytic conditions. For studying automorphic forms, it is important to know the representations of the group $\operatorname {GL} (2,\mathbb {A} _{\mathbb {Q} }).$ It is also possible to study automorphic L-functions, which can be described as integrals over $\operatorname {GL} (2,\mathbb {A} _{\mathbb {Q} }).$[33]
Generalise even further is possible by replacing $\mathbb {Q} $ with a number field and $\operatorname {GL} (2)$ with an arbitrary reductive algebraic group.
Further applications
A generalisation of Artin reciprocity law leads to the connection of representations of $\operatorname {GL} (2,\mathbb {A} _{K})$ and of Galois representations of $K$ (Langlands program).
The idele class group is a key object of class field theory, which describes abelian extensions of the field. The product of the local reciprocity maps in local class field theory gives a homomorphism of the idele group to the Galois group of the maximal abelian extension of the global field. The Artin reciprocity law, which is a sweeping generalisation of the Gauss quadratic reciprocity law, states that the product vanishes on the multiplicative group of the number field. Thus, the global reciprocity map of the idele class group to the abelian part of the absolute Galois group of the field will be obtained.
The self-duality of the adele ring of the function field of a curve over a finite field easily implies the Riemann–Roch theorem and the duality theory for the curve.
References
1. Groechenig, Michael (August 2017). "Adelic Descent Theory". Compositio Mathematica. 153 (8): 1706–1746. arXiv:1511.06271. doi:10.1112/S0010437X17007217. ISSN 0010-437X. S2CID 54016389.
2. Sutherland, Andrew (1 December 2015). 18.785 Number theory I Lecture #22 (PDF). MIT. p. 4.
3. "ring of adeles in nLab". ncatlab.org.
4. Geometric Class Field Theory, notes by Tony Feng of a lecture of Bhargav Bhatt (PDF).
5. Weil uniformization theorem, nlab article.
6. Cassels & Fröhlich 1967.
7. Tate, John (1968), "Residues of differentials on curves" (PDF), Annales Scientifiques de l'École Normale Supérieure, 1: 149–159, doi:10.24033/asens.1162.
8. This proof can be found in Cassels & Fröhlich 1967, p. 64.
9. The definitions are based on Weil 1967, p. 60.
10. See Weil 1967, p. 64 or Cassels & Fröhlich 1967, p. 74.
11. For proof see Deitmar 2010, p. 124, theorem 5.2.1.
12. See Cassels & Fröhlich 1967, p. 64, Theorem, or Weil 1967, p. 64, Theorem 2.
13. The next statement can be found in Neukirch 2007, p. 383.
14. See Deitmar 2010, p. 126, Theorem 5.2.2 for the rational case.
15. This section is based on Weil 1967, p. 71.
16. A proof of this statement can be found in Weil 1967, p. 71.
17. A proof of this statement can be found in Weil 1967, p. 72.
18. For a proof see Neukirch 2007, p. 388.
19. This statement can be found in Cassels & Fröhlich 1967, p. 69.
20. $\mathbb {A} _{K}^{1}$ is also used for the set of the $1$-idele but $I_{K}^{1}$ is used in this example.
21. There are many proofs for this result. The one shown below is based on Neukirch 2007, p. 195.
22. For a proof see Cassels & Fröhlich 1967, p. 66.
23. This proof can be found in Weil 1967, p. 76 or in Cassels & Fröhlich 1967, p. 70.
24. Part of Theorem 5.3.3 in Deitmar 2010.
25. The general proof of this theorem for any global field is given in Weil 1967, p. 77.
26. For more information, see Cassels & Fröhlich 1967, p. 71.
27. A proof can be found in Weil 1967, p. 78 or in Cassels & Fröhlich 1967, p. 72.
28. A proof can be found in Cassels & Fröhlich 1967, p. 48.
29. A proof can be found in Cassels & Fröhlich 1967, p. 67
30. A proof can be found in Weil 1967, p. 66.
31. For more see Deitmar 2010, p. 129.
32. A proof can be found Deitmar 2010, p. 128, Theorem 5.3.4. See also p. 139 for more information on Tate's thesis.
33. For further information see Chapters 7 and 8 in Deitmar 2010.
Sources
• Cassels, John; Fröhlich, Albrecht (1967). Algebraic number theory: proceedings of an instructional conference, organized by the London Mathematical Society, (a NATO Advanced Study Institute). Vol. XVIII. London: Academic Press. ISBN 978-0-12-163251-9. 366 pages.
• Neukirch, Jürgen (2007). Algebraische Zahlentheorie, unveränd. nachdruck der 1. aufl. edn (in German). Vol. XIII. Berlin: Springer. ISBN 9783540375470. 595 pages.
• Weil, André (1967). Basic number theory. Vol. XVIII. Berlin; Heidelberg; New York: Springer. ISBN 978-3-662-00048-9. 294 pages.
• Deitmar, Anton (2010). Automorphe Formen (in German). Vol. VIII. Berlin; Heidelberg (u.a.): Springer. ISBN 978-3-642-12389-4. 250 pages.
• Lang, Serge (1994). Algebraic number theory, Graduate Texts in Mathematics 110 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-94225-4.
External links
• What problem do the adeles solve?
• Some good books on adeles
|
Wikipedia
|
Dual number
In algebra, the dual numbers are a hypercomplex number system first introduced in the 19th century. They are expressions of the form a + bε, where a and b are real numbers, and ε is a symbol taken to satisfy $\varepsilon ^{2}=0$ with $\varepsilon \neq 0$.
Dual numbers can be added component-wise, and multiplied by the formula
$(a+b\varepsilon )(c+d\varepsilon )=ac+(ad+bc)\varepsilon ,$
which follows from the property ε2 = 0 and the fact that multiplication is a bilinear operation.
The dual numbers form a commutative algebra of dimension two over the reals, and also an Artinian local ring. They are one of the simplest examples of a ring that has nonzero nilpotent elements.
History
Dual numbers were introduced in 1873 by William Clifford, and were used at the beginning of the twentieth century by the German mathematician Eduard Study, who used them to represent the dual angle which measures the relative position of two skew lines in space. Study defined a dual angle as θ + dε, where θ is the angle between the directions of two lines in three-dimensional space and d is a distance between them. The n-dimensional generalization, the Grassmann number, was introduced by Hermann Grassmann in the late 19th century.
Modern definition
In modern algebra, the algebra of dual numbers is often defined as the quotient of a polynomial ring over the real numbers $(\mathbb {R} )$ by the principal ideal generated by the square of the indeterminate, that is
$\mathbb {R} [X]/\left\langle X^{2}\right\rangle .$
It may also be defined as the exterior algebra of a one-dimensional vector space with $\varepsilon $ as its basis element.
Division
Division of dual numbers is defined when the real part of the denominator is non-zero. The division process is analogous to complex division in that the denominator is multiplied by its conjugate in order to cancel the non-real parts.
Therefore, to divide an equation of the form
${\frac {a+b\varepsilon }{c+d\varepsilon }}$
we multiply the numerator and denominator by the conjugate of the denominator:
${\begin{aligned}{\frac {a+b\varepsilon }{c+d\varepsilon }}&={\frac {(a+b\varepsilon )(c-d\varepsilon )}{(c+d\varepsilon )(c-d\varepsilon )}}\\[5pt]&={\frac {ac-ad\varepsilon +bc\varepsilon -bd\varepsilon ^{2}}{c^{2}+cd\varepsilon -cd\varepsilon -d^{2}\varepsilon ^{2}}}\\[5pt]&={\frac {ac-ad\varepsilon +bc\varepsilon -0}{c^{2}-0}}\\[5pt]&={\frac {ac+\varepsilon (bc-ad)}{c^{2}}}\\[5pt]&={\frac {a}{c}}+{\frac {bc-ad}{c^{2}}}\varepsilon \end{aligned}}$
which is defined when c is non-zero.
If, on the other hand, c is zero while d is not, then the equation
${a+b\varepsilon =(x+y\varepsilon )d\varepsilon }={xd\varepsilon +0}$
1. has no solution if a is nonzero
2. is otherwise solved by any dual number of the form b/d + yε.
This means that the non-real part of the "quotient" is arbitrary and division is therefore not defined for purely nonreal dual numbers. Indeed, they are (trivially) zero divisors and clearly form an ideal of the associative algebra (and thus ring) of the dual numbers.
Matrix representation
The dual number $a+b\epsilon $ can be represented by the square matrix ${\begin{pmatrix}a&b\\0&a\end{pmatrix}}$. In this representation the matrix ${\begin{pmatrix}0&1\\0&0\end{pmatrix}}$ squares to the zero matrix, corresponding to the dual number $\varepsilon $.
There are other ways to represent dual numbers as square matrices. They consist of representing the dual number $1$ by the identity matrix, and $\epsilon $ by any matrix whose square is the zero matrix; that is, in the case of 2×2 matrices, any nonzero matrix of the form
${\begin{pmatrix}a&b\\c&-a\end{pmatrix}}$
with $a^{2}+bc=0.$[1]
Differentiation
One application of dual numbers is automatic differentiation. Any polynomial
$P(x)=p_{0}+p_{1}x+p_{2}x^{2}+\cdots +p_{n}x^{n}$
with real coefficients can be extended to a function of a dual-number-valued argument,
${\begin{aligned}P(a+b\varepsilon )&=p_{0}+p_{1}(a+b\varepsilon )+\cdots +p_{n}(a+b\varepsilon )^{n}\\[2mu]&=p_{0}+p_{1}a+p_{2}a^{2}+\cdots +p_{n}a^{n}+p_{1}b\varepsilon +2p_{2}ab\varepsilon +\cdots +np_{n}a^{n-1}b\varepsilon \\[5mu]&=P(a)+bP'(a)\varepsilon ,\end{aligned}}$
where $P'$ is the derivative of $P.$
More generally, any (analytic) real function can be extended to the dual numbers via its Taylor series:
$f(a+b\varepsilon )=\sum _{n=0}^{\infty }{\frac {f^{(n)}(a)b^{n}\varepsilon ^{n}}{n!}}=f(a)+bf'(a)\varepsilon ,$
since all terms involving ε2 or greater powers are trivially 0 by the definition of ε.
By computing compositions of these functions over the dual numbers and examining the coefficient of ε in the result we find we have automatically computed the derivative of the composition.
A similar method works for polynomials of n variables, using the exterior algebra of an n-dimensional vector space.
Geometry
The "unit circle" of dual numbers consists of those with a = ±1 since these satisfy zz* = 1 where z* = a − bε. However, note that
$e^{b\varepsilon }=\sum _{n=0}^{\infty }{\frac {\left(b\varepsilon \right)^{n}}{n!}}=1+b\varepsilon ,$
so the exponential map applied to the ε-axis covers only half the "circle".
Let z = a + bε. If a ≠ 0 and m = b/a, then z = a(1 + mε) is the polar decomposition of the dual number z, and the slope m is its angular part. The concept of a rotation in the dual number plane is equivalent to a vertical shear mapping since (1 + pε)(1 + qε) = 1 + (p + q)ε.
In absolute space and time the Galilean transformation
$\left(t',x'\right)=(t,x){\begin{pmatrix}1&v\\0&1\end{pmatrix}}\,,$
that is
$t'=t,\quad x'=vt+x,$
relates the resting coordinates system to a moving frame of reference of velocity v. With dual numbers t + xε representing events along one space dimension and time, the same transformation is effected with multiplication by 1 + vε.
Cycles
Given two dual numbers p and q, they determine the set of z such that the difference in slopes ("Galilean angle") between the lines from z to p and q is constant. This set is a cycle in the dual number plane; since the equation setting the difference in slopes of the lines to a constant is a quadratic equation in the real part of z, a cycle is a parabola. The "cyclic rotation" of the dual number plane occurs as a motion of its projective line. According to Isaak Yaglom,[2]: 92–93 the cycle Z = {z : y = αx2} is invariant under the composition of the shear
$x_{1}=x,\quad y_{1}=vx+y$
with the translation
$x'=x_{1}={\frac {v}{2a}},\quad y'=y_{1}+{\frac {v^{2}}{4a}}.$
Applications in mechanics
Dual numbers find applications in mechanics, notably for kinematic synthesis. For example, the dual numbers make it possible to transform the input/output equations of a four-bar spherical linkage, which includes only rotoid joints, into a four-bar spatial mechanism (rotoid, rotoid, rotoid, cylindrical). The dualized angles are made of a primitive part, the angles, and a dual part, which has units of length.[3] See screw theory for more.
Generalizations
This construction can be carried out more generally: for a commutative ring R one can define the dual numbers over R as the quotient of the polynomial ring R[X] by the ideal (X2): the image of X then has square equal to zero and corresponds to the element ε from above.
Arbitrary module of elements of zero square
There is a more general construction of the dual numbers. Given a commutative ring $R$ and a module $M$, there is a ring $R[M]$ called the ring of dual numbers which has the following structures:
It is the $R$-module $R\oplus M$ with the multiplication defined by $(r,i)\cdot \left(r',i'\right)=\left(rr',ri'+r'i\right)$ for $r,r'\in R$ and $i,i'\in I.$
The algebra of dual numbers is the special case where $M=R$ and $\varepsilon =(0,1).$
Superspace
Dual numbers find applications in physics, where they constitute one of the simplest non-trivial examples of a superspace. Equivalently, they are supernumbers with just one generator; supernumbers generalize the concept to n distinct generators ε, each anti-commuting, possibly taking n to infinity. Superspace generalizes supernumbers slightly, by allowing multiple commuting dimensions.
The motivation for introducing dual numbers into physics follows from the Pauli exclusion principle for fermions. The direction along ε is termed the "fermionic" direction, and the real component is termed the "bosonic" direction. The fermionic direction earns this name from the fact that fermions obey the Pauli exclusion principle: under the exchange of coordinates, the quantum mechanical wave function changes sign, and thus vanishes if two coordinates are brought together; this physical idea is captured by the algebraic relation ε2 = 0.
Projective line
The idea of a projective line over dual numbers was advanced by Grünwald[4] and Corrado Segre.[5]
Just as the Riemann sphere needs a north pole point at infinity to close up the complex projective line, so a line at infinity succeeds in closing up the plane of dual numbers to a cylinder.[2]: 149–153
Suppose D is the ring of dual numbers x + yε and U is the subset with x ≠ 0. Then U is the group of units of D. Let B = {(a, b) ∈ D × D : a ∈ U or b ∈ U}. A relation is defined on B as follows: (a, b) ~ (c, d) when there is a u in U such that ua = c and ub = d. This relation is in fact an equivalence relation. The points of the projective line over D are equivalence classes in B under this relation: P(D) = B/~. They are represented with projective coordinates [a, b].
Consider the embedding D → P(D) by z → [z, 1]. Then points [1, n], for n2 = 0, are in P(D) but are not the image of any point under the embedding. P(D) is mapped onto a cylinder by projection: Take a cylinder tangent to the double number plane on the line {yε : y ∈ $\mathbb {R} $}, ε2 = 0. Now take the opposite line on the cylinder for the axis of a pencil of planes. The planes intersecting the dual number plane and cylinder provide a correspondence of points between these surfaces. The plane parallel to the dual number plane corresponds to points [1, n], n2 = 0 in the projective line over dual numbers.
See also
• Smooth infinitesimal analysis
• Perturbation theory
• Infinitesimal
• Screw theory
• Dual-complex number
• Laguerre transformations
• Grassmann number
• Automatic differentiation
References
1. Abstract Algebra/2x2 real matrices at Wikibooks
2. Yaglom, I. M. (1979). A Simple Non-Euclidean Geometry and its Physical Basis. Springer. ISBN 0-387-90332-1. MR 0520230.
3. Angeles, Jorge (1998), Angeles, Jorge; Zakhariev, Evtim (eds.), "The Application of Dual Algebra to Kinematic Analysis", Computational Methods in Mechanical Systems: Mechanism Analysis, Synthesis, and Optimization, NATO ASI Series, Springer Berlin Heidelberg, vol. 161, pp. 3–32, doi:10.1007/978-3-662-03729-4_1, ISBN 9783662037294
4. Grünwald, Josef (1906). "Über duale Zahlen und ihre Anwendung in der Geometrie". Monatshefte für Mathematik. 17: 81–136. doi:10.1007/BF01697639. S2CID 119840611.
5. Segre, Corrado (1912). "XL. Le geometrie proiettive nei campi di numeri duali". Opere. Also in Atti della Reale Accademia della Scienze di Torino 47.
Further reading
• Bencivenga, Ulderico (1946). "Sulla rappresentazione geometrica delle algebre doppie dotate di modulo" [On the geometric representation of double algebras with modulus]. Atti della Reale Accademia delle Scienze e Belle-Lettere di Napoli. 3 (in Italian). 2 (7). MR 0021123.
• Clifford, William Kingdon (1873). "Preliminary Sketch of Bi-quaternions". Proceedings of the London Mathematical Society. 4: 381–395.
• Harkin, Anthony A.; Harkin, Joseph B. (April 2004). "Geometry of Generalized Complex Numbers" (PDF). Mathematics Magazine. 77 (2): 118–129. doi:10.1080/0025570X.2004.11953236. S2CID 7837108. Archived (PDF) from the original on 2022-10-09.
• Miller, William; Boehning, Rochelle (1968). "Gaussian, Parabolic and Hyperbolic Numbers". The Mathematics Teacher. 61 (4): 377–382. doi:10.5951/MT.61.4.0377.
• Study, Eduard (1903). Geometrie der Dynamen. B. G. Teubner. p. 196. From Cornell Historical Mathematical Monographs at Cornell University.
• Yaglom, I. M. (1968). Complex Numbers in Geometry. Translated from Russian by Eric J. F. Primrose. New York and London: Academic Press. p. 12–18.
• Brand, Louis (1947). Vector and tensor analysis. New York: John Wiley & Sons.
• Fischer, Ian S. (1999). Dual number methods in kinematics, static and dynamics. Boca Raton: CRC Press.
• Bertram, W. (2008). Differential Geometry, Lie Groups and Symmetric Spaces over General Base Fields and Rings. Memoirs of the AMS. Vol. 192. Providence, Rhode Island: Amer. Math. Soc.
• ""Higher" tangent space". math.stackexchange.com.
Number systems
Sets of definable numbers
• Natural numbers ($\mathbb {N} $)
• Integers ($\mathbb {Z} $)
• Rational numbers ($\mathbb {Q} $)
• Constructible numbers
• Algebraic numbers ($\mathbb {A} $)
• Closed-form numbers
• Periods
• Computable numbers
• Arithmetical numbers
• Set-theoretically definable numbers
• Gaussian integers
Composition algebras
• Division algebras: Real numbers ($\mathbb {R} $)
• Complex numbers ($\mathbb {C} $)
• Quaternions ($\mathbb {H} $)
• Octonions ($\mathbb {O} $)
Split
types
• Over $\mathbb {R} $:
• Split-complex numbers
• Split-quaternions
• Split-octonions
Over $\mathbb {C} $:
• Bicomplex numbers
• Biquaternions
• Bioctonions
Other hypercomplex
• Dual numbers
• Dual quaternions
• Dual-complex numbers
• Hyperbolic quaternions
• Sedenions ($\mathbb {S} $)
• Split-biquaternions
• Multicomplex numbers
• Geometric algebra/Clifford algebra
• Algebra of physical space
• Spacetime algebra
Other types
• Cardinal numbers
• Extended natural numbers
• Irrational numbers
• Fuzzy numbers
• Hyperreal numbers
• Levi-Civita field
• Surreal numbers
• Transcendental numbers
• Ordinal numbers
• p-adic numbers (p-adic solenoids)
• Supernatural numbers
• Profinite integers
• Superreal numbers
• Normal numbers
• Classification
• List
Infinitesimals
History
• Adequality
• Leibniz's notation
• Integral symbol
• Criticism of nonstandard analysis
• The Analyst
• The Method of Mechanical Theorems
• Cavalieri's principle
Related branches
• Nonstandard analysis
• Nonstandard calculus
• Internal set theory
• Synthetic differential geometry
• Smooth infinitesimal analysis
• Constructive nonstandard analysis
• Infinitesimal strain theory (physics)
Formalizations
• Differentials
• Hyperreal numbers
• Dual numbers
• Surreal numbers
Individual concepts
• Standard part function
• Transfer principle
• Hyperinteger
• Increment theorem
• Monad
• Internal set
• Levi-Civita field
• Hyperfinite set
• Law of continuity
• Overspill
• Microcontinuity
• Transcendental law of homogeneity
Mathematicians
• Gottfried Wilhelm Leibniz
• Abraham Robinson
• Pierre de Fermat
• Augustin-Louis Cauchy
• Leonhard Euler
Textbooks
• Analyse des Infiniment Petits
• Elementary Calculus
• Cours d'Analyse
|
Wikipedia
|
Endomorphism ring
In mathematics, the endomorphisms of an abelian group X form a ring. This ring is called the endomorphism ring of X, denoted by End(X); the set of all homomorphisms of X into itself. Addition of endomorphisms arises naturally in a pointwise manner and multiplication via endomorphism composition. Using these operations, the set of endomorphisms of an abelian group forms a (unital) ring, with the zero map $ 0:x\mapsto 0$ as additive identity and the identity map $ 1:x\mapsto x$ as multiplicative identity.[1][2]
The functions involved are restricted to what is defined as a homomorphism in the context, which depends upon the category of the object under consideration. The endomorphism ring consequently encodes several internal properties of the object. As the resulting object is often an algebra over some ring R, this may also be called the endomorphism algebra.
An abelian group is the same thing as a module over the ring of integers, which is the initial object in the category of rings. In a similar fashion, if R is any commutative ring, the endomorphisms of an R-module form an algebra over R by the same axioms and derivation. In particular, if R is a field, its modules M are vector spaces and their endomorphism rings are algebras over the field R.
Description
Let (A, +) be an abelian group and we consider the group homomorphisms from A into A. Then addition of two such homomorphisms may be defined pointwise to produce another group homomorphism. Explicitly, given two such homomorphisms f and g, the sum of f and g is the homomorphism f + g : x ↦ f(x) + g(x). Under this operation End(A) is an abelian group. With the additional operation of composition of homomorphisms, End(A) is a ring with multiplicative identity. This composition is explicitly fg : x ↦ f(g(x)). The multiplicative identity is the identity homomorphism on A.
If the set A does not form an abelian group, then the above construction is not necessarily additive, as then the sum of two homomorphisms need not be a homomorphism.[3] This set of endomorphisms is a canonical example of a near-ring that is not a ring.
Properties
• Endomorphism rings always have additive and multiplicative identities, respectively the zero map and identity map.
• Endomorphism rings are associative, but typically non-commutative.
• If a module is simple, then its endomorphism ring is a division ring (this is sometimes called Schur's lemma).[4]
• A module is indecomposable if and only if its endomorphism ring does not contain any non-trivial idempotent elements.[5] If the module is an injective module, then indecomposability is equivalent to the endomorphism ring being a local ring.[6]
• For a semisimple module, the endomorphism ring is a von Neumann regular ring.
• The endomorphism ring of a nonzero right uniserial module has either one or two maximal right ideals. If the module is Artinian, Noetherian, projective or injective, then the endomorphism ring has a unique maximal ideal, so that it is a local ring.
• The endomorphism ring of an Artinian uniform module is a local ring.[7]
• The endomorphism ring of a module with finite composition length is a semiprimary ring.
• The endomorphism ring of a continuous module or discrete module is a clean ring.[8]
• If an R module is finitely generated and projective (that is, a progenerator), then the endomorphism ring of the module and R share all Morita invariant properties. A fundamental result of Morita theory is that all rings equivalent to R arise as endomorphism rings of progenerators.
Examples
• In the category of R modules the endomorphism ring of an R-module M will only use the R module homomorphisms, which are typically a proper subset of the abelian group homomorphisms.[9] When M is a finitely generated projective module, the endomorphism ring is central to Morita equivalence of module categories.
• For any abelian group $A$, $M_{n}(\operatorname {End} (A))\cong \operatorname {End} (A^{n})$, since any matrix in $M_{n}(\operatorname {End} (A))$ carries a natural homomorphism structure of $A^{n}$ as follows:
${\begin{pmatrix}\varphi _{11}&\cdots &\varphi _{1n}\\\vdots &&\vdots \\\varphi _{n1}&\cdots &\varphi _{nn}\end{pmatrix}}{\begin{pmatrix}a_{1}\\\vdots \\a_{n}\end{pmatrix}}={\begin{pmatrix}\sum _{i=1}^{n}\varphi _{1i}(a_{i})\\\vdots \\\sum _{i=1}^{n}\varphi _{ni}(a_{i})\end{pmatrix}}.$
One can use this isomorphism to construct a lot of non-commutative endomorphism rings. For example: $\operatorname {End} (\mathbb {Z} \times \mathbb {Z} )\cong M_{2}(\mathbb {Z} )$, since $\operatorname {End} (\mathbb {Z} )\cong \mathbb {Z} $.
Also, when $R=K$ is a field, there is a canonical isomorphism $\operatorname {End} (K)\cong K$, so $\operatorname {End} (K^{n})\cong M_{n}(K)$, that is, the endomorphism ring of a $K$-vector space is identified with the ring of n-by-n matrices with entries in $K$.[10] More generally, the endomorphism algebra of the free module $M=R^{n}$ is naturally $n$-by-$n$ matrices with entries in the ring $R$.
• As a particular example of the last point, for any ring R with unity, End(RR) = R, where the elements of R act on R by left multiplication.
• In general, endomorphism rings can be defined for the objects of any preadditive category.
Notes
1. Fraleigh (1976, p. 211)
2. Passman (1991, pp. 4–5)
3. Dummit & Foote, p. 347)
4. Jacobson 2009, p. 118.
5. Jacobson 2009, p. 111, Prop. 3.1.
6. Wisbauer 1991, p. 163.
7. Wisbauer 1991, p. 263.
8. Camillo et al. 2006.
9. Abelian groups may also be viewed as modules over the ring of integers.
10. Drozd & Kirichenko 1994, pp. 23–31.
References
• Camillo, V. P.; Khurana, D.; Lam, T. Y.; Nicholson, W. K.; Zhou, Y. (2006), "Continuous modules are clean", J. Algebra, 304 (1): 94–111, doi:10.1016/j.jalgebra.2006.06.032, ISSN 0021-8693, MR 2255822
• Drozd, Yu. A.; Kirichenko, V.V. (1994), Finite Dimensional Algebras, Berlin: Springer-Verlag, ISBN 3-540-53380-X
• Dummit, David; Foote, Richard, Algebra
• Fraleigh, John B. (1976), A First Course In Abstract Algebra (2nd ed.), Reading: Addison-Wesley, ISBN 0-201-01984-1
• "Endomorphism ring", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Jacobson, Nathan (2009), Basic algebra, vol. 2 (2nd ed.), Dover, ISBN 978-0-486-47187-7
• Passman, Donald S. (1991), A Course in Ring Theory, Pacific Grove: Wadsworth & Brooks/Cole, ISBN 0-534-13776-8
• Wisbauer, Robert (1991), Foundations of module and ring theory, Algebra, Logic and Applications, vol. 3 (Revised and translated from the 1988 German ed.), Philadelphia, PA: Gordon and Breach Science Publishers, pp. xii+606, ISBN 2-88124-805-5, MR 1144522 A handbook for study and research
|
Wikipedia
|
Generic matrix ring
In algebra, a generic matrix ring is a sort of a universal matrix ring.
Definition
We denote by $F_{n}$ a generic matrix ring of size n with variables $X_{1},\dots X_{m}$. It is characterized by the universal property: given a commutative ring R and n-by-n matrices $A_{1},\dots ,A_{m}$ over R, any mapping $X_{i}\mapsto A_{i}$ extends to the ring homomorphism (called evaluation) $F_{n}\to M_{n}(R)$.
Explicitly, given a field k, it is the subalgebra $F_{n}$ of the matrix ring $M_{n}(k[(X_{l})_{ij}\mid 1\leq l\leq m,\ 1\leq i,j\leq n])$ generated by n-by-n matrices $X_{1},\dots ,X_{m}$, where $(X_{l})_{ij}$ are matrix entries and commute by definition. For example, if m = 1 then $F_{1}$ is a polynomial ring in one variable.
For example, a central polynomial is an element of the ring $F_{n}$ that will map to a central element under an evaluation. (In fact, it is in the invariant ring $k[(X_{l})_{ij}]^{\operatorname {GL} _{n}(k)}$ since it is central and invariant.[1])
By definition, $F_{n}$ is a quotient of the free ring $k\langle t_{1},\dots ,t_{m}\rangle $ with $t_{i}\mapsto X_{i}$ by the ideal consisting of all p that vanish identically on all n-by-n matrices over k.
Geometric perspective
The universal property means that any ring homomorphism from $k\langle t_{1},\dots ,t_{m}\rangle $ to a matrix ring factors through $F_{n}$. This has a following geometric meaning. In algebraic geometry, the polynomial ring $k[t,\dots ,t_{m}]$ is the coordinate ring of the affine space $k^{m}$, and to give a point of $k^{m}$ is to give a ring homomorphism (evaluation) $k[t,\dots ,t_{m}]\to k$ (either by the Hilbert nullstellensatz or by the scheme theory). The free ring $k\langle t_{1},\dots ,t_{m}\rangle $ plays the role of the coordinate ring of the affine space in the noncommutative algebraic geometry (i.e., we don't demand free variables to commute) and thus a generic matrix ring of size n is the coordinate ring of a noncommutative affine variety whose points are the Spec's of matrix rings of size n (see below for a more concrete discussion.)
The maximal spectrum of a generic matrix ring
For simplicity, assume k is algebraically closed. Let A be an algebra over k and let $\operatorname {Spec} _{n}(A)$ denote the set of all maximal ideals ${\mathfrak {m}}$ in A such that $A/{\mathfrak {m}}\approx M_{n}(k)$. If A is commutative, then $\operatorname {Spec} _{1}(A)$ is the maximal spectrum of A and $\operatorname {Spec} _{n}(A)$ is empty for any $n>1$.
References
1. Artin 1999, Proposition V.15.2.
• Artin, Michael (1999). "Noncommutative Rings" (PDF).
• Cohn, Paul M. (2003). Further algebra and applications (Revised ed. of Algebra, 2nd ed.). London: Springer-Verlag. ISBN 1-85233-667-6. Zbl 1006.00001.
|
Wikipedia
|
Period (algebraic geometry)
In algebraic geometry, a period is a number that can be expressed as an integral of an algebraic function over an algebraic domain. Sums and products of periods remain periods, so the periods form a ring.
For a more frequently used sense of the word "period" in mathematics, see Periodic function.
Maxim Kontsevich and Don Zagier gave a survey of periods and introduced some conjectures about them.[1] Periods also arise in computing the integrals that arise from Feynman diagrams, and there has been intensive work trying to understand the connections.[2]
Definition
A real number is a period if it is of the form
$\int _{P(x,y,z,\ldots )\geq 0}Q(x,y,z,\ldots )\mathrm {d} x\mathrm {d} y\mathrm {d} z\ldots $
where $P$ is a polynomial and $Q$ a rational function on $\mathbb {R} ^{n}$ with rational coefficients. A complex number is a period if its real and imaginary parts are periods.[3]
An alternative definition allows $P$ and $Q$ to be algebraic functions;[4] this looks more general, but is equivalent. The coefficients of the rational functions and polynomials can also be generalised to algebraic numbers because irrational algebraic numbers are expressible in terms of areas of suitable domains.
In the other direction, $Q$ can be restricted to be the constant function $1$ or $-1$, by replacing the integrand with an integral of $\pm 1$ over a region defined by a polynomial in additional variables. In other words, a (nonnegative) period is the volume of a region in $\mathbb {R} ^{n}$ defined by a polynomial inequality.
Examples
Besides the algebraic numbers, the following numbers are known to be periods:
• The natural logarithm of any positive algebraic number a, which is $\int _{1}^{a}{\frac {1}{x}}\ \mathrm {d} x$
• π $=\int _{0}^{1}{\frac {4}{x^{2}+1}}\ \mathrm {d} x$
• Elliptic integrals with rational arguments
• All zeta constants (the Riemann zeta function of an integer) and multiple zeta values
• Special values of hypergeometric functions at algebraic arguments
• Γ(p/q)q for natural numbers p and q.
An example of a real number that is not a period is given by Chaitin's constant Ω. Any other non-computable number also gives an example of a real number that is not a period. Currently there are no natural examples of computable numbers that have been proved not to be periods, however it is possible to construct artificial examples.[5] Plausible candidates for numbers that are not periods include e, 1/π, and the Euler–Mascheroni constant γ.
Properties and motivation
The periods are intended to bridge the gap between the algebraic numbers and the transcendental numbers. The class of algebraic numbers is too narrow to include many common mathematical constants, while the set of transcendental numbers is not countable, and its members are not generally computable.
The set of all periods is countable, and all periods are computable,[6] and in particular definable.
Conjectures
Many of the constants known to be periods are also given by integrals of transcendental functions. Kontsevich and Zagier note that there "seems to be no universal rule explaining why certain infinite sums or integrals of transcendental functions are periods".
Kontsevich and Zagier conjectured that, if a period is given by two different integrals, then each integral can be transformed into the other using only the linearity of integrals (in both the integrand and the domain), changes of variables, and the Newton–Leibniz formula
$\int _{a}^{b}f'(x)\,dx=f(b)-f(a)$
(or, more generally, the Stokes formula).
A useful property of algebraic numbers is that equality between two algebraic expressions can be determined algorithmically. The conjecture of Kontsevich and Zagier would imply that equality of periods is also decidable: inequality of computable reals is known recursively enumerable; and conversely if two integrals agree, then an algorithm could confirm so by trying all possible ways to transform one of them into the other one.
It is conjectured that Euler's number e and the Euler–Mascheroni constant γ are not periods.
Generalizations
The periods can be extended to exponential periods by permitting the integrand $Q$ to be the product of an algebraic function and the exponential of an algebraic function. This extension includes all algebraic powers of e, the gamma function of rational arguments, and values of Bessel functions.
Kontsevich and Zagier suggest that there are "indications" that periods can be naturally generalized even further, to include Euler's constant γ. With this inclusion, "all classical constants are periods in the appropriate sense".
See also
• Jacobian variety
• Gauss–Manin connection
• Mixed motives (math)
• Tannakian formalism
References
• Kontsevich, Maxim; Zagier, Don (2001). "Periods" (PDF). In Engquist, Björn; Schmid, Wilfried (eds.). Mathematics unlimited—2001 and beyond. Berlin, New York City: Springer. pp. 771–808. ISBN 9783540669135. MR 1852188.
• Marcolli, Matilde (2010). "Feynman integrals and motives". European Congress of Mathematics. Eur. Math. Soc. Zürich. pp. 293–332. arXiv:0907.0321.
Footnotes
1. Kontsevich & Zagier 2001.
2. Marcolli 2010.
3. Kontsevich & Zagier 2001, p. 3.
4. Weisstein, Eric W. "Periods". WolframMathWorld (Wolfram Research). Retrieved 2019-06-19.
5. Yoshinaga, Masahiko (2008-05-03). "Periods and elementary real numbers". arXiv:0805.0349 [math.AG].
6. Tent, Katrin; Ziegler, Martin (2010). "Computable functions of reals" (PDF). Münster Journal of Mathematics. 3: 43–66.
Further reading
• Belkale, Prakash; Brosnan, Patrick (2003), "Periods and Igusa local zeta functions", International Mathematics Research Notices, 2003 (49): 2655–2670, doi:10.1155/S107379280313142X, ISSN 1073-7928, MR 2012522
• Waldschmidt, Michel (2006), "Transcendence of periods: the state of the art" (PDF), Pure and Applied Mathematics Quarterly, 2 (2): 435–463, doi:10.4310/PAMQ.2006.v2.n2.a3, ISSN 1558-8599, MR 2251476
External links
• PlanetMath: Period
Number systems
Sets of definable numbers
• Natural numbers ($\mathbb {N} $)
• Integers ($\mathbb {Z} $)
• Rational numbers ($\mathbb {Q} $)
• Constructible numbers
• Algebraic numbers ($\mathbb {A} $)
• Closed-form numbers
• Periods
• Computable numbers
• Arithmetical numbers
• Set-theoretically definable numbers
• Gaussian integers
Composition algebras
• Division algebras: Real numbers ($\mathbb {R} $)
• Complex numbers ($\mathbb {C} $)
• Quaternions ($\mathbb {H} $)
• Octonions ($\mathbb {O} $)
Split
types
• Over $\mathbb {R} $:
• Split-complex numbers
• Split-quaternions
• Split-octonions
Over $\mathbb {C} $:
• Bicomplex numbers
• Biquaternions
• Bioctonions
Other hypercomplex
• Dual numbers
• Dual quaternions
• Dual-complex numbers
• Hyperbolic quaternions
• Sedenions ($\mathbb {S} $)
• Split-biquaternions
• Multicomplex numbers
• Geometric algebra/Clifford algebra
• Algebra of physical space
• Spacetime algebra
Other types
• Cardinal numbers
• Extended natural numbers
• Irrational numbers
• Fuzzy numbers
• Hyperreal numbers
• Levi-Civita field
• Surreal numbers
• Transcendental numbers
• Ordinal numbers
• p-adic numbers (p-adic solenoids)
• Supernatural numbers
• Profinite integers
• Superreal numbers
• Normal numbers
• Classification
• List
|
Wikipedia
|
Ring of polynomial functions
In mathematics, the ring of polynomial functions on a vector space V over a field k gives a coordinate-free analog of a polynomial ring. It is denoted by k[V]. If V is finite dimensional and is viewed as an algebraic variety, then k[V] is precisely the coordinate ring of V.
The explicit definition of the ring can be given as follows. If $k[t_{1},\dots ,t_{n}]$ is a polynomial ring, then we can view $t_{i}$ as coordinate functions on $k^{n}$; i.e., $t_{i}(x)=x_{i}$ when $x=(x_{1},\dots ,x_{n}).$ This suggests the following: given a vector space V, let k[V] be the commutative k-algebra generated by the dual space $V^{*}$, which is a subring of the ring of all functions $V\to k$. If we fix a basis for V and write $t_{i}$ for its dual basis, then k[V] consists of polynomials in $t_{i}$.
If k is infinite, then k[V] is the symmetric algebra of the dual space $V^{*}$.
In applications, one also defines k[V] when V is defined over some subfield of k (e.g., k is the complex field and V is a real vector space.) The same definition still applies.
Throughout the article, for simplicity, the base field k is assumed to be infinite.
Relation with polynomial ring
Let $A=K[x]$ be the set of all polynomials over a field K and B be the set of all polynomial functions in one variable over K. Both A and B are algebras over K given by the standard multiplication and addition of polynomials and functions. We can map each $f$ in A to ${\hat {f}}$ in B by the rule ${\hat {f}}(t)=f(t)$. A routine check shows that the mapping $f\mapsto {\hat {f}}$ is a homomorphism of the algebras A and B. This homomorphism is an isomorphism if and only if K is an infinite field. For example, if K is a finite field then let $p(x)=\prod \limits _{t\in K}(x-t)$. p is a nonzero polynomial in K[x], however $p(t)=0$ for all t in K, so ${\hat {p}}=0$ is the zero function and our homomorphism is not an isomorphism (and, actually, the algebras are not isomorphic, since the algebra of polynomials is infinite while that of polynomial functions is finite).
If K is infinite then choose a polynomial f such that ${\hat {f}}=0$. We want to show this implies that $f=0$. Let $\deg f=n$ and let $t_{0},t_{1},\dots ,t_{n}$ be n +1 distinct elements of K. Then $f(t_{i})=0$ for $0\leq i\leq n$ and by Lagrange interpolation we have $f=0$. Hence the mapping $f\mapsto {\hat {f}}$ is injective. Since this mapping is clearly surjective, it is bijective and thus an algebra isomorphism of A and B.
Symmetric multilinear maps
Let k be an infinite field of characteristic zero (or at least very large) and V a finite-dimensional vector space.
Let $S^{q}(V)$ denote the vector space of multilinear functionals $\textstyle \lambda :\prod _{1}^{q}V\to k$ :\prod _{1}^{q}V\to k} that are symmetric; $\lambda (v_{1},\dots ,v_{q})$ is the same for all permutations of $v_{i}$'s.
Any λ in $S^{q}(V)$ gives rise to a homogeneous polynomial function f of degree q: we just let $f(v)=\lambda (v,\dots ,v).$ To see that f is a polynomial function, choose a basis $e_{i},\,1\leq i\leq n$ of V and $t_{i}$ its dual. Then
$\lambda (v_{1},\dots ,v_{q})=\sum _{i_{1},\dots ,i_{q}=1}^{n}\lambda (e_{i_{1}},\dots ,e_{i_{q}})t_{i_{1}}(v_{1})\cdots t_{i_{q}}(v_{q})$,
which implies f is a polynomial in the ti's.
Thus, there is a well-defined linear map:
$\phi :S^{q}(V)\to k[V]_{q},\,\phi (\lambda )(v)=\lambda (v,\cdots ,v).$
We show it is an isomorphism. Choosing a basis as before, any homogeneous polynomial function f of degree q can be written as:
$f=\sum _{i_{1},\dots ,i_{q}=1}^{n}a_{i_{1}\cdots i_{q}}t_{i_{1}}\cdots t_{i_{q}}$
where $a_{i_{1}\cdots i_{q}}$ are symmetric in $i_{1},\dots ,i_{q}$. Let
$\psi (f)(v_{1},\dots ,v_{q})=\sum _{i_{1},\cdots ,i_{q}=1}^{n}a_{i_{1}\cdots i_{q}}t_{i_{1}}(v_{1})\cdots t_{i_{q}}(v_{q}).$
Clearly, $\phi \circ \psi $ is the identity; in particular, φ is surjective. To see φ is injective, suppose φ(λ) = 0. Consider
$\phi (\lambda )(t_{1}v_{1}+\cdots +t_{q}v_{q})=\lambda (t_{1}v_{1}+\cdots +t_{q}v_{q},...,t_{1}v_{1}+\cdots +t_{q}v_{q})$,
which is zero. The coefficient of t1t2 … tq in the above expression is q! times λ(v1, …, vq); it follows that λ = 0.
Note: φ is independent of a choice of basis; so the above proof shows that ψ is also independent of a basis, the fact not a priori obvious.
Example: A bilinear functional gives rise to a quadratic form in a unique way and any quadratic form arises in this way.
Taylor series expansion
Main article: Taylor series
Given a smooth function, locally, one can get a partial derivative of the function from its Taylor series expansion and, conversely, one can recover the function from the series expansion. This fact continues to hold for polynomials functions on a vector space. If f is in k[V], then we write: for x, y in V,
$f(x+y)=\sum _{n=0}^{\infty }g_{n}(x,y)$
where gn(x, y) are homogeneous of degree n in y, and only finitely many of them are nonzero. We then let
$(P_{y}f)(x)=g_{1}(x,y),$
resulting in the linear endomorphism Py of k[V]. It is called the polarization operator. We then have, as promised:
Theorem — For each f in k[V] and x, y in V,
$f(x+y)=\sum _{n=0}^{\infty }{1 \over n!}P_{y}^{n}f(x)$.
Proof: We first note that (Py f) (x) is the coefficient of t in f(x + t y); in other words, since g0(x, y) = g0(x, 0) = f(x),
$P_{y}f(x)=\left.{d \over dt}\right|_{t=0}f(x+ty)$
where the right-hand side is, by definition,
$\left.{f(x+ty)-f(x) \over t}\right|_{t=0}.$
The theorem follows from this. For example, for n = 2, we have:
$P_{y}^{2}f(x)=\left.{\partial \over \partial t_{1}}\right|_{t_{1}=0}P_{y}f(x+t_{1}y)=\left.{\partial \over \partial t_{1}}\right|_{t_{1}=0}\left.{\partial \over \partial t_{2}}\right|_{t_{2}=0}f(x+(t_{1}+t_{2})y)=2!g_{2}(x,y).$
The general case is similar. $\square $
Operator product algebra
When the polynomials are valued not over a field k, but over some algebra, then one may define additional structure. Thus, for example, one may consider the ring of functions over GL(n,m), instead of for k = GL(1,m). In this case, one may impose an additional axiom.
The operator product algebra is an associative algebra of the form
$A^{i}(x)B^{j}(y)=\sum _{k}f_{k}^{ij}(x,y,z)C^{k}(z)$
The structure constants $f_{k}^{ij}(x,y,z)$ are required to be single-valued functions, rather than sections of some vector bundle. The fields (or operators) $A^{i}(x)$ are required to span the ring of functions. In practical calculations, it is usually required that the sums be analytic within some radius of convergence; typically with a radius of convergence of $|x-y|$. Thus, the ring of functions can be taken to be the ring of polynomial functions.
The above can be considered to be an additional requirement imposed on the ring; it is sometimes called the bootstrap. In physics, a special case of the operator product algebra is known as the operator product expansion.
See also
• Algebraic geometry of projective spaces
• Polynomial ring
• Symmetric algebra
• Zariski tangent space
Notes
References
• Kobayashi, S.; Nomizu, K. (1963), Foundations of Differential Geometry, Vol. 2 (new ed.), Wiley-Interscience (published 2004).
|
Wikipedia
|
Integer
An integer is the number zero (0), a positive natural number (1, 2, 3, etc.) or a negative integer with a minus sign (−1, −2, −3, etc.).[1] The negative numbers are the additive inverses of the corresponding positive numbers.[2] In the language of mathematics, the set of integers is often denoted by the boldface Z or blackboard bold $\mathbb {Z} $.[3][4]
For computer representation, see Integer (computer science). For the generalization in algebraic number theory, see Algebraic integer.
Algebraic structure → Group theory
Group theory
Basic notions
• Subgroup
• Normal subgroup
• Quotient group
• (Semi-)direct product
Group homomorphisms
• kernel
• image
• direct sum
• wreath product
• simple
• finite
• infinite
• continuous
• multiplicative
• additive
• cyclic
• abelian
• dihedral
• nilpotent
• solvable
• action
• Glossary of group theory
• List of group theory topics
Finite groups
• Cyclic group Zn
• Symmetric group Sn
• Alternating group An
• Dihedral group Dn
• Quaternion group Q
• Cauchy's theorem
• Lagrange's theorem
• Sylow theorems
• Hall's theorem
• p-group
• Elementary abelian group
• Frobenius group
• Schur multiplier
Classification of finite simple groups
• cyclic
• alternating
• Lie type
• sporadic
• Discrete groups
• Lattices
• Integers ($\mathbb {Z} $)
• Free group
Modular groups
• PSL(2, $\mathbb {Z} $)
• SL(2, $\mathbb {Z} $)
• Arithmetic group
• Lattice
• Hyperbolic group
Topological and Lie groups
• Solenoid
• Circle
• General linear GL(n)
• Special linear SL(n)
• Orthogonal O(n)
• Euclidean E(n)
• Special orthogonal SO(n)
• Unitary U(n)
• Special unitary SU(n)
• Symplectic Sp(n)
• G2
• F4
• E6
• E7
• E8
• Lorentz
• Poincaré
• Conformal
• Diffeomorphism
• Loop
Infinite dimensional Lie group
• O(∞)
• SU(∞)
• Sp(∞)
Algebraic groups
• Linear algebraic group
• Reductive group
• Abelian variety
• Elliptic curve
The set of natural numbers $\mathbb {N} $ is a subset of $\mathbb {Z} $, which in turn is a subset of the set of all rational numbers $\mathbb {Q} $, itself a subset of the real numbers $\mathbb {R} $.[lower-alpha 1] Like the natural numbers, $\mathbb {Z} $ is countably infinite. An integer may be regarded as a real number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, 5+1/2, and √2 are not.[8]
The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic integers. In fact, (rational) integers are algebraic integers that are also rational numbers.
History
The word integer comes from the Latin integer meaning "whole" or (literally) "untouched", from in ("not") plus tangere ("to touch"). "Entire" derives from the same origin via the French word entier, which means both entire and integer.[9] Historically the term was used for a number that was a multiple of 1,[10][11] or to the whole part of a mixed number.[12][13] Only positive integers were considered, making the term synonymous with the natural numbers. The definition of integer expanded over time to include negative numbers as their usefulness was recognized.[14] For example Leonhard Euler in his 1765 Elements of Algebra defined integers to include both positive and negative numbers.[15] However, European mathematicians, for the most part, resisted the concept of negative numbers until the middle of the 19th century.[14]
The use of the letter Z to denote the set of integers comes from the German word Zahlen ("numbers")[3][4] and has been attributed to David Hilbert.[16] The earliest known use of the notation in a textbook occurs in Algébre written by the collective Nicolas Bourbaki, dating to 1947.[3][17] The notation was not adopted immediately, for example another textbook used the letter J[18] and a 1960 paper used Z to denote the non-negative integers.[19] But by 1961, Z was generally used by modern algebra texts to denote the positive and negative integers.[20]
The symbol $\mathbb {Z} $ is often annotated to denote various sets, with varying usage amongst different authors: $\mathbb {Z} ^{+}$,$\mathbb {Z} _{+}$ or $\mathbb {Z} ^{>}$ for the positive integers, $\mathbb {Z} ^{0+}$ or $\mathbb {Z} ^{\geq }$ for non-negative integers, and $\mathbb {Z} ^{\neq }$ for non-zero integers. Some authors use $\mathbb {Z} ^{*}$ for non-zero integers, while others use it for non-negative integers, or for {–1, 1} (the group of units of $\mathbb {Z} $). Additionally, $\mathbb {Z} _{p}$ is used to denote either the set of integers modulo p (i.e., the set of congruence classes of integers), or the set of p-adic integers.[21][22]
The whole numbers were synonymous with the integers up until the early 1950s.[23][24][25] In the late 1950s, as part of the New Math movement,[26] American elementary school teachers began teaching that "whole numbers" referred to the natural numbers, excluding negative numbers, while "integer" included the negative numbers.[27][28] "Whole number" remains ambiguous to the present day.[29]
Algebraic properties
Algebraic structure → Ring theory
Ring theory
Basic concepts
Rings
• Subrings
• Ideal
• Quotient ring
• Fractional ideal
• Total ring of fractions
• Product of rings
• Free product of associative algebras
• Tensor product of algebras
Ring homomorphisms
• Kernel
• Inner automorphism
• Frobenius endomorphism
Algebraic structures
• Module
• Associative algebra
• Graded ring
• Involutive ring
• Category of rings
• Initial ring $\mathbb {Z} $
• Terminal ring $0=\mathbb {Z} _{1}$
Related structures
• Field
• Finite field
• Non-associative ring
• Lie ring
• Jordan ring
• Semiring
• Semifield
Commutative algebra
Commutative rings
• Integral domain
• Integrally closed domain
• GCD domain
• Unique factorization domain
• Principal ideal domain
• Euclidean domain
• Field
• Finite field
• Composition ring
• Polynomial ring
• Formal power series ring
Algebraic number theory
• Algebraic number field
• Ring of integers
• Algebraic independence
• Transcendental number theory
• Transcendence degree
p-adic number theory and decimals
• Direct limit/Inverse limit
• Zero ring $\mathbb {Z} _{1}$
• Integers modulo pn $\mathbb {Z} /p^{n}\mathbb {Z} $
• Prüfer p-ring $\mathbb {Z} (p^{\infty })$
• Base-p circle ring $\mathbb {T} $
• Base-p integers $\mathbb {Z} $
• p-adic rationals $\mathbb {Z} [1/p]$
• Base-p real numbers $\mathbb {R} $
• p-adic integers $\mathbb {Z} _{p}$
• p-adic numbers $\mathbb {Q} _{p}$
• p-adic solenoid $\mathbb {T} _{p}$
Algebraic geometry
• Affine variety
Noncommutative algebra
Noncommutative rings
• Division ring
• Semiprimitive ring
• Simple ring
• Commutator
Noncommutative algebraic geometry
Free algebra
Clifford algebra
• Geometric algebra
Operator algebra
Like the natural numbers, $\mathbb {Z} $ is closed under the operations of addition and multiplication, that is, the sum and product of any two integers is an integer. However, with the inclusion of the negative natural numbers (and importantly, 0), $\mathbb {Z} $, unlike the natural numbers, is also closed under subtraction.[30]
The integers form a unital ring which is the most basic one, in the following sense: for any unital ring, there is a unique ring homomorphism from the integers into this ring. This universal property, namely to be an initial object in the category of rings, characterizes the ring $\mathbb {Z} $.
$\mathbb {Z} $ is not closed under division, since the quotient of two integers (e.g., 1 divided by 2) need not be an integer. Although the natural numbers are closed under exponentiation, the integers are not (since the result can be a fraction when the exponent is negative).
The following table lists some of the basic properties of addition and multiplication for any integers a, b and c:
Properties of addition and multiplication on integers
Addition Multiplication
Closure: a + b is an integer a × b is an integer
Associativity: a + (b + c) = (a + b) + c a × (b × c) = (a × b) × c
Commutativity: a + b = b + a a × b = b × a
Existence of an identity element: a + 0 = a a × 1 = a
Existence of inverse elements: a + (−a) = 0 The only invertible integers (called units) are −1 and 1.
Distributivity: a × (b + c) = (a × b) + (a × c) and (a + b) × c = (a × c) + (b × c)
No zero divisors: If a × b = 0, then a = 0 or b = 0 (or both)
The first five properties listed above for addition say that $\mathbb {Z} $, under addition, is an abelian group. It is also a cyclic group, since every non-zero integer can be written as a finite sum 1 + 1 + ... + 1 or (−1) + (−1) + ... + (−1). In fact, $\mathbb {Z} $ under addition is the only infinite cyclic group—in the sense that any infinite cyclic group is isomorphic to $\mathbb {Z} $.
The first four properties listed above for multiplication say that $\mathbb {Z} $ under multiplication is a commutative monoid. However, not every integer has a multiplicative inverse (as is the case of the number 2), which means that $\mathbb {Z} $ under multiplication is not a group.
All the rules from the above property table (except for the last), when taken together, say that $\mathbb {Z} $ together with addition and multiplication is a commutative ring with unity. It is the prototype of all objects of such algebraic structure. Only those equalities of expressions are true in $\mathbb {Z} $ for all values of variables, which are true in any unital commutative ring. Certain non-zero integers map to zero in certain rings.
The lack of zero divisors in the integers (last property in the table) means that the commutative ring $\mathbb {Z} $ is an integral domain.
The lack of multiplicative inverses, which is equivalent to the fact that $\mathbb {Z} $ is not closed under division, means that $\mathbb {Z} $ is not a field. The smallest field containing the integers as a subring is the field of rational numbers. The process of constructing the rationals from the integers can be mimicked to form the field of fractions of any integral domain. And back, starting from an algebraic number field (an extension of rational numbers), its ring of integers can be extracted, which includes $\mathbb {Z} $ as its subring.
Although ordinary division is not defined on $\mathbb {Z} $, the division "with remainder" is defined on them. It is called Euclidean division, and possesses the following important property: given two integers a and b with b ≠ 0, there exist unique integers q and r such that a = q × b + r and 0 ≤ r < |b|, where |b| denotes the absolute value of b. The integer q is called the quotient and r is called the remainder of the division of a by b. The Euclidean algorithm for computing greatest common divisors works by a sequence of Euclidean divisions.
The above says that $\mathbb {Z} $ is a Euclidean domain. This implies that $\mathbb {Z} $ is a principal ideal domain, and any positive integer can be written as the products of primes in an essentially unique way.[31] This is the fundamental theorem of arithmetic.
Order-theoretic properties
$\mathbb {Z} $ is a totally ordered set without upper or lower bound. The ordering of $\mathbb {Z} $ is given by: :... −3 < −2 < −1 < 0 < 1 < 2 < 3 < ... An integer is positive if it is greater than zero, and negative if it is less than zero. Zero is defined as neither negative nor positive.
The ordering of integers is compatible with the algebraic operations in the following way:
1. if a < b and c < d, then a + c < b + d
2. if a < b and 0 < c, then ac < bc.
Thus it follows that $\mathbb {Z} $ together with the above ordering is an ordered ring.
The integers are the only nontrivial totally ordered abelian group whose positive elements are well-ordered.[32] This is equivalent to the statement that any Noetherian valuation ring is either a field—or a discrete valuation ring.
Construction
Traditional development
In elementary school teaching, integers are often intuitively defined as the union of the (positive) natural numbers, zero, and the negations of the natural numbers. This can be formalized as follows.[33] First construct the set of natural numbers according to the Peano axioms, call this $P$. Then construct a set $P^{-}$ which is disjoint from $P$ and in one-to-one correspondence with $P$ via a function $\psi $. For example, take $P^{-}$ to be the ordered pairs $(1,n)$ with the mapping $\psi =n\mapsto (1,n)$. Finally let 0 be some object not in $P$ or $P^{-}$, for example the ordered pair $(0,0)$. Then the integers are defined to be the union $P\cup P^{-}\cup \{0\}$.
The traditional arithmetic operations can then be defined on the integers in a piecewise fashion, for each of positive numbers, negative numbers, and zero. For example negation is defined as follows:
$-x={\begin{cases}\psi (x),&{\text{if }}x\in P\\\psi ^{-1}(x),&{\text{if }}x\in P^{-}\\0,&{\text{if }}x=0\end{cases}}$
The traditional style of definition leads to many different cases (each arithmetic operation needs to be defined on each combination of types of integer) and makes it tedious to prove that integers obey the various laws of arithmetic.[34]
Equivalence classes of ordered pairs
In modern set-theoretic mathematics, a more abstract construction[35][36] allowing one to define arithmetical operations without any case distinction is often used instead.[37] The integers can thus be formally constructed as the equivalence classes of ordered pairs of natural numbers (a,b).[38]
The intuition is that (a,b) stands for the result of subtracting b from a.[38] To confirm our expectation that 1 − 2 and 4 − 5 denote the same number, we define an equivalence relation ~ on these pairs with the following rule:
$(a,b)\sim (c,d)$
precisely when
$a+d=b+c.$
Addition and multiplication of integers can be defined in terms of the equivalent operations on the natural numbers;[38] by using [(a,b)] to denote the equivalence class having (a,b) as a member, one has:
$[(a,b)]+[(c,d)]:=[(a+c,b+d)].$
$[(a,b)]\cdot [(c,d)]:=[(ac+bd,ad+bc)].$
The negation (or additive inverse) of an integer is obtained by reversing the order of the pair:
$-[(a,b)]:=[(b,a)].$
Hence subtraction can be defined as the addition of the additive inverse:
$[(a,b)]-[(c,d)]:=[(a+d,b+c)].$
The standard ordering on the integers is given by:
$[(a,b)]<[(c,d)]$ if and only if $a+d<b+c.$
It is easily verified that these definitions are independent of the choice of representatives of the equivalence classes.
Every equivalence class has a unique member that is of the form (n,0) or (0,n) (or both at once). The natural number n is identified with the class [(n,0)] (i.e., the natural numbers are embedded into the integers by map sending n to [(n,0)]), and the class [(0,n)] is denoted −n (this covers all remaining classes, and gives the class [(0,0)] a second time since −0 = 0.
Thus, [(a,b)] is denoted by
${\begin{cases}a-b,&{\mbox{if }}a\geq b\\-(b-a),&{\mbox{if }}a<b.\end{cases}}$
If the natural numbers are identified with the corresponding integers (using the embedding mentioned above), this convention creates no ambiguity.
This notation recovers the familiar representation of the integers as {..., −2, −1, 0, 1, 2, ...} .
Some examples are:
${\begin{aligned}0&=[(0,0)]&=[(1,1)]&=\cdots &&=[(k,k)]\\1&=[(1,0)]&=[(2,1)]&=\cdots &&=[(k+1,k)]\\-1&=[(0,1)]&=[(1,2)]&=\cdots &&=[(k,k+1)]\\2&=[(2,0)]&=[(3,1)]&=\cdots &&=[(k+2,k)]\\-2&=[(0,2)]&=[(1,3)]&=\cdots &&=[(k,k+2)].\end{aligned}}$
Other approaches
In theoretical computer science, other approaches for the construction of integers are used by automated theorem provers and term rewrite engines. Integers are represented as algebraic terms built using a few basic operations (e.g., zero, succ, pred) and, possibly, using natural numbers, which are assumed to be already constructed (using, say, the Peano approach).
There exist at least ten such constructions of signed integers.[39] These constructions differ in several ways: the number of basic operations used for the construction, the number (usually, between 0 and 2) and the types of arguments accepted by these operations; the presence or absence of natural numbers as arguments of some of these operations, and the fact that these operations are free constructors or not, i.e., that the same integer can be represented using only one or many algebraic terms.
The technique for the construction of integers presented in the previous section corresponds to the particular case where there is a single basic operation pair$(x,y)$ that takes as arguments two natural numbers $x$ and $y$, and returns an integer (equal to $x-y$). This operation is not free since the integer 0 can be written pair(0,0), or pair(1,1), or pair(2,2), etc. This technique of construction is used by the proof assistant Isabelle; however, many other tools use alternative construction techniques, notable those based upon free constructors, which are simpler and can be implemented more efficiently in computers.
Computer science
An integer is often a primitive data type in computer languages. However, integer data types can only represent a subset of all integers, since practical computers are of finite capacity. Also, in the common two's complement representation, the inherent definition of sign distinguishes between "negative" and "non-negative" rather than "negative, positive, and 0". (It is, however, certainly possible for a computer to determine whether an integer value is truly positive.) Fixed length integer approximation data types (or subsets) are denoted int or Integer in several programming languages (such as Algol68, C, Java, Delphi, etc.).
Variable-length representations of integers, such as bignums, can store any integer that fits in the computer's memory. Other integer data types are implemented with a fixed size, usually a number of bits which is a power of 2 (4, 8, 16, etc.) or a memorable number of decimal digits (e.g., 9 or 10).
Cardinality
The cardinality of the set of integers is equal to ℵ0 (aleph-null). This is readily demonstrated by the construction of a bijection, that is, a function that is injective and surjective from $\mathbb {Z} $ to $\mathbb {N} =\{0,1,2,...\}.$ Such a function may be defined as
$f(x)={\begin{cases}-2x,&{\mbox{if }}x\leq 0\\2x-1,&{\mbox{if }}x>0,\end{cases}}$
with graph (set of the pairs $(x,f(x))$ is
{... (−4,8), (−3,6), (−2,4), (−1,2), (0,0), (1,1), (2,3), (3,5), ...}.
Its inverse function is defined by
${\begin{cases}g(2x)=-x\\g(2x-1)=x,\end{cases}}$
with graph
{(0, 0), (1, 1), (2, −1), (3, 2), (4, −2), (5, −3), ...}.
See also
• Canonical factorization of a positive integer
• Hyperinteger
• Integer complexity
• Integer lattice
• Integer part
• Integer sequence
• Integer-valued function
• Mathematical symbols
• Parity (mathematics)
• Profinite integer
Number systems
Complex $:\;\mathbb {C} $ :\;\mathbb {C} }
Real $:\;\mathbb {R} $ :\;\mathbb {R} }
Rational $:\;\mathbb {Q} $ :\;\mathbb {Q} }
Integer $:\;\mathbb {Z} $ :\;\mathbb {Z} }
Natural $:\;\mathbb {N} $ :\;\mathbb {N} }
Zero: 0
One: 1
Prime numbers
Composite numbers
Negative integers
Fraction
Finite decimal
Dyadic (finite binary)
Repeating decimal
Irrational
Algebraic irrational
Transcendental
Imaginary
Footnotes
1. More precisely, each system is embedded in the next, isomorphically mapped to a subset.[5] The commonly-assumed set-theoretic containment may be obtained by constructing the reals, discarding any earlier constructions, and defining the other sets as subsets of the reals.[6] Such a convention is "a matter of choice", yet not.[7]
References
1. Science and Technology Encyclopedia. University of Chicago Press. September 2000. p. 280. ISBN 978-0-226-74267-0.
2. "Integers: Introduction to the concept, with activities comparing temperatures and money. | Unit 1". OER Commons.
3. Miller, Jeff (29 August 2010). "Earliest Uses of Symbols of Number Theory". Archived from the original on 31 January 2010. Retrieved 20 September 2010.
4. Peter Jephson Cameron (1998). Introduction to Algebra. Oxford University Press. p. 4. ISBN 978-0-19-850195-4. Archived from the original on 8 December 2016. Retrieved 15 February 2016.
5. Partee, Barbara H.; Meulen, Alice ter; Wall, Robert E. (30 April 1990). Mathematical Methods in Linguistics. Springer Science & Business Media. pp. 78–82. ISBN 978-90-277-2245-4. The natural numbers are not themselves a subset of this set-theoretic representation of the integers. Rather, the set of all integers contains a subset consisting of the positive integers and zero which is isomorphic to the set of natural numbers.
6. Wohlgemuth, Andrew (10 June 2014). Introduction to Proof in Abstract Mathematics. Courier Corporation. p. 237. ISBN 978-0-486-14168-8.
7. Polkinghorne, John (19 May 2011). Meaning in Mathematics. OUP Oxford. p. 68. ISBN 978-0-19-162189-5.
8. Prep, Kaplan Test (4 June 2019). GMAT Complete 2020: The Ultimate in Comprehensive Self-Study for GMAT. Simon and Schuster. ISBN 978-1-5062-4844-8.
9. Evans, Nick (1995). "A-Quantifiers and Scope". In Bach, Emmon W. (ed.). Quantification in Natural Languages. Dordrecht, The Netherlands; Boston, MA: Kluwer Academic Publishers. p. 262. ISBN 978-0-7923-3352-4.
10. Smedley, Edward; Rose, Hugh James; Rose, Henry John (1845). Encyclopædia Metropolitana. B. Fellowes. p. 537. An integer is a multiple of unity
11. Encyclopaedia Britannica 1771, p. 367
12. Pisano, Leonardo; Boncompagni, Baldassarre (transliteration) (1202). Incipit liber Abbaci compositus to Lionardo filio Bonaccii Pisano in year Mccij [The Book of Calculation] (Manuscript) (in Latin). Translated by Sigler, Laurence E. Museo Galileo. p. 30. Nam rupti uel fracti semper ponendi sunt post integra, quamuis prius integra quam rupti pronuntiari debeant. [And the fractions are always put after the whole, thus first the integer is written, and then the fraction]
13. Encyclopaedia Britannica 1771, p. 83
14. Martinez, Alberto (2014). Negative Math. Princeton University Press. pp. 80–109.
15. Euler, Leonhard (1771). Vollstandige Anleitung Zur Algebra [Complete Introduction to Algebra] (in German). Vol. 1. p. 10. Alle diese Zahlen, so wohl positive als negative, führen den bekannten Nahmen der gantzen Zahlen, welche also entweder größer oder kleiner sind als nichts. Man nennt dieselbe gantze Zahlen, um sie von den gebrochenen, und noch vielerley andern Zahlen, wovon unten gehandelt werden wird, zu unterscheiden. [All these numbers, both positive and negative, are called whole numbers, which are either greater or lesser than nothing. We call them whole numbers, to distinguish them from fractions, and from several other kinds of numbers of which we shall hereafter speak.]
16. The University of Leeds Review. Vol. 31–32. University of Leeds. 1989. p. 46. Incidentally, Z comes from "Zahl": the notation was created by Hilbert.
17. Bourbaki, Nicolas (1951). Algèbre, Chapter 1 (in French) (2nd ed.). Paris: Hermann. p. 27. Le symétrisé de N se note Z; ses éléments sont appelés entiers rationnels. [The group of differences of N is denoted by Z; its elements are called the rational integers.]
18. Birkhoff, Garrett (1948). Lattice Theory (Revised ed.). American Mathematical Society. p. 63. the set J of all integers
19. Society, Canadian Mathematical (1960). Canadian Journal of Mathematics. Canadian Mathematical Society. p. 374. Consider the set Z of non-negative integers
20. Bezuszka, Stanley (1961). Contemporary Progress in Mathematics: Teacher Supplement [to] Part 1 and Part 2. Boston College. p. 69. Modern Algebra texts generally designate the set of integers by the capital letter Z.
21. Keith Pledger and Dave Wilkins, "Edexcel AS and A Level Modular Mathematics: Core Mathematics 1" Pearson 2008
22. LK Turner, FJ BUdden, D Knighton, "Advanced Mathematics", Book 2, Longman 1975.
23. Mathews, George Ballard (1892). Theory of Numbers. Deighton, Bell and Company. p. 2.
24. Betz, William (1934). Junior Mathematics for Today. Ginn. The whole numbers, or integers, when arranged in their natural order, such as 1, 2, 3, are called consecutive integers.
25. Peck, Lyman C. (1950). Elements of Algebra. McGraw-Hill. p. 3. The numbers which so arise are called positive whole numbers, or positive integers.
26. Hayden, Robert (1981). A history of the "new math" movement in the United States (PhD). Iowa State University. p. 145. doi:10.31274/rtd-180813-5631. A much more influential force in bringing news of the "new math" to high school teachers and administrators was the National Council of Teachers of Mathematics (NCTM).
27. The Growth of Mathematical Ideas, Grades K-12: 24th Yearbook. National Council of Teachers of Mathematics. 1959. p. 14. ISBN 9780608166186.
28. Deans, Edwina (1963). Elementary School Mathematics: New Directions. U.S. Department of Health, Education, and Welfare, Office of Education. p. 42.
29. "entry: whole number". The American Heritage Dictionary. HarperCollins.
30. "Integer | mathematics". Encyclopedia Britannica. Retrieved 11 August 2020.
31. Lang, Serge (1993). Algebra (3rd ed.). Addison-Wesley. pp. 86–87. ISBN 978-0-201-55540-0.
32. Warner, Seth (2012). Modern Algebra. Dover Books on Mathematics. Courier Corporation. Theorem 20.14, p. 185. ISBN 978-0-486-13709-4. Archived from the original on 6 September 2015. Retrieved 29 April 2015..
33. Mendelson, Elliott (1985). Number systems and the foundations of analysis. Malabar, Fla. : R.E. Krieger Pub. Co. p. 153. ISBN 978-0-89874-818-5.
34. Mendelson, Elliott (2008). Number Systems and the Foundations of Analysis. Dover Books on Mathematics. Courier Dover Publications. p. 86. ISBN 978-0-486-45792-5. Archived from the original on 8 December 2016. Retrieved 15 February 2016..
35. Ivorra Castillo: Álgebra
36. Kramer, Jürg; von Pippich, Anna-Maria (2017). From Natural Numbers to Quaternions (1st ed.). Switzerland: Springer Cham. pp. 78–81. doi:10.1007/978-3-319-69429-0. ISBN 978-3-319-69427-6.
37. Frobisher, Len (1999). Learning to Teach Number: A Handbook for Students and Teachers in the Primary School. The Stanley Thornes Teaching Primary Maths Series. Nelson Thornes. p. 126. ISBN 978-0-7487-3515-0. Archived from the original on 8 December 2016. Retrieved 15 February 2016..
38. Campbell, Howard E. (1970). The structure of arithmetic. Appleton-Century-Crofts. p. 83. ISBN 978-0-390-16895-5.
39. Garavel, Hubert (2017). On the Most Suitable Axiomatization of Signed Integers. Post-proceedings of the 23rd International Workshop on Algebraic Development Techniques (WADT'2016). Lecture Notes in Computer Science. Vol. 10644. Springer. pp. 120–134. doi:10.1007/978-3-319-72044-9_9. ISBN 978-3-319-72043-2. Archived from the original on 26 January 2018. Retrieved 25 January 2018.
Sources
• Bell, E.T. (1986). Men of Mathematics. New York: Simon & Schuster. ISBN 0-671-46400-0.)
• Herstein, I.N. (1975). Topics in Algebra (2nd ed.). Wiley. ISBN 0-471-01090-1.
• Mac Lane, Saunders; Birkhoff, Garrett (1999). Algebra (3rd ed.). American Mathematical Society. ISBN 0-8218-1646-2.
• A Society of Gentlemen in Scotland (1771). Encyclopaedia Britannica. Edinburgh.
External links
Look up integer in Wiktionary, the free dictionary.
• "Integer", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• The Positive Integers – divisor tables and numeral representation tools
• On-Line Encyclopedia of Integer Sequences cf OEIS
• Weisstein, Eric W. "Integer". MathWorld.
This article incorporates material from Integer on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
Integers
0s
• 0
• 1
• 2
• 3
• 4
• 5
• 6
• 7
• 8
• 9
• 10
• 11
• 12
• 13
• 14
• 15
• 16
• 17
• 18
• 19
• 20
• 21
• 22
• 23
• 24
• 25
• 26
• 27
• 28
• 29
• 30
• 31
• 32
• 33
• 34
• 35
• 36
• 37
• 38
• 39
• 40
• 41
• 42
• 43
• 44
• 45
• 46
• 47
• 48
• 49
• 50
• 51
• 52
• 53
• 54
• 55
• 56
• 57
• 58
• 59
• 60
• 61
• 62
• 63
• 64
• 65
• 66
• 67
• 68
• 69
• 70
• 71
• 72
• 73
• 74
• 75
• 76
• 77
• 78
• 79
• 80
• 81
• 82
• 83
• 84
• 85
• 86
• 87
• 88
• 89
• 90
• 91
• 92
• 93
• 94
• 95
• 96
• 97
• 98
• 99
100s
• 100
• 101
• 102
• 103
• 104
• 105
• 106
• 107
• 108
• 109
• 110
• 111
• 112
• 113
• 114
• 115
• 116
• 117
• 118
• 119
• 120
• 121
• 122
• 123
• 124
• 125
• 126
• 127
• 128
• 129
• 130
• 131
• 132
• 133
• 134
• 135
• 136
• 137
• 138
• 139
• 140
• 141
• 142
• 143
• 144
• 145
• 146
• 147
• 148
• 149
• 150
• 151
• 152
• 153
• 154
• 155
• 156
• 157
• 158
• 159
• 160
• 161
• 162
• 163
• 164
• 165
• 166
• 167
• 168
• 169
• 170
• 171
• 172
• 173
• 174
• 175
• 176
• 177
• 178
• 179
• 180
• 181
• 182
• 183
• 184
• 185
• 186
• 187
• 188
• 189
• 190
• 191
• 192
• 193
• 194
• 195
• 196
• 197
• 198
• 199
200s
• 200
• 201
• 202
• 203
• 204
• 205
• 206
• 207
• 208
• 209
• 210
• 211
• 212
• 213
• 214
• 215
• 216
• 217
• 218
• 219
• 220
• 221
• 222
• 223
• 224
• 225
• 226
• 227
• 228
• 229
• 230
• 231
• 232
• 233
• 234
• 235
• 236
• 237
• 238
• 239
• 240
• 241
• 242
• 243
• 244
• 245
• 246
• 247
• 248
• 249
• 250
• 251
• 252
• 253
• 254
• 255
• 256
• 257
• 258
• 259
• 260
• 261
• 262
• 263
• 264
• 265
• 266
• 267
• 268
• 269
• 270
• 271
• 272
• 273
• 274
• 275
• 276
• 277
• 278
• 279
• 280
• 281
• 282
• 283
• 284
• 285
• 286
• 287
• 288
• 289
• 290
• 291
• 292
• 293
• 294
• 295
• 296
• 297
• 298
• 299
300s
• 300
• 301
• 302
• 303
• 304
• 305
• 306
• 307
• 308
• 309
• 310
• 311
• 312
• 313
• 314
• 315
• 316
• 317
• 318
• 319
• 320
• 321
• 322
• 323
• 324
• 325
• 326
• 327
• 328
• 329
• 330
• 331
• 332
• 333
• 334
• 335
• 336
• 337
• 338
• 339
• 340
• 341
• 342
• 343
• 344
• 345
• 346
• 347
• 348
• 349
• 350
• 351
• 352
• 353
• 354
• 355
• 356
• 357
• 358
• 359
• 360
• 361
• 362
• 363
• 364
• 365
• 366
• 367
• 368
• 369
• 370
• 371
• 372
• 373
• 374
• 375
• 376
• 377
• 378
• 379
• 380
• 381
• 382
• 383
• 384
• 385
• 386
• 387
• 388
• 389
• 390
• 391
• 392
• 393
• 394
• 395
• 396
• 397
• 398
• 399
400s
• 400
• 401
• 402
• 403
• 404
• 405
• 406
• 407
• 408
• 409
• 410
• 411
• 412
• 413
• 414
• 415
• 416
• 417
• 418
• 419
• 420
• 421
• 422
• 423
• 424
• 425
• 426
• 427
• 428
• 429
• 430
• 431
• 432
• 433
• 434
• 435
• 436
• 437
• 438
• 439
• 440
• 441
• 442
• 443
• 444
• 445
• 446
• 447
• 448
• 449
• 450
• 451
• 452
• 453
• 454
• 455
• 456
• 457
• 458
• 459
• 460
• 461
• 462
• 463
• 464
• 465
• 466
• 467
• 468
• 469
• 470
• 471
• 472
• 473
• 474
• 475
• 476
• 477
• 478
• 479
• 480
• 481
• 482
• 483
• 484
• 485
• 486
• 487
• 488
• 489
• 490
• 491
• 492
• 493
• 494
• 495
• 496
• 497
• 498
• 499
500s
• 500
• 501
• 502
• 503
• 504
• 505
• 506
• 507
• 508
• 509
• 510
• 511
• 512
• 513
• 514
• 515
• 516
• 517
• 518
• 519
• 520
• 521
• 522
• 523
• 524
• 525
• 526
• 527
• 528
• 529
• 530
• 531
• 532
• 533
• 534
• 535
• 536
• 537
• 538
• 539
• 540
• 541
• 542
• 543
• 544
• 545
• 546
• 547
• 548
• 549
• 550
• 551
• 552
• 553
• 554
• 555
• 556
• 557
• 558
• 559
• 560
• 561
• 562
• 563
• 564
• 565
• 566
• 567
• 568
• 569
• 570
• 571
• 572
• 573
• 574
• 575
• 576
• 577
• 578
• 579
• 580
• 581
• 582
• 583
• 584
• 585
• 586
• 587
• 588
• 589
• 590
• 591
• 592
• 593
• 594
• 595
• 596
• 597
• 598
• 599
600s
• 600
• 601
• 602
• 603
• 604
• 605
• 606
• 607
• 608
• 609
• 610
• 611
• 612
• 613
• 614
• 615
• 616
• 617
• 618
• 619
• 620
• 621
• 622
• 623
• 624
• 625
• 626
• 627
• 628
• 629
• 630
• 631
• 632
• 633
• 634
• 635
• 636
• 637
• 638
• 639
• 640
• 641
• 642
• 643
• 644
• 645
• 646
• 647
• 648
• 649
• 650
• 651
• 652
• 653
• 654
• 655
• 656
• 657
• 658
• 659
• 660
• 661
• 662
• 663
• 664
• 665
• 666
• 667
• 668
• 669
• 670
• 671
• 672
• 673
• 674
• 675
• 676
• 677
• 678
• 679
• 680
• 681
• 682
• 683
• 684
• 685
• 686
• 687
• 688
• 689
• 690
• 691
• 692
• 693
• 694
• 695
• 696
• 697
• 698
• 699
700s
• 700
• 701
• 702
• 703
• 704
• 705
• 706
• 707
• 708
• 709
• 710
• 711
• 712
• 713
• 714
• 715
• 716
• 717
• 718
• 719
• 720
• 721
• 722
• 723
• 724
• 725
• 726
• 727
• 728
• 729
• 730
• 731
• 732
• 733
• 734
• 735
• 736
• 737
• 738
• 739
• 740
• 741
• 742
• 743
• 744
• 745
• 746
• 747
• 748
• 749
• 750
• 751
• 752
• 753
• 754
• 755
• 756
• 757
• 758
• 759
• 760
• 761
• 762
• 763
• 764
• 765
• 766
• 767
• 768
• 769
• 770
• 771
• 772
• 773
• 774
• 775
• 776
• 777
• 778
• 779
• 780
• 781
• 782
• 783
• 784
• 785
• 786
• 787
• 788
• 789
• 790
• 791
• 792
• 793
• 794
• 795
• 796
• 797
• 798
• 799
800s
• 800
• 801
• 802
• 803
• 804
• 805
• 806
• 807
• 808
• 809
• 810
• 811
• 812
• 813
• 814
• 815
• 816
• 817
• 818
• 819
• 820
• 821
• 822
• 823
• 824
• 825
• 826
• 827
• 828
• 829
• 830
• 831
• 832
• 833
• 834
• 835
• 836
• 837
• 838
• 839
• 840
• 841
• 842
• 843
• 844
• 845
• 846
• 847
• 848
• 849
• 850
• 851
• 852
• 853
• 854
• 855
• 856
• 857
• 858
• 859
• 860
• 861
• 862
• 863
• 864
• 865
• 866
• 867
• 868
• 869
• 870
• 871
• 872
• 873
• 874
• 875
• 876
• 877
• 878
• 879
• 880
• 881
• 882
• 883
• 884
• 885
• 886
• 887
• 888
• 889
• 890
• 891
• 892
• 893
• 894
• 895
• 896
• 897
• 898
• 899
900s
• 900
• 901
• 902
• 903
• 904
• 905
• 906
• 907
• 908
• 909
• 910
• 911
• 912
• 913
• 914
• 915
• 916
• 917
• 918
• 919
• 920
• 921
• 922
• 923
• 924
• 925
• 926
• 927
• 928
• 929
• 930
• 931
• 932
• 933
• 934
• 935
• 936
• 937
• 938
• 939
• 940
• 941
• 942
• 943
• 944
• 945
• 946
• 947
• 948
• 949
• 950
• 951
• 952
• 953
• 954
• 955
• 956
• 957
• 958
• 959
• 960
• 961
• 962
• 963
• 964
• 965
• 966
• 967
• 968
• 969
• 970
• 971
• 972
• 973
• 974
• 975
• 976
• 977
• 978
• 979
• 980
• 981
• 982
• 983
• 984
• 985
• 986
• 987
• 988
• 989
• 990
• 991
• 992
• 993
• 994
• 995
• 996
• 997
• 998
• 999
≥1000
• 1000
• 2000
• 3000
• 4000
• 5000
• 6000
• 7000
• 8000
• 9000
• 10,000
• 20,000
• 30,000
• 40,000
• 50,000
• 60,000
• 70,000
• 80,000
• 90,000
• 100,000
• 1,000,000
• 10,000,000
• 100,000,000
• 1,000,000,000
Number systems
Sets of definable numbers
• Natural numbers ($\mathbb {N} $)
• Integers ($\mathbb {Z} $)
• Rational numbers ($\mathbb {Q} $)
• Constructible numbers
• Algebraic numbers ($\mathbb {A} $)
• Closed-form numbers
• Periods
• Computable numbers
• Arithmetical numbers
• Set-theoretically definable numbers
• Gaussian integers
Composition algebras
• Division algebras: Real numbers ($\mathbb {R} $)
• Complex numbers ($\mathbb {C} $)
• Quaternions ($\mathbb {H} $)
• Octonions ($\mathbb {O} $)
Split
types
• Over $\mathbb {R} $:
• Split-complex numbers
• Split-quaternions
• Split-octonions
Over $\mathbb {C} $:
• Bicomplex numbers
• Biquaternions
• Bioctonions
Other hypercomplex
• Dual numbers
• Dual quaternions
• Dual-complex numbers
• Hyperbolic quaternions
• Sedenions ($\mathbb {S} $)
• Split-biquaternions
• Multicomplex numbers
• Geometric algebra/Clifford algebra
• Algebra of physical space
• Spacetime algebra
Other types
• Cardinal numbers
• Extended natural numbers
• Irrational numbers
• Fuzzy numbers
• Hyperreal numbers
• Levi-Civita field
• Surreal numbers
• Transcendental numbers
• Ordinal numbers
• p-adic numbers (p-adic solenoids)
• Supernatural numbers
• Profinite integers
• Superreal numbers
• Normal numbers
• Classification
• List
Rational numbers
• Integer
• Dedekind cut
• Dyadic rational
• Half-integer
• Superparticular ratio
Authority control: National
• Germany
• Japan
• Czech Republic
|
Wikipedia
|
Affine variety
In algebraic geometry, an affine algebraic set is the set of the common zeros over an algebraically closed field k of some family of polynomials in the polynomial ring $k[X_{1},\ldots ,X_{n}].$ An affine variety or affine algebraic variety, is an affine algebraic set such that the ideal generated by the defining polynomials is prime.
Some texts call variety any algebraic set, and irreducible variety an algebraic set whose defining ideal is prime (affine variety in the above sense).
In some contexts (see, for example, Hilbert's Nullstellensatz), it is useful to distinguish the field k in which the coefficients are considered, from the algebraically closed field K (containing k) over which the common zeros are considered (that is, the points of the affine algebric set are in Kn). In this case, the variety is said defined over k, and the points of the variety that belong to kn are said k-rational or rational over k. In the common case where k is the field of real numbers, a k-rational point is called a real point.[1] When the field k is not specified, a rational point is a point that is rational over the rational numbers. For example, Fermat's Last Theorem asserts that the affine algebraic variety (it is a curve) defined by xn + yn − 1 = 0 has no rational points for any integer n greater than two.
Introduction
An affine algebraic set is the set of solutions in an algebraically closed field k of a system of polynomial equations with coefficients in k. More precisely, if $f_{1},\ldots ,f_{m}$ are polynomials with coefficients in k, they define an affine algebraic set
$V(f_{1},\ldots ,f_{m})=\left\{(a_{1},\ldots ,a_{n})\in k^{n}\;|\;f_{1}(a_{1},\ldots ,a_{n})=\ldots =f_{m}(a_{1},\ldots ,a_{n})=0\right\}.$
An affine (algebraic) variety is an affine algebraic set which is not the union of two proper affine algebraic subsets. Such an affine algebraic set is often said to be irreducible.
If X is an affine algebraic set, and I is the ideal of all polynomials that are zero on X, then the quotient ring $R=k[x_{1},\ldots ,x_{n}]/I$ is called the coordinate ring of X. If X is an affine variety, then I is prime, so the coordinate ring is an integral domain. The elements of the coordinate ring R are also called the regular functions or the polynomial functions on the variety. They form the ring of regular functions on the variety, or, simply, the ring of the variety; in other words (see #Structure sheaf), it is the space of global sections of the structure sheaf of X.
The dimension of a variety is an integer associated to every variety, and even to every algebraic set, whose importance relies on the large number of its equivalent definitions (see Dimension of an algebraic variety).
Examples
• The complement of a hypersurface in an affine variety X (that is X - { f = 0 } for some polynomial f) is affine. Its defining equations are obtained by saturating by f the defining ideal of X. The coordinate ring is thus the localization $k[X][f^{-1}]$.
• In particular, $\mathbb {C} -0$ (the affine line with the origin removed) is affine.
• On the other hand, $\mathbb {C} ^{2}-0$ (the affine plane with the origin removed) is not an affine variety; cf. Hartogs' extension theorem.
• The subvarieties of codimension one in the affine space $k^{n}$ are exactly the hypersurfaces, that is the varieties defined by a single polynomial.
• The normalization of an irreducible affine variety is affine; the coordinate ring of the normalization is the integral closure of the coordinate ring of the variety. (Similarly, the normalization of a projective variety is a projective variety.)
Rational points
Main article: rational point
For an affine variety $V\subseteq K^{n}$ over an algebraically closed field K, and a subfield k of K, a k-rational point of V is a point $p\in V\cap k^{n}.$ That is, a point of V whose coordinates are elements of k. The collection of k-rational points of an affine variety V is often denoted $V(k).$ Often, if the base field is the complex numbers C, points which are R-rational (where R is the real numbers) are called real points of the variety, and Q-rational points (Q the rational numbers) are often simply called rational points.
For instance, (1, 0) is a Q-rational and an R-rational point of the variety $V=V(x^{2}+y^{2}-1)\subseteq \mathbf {C} ^{2},$ as it is in V and all its coordinates are integers. The point (√2/2, √2/2) is a real point of V that is not Q-rational, and $(i,{\sqrt {2}})$ is a point of V that is not R-rational. This variety is called a circle, because the set of its R-rational points is the unit circle. It has infinitely many Q-rational points that are the points
$\left({\frac {1-t^{2}}{1+t^{2}}},{\frac {2t}{1+t^{2}}}\right)$
where t is a rational number.
The circle $V(x^{2}+y^{2}-3)\subseteq \mathbf {C} ^{2}$ is an example of an algebraic curve of degree two that has no Q-rational point. This can be deduced from the fact that, modulo 4, the sum of two squares cannot be 3.
It can be proved that an algebraic curve of degree two with a Q-rational point has infinitely many other Q-rational points; each such point is the second intersection point of the curve and a line with a rational slope passing through the rational point.
The complex variety $V(x^{2}+y^{2}+1)\subseteq \mathbf {C} ^{2}$ has no R-rational points, but has many complex points.
If V is an affine variety in C2 defined over the complex numbers C, the R-rational points of V can be drawn on a piece of paper or by graphing software. The figure on the right shows the R-rational points of $V(y^{2}-x^{3}+x^{2}+16x)\subseteq \mathbf {C} ^{2}.$
Singular points and tangent space
Let V be an affine variety defined by the polynomials $f_{1},\dots ,f_{r}\in k[x_{1},\dots ,x_{n}],$ and $a=(a_{1},\dots ,a_{n})$ be a point of V.
The Jacobian matrix JV(a) of V at a is the matrix of the partial derivatives
${\frac {\partial f_{j}}{\partial {x_{i}}}}(a_{1},\dots ,a_{n}).$
The point a is regular if the rank of JV(a) equals the codimension of V, and singular otherwise.
If a is regular, the tangent space to V at a is the affine subspace of $k^{n}$ defined by the linear equations[2]
$\sum _{i=1}^{n}{\frac {\partial f_{j}}{\partial {x_{i}}}}(a_{1},\dots ,a_{n})(x_{i}-a_{i})=0,\quad j=1,\dots ,r.$
If the point is singular, the affine subspace defined by these equations is also called a tangent space by some authors, while other authors say that there is no tangent space at a singular point.[3] A more intrinsic definition, which does not use coordinates is given by Zariski tangent space.
The Zariski topology
Main article: Zariski topology
The affine algebraic sets of kn form the closed sets of a topology on kn, called the Zariski topology. This follows from the fact that $V(0)=k^{n},$ $V(1)=\emptyset ,$ $V(S)\cup V(T)=V(ST),$ and $V(S)\cap V(T)=V(S,T)$ (in fact, a countable intersection of affine algebraic sets is an affine algebraic set).
The Zariski topology can also be described by way of basic open sets, where Zariski-open sets are countable unions of sets of the form $U_{f}=\{p\in k^{n}:f(p)\neq 0\}$ for $f\in k[x_{1},\ldots ,x_{n}].$ These basic open sets are the complements in kn of the closed sets $V(f)=D_{f}=\{p\in k^{n}:f(p)=0\},$ zero loci of a single polynomial. If k is Noetherian (for instance, if k is a field or a principal ideal domain), then every ideal of k is finitely-generated, so every open set is a finite union of basic open sets.
If V is an affine subvariety of kn the Zariski topology on V is simply the subspace topology inherited from the Zariski topology on kn.
Geometry–algebra correspondence
The geometric structure of an affine variety is linked in a deep way to the algebraic structure of its coordinate ring. Let I and J be ideals of k[V], the coordinate ring of an affine variety V. Let I(V) be the set of all polynomials in $k[x_{1},\ldots ,x_{n}],$ which vanish on V, and let ${\sqrt {I}}$ denote the radical of the ideal I, the set of polynomials f for which some power of f is in I. The reason that the base field is required to be algebraically closed is that affine varieties automatically satisfy Hilbert's nullstellensatz: for an ideal J in $k[x_{1},\ldots ,x_{n}],$ where k is an algebraically closed field, $I(V(J))={\sqrt {J}}.$
Radical ideals (ideals which are their own radical) of k[V] correspond to algebraic subsets of V. Indeed, for radical ideals I and J, $I\subseteq J$ if and only if $V(J)\subseteq V(I).$ Hence V(I)=V(J) if and only if I=J. Furthermore, the function taking an affine algebraic set W and returning I(W), the set of all functions which also vanish on all points of W, is the inverse of the function assigning an algebraic set to a radical ideal, by the nullstellensatz. Hence the correspondence between affine algebraic sets and radical ideals is a bijection. The coordinate ring of an affine algebraic set is reduced (nilpotent-free), as an ideal I in a ring R is radical if and only if the quotient ring R/I is reduced.
Prime ideals of the coordinate ring correspond to affine subvarieties. An affine algebraic set V(I) can be written as the union of two other algebraic sets if and only if I=JK for proper ideals J and K not equal to I (in which case $V(I)=V(J)\cup V(K)$). This is the case if and only if I is not prime. Affine subvarieties are precisely those whose coordinate ring is an integral domain. This is because an ideal is prime if and only if the quotient of the ring by the ideal is an integral domain.
Maximal ideals of k[V] correspond to points of V. If I and J are radical ideals, then $V(J)\subseteq V(I)$ if and only if $I\subseteq J.$ As maximal ideals are radical, maximal ideals correspond to minimal algebraic sets (those which contain no proper algebraic subsets), which are points in V. If V is an affine variety with coordinate ring $R=k[x_{1},\ldots ,x_{n}]/\langle f_{1},\ldots ,f_{m}\rangle ,$ this correspondence becomes explicit through the map $(a_{1},\ldots ,a_{n})\mapsto \langle {\overline {x_{1}-a_{1}}},\ldots ,{\overline {x_{n}-a_{n}}}\rangle ,$ where ${\overline {x_{i}-a_{i}}}$ denotes the image in the quotient algebra R of the polynomial $x_{i}-a_{i}.$ An algebraic subset is a point if and only if the coordinate ring of the subset is a field, as the quotient of a ring by a maximal ideal is a field.
The following table summarises this correspondence, for algebraic subsets of an affine variety and ideals of the corresponding coordinate ring:
Type of algebraic setType of idealType of coordinate ring
affine algebraic subsetradical idealreduced ring
affine subvarietyprime idealintegral domain
pointmaximal idealfield
Products of affine varieties
A product of affine varieties can be defined using the isomorphism An × Am = An+m, then embedding the product in this new affine space. Let An and Am have coordinate rings k[x1,..., xn] and k[y1,..., ym] respectively, so that their product An+m has coordinate ring k[x1,..., xn, y1,..., ym]. Let V = V( f1,..., fN) be an algebraic subset of An, and W = V( g1,..., gM) an algebraic subset of Am. Then each fi is a polynomial in k[x1,..., xn], and each gj is in k[y1,..., ym]. The product of V and W is defined as the algebraic set V × W = V( f1,..., fN, g1,..., gM) in An+m. The product is irreducible if each V, W is irreducible.[4]
The Zariski topology on An × Am is not the topological product of the Zariski topologies on the two spaces. Indeed, the product topology is generated by products of the basic open sets Uf = An − V( f ) and Tg = Am − V( g ). Hence, polynomials that are in k[x1,..., xn, y1,..., ym] but cannot be obtained as a product of a polynomial in k[x1,..., xn] with a polynomial in k[y1,..., ym] will define algebraic sets that are in the Zariski topology on An × Am , but not in the product topology.
Morphisms of affine varieties
Main article: Morphism of algebraic varieties
A morphism, or regular map, of affine varieties is a function between affine varieties which is polynomial in each coordinate: more precisely, for affine varieties V ⊆ kn and W ⊆ km, a morphism from V to W is a map φ : V → W of the form φ(a1, ..., an) = (f1(a1, ..., an), ..., fm(a1, ..., an)), where fi ∈ k[X1, ..., Xn] for each i = 1, ..., m. These are the morphisms in the category of affine varieties.
There is a one-to-one correspondence between morphisms of affine varieties over an algebraically closed field k, and homomorphisms of coordinate rings of affine varieties over k going in the opposite direction. Because of this, along with the fact that there is a one-to-one correspondence between affine varieties over k and their coordinate rings, the category of affine varieties over k is dual to the category of coordinate rings of affine varieties over k. The category of coordinate rings of affine varieties over k is precisely the category of finitely-generated, nilpotent-free algebras over k.
More precisely, for each morphism φ : V → W of affine varieties, there is a homomorphism φ# : k[W] → k[V] between the coordinate rings (going in the opposite direction), and for each such homomorphism, there is a morphism of the varieties associated to the coordinate rings. This can be shown explicitly: let V ⊆ kn and W ⊆ km be affine varieties with coordinate rings k[V] = k[X1, ..., Xn] / I and k[W] = k[Y1, ..., Ym] / J respectively. Let φ : V → W be a morphism. Indeed, a homomorphism between polynomial rings θ : k[Y1, ..., Ym] / J → k[X1, ..., Xn] / I factors uniquely through the ring k[X1, ..., Xn], and a homomorphism ψ : k[Y1, ..., Ym] / J → k[X1, ..., Xn] is determined uniquely by the images of Y1, ..., Ym. Hence, each homomorphism φ# : k[W] → k[V] corresponds uniquely to a choice of image for each Yi. Then given any morphism φ = (f1, ..., fm) from V to W, a homomorphism can be constructed φ# : k[W] → k[V] which sends Yi to ${\overline {f_{i}}},$ where ${\overline {f_{i}}}$ is the equivalence class of fi in k[V].
Similarly, for each homomorphism of the coordinate rings, a morphism of the affine varieties can be constructed in the opposite direction. Mirroring the paragraph above, a homomorphism φ# : k[W] → k[V] sends Yi to a polynomial $f_{i}(X_{1},\dots ,X_{n})$ in k[V]. This corresponds to the morphism of varieties φ : V → W defined by φ(a1, ... , an) = (f1(a1, ..., an), ..., fm(a1, ..., an)).
Structure sheaf
Equipped with the structure sheaf described below, an affine variety is a locally ringed space.
Given an affine variety X with coordinate ring A, the sheaf of k-algebras ${\mathcal {O}}_{X}$ is defined by letting ${\mathcal {O}}_{X}(U)=\Gamma (U,{\mathcal {O}}_{X})$ be the ring of regular functions on U.
Let D(f) = { x | f(x) ≠ 0 } for each f in A. They form a base for the topology of X and so ${\mathcal {O}}_{X}$ is determined by its values on the open sets D(f). (See also: sheaf of modules#Sheaf associated to a module.)
The key fact, which relies on Hilbert nullstellensatz in the essential way, is the following:
Claim — $\Gamma (D(f),{\mathcal {O}}_{X})=A[f^{-1}]$ for any f in A.
Proof:[5] The inclusion ⊃ is clear. For the opposite, let g be in the left-hand side and $J=\{h\in A|hg\in A\}$, which is an ideal. If x is in D(f), then, since g is regular near x, there is some open affine neighborhood D(h) of x such that $g\in k[D(h)]=A[h^{-1}]$; that is, hm g is in A and thus x is not in V(J). In other words, $V(J)\subset \{x|f(x)=0\}$ and thus the Hilbert nullstellensatz implies f is in the radical of J; i.e., $f^{n}g\in A$. $\square $
The claim, first of all, implies that X is a "locally ringed" space since
${\mathcal {O}}_{X,x}=\varinjlim _{f(x)\neq 0}A[f^{-1}]=A_{{\mathfrak {m}}_{x}}$
where ${\mathfrak {m}}_{x}=\{f\in A|f(x)=0\}$. Secondly, the claim implies that ${\mathcal {O}}_{X}$ is a sheaf; indeed, it says if a function is regular (pointwise) on D(f), then it must be in the coordinate ring of D(f); that is, "regular-ness" can be patched together.
Hence, $(X,{\mathcal {O}}_{X})$ is a locally ringed space.
Serre's theorem on affineness
Main article: Serre's theorem on affineness
A theorem of Serre gives a cohomological characterization of an affine variety; it says an algebraic variety is affine if and only if $H^{i}(X,F)=0$ for any $i>0$ and any quasi-coherent sheaf F on X. (cf. Cartan's theorem B.) This makes the cohomological study of an affine variety non-existent, in a sharp contrast to the projective case in which cohomology groups of line bundles are of central interest.
Affine algebraic groups
Main article: linear algebraic group
An affine variety G over an algebraically closed field k is called an affine algebraic group if it has:
• A multiplication μ: G × G → G, which is a regular morphism that follows the associativity axiom—that is, such that μ(μ(f, g), h) = μ(f, μ(g, h)) for all points f, g and h in G;
• An identity element e such that μ(e, g) = μ(g, e) = g for every g in G;
• An inverse morphism, a regular bijection ι: G → G such that μ(ι(g), g) = μ(g, ι(g)) = e for every g in G.
Together, these define a group structure on the variety. The above morphisms are often written using ordinary group notation: μ(f, g) can be written as f + g, f⋅g, or fg; the inverse ι(g) can be written as −g or g−1. Using the multiplicative notation, the associativity, identity and inverse laws can be rewritten as: f(gh) = (fg)h, ge = eg = g and gg−1 = g−1g = e.
The most prominent example of an affine algebraic group is GLn(k), the general linear group of degree n. This is the group of linear transformations of the vector space kn; if a basis of kn, is fixed, this is equivalent to the group of n×n invertible matrices with entries in k. It can be shown that any affine algebraic group is isomorphic to a subgroup of GLn(k). For this reason, affine algebraic groups are often called linear algebraic groups.
Affine algebraic groups play an important role in the classification of finite simple groups, as the groups of Lie type are all sets of Fq-rational points of an affine algebraic group, where Fq is a finite field.
Generalizations
• If an author requires the base field of an affine variety to be algebraically closed (as this article does), then irreducible affine algebraic sets over non-algebraically closed fields are a generalization of affine varieties. This generalization notably includes affine varieties over the real numbers.
• An affine variety plays a role of a local chart for algebraic varieties; that is to say, general algebraic varieties such as projective varieties are obtained by gluing affine varieties. Linear structures that are attached to varieties are also (trivially) affine varieties; e.g., tangent spaces, fibers of algebraic vector bundles.
• An affine variety is a special case of an affine scheme, a locally-ringed space which is isomorphic to the spectrum of a commutative ring (up to an equivalence of categories). Each affine variety has an affine scheme associated to it: if V(I) is an affine variety in kn with coordinate ring R = k[x1, ..., xn] / I, then the scheme corresponding to V(I) is Spec(R), the set of prime ideals of R. The affine scheme has "classical points" which correspond with points of the variety (and hence maximal ideals of the coordinate ring of the variety), and also a point for each closed subvariety of the variety (these points correspond to prime, non-maximal ideals of the coordinate ring). This creates a more well-defined notion of the "generic point" of an affine variety, by assigning to each closed subvariety an open point which is dense in the subvariety. More generally, an affine scheme is an affine variety if it is reduced, irreducible, and of finite type over an algebraically closed field k.
Notes
1. Reid (1988)
2. Milne (2017), Ch. 5
3. Reid (1988), p. 94.
4. This is because, over an algebraically closed field, the tensor product of integral domains is an integral domain; see integral domain#Properties.
5. Mumford 1999, Ch. I, § 4. Proposition 1.
See also
• Algebraic variety
• Affine scheme
• Representations on coordinate rings
References
The original article was written as a partial human translation of the corresponding French article.
• Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157
• Fulton, William (1969). Algebraic Curves (PDF). Addison-Wesley. ISBN 0-201-510103.
• Milne, J.S. (2017). "Algebraic Geometry" (PDF). www.jmilne.org. Retrieved 16 July 2021.
• Milne, Lectures on Étale cohomology
• Mumford, David (1999). The Red Book of Varieties and Schemes: Includes the Michigan Lectures (1974) on Curves and Their Jacobians. Lecture Notes in Mathematics. Vol. 1358 (2nd ed.). Springer-Verlag. doi:10.1007/b62130. ISBN 354063293X.
• Reid, Miles (1988). Undergraduate Algebraic Geometry. Cambridge University Press. ISBN 0-521-35662-8.
|
Wikipedia
|
Ring of sets
In mathematics, there are two different notions of a ring of sets, both referring to certain families of sets.
Not to be confused with Ring (mathematics).
In order theory, a nonempty family of sets ${\mathcal {R}}$ is called a ring (of sets) if it is closed under union and intersection.[1] That is, the following two statements are true for all sets $A$ and $B$,
1. $A,B\in {\mathcal {R}}$ implies $A\cup B\in {\mathcal {R}}$ and
2. $A,B\in {\mathcal {R}}$ implies $A\cap B\in {\mathcal {R}}.$
In measure theory, a nonempty family of sets ${\mathcal {R}}$ is called a ring (of sets) if it is closed under union and relative complement (set-theoretic difference).[2] That is, the following two statements are true for all sets $A$ and $B$,
1. $A,B\in {\mathcal {R}}$ implies $A\cup B\in {\mathcal {R}}$ and
2. $A,B\in {\mathcal {R}}$ implies $A\setminus B\in {\mathcal {R}}.$
This implies that a ring in the measure-theoretic sense always contains the empty set. Furthermore, for all sets A and B,
$A\cap B=A\setminus (A\setminus B),$
which shows that a family of sets closed under relative complement is also closed under intersection, so that a ring in the measure-theoretic sense is also a ring in the order-theoretic sense.
Examples
If X is any set, then the power set of X (the family of all subsets of X) forms a ring of sets in either sense.
If (X, ≤) is a partially ordered set, then its upper sets (the subsets of X with the additional property that if x belongs to an upper set U and x ≤ y, then y must also belong to U) are closed under both intersections and unions. However, in general it will not be closed under differences of sets.
The open sets and closed sets of any topological space are closed under both unions and intersections.[1]
On the real line R, the family of sets consisting of the empty set and all finite unions of half-open intervals of the form (a, b], with a, b ∈ R is a ring in the measure-theoretic sense.
If T is any transformation defined on a space, then the sets that are mapped into themselves by T are closed under both unions and intersections.[1]
If two rings of sets are both defined on the same elements, then the sets that belong to both rings themselves form a ring of sets.[1]
Related structures
A ring of sets in the order-theoretic sense forms a distributive lattice in which the intersection and union operations correspond to the lattice's meet and join operations, respectively. Conversely, every distributive lattice is isomorphic to a ring of sets; in the case of finite distributive lattices, this is Birkhoff's representation theorem and the sets may be taken as the lower sets of a partially ordered set.[1]
A family of sets closed under union and relative complement is also closed under symmetric difference and intersection. Conversely, every family of sets closed under both symmetric difference and intersection is also closed under union and relative complement. This is due to the identities
1. $A\cup B=(A\,\triangle \,B)\,\triangle \,(A\cap B)$ and
2. $A\setminus B=A\,\triangle \,(A\cap B).$
Symmetric difference and intersection together give a ring in the measure-theoretic sense the structure of a boolean ring.
In the measure-theoretic sense, a σ-ring is a ring closed under countable unions, and a δ-ring is a ring closed under countable intersections. Explicitly, a σ-ring over $X$ is a set ${\mathcal {F}}$ such that for any sequence $\{A_{k}\}_{k=1}^{\infty }\subseteq {\mathcal {F}},$ we have $ \bigcup _{k=1}^{\infty }A_{k}\in {\mathcal {F}}.$
Given a set $X,$ a field of sets − also called an algebra over $X$ − is a ring that contains $X.$ This definition entails that an algebra is closed under absolute complement $A^{c}=X\setminus A.$ A σ-algebra is an algebra that is also closed under countable unions, or equivalently a σ-ring that contains $X.$ In fact, by de Morgan's laws, a δ-ring that contains $X$ is necessarily a σ-algebra as well. Fields of sets, and especially σ-algebras, are central to the modern theory of probability and the definition of measures.
A semiring (of sets) is a family of sets ${\mathcal {S}}$ with the properties
1. $\varnothing \in {\mathcal {S}},$
• If (3) holds, then $\varnothing \in {\mathcal {S}}$ if and only if ${\mathcal {S}}\neq \varnothing .$
2. $A,B\in {\mathcal {S}}$ implies $A\cap B\in {\mathcal {S}},$ and
3. $A,B\in {\mathcal {S}}$ implies $A\setminus B=\bigcup _{i=1}^{n}C_{i}$ for some disjoint $C_{1},\ldots ,C_{n}\in {\mathcal {S}}.$
Every ring (in the measure theory sense) is a semi-ring. On the other hand, ${\mathcal {S}}:=\{\emptyset ,\{x\},\{y\}\}$ on $X=\{x,y\}$ is a semi-ring but not a ring, since it is not closed under unions.
A semialgebra[3] or elementary family [4] is a collection ${\mathcal {S}}$ of subsets of $X$ satisfying the semiring properties except with (3) replaced with:
• If $E\in {\mathcal {S}}$ then there exists a finite number of mutually disjoint sets $C_{1},\ldots ,C_{n}\in {\mathcal {S}}$ such that $X\setminus E=\bigcup _{i=1}^{n}C_{i}.$
This condition is stronger than (3), which can be seen as follows. If ${\mathcal {S}}$ is a semialgebra and $E,F\in {\mathcal {S}}$, then we can write $F^{c}=F_{1}\cup ...\cup F_{n}$ for disjoint $F_{i}\in S$. Then:
$E\setminus F=E\cap F^{c}=E\cap (F_{1}\cup ...\cup F_{n})=(E\cap F_{1})\cup ...\cup (E\cap F_{n})$
and every $E\cap F_{i}\in S$ since it is closed under intersection, and disjoint since they are contained in the disjoint $F_{i}$'s. Moreover the condition is strictly stronger: any $S$ that is both a ring and a semialgebra is an algebra, hence any ring that is not an algebra is also not a semialgebra (e.g. the collection of finite sets on an infinite set $X$).
See also
• Algebra of sets – Identities and relationships involving sets
• δ-ring – Ring closed under countable intersections
• Field of sets – Algebraic concept in measure theory, also referred to as an algebra of sets
• 𝜆-system (Dynkin system) – Family closed under complements and countable disjoint unions
• Monotone class – theoremPages displaying wikidata descriptions as a fallbackPages displaying short descriptions with no spaces
• π-system – Family of sets closed under intersection
• σ-algebra – Algebric structure of set algebra
• 𝜎-ideal – Family closed under subsets and countable unions
• 𝜎-ring – Ring closed under countable unions
Families ${\mathcal {F}}$ of sets over $\Omega $
Is necessarily true of ${\mathcal {F}}\colon $
or, is ${\mathcal {F}}$ closed under:
Directed
by $\,\supseteq $
$A\cap B$ $A\cup B$ $B\setminus A$ $\Omega \setminus A$ $A_{1}\cap A_{2}\cap \cdots $ $A_{1}\cup A_{2}\cup \cdots $ $\Omega \in {\mathcal {F}}$ $\varnothing \in {\mathcal {F}}$ F.I.P.
π-system
Semiring Never
Semialgebra (Semifield) Never
Monotone class only if $A_{i}\searrow $only if $A_{i}\nearrow $
𝜆-system (Dynkin System) only if
$A\subseteq B$
only if $A_{i}\nearrow $ or
they are disjoint
Never
Ring (Order theory)
Ring (Measure theory) Never
δ-Ring Never
𝜎-Ring Never
Algebra (Field) Never
𝜎-Algebra (𝜎-Field) Never
Dual ideal
Filter NeverNever$\varnothing \not \in {\mathcal {F}}$
Prefilter (Filter base) NeverNever$\varnothing \not \in {\mathcal {F}}$
Filter subbase NeverNever$\varnothing \not \in {\mathcal {F}}$
Open Topology
(even arbitrary $\cup $)
Never
Closed Topology
(even arbitrary $\cap $)
Never
Is necessarily true of ${\mathcal {F}}\colon $
or, is ${\mathcal {F}}$ closed under:
directed
downward
finite
intersections
finite
unions
relative
complements
complements
in $\Omega $
countable
intersections
countable
unions
contains $\Omega $ contains $\varnothing $ Finite
Intersection
Property
Additionally, a semiring is a π-system where every complement $B\setminus A$ is equal to a finite disjoint union of sets in ${\mathcal {F}}.$
A semialgebra is a semiring that contains $\Omega .$
$A,B,A_{1},A_{2},\ldots $ are arbitrary elements of ${\mathcal {F}}$ and it is assumed that ${\mathcal {F}}\neq \varnothing .$
References
1. Birkhoff, Garrett (1937), "Rings of sets", Duke Mathematical Journal, 3 (3): 443–454, doi:10.1215/S0012-7094-37-00334-X, MR 1546000.
2. De Barra, Gar (2003), Measure Theory and Integration, Horwood Publishing, p. 13, ISBN 9781904275046.
3. Durrett 2019, pp. 3–4.
4. Folland 1999, p. 23.
Sources
• Durrett, Richard (2019). Probability: Theory and Examples (PDF). Cambridge Series in Statistical and Probabilistic Mathematics. Vol. 49 (5th ed.). Cambridge New York, NY: Cambridge University Press. ISBN 978-1-108-47368-2. OCLC 1100115281. Retrieved November 5, 2020.
• Folland, Gerald B. (1999). Real Analysis: Modern Techniques and Their Applications (2nd ed.). John Wiley & Sons. ISBN 0-471-31716-0.
External links
• Ring of sets at Encyclopedia of Mathematics
|
Wikipedia
|
Ring of symmetric functions
In algebra and in particular in algebraic combinatorics, the ring of symmetric functions is a specific limit of the rings of symmetric polynomials in n indeterminates, as n goes to infinity. This ring serves as universal structure in which relations between symmetric polynomials can be expressed in a way independent of the number n of indeterminates (but its elements are neither polynomials nor functions). Among other things, this ring plays an important role in the representation theory of the symmetric group.
This article is about the ring of symmetric functions in algebraic combinatorics. For general properties of symmetric functions, see symmetric function.
The ring of symmetric functions can be given a coproduct and a bilinear form making it into a positive selfadjoint graded Hopf algebra that is both commutative and cocommutative.
Symmetric polynomials
Main article: Symmetric polynomial
The study of symmetric functions is based on that of symmetric polynomials. In a polynomial ring in some finite set of indeterminates, a polynomial is called symmetric if it stays the same whenever the indeterminates are permuted in any way. More formally, there is an action by ring automorphisms of the symmetric group Sn on the polynomial ring in n indeterminates, where a permutation acts on a polynomial by simultaneously substituting each of the indeterminates for another according to the permutation used. The invariants for this action form the subring of symmetric polynomials. If the indeterminates are X1, ..., Xn, then examples of such symmetric polynomials are
$X_{1}+X_{2}+\cdots +X_{n},\,$
$X_{1}^{3}+X_{2}^{3}+\cdots +X_{n}^{3},\,$
and
$X_{1}X_{2}\cdots X_{n}.\,$
A somewhat more complicated example is X13X2X3 + X1X23X3 + X1X2X33 + X13X2X4 + X1X23X4 + X1X2X43 + ... where the summation goes on to include all products of the third power of some variable and two other variables. There are many specific kinds of symmetric polynomials, such as elementary symmetric polynomials, power sum symmetric polynomials, monomial symmetric polynomials, complete homogeneous symmetric polynomials, and Schur polynomials.
The ring of symmetric functions
Most relations between symmetric polynomials do not depend on the number n of indeterminates, other than that some polynomials in the relation might require n to be large enough in order to be defined. For instance the Newton's identity for the third power sum polynomial p3 leads to
$p_{3}(X_{1},\ldots ,X_{n})=e_{1}(X_{1},\ldots ,X_{n})^{3}-3e_{2}(X_{1},\ldots ,X_{n})e_{1}(X_{1},\ldots ,X_{n})+3e_{3}(X_{1},\ldots ,X_{n}),$
where the $e_{i}$ denote elementary symmetric polynomials; this formula is valid for all natural numbers n, and the only notable dependency on it is that ek(X1,...,Xn) = 0 whenever n < k. One would like to write this as an identity
$p_{3}=e_{1}^{3}-3e_{2}e_{1}+3e_{3}$
that does not depend on n at all, and this can be done in the ring of symmetric functions. In that ring there are nonzero elements ek for all integers k ≥ 1, and any element of the ring can be given by a polynomial expression in the elements ek.
Definitions
A ring of symmetric functions can be defined over any commutative ring R, and will be denoted ΛR; the basic case is for R = Z. The ring ΛR is in fact a graded R-algebra. There are two main constructions for it; the first one given below can be found in (Stanley, 1999), and the second is essentially the one given in (Macdonald, 1979).
As a ring of formal power series
The easiest (though somewhat heavy) construction starts with the ring of formal power series $R[[X_{1},X_{2},...]]$ over R in infinitely (countably) many indeterminates; the elements of this power series ring are formal infinite sums of terms, each of which consists of a coefficient from R multiplied by a monomial, where each monomial is a product of finitely many finite powers of indeterminates. One defines ΛR as its subring consisting of those power series S that satisfy
1. S is invariant under any permutation of the indeterminates, and
2. the degrees of the monomials occurring in S are bounded.
Note that because of the second condition, power series are used here only to allow infinitely many terms of a fixed degree, rather than to sum terms of all possible degrees. Allowing this is necessary because an element that contains for instance a term X1 should also contain a term Xi for every i > 1 in order to be symmetric. Unlike the whole power series ring, the subring ΛR is graded by the total degree of monomials: due to condition 2, every element of ΛR is a finite sum of homogeneous elements of ΛR (which are themselves infinite sums of terms of equal degree). For every k ≥ 0, the element ek ∈ ΛR is defined as the formal sum of all products of k distinct indeterminates, which is clearly homogeneous of degree k.
As an algebraic limit
Another construction of ΛR takes somewhat longer to describe, but better indicates the relationship with the rings R[X1,...,Xn]Sn of symmetric polynomials in n indeterminates. For every n there is a surjective ring homomorphism ρn from the analogous ring R[X1,...,Xn+1]Sn+1 with one more indeterminate onto R[X1,...,Xn]Sn, defined by setting the last indeterminate Xn+1 to 0. Although ρn has a non-trivial kernel, the nonzero elements of that kernel have degree at least $n+1$ (they are multiples of X1X2...Xn+1). This means that the restriction of ρn to elements of degree at most n is a bijective linear map, and ρn(ek(X1,...,Xn+1)) = ek(X1,...,Xn) for all k ≤ n. The inverse of this restriction can be extended uniquely to a ring homomorphism φn from R[X1,...,Xn]Sn to R[X1,...,Xn+1]Sn+1, as follows for instance from the fundamental theorem of symmetric polynomials. Since the images φn(ek(X1,...,Xn)) = ek(X1,...,Xn+1) for k = 1,...,n are still algebraically independent over R, the homomorphism φn is injective and can be viewed as a (somewhat unusual) inclusion of rings; applying φn to a polynomial amounts to adding all monomials containing the new indeterminate obtained by symmetry from monomials already present. The ring ΛR is then the "union" (direct limit) of all these rings subject to these inclusions. Since all φn are compatible with the grading by total degree of the rings involved, ΛR obtains the structure of a graded ring.
This construction differs slightly from the one in (Macdonald, 1979). That construction only uses the surjective morphisms ρn without mentioning the injective morphisms φn: it constructs the homogeneous components of ΛR separately, and equips their direct sum with a ring structure using the ρn. It is also observed that the result can be described as an inverse limit in the category of graded rings. That description however somewhat obscures an important property typical for a direct limit of injective morphisms, namely that every individual element (symmetric function) is already faithfully represented in some object used in the limit construction, here a ring R[X1,...,Xd]Sd. It suffices to take for d the degree of the symmetric function, since the part in degree d of that ring is mapped isomorphically to rings with more indeterminates by φn for all n ≥ d. This implies that for studying relations between individual elements, there is no fundamental difference between symmetric polynomials and symmetric functions.
Defining individual symmetric functions
The name "symmetric function" for elements of ΛR is a misnomer: in neither construction are the elements functions, and in fact, unlike symmetric polynomials, no function of independent variables can be associated to such elements (for instance e1 would be the sum of all infinitely many variables, which is not defined unless restrictions are imposed on the variables). However the name is traditional and well established; it can be found both in (Macdonald, 1979), which says (footnote on p. 12)
The elements of Λ (unlike those of Λn) are no longer polynomials: they are formal infinite sums of monomials. We have therefore reverted to the older terminology of symmetric functions.
(here Λn denotes the ring of symmetric polynomials in n indeterminates), and also in (Stanley, 1999).
To define a symmetric function one must either indicate directly a power series as in the first construction, or give a symmetric polynomial in n indeterminates for every natural number n in a way compatible with the second construction. An expression in an unspecified number of indeterminates may do both, for instance
$e_{2}=\sum _{i<j}X_{i}X_{j}\,$
can be taken as the definition of an elementary symmetric function if the number of indeterminates is infinite, or as the definition of an elementary symmetric polynomial in any finite number of indeterminates. Symmetric polynomials for the same symmetric function should be compatible with the homomorphisms ρn (decreasing the number of indeterminates is obtained by setting some of them to zero, so that the coefficients of any monomial in the remaining indeterminates is unchanged), and their degree should remain bounded. (An example of a family of symmetric polynomials that fails both conditions is $\textstyle \prod _{i=1}^{n}X_{i}$; the family $\textstyle \prod _{i=1}^{n}(X_{i}+1)$ fails only the second condition.) Any symmetric polynomial in n indeterminates can be used to construct a compatible family of symmetric polynomials, using the homomorphisms ρi for i < n to decrease the number of indeterminates, and φi for i ≥ n to increase the number of indeterminates (which amounts to adding all monomials in new indeterminates obtained by symmetry from monomials already present).
The following are fundamental examples of symmetric functions.
• The monomial symmetric functions mα. Suppose α = (α1,α2,...) is a sequence of non-negative integers, only finitely many of which are non-zero. Then we can consider the monomial defined by α: Xα = X1α1X2α2X3α3.... Then mα is the symmetric function determined by Xα, i.e. the sum of all monomials obtained from Xα by symmetry. For a formal definition, define β ~ α to mean that the sequence β is a permutation of the sequence α and set
$m_{\alpha }=\sum \nolimits _{\beta \sim \alpha }X^{\beta }.$
This symmetric function corresponds to the monomial symmetric polynomial mα(X1,...,Xn) for any n large enough to have the monomial Xα. The distinct monomial symmetric functions are parametrized by the integer partitions (each mα has a unique representative monomial Xλ with the parts λi in weakly decreasing order). Since any symmetric function containing any of the monomials of some mα must contain all of them with the same coefficient, each symmetric function can be written as an R-linear combination of monomial symmetric functions, and the distinct monomial symmetric functions therefore form a basis of ΛR as an R-module.
• The elementary symmetric functions ek, for any natural number k; one has ek = mα where $\textstyle X^{\alpha }=\prod _{i=1}^{k}X_{i}$. As a power series, this is the sum of all distinct products of k distinct indeterminates. This symmetric function corresponds to the elementary symmetric polynomial ek(X1,...,Xn) for any n ≥ k.
• The power sum symmetric functions pk, for any positive integer k; one has pk = m(k), the monomial symmetric function for the monomial X1k. This symmetric function corresponds to the power sum symmetric polynomial pk(X1,...,Xn) = X1k + ... + Xnk for any n ≥ 1.
• The complete homogeneous symmetric functions hk, for any natural number k; hk is the sum of all monomial symmetric functions mα where α is a partition of k. As a power series, this is the sum of all monomials of degree k, which is what motivates its name. This symmetric function corresponds to the complete homogeneous symmetric polynomial hk(X1,...,Xn) for any n ≥ k.
• The Schur functions sλ for any partition λ, which corresponds to the Schur polynomial sλ(X1,...,Xn) for any n large enough to have the monomial Xλ.
There is no power sum symmetric function p0: although it is possible (and in some contexts natural) to define $\textstyle p_{0}(X_{1},\ldots ,X_{n})=\sum _{i=1}^{n}X_{i}^{0}=n$ as a symmetric polynomial in n variables, these values are not compatible with the morphisms ρn. The "discriminant" $\textstyle (\prod _{i<j}(X_{i}-X_{j}))^{2}$ is another example of an expression giving a symmetric polynomial for all n, but not defining any symmetric function. The expressions defining Schur polynomials as a quotient of alternating polynomials are somewhat similar to that for the discriminant, but the polynomials sλ(X1,...,Xn) turn out to be compatible for varying n, and therefore do define a symmetric function.
A principle relating symmetric polynomials and symmetric functions
For any symmetric function P, the corresponding symmetric polynomials in n indeterminates for any natural number n may be designated by P(X1,...,Xn). The second definition of the ring of symmetric functions implies the following fundamental principle:
If P and Q are symmetric functions of degree d, then one has the identity $P=Q$ of symmetric functions if and only if one has the identity P(X1,...,Xd) = Q(X1,...,Xd) of symmetric polynomials in d indeterminates. In this case one has in fact P(X1,...,Xn) = Q(X1,...,Xn) for any number n of indeterminates.
This is because one can always reduce the number of variables by substituting zero for some variables, and one can increase the number of variables by applying the homomorphisms φn; the definition of those homomorphisms assures that φn(P(X1,...,Xn)) = P(X1,...,Xn+1) (and similarly for Q) whenever n ≥ d. See a proof of Newton's identities for an effective application of this principle.
Properties of the ring of symmetric functions
Identities
The ring of symmetric functions is a convenient tool for writing identities between symmetric polynomials that are independent of the number of indeterminates: in ΛR there is no such number, yet by the above principle any identity in ΛR automatically gives identities the rings of symmetric polynomials over R in any number of indeterminates. Some fundamental identities are
$\sum _{i=0}^{k}(-1)^{i}e_{i}h_{k-i}=0=\sum _{i=0}^{k}(-1)^{i}h_{i}e_{k-i}\quad {\mbox{for all }}k>0,$
which shows a symmetry between elementary and complete homogeneous symmetric functions; these relations are explained under complete homogeneous symmetric polynomial.
$ke_{k}=\sum _{i=1}^{k}(-1)^{i-1}p_{i}e_{k-i}\quad {\mbox{for all }}k\geq 0,$
the Newton identities, which also have a variant for complete homogeneous symmetric functions:
$kh_{k}=\sum _{i=1}^{k}p_{i}h_{k-i}\quad {\mbox{for all }}k\geq 0.$
Structural properties of ΛR
Important properties of ΛR include the following.
1. The set of monomial symmetric functions parametrized by partitions form a basis of ΛR as a graded R-module, those parametrized by partitions of d being homogeneous of degree d; the same is true for the set of Schur functions (also parametrized by partitions).
2. ΛR is isomorphic as a graded R-algebra to a polynomial ring R[Y1,Y2, ...] in infinitely many variables, where Yi is given degree i for all i > 0, one isomorphism being the one that sends Yi to ei ∈ ΛR for every i.
3. There is an involutory automorphism ω of ΛR that interchanges the elementary symmetric functions ei and the complete homogeneous symmetric function hi for all i. It also sends each power sum symmetric function pi to (−1)i−1pi, and it permutes the Schur functions among each other, interchanging sλ and sλt where λt is the transpose partition of λ.
Property 2 is the essence of the fundamental theorem of symmetric polynomials. It immediately implies some other properties:
• The subring of ΛR generated by its elements of degree at most n is isomorphic to the ring of symmetric polynomials over R in n variables;
• The Hilbert–Poincaré series of ΛR is $\textstyle \prod _{i=1}^{\infty }{\frac {1}{1-t^{i}}}$, the generating function of the integer partitions (this also follows from property 1);
• For every n > 0, the R-module formed by the homogeneous part of ΛR of degree n, modulo its intersection with the subring generated by its elements of degree strictly less than n, is free of rank 1, and (the image of) en is a generator of this R-module;
• For every family of symmetric functions (fi)i>0 in which fi is homogeneous of degree i and gives a generator of the free R-module of the previous point (for all i), there is an alternative isomorphism of graded R-algebras from R[Y1,Y2, ...] as above to ΛR that sends Yi to fi; in other words, the family (fi)i>0 forms a set of free polynomial generators of ΛR.
This final point applies in particular to the family (hi)i>0 of complete homogeneous symmetric functions. If R contains the field $\mathbb {Q} $ of rational numbers, it applies also to the family (pi)i>0 of power sum symmetric functions. This explains why the first n elements of each of these families define sets of symmetric polynomials in n variables that are free polynomial generators of that ring of symmetric polynomials.
The fact that the complete homogeneous symmetric functions form a set of free polynomial generators of ΛR already shows the existence of an automorphism ω sending the elementary symmetric functions to the complete homogeneous ones, as mentioned in property 3. The fact that ω is an involution of ΛR follows from the symmetry between elementary and complete homogeneous symmetric functions expressed by the first set of relations given above.
The ring of symmetric functions ΛZ is the Exp ring of the integers Z. It is also a lambda-ring in a natural fashion; in fact it is the universal lambda-ring in one generator.
Generating functions
The first definition of ΛR as a subring of $R[[X_{1},X_{2},...]]$ allows the generating functions of several sequences of symmetric functions to be elegantly expressed. Contrary to the relations mentioned earlier, which are internal to ΛR, these expressions involve operations taking place in R[[X1,X2,...;t]] but outside its subring ΛR[[t]], so they are meaningful only if symmetric functions are viewed as formal power series in indeterminates Xi. We shall write "(X)" after the symmetric functions to stress this interpretation.
The generating function for the elementary symmetric functions is
$E(t)=\sum _{k\geq 0}e_{k}(X)t^{k}=\prod _{i=1}^{\infty }(1+X_{i}t).$
Similarly one has for complete homogeneous symmetric functions
$H(t)=\sum _{k\geq 0}h_{k}(X)t^{k}=\prod _{i=1}^{\infty }\left(\sum _{k\geq 0}(X_{i}t)^{k}\right)=\prod _{i=1}^{\infty }{\frac {1}{1-X_{i}t}}.$
The obvious fact that $E(-t)H(t)=1=E(t)H(-t)$ explains the symmetry between elementary and complete homogeneous symmetric functions. The generating function for the power sum symmetric functions can be expressed as
$P(t)=\sum _{k>0}p_{k}(X)t^{k}=\sum _{k>0}\sum _{i=1}^{\infty }(X_{i}t)^{k}=\sum _{i=1}^{\infty }{\frac {X_{i}t}{1-X_{i}t}}={\frac {tE'(-t)}{E(-t)}}={\frac {tH'(t)}{H(t)}}$
((Macdonald, 1979) defines P(t) as Σk>0 pk(X)tk−1, and its expressions therefore lack a factor t with respect to those given here). The two final expressions, involving the formal derivatives of the generating functions E(t) and H(t), imply Newton's identities and their variants for the complete homogeneous symmetric functions. These expressions are sometimes written as
$P(t)=-t{\frac {d}{dt}}\log(E(-t))=t{\frac {d}{dt}}\log(H(t)),$
which amounts to the same, but requires that R contain the rational numbers, so that the logarithm of power series with constant term 1 is defined (by $\textstyle \log(1-tS)=-\sum _{i>0}{\frac {1}{i}}(tS)^{i}$).
Specializations
Let $\Lambda $ be the ring of symmetric functions and $R$ a commutative algebra with unit element. An algebra homomorphism $\varphi :\Lambda \to R,\quad f\mapsto f(\varphi )$ :\Lambda \to R,\quad f\mapsto f(\varphi )} is called a specialization.[1]
Example:
• Given some real numbers $a_{1},\dots ,a_{k}$ and $f(x_{1},x_{2},\dots ,)\in \Lambda $, then the substitution $x_{1}=a_{1},\dots ,x_{k}=a_{k}$ and $x_{j}=0,\forall j>k$ is a specialization.
• Let $f\in \Lambda $, then $\operatorname {ps} (f):=f(1,q,q^{2},q^{3},\dots )$ is called principal specialization.
See also
• Newton's identities
• Quasisymmetric function
References
1. Stanley, Richard P.; Fomin, Sergey P. Enumerative Combinatorics. Vol. 2. Cambridge University Press.
• Macdonald, I. G. Symmetric functions and Hall polynomials. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 1979. viii+180 pp. ISBN 0-19-853530-9 MR553598
• Macdonald, I. G. Symmetric functions and Hall polynomials. Second edition. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1995. x+475 pp. ISBN 0-19-853489-2 MR1354144
• Stanley, Richard P. Enumerative Combinatorics, Vol. 2, Cambridge University Press, 1999. ISBN 0-521-56069-1 (hardback) ISBN 0-521-78987-7 (paperback).
|
Wikipedia
|
Ring spectrum
In stable homotopy theory, a ring spectrum is a spectrum E together with a multiplication map
μ: E ∧ E → E
For the concept of spectrum of a ring in algebraic geometry, see spectrum of a ring.
and a unit map
η: S → E,
where S is the sphere spectrum. These maps have to satisfy associativity and unitality conditions up to homotopy, much in the same way as the multiplication of a ring is associative and unital. That is,
μ (id ∧ μ) ∼ μ (μ ∧ id)
and
μ (id ∧ η) ∼ id ∼ μ(η ∧ id).
Examples of ring spectra include singular homology with coefficients in a ring, complex cobordism, K-theory, and Morava K-theory.
See also
• Highly structured ring spectrum
References
• Adams, J. Frank (1974), Stable homotopy and generalised homology, Chicago Lectures in Mathematics, University of Chicago Press, ISBN 0-226-00523-2, MR 0402720
|
Wikipedia
|
Ring theory
In algebra, ring theory is the study of rings[1]—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their representations, or, in different language, modules, special classes of rings (group rings, division rings, universal enveloping algebras), as well as an array of properties that proved to be of interest both within the theory itself and for its applications, such as homological properties and polynomial identities.
Algebraic structures
Group-like
• Group
• Semigroup / Monoid
• Rack and quandle
• Quasigroup and loop
• Abelian group
• Magma
• Lie group
Group theory
Ring-like
• Ring
• Rng
• Semiring
• Near-ring
• Commutative ring
• Domain
• Integral domain
• Field
• Division ring
• Lie ring
Ring theory
Lattice-like
• Lattice
• Semilattice
• Complemented lattice
• Total order
• Heyting algebra
• Boolean algebra
• Map of lattices
• Lattice theory
Module-like
• Module
• Group with operators
• Vector space
• Linear algebra
Algebra-like
• Algebra
• Associative
• Non-associative
• Composition algebra
• Lie algebra
• Graded
• Bialgebra
• Hopf algebra
Algebraic structure → Ring theory
Ring theory
Basic concepts
Rings
• Subrings
• Ideal
• Quotient ring
• Fractional ideal
• Total ring of fractions
• Product of rings
• Free product of associative algebras
• Tensor product of algebras
Ring homomorphisms
• Kernel
• Inner automorphism
• Frobenius endomorphism
Algebraic structures
• Module
• Associative algebra
• Graded ring
• Involutive ring
• Category of rings
• Initial ring $\mathbb {Z} $
• Terminal ring $0=\mathbb {Z} _{1}$
Related structures
• Field
• Finite field
• Non-associative ring
• Lie ring
• Jordan ring
• Semiring
• Semifield
Commutative algebra
Commutative rings
• Integral domain
• Integrally closed domain
• GCD domain
• Unique factorization domain
• Principal ideal domain
• Euclidean domain
• Field
• Finite field
• Composition ring
• Polynomial ring
• Formal power series ring
Algebraic number theory
• Algebraic number field
• Ring of integers
• Algebraic independence
• Transcendental number theory
• Transcendence degree
p-adic number theory and decimals
• Direct limit/Inverse limit
• Zero ring $\mathbb {Z} _{1}$
• Integers modulo pn $\mathbb {Z} /p^{n}\mathbb {Z} $
• Prüfer p-ring $\mathbb {Z} (p^{\infty })$
• Base-p circle ring $\mathbb {T} $
• Base-p integers $\mathbb {Z} $
• p-adic rationals $\mathbb {Z} [1/p]$
• Base-p real numbers $\mathbb {R} $
• p-adic integers $\mathbb {Z} _{p}$
• p-adic numbers $\mathbb {Q} _{p}$
• p-adic solenoid $\mathbb {T} _{p}$
Algebraic geometry
• Affine variety
Noncommutative algebra
Noncommutative rings
• Division ring
• Semiprimitive ring
• Simple ring
• Commutator
Noncommutative algebraic geometry
Free algebra
Clifford algebra
• Geometric algebra
Operator algebra
Commutative rings are much better understood than noncommutative ones. Algebraic geometry and algebraic number theory, which provide many natural examples of commutative rings, have driven much of the development of commutative ring theory, which is now, under the name of commutative algebra, a major area of modern mathematics. Because these three fields (algebraic geometry, algebraic number theory and commutative algebra) are so intimately connected it is usually difficult and meaningless to decide which field a particular result belongs to. For example, Hilbert's Nullstellensatz is a theorem which is fundamental for algebraic geometry, and is stated and proved in terms of commutative algebra. Similarly, Fermat's Last Theorem is stated in terms of elementary arithmetic, which is a part of commutative algebra, but its proof involves deep results of both algebraic number theory and algebraic geometry.
Noncommutative rings are quite different in flavour, since more unusual behavior can arise. While the theory has developed in its own right, a fairly recent trend has sought to parallel the commutative development by building the theory of certain classes of noncommutative rings in a geometric fashion as if they were rings of functions on (non-existent) 'noncommutative spaces'. This trend started in the 1980s with the development of noncommutative geometry and with the discovery of quantum groups. It has led to a better understanding of noncommutative rings, especially noncommutative Noetherian rings.[2]
For the definitions of a ring and basic concepts and their properties, see Ring (mathematics). The definitions of terms used throughout ring theory may be found in Glossary of ring theory.
Commutative rings
Main article: Commutative algebra
A ring is called commutative if its multiplication is commutative. Commutative rings resemble familiar number systems, and various definitions for commutative rings are designed to formalize properties of the integers. Commutative rings are also important in algebraic geometry. In commutative ring theory, numbers are often replaced by ideals, and the definition of the prime ideal tries to capture the essence of prime numbers. Integral domains, non-trivial commutative rings where no two non-zero elements multiply to give zero, generalize another property of the integers and serve as the proper realm to study divisibility. Principal ideal domains are integral domains in which every ideal can be generated by a single element, another property shared by the integers. Euclidean domains are integral domains in which the Euclidean algorithm can be carried out. Important examples of commutative rings can be constructed as rings of polynomials and their factor rings. Summary: Euclidean domain ⊂ principal ideal domain ⊂ unique factorization domain ⊂ integral domain ⊂ commutative ring.
Algebraic geometry
Main article: Algebraic geometry
Algebraic geometry is in many ways the mirror image of commutative algebra. This correspondence started with Hilbert's Nullstellensatz that establishes a one-to-one correspondence between the points of an algebraic variety, and the maximal ideals of its coordinate ring. This correspondence has been enlarged and systematized for translating (and proving) most geometrical properties of algebraic varieties into algebraic properties of associated commutative rings. Alexander Grothendieck completed this by introducing schemes, a generalization of algebraic varieties, which may be built from any commutative ring. More precisely, the spectrum of a commutative ring is the space of its prime ideals equipped with Zariski topology, and augmented with a sheaf of rings. These objects are the "affine schemes" (generalization of affine varieties), and a general scheme is then obtained by "gluing together" (by purely algebraic methods) several such affine schemes, in analogy to the way of constructing a manifold by gluing together the charts of an atlas.
Noncommutative rings
Main articles: Noncommutative ring, Noncommutative algebraic geometry, and Noncommutative geometry
Noncommutative rings resemble rings of matrices in many respects. Following the model of algebraic geometry, attempts have been made recently at defining noncommutative geometry based on noncommutative rings. Noncommutative rings and associative algebras (rings that are also vector spaces) are often studied via their categories of modules. A module over a ring is an abelian group that the ring acts on as a ring of endomorphisms, very much akin to the way fields (integral domains in which every non-zero element is invertible) act on vector spaces. Examples of noncommutative rings are given by rings of square matrices or more generally by rings of endomorphisms of abelian groups or modules, and by monoid rings.
Representation theory
Main article: Representation theory
Representation theory is a branch of mathematics that draws heavily on non-commutative rings. It studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and the algebraic operations in terms of matrix addition and matrix multiplication, which is non-commutative. The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras. The most prominent of these (and historically the first) is the representation theory of groups, in which elements of a group are represented by invertible matrices in such a way that the group operation is matrix multiplication.
Some relevant theorems
General
• Isomorphism theorems for rings
• Nakayama's lemma
Structure theorems
• The Artin–Wedderburn theorem determines the structure of semisimple rings
• The Jacobson density theorem determines the structure of primitive rings
• Goldie's theorem determines the structure of semiprime Goldie rings
• The Zariski–Samuel theorem determines the structure of a commutative principal ideal ring
• The Hopkins–Levitzki theorem gives necessary and sufficient conditions for a Noetherian ring to be an Artinian ring
• Morita theory consists of theorems determining when two rings have "equivalent" module categories
• Cartan–Brauer–Hua theorem gives insight on the structure of division rings
• Wedderburn's little theorem states that finite domains are fields
Other
• The Skolem–Noether theorem characterizes the automorphisms of simple rings
Structures and invariants of rings
Dimension of a commutative ring
Main article: Dimension theory (algebra)
In this section, R denotes a commutative ring. The Krull dimension of R is the supremum of the lengths n of all the chains of prime ideals ${\mathfrak {p}}_{0}\subsetneq {\mathfrak {p}}_{1}\subsetneq \cdots \subsetneq {\mathfrak {p}}_{n}$. It turns out that the polynomial ring $k[t_{1},\cdots ,t_{n}]$ over a field k has dimension n. The fundamental theorem of dimension theory states that the following numbers coincide for a noetherian local ring $(R,{\mathfrak {m}})$:[3]
• The Krull dimension of R.
• The minimum number of the generators of the ${\mathfrak {m}}$-primary ideals.
• The dimension of the graded ring $\textstyle \operatorname {gr} _{\mathfrak {m}}(R)=\bigoplus _{k\geq 0}{\mathfrak {m}}^{k}/{{\mathfrak {m}}^{k+1}}$ (equivalently, 1 plus the degree of its Hilbert polynomial).
A commutative ring R is said to be catenary if for every pair of prime ideals ${\mathfrak {p}}\subset {\mathfrak {p}}'$, there exists a finite chain of prime ideals ${\mathfrak {p}}={\mathfrak {p}}_{0}\subsetneq \cdots \subsetneq {\mathfrak {p}}_{n}={\mathfrak {p}}'$ that is maximal in the sense that it is impossible to insert an additional prime ideal between two ideals in the chain, and all such maximal chains between ${\mathfrak {p}}$ and ${\mathfrak {p}}'$ have the same length. Practically all noetherian rings that appear in applications are catenary. Ratliff proved that a noetherian local integral domain R is catenary if and only if for every prime ideal ${\mathfrak {p}}$,
$\operatorname {dim} R=\operatorname {ht} {\mathfrak {p}}+\operatorname {dim} R/{\mathfrak {p}}$
where $\operatorname {ht} {\mathfrak {p}}$ is the height of ${\mathfrak {p}}$.[4]
If R is an integral domain that is a finitely generated k-algebra, then its dimension is the transcendence degree of its field of fractions over k. If S is an integral extension of a commutative ring R, then S and R have the same dimension.
Closely related concepts are those of depth and global dimension. In general, if R is a noetherian local ring, then the depth of R is less than or equal to the dimension of R. When the equality holds, R is called a Cohen–Macaulay ring. A regular local ring is an example of a Cohen–Macaulay ring. It is a theorem of Serre that R is a regular local ring if and only if it has finite global dimension and in that case the global dimension is the Krull dimension of R. The significance of this is that a global dimension is a homological notion.
Morita equivalence
Two rings R, S are said to be Morita equivalent if the category of left modules over R is equivalent to the category of left modules over S. In fact, two commutative rings which are Morita equivalent must be isomorphic, so the notion does not add anything new to the category of commutative rings. However, commutative rings can be Morita equivalent to noncommutative rings, so Morita equivalence is coarser than isomorphism. Morita equivalence is especially important in algebraic topology and functional analysis.
Finitely generated projective module over a ring and Picard group
Let R be a commutative ring and $\mathbf {P} (R)$ the set of isomorphism classes of finitely generated projective modules over R; let also $\mathbf {P} _{n}(R)$ subsets consisting of those with constant rank n. (The rank of a module M is the continuous function $\operatorname {Spec} R\to \mathbb {Z} ,\,{\mathfrak {p}}\mapsto \dim M\otimes _{R}k({\mathfrak {p}})$.[5]) $\mathbf {P} _{1}(R)$ is usually denoted by Pic(R). It is an abelian group called the Picard group of R.[6] If R is an integral domain with the field of fractions F of R, then there is an exact sequence of groups:[7]
$1\to R^{*}\to F^{*}{\overset {f\mapsto fR}{\to }}\operatorname {Cart} (R)\to \operatorname {Pic} (R)\to 1$
where $\operatorname {Cart} (R)$ is the set of fractional ideals of R. If R is a regular domain (i.e., regular at any prime ideal), then Pic(R) is precisely the divisor class group of R.[8]
For example, if R is a principal ideal domain, then Pic(R) vanishes. In algebraic number theory, R will be taken to be the ring of integers, which is Dedekind and thus regular. It follows that Pic(R) is a finite group (finiteness of class number) that measures the deviation of the ring of integers from being a PID.
One can also consider the group completion of $\mathbf {P} (R)$; this results in a commutative ring K0(R). Note that K0(R) = K0(S) if two commutative rings R, S are Morita equivalent.
See also: Algebraic K-theory
Structure of noncommutative rings
Main article: Noncommutative ring
The structure of a noncommutative ring is more complicated than that of a commutative ring. For example, there exist simple rings that contain no non-trivial proper (two-sided) ideals, yet contain non-trivial proper left or right ideals. Various invariants exist for commutative rings, whereas invariants of noncommutative rings are difficult to find. As an example, the nilradical of a ring, the set of all nilpotent elements, is not necessarily an ideal unless the ring is commutative. Specifically, the set of all nilpotent elements in the ring of all n × n matrices over a division ring never forms an ideal, irrespective of the division ring chosen. There are, however, analogues of the nilradical defined for noncommutative rings, that coincide with the nilradical when commutativity is assumed.
The concept of the Jacobson radical of a ring; that is, the intersection of all right (left) annihilators of simple right (left) modules over a ring, is one example. The fact that the Jacobson radical can be viewed as the intersection of all maximal right (left) ideals in the ring, shows how the internal structure of the ring is reflected by its modules. It is also a fact that the intersection of all maximal right ideals in a ring is the same as the intersection of all maximal left ideals in the ring, in the context of all rings; irrespective of whether the ring is commutative.
Noncommutative rings are an active area of research due to their ubiquity in mathematics. For instance, the ring of n-by-n matrices over a field is noncommutative despite its natural occurrence in geometry, physics and many parts of mathematics. More generally, endomorphism rings of abelian groups are rarely commutative, the simplest example being the endomorphism ring of the Klein four-group.
One of the best-known strictly noncommutative ring is the quaternions.
Applications
The ring of integers of a number field
Main article: Ring of integers
The coordinate ring of an algebraic variety
If X is an affine algebraic variety, then the set of all regular functions on X forms a ring called the coordinate ring of X. For a projective variety, there is an analogous ring called the homogeneous coordinate ring. Those rings are essentially the same things as varieties: they correspond in essentially a unique way. This may be seen via either Hilbert's Nullstellensatz or scheme-theoretic constructions (i.e., Spec and Proj).
Ring of invariants
A basic (and perhaps the most fundamental) question in the classical invariant theory is to find and study polynomials in the polynomial ring $k[V]$ that are invariant under the action of a finite group (or more generally reductive) G on V. The main example is the ring of symmetric polynomials: symmetric polynomials are polynomials that are invariant under permutation of variable. The fundamental theorem of symmetric polynomials states that this ring is $R[\sigma _{1},\ldots ,\sigma _{n}]$ where $\sigma _{i}$ are elementary symmetric polynomials.
History
Commutative ring theory originated in algebraic number theory, algebraic geometry, and invariant theory. Central to the development of these subjects were the rings of integers in algebraic number fields and algebraic function fields, and the rings of polynomials in two or more variables. Noncommutative ring theory began with attempts to extend the complex numbers to various hypercomplex number systems. The genesis of the theories of commutative and noncommutative rings dates back to the early 19th century, while their maturity was achieved only in the third decade of the 20th century.
More precisely, William Rowan Hamilton put forth the quaternions and biquaternions; James Cockle presented tessarines and coquaternions; and William Kingdon Clifford was an enthusiast of split-biquaternions, which he called algebraic motors. These noncommutative algebras, and the non-associative Lie algebras, were studied within universal algebra before the subject was divided into particular mathematical structure types. One sign of re-organization was the use of direct sums to describe algebraic structure.
The various hypercomplex numbers were identified with matrix rings by Joseph Wedderburn (1908) and Emil Artin (1928). Wedderburn's structure theorems were formulated for finite-dimensional algebras over a field while Artin generalized them to Artinian rings.
In 1920, Emmy Noether, in collaboration with W. Schmeidler, published a paper about the theory of ideals in which they defined left and right ideals in a ring. The following year she published a landmark paper called Idealtheorie in Ringbereichen, analyzing ascending chain conditions with regard to (mathematical) ideals. Noted algebraist Irving Kaplansky called this work "revolutionary";[9] the publication gave rise to the term "Noetherian ring", and several other mathematical objects being called Noetherian.[9][10]
Notes
1. Ring theory may include also the study of rngs.
2. Goodearl & Warfield (1989).
3. Matsumura 1989, Theorem 13.4
4. Matsumura 1989, Theorem 31.4
5. Weibel 2013, Ch I, Definition 2.2.3
6. Weibel 2013, Definition preceding Proposition 3.2 in Ch I
7. Weibel 2013, Ch I, Proposition 3.5
8. Weibel 2013, Ch I, Corollary 3.8.1
9. Kimberling 1981, p. 18.
10. Dick, Auguste (1981), Emmy Noether: 1882–1935, translated by Blocher, H. I., Birkhäuser, ISBN 3-7643-3019-8, p. 44–45.
References
• Allenby, R. B. J. T. (1991), Rings, Fields and Groups (Second ed.), Edward Arnold, London, p. xxvi+383, ISBN 0-7131-3476-3, MR 1144518
• Blyth, T.S.; Robertson, E.F. (1985), Groups, Rings and Fields: Algebra through practice, Book 3, Cambridge: Cambridge University Press, ISBN 0-521-27288-2
• Faith, Carl (1999), Rings and Things and a Fine Array of Twentieth Century Associative Algebra, Mathematical Surveys and Monographs, vol. 65, Providence, RI: American Mathematical Society, ISBN 0-8218-0993-8, MR 1657671
• Goodearl, K. R.; Warfield, R. B., Jr. (1989), An Introduction to Noncommutative Noetherian Rings, London Mathematical Society Student Texts, vol. 16, Cambridge: Cambridge University Press, ISBN 0-521-36086-2, MR 1020298{{citation}}: CS1 maint: multiple names: authors list (link)
• Judson, Thomas W. (1997), Abstract Algebra: Theory and Applications
• Kimberling, Clark (1981), "Emmy Noether and Her Influence", in Brewer, James W; Smith, Martha K (eds.), Emmy Noether: A Tribute to Her Life and Work, Marcel Dekker, pp. 3–61
• Lam, T. Y. (1999), Lectures on Modules and Rings, Graduate Texts in Mathematics, vol. 189, New York: Springer-Verlag, doi:10.1007/978-1-4612-0525-8, ISBN 0-387-98428-3, MR 1653294
• Lam, T. Y. (2001), A First Course in Noncommutative Rings, Graduate Texts in Mathematics, vol. 131 (Second ed.), New York: Springer-Verlag, doi:10.1007/978-1-4419-8616-0, ISBN 0-387-95183-0, MR 1838439
• Lam, T. Y. (2003), Exercises in Classical Ring Theory, Problem Books in Mathematics (Second ed.), New York: Springer-Verlag, ISBN 0-387-00500-5, MR 2003255
• Matsumura, Hideyuki (1989), Commutative Ring Theory, Cambridge Studies in Advanced Mathematics, vol. 8 (Second ed.), Cambridge, UK.: Cambridge University Press, ISBN 0-521-36764-6, MR 1011461
• McConnell, J. C.; Robson, J. C. (2001), Noncommutative Noetherian Rings, Graduate Studies in Mathematics, vol. 30, Providence, RI: American Mathematical Society, doi:10.1090/gsm/030, ISBN 0-8218-2169-5, MR 1811901
• O'Connor, J. J.; Robertson, E. F. (September 2004), "The development of ring theory", MacTutor History of Mathematics Archive
• Pierce, Richard S. (1982), Associative Algebras, Graduate Texts in Mathematics, vol. 88, New York: Springer-Verlag, ISBN 0-387-90693-2, MR 0674652
• Rowen, Louis H. (1988), Ring Theory, Vol. I, Pure and Applied Mathematics, vol. 127, Boston, MA: Academic Press, ISBN 0-12-599841-4, MR 0940245. Vol. II, Pure and Applied Mathematics 128, ISBN 0-12-599842-2.
• Weibel, Charles A. (2013), The K-book: An introduction to algebraic K-theory, Graduate Studies in Mathematics, vol. 145, Providence, RI: American Mathematical Society, ISBN 978-0-8218-9132-2, MR 3076731
|
Wikipedia
|
Torus
In geometry, a torus (PL: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanar with the circle. The main types of toruses include ring toruses, horn toruses, and spindle toruses. A ring torus is sometimes colloquially referred to as a donut or doughnut.
This article is about the mathematical surface. For the volume, see Solid torus. For other uses, see Torus (disambiguation).
If the axis of revolution does not touch the circle, the surface has a ring shape and is called a torus of revolution, also known as a ring torus. If the axis of revolution is tangent to the circle, the surface is a horn torus. If the axis of revolution passes twice through the circle, the surface is a spindle torus (or self-crossing torus or self-intersecting torus). If the axis of revolution passes through the center of the circle, the surface is a degenerate torus, a double-covered sphere. If the revolved curve is not a circle, the surface is called a toroid, as in a square toroid.
Real-world objects that approximate a torus of revolution include swim rings, inner tubes and ringette rings.
A torus should not be confused with a solid torus, which is formed by rotating a disk, rather than a circle, around an axis. A solid torus is a torus plus the volume inside the torus. Real-world objects that approximate a solid torus include O-rings, non-inflatable lifebuoys, ring doughnuts, and bagels.
In topology, a ring torus is homeomorphic to the Cartesian product of two circles: $S^{1}\times S^{1}$, and the latter is taken to be the definition in that context. It is a compact 2-manifold of genus 1. The ring torus is one way to embed this space into Euclidean space, but another way to do this is the Cartesian product of the embedding of $S^{1}$ in the plane with itself. This produces a geometric object called the Clifford torus, a surface in 4-space.
In the field of topology, a torus is any topological space that is homeomorphic to a torus.[1] The surface of a coffee cup and a doughnut are both topological tori with genus one.
An example of a torus can be constructed by taking a rectangular strip of flexible material such as rubber, and joining the top edge to the bottom edge, and the left edge to the right edge, without any half-twists (compare Möbius strip).
Geometry
Bottom-halves and
vertical cross-sections
R > r: ring torus or anchor ring
R=r: horn torus
R < r: self-intersecting spindle torus
A torus can be defined parametrically by:[2]
${\begin{aligned}x(\theta ,\varphi )&=(R+r\cos \theta )\cos {\varphi }\\y(\theta ,\varphi )&=(R+r\cos \theta )\sin {\varphi }\\z(\theta ,\varphi )&=r\sin \theta \\{\text{with:}}~\theta ,\varphi \in [0,2\pi )\end{aligned}}$
where
• θ, φ are angles which make a full circle, so their values start and end at the same point,
• R is the distance from the center of the tube to the center of the torus,
• r is the radius of the tube.
Angle θ represents rotation around the tube, whereas φ represents rotation around the torus' axis of revolution. R is known as the "major radius" and r is known as the "minor radius".[3] The ratio R divided by r is known as the "aspect ratio". The typical doughnut confectionery has an aspect ratio of about 3 to 2.
An implicit equation in Cartesian coordinates for a torus radially symmetric about the z-axis is
$\left({\sqrt {x^{2}+y^{2}}}-R\right)^{2}+z^{2}=r^{2},$
or the solution of f(x, y, z) = 0, where
$f(x,y,z)=\left({\sqrt {x^{2}+y^{2}}}-R\right)^{2}+z^{2}-r^{2}.$
Algebraically eliminating the square root gives a quartic equation,
$\left(x^{2}+y^{2}+z^{2}+R^{2}-r^{2}\right)^{2}=4R^{2}\left(x^{2}+y^{2}\right).$
The three classes of standard tori correspond to the three possible aspect ratios between R and r:
• When R > r, the surface will be the familiar ring torus or anchor ring.
• R = r corresponds to the horn torus, which in effect is a torus with no "hole".
• R < r describes the self-intersecting spindle torus; its inner shell is a lemon and its outer shell is an apple
• When R = 0, the torus degenerates to the sphere.
When R ≥ r, the interior
$\left({\sqrt {x^{2}+y^{2}}}-R\right)^{2}+z^{2}<r^{2}$
of this torus is diffeomorphic (and, hence, homeomorphic) to a product of a Euclidean open disk and a circle. The volume of this solid torus and the surface area of its torus are easily computed using Pappus's centroid theorem, giving:[4]
${\begin{aligned}A&=\left(2\pi r\right)\left(2\pi R\right)=4\pi ^{2}Rr\\V&=\left(\pi r^{2}\right)\left(2\pi R\right)=2\pi ^{2}Rr^{2}.\end{aligned}}$
These formulas are the same as for a cylinder of length 2πR and radius r, obtained from cutting the tube along the plane of a small circle, and unrolling it by straightening out (rectifying) the line running around the center of the tube. The losses in surface area and volume on the inner side of the tube exactly cancel out the gains on the outer side.
Expressing the surface area and the volume by the distance p of an outermost point on the surface of the torus to the center, and the distance q of an innermost point to the center (so that R = p + q/2 and r = p − q/2), yields
${\begin{aligned}A&=4\pi ^{2}\left({\frac {p+q}{2}}\right)\left({\frac {p-q}{2}}\right)=\pi ^{2}(p+q)(p-q)\\V&=2\pi ^{2}\left({\frac {p+q}{2}}\right)\left({\frac {p-q}{2}}\right)^{2}={\tfrac {1}{4}}\pi ^{2}(p+q)(p-q)^{2}\end{aligned}}$
As a torus is the product of two circles, a modified version of the spherical coordinate system is sometimes used. In traditional spherical coordinates there are three measures, R, the distance from the center of the coordinate system, and θ and φ, angles measured from the center point.
As a torus has, effectively, two center points, the centerpoints of the angles are moved; φ measures the same angle as it does in the spherical system, but is known as the "toroidal" direction. The center point of θ is moved to the center of r, and is known as the "poloidal" direction. These terms were first used in a discussion of the Earth's magnetic field, where "poloidal" was used to denote "the direction toward the poles".[5]
In modern use, toroidal and poloidal are more commonly used to discuss magnetic confinement fusion devices.
Topology
Topologically, a torus is a closed surface defined as the product of two circles: S1 × S1. This can be viewed as lying in C2 and is a subset of the 3-sphere S3 of radius √2. This topological torus is also often called the Clifford torus. In fact, S3 is filled out by a family of nested tori in this manner (with two degenerate circles), a fact which is important in the study of S3 as a fiber bundle over S2 (the Hopf bundle).
The surface described above, given the relative topology from $\mathbb {R} ^{3}$, is homeomorphic to a topological torus as long as it does not intersect its own axis. A particular homeomorphism is given by stereographically projecting the topological torus into $\mathbb {R} ^{3}$ from the north pole of S3.
The torus can also be described as a quotient of the Cartesian plane under the identifications
$(x,y)\sim (x+1,y)\sim (x,y+1),\,$
or, equivalently, as the quotient of the unit square by pasting the opposite edges together, described as a fundamental polygon ABA−1B−1.
The fundamental group of the torus is just the direct product of the fundamental group of the circle with itself:
$\pi _{1}(\mathbb {T} ^{2})=\pi _{1}(\mathbb {S} ^{1})\times \pi _{1}(\mathbb {S} ^{1})\cong \mathbb {Z} \times \mathbb {Z} .$
Intuitively speaking, this means that a closed path that circles the torus' "hole" (say, a circle that traces out a particular latitude) and then circles the torus' "body" (say, a circle that traces out a particular longitude) can be deformed to a path that circles the body and then the hole. So, strictly 'latitudinal' and strictly 'longitudinal' paths commute. An equivalent statement may be imagined as two shoelaces passing through each other, then unwinding, then rewinding.
If a torus is punctured and turned inside out then another torus results, with lines of latitude and longitude interchanged. This is equivalent to building a torus from a cylinder, by joining the circular ends together, in two ways: around the outside like joining two ends of a garden hose, or through the inside like rolling a sock (with the toe cut off). Additionally, if the cylinder was made by gluing two opposite sides of a rectangle together, choosing the other two sides instead will cause the same reversal of orientation.
The first homology group of the torus is isomorphic to the fundamental group (this follows from Hurewicz theorem since the fundamental group is abelian).
Two-sheeted cover
The 2-torus double-covers the 2-sphere, with four ramification points. Every conformal structure on the 2-torus can be represented as a two-sheeted cover of the 2-sphere. The points on the torus corresponding to the ramification points are the Weierstrass points. In fact, the conformal type of the torus is determined by the cross-ratio of the four points.
n-dimensional torus
The torus has a generalization to higher dimensions, the n-dimensional torus, often called the n-torus or hypertorus for short. (This is the more typical meaning of the term "n-torus", the other referring to n holes or of genus n.[6]) Recalling that the torus is the product space of two circles, the n-dimensional torus is the product of n circles. That is:
$\mathbb {T} ^{n}=\underbrace {\mathbb {S} ^{1}\times \cdots \times \mathbb {S} ^{1}} _{n}.$
The standard 1-torus is just the circle: $\mathbb {T} ^{1}=\mathbb {S} ^{1}$. The torus discussed above is the standard 2-torus, $\mathbb {T} ^{2}$. And similar to the 2-torus, the n-torus, $\mathbb {T} ^{n}$ can be described as a quotient of $\mathbb {R} ^{n}$ under integral shifts in any coordinate. That is, the n-torus is $\mathbb {R} ^{n}$ modulo the action of the integer lattice $\mathbb {Z} ^{n}$ (with the action being taken as vector addition). Equivalently, the n-torus is obtained from the n-dimensional hypercube by gluing the opposite faces together.
An n-torus in this sense is an example of an n-dimensional compact manifold. It is also an example of a compact abelian Lie group. This follows from the fact that the unit circle is a compact abelian Lie group (when identified with the unit complex numbers with multiplication). Group multiplication on the torus is then defined by coordinate-wise multiplication.
Toroidal groups play an important part in the theory of compact Lie groups. This is due in part to the fact that in any compact Lie group G one can always find a maximal torus; that is, a closed subgroup which is a torus of the largest possible dimension. Such maximal tori T have a controlling role to play in theory of connected G. Toroidal groups are examples of protori, which (like tori) are compact connected abelian groups, which are not required to be manifolds.
Automorphisms of T are easily constructed from automorphisms of the lattice $\mathbb {Z} ^{n}$, which are classified by invertible integral matrices of size n with an integral inverse; these are just the integral matrices with determinant ±1. Making them act on $\mathbb {R} ^{n}$ in the usual way, one has the typical toral automorphism on the quotient.
The fundamental group of an n-torus is a free abelian group of rank n. The k-th homology group of an n-torus is a free abelian group of rank n choose k. It follows that the Euler characteristic of the n-torus is 0 for all n. The cohomology ring H•($\mathbb {T} ^{n}$, Z) can be identified with the exterior algebra over the Z-module $\mathbb {Z} ^{n}$ whose generators are the duals of the n nontrivial cycles.
See also: Quasitoric manifold
Configuration space
As the n-torus is the n-fold product of the circle, the n-torus is the configuration space of n ordered, not necessarily distinct points on the circle. Symbolically, $\mathbb {T} ^{n}=(\mathbb {S} ^{1})^{n}$. The configuration space of unordered, not necessarily distinct points is accordingly the orbifold $\mathbb {T} ^{n}/\mathbb {S} _{n}$, which is the quotient of the torus by the symmetric group on n letters (by permuting the coordinates).
For n = 2, the quotient is the Möbius strip, the edge corresponding to the orbifold points where the two coordinates coincide. For n = 3 this quotient may be described as a solid torus with cross-section an equilateral triangle, with a twist; equivalently, as a triangular prism whose top and bottom faces are connected with a 1/3 twist (120°): the 3-dimensional interior corresponds to the points on the 3-torus where all 3 coordinates are distinct, the 2-dimensional face corresponds to points with 2 coordinates equal and the 3rd different, while the 1-dimensional edge corresponds to points with all 3 coordinates identical.
These orbifolds have found significant applications to music theory in the work of Dmitri Tymoczko and collaborators (Felipe Posada, Michael Kolinas, et al.), being used to model musical triads.[7][8]
Flat torus
A flat torus is a torus with the metric inherited from its representation as the quotient, $\mathbb {R} ^{2}$/L, where L is a discrete subgroup of $\mathbb {R} ^{2}$ isomorphic to $\mathbb {Z} ^{2}$. This gives the quotient the structure of a Riemannian manifold. Perhaps the simplest example of this is when L = $\mathbb {Z} ^{2}$: $\mathbb {R} ^{2}/\mathbb {Z} ^{2}$, which can also be described as the Cartesian plane under the identifications (x, y) ~ (x + 1, y) ~ (x, y + 1). This particular flat torus (and any uniformly scaled version of it) is known as the "square" flat torus.
This metric of the square flat torus can also be realised by specific embeddings of the familiar 2-torus into Euclidean 4-space or higher dimensions. Its surface has zero Gaussian curvature everywhere. Its surface is flat in the same sense that the surface of a cylinder is flat. In 3 dimensions, one can bend a flat sheet of paper into a cylinder without stretching the paper, but this cylinder cannot be bent into a torus without stretching the paper (unless some regularity and differentiability conditions are given up, see below).
A simple 4-dimensional Euclidean embedding of a rectangular flat torus (more general than the square one) is as follows:
$(x,y,z,w)=(R\cos u,R\sin u,P\cos v,P\sin v)$
where R and P are positive constants determining the aspect ratio. It is diffeomorphic to a regular torus but not isometric. It can not be analytically embedded (smooth of class Ck, 2 ≤ k ≤ ∞) into Euclidean 3-space. Mapping it into 3-space requires one to stretch it, in which case it looks like a regular torus. For example, in the following map:
$(x,y,z)=((R+P\sin v)\cos u,(R+P\sin v)\sin u,P\cos v).$
If R and P in the above flat torus parametrization form a unit vector (R, P) = (cos(η), sin(η)) then u, v, and 0 < η < π/2 parameterize the unit 3-sphere as Hopf coordinates. In particular, for certain very specific choices of a square flat torus in the 3-sphere S3, where η = π/4 above, the torus will partition the 3-sphere into two congruent solid tori subsets with the aforesaid flat torus surface as their common boundary. One example is the torus T defined by
$T=\left\{(x,y,z,w)\in \mathbb {S} ^{3}\mid x^{2}+y^{2}={\frac {1}{2}},\ z^{2}+w^{2}={\frac {1}{2}}\right\}.$
Other tori in S3 having this partitioning property include the square tori of the form Q⋅T, where Q is a rotation of 4-dimensional space $\mathbb {R} ^{4}$, or in other words Q is a member of the Lie group SO(4).
It is known that there exists no C2 (twice continuously differentiable) embedding of a flat torus into 3-space. (The idea of the proof is to take a large sphere containing such a flat torus in its interior, and shrink the radius of the sphere until it just touches the torus for the first time. Such a point of contact must be a tangency. But that would imply that part of the torus, since it has zero curvature everywhere, must lie strictly outside the sphere, which is a contradiction.) On the other hand, according to the Nash-Kuiper theorem, which was proven in the 1950s, an isometric C1 embedding exists. This is solely an existence proof and does not provide explicit equations for such an embedding.
In April 2012, an explicit C1 (continuously differentiable) embedding of a flat torus into 3-dimensional Euclidean space $\mathbb {R} ^{3}$ was found.[9][10][11][12] It is a flat torus in the sense that as metric spaces, it is isometric to a flat square torus. It is similar in structure to a fractal as it is constructed by repeatedly corrugating an ordinary torus. Like fractals, it has no defined Gaussian curvature. However, unlike fractals, it does have defined surface normals, yielding a so-called "smooth fractal". The key to obtain the smoothness of this corrugated torus is to have the amplitudes of successive corrugations decreasing faster than their "wavelengths".[13] (These infinitely recursive corrugations are used only for embedding into three dimensions; they are not an intrinsic feature of the flat torus.) This is the first time that any such embedding was defined by explicit equations or depicted by computer graphics.
Genus g surface
Main article: Genus g surface
In the theory of surfaces there is another object, the "genus" g surface. Instead of the product of n circles, a genus g surface is the connected sum of g two-tori. To form a connected sum of two surfaces, remove from each the interior of a disk and "glue" the surfaces together along the boundary circles. To form the connected sum of more than two surfaces, sum two of them at a time until they are all connected. In this sense, a genus g surface resembles the surface of g doughnuts stuck together side by side, or a 2-sphere with g handles attached.
As examples, a genus zero surface (without boundary) is the two-sphere while a genus one surface (without boundary) is the ordinary torus. The surfaces of higher genus are sometimes called n-holed tori (or, rarely, n-fold tori). The terms double torus and triple torus are also occasionally used.
The classification theorem for surfaces states that every compact connected surface is topologically equivalent to either the sphere or the connect sum of some number of tori, disks, and real projective planes.
genus two
genus three
Toroidal polyhedra
Further information: Toroidal polyhedron
Polyhedra with the topological type of a torus are called toroidal polyhedra, and have Euler characteristic V − E + F = 0. For any number of holes, the formula generalizes to V − E + F = 2 − 2N, where N is the number of holes.
The term "toroidal polyhedron" is also used for higher-genus polyhedra and for immersions of toroidal polyhedra.
Automorphisms
The homeomorphism group (or the subgroup of diffeomorphisms) of the torus is studied in geometric topology. Its mapping class group (the connected components of the homeomorphism group) is surjective onto the group $\operatorname {GL} (n,\mathbb {Z} )$ of invertible integer matrices, which can be realized as linear maps on the universal covering space $\mathbb {R} ^{n}$ that preserve the standard lattice $\mathbb {Z} ^{n}$ (this corresponds to integer coefficients) and thus descend to the quotient.
At the level of homotopy and homology, the mapping class group can be identified as the action on the first homology (or equivalently, first cohomology, or on the fundamental group, as these are all naturally isomorphic; also the first cohomology group generates the cohomology algebra:
$\operatorname {MCG} _{\operatorname {Ho} }(\mathbb {T} ^{n})=\operatorname {Aut} (\pi _{1}(X))=\operatorname {Aut} (\mathbb {Z} ^{n})=\operatorname {GL} (n,\mathbb {Z} ).$
Since the torus is an Eilenberg–MacLane space K(G, 1), its homotopy equivalences, up to homotopy, can be identified with automorphisms of the fundamental group); all homotopy equivalences of the torus can be realized by homeomorphisms – every homotopy equivalence is homotopic to a homeomorphism.
Thus the short exact sequence of the mapping class group splits (an identification of the torus as the quotient of $\mathbb {R} ^{n}$ gives a splitting, via the linear maps, as above):
$1\to \operatorname {Homeo} _{0}(\mathbb {T} ^{n})\to \operatorname {Homeo} (\mathbb {T} ^{n})\to \operatorname {MCG} _{\operatorname {TOP} }(\mathbb {T} ^{n})\to 1.$
The mapping class group of higher genus surfaces is much more complicated, and an area of active research.
Coloring a torus
The torus's chromatic number is seven, meaning every graph that can be embedded on the torus has a chromatic number of at most seven. (Since the complete graph ${\mathsf {K_{7}}}$ can be embedded on the torus, and $\chi ({\mathsf {K_{7}}})=7$, the upper bound is tight.) Equivalently, in a torus divided into regions, it is always possible to color the regions using no more than seven colors so that no neighboring regions are the same color. (Contrast with the four color theorem for the plane.)
de Bruijn torus
In combinatorial mathematics, a de Bruijn torus is an array of symbols from an alphabet (often just 0 and 1) that contains every m-by-n matrix exactly once. It is a torus because the edges are considered wraparound for the purpose of finding matrices. Its name comes from the De Bruijn sequence, which can be considered a special case where n is 1 (one dimension).
Cutting a torus
A solid torus of revolution can be cut by n (> 0) planes into maximally
${\begin{pmatrix}n+2\\n-1\end{pmatrix}}+{\begin{pmatrix}n\\n-1\end{pmatrix}}={\tfrac {1}{6}}(n^{3}+3n^{2}+8n)$
parts.[14]
The first 11 numbers of parts, for 0 ≤ n ≤ 10 (including the case of n = 0, not covered by the above formulas), are as follows:
1, 2, 6, 13, 24, 40, 62, 91, 128, 174, 230, ... (sequence A003600 in the OEIS).
See also
• 3-torus
• Algebraic torus
• Angenent torus
• Annulus (geometry)
• Clifford torus
• Complex torus
• Dupin cyclide
• Elliptic curve
• Irrational winding of a torus
• Joint European Torus
• Klein bottle
• Loewner's torus inequality
• Maximal torus
• Period lattice
• Real projective plane
• Sphere
• Spiric section
• Surface (topology)
• Toric lens
• Toric section
• Toric variety
• Toroid
• Toroidal and poloidal
• Torus-based cryptography
• Torus knot
• Umbilic torus
• Villarceau circles
Notes
• Nociones de Geometría Analítica y Álgebra Lineal, ISBN 978-970-10-6596-9, Author: Kozak Ana Maria, Pompeya Pastorelli Sonia, Verdanega Pedro Emilio, Editorial: McGraw-Hill, Edition 2007, 744 pages, language: Spanish
• Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 0-521-79540-0.
• V. V. Nikulin, I. R. Shafarevich. Geometries and Groups. Springer, 1987. ISBN 3-540-15281-4, ISBN 978-3-540-15281-1.
• "Tore (notion géométrique)" at Encyclopédie des Formes Mathématiques Remarquables
References
1. Gallier, Jean; Xu, Dianna (2013). A Guide to the Classification Theorem for Compact Surfaces. Geometry and Computing. Vol. 9. Springer, Heidelberg. doi:10.1007/978-3-642-34364-3. ISBN 978-3-642-34363-6. MR 3026641.
2. "Equations for the Standard Torus". Geom.uiuc.edu. 6 July 1995. Archived from the original on 29 April 2012. Retrieved 21 July 2012.
3. "Torus". Spatial Corp. Archived from the original on 13 December 2014. Retrieved 16 November 2014.
4. Weisstein, Eric W. "Torus". MathWorld.
5. "poloidal". Oxford English Dictionary Online. Oxford University Press. Retrieved 10 August 2007.
6. Weisstein, Eric W. "Torus". mathworld.wolfram.com. Retrieved 27 July 2021.
7. Tymoczko, Dmitri (7 July 2006). "The Geometry of Musical Chords" (PDF). Science. 313 (5783): 72–74. Bibcode:2006Sci...313...72T. CiteSeerX 10.1.1.215.7449. doi:10.1126/science.1126287. PMID 16825563. S2CID 2877171. Archived (PDF) from the original on 25 July 2011.
8. Phillips, Tony (October 2006). "Take on Math in the Media". American Mathematical Society. Archived from the original on 5 October 2008.
9. Filippelli, Gianluigi (27 April 2012). "Doc Madhattan: A flat torus in three dimensional space". Proceedings of the National Academy of Sciences. 109 (19): 7218–7223. doi:10.1073/pnas.1118478109. PMC 3358891. PMID 22523238. Archived from the original on 25 June 2012. Retrieved 21 July 2012.
10. Enrico de Lazaro (18 April 2012). "Mathematicians Produce First-Ever Image of Flat Torus in 3D | Mathematics". Sci-News.com. Archived from the original on 1 June 2012. Retrieved 21 July 2012.
11. "Mathematics: first-ever image of a flat torus in 3D – CNRS Web site – CNRS". Archived from the original on 5 July 2012. Retrieved 21 July 2012.
12. "Flat tori finally visualized!". Math.univ-lyon1.fr. 18 April 2012. Archived from the original on 18 June 2012. Retrieved 21 July 2012.
13. Hoang, Lê Nguyên (2016). "The Tortuous Geometry of the Flat Torus". Science4All. Retrieved 1 November 2022.
14. Weisstein, Eric W. "Torus Cutting". MathWorld.
External links
Look up torus in Wiktionary, the free dictionary.
Wikimedia Commons has media related to:
Torus (category)
• Creation of a torus at cut-the-knot
• "4D torus" Fly-through cross-sections of a four-dimensional torus
• "Relational Perspective Map" Visualizing high dimensional data with flat torus
• Polydoes, doughnut-shaped polygons
• Archived at Ghostarchive and the Wayback Machine: Séquin, Carlo H (27 January 2014). "Topology of a Twisted Torus – Numberphile" (video). Brady Haran.
• Anders Sandberg (4 February 2014). "Torus Earth". Retrieved 24 July 2019.
Compact topological surfaces and their immersions in 3D
Without boundary
Orientable
• Sphere (genus 0)
• Torus (genus 1)
• Number 8 (genus 2)
• Pretzel (genus 3) ...
Non-orientable
• Real projective plane
• genus 1; Boy's surface
• Roman surface
• Klein bottle (genus 2)
• Dyck's surface (genus 3) ...
With boundary
• Disk
• Semisphere
• Ribbon
• Annulus
• Cylinder
• Möbius strip
• Cross-cap
• Sphere with three holes ...
Related
notions
Properties
• Connectedness
• Compactness
• Triangulatedness or smoothness
• Orientability
Characteristics
• Number of boundary components
• Genus
• Euler characteristic
Operations
• Connected sum
• Making a hole
• Gluing a handle
• Gluing a cross-cap
• Immersion
Authority control: National
• Germany
• Poland
|
Wikipedia
|
Ring (mathematics)
In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series.
Algebraic structures
Group-like
• Group
• Semigroup / Monoid
• Rack and quandle
• Quasigroup and loop
• Abelian group
• Magma
• Lie group
Group theory
Ring-like
• Ring
• Rng
• Semiring
• Near-ring
• Commutative ring
• Domain
• Integral domain
• Field
• Division ring
• Lie ring
Ring theory
Lattice-like
• Lattice
• Semilattice
• Complemented lattice
• Total order
• Heyting algebra
• Boolean algebra
• Map of lattices
• Lattice theory
Module-like
• Module
• Group with operators
• Vector space
• Linear algebra
Algebra-like
• Algebra
• Associative
• Non-associative
• Composition algebra
• Lie algebra
• Graded
• Bialgebra
• Hopf algebra
Algebraic structure → Ring theory
Ring theory
Basic concepts
Rings
• Subrings
• Ideal
• Quotient ring
• Fractional ideal
• Total ring of fractions
• Product of rings
• Free product of associative algebras
• Tensor product of algebras
Ring homomorphisms
• Kernel
• Inner automorphism
• Frobenius endomorphism
Algebraic structures
• Module
• Associative algebra
• Graded ring
• Involutive ring
• Category of rings
• Initial ring $\mathbb {Z} $
• Terminal ring $0=\mathbb {Z} _{1}$
Related structures
• Field
• Finite field
• Non-associative ring
• Lie ring
• Jordan ring
• Semiring
• Semifield
Commutative algebra
Commutative rings
• Integral domain
• Integrally closed domain
• GCD domain
• Unique factorization domain
• Principal ideal domain
• Euclidean domain
• Field
• Finite field
• Composition ring
• Polynomial ring
• Formal power series ring
Algebraic number theory
• Algebraic number field
• Ring of integers
• Algebraic independence
• Transcendental number theory
• Transcendence degree
p-adic number theory and decimals
• Direct limit/Inverse limit
• Zero ring $\mathbb {Z} _{1}$
• Integers modulo pn $\mathbb {Z} /p^{n}\mathbb {Z} $
• Prüfer p-ring $\mathbb {Z} (p^{\infty })$
• Base-p circle ring $\mathbb {T} $
• Base-p integers $\mathbb {Z} $
• p-adic rationals $\mathbb {Z} [1/p]$
• Base-p real numbers $\mathbb {R} $
• p-adic integers $\mathbb {Z} _{p}$
• p-adic numbers $\mathbb {Q} _{p}$
• p-adic solenoid $\mathbb {T} _{p}$
Algebraic geometry
• Affine variety
Noncommutative algebra
Noncommutative rings
• Division ring
• Semiprimitive ring
• Simple ring
• Commutator
Noncommutative algebraic geometry
Free algebra
Clifford algebra
• Geometric algebra
Operator algebra
Formally, a ring is an abelian group whose operation is called addition, with a second binary operation called multiplication that is associative, is distributive over the addition operation, and has a multiplicative identity element. (Some authors use the term "rng" with a missing "i" to refer to the more general structure that omits this last requirement; see § Notes on the definition.)
Whether a ring is commutative (that is, whether the order in which two elements are multiplied might change the result) has profound implications on its behavior. Commutative algebra, the theory of commutative rings, is a major branch of ring theory. Its development has been greatly influenced by problems and ideas of algebraic number theory and algebraic geometry. The simplest commutative rings are those that admit division by non-zero elements; such rings are called fields.
Examples of commutative rings include the set of integers with their standard addition and multiplication, the set of polynomials with their addition and multiplication, the coordinate ring of an affine algebraic variety, and the ring of integers of a number field. Examples of noncommutative rings include the ring of n × n real square matrices with n ≥ 2, group rings in representation theory, operator algebras in functional analysis, rings of differential operators, and cohomology rings in topology.
The conceptualization of rings spanned the 1870s to the 1920s, with key contributions by Dedekind, Hilbert, Fraenkel, and Noether. Rings were first formalized as a generalization of Dedekind domains that occur in number theory, and of polynomial rings and rings of invariants that occur in algebraic geometry and invariant theory. They later proved useful in other branches of mathematics such as geometry and analysis.
Definition
A ring is a set R equipped with two binary operations[lower-alpha 1] + (addition) and ⋅ (multiplication) satisfying the following three sets of axioms, called the ring axioms[1][2][3]
1. R is an abelian group under addition, meaning that:
• (a + b) + c = a + (b + c) for all a, b, c in R (that is, + is associative).
• a + b = b + a for all a, b in R (that is, + is commutative).
• There is an element 0 in R such that a + 0 = a for all a in R (that is, 0 is the additive identity).
• For each a in R there exists −a in R such that a + (−a) = 0 (that is, −a is the additive inverse of a).
2. R is a monoid under multiplication, meaning that:
• (a · b) · c = a · (b · c) for all a, b, c in R (that is, ⋅ is associative).
• There is an element 1 in R such that a · 1 = a and 1 · a = a for all a in R (that is, 1 is the multiplicative identity). [lower-alpha 2]
3. Multiplication is distributive with respect to addition, meaning that:
• a · (b + c) = (a · b) + (a · c) for all a, b, c in R (left distributivity).
• (b + c) · a = (b · a) + (c · a) for all a, b, c in R (right distributivity).
Notes on the definition
In the terminology of this article, a ring is defined to have a multiplicative identity, while a structure with the same axiomatic definition but without the requirement for a multiplicative identity is instead called a rng (IPA: /rʊŋ/). For example, the set of even integers with the usual + and ⋅ is a rng, but not a ring. As explained in § History below, many authors apply the term "ring" without requiring a multiplicative identity.
The multiplication symbol ⋅ is usually omitted; for example, xy means x · y.
Although ring addition is commutative, ring multiplication is not required to be commutative: ab need not necessarily equal ba. Rings that also satisfy commutativity for multiplication (such as the ring of integers) are called commutative rings. Books on commutative algebra or algebraic geometry often adopt the convention that ring means commutative ring, to simplify terminology.
In a ring, multiplicative inverses are not required to exist. A nonzero commutative ring in which every nonzero element has a multiplicative inverse is called a field.
The additive group of a ring is the underlying set equipped with only the operation of addition. Although the definition requires that the additive group be abelian, this can be inferred from the other ring axioms.[4] The proof makes use of the "1", and does not work in a rng. (For a rng, omitting the axiom of commutativity of addition leaves it inferable from the remaining rng assumptions only for elements that are products: ab + cd = cd + ab.)
Although most modern authors use the term "ring" as defined here, there are a few who use the term to refer to more general structures in which there is no requirement for multiplication to be associative.[5] For these authors, every algebra is a "ring".
Illustration
The most familiar example of a ring is the set of all integers $\mathbb {Z} ,$ consisting of the numbers
$\dots ,-5,-4,-3,-2,-1,0,1,2,3,4,5,\dots $
The axioms of a ring were elaborated as a generalization of familiar properties of addition and multiplication of integers.
Some properties
Some basic properties of a ring follow immediately from the axioms:
• The additive identity is unique.
• The additive inverse of each element is unique.
• The multiplicative identity is unique.
• For any element x in a ring R, one has x0 = 0 = 0x (zero is an absorbing element with respect to multiplication) and (–1)x = –x.
• If 0 = 1 in a ring R (or more generally, 0 is a unit element), then R has only one element, and is called the zero ring.
• If a ring R contains the zero ring as a subring, then R itself is the zero ring.[6]
• The binomial formula holds for any x and y satisfying xy = yx.
Example: Integers modulo 4
See also: Modular arithmetic
Equip the set $\mathbb {Z} /4\mathbb {Z} =\left\{{\overline {0}},{\overline {1}},{\overline {2}},{\overline {3}}\right\}$ with the following operations:
• The sum ${\overline {x}}+{\overline {y}}$ in $\mathbb {Z} /4\mathbb {Z} $ is the remainder when the integer x + y is divided by 4 (as x + y is always smaller than 8, this remainder is either x + y or x + y − 4). For example, ${\overline {2}}+{\overline {3}}={\overline {1}}$ and ${\overline {3}}+{\overline {3}}={\overline {2}}.$
• The product ${\overline {x}}\cdot {\overline {y}}$ in $\mathbb {Z} /4\mathbb {Z} $ is the remainder when the integer xy is divided by 4. For example, ${\overline {2}}\cdot {\overline {3}}={\overline {2}}$ and ${\overline {3}}\cdot {\overline {3}}={\overline {1}}.$
Then $\mathbb {Z} /4\mathbb {Z} $ is a ring: each axiom follows from the corresponding axiom for $\mathbb {Z} .$ If x is an integer, the remainder of x when divided by 4 may be considered as an element of $\mathbb {Z} /4\mathbb {Z} ,$ and this element is often denoted by "x mod 4" or ${\overline {x}},$ which is consistent with the notation for 0, 1, 2, 3. The additive inverse of any ${\overline {x}}$ in $\mathbb {Z} /4\mathbb {Z} $ is $-{\overline {x}}={\overline {-x}}.$ For example, $-{\overline {3}}={\overline {-3}}={\overline {1}}.$
Example: 2-by-2 matrices
The set of 2-by-2 square matrices with entries in a field F is[7][8][9][10]
$\operatorname {M} _{2}(F)=\left\{\left.{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\right|\ a,b,c,d\in F\right\}.$
With the operations of matrix addition and matrix multiplication, $\operatorname {M} _{2}(F)$ satisfies the above ring axioms. The element $\left({\begin{smallmatrix}1&0\\0&1\end{smallmatrix}}\right)$ is the multiplicative identity of the ring. If $A=\left({\begin{smallmatrix}0&1\\1&0\end{smallmatrix}}\right)$ and $B=\left({\begin{smallmatrix}0&1\\0&0\end{smallmatrix}}\right),$ then $AB=\left({\begin{smallmatrix}0&0\\0&1\end{smallmatrix}}\right)$ while $BA=\left({\begin{smallmatrix}1&0\\0&0\end{smallmatrix}}\right);$ this example shows that the ring is noncommutative.
More generally, for any ring R, commutative or not, and any nonnegative integer n, the square matrices of dimension n with entries in R form a ring: see Matrix ring.
History
See also: Ring theory § History
Dedekind
The study of rings originated from the theory of polynomial rings and the theory of algebraic integers.[11] In 1871, Richard Dedekind defined the concept of the ring of integers of a number field.[12] In this context, he introduced the terms "ideal" (inspired by Ernst Kummer's notion of ideal number) and "module" and studied their properties. Dedekind did not use the term "ring" and did not define the concept of a ring in a general setting.
Hilbert
The term "Zahlring" (number ring) was coined by David Hilbert in 1892 and published in 1897.[13] In 19th century German, the word "Ring" could mean "association", which is still used today in English in a limited sense (for example, spy ring),[14] so if that were the etymology then it would be similar to the way "group" entered mathematics by being a non-technical word for "collection of related things". According to Harvey Cohn, Hilbert used the term for a ring that had the property of "circling directly back" to an element of itself (in the sense of an equivalence).[15] Specifically, in a ring of algebraic integers, all high powers of an algebraic integer can be written as an integral combination of a fixed set of lower powers, and thus the powers "cycle back". For instance, if $a^{3}-4a+1=0$ then:
${\begin{aligned}a^{3}&=4a-1,\\a^{4}&=4a^{2}-a,\\a^{5}&=-a^{2}+16a-4,\\a^{6}&=16a^{2}-8a+1,\\a^{7}&=-8a^{2}+65a-16,\\\vdots \ &\qquad \vdots \end{aligned}}$
and so on; in general, an is going to be an integral linear combination of 1, a, and a2.
Fraenkel and Noether
The first axiomatic definition of a ring was given by Adolf Fraenkel in 1915,[16][17] but his axioms were stricter than those in the modern definition. For instance, he required every non-zero-divisor to have a multiplicative inverse.[18] In 1921, Emmy Noether gave a modern axiomatic definition of commutative rings (with and without 1) and developed the foundations of commutative ring theory in her paper Idealtheorie in Ringbereichen.[19]
Multiplicative identity and the term "ring"
Fraenkel's axioms for a "ring" included that of a multiplicative identity,[20] whereas Noether's did not.[19]
Most or all books on algebra[21][22] up to around 1960 followed Noether's convention of not requiring a 1 for a "ring". Starting in the 1960s, it became increasingly common to see books including the existence of 1 in the definition of "ring", especially in advanced books by notable authors such as Artin,[23] Bourbaki,[24] Eisenbud,[25] and Lang.[3] There are also books published as late as 2022 that use the term without the requirement for a 1.[26][27][28][29]
Gardner and Wiegandt assert that, when dealing with several objects in the category of rings (as opposed to working with a fixed ring), if one requires all rings to have a 1, then some consequences include the lack of existence of infinite direct sums of rings, and that proper direct summands of rings are not subrings. They conclude that "in many, maybe most, branches of ring theory the requirement of the existence of a unity element is not sensible, and therefore unacceptable."[30] Poonen makes the counterargument that the natural notion for rings is the direct product rather than the direct sum. He further argues that rings without a multiplicative identity are not totally associative (the product of any finite sequence of ring elements, including the empty sequence, is well-defined, independent of the order of operations) and writes "the natural extension of associativity demands that rings should contain an empty product, so it is natural to require rings to have a 1".[31]
Authors who follow either convention for the use of the term "ring" may use one of the following terms to refer to objects satisfying the other convention:
• to include a requirement for a multiplicative identity: "unital ring", "unitary ring", "unit ring", "ring with unity", "ring with identity", "ring with a unit",[32] or "ring with 1".[33]
• to omit a requirement for a multiplicative identity: "rng"[34] or "pseudo-ring",[35] although the latter may be confusing because it also has other meanings.
Basic examples
See also: Associative algebra § Examples
Commutative rings
• The prototypical example is the ring of integers with the two operations of addition and multiplication.
• The rational, real and complex numbers are commutative rings of a type called fields.
• A unital associative algebra over a commutative ring R is itself a ring as well as an R-module. Some examples:
• The algebra R[X] of polynomials with coefficients in R.
• The algebra $R[[X_{1},\dots ,X_{n}]]$ of formal power series with coefficients in R.
• The set of all continuous real-valued functions defined on the real line forms a commutative $\mathbb {R} $-algebra. The operations are pointwise addition and multiplication of functions.
• Let X be a set, and let R be a ring. Then the set of all functions from X to R forms a ring, which is commutative if R is commutative.
• The ring of quadratic integers, the integral closure of $\mathbb {Z} $ in a quadratic extension of $\mathbb {Q} .$ It is a subring of the ring of all algebraic integers.
• The ring of profinite integers ${\widehat {\mathbb {Z} }},$ the (infinite) product of the rings of p-adic integers $\mathbb {Z} _{p}$ over all prime numbers p.
• The Hecke ring, the ring generated by Hecke operators.
• If S is a set, then the power set of S becomes a ring if we define addition to be the symmetric difference of sets and multiplication to be intersection. This is an example of a Boolean ring.
Noncommutative rings
• For any ring R and any natural number n, the set of all square n-by-n matrices with entries from R, forms a ring with matrix addition and matrix multiplication as operations. For n = 1, this matrix ring is isomorphic to R itself. For n > 1 (and R not the zero ring), this matrix ring is noncommutative.
• If G is an abelian group, then the endomorphisms of G form a ring, the endomorphism ring End(G) of G. The operations in this ring are addition and composition of endomorphisms. More generally, if V is a left module over a ring R, then the set of all R-linear maps forms a ring, also called the endomorphism ring and denoted by EndR(V).
• The endomorphism ring of an elliptic curve. It is a commutative ring if the elliptic curve is defined over a field of characteristic zero.
• If G is a group and R is a ring, the group ring of G over R is a free module over R having G as basis. Multiplication is defined by the rules that the elements of G commute with the elements of R and multiply together as they do in the group G.
• The ring of differential operators (depending on the context). In fact, many rings that appear in analysis are noncommutative. For example, most Banach algebras are noncommutative.
Non-rings
• The set of natural numbers $\mathbb {N} $ with the usual operations is not a ring, since $(\mathbb {N} ,+)$ is not even a group (not all the elements are invertible with respect to addition – for instance, there is no natural number which can be added to 3 to get 0 as a result). There is a natural way to enlarge it to a ring, by including negative numbers to produce the ring of integers $\mathbb {Z} .$ The natural numbers (including 0) form an algebraic structure known as a semiring (which has all of the axioms of a ring excluding that of an additive inverse).
• Let R be the set of all continuous functions on the real line that vanish outside a bounded interval that depends on the function, with addition as usual but with multiplication defined as convolution:
$(f*g)(x)=\int _{-\infty }^{\infty }f(y)g(x-y)\,dy.$
Then R is a rng, but not a ring: the Dirac delta function has the property of a multiplicative identity, but it is not a function and hence is not an element of R.
Basic concepts
Products and powers
For each nonnegative integer n, given a sequence $(a_{1},\dots ,a_{n})$ of n elements of R, one can define the product $P_{n}=\prod _{i=1}^{n}a_{i}$ recursively: let P0 = 1 and let Pm = Pm−1am for 1 ≤ m ≤ n.
As a special case, one can define nonnegative integer powers of an element a of a ring: a0 = 1 and an = an−1a for n ≥ 1. Then am+n = aman for all m, n ≥ 0.
Elements in a ring
A left zero divisor of a ring R is an element a in the ring such that there exists a nonzero element b of R such that ab = 0.[lower-alpha 3] A right zero divisor is defined similarly.
A nilpotent element is an element a such that an = 0 for some n > 0. One example of a nilpotent element is a nilpotent matrix. A nilpotent element in a nonzero ring is necessarily a zero divisor.
An idempotent $e$ is an element such that e2 = e. One example of an idempotent element is a projection in linear algebra.
A unit is an element a having a multiplicative inverse; in this case the inverse is unique, and is denoted by a–1. The set of units of a ring is a group under ring multiplication; this group is denoted by R× or R* or U(R). For example, if R is the ring of all square matrices of size n over a field, then R× consists of the set of all invertible matrices of size n, and is called the general linear group.
Subring
Main article: Subring
A subset S of R is called a subring if any one of the following equivalent conditions holds:
• the addition and multiplication of R restrict to give operations S × S → S making S a ring with the same multiplicative identity as R.
• 1 ∈ S; and for all x, y in S, the elements xy, x + y, and −x are in S.
• S can be equipped with operations making it a ring such that the inclusion map S → R is a ring homomorphism.
For example, the ring $\mathbb {Z} $ of integers is a subring of the field of real numbers and also a subring of the ring of polynomials $\mathbb {Z} [X]$ (in both cases, $\mathbb {Z} $ contains 1, which is the multiplicative identity of the larger rings). On the other hand, the subset of even integers $2\mathbb {Z} $ does not contain the identity element 1 and thus does not qualify as a subring of $\mathbb {Z} ;$ ;} one could call $2\mathbb {Z} $ a subrng, however.
An intersection of subrings is a subring. Given a subset E of R, the smallest subring of R containing E is the intersection of all subrings of R containing E, and it is called the subring generated by E.
For a ring R, the smallest subring of R is called the characteristic subring of R. It can be generated through addition of copies of 1 and −1. It is possible that n · 1 = 1 + 1 + ... + 1 (n times) can be zero. If n is the smallest positive integer such that this occurs, then n is called the characteristic of R. In some rings, n · 1 is never zero for any positive integer n, and those rings are said to have characteristic zero.
Given a ring R, let Z(R) denote the set of all elements x in R such that x commutes with every element in R: xy = yx for any y in R. Then Z(R) is a subring of R, called the center of R. More generally, given a subset X of R, let S be the set of all elements in R that commute with every element in X. Then S is a subring of R, called the centralizer (or commutant) of X. The center is the centralizer of the entire ring R. Elements or subsets of the center are said to be central in R; they (each individually) generate a subring of the center.
Ideal
Main article: Ideal (ring theory)
Let R be a ring. A left ideal of R is a nonempty subset I of R such that for any x, y in I and r in R, the elements x + y and rx are in I. If R I denotes the R-span of I, that is, the set of finite sums
$r_{1}x_{1}+\cdots +r_{n}x_{n}\quad {\textrm {such}}\;{\textrm {that}}\;r_{i}\in R\;{\textrm {and}}\;x_{i}\in I,$
then I is a left ideal if $RI\subseteq I.$ Similarly, a right ideal is a subset I such that $IR\subseteq I.$ A subset I is said to be a two-sided ideal or simply ideal if it is both a left ideal and right ideal. A one-sided or two-sided ideal is then an additive subgroup of R. If E is a subset of R, then R E is a left ideal, called the left ideal generated by E; it is the smallest left ideal containing E. Similarly, one can consider the right ideal or the two-sided ideal generated by a subset of R.
If x is in R, then Rx and xR are left ideals and right ideals, respectively; they are called the principal left ideals and right ideals generated by x. The principal ideal RxR is written as (x). For example, the set of all positive and negative multiples of 2 along with 0 form an ideal of the integers, and this ideal is generated by the integer 2. In fact, every ideal of the ring of integers is principal.
Like a group, a ring is said to be simple if it is nonzero and it has no proper nonzero two-sided ideals. A commutative simple ring is precisely a field.
Rings are often studied with special conditions set upon their ideals. For example, a ring in which there is no strictly increasing infinite chain of left ideals is called a left Noetherian ring. A ring in which there is no strictly decreasing infinite chain of left ideals is called a left Artinian ring. It is a somewhat surprising fact that a left Artinian ring is left Noetherian (the Hopkins–Levitzki theorem). The integers, however, form a Noetherian ring which is not Artinian.
For commutative rings, the ideals generalize the classical notion of divisibility and decomposition of an integer into prime numbers in algebra. A proper ideal P of R is called a prime ideal if for any elements $x,y\in R$ we have that $xy\in P$ implies either $x\in P$ or $y\in P.$ Equivalently, P is prime if for any ideals $I,J$ we have that $IJ\subseteq P$ implies either $I\subseteq P$ or $J\subseteq P.$ This latter formulation illustrates the idea of ideals as generalizations of elements.
Homomorphism
Main article: Ring homomorphism
A homomorphism from a ring (R, +, ⋅) to a ring (S, ‡, ∗) is a function f from R to S that preserves the ring operations; namely, such that, for all a, b in R the following identities hold:
${\begin{aligned}&f(a+b)=f(a)\ddagger f(b)\\&f(a\cdot b)=f(a)*f(b)\\&f(1_{R})=1_{S}\end{aligned}}$
If one is working with rngs, then the third condition is dropped.
A ring homomorphism f is said to be an isomorphism if there exists an inverse homomorphism to f (that is, a ring homomorphism that is an inverse function). Any bijective ring homomorphism is a ring isomorphism. Two rings R, S are said to be isomorphic if there is an isomorphism between them and in that case one writes $R\simeq S.$ A ring homomorphism between the same ring is called an endomorphism, and an isomorphism between the same ring an automorphism.
Examples:
• The function that maps each integer x to its remainder modulo 4 (a number in { 0, 1, 2, 3 }) is a homomorphism from the ring $\mathbb {Z} $ to the quotient ring $\mathbb {Z} /4\mathbb {Z} $ ("quotient ring" is defined below).
• If u is a unit element in a ring R, then $R\to R,x\mapsto uxu^{-1}$ is a ring homomorphism, called an inner automorphism of R.
• Let R be a commutative ring of prime characteristic p. Then $x\to x^{p}$ is a ring endomorphism of R called the Frobenius homomorphism.
• The Galois group of a field extension L/K is the set of all automorphisms of L whose restrictions to n are the identity.
• For any ring R, there are a unique ring homomorphism $\mathbb {Z} \mapsto R$ and a unique ring homomorphism R → 0.
• An epimorphism (that is, right-cancelable morphism) of rings need not be surjective. For example, the unique map $\mathbb {Z} \to \mathbb {Q} $ is an epimorphism.
• An algebra homomorphism from a k-algebra to the endomorphism algebra of a vector space over k is called a representation of the algebra.
Given a ring homomorphism f : R → S, the set of all elements mapped to 0 by f is called the kernel of f. The kernel is a two-sided ideal of R. The image of f, on the other hand, is not always an ideal, but it is always a subring of S.
To give a ring homomorphism from a commutative ring R to a ring A with image contained in the center of A is the same as to give a structure of an algebra over R to A (which in particular gives a structure of an A-module).
Quotient ring
Main article: Quotient ring
The notion of quotient ring is analogous to the notion of a quotient group. Given a ring (R, +, ⋅ ) and a two-sided ideal I of (R, +, ⋅ ), view I as subgroup of (R, +); then the quotient ring R/I is the set of cosets of I together with the operations
${\begin{aligned}&(a+I)+(b+I)=(a+b)+I,\\&(a+I)(b+I)=(ab)+I.\end{aligned}}$
for all a, b in R. The ring R/I is also called a factor ring.
As with a quotient group, there is a canonical homomorphism p : R → R/I, given by $x\mapsto x+I.$ It is surjective and satisfies the following universal property:
• If f : R → S is a ring homomorphism such that f(I) = 0, then there is a unique homomorphism ${\overline {f}}:R/I\to S$ such that $f={\overline {f}}\circ p.$
For any ring homomorphism f : R → S, invoking the universal property with I = ker f produces a homomorphism ${\overline {f}}:R/\ker f\to S$ that gives an isomorphism from R/ker f to the image of f.
Module
Main article: Module (mathematics)
The concept of a module over a ring generalizes the concept of a vector space (over a field) by generalizing from multiplication of vectors with elements of a field (scalar multiplication) to multiplication with elements of a ring. More precisely, given a ring R, an R-module M is an abelian group equipped with an operation R × M → M (associating an element of M to every pair of an element of R and an element of M) that satisfies certain axioms. This operation is commonly denoted by juxtaposition and called multiplication. The axioms of modules are the following: for all a, b in R and all x, y in M,
M is an abelian group under addition.
${\begin{aligned}&a(x+y)=ax+ay\\&(a+b)x=ax+bx\\&1x=x\\&(ab)x=a(bx)\end{aligned}}$
When the ring is noncommutative these axioms define left modules; right modules are defined similarly by writing xa instead of ax. This is not only a change of notation, as the last axiom of right modules (that is x(ab) = (xa)b) becomes (ab)x = b(ax), if left multiplication (by ring elements) is used for a right module.
Basic examples of modules are ideals, including the ring itself.
Although similarly defined, the theory of modules is much more complicated than that of vector space, mainly, because, unlike vector spaces, modules are not characterized (up to an isomorphism) by a single invariant (the dimension of a vector space). In particular, not all modules have a basis.
The axioms of modules imply that (−1)x = −x, where the first minus denotes the additive inverse in the ring and the second minus the additive inverse in the module. Using this and denoting repeated addition by a multiplication by a positive integer allows identifying abelian groups with modules over the ring of integers.
Any ring homomorphism induces a structure of a module: if f : R → S is a ring homomorphism, then S is a left module over R by the multiplication: rs = f(r)s. If R is commutative or if f(R) is contained in the center of S, the ring S is called a R-algebra. In particular, every ring is an algebra over the integers.
Constructions
Direct product
Main article: Direct product of rings
Let R and S be rings. Then the product R × S can be equipped with the following natural ring structure:
${\begin{aligned}&(r_{1},s_{1})+(r_{2},s_{2})=(r_{1}+r_{2},s_{1}+s_{2})\\&(r_{1},s_{1})\cdot (r_{2},s_{2})=(r_{1}\cdot r_{2},s_{1}\cdot s_{2})\end{aligned}}$
for all r1, r2 in R and s1, s2 in S. The ring R × S with the above operations of addition and multiplication and the multiplicative identity (1, 1) is called the direct product of R with S. The same construction also works for an arbitrary family of rings: if Ri are rings indexed by a set I, then $ \prod _{i\in I}R_{i}$ is a ring with componentwise addition and multiplication.
Let R be a commutative ring and ${\mathfrak {a}}_{1},\cdots ,{\mathfrak {a}}_{n}$ be ideals such that ${\mathfrak {a}}_{i}+{\mathfrak {a}}_{j}=(1)$ whenever i ≠ j. Then the Chinese remainder theorem says there is a canonical ring isomorphism:
$R/ \bigcap _{i=1}^{n}{{\mathfrak {a}}_{i}}}\simeq \prod _{i=1}^{n}{R/{\mathfrak {a}}_{i}},\qquad x{\bmod \bigcap _{i=1}^{n}{\mathfrak {a}}_{i}}}\mapsto (x{\bmod {\mathfrak {a}}}_{1},\ldots ,x{\bmod {\mathfrak {a}}}_{n}).$
A "finite" direct product may also be viewed as a direct sum of ideals.[36] Namely, let $R_{i},1\leq i\leq n$ be rings, $ R_{i}\to R=\prod R_{i}$ the inclusions with the images ${\mathfrak {a}}_{i}$ (in particular ${\mathfrak {a}}_{i}$ are rings though not subrings). Then ${\mathfrak {a}}_{i}$ are ideals of R and
$R={\mathfrak {a}}_{1}\oplus \cdots \oplus {\mathfrak {a}}_{n},\quad {\mathfrak {a}}_{i}{\mathfrak {a}}_{j}=0,i\neq j,\quad {\mathfrak {a}}_{i}^{2}\subseteq {\mathfrak {a}}_{i}$
as a direct sum of abelian groups (because for abelian groups finite products are the same as direct sums). Clearly the direct sum of such ideals also defines a product of rings that is isomorphic to R. Equivalently, the above can be done through central idempotents. Assume that R has the above decomposition. Then we can write
$1=e_{1}+\cdots +e_{n},\quad e_{i}\in {\mathfrak {a}}_{i}.$
By the conditions on ${\mathfrak {a}}_{i},$ one has that ei are central idempotents and eiej = 0, i ≠ j (orthogonal). Again, one can reverse the construction. Namely, if one is given a partition of 1 in orthogonal central idempotents, then let ${\mathfrak {a}}_{i}=Re_{i},$ which are two-sided ideals. If each ei is not a sum of orthogonal central idempotents,[lower-alpha 4] then their direct sum is isomorphic to R.
An important application of an infinite direct product is the construction of a projective limit of rings (see below). Another application is a restricted product of a family of rings (cf. adele ring).
Polynomial ring
Main article: Polynomial ring
Given a symbol t (called a variable) and a commutative ring R, the set of polynomials
$R[t]=\left\{a_{n}t^{n}+a_{n-1}t^{n-1}+\dots +a_{1}t+a_{0}\mid n\geq 0,a_{j}\in R\right\}$
forms a commutative ring with the usual addition and multiplication, containing R as a subring. It is called the polynomial ring over R. More generally, the set $R\left[t_{1},\ldots ,t_{n}\right]$ of all polynomials in variables $t_{1},\ldots ,t_{n}$ forms a commutative ring, containing $R\left[t_{i}\right]$ as subrings.
If R is an integral domain, then R[t] is also an integral domain; its field of fractions is the field of rational functions. If R is a Noetherian ring, then R[t] is a Noetherian ring. If R is a unique factorization domain, then R[t] is a unique factorization domain. Finally, R is a field if and only if R[t] is a principal ideal domain.
Let $R\subseteq S$ be commutative rings. Given an element x of S, one can consider the ring homomorphism
$R[t]\to S,\quad f\mapsto f(x)$
(that is, the substitution). If S = R[t] and x = t, then f(t) = f. Because of this, the polynomial f is often also denoted by f(t). The image of the map $f\mapsto f(x)$ is denoted by R[x]; it is the same thing as the subring of S generated by R and x.
Example: $k\left[t^{2},t^{3}\right]$ denotes the image of the homomorphism
$k[x,y]\to k[t],\,f\mapsto f\left(t^{2},t^{3}\right).$
In other words, it is the subalgebra of k[t] generated by t2 and t3.
Example: let f be a polynomial in one variable, that is, an element in a polynomial ring R. Then f(x + h) is an element in R[h] and f(x + h) – f(x) is divisible by h in that ring. The result of substituting zero to h in (f(x + h) – f(x)) / h is f' (x), the derivative of f at x.
The substitution is a special case of the universal property of a polynomial ring. The property states: given a ring homomorphism $\phi :R\to S$ and an element x in S there exists a unique ring homomorphism ${\overline {\phi }}:R[t]\to S$ such that ${\overline {\phi }}(t)=x$ and ${\overline {\phi }}$ restricts to ϕ.[37] For example, choosing a basis, a symmetric algebra satisfies the universal property and so is a polynomial ring.
To give an example, let S be the ring of all functions from R to itself; the addition and the multiplication are those of functions. Let x be the identity function. Each r in R defines a constant function, giving rise to the homomorphism R → S. The universal property says that this map extends uniquely to
$R[t]\to S,\quad f\mapsto {\overline {f}}$
(t maps to x) where ${\overline {f}}$ is the polynomial function defined by f. The resulting map is injective if and only if R is infinite.
Given a non-constant monic polynomial f in R[t], there exists a ring S containing R such that f is a product of linear factors in S[t].[38]
Let k be an algebraically closed field. The Hilbert's Nullstellensatz (theorem of zeros) states that there is a natural one-to-one correspondence between the set of all prime ideals in $k\left[t_{1},\ldots ,t_{n}\right]$ and the set of closed subvarieties of kn. In particular, many local problems in algebraic geometry may be attacked through the study of the generators of an ideal in a polynomial ring. (cf. Gröbner basis.)
There are some other related constructions. A formal power series ring $R[\![t]\!]$ consists of formal power series
$\sum _{0}^{\infty }a_{i}t^{i},\quad a_{i}\in R$
together with multiplication and addition that mimic those for convergent series. It contains R[t] as a subring. A formal power series ring does not have the universal property of a polynomial ring; a series may not converge after a substitution. The important advantage of a formal power series ring over a polynomial ring is that it is local (in fact, complete).
Matrix ring and endomorphism ring
Main articles: Matrix ring and Endomorphism ring
Let R be a ring (not necessarily commutative). The set of all square matrices of size n with entries in R forms a ring with the entry-wise addition and the usual matrix multiplication. It is called the matrix ring and is denoted by Mn(R). Given a right R-module U, the set of all R-linear maps from U to itself forms a ring with addition that is of function and multiplication that is of composition of functions; it is called the endomorphism ring of U and is denoted by EndR(U).
As in linear algebra, a matrix ring may be canonically interpreted as an endomorphism ring: $\operatorname {End} _{R}(R^{n})\simeq \operatorname {M} _{n}(R).$ This is a special case of the following fact: If $f:\oplus _{1}^{n}U\to \oplus _{1}^{n}U$ is an R-linear map, then f may be written as a matrix with entries fij in S = EndR(U), resulting in the ring isomorphism:
$\operatorname {End} _{R}(\oplus _{1}^{n}U)\to \operatorname {M} _{n}(S),\quad f\mapsto (f_{ij}).$
Any ring homomorphism R → S induces Mn(R) → Mn(S).[39]
Schur's lemma says that if U is a simple right R-module, then EndR(U) is a division ring.[40] If $U=\bigoplus _{i=1}^{r}U_{i}^{\oplus m_{i}}$ is a direct sum of mi-copies of simple R-modules $U_{i},$ then
$\operatorname {End} _{R}(U)\simeq \prod _{i=1}^{r}\operatorname {M} _{m_{i}}(\operatorname {End} _{R}(U_{i})).$
The Artin–Wedderburn theorem states any semisimple ring (cf. below) is of this form.
A ring R and the matrix ring Mn(R) over it are Morita equivalent: the category of right modules of R is equivalent to the category of right modules over Mn(R).[39] In particular, two-sided ideals in R correspond in one-to-one to two-sided ideals in Mn(R).
Limits and colimits of rings
Let Ri be a sequence of rings such that Ri is a subring of Ri + 1 for all i. Then the union (or filtered colimit) of Ri is the ring $\varinjlim R_{i}$ defined as follows: it is the disjoint union of all Ri's modulo the equivalence relation x ~ y if and only if x = y in Ri for sufficiently large i.
Examples of colimits:
• A polynomial ring in infinitely many variables: $R[t_{1},t_{2},\cdots ]=\varinjlim R[t_{1},t_{2},\cdots ,t_{m}].$
• The algebraic closure of finite fields of the same characteristic ${\overline {\mathbf {F} }}_{p}=\varinjlim \mathbf {F} _{p^{m}}.$
• The field of formal Laurent series over a field k: $k(\!(t)\!)=\varinjlim t^{-m}k[\![t]\!]$ (it is the field of fractions of the formal power series ring $k[\![t]\!].$)
• The function field of an algebraic variety over a field k is $\varinjlim k[U]$ where the limit runs over all the coordinate rings k[U] of nonempty open subsets U (more succinctly it is the stalk of the structure sheaf at the generic point.)
Any commutative ring is the colimit of finitely generated subrings.
A projective limit (or a filtered limit) of rings is defined as follows. Suppose we're given a family of rings Ri, i running over positive integers, say, and ring homomorphisms Rj → Ri, j ≥ i such that Ri → Ri are all the identities and Rk → Rj → Ri is Rk → Ri whenever k ≥ j ≥ i. Then $\varprojlim R_{i}$ is the subring of $\textstyle \prod R_{i}$ consisting of (xn) such that xj maps to xi under Rj → Ri, j ≥ i.
For an example of a projective limit, see § Completion.
Localization
The localization generalizes the construction of the field of fractions of an integral domain to an arbitrary ring and modules. Given a (not necessarily commutative) ring R and a subset S of R, there exists a ring $R[S^{-1}]$ together with the ring homomorphism $R\to R\left[S^{-1}\right]$ that "inverts" S; that is, the homomorphism maps elements in S to unit elements in $R\left[S^{-1}\right],$ and, moreover, any ring homomorphism from R that "inverts" S uniquely factors through $R\left[S^{-1}\right].$[41] The ring $R\left[S^{-1}\right]$ is called the localization of R with respect to S. For example, if R is a commutative ring and f an element in R, then the localization $R\left[f^{-1}\right]$ consists of elements of the form $r/f^{n},\,r\in R,\,n\geq 0$ (to be precise, $R\left[f^{-1}\right]=R[t]/(tf-1).$)[42]
The localization is frequently applied to a commutative ring R with respect to the complement of a prime ideal (or a union of prime ideals) in R. In that case $S=R-{\mathfrak {p}},$ one often writes $R_{\mathfrak {p}}$ for $R\left[S^{-1}\right].$ $R_{\mathfrak {p}}$ is then a local ring with the maximal ideal ${\mathfrak {p}}R_{\mathfrak {p}}.$ This is the reason for the terminology "localization". The field of fractions of an integral domain R is the localization of R at the prime ideal zero. If ${\mathfrak {p}}$ is a prime ideal of a commutative ring R, then the field of fractions of $R/{\mathfrak {p}}$ is the same as the residue field of the local ring $R_{\mathfrak {p}}$ and is denoted by $k({\mathfrak {p}}).$
If M is a left R-module, then the localization of M with respect to S is given by a change of rings $M\left[S^{-1}\right]=R\left[S^{-1}\right]\otimes _{R}M.$
The most important properties of localization are the following: when R is a commutative ring and S a multiplicatively closed subset
• ${\mathfrak {p}}\mapsto {\mathfrak {p}}\left[S^{-1}\right]$ is a bijection between the set of all prime ideals in R disjoint from S and the set of all prime ideals in $R\left[S^{-1}\right].$[43]
• $R\left[S^{-1}\right]=\varinjlim R\left[f^{-1}\right],$ f running over elements in S with partial ordering given by divisibility.[44]
• The localization is exact:
$0\to M'\left[S^{-1}\right]\to M\left[S^{-1}\right]\to M''\left[S^{-1}\right]\to 0$
is exact over $R\left[S^{-1}\right]$ whenever $0\to M'\to M\to M''\to 0$ is exact over R.
• Conversely, if $0\to M'_{\mathfrak {m}}\to M_{\mathfrak {m}}\to M''_{\mathfrak {m}}\to 0$ is exact for any maximal ideal ${\mathfrak {m}},$ then $0\to M'\to M\to M''\to 0$ is exact.
• A remark: localization is no help in proving a global existence. One instance of this is that if two modules are isomorphic at all prime ideals, it does not follow that they are isomorphic. (One way to explain this is that the localization allows one to view a module as a sheaf over prime ideals and a sheaf is inherently a local notion.)
In category theory, a localization of a category amounts to making some morphisms isomorphisms. An element in a commutative ring R may be thought of as an endomorphism of any R-module. Thus, categorically, a localization of R with respect to a subset S of R is a functor from the category of R-modules to itself that sends elements of S viewed as endomorphisms to automorphisms and is universal with respect to this property. (Of course, R then maps to $R\left[S^{-1}\right]$ and R-modules map to $R\left[S^{-1}\right]$-modules.)
Completion
Let R be a commutative ring, and let I be an ideal of R. The completion of R at I is the projective limit ${\hat {R}}=\varprojlim R/I^{n};$ it is a commutative ring. The canonical homomorphisms from R to the quotients $R/I^{n}$ induce a homomorphism $R\to {\hat {R}}.$ The latter homomorphism is injective if R is a Noetherian integral domain and I is a proper ideal, or if R is a Noetherian local ring with maximal ideal I, by Krull's intersection theorem.[45] The construction is especially useful when I is a maximal ideal.
The basic example is the completion of $\mathbb {Z} $ at the principal ideal (p) generated by a prime number p; it is called the ring of p-adic integers and is denoted $\mathbb {Z} _{p}.$ The completion can in this case be constructed also from the p-adic absolute value on $\mathbb {Q} .$ The p-adic absolute value on $\mathbb {Q} $ is a map $x\mapsto |x|$ from $\mathbb {Q} $ to $\mathbb {R} $ given by $|n|_{p}=p^{-v_{p}(n)}$ where $v_{p}(n)$ denotes the exponent of p in the prime factorization of a nonzero integer n into prime numbers (we also put $|0|_{p}=0$ and $|m/n|_{p}=|m|_{p}/|n|_{p}$). It defines a distance function on $\mathbb {Q} $ and the completion of $\mathbb {Q} $ as a metric space is denoted by $\mathbb {Q} _{p}.$ It is again a field since the field operations extend to the completion. The subring of $\mathbb {Q} _{p}$ consisting of elements x with $|x|_{p}\leq 1$ is isomorphic to $\mathbb {Z} _{p}.$
Similarly, the formal power series ring R[{[t]}] is the completion of R[t] at (t) (see also Hensel's lemma)
A complete ring has much simpler structure than a commutative ring. This owns to the Cohen structure theorem, which says, roughly, that a complete local ring tends to look like a formal power series ring or a quotient of it. On the other hand, the interaction between the integral closure and completion has been among the most important aspects that distinguish modern commutative ring theory from the classical one developed by the likes of Noether. Pathological examples found by Nagata led to the reexamination of the roles of Noetherian rings and motivated, among other things, the definition of excellent ring.
Rings with generators and relations
The most general way to construct a ring is by specifying generators and relations. Let F be a free ring (that is, free algebra over the integers) with the set X of symbols, that is, F consists of polynomials with integral coefficients in noncommuting variables that are elements of X. A free ring satisfies the universal property: any function from the set X to a ring R factors through F so that F → R is the unique ring homomorphism. Just as in the group case, every ring can be represented as a quotient of a free ring.[46]
Now, we can impose relations among symbols in X by taking a quotient. Explicitly, if E is a subset of F, then the quotient ring of F by the ideal generated by E is called the ring with generators X and relations E. If we used a ring, say, A as a base ring instead of $\mathbb {Z} ,$ then the resulting ring will be over A. For example, if $E=\{xy-yx\mid x,y\in X\},$ then the resulting ring will be the usual polynomial ring with coefficients in A in variables that are elements of X (It is also the same thing as the symmetric algebra over A with symbols X.)
In the category-theoretic terms, the formation $S\mapsto {\text{the free ring generated by the set }}S$ is the left adjoint functor of the forgetful functor from the category of rings to Set (and it is often called the free ring functor.)
Let A, B be algebras over a commutative ring R. Then the tensor product of R-modules $A\otimes _{R}B$ is an R-algebra with multiplication characterized by $(x\otimes u)(y\otimes v)=xy\otimes uv.$
See also: Tensor product of algebras and Change of rings
Special kinds of rings
Domains
A nonzero ring with no nonzero zero-divisors is called a domain. A commutative domain is called an integral domain. The most important integral domains are principal ideal domains, PIDs for short, and fields. A principal ideal domain is an integral domain in which every ideal is principal. An important class of integral domains that contain a PID is a unique factorization domain (UFD), an integral domain in which every nonunit element is a product of prime elements (an element is prime if it generates a prime ideal.) The fundamental question in algebraic number theory is on the extent to which the ring of (generalized) integers in a number field, where an "ideal" admits prime factorization, fails to be a PID.
Among theorems concerning a PID, the most important one is the structure theorem for finitely generated modules over a principal ideal domain. The theorem may be illustrated by the following application to linear algebra.[47] Let V be a finite-dimensional vector space over a field k and f : V → V a linear map with minimal polynomial q. Then, since k[t] is a unique factorization domain, q factors into powers of distinct irreducible polynomials (that is, prime elements):
$q=p_{1}^{e_{1}}\ldots p_{s}^{e_{s}}.$
Letting $t\cdot v=f(v),$ we make V a k[t]-module. The structure theorem then says V is a direct sum of cyclic modules, each of which is isomorphic to the module of the form $k[t]/\left(p_{i}^{k_{j}}\right).$ Now, if $p_{i}(t)=t-\lambda _{i},$ then such a cyclic module (for pi) has a basis in which the restriction of f is represented by a Jordan matrix. Thus, if, say, k is algebraically closed, then all pi's are of the form t – λi and the above decomposition corresponds to the Jordan canonical form of f.
In algebraic geometry, UFDs arise because of smoothness. More precisely, a point in a variety (over a perfect field) is smooth if the local ring at the point is a regular local ring. A regular local ring is a UFD.[48]
The following is a chain of class inclusions that describes the relationship between rings, domains and fields:
rngs ⊃ rings ⊃ commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed fields
Division ring
A division ring is a ring such that every non-zero element is a unit. A commutative division ring is a field. A prominent example of a division ring that is not a field is the ring of quaternions. Any centralizer in a division ring is also a division ring. In particular, the center of a division ring is a field. It turned out that every finite domain (in particular finite division ring) is a field; in particular commutative (the Wedderburn's little theorem).
Every module over a division ring is a free module (has a basis); consequently, much of linear algebra can be carried out over a division ring instead of a field.
The study of conjugacy classes figures prominently in the classical theory of division rings; see, for example, the Cartan–Brauer–Hua theorem.
A cyclic algebra, introduced by L. E. Dickson, is a generalization of a quaternion algebra.
Semisimple rings
Main article: Semisimple module
A semisimple module is a direct sum of simple modules. A semisimple ring is a ring that is semisimple as a left module (or right module) over itself.
Examples
• A division ring is semisimple (and simple).
• For any division ring D and positive integer n, the matrix ring Mn(D) is semisimple (and simple).
• For a field k and finite group G, the group ring kG is semisimple if and only if the characteristic of k does not divide the order of G (Maschke's theorem).
• Clifford algebras are semisimple.
The Weyl algebra over a field is a simple ring, but it is not semisimple. The same holds for a ring of differential operators in many variables.
Properties
Any module over a semisimple ring is semisimple. (Proof: A free module over a semisimple ring is semisimple and any module is a quotient of a free module.)
For a ring R, the following are equivalent:
• R is semisimple.
• R is artinian and semiprimitive.
• R is a finite direct product $ \prod _{i=1}^{r}\operatorname {M} _{n_{i}}(D_{i})$ where each ni is a positive integer, and each Di is a division ring (Artin–Wedderburn theorem).
Semisimplicity is closely related to separability. A unital associative algebra A over a field k is said to be separable if the base extension $A\otimes _{k}F$ is semisimple for every field extension F / k. If A happens to be a field, then this is equivalent to the usual definition in field theory (cf. separable extension.)
Central simple algebra and Brauer group
Main article: Central simple algebra
For a field k, a k-algebra is central if its center is k and is simple if it is a simple ring. Since the center of a simple k-algebra is a field, any simple k-algebra is a central simple algebra over its center. In this section, a central simple algebra is assumed to have finite dimension. Also, we mostly fix the base field; thus, an algebra refers to a k-algebra. The matrix ring of size n over a ring R will be denoted by Rn.
The Skolem–Noether theorem states any automorphism of a central simple algebra is inner.
Two central simple algebras A and B are said to be similar if there are integers n and m such that $A\otimes _{k}k_{n}\approx B\otimes _{k}k_{m}.$[49] Since $k_{n}\otimes _{k}k_{m}\simeq k_{nm},$ the similarity is an equivalence relation. The similarity classes [A] with the multiplication $[A][B]=\left[A\otimes _{k}B\right]$ form an abelian group called the Brauer group of k and is denoted by Br(k). By the Artin–Wedderburn theorem, a central simple algebra is the matrix ring of a division ring; thus, each similarity class is represented by a unique division ring.
For example, Br(k) is trivial if k is a finite field or an algebraically closed field (more generally quasi-algebraically closed field; cf. Tsen's theorem). $\operatorname {Br} (\mathbb {R} )$ has order 2 (a special case of the theorem of Frobenius). Finally, if k is a nonarchimedean local field (for example, $\mathbb {Q} _{p}$), then $\operatorname {Br} (k)=\mathbb {Q} /\mathbb {Z} $ through the invariant map.
Now, if F is a field extension of k, then the base extension $-\otimes _{k}F$ induces Br(k) → Br(F). Its kernel is denoted by Br(F/k). It consists of [A] such that $A\otimes _{k}F$ is a matrix ring over F (that is, A is split by F.) If the extension is finite and Galois, then Br(F/k) is canonically isomorphic to $H^{2}\left(\operatorname {Gal} (F/k),k^{*}\right).$[50]
Azumaya algebras generalize the notion of central simple algebras to a commutative local ring.
Valuation ring
Main article: Valuation ring
If K is a field, a valuation v is a group homomorphism from the multiplicative group K∗ to a totally ordered abelian group G such that, for any f, g in K with f + g nonzero, v(f + g) ≥ min{v(f), v(g)}. The valuation ring of v is the subring of K consisting of zero and all nonzero f such that v(f) ≥ 0.
Examples:
• The field of formal Laurent series $k(\!(t)\!)$ over a field k comes with the valuation v such that v(f) is the least degree of a nonzero term in f; the valuation ring of v is the formal power series ring $k[\![t]\!].$
• More generally, given a field k and a totally ordered abelian group G, let $k(\!(G)\!)$ be the set of all functions from G to k whose supports (the sets of points at which the functions are nonzero) are well ordered. It is a field with the multiplication given by convolution:
$(f*g)(t)=\sum _{s\in G}f(s)g(t-s).$
It also comes with the valuation v such that v(f) is the least element in the support of f. The subring consisting of elements with finite support is called the group ring of G (which makes sense even if G is not commutative). If G is the ring of integers, then we recover the previous example (by identifying f with the series whose n-th coefficient is f(n).)
Rings with extra structure
A ring may be viewed as an abelian group (by using the addition operation), with extra structure: namely, ring multiplication. In the same way, there are other mathematical objects which may be considered as rings with extra structure. For example:
• An associative algebra is a ring that is also a vector space over a field n such that the scalar multiplication is compatible with the ring multiplication. For instance, the set of n-by-n matrices over the real field $\mathbb {R} $ has dimension n2 as a real vector space.
• A ring R is a topological ring if its set of elements R is given a topology which makes the addition map ( $+:R\times R\to R\,$) and the multiplication map $\cdot :R\times R\to R$ to be both continuous as maps between topological spaces (where X × X inherits the product topology or any other product in the category). For example, n-by-n matrices over the real numbers could be given either the Euclidean topology, or the Zariski topology, and in either case one would obtain a topological ring.
• A λ-ring is a commutative ring R together with operations λn: R → R that are like n-th exterior powers:
$\lambda ^{n}(x+y)=\sum _{0}^{n}\lambda ^{i}(x)\lambda ^{n-i}(y).$
For example, $\mathbb {Z} $ is a λ-ring with $\lambda ^{n}(x)={\binom {x}{n}},$ the binomial coefficients. The notion plays a central rule in the algebraic approach to the Riemann–Roch theorem.
• A totally ordered ring is a ring with a total ordering that is compatible with ring operations.
Some examples of the ubiquity of rings
Many different kinds of mathematical objects can be fruitfully analyzed in terms of some associated ring.
Cohomology ring of a topological space
To any topological space X one can associate its integral cohomology ring
$H^{*}(X,\mathbb {Z} )=\bigoplus _{i=0}^{\infty }H^{i}(X,\mathbb {Z} ),$
a graded ring. There are also homology groups $H_{i}(X,\mathbb {Z} )$ of a space, and indeed these were defined first, as a useful tool for distinguishing between certain pairs of topological spaces, like the spheres and tori, for which the methods of point-set topology are not well-suited. Cohomology groups were later defined in terms of homology groups in a way which is roughly analogous to the dual of a vector space. To know each individual integral homology group is essentially the same as knowing each individual integral cohomology group, because of the universal coefficient theorem. However, the advantage of the cohomology groups is that there is a natural product, which is analogous to the observation that one can multiply pointwise a k-multilinear form and an l-multilinear form to get a (k + l)-multilinear form.
The ring structure in cohomology provides the foundation for characteristic classes of fiber bundles, intersection theory on manifolds and algebraic varieties, Schubert calculus and much more.
Burnside ring of a group
To any group is associated its Burnside ring which uses a ring to describe the various ways the group can act on a finite set. The Burnside ring's additive group is the free abelian group whose basis is the set of transitive actions of the group and whose addition is the disjoint union of the action. Expressing an action in terms of the basis is decomposing an action into its transitive constituents. The multiplication is easily expressed in terms of the representation ring: the multiplication in the Burnside ring is formed by writing the tensor product of two permutation modules as a permutation module. The ring structure allows a formal way of subtracting one action from another. Since the Burnside ring is contained as a finite index subring of the representation ring, one can pass easily from one to the other by extending the coefficients from integers to the rational numbers.
Representation ring of a group ring
To any group ring or Hopf algebra is associated its representation ring or "Green ring". The representation ring's additive group is the free abelian group whose basis are the indecomposable modules and whose addition corresponds to the direct sum. Expressing a module in terms of the basis is finding an indecomposable decomposition of the module. The multiplication is the tensor product. When the algebra is semisimple, the representation ring is just the character ring from character theory, which is more or less the Grothendieck group given a ring structure.
Function field of an irreducible algebraic variety
To any irreducible algebraic variety is associated its function field. The points of an algebraic variety correspond to valuation rings contained in the function field and containing the coordinate ring. The study of algebraic geometry makes heavy use of commutative algebra to study geometric concepts in terms of ring-theoretic properties. Birational geometry studies maps between the subrings of the function field.
Face ring of a simplicial complex
Every simplicial complex has an associated face ring, also called its Stanley–Reisner ring. This ring reflects many of the combinatorial properties of the simplicial complex, so it is of particular interest in algebraic combinatorics. In particular, the algebraic geometry of the Stanley–Reisner ring was used to characterize the numbers of faces in each dimension of simplicial polytopes.
Category-theoretic description
Every ring can be thought of as a monoid in Ab, the category of abelian groups (thought of as a monoidal category under the tensor product of $\mathbb {Z} $-modules). The monoid action of a ring R on an abelian group is simply an R-module. Essentially, an R-module is a generalization of the notion of a vector space – where rather than a vector space over a field, one has a "vector space over a ring".
Let (A, +) be an abelian group and let End(A) be its endomorphism ring (see above). Note that, essentially, End(A) is the set of all morphisms of A, where if f is in End(A), and g is in End(A), the following rules may be used to compute f + g and f ⋅ g:
${\begin{aligned}&(f+g)(x)=f(x)+g(x)\\&(f\cdot g)(x)=f(g(x)),\end{aligned}}$
where + as in f(x) + g(x) is addition in A, and function composition is denoted from right to left. Therefore, associated to any abelian group, is a ring. Conversely, given any ring, (R, +, ⋅ ), (R, +) is an abelian group. Furthermore, for every r in R, right (or left) multiplication by r gives rise to a morphism of (R, +), by right (or left) distributivity. Let A = (R, +). Consider those endomorphisms of A, that "factor through" right (or left) multiplication of R. In other words, let EndR(A) be the set of all morphisms m of A, having the property that m(r ⋅ x) = r ⋅ m(x). It was seen that every r in R gives rise to a morphism of A: right multiplication by r. It is in fact true that this association of any element of R, to a morphism of A, as a function from R to EndR(A), is an isomorphism of rings. In this sense, therefore, any ring can be viewed as the endomorphism ring of some abelian X-group (by X-group, it is meant a group with X being its set of operators).[51] In essence, the most general form of a ring, is the endomorphism group of some abelian X-group.
Any ring can be seen as a preadditive category with a single object. It is therefore natural to consider arbitrary preadditive categories to be generalizations of rings. And indeed, many definitions and theorems originally given for rings can be translated to this more general context. Additive functors between preadditive categories generalize the concept of ring homomorphism, and ideals in additive categories can be defined as sets of morphisms closed under addition and under composition with arbitrary morphisms.
Generalization
Algebraists have defined structures more general than rings by weakening or dropping some of ring axioms.
Rng
A rng is the same as a ring, except that the existence of a multiplicative identity is not assumed.[52]
Nonassociative ring
A nonassociative ring is an algebraic structure that satisfies all of the ring axioms except the associative property and the existence of a multiplicative identity. A notable example is a Lie algebra. There exists some structure theory for such algebras that generalizes the analogous results for Lie algebras and associative algebras.
Semiring
A semiring (sometimes rig) is obtained by weakening the assumption that (R, +) is an abelian group to the assumption that (R, +) is a commutative monoid, and adding the axiom that 0 ⋅ a = a ⋅ 0 = 0 for all a in R (since it no longer follows from the other axioms).
Examples:
• the non-negative integers $\{0,1,2,\ldots \}$ with ordinary addition and multiplication;
• the tropical semiring.
Other ring-like objects
Ring object in a category
Let C be a category with finite products. Let pt denote a terminal object of C (an empty product). A ring object in C is an object R equipped with morphisms $R\times R\;{\stackrel {a}{\to }}\,R$ (addition), $R\times R\;{\stackrel {m}{\to }}\,R$ (multiplication), $\operatorname {pt} {\stackrel {0}{\to }}\,R$ (additive identity), $R\;{\stackrel {i}{\to }}\,R$ (additive inverse), and $\operatorname {pt} {\stackrel {1}{\to }}\,R$ (multiplicative identity) satisfying the usual ring axioms. Equivalently, a ring object is an object R equipped with a factorization of its functor of points $h_{R}=\operatorname {Hom} (-,R):C^{\operatorname {op} }\to \mathbf {Sets} $ through the category of rings: $C^{\operatorname {op} }\to \mathbf {Rings} {\stackrel {\textrm {forgetful}}{\longrightarrow }}\mathbf {Sets} .$
Ring scheme
In algebraic geometry, a ring scheme over a base scheme S is a ring object in the category of S-schemes. One example is the ring scheme Wn over $\operatorname {Spec} \mathbb {Z} $, which for any commutative ring A returns the ring Wn(A) of p-isotypic Witt vectors of length n over A.[53]
Ring spectrum
In algebraic topology, a ring spectrum is a spectrum X together with a multiplication $\mu :X\wedge X\to X$ and a unit map S → X from the sphere spectrum S, such that the ring axiom diagrams commute up to homotopy. In practice, it is common to define a ring spectrum as a monoid object in a good category of spectra such as the category of symmetric spectra.
See also
Wikibooks has a book on the topic of: Abstract Algebra/Rings
• Algebra over a commutative ring
• Categorical ring
• Category of rings
• Glossary of ring theory
• Nonassociative ring
• Ring of sets
• Semiring
• Spectrum of a ring
• Simplicial commutative ring
Special types of rings:
• Boolean ring
• Dedekind ring
• Differential ring
• Exponential ring
• Finite ring
• Lie ring
• Local ring
• Noetherian and artinian rings
• Ordered ring
• Poisson ring
• Reduced ring
• Regular ring
• Ring of periods
• SBI ring
• Valuation ring and discrete valuation ring
Notes
1. This means that each operation is defined and produces a unique result in R for each ordered pair of elements of R.
2. The existence of 1 is not assumed by some authors; here, the term rng is used if existence of a multiplicative identity is not assumed. See next subsection.
3. Some other authors such as Lang further require a zero divisor to be nonzero.
4. Such a central idempotent is called centrally primitive.
Citations
1. Bourbaki 1989, p. 96, Ch 1, §8.1.
2. Saunders Mac Lane; Garrett Birkhoff (1967). Algebra. AMS Chelsea. p. 85.
3. Lang (2002), p. 83.
4. Isaacs 1994, p. 160.
5. "Non-associative rings and algebras". Encyclopedia of Mathematics.
6. Isaacs 1994, p. 161.
7. Lam, A first course on noncommutative rings, 2nd edition, Springer, 2001; Theorem 3.1.
8. Lang, Undergraduate algebra, Springer, 2005; V.§3.
9. Serre, Lie algebras and Lie groups, 2nd edition, corrected 5th printing, Springer, 2006; p. 3.
10. Serre, Local fields, Springer, 1979; p. 158.
11. "The development of Ring Theory".
12. Kleiner 1998, p. 27. sfn error: no target: CITEREFKleiner1998 (help)
13. Hilbert 1897.
14. "Why is a ring called a "ring"? – MathOverflow".
15. Cohn, Harvey (1980), Advanced Number Theory, New York: Dover Publications, p. 49, ISBN 978-0-486-64023-5
16. Fraenkel 1915, pp. 143–145.
17. Jacobson 2009, p. 86, footnote 1.
18. Fraenkel 1915, p. 144, axiom R8).
19. Noether 1921, p. 29.
20. Fraenkel 1915, p. 144, axiom R7).
21. van der Waerden 1930.
22. Zariski & Samuel 1958.
23. Artin 2018, p. 346.
24. Bourbaki 1989, p. 96.
25. Eisenbud, p. 11. sfn error: no target: CITEREFEisenbud (help)
26. Gallian 2006, p. 235.
27. Hungerford 1997, p. 42.
28. Warner 1965, p. 188.
29. Garling, D. J. H. (2022). Galois Theory and Its Algebraic Background (2nd ed.). Cambridge: Cambridge University Press. ISBN 978-1-108-83892-4.
30. Gardner & Wiegandt 2003.
31. Poonen 2018.
32. Wilder 1965, p. 176.
33. Rotman 1998, p. 7.
34. Jacobson 2009, p. 155.
35. Bourbaki 1989, p. 98.
36. Cohn 2003, Theorem 4.5.1.
37. Jacobson 1974, Theorem 2.10. sfn error: no target: CITEREFJacobson1974 (help)
38. Bourbaki 1964, Ch 5. §1, Lemma 2.
39. Cohn 2003, 4.4.
40. Lang 2002, Ch. XVII. Proposition 1.1.
41. Cohn 1995, Proposition 1.3.1.
42. Eisenbud 2004, Exercise 2.2. sfn error: no target: CITEREFEisenbud2004 (help)
43. Milne 2012, Proposition 6.4. sfn error: no target: CITEREFMilne2012 (help)
44. Milne 2012, end of Chapter 7. sfn error: no target: CITEREFMilne2012 (help)
45. Atiyah & Macdonald 1969, Theorem 10.17 and its corollaries.
46. Cohn 1995, pg. 242.
47. Lang 2002, Ch XIV, §2.
48. Weibel, Ch 1, Theorem 3.8.
49. Milne & CFT, Ch IV, §2. sfn error: no target: CITEREFMilneCFT (help)
50. Serre, J-P., Applications algébriques de la cohomologie des groupes, I, II, Séminaire Henri Cartan, 1950/51
51. Jacobson 2009, p. 162, Theorem 3.2.
52. Jacobson 2009.
53. Serre, p. 44.
References
General references
• Artin, Michael (2018). Algebra (2nd ed.). Pearson.
• Atiyah, Michael; Macdonald, Ian G. (1969). Introduction to commutative algebra. Addison–Wesley.
• Bourbaki, N. (1964). Algèbre commutative. Hermann.
• Bourbaki, N. (1989). Algebra I, Chapters 1–3. Springer.
• Cohn, Paul Moritz (2003), Basic algebra: groups, rings, and fields, Springer, ISBN 978-1-85233-587-8.
• Eisenbud, David (1995). Commutative algebra with a view toward algebraic geometry. Springer.
• Gallian, Joseph A. (2006). Contemporary Abstract Algebra, Sixth Edition. Houghton Mifflin. ISBN 9780618514717.
• Gardner, J.W.; Wiegandt, R. (2003). Radical Theory of Rings. Chapman & Hall/CRC Pure and Applied Mathematics. ISBN 0824750330.
• Herstein, I. N. (1994) [reprint of the 1968 original]. Noncommutative rings. Carus Mathematical Monographs. Vol. 15. With an afterword by Lance W. Small. Mathematical Association of America. ISBN 0-88385-015-X.
• Hungerford, Thomas W. (1997). Abstract Algebra: an Introduction, Second Edition. Brooks/Cole. ISBN 9780030105593.
• Jacobson, Nathan (2009). Basic algebra. Vol. 1 (2nd ed.). Dover. ISBN 978-0-486-47189-1.
• Jacobson, Nathan (1964). "Structure of rings". American Mathematical Society Colloquium Publications (Revised ed.). 37.
• Jacobson, Nathan (1943). "The Theory of Rings". American Mathematical Society Mathematical Surveys. I.
• Kaplansky, Irving (1974), Commutative rings (Revised ed.), University of Chicago Press, ISBN 0-226-42454-5, MR 0345945.
• Lam, Tsit Yuen (2001). A first course in noncommutative rings. Graduate Texts in Mathematics. Vol. 131 (2nd ed.). Springer. ISBN 0-387-95183-0.
• Lam, Tsit Yuen (2003). Exercises in classical ring theory. Problem Books in Mathematics (2nd ed.). Springer. ISBN 0-387-00500-5.
• Lam, Tsit Yuen (1999). Lectures on modules and rings. Graduate Texts in Mathematics. Vol. 189. Springer. ISBN 0-387-98428-3.
• Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, vol. 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556, Zbl 0984.00001.
• Matsumura, Hideyuki (1989). Commutative Ring Theory. Cambridge Studies in Advanced Mathematics (2nd ed.). Cambridge University Press. ISBN 978-0-521-36764-6.
• Milne, J. "A primer of commutative algebra".
• Rotman, Joseph (1998), Galois Theory (2nd ed.), Springer, ISBN 0-387-98541-7.
• van der Waerden, Bartel Leendert (1930), Moderne Algebra. Teil I, Die Grundlehren der mathematischen Wissenschaften, vol. 33, Springer, ISBN 978-3-540-56799-8, MR 0009016.
• Warner, Seth (1965). Modern Algebra. Dover. ISBN 9780486663418.
• Wilder, Raymond Louis (1965). Introduction to Foundations of Mathematics. Wiley.
• Zariski, Oscar; Samuel, Pierre (1958). Commutative Algebra. Vol. 1. Van Nostrand.
Special references
• Balcerzyk, Stanisław; Józefiak, Tadeusz (1989), Commutative Noetherian and Krull rings, Mathematics and its Applications, Chichester: Ellis Horwood Ltd., ISBN 978-0-13-155615-7
• Balcerzyk, Stanisław; Józefiak, Tadeusz (1989), Dimension, multiplicity and homological methods, Mathematics and its Applications, Chichester: Ellis Horwood Ltd., ISBN 978-0-13-155623-2
• Ballieu, R. (1947). "Anneaux finis; systèmes hypercomplexes de rang trois sur un corps commutatif". Ann. Soc. Sci. Bruxelles. I (61): 222–227.
• Berrick, A. J.; Keating, M. E. (2000). An Introduction to Rings and Modules with K-Theory in View. Cambridge University Press.
• Cohn, Paul Moritz (1995), Skew Fields: Theory of General Division Rings, Encyclopedia of Mathematics and its Applications, vol. 57, Cambridge University Press, ISBN 9780521432177
• Eisenbud, David (1995), Commutative algebra. With a view toward algebraic geometry., Graduate Texts in Mathematics, vol. 150, Springer, ISBN 978-0-387-94268-1, MR 1322960
• Gilmer, R.; Mott, J. (1973). "Associative Rings of Order". Proc. Japan Acad. 49: 795–799. doi:10.3792/pja/1195519146.
• Harris, J. W.; Stocker, H. (1998). Handbook of Mathematics and Computational Science. Springer.
• Isaacs, I. M. (1994). Algebra: A Graduate Course. AMS. ISBN 978-0-8218-4799-2.
• Jacobson, Nathan (1945), "Structure theory of algebraic algebras of bounded degree", Annals of Mathematics, Annals of Mathematics, 46 (4): 695–707, doi:10.2307/1969205, ISSN 0003-486X, JSTOR 1969205
• Knuth, D. E. (1998). The Art of Computer Programming. Vol. 2: Seminumerical Algorithms (3rd ed.). Addison–Wesley.
• Korn, G. A.; Korn, T. M. (2000). Mathematical Handbook for Scientists and Engineers. Dover. ISBN 9780486411477.
• Milne, J. "Class field theory".
• Nagata, Masayoshi (1962) [1975 reprint], Local rings, Interscience Tracts in Pure and Applied Mathematics, vol. 13, Interscience Publishers, ISBN 978-0-88275-228-0, MR 0155856
• Pierce, Richard S. (1982). Associative algebras. Graduate Texts in Mathematics. Vol. 88. Springer. ISBN 0-387-90693-2.
• Poonen, Bjorn (2018), Why all rings should have a 1 (PDF), arXiv:1404.0135, archived (PDF) from the original on 2015-04-24
• Serre, Jean-Pierre (1979), Local fields, Graduate Texts in Mathematics, vol. 67, Springer
• Springer, Tonny A. (1977), Invariant theory, Lecture Notes in Mathematics, vol. 585, Springer, ISBN 9783540373704
• Weibel, Charles. "The K-book: An introduction to algebraic K-theory".
• Zariski, Oscar; Samuel, Pierre (1975). Commutative algebra. Graduate Texts in Mathematics. Vol. 28–29. Springer. ISBN 0-387-90089-6.
Primary sources
• Fraenkel, A. (1915). "Über die Teiler der Null und die Zerlegung von Ringen". J. Reine Angew. Math. 1915 (145): 139–176. doi:10.1515/crll.1915.145.139. S2CID 118962421.
• Hilbert, David (1897). "Die Theorie der algebraischen Zahlkörper". Jahresbericht der Deutschen Mathematiker-Vereinigung. 4.
• Noether, Emmy (1921). "Idealtheorie in Ringbereichen". Math. Annalen. 83 (1–2): 24–66. doi:10.1007/bf01464225. S2CID 121594471.
Historical references
• History of ring theory at the MacTutor Archive
• Garrett Birkhoff and Saunders Mac Lane (1996) A Survey of Modern Algebra, 5th ed. New York: Macmillan.
• Bronshtein, I. N. and Semendyayev, K. A. (2004) Handbook of Mathematics, 4th ed. New York: Springer-Verlag ISBN 3-540-43491-7.
• Faith, Carl (1999) Rings and things and a fine array of twentieth century associative algebra. Mathematical Surveys and Monographs, 65. American Mathematical Society ISBN 0-8218-0993-8.
• Itô, K. editor (1986) "Rings." §368 in Encyclopedic Dictionary of Mathematics, 2nd ed., Vol. 2. Cambridge, MA: MIT Press.
• Israel Kleiner (1996) "The Genesis of the Abstract Ring Concept", American Mathematical Monthly 103: 417–424 doi:10.2307/2974935
• Kleiner, I. (1998) "From numbers to rings: the early history of ring theory", Elemente der Mathematik 53: 18–35.
• B. L. van der Waerden (1985) A History of Algebra, Springer-Verlag,
Authority control
International
• FAST
National
• Spain
• France
• BnF data
• Germany
• Israel
• United States
• Czech Republic
|
Wikipedia
|
Approximation property (ring theory)
In algebra, a commutative Noetherian ring A is said to have the approximation property with respect to an ideal I if each finite system of polynomial equations with coefficients in A has a solution in A if and only if it has a solution in the I-adic completion of A.[1][2] The notion of the approximation property is due to Michael Artin.
This article is about the concept in ring theory. For the concept in functional analysis, see approximation property.
See also
• Artin approximation theorem
• Popescu's theorem
Notes
1. Rotthaus, Christel (1997). "Excellent Rings, Henselian Rings, and the Approximation Property". Rocky Mountain Journal of Mathematics. 27 (1): 317–334. doi:10.1216/rmjm/1181071964. JSTOR 44238106.
2. "Tag 07BW: Smoothing Ring Maps". The Stacks Project. Columbia University, Department of Mathematics. Retrieved 2018-02-19.
References
• Popescu, Dorin (1986). "General Néron desingularization and approximation". Nagoya Mathematical Journal. 104: 85–115. doi:10.1017/S0027763000022698.
• Rotthaus, Christel (1987). "On the approximation property of excellent rings". Inventiones Mathematicae. 88: 39–63. doi:10.1007/BF01405090.
• Artin, M (1969). "Algebraic approximation of structures over complete local rings". Publications Mathématiques de l'IHÉS. 36: 23–58. doi:10.1007/BF02684596. ISSN 0073-8301.
• Artin, M (1968). "On the solutions of analytic equations". Inventiones Mathematicae. 5 (4): 277–291. doi:10.1007/BF01389777. ISSN 0020-9910.
|
Wikipedia
|
Ringed topos
In mathematics, a ringed topos is a generalization of a ringed space; that is, the notion is obtained by replacing a "topological space" by a "topos". The notion of a ringed topos has applications to deformation theory in algebraic geometry (cf. cotangent complex) and the mathematical foundation of quantum mechanics. In the latter subject, a Bohr topos is a ringed topos that plays the role of a quantum phase space.[1][2]
The definition of a topos-version of a "locally ringed space" is not straightforward, as the meaning of "local" in this context is not obvious. One can introduce the notion of a locally ringed topos by introducing a sort of geometric conditions of local rings (see SGA4, Exposé IV, Exercise 13.9), which is equivalent to saying that all the stalks of the structure ring object are local rings when there are enough points.
Morphisms
A morphism $(T,{\mathcal {O}}_{T})\to (T',{\mathcal {O}}_{T'})$ of ringed topoi is a pair consisting of a topos morphism $f:T\to T'$ and a ring homomorphism ${\mathcal {O}}_{T'}\to f_{*}{\mathcal {O}}_{T}$.
If one replaces a "topos" by an ∞-topos, then one gets the notion of a ringed ∞-topos.
Examples
Ringed topos of a topological space
One of the key motivating examples of a ringed topos comes from topology. Consider the site ${\text{Open}}(X)$ of a topological space $X$, and the sheaf of continuous functions
$C_{X}^{0}:{\text{Open}}(X)^{op}\to {\text{CRing}}$
sending an object $U\in {\text{Open}}(X)$, an open subset of $X$, to the ring of continuous functions $C_{X}^{0}(U)$ on $U$. Then, the pair $({\text{Sh}}({\text{Open}}(X)),C_{X}^{0})$ forms a ringed topos. Note this can be generalized to any ringed space $(X,{\mathcal {O}}_{X})$ where
${\mathcal {O}}_{X}:{\text{Open}}(X)^{op}\to {\text{Rings}}$
so the pair $({\text{Sh}}({\text{Open}}(X)),{\mathcal {O}}_{X})$ is a ringed topos.
Ringed topos of a scheme
Another key example is the ringed topos associated to a scheme $(X,{\mathcal {O}}_{X})$, which is again the ringed topos associated to the underlying locally ringed space.
Relation with functor of points
Recall that the functor of points view of scheme theory defines a scheme $X$ as a functor $X:{\text{CAlg}}\to {\text{Sets}}$ which satisfies a sheaf condition and gluing condition.[3] That is, for any open cover ${\text{Spec}}(R_{f_{i}})\to {\text{Spec}}(R)$ of affine schemes, there is the following exact sequence
$X(R)\to \prod X(R_{f_{i}})\rightrightarrows \prod X(R_{f_{i}f_{j}})$
Also, there must exist open affine subfunctors
$U_{i}={\text{Spec}}(A_{i})={\text{Hom}}_{\text{CAlg}}(A_{i},-)$
covering $X$, meaning for any $\xi \in X(R)$, there is a $\xi |_{U_{i}}\in U_{i}(R)$. Then, there is a topos associated to $X$ whose underlying site is the site of open subfunctors. This site is isomorphic to the site associated to the underlying topological space of the ringed space corresponding to the scheme. Then, topos theory gives a way to construct scheme theory without having to use locally ringed spaces using the associated locally ringed topos.
Ringed topos of sets
The category of sets is equivalent to the category of sheaves on the category with one object and only the identity morphism, so ${\text{Sh}}(*)\cong {\text{Sets}}$. Then, given any ring $A$, there is an associated sheaf ${\text{Hom}}_{Sets}(-,A):{\text{Sets}}^{op}\to {\text{Rings}}$. This can be used to find toy examples of morphisms of ringed topoi.
Notes
1. Schreiber, Urs (2011-07-25). "Bohr toposes". The n-Category Café. Retrieved 2018-02-19.
2. Heunen, Chris; Landsman, Nicolaas P.; Spitters, Bas (2009-10-01). "A Topos for Algebraic Quantum Theory". Communications in Mathematical Physics. 291 (1): 63–110. arXiv:0709.4364. Bibcode:2009CMaPh.291...63H. doi:10.1007/s00220-009-0865-6. ISSN 0010-3616.
3. "Section 26.15 (01JF): A representability criterion—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-04-28.
References
• The standard reference is the fourth volume of the Séminaire de Géométrie Algébrique du Bois Marie.
• Francis, J. Derived Algebraic Geometry Over ${\mathcal {E}}_{n}$-Rings
• Grothendieck Duality for Derived Stacks
• Ringed topos at the nLab
• Locally ringed topos at the nLab
|
Wikipedia
|
Heawood conjecture
In graph theory, the Heawood conjecture or Ringel–Youngs theorem gives a lower bound for the number of colors that are necessary for graph coloring on a surface of a given genus. For surfaces of genus 0, 1, 2, 3, 4, 5, 6, 7, ..., the required number of colors is 4, 7, 8, 9, 10, 11, 12, 12, .... OEIS: A000934, the chromatic number or Heawood number.
The conjecture was formulated in 1890 by Percy John Heawood and proven in 1968 by Gerhard Ringel and Ted Youngs. One case, the non-orientable Klein bottle, proved an exception to the general formula. An entirely different approach was needed for the much older problem of finding the number of colors needed for the plane or sphere, solved in 1976 as the four color theorem by Haken and Appel. On the sphere the lower bound is easy, whereas for higher genera the upper bound is easy and was proved in Heawood's original short paper that contained the conjecture. In other words, Ringel, Youngs and others had to construct extreme examples for every genus g = 1,2,3,.... If g = 12s + k, the genera fall into 12 cases according as k = 0,1,2,3,4,5,6,7,8,9,10,11. To simplify, suppose that case k has been established if only a finite number of g's of the form 12s + k are in doubt. Then the years in which the twelve cases were settled and by whom are the following:
• 1954, Ringel: case 5
• 1961, Ringel: cases 3,7,10
• 1963, Terry, Welch, Youngs: cases 0,4
• 1964, Gustin, Youngs: case 1
• 1965, Gustin: case 9
• 1966, Youngs: case 6
• 1967, Ringel, Youngs: cases 2,8,11
The last seven sporadic exceptions were settled as follows:
• 1967, Mayer: cases 18, 20, 23
• 1968, Ringel, Youngs: cases 30, 35, 47, 59, and the conjecture was proved.
Formal statement
Percy John Heawood conjectured in 1890 that for a given genus g > 0, the minimum number of colors necessary to color all graphs drawn on an orientable surface of that genus (or equivalently to color the regions of any partition of the surface into simply connected regions) is given by
$\gamma (g)=\left\lfloor {\frac {7+{\sqrt {1+48g}}}{2}}\right\rfloor ,$
where $\left\lfloor x\right\rfloor $ is the floor function.
Replacing the genus by the Euler characteristic, we obtain a formula that covers both the orientable and non-orientable cases,
$\gamma (\chi )=\left\lfloor {\frac {7+{\sqrt {49-24\chi }}}{2}}\right\rfloor .$
This relation holds, as Ringel and Youngs showed, for all surfaces except for the Klein bottle. Philip Franklin (1930) proved that the Klein bottle requires at most 6 colors, rather than 7 as predicted by the formula. The Franklin graph can be drawn on the Klein bottle in a way that forms six mutually-adjacent regions, showing that this bound is tight.
The upper bound, proved in Heawood's original short paper, is based on a greedy coloring algorithm. By manipulating the Euler characteristic, one can show that every graph embedded in the given surface must have at least one vertex of degree less than the given bound. If one removes this vertex, and colors the rest of the graph, the small number of edges incident to the removed vertex ensures that it can be added back to the graph and colored without increasing the needed number of colors beyond the bound. In the other direction, the proof is more difficult, and involves showing that in each case (except the Klein bottle) a complete graph with a number of vertices equal to the given number of colors can be embedded on the surface.
Example
The torus has g = 1, so χ = 0. Therefore, as the formula states, any subdivision of the torus into regions can be colored using at most seven colors. The illustration shows a subdivision of the torus in which each of seven regions are adjacent to each other region; this subdivision shows that the bound of seven on the number of colors is tight for this case. The boundary of this subdivision forms an embedding of the Heawood graph onto the torus.
References
1. Grünbaum, Branko; Szilassi, Lajos (2009), "Geometric Realizations of Special Toroidal Complexes", Contributions to Discrete Mathematics, 4 (1): 21–39, doi:10.11575/cdm.v4i1.61986, ISSN 1715-0868
• Franklin, P. (1934). "A six color problem". MIT Journal of Mathematics and Physics. 13 (1–4): 363–379. doi:10.1002/sapm1934131363. hdl:2027/mdp.39015019892200.
• Heawood, P. J. (1890). "Map colour theorem". Quarterly Journal of Mathematics. 24: 332–338.
• Ringel, G.; Youngs, J. W. T. (1968). "Solution of the Heawood map-coloring problem". Proceedings of the National Academy of Sciences of the United States of America. 60 (2): 438–445. Bibcode:1968PNAS...60..438R. doi:10.1073/pnas.60.2.438. MR 0228378. PMC 225066. PMID 16591648.
External links
• Weisstein, Eric W. "Heawood Conjecture". MathWorld.
|
Wikipedia
|
Ringing (signal)
In electronics, signal processing, and video, ringing is oscillation of a signal, particularly in the step response (the response to a sudden change in input). Often ringing is undesirable, but not always, as in the case of resonant inductive coupling. It is also known as hunting.[1] It is closely related to overshoot, often instigated as damping response following overshoot or undershoot, and thus the terms are at times conflated.
It is also known as ripple, particularly in electricity or in frequency domain response.
Electricity
In electrical circuits, ringing is an unwanted oscillation of a voltage or current. It happens when an electrical pulse causes the parasitic capacitances and inductances in the circuit (i.e. those that are not part of the design, but just by-products of the materials used to construct the circuit) to resonate at their characteristic frequency.[2] Ringing artifacts are also present in square waves; see Gibbs phenomenon.
Ringing is undesirable because it causes extra current to flow, thereby wasting energy and causing extra heating of the components; it can cause unwanted electromagnetic radiation to be emitted; it can delay arrival at a desired final state (increase settling time); and it may cause unwanted triggering of bistable elements in digital circuits. Ringy communications circuits may suffer falsing.
Ringing can be due to signal reflection, in which case it may be minimized by impedance matching.
Video
In video circuits, electrical ringing causes closely spaced repeated ghosts of a vertical or diagonal edge where dark changes to light or vice versa, going from left to right. In a CRT the electron beam upon changing from dark to light or vice versa instead of changing quickly to the desired intensity and staying there, overshoots and undershoots a few times. This bouncing could occur anywhere in the electronics or cabling and is often caused by or accentuated by a too high setting of the sharpness control.
Audio
Ringing can affect audio equipment in a number of ways. Audio amplifiers can produce ringing depending on their design, although the transients that can produce such ringing rarely occur in audio signals.
Transducers (i.e., microphones and loudspeakers) can also ring. Mechanical ringing is more of a problem with loudspeakers as the moving masses are larger and less easily damped, but unless extreme they are difficult to audibly identify.
In digital audio, ringing can occur as a result of filters such as brickwall filters. Here, the ringing occurs before the transient as well as after.
Signal processing
In signal processing, "ringing" may refer to ringing artifacts: spurious signals near sharp transitions. These have a number of causes, and occur for instance in JPEG compression and as pre-echo in some audio compression.
See also
• Microphonics
• Ripple (electrical)
• Impedance matching
References
1. Oxford English Dictionary (2nd ed.). Oxford University Press. 1989. f. The action of a machine, instrument, system, etc., that is hunting (see hunt v. 7b); an undesirable oscillation about an equilibrium speed, position, or state.
2. Johnson, H. and Graham, M. High-Speed Digital Design: A Handbook of Black Magic. 1993. pp. 88–90
External links
• Microphony with older video cameras
|
Wikipedia
|
Ring of integers
In mathematics, the ring of integers of an algebraic number field $K$ is the ring of all algebraic integers contained in $K$.[1] An algebraic integer is a root of a monic polynomial with integer coefficients: $x^{n}+c_{n-1}x^{n-1}+\cdots +c_{0}$.[2] This ring is often denoted by $O_{K}$ or ${\mathcal {O}}_{K}$. Since any integer belongs to $K$ and is an integral element of $K$, the ring $\mathbb {Z} $ is always a subring of $O_{K}$.
Algebraic structure → Ring theory
Ring theory
Basic concepts
Rings
• Subrings
• Ideal
• Quotient ring
• Fractional ideal
• Total ring of fractions
• Product of rings
• Free product of associative algebras
• Tensor product of algebras
Ring homomorphisms
• Kernel
• Inner automorphism
• Frobenius endomorphism
Algebraic structures
• Module
• Associative algebra
• Graded ring
• Involutive ring
• Category of rings
• Initial ring $\mathbb {Z} $
• Terminal ring $0=\mathbb {Z} _{1}$
Related structures
• Field
• Finite field
• Non-associative ring
• Lie ring
• Jordan ring
• Semiring
• Semifield
Commutative algebra
Commutative rings
• Integral domain
• Integrally closed domain
• GCD domain
• Unique factorization domain
• Principal ideal domain
• Euclidean domain
• Field
• Finite field
• Composition ring
• Polynomial ring
• Formal power series ring
Algebraic number theory
• Algebraic number field
• Ring of integers
• Algebraic independence
• Transcendental number theory
• Transcendence degree
p-adic number theory and decimals
• Direct limit/Inverse limit
• Zero ring $\mathbb {Z} _{1}$
• Integers modulo pn $\mathbb {Z} /p^{n}\mathbb {Z} $
• Prüfer p-ring $\mathbb {Z} (p^{\infty })$
• Base-p circle ring $\mathbb {T} $
• Base-p integers $\mathbb {Z} $
• p-adic rationals $\mathbb {Z} [1/p]$
• Base-p real numbers $\mathbb {R} $
• p-adic integers $\mathbb {Z} _{p}$
• p-adic numbers $\mathbb {Q} _{p}$
• p-adic solenoid $\mathbb {T} _{p}$
Algebraic geometry
• Affine variety
Noncommutative algebra
Noncommutative rings
• Division ring
• Semiprimitive ring
• Simple ring
• Commutator
Noncommutative algebraic geometry
Free algebra
Clifford algebra
• Geometric algebra
Operator algebra
The ring of integers $\mathbb {Z} $ is the simplest possible ring of integers.[lower-alpha 1] Namely, $\mathbb {Z} =O_{\mathbb {Q} }$ where $\mathbb {Q} $ is the field of rational numbers.[3] And indeed, in algebraic number theory the elements of $\mathbb {Z} $ are often called the "rational integers" because of this.
The next simplest example is the ring of Gaussian integers $\mathbb {Z} [i]$, consisting of complex numbers whose real and imaginary parts are integers. It is the ring of integers in the number field $\mathbb {Q} (i)$ of Gaussian rationals, consisting of complex numbers whose real and imaginary parts are rational numbers. Like the rational integers, $\mathbb {Z} [i]$ is a Euclidean domain.
The ring of integers of an algebraic number field is the unique maximal order in the field. It is always a Dedekind domain.[4]
Properties
The ring of integers OK is a finitely-generated Z-module. Indeed, it is a free Z-module, and thus has an integral basis, that is a basis b1, ..., bn ∈ OK of the Q-vector space K such that each element x in OK can be uniquely represented as
$x=\sum _{i=1}^{n}a_{i}b_{i},$
with ai ∈ Z.[5] The rank n of OK as a free Z-module is equal to the degree of K over Q.
Examples
Computational tool
A useful tool for computing the integral closure of the ring of integers in an algebraic field K/Q is the discriminant. If K is of degree n over Q, and $\alpha _{1},\ldots ,\alpha _{n}\in {\mathcal {O}}_{K}$ form a basis of K over Q, set $d=\Delta _{K/\mathbb {Q} }(\alpha _{1},\ldots ,\alpha _{n})$. Then, ${\mathcal {O}}_{K}$ is a submodule of the Z-module spanned by $\alpha _{1}/d,\ldots ,\alpha _{n}/d$.[6] pg. 33 In fact, if d is square-free, then $\alpha _{1},\ldots ,\alpha _{n}$ forms an integral basis for ${\mathcal {O}}_{K}$.[6] pg. 35
Cyclotomic extensions
If p is a prime, ζ is a pth root of unity and K = Q(ζ ) is the corresponding cyclotomic field, then an integral basis of OK = Z[ζ] is given by (1, ζ, ζ 2, ..., ζ p−2).[7]
Quadratic extensions
If $d$ is a square-free integer and $K=\mathbb {Q} ({\sqrt {d}}\,)$ is the corresponding quadratic field, then ${\mathcal {O}}_{K}$ is a ring of quadratic integers and its integral basis is given by (1, (1 + √d) /2) if d ≡ 1 (mod 4) and by (1, √d) if d ≡ 2, 3 (mod 4).[8] This can be found by computing the minimal polynomial of an arbitrary element $a+b{\sqrt {d}}\in \mathbf {Q} ({\sqrt {d}})$ where $a,b\in \mathbf {Q} $.
Multiplicative structure
In a ring of integers, every element has a factorization into irreducible elements, but the ring need not have the property of unique factorization: for example, in the ring of integers Z[√−5], the element 6 has two essentially different factorizations into irreducibles:[4][9]
$6=2\cdot 3=(1+{\sqrt {-5}})(1-{\sqrt {-5}}).$
A ring of integers is always a Dedekind domain, and so has unique factorization of ideals into prime ideals.[10]
The units of a ring of integers OK is a finitely generated abelian group by Dirichlet's unit theorem. The torsion subgroup consists of the roots of unity of K. A set of torsion-free generators is called a set of fundamental units.[11]
Generalization
One defines the ring of integers of a non-archimedean local field F as the set of all elements of F with absolute value ≤ 1; this is a ring because of the strong triangle inequality.[12] If F is the completion of an algebraic number field, its ring of integers is the completion of the latter's ring of integers. The ring of integers of an algebraic number field may be characterised as the elements which are integers in every non-archimedean completion.[3]
For example, the p-adic integers Zp are the ring of integers of the p-adic numbers Qp .
See also
• Minimal polynomial (field theory)
• Integral closure – gives a technique for computing integral closures
Notes
1. The ring of integers, without specifying the field, refers to the ring $\mathbb {Z} $ of "ordinary" integers, the prototypical object for all those rings. It is a consequence of the ambiguity of the word "integer" in abstract algebra.
Citations
1. Alaca & Williams 2003, p. 110, Defs. 6.1.2-3.
2. Alaca & Williams 2003, p. 74, Defs. 4.1.1-2.
3. Cassels 1986, p. 192.
4. Samuel 1972, p. 49.
5. Cassels (1986) p. 193
6. Baker. "Algebraic Number Theory" (PDF). pp. 33–35.
7. Samuel 1972, p. 43.
8. Samuel 1972, p. 35.
9. Artin, Michael (2011). Algebra. Prentice Hall. p. 360. ISBN 978-0-13-241377-0.
10. Samuel 1972, p. 50.
11. Samuel 1972, pp. 59–62.
12. Cassels 1986, p. 41.
References
• Alaca, Saban; Williams, Kenneth S. (2003). Introductory Algebraic Number Theory. Cambridge University Press. ISBN 9780511791260.
• Cassels, J.W.S. (1986). Local fields. London Mathematical Society Student Texts. Vol. 3. Cambridge: Cambridge University Press. ISBN 0-521-31525-5. Zbl 0595.12006.
• Neukirch, Jürgen (1999). Algebraische Zahlentheorie. Grundlehren der mathematischen Wissenschaften. Vol. 322. Berlin: Springer-Verlag. ISBN 978-3-540-65399-8. MR 1697859. Zbl 0956.11021.
• Samuel, Pierre (1972). Algebraic number theory. Hermann/Kershaw.
|
Wikipedia
|
Ring of modular forms
In mathematics, the ring of modular forms associated to a subgroup Γ of the special linear group SL(2, Z) is the graded ring generated by the modular forms of Γ. The study of rings of modular forms describes the algebraic structure of the space of modular forms.
Definition
Let Γ be a subgroup of SL(2, Z) that is of finite index and let Mk(Γ) be the vector space of modular forms of weight k. The ring of modular forms of Γ is the graded ring $ M(\Gamma )=\bigoplus _{k\geq 0}M_{k}(\Gamma )$.[1]
Example
The ring of modular forms of the full modular group SL(2, Z) is freely generated by the Eisenstein series E4 and E6. In other words, Mk(Γ) is isomorphic as a $\mathbb {C} $-algebra to $\mathbb {C} [E_{4},E_{6}]$, which is the polynomial ring of two variables over the complex numbers.[1]
Properties
The ring of modular forms is a graded Lie algebra since the Lie bracket $[f,g]=kfg'-\ell f'g$ of modular forms f and g of respective weights k and ℓ is a modular form of weight k + ℓ + 2.[1] A bracket can be defined for the n-th derivative of modular forms and such a bracket is called a Rankin–Cohen bracket.[1]
Congruence subgroups of SL(2, Z)
In 1973, Pierre Deligne and Michael Rapoport showed that the ring of modular forms M(Γ) is finitely generated when Γ is a congruence subgroup of SL(2, Z).[2]
In 2003, Lev Borisov and Paul Gunnells showed that the ring of modular forms M(Γ) is generated in weight at most 3 when $\Gamma $ is the congruence subgroup $\Gamma _{1}(N)$ of prime level N in SL(2, Z) using the theory of toric modular forms.[3] In 2014, Nadim Rustom extended the result of Borisov and Gunnells for $\Gamma _{1}(N)$ to all levels N and also demonstrated that the ring of modular forms for the congruence subgroup $\Gamma _{0}(N)$ is generated in weight at most 6 for some levels N.[4]
In 2015, John Voight and David Zureick-Brown generalized these results: they proved that the graded ring of modular forms of even weight for any congruence subgroup Γ of SL(2, Z) is generated in weight at most 6 with relations generated in weight at most 12.[5] Building on this work, in 2016, Aaron Landesman, Peter Ruhm, and Robin Zhang showed that the same bounds hold for the full ring (all weights), with the improved bounds of 5 and 10 when Γ has some nonzero odd weight modular form.[6]
General Fuchsian groups
A Fuchsian group Γ corresponds to the orbifold obtained from the quotient $\Gamma \backslash \mathbb {H} $ of the upper half-plane $\mathbb {H} $. By a stacky generalization of Riemann's existence theorem, there is a correspondence between the ring of modular forms of Γ and the a particular section ring closely related to the canonical ring of a stacky curve.[5]
There is a general formula for the weights of generators and relations of rings of modular forms due to the work of Voight and Zureick-Brown and the work of Landesman, Ruhm, and Zhang. Let $e_{i}$ be the stabilizer orders of the stacky points of the stacky curve (equivalently, the cusps of the orbifold $\Gamma \backslash \mathbb {H} $) associated to Γ. If Γ has no nonzero odd weight modular forms, then the ring of modular forms is generated in weight at most $6\max(1,e_{1},e_{2},\ldots ,e_{r})$ and has relations generated in weight at most $12\max(1,e_{1},e_{2},\ldots ,e_{r})$.[5] If Γ has a nonzero odd weight modular form, then the ring of modular forms is generated in weight at most $\max(5,e_{1},e_{2},\ldots ,e_{r})$ and has relations generated in weight at most $2\max(5,e_{1},e_{2},\ldots ,e_{r})$.[6]
Applications
In string theory and supersymmetric gauge theory, the algebraic structure of the ring of modular forms can be used to study the structure of the Higgs vacua of four-dimensional gauge theories with N = 1 supersymmetry.[7] The stabilizers of superpotentials in N = 4 supersymmetric Yang–Mills theory are rings of modular forms of the congruence subgroup Γ(2) of SL(2, Z).[7][8]
References
1. Zagier, Don (2008). "Elliptic Modular Forms and Their Applications" (PDF). In Bruinier, Jan Hendrik; van der Geer, Gerard; Harder, Günter; Zagier, Don (eds.). The 1-2-3 of Modular Forms. Universitext. Springer-Verlag. pp. 1–103. doi:10.1007/978-3-540-74119-0_1. ISBN 978-3-540-74119-0.
2. Deligne, Pierre; Rapoport, Michael (2009) [1973]. "Les schémas de modules de courbes elliptiques". Modular functions of one variable, II. Lecture Notes in Mathematics. Vol. 349. Springer. pp. 143–316. ISBN 9783540378556.
3. Borisov, Lev A.; Gunnells, Paul E. (2003). "Toric modular forms of higher weight". J. Reine Angew. Math. 560: 43–64. arXiv:math/0203242. Bibcode:2002math......3242B.
4. Rustom, Nadim (2014). "Generators of graded rings of modular forms". Journal of Number Theory. 138: 97–118. arXiv:1209.3864. doi:10.1016/j.jnt.2013.12.008. S2CID 119317127.
5. Voight, John; Zureick-Brown, David (2015). The canonical ring of a stacky curve. Memoirs of the American Mathematical Society. arXiv:1501.04657. Bibcode:2015arXiv150104657V.
6. Landesman, Aaron; Ruhm, Peter; Zhang, Robin (2016). "Spin canonical rings of log stacky curves". Annales de l'Institut Fourier. 66 (6): 2339–2383. arXiv:1507.02643. doi:10.5802/aif.3065. S2CID 119326707.
7. Bourget, Antoine; Troost, Jan (2017). "Permutations of massive vacua" (PDF). Journal of High Energy Physics. 2017 (42): 42. arXiv:1702.02102. Bibcode:2017JHEP...05..042B. doi:10.1007/JHEP05(2017)042. ISSN 1029-8479. S2CID 119225134.
8. Ritz, Adam (2006). "Central charges, S-duality and massive vacua of N = 1* super Yang-Mills". Physics Letters B. 641 (3–4): 338–341. arXiv:hep-th/0606050. doi:10.1016/j.physletb.2006.08.066. S2CID 13895731.
Major mathematics areas
• History
• Timeline
• Future
• Outline
• Lists
• Glossary
Foundations
• Category theory
• Information theory
• Mathematical logic
• Philosophy of mathematics
• Set theory
• Type theory
Algebra
• Abstract
• Commutative
• Elementary
• Group theory
• Linear
• Multilinear
• Universal
• Homological
Analysis
• Calculus
• Real analysis
• Complex analysis
• Hypercomplex analysis
• Differential equations
• Functional analysis
• Harmonic analysis
• Measure theory
Discrete
• Combinatorics
• Graph theory
• Order theory
Geometry
• Algebraic
• Analytic
• Arithmetic
• Differential
• Discrete
• Euclidean
• Finite
Number theory
• Arithmetic
• Algebraic number theory
• Analytic number theory
• Diophantine geometry
Topology
• General
• Algebraic
• Differential
• Geometric
• Homotopy theory
Applied
• Engineering mathematics
• Mathematical biology
• Mathematical chemistry
• Mathematical economics
• Mathematical finance
• Mathematical physics
• Mathematical psychology
• Mathematical sociology
• Mathematical statistics
• Probability
• Statistics
• Systems science
• Control theory
• Game theory
• Operations research
Computational
• Computer science
• Theory of computation
• Computational complexity theory
• Numerical analysis
• Optimization
• Computer algebra
Related topics
• Mathematicians
• lists
• Informal mathematics
• Films about mathematicians
• Recreational mathematics
• Mathematics and art
• Mathematics education
• Mathematics portal
• Category
• Commons
• WikiProject
|
Wikipedia
|
Riordan array
A (proper) Riordan array is an infinite lower triangular matrix, $D$, constructed out of two formal power series, $d(t)$ of order 0 and $h(t)$ of order 1, in such a way that $d_{n,k}=[t^{n}]d(t)h(t)^{k}$.
A Riordan array is an element of the Riordan group.[1] It was created by mathematician Louis W. Shapiro and named after mathematician John Riordan.[1]
The study of Riordan arrays is a growing field that is both being influenced by, and continuing its contributions to, other fields such as combinatorics, group theory, matrix theory, number theory, probability, sequences and series, Lie groups and Lie algebras, orthogonal polynomials, graph theory, networks, Beal conjecture, Riemann hypothesis, unimodal sequences, combinatorial identities, elliptic curves, numerical approximation, asymptotics, and data analysis. Riordan arrays is also a powerful unifying concept, binding together important tools: generating functions, computer algebra systems, formal languages, path model, and so on.[2]
Details follow.
A formal power series $a(x)=a_{0}+a_{1}x+a_{2}x^{2}+\cdots =\sum _{j\geq 0}a_{j}x^{j}\in \mathbb {C} [[x]]$ is said to have order $r$ if $a_{0}=...=a_{r-1}=0\neq a_{r}.$ Write ${\mathcal {F}}_{r}$ for the formal power series of order $r.$ A power series $a(x)$ of order 0 has a multiplicative inverse (i.e. $1/a(x)$ is a power series) iff it has order 0, i.e. iff it lies in ${\mathcal {F}}_{0}$; it has a composition inverse that is there exists a power series ${\bar {a}}$ such that ${\bar {a}}(a(x))=x$ iff it has order 1, i.e. iff it lies in ${\mathcal {F}}_{1}.$
As mentioned a Riordan array is defined usually as a pair of power series $(a(t),b(t))\in {\mathcal {F}}_{0}\times {\mathcal {F}}_{1}.$ The 'array' part in its name stems from the fact that one associates to $(a(t),b(t))$ the array of complex numbers defined by $d_{n,k}:=[t^{n}]a(t)b(t)^{k},$ $n,k\in \mathbb {N} =\mathbb {Z} _{\geq 0}.$ (Here $[t^{n}]...$ means coefficient of $t^{n}$ in $\cdots $) . So column $k$ of the array simply consists of the sequence of coefficients of the power series $a(t)b(t)^{k};$ in particular column 0 determines and is determined by the power series $a(t).$ As $a(t)$ is of order 0, it has a multiplicative inverse and it follows that from the array's column 1 we can recover $b(t)$ as $b(t)=a(t)^{-1}a(t)b(t).$ Since $b(t)=b_{1}t+b_{2}t^{2}+\cdots $ has order 1, $b(t)^{k}$ has order $k$ and so has $a(t)b(t)^{k}.$ It follows that the array $d_{n,k}$ is infinite triangular exhibiting a geometric progression $(d_{k,k})_{k\geq 0}=(a_{0}b_{1}^{k})_{k\geq 0}$ on its main diagonal. It also follows that the map associating to a pair of power series $(a(t),b(t))\in {\mathcal {F}}_{0}\times {\mathcal {F}}_{1}$ its triangular array is injective.
An example for a Riordan array is given by the pair of power series $({\frac {1}{1-x}},{\frac {x}{1-x}})=(\sum _{j\geq 0}x^{j},\sum _{j\geq 0}x^{j+1})\in {\mathcal {F}}_{0}\times {\mathcal {F}}_{1}.$ It is not difficult to show that this pair generates the infinite triangular array of binomial coefficients $d_{n,k}={n \choose k},$ also called Pascal matrix
$P=\left({\begin{array}{ccccccc}1&&&&&&\cdots \\1&1&&&&&\\1&2&1&&&&\\1&3&3&1&&&\\1&4&6&4&1&&\\&&&\cdots &&&\cdots \end{array}}\right)$
Proof. If $q(x)=\sum _{j\geq 0}q_{j}x^{j}$ is a power series with associated coefficient sequence $(q_{0},q_{1},q_{2},...),$ then, by Cauchy multiplication of power series, $q(x){\frac {x}{1-x}}=\sum _{j\geq 0}(0+q_{0}+q_{1}+\cdots +q_{j-1})x^{j}.$ So the latter series has as coefficient sequence $(0,q_{0},q_{0}+q_{1},q_{0}+q_{1}+q_{2},....)$ and hence $[t^{n}]q(x){\frac {x}{1-x}}=q_{0}+\cdots +q_{n-1}.$ Fix any $k\in \mathbb {Z} _{\geq 0}.$ If $q_{n}={n \choose k},$ so that $(q_{n})_{n\geq 0}$ represents column $k$ of the Pascal array, then $\sum _{j=0}^{n-1}q_{j}=\sum _{j=0}^{n-1}{j \choose k}={n \choose k+1}.$ This argument allows to see by induction on $k$ that ${\frac {1}{1-x}}({\frac {x}{1-x}})^{k}$ has column $k$ of the Pascal array as coefficient sequence. $\Box $
We are going to prove some much used facts about Riordan arrays. Note that the matrix multiplication rules applied to infinite triangular matrices lead to finite sums only and the product of two infinite triangular matrices is infinite triangular. The next two theorems were discovered essentially by Shapiro and coworkers,[1] who say they modified work they found in papers by Gian-Carlo Rota and the book of Roman [3]
Theorem. a. Let $(a(x),b(x))$ and $(c(x),d(x))$ be Riordan arrays, viewed as infinite lower triangular matrices. Then the product of these matrices is the array associated to the pair $(a(x)c(b(x)),d(b(x)))$ of formal power series which itself is a Riordan array.
b. This fact justifies to define a multiplication `$*$' of Riordan arrays viewed as pairs of power series by
$(a(x),b(x))*(c(x),d(x))=(a(x)c(b(x)),d(b(x))$
Proof. Since $a(x),c(x)$ have order 0 it is clear that $a(x)c(b(x))$ has order 0. Similarly $b(x),d(x)\in {\mathcal {F}}_{1}$ implies $d(b(x))\in {\mathcal {F}}_{1}.$ So $(a(x)c(b(x)),d(b(x)))$ is a Riordan array. Define a matrix $M$ as the Riordan array $(a(x),b(x)).$ By definitions its $j$-th column $M_{*,j}$ is the sequence of coefficients of the power series $a(x)b(x)^{j}.$ If we multiply this matrix from the right with the sequence $(r_{0},r_{1},r_{2},...)^{T}$ we get as a result a linear combination of columns of $M$ which we can read as a linear combination of power series, namely $\sum _{\nu \geq 0}r_{\nu }M_{*,\nu }=\sum _{\nu \geq 0}r_{\nu }a(x)b(x)^{\nu }=a(x)\sum _{\nu \geq 0}r_{\nu }b(x)^{\nu }.$ Thus, viewing sequence $(r_{0},r_{1},r_{2},...)^{T}$ as codified by the power series $r(x),$ we showed $(a(x),b(x))*r(x)=a(x)r(b(x)).$ Here the $*$ is the symbol for indicating correspondence on the power series level with matrix multiplication. We multiplied a Riordan array $(a(x),b(x))$ with a single power series. Let now $(c(x),d(x))$ be another Riordan array viewn as a matrix. One can form the product $(a(x),b(x))(c(x),d(x)).$ The $j$-th column of this product is just $(a(x),b(x))$ multiplied with the $j$-th column of $(c(x),d(x)).$ Since the latter corresponds to the power series $c(x)d(x)^{j},$ it follows by above, that the $j$-th column of $(a(x),b(x))(c(x),d(x))$ corresponds to $a(x)c(b(x))d(b(x))^{j}.$ As this holds for all column indices $j$ occurring in $(c(x),d(x))$ we have shown part a. Part b is now clear. $\Box $
Theorem. The family of Riordan arrays endowed with the product '$*$' defined above forms a group: the Riordan group.[1]
Proof. The associativity of the multiplication `$*$' follows from associativity of matrix multiplication. Next note $(1,x)*(c(x),d(x))=(1\cdot c(x),d(x))=(c(x),d(x)).$ So $(1,x)$ is a left neutral element. Finally we claim that $(c({\bar {d}}(x))^{-1},{\bar {d}}(x))$ is the left inverse to the power series $(c(x),d(x)).$ For this check the computation $(c({\bar {d}}(x))^{-1},{\bar {d}}(x))*(c(x),d(x))$ $=((c({\bar {d}}(x))^{-1}c(d(x)),d({\bar {d}}(x)))=(1,x).$ As is well known, an associative structure in which a left neutral element exists and for each element a left inverse is a group. $\Box $
Of course not all invertible infinite lower triangular arrays are Riordan arrays. Here is a useful characterization for the arrays that are Riordan. The following result seems to be due to Rogers [4]
Theorem. An infinite lower triangular array $D=(d_{n,k})_{n,k\geq 0}$ is a Riordan array if and only if there exist a sequence traditionally called the $A$-sequence, $A=(a_{0}\neq 0,a_{1},...)$ such that
$*_{1}:d_{n+1,k+1}=a_{0}d_{n,k}+a_{1}d_{n,k+1}+a_{2}d_{n,k+2}+\cdots =\sum _{j\geq 0}a_{j}d_{n,k+j}$
Proof.[5] $\Rightarrow :$ :} Let $D$ be the Riordan array stemming from $(d(t),h(t)).$ Since $d(t)\in {\mathcal {F}}_{0},$ $d_{0,0}\neq 0.$ Since $h(t)$ has order 1, it follows that $(d(t)h(t)/t,h(t))$ is a Riordan array and by the group property there exists a Riordan array $(A(t),B(t))$ such that $(d(t),h(t))*(A(t),B(t))=(d(t)h(t)/t,h(t)).$ Computing the left hand side yields $(d(t)A(h(t)),B(h(t))$ and so comparison yields $B(h(t))=h(t).$ Of course $B(t)=t$ is a solution to this equation; it is unique because $B$ is composition invertible. So we can rewrite the equation as $(d(t),h(t))*(A(t),t)=(d(t)h(t)/t,h(t)).$
Now from the matrix multiplication law, the $n,k$-entry of the left hand side of this latter equation is
$\sum _{j\geq 0}d_{n,j}(A(t),t)_{j,k}=\sum \limits _{j\geq 0}d_{n,j}[t^{j}]A(t)t^{k}=\sum \limits _{j\geq 0}d_{n,j}[t^{j-k}]A(t)=\sum \limits _{j\geq 0}d_{n,j}a_{j-k}=\sum \limits _{j\geq 0}a_{j}d_{n,k+j}.$
At the other hand the $n,k$-entry of the rhs of the equation above is
$t^{[n]}{\frac {1}{t}}d(t)h(t)h(t)^{k}=t^{[n+1]}d(t)h(t)^{k+1}=d_{n+1,k+1},$
so that i results. From $*_{1}$ we also get $d_{n+1,n+1}=a_{0}d_{n,n}$ for all $n\geq 0$ and since we know that the diagonal elements are nonzero, we have $a_{0}\neq 0.$ Note that using equation $*_{1}$ one can compute all entries knowing the entries $(d_{n,0})_{n\geq 0}.$
$\Leftarrow :$ :} Now assume we know of a triangular array the equations $*_{1}$ for some sequence $(a_{j})_{j\geq 0}.$ Let $A(t)$ be the generating function of that sequence and define $h(t)$ from the equation $tA(h(t))=h(t).$ Check that it is possible to solve the resulting equations for the coefficients of $h;$ and since $a_{0}\neq 0$ one gets that $h(t)$ has order 1. Let $d(t)$ be the generating function of the sequence $(d_{0,0},d_{1,0},d_{2,0},...).$ Then for the pair $(d(t),h(t))$ we find $(d(t),h(t))*(A(t),t)=(d(t)A(h(t)),h(t))=(d(t)h(t)/t,h(t)).$ This is precisely the same equations we have found in the first part of the proof and going through its reasoning we find equations like in $*_{1}$. Since $d(t)$ (or the sequence of its coefficients) determines the other entries we find that the array we started with is the array we deduced. So the array in $*_{1}$ is a Riordan array. $\Box $
Clearly the $A$-sequence alone does not deliver all the information about a Riordan array. Besides the $A$-sequence it is the $Z$-sequence that has shown to be surprisingly useful.[6]
Theorem. Let $(d_{n,k})_{n,k\geq 0}$ be an infinite lower triangular array whose diagonal sequence $(d_{n,n})_{n\geq 0}$ does not contain zeros. Then there exists a unique sequence $Z=(z_{0},z_{1},z_{2},...)$ such that
$d_{n+1,0}=z_{0}d_{n,0}+z_{1}d_{n,1}+z_{2}d_{n,2}+\cdots =\sum \limits _{j\geq 0}z_{j}d_{n,j},$ \quad $n=0,1,2,3,...$
Proof. The proof is simple: By triangularity of the array, the equation claimed is equivalent to $d_{n+1,0}=\sum _{j=0}^{n}z_{j}d_{n,j}.$ For $n=0,$ this equation is $d_{1,0}=z_{0}d_{0,0}$ and, as $d_{0,0}\neq 0,$ it allows computing $z_{0}$ uniquely. In general if $z_{0},z_{1},...,z_{n-1}$ are known already then $d_{n+1,0}-\sum _{j=0}^{n-1}z_{j}d_{n,j}=z_{n}d_{n,n}$ allows to compute $z_{n}$ uniquely. $\Box $
Very recently there appeared the book[7] which should be a valuable source for further information.
References
1. Shapiro, Louis W.; Getu, Seyoum; Woan, Wen-Jin; Woodson, Leon C. (November 1991). "The Riordan group". Discrete Applied Mathematics. 34 (1?3): 229?239. doi:10.1016/0166-218X(91)90088-E.
2. "6th International Conference on Riordan Arrays and Related Topics". 6th International Conference on Riordan Arrays and Related Topics.
3. Roman, S. (1984). The Umbral Calculus. New York: Academic Press.
4. Rogers, D. G. (1978). "Pascal triangles, Catalan numbers, and renewal arrays". Discrete Math. 22: 301–310.
5. He, T.X.; Sprugnoli, R. (2009). "Sequence characterization of Riordan Arrays". Discrete Mathematics. 309: 3962–3974.
6. Merlini, D.; Rogers, D.G.; Sprugnoli, R.; Verri, M.C., M.C. (1997). "On Some Alternative Characterizations of Riordan Arrays". Can. J. Math. 49 (2): 301–320.
7. Shapiro, L.; Sprugnoli, R; Barry, P.; Cheon, G.S.; He, T.X.; Merlini, D.; Wang, W. (2022). The Riordan Group and Applications. Springer. ISBN 978-3-030-94150-5.
|
Wikipedia
|
Rips machine
In geometric group theory, the Rips machine is a method of studying the action of groups on R-trees. It was introduced in unpublished work of Eliyahu Rips in about 1991.
An R-tree is a uniquely arcwise-connected metric space in which every arc is isometric to some real interval. Rips proved the conjecture of Morgan and Shalen[1] that any finitely generated group acting freely on an R-tree is a free product of free abelian and surface groups.[2]
Actions of surface groups on R-trees
By Bass–Serre theory, a group acting freely on a simplicial tree is free. This is no longer true for R-trees, as Morgan and Shalen showed that the fundamental groups of surfaces of Euler characteristic less than −1 also act freely on a R-trees.[1] They proved that the fundamental group of a connected closed surface S acts freely on an R-tree if and only if S is not one of the 3 nonorientable surfaces of Euler characteristic ≥−1.
Applications
The Rips machine assigns to a stable isometric action of a finitely generated group G a certain "normal form" approximation of that action by a stable action of G on a simplicial tree and hence a splitting of G in the sense of Bass–Serre theory. Group actions on real trees arise naturally in several contexts in geometric topology: for example as boundary points of the Teichmüller space[3] (every point in the Thurston boundary of the Teichmüller space is represented by a measured geodesic lamination on the surface; this lamination lifts to the universal cover of the surface and a naturally dual object to that lift is an $\mathbb {R} $-tree endowed with an isometric action of the fundamental group of the surface), as Gromov-Hausdorff limits of, appropriately rescaled, Kleinian group actions,[4][5] and so on. The use of $\mathbb {R} $-trees machinery provides substantial shortcuts in modern proofs of Thurston's Hyperbolization Theorem for Haken 3-manifolds.[5][6] Similarly, $\mathbb {R} $-trees play a key role in the study of Culler-Vogtmann's Outer space[7][8] as well as in other areas of geometric group theory; for example, asymptotic cones of groups often have a tree-like structure and give rise to group actions on real trees.[9][10] The use of $\mathbb {R} $-trees, together with Bass–Serre theory, is a key tool in the work of Sela on solving the isomorphism problem for (torsion-free) word-hyperbolic groups, Sela's version of the JSJ-decomposition theory and the work of Sela on the Tarski Conjecture for free groups and the theory of limit groups.[11][12]
References
1. Morgan, John W.; Shalen, Peter B. (1991), "Free actions of surface groups on R-trees", Topology, 30 (2): 143–154, doi:10.1016/0040-9383(91)90002-L, ISSN 0040-9383, MR 1098910
2. Bestvina, Mladen; Feighn, Mark (1995), "Stable actions of groups on real trees", Inventiones Mathematicae, 121 (2): 287–321, doi:10.1007/BF01884300, ISSN 0020-9910, MR 1346208, S2CID 122048815
3. Skora, Richard (1990), "Splittings of surfaces", Bulletin of the American Mathematical Society, New Series, 23 (1): 85–90, doi:10.1090/S0273-0979-1990-15907-5
4. Bestvina, Mladen (1988), "Degenerations of the hyperbolic space", Duke Mathematical Journal, 56 (1): 143–161, doi:10.1215/S0012-7094-88-05607-4
5. Kapovich, Michael (2001), Hyperbolic manifolds and discrete groups, Progress in Mathematics, vol. 183, Birkhäuser, Boston, MA, doi:10.1007/978-0-8176-4913-5, ISBN 0-8176-3904-7
6. Otal, Jean-Pierre (2001), The hyperbolization theorem for fibered 3-manifolds, SMF/AMS Texts and Monographs, vol. 7, American Mathematical Society, Providence, RI and Société Mathématique de France, Paris, ISBN 0-8218-2153-9
7. Cohen, Marshall; Lustig, Martin (1995), "Very small group actions on $\mathbb {R} $-trees and Dehn twist automorphisms", Topology, 34 (3): 575–617, doi:10.1016/0040-9383(94)00038-M
8. Levitt, Gilbert; Lustig, Martin (2003), "Irreducible automorphisms of Fn have north-south dynamics on compactified outer space", Journal de l'Institut de Mathématiques de Jussieu, 2 (1): 59–72, doi:10.1017/S1474748003000033, S2CID 120675231
9. Druţu, Cornelia; Sapir, Mark (2005), "Tree-graded spaces and asymptotic cones of groups (With an appendix by Denis Osin and Mark Sapir.)", Topology, 44 (5): 959–1058, doi:10.1016/j.top.2005.03.003
10. Druţu, Cornelia; Sapir, Mark (2008), "Groups acting on tree-graded spaces and splittings of relatively hyperbolic groups", Advances in Mathematics, 217 (3): 1313–1367, doi:10.1016/j.aim.2007.08.012
11. Sela, Zlil (2002), "Diophantine geometry over groups and the elementary theory of free and hyperbolic groups", Proceedings of the International Congress of Mathematicians, vol. II, Beijing: Higher Education Press, Beijing, pp. 87–92, ISBN 7-04-008690-5
12. Sela, Zlil (2001), "Diophantine geometry over groups. I. Makanin-Razborov diagrams", Publications Mathématiques de l'Institut des Hautes Études Scientifiques, 93: 31–105, doi:10.1007/s10240-001-8188-y
Further reading
• Gaboriau, D.; Levitt, G.; Paulin, F. (1994), "Pseudogroups of isometries of R and Rips' theorem on free actions on R-trees", Israel Journal of Mathematics, 87 (1): 403–428, doi:10.1007/BF02773004, ISSN 0021-2172, MR 1286836, S2CID 122353183
• Kapovich, Michael (2009) [2001], Hyperbolic manifolds and discrete groups, Modern Birkhäuser Classics, Boston, MA: Birkhäuser Boston, doi:10.1007/978-0-8176-4913-5, ISBN 978-0-8176-4912-8, MR 1792613
• Shalen, Peter B. (1987), "Dendrology of groups: an introduction", in Gersten, S. M. (ed.), Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Berlin, New York: Springer-Verlag, pp. 265–319, ISBN 978-0-387-96618-2, MR 0919830
|
Wikipedia
|
Risch algorithm
In symbolic computation, the Risch algorithm is a method of indefinite integration used in some computer algebra systems to find antiderivatives. It is named after the American mathematician Robert Henry Risch, a specialist in computer algebra who developed it in 1968.
Part of a series of articles about
Calculus
• Fundamental theorem
• Limits
• Continuity
• Rolle's theorem
• Mean value theorem
• Inverse function theorem
Differential
Definitions
• Derivative (generalizations)
• Differential
• infinitesimal
• of a function
• total
Concepts
• Differentiation notation
• Second derivative
• Implicit differentiation
• Logarithmic differentiation
• Related rates
• Taylor's theorem
Rules and identities
• Sum
• Product
• Chain
• Power
• Quotient
• L'Hôpital's rule
• Inverse
• General Leibniz
• Faà di Bruno's formula
• Reynolds
Integral
• Lists of integrals
• Integral transform
• Leibniz integral rule
Definitions
• Antiderivative
• Integral (improper)
• Riemann integral
• Lebesgue integration
• Contour integration
• Integral of inverse functions
Integration by
• Parts
• Discs
• Cylindrical shells
• Substitution (trigonometric, tangent half-angle, Euler)
• Euler's formula
• Partial fractions
• Changing order
• Reduction formulae
• Differentiating under the integral sign
• Risch algorithm
Series
• Geometric (arithmetico-geometric)
• Harmonic
• Alternating
• Power
• Binomial
• Taylor
Convergence tests
• Summand limit (term test)
• Ratio
• Root
• Integral
• Direct comparison
• Limit comparison
• Alternating series
• Cauchy condensation
• Dirichlet
• Abel
Vector
• Gradient
• Divergence
• Curl
• Laplacian
• Directional derivative
• Identities
Theorems
• Gradient
• Green's
• Stokes'
• Divergence
• generalized Stokes
Multivariable
Formalisms
• Matrix
• Tensor
• Exterior
• Geometric
Definitions
• Partial derivative
• Multiple integral
• Line integral
• Surface integral
• Volume integral
• Jacobian
• Hessian
Advanced
• Calculus on Euclidean space
• Generalized functions
• Limit of distributions
Specialized
• Fractional
• Malliavin
• Stochastic
• Variations
Miscellaneous
• Precalculus
• History
• Glossary
• List of topics
• Integration Bee
• Mathematical analysis
• Nonstandard analysis
The algorithm transforms the problem of integration into a problem in algebra. It is based on the form of the function being integrated and on methods for integrating rational functions, radicals, logarithms, and exponential functions. Risch called it a decision procedure, because it is a method for deciding whether a function has an elementary function as an indefinite integral, and if it does, for determining that indefinite integral. However, the algorithm does not always succeed in identifying whether or not the antiderivative of a given function in fact can be expressed in terms of elementary functions.
The complete description of the Risch algorithm takes over 100 pages.[1] The Risch–Norman algorithm is a simpler, faster, but less powerful variant that was developed in 1976 by Arthur Norman.
Some significant progress has been made in computing the logarithmic part of a mixed transcendental-algebraic integral by Brian L. Miller.[2]
Description
The Risch algorithm is used to integrate elementary functions. These are functions obtained by composing exponentials, logarithms, radicals, trigonometric functions, and the four arithmetic operations (+ − × ÷). Laplace solved this problem for the case of rational functions, as he showed that the indefinite integral of a rational function is a rational function and a finite number of constant multiples of logarithms of rational functions . The algorithm suggested by Laplace is usually described in calculus textbooks; as a computer program, it was finally implemented in the 1960s.
Liouville formulated the problem that is solved by the Risch algorithm. Liouville proved by analytical means that if there is an elementary solution g to the equation g′ = f then there exist constants αi and functions ui and v in the field generated by f such that the solution is of the form
$g=v+\sum _{i<n}\alpha _{i}\ln(u_{i})$
Risch developed a method that allows one to consider only a finite set of functions of Liouville's form.
The intuition for the Risch algorithm comes from the behavior of the exponential and logarithm functions under differentiation. For the function f eg, where f and g are differentiable functions, we have
$\left(f\cdot e^{g}\right)^{\prime }=\left(f^{\prime }+f\cdot g^{\prime }\right)\cdot e^{g},\,$
so if eg were in the result of an indefinite integration, it should be expected to be inside the integral. Also, as
$\left(f\cdot (\ln g)^{n}\right)^{\prime }=f^{\prime }\left(\ln g\right)^{n}+nf{\frac {g^{\prime }}{g}}\left(\ln g\right)^{n-1}$
then if (ln g)n were in the result of an integration, then only a few powers of the logarithm should be expected.
Problem examples
Finding an elementary antiderivative is very sensitive to details. For instance, the following algebraic function (posted to sci.math.symbolic by Henri Cohen in 1993[3]) has an elementary antiderivative, as Wolfram Mathematica since version 13 shows (however, Mathematica does not use the Risch algorithm to compute this integral):[4][5]
$f(x)={\frac {x}{\sqrt {x^{4}+10x^{2}-96x-71}}},$
namely:
${\begin{aligned}F(x)=-{\frac {1}{8}}\ln &\,{\Big (}(x^{6}+15x^{4}-80x^{3}+27x^{2}-528x+781){\sqrt {x^{4}+10x^{2}-96x-71}}{\Big .}\\&{}-{\Big .}(x^{8}+20x^{6}-128x^{5}+54x^{4}-1408x^{3}+3124x^{2}+10001){\Big )}+C.\end{aligned}}$
But if the constant term 71 is changed to 72, it is not possible to represent the antiderivative in terms of elementary functions,[6] as FriCAS also shows. Some computer algebra systems may here return an antiderivative in terms of non-elementary functions (i.e. elliptic integrals), which are outside the scope of the Risch algorithm. This integral was solved by Chebyshev (and in what cases it is elementary),[7] but the strict proof for it was ultimately done by Zolotarev.[6]
The following is a more complex example that involves both algebraic and transcendental functions:[8]
$f(x)={\frac {x^{2}+2x+1+(3x+1){\sqrt {x+\ln x}}}{x\,{\sqrt {x+\ln x}}\left(x+{\sqrt {x+\ln x}}\right)}}.$
In fact, the antiderivative of this function has a fairly short form that can be found using substitution $u=x+{\sqrt {x+\ln x}}$ (SymPy can solve it while FriCAS fails with "implementation incomplete (constant residues)" error in Risch algorithm):
$F(x)=2\left({\sqrt {x+\ln x}}+\ln \left(x+{\sqrt {x+\ln x}}\right)\right)+C.$
Some Davenport "theorems" are still being clarified. For example in 2020 a counterexample to such a "theorem" was found, where it turns out that an elementary antiderivative exists after all.[9]
Implementation
Transforming Risch's theoretical algorithm into an algorithm that can be effectively executed by a computer was a complex task which took a long time.
The case of the purely transcendental functions (which do not involve roots of polynomials) is relatively easy and was implemented early in most computer algebra systems. The first implementation was done by Joel Moses in Macsyma soon after the publication of Risch's paper.[10]
The case of purely algebraic functions was solved and implemented in Reduce by James H. Davenport, though for simplicity it could only deal with square roots and repeated square roots and not general Radicals or other non-quadratic algebraic relations between variables.[11]
The general case was solved and almost fully implemented in Scratchpad, a precursor of Axiom, by Manuel Bronstein, and is now being developed in Axiom's fork, FriCAS.[12] However, the implementation did not include some of the branches for special cases completely.[13] Currently, there is no known full implementation of the Risch algorithm.[14]
Decidability
The Risch algorithm applied to general elementary functions is not an algorithm but a semi-algorithm because it needs to check, as a part of its operation, if certain expressions are equivalent to zero (constant problem), in particular in the constant field. For expressions that involve only functions commonly taken to be elementary it is not known whether an algorithm performing such a check exists or not (current computer algebra systems use heuristics); moreover, if one adds the absolute value function to the list of elementary functions, it is known that no such algorithm exists; see Richardson's theorem.
Note that this issue also arises in the polynomial division algorithm; this algorithm will fail if it cannot correctly determine whether coefficients vanish identically.[15] Virtually every non-trivial algorithm relating to polynomials uses the polynomial division algorithm, the Risch algorithm included. If the constant field is computable, i.e., for elements not dependent on x, the problem of zero-equivalence is decidable, then the Risch algorithm is a complete algorithm. Examples of computable constant fields are Q and Q(y), i.e., rational numbers and rational functions in y with rational number coefficients, respectively, where y is an indeterminate that does not depend on x.
This is also an issue in the Gaussian elimination matrix algorithm (or any algorithm that can compute the nullspace of a matrix), which is also necessary for many parts of the Risch algorithm. Gaussian elimination will produce incorrect results if it cannot correctly determine if a pivot is identically zero.
See also
• Axiom (computer algebra system)
• Closed-form expression
• Incomplete gamma function
• Lists of integrals
• Liouville's theorem (differential algebra)
• Nonelementary integral
• Symbolic integration
Notes
1. Geddes, Czapor & Labahn 1992.
2. Miller, Brian L. (May 2012). "On the integration of elementary functions: Computing the logarithmic part". {{cite journal}}: Cite journal requires |journal= (help)
3. Cohen, Henri (December 21, 1993). "A Christmas present for your favorite CAS".{{cite web}}: CS1 maint: url-status (link)
4. "Wolfram Cloud". Wolfram Cloud. Retrieved December 11, 2021.
5. This example was posted by Manuel Bronstein to the Usenet forum comp.soft-sys.math.maple on November 24, 2000.
6. Zolotareff, G. (December 1, 1872). "Sur la méthode d'intégration de M. Tchébychef". Mathematische Annalen (in French). 5 (4): 560–580. doi:10.1007/BF01442910. ISSN 1432-1807. S2CID 123629827.
7. Chebyshev, P. L. (1899–1907). Oeuvres de P.L. Tchebychef (in French). University of California Berkeley. St. Petersbourg, Commissionaires de l'Academie imperiale des sciences.
8. Bronstein 1998.
9. Masser, David; Zannier, Umberto (December 2020). "Torsion points, Pell's equation, and integration in elementary terms". Acta Mathematica. 225 (2): 227–312. doi:10.4310/ACTA.2020.v225.n2.a2. ISSN 1871-2509. S2CID 221405883.
10. Moses 2012.
11. Davenport 1981.
12. Bronstein 1990.
13. Bronstein, Manuel (September 5, 2003). "Manuel Bronstein on Axiom's Integration Capabilities". groups.google.com. Retrieved February 10, 2023.
14. "integration - Does there exist a complete implementation of the Risch algorithm?". MathOverflow. October 15, 2020. Retrieved February 10, 2023.
15. "Mathematica 7 Documentation: PolynomialQuotient". Section: Possible Issues. Retrieved July 17, 2010.
References
• Bronstein, Manuel (1990). "Integration of elementary functions". Journal of Symbolic Computation. 9 (2): 117–173. doi:10.1016/s0747-7171(08)80027-2.
• Bronstein, Manuel (1998). "Symbolic Integration Tutorial" (PDF). {{cite journal}}: Cite journal requires |journal= (help)
• Bronstein, Manuel (2005). Symbolic Integration I. Springer. ISBN 3-540-21493-3.
• Davenport, James H. (1981). On the integration of algebraic functions. Lecture Notes in Computer Science. Vol. 102. Springer. ISBN 978-3-540-10290-8.
• Geddes, Keith O.; Czapor, Stephen R.; Labahn, George (1992). Algorithms for computer algebra. Boston, MA: Kluwer Academic Publishers. pp. xxii+585. Bibcode:1992afca.book.....G. doi:10.1007/b102438. ISBN 0-7923-9259-0.
• Moses, Joel (2012). "Macsyma: A personal history". Journal of Symbolic Computation. 47 (2): 123–130. doi:10.1016/j.jsc.2010.08.018.
• Risch, R. H. (1969). "The problem of integration in finite terms". Transactions of the American Mathematical Society. American Mathematical Society. 139: 167–189. doi:10.2307/1995313. JSTOR 1995313.
• Risch, R. H. (1970). "The solution of the problem of integration in finite terms". Bulletin of the American Mathematical Society. 76 (3): 605–608. doi:10.1090/S0002-9904-1970-12454-5.
• Rosenlicht, Maxwell (1972). "Integration in finite terms". American Mathematical Monthly. Mathematical Association of America. 79 (9): 963–972. doi:10.2307/2318066. JSTOR 2318066.
External links
• Bhatt, Bhuvanesh. "Risch Algorithm". MathWorld.
Integrals
Types of integrals
• Riemann integral
• Lebesgue integral
• Burkill integral
• Bochner integral
• Daniell integral
• Darboux integral
• Henstock–Kurzweil integral
• Haar integral
• Hellinger integral
• Khinchin integral
• Kolmogorov integral
• Lebesgue–Stieltjes integral
• Pettis integral
• Pfeffer integral
• Riemann–Stieltjes integral
• Regulated integral
Integration techniques
• Substitution
• Trigonometric
• Euler
• Weierstrass
• By parts
• Partial fractions
• Euler's formula
• Inverse functions
• Changing order
• Reduction formulas
• Parametric derivatives
• Differentiation under the integral sign
• Laplace transform
• Contour integration
• Laplace's method
• Numerical integration
• Simpson's rule
• Trapezoidal rule
• Risch algorithm
Improper integrals
• Gaussian integral
• Dirichlet integral
• Fermi–Dirac integral
• complete
• incomplete
• Bose–Einstein integral
• Frullani integral
• Common integrals in quantum field theory
Stochastic integrals
• Itô integral
• Russo–Vallois integral
• Stratonovich integral
• Skorokhod integral
Miscellaneous
• Basel problem
• Euler–Maclaurin formula
• Gabriel's horn
• Integration Bee
• Proof that 22/7 exceeds π
• Volumes
• Washers
• Shells
|
Wikipedia
|
Rising sun lemma
In mathematical analysis, the rising sun lemma is a lemma due to Frigyes Riesz, used in the proof of the Hardy–Littlewood maximal theorem. The lemma was a precursor in one dimension of the Calderón–Zygmund lemma.[1]
The lemma is stated as follows:[2]
Suppose g is a real-valued continuous function on the interval [a,b] and S is the set of x in [a,b] such that there exists a y∈(x,b] with g(y) > g(x). (Note that b cannot be in S, though a may be.) Define E = S ∩ (a,b).
Then E is an open set, and it may be written as a countable union of disjoint intervals
$E=\bigcup _{k}(a_{k},b_{k})$
such that g(ak) = g(bk), unless ak = a ∈ S for some k, in which case g(a) < g(bk) for that one k. Furthermore, if x ∈ (ak,bk), then g(x) < g(bk).
The colorful name of the lemma comes from imagining the graph of the function g as a mountainous landscape, with the sun shining horizontally from the right. The set E consist of points that are in the shadow.
Proof
We need a lemma: Suppose [c,d) ⊂ S, but d ∉ S. Then g(c) < g(d). To prove this, suppose g(c) ≥ g(d). Then g achieves its maximum on [c,d] at some point z < d. Since z ∈ S, there is a y in (z,b] with g(z) < g(y). If y ≤ d, then g would not reach its maximum on [c,d] at z. Thus, y ∈ (d,b], and g(d) ≤ g(z) < g(y). This means that d ∈ S, which is a contradiction, thus establishing the lemma.
The set E is open, so it is composed of a countable union of disjoint intervals (ak,bk).
It follows immediately from the lemma that g(x) < g(bk) for x in (ak,bk). Since g is continuous, we must also have g(ak) ≤ g(bk).
If ak ≠ a or a ∉ S, then ak ∉ S, so g(ak) ≥ g(bk), for otherwise ak ∈ S. Thus, g(ak) = g(bk) in these cases.
Finally, if ak = a ∈ S, the lemma tells us that g(a) < g(bk).
Notes
1. Stein 1998
2. See:
• Riesz 1932
• Zygmund 1977, p. 31
• Tao 2011, pp. 118–119
• Duren 2000, Appendix B
References
• Duren, Peter L. (2000), Theory of Hp Spaces, New York: Dover Publications, ISBN 0-486-41184-2
• Garling, D.J.H. (2007), Inequalities: a journey into linear analysis, Cambridge University Press, ISBN 978-0-521-69973-0
• Korenovskyy, A. A.; A. K. Lerner; A. M. Stokolos (November 2004), "On a multidimensional form of F. Riesz's "rising sun" lemma", Proceedings of the American Mathematical Society, 133 (5): 1437–1440, doi:10.1090/S0002-9939-04-07653-1
• Riesz, Frédéric (1932), "Sur un Théorème de Maximum de Mm. Hardy et Littlewood", Journal of the London Mathematical Society, 7 (1): 10–13, doi:10.1112/jlms/s1-7.1.10, archived from the original on 2013-04-15, retrieved 2008-07-21
• Stein, Elias (1998), "Singular integrals: The Roles of Calderón and Zygmund" (PDF), Notices of the American Mathematical Society, 45 (9): 1130–1140.
• Tao, Terence (2011), An Introduction to Measure Theory, Graduate Studies in Mathematics, vol. 126, American Mathematical Society, ISBN 978-0821869192
• Zygmund, Antoni (1977), Trigonometric Series. Vol. I, II (2nd ed.), Cambridge University Press, ISBN 0-521-07477-0
|
Wikipedia
|
Risk difference
The risk difference (RD), excess risk, or attributable risk is the difference between the risk of an outcome in the exposed group and the unexposed group. It is computed as $I_{e}-I_{u}$, where $I_{e}$is the incidence in the exposed group, and $I_{u}$ is the incidence in the unexposed group. If the risk of an outcome is increased by the exposure, the term absolute risk increase (ARI) is used, and computed as $I_{e}-I_{u}$. Equivalently, if the risk of an outcome is decreased by the exposure, the term absolute risk reduction (ARR) is used, and computed as $I_{u}-I_{e}$.[1][2]
The inverse of the absolute risk reduction is the number needed to treat, and the inverse of the absolute risk increase is the number needed to harm.[1]
Usage in reporting
It is recommended to use absolute measurements, such as risk difference, alongside the relative measurements, when presenting the results of randomized controlled trials.[3] Their utility can be illustrated by the following example of a hypothetical drug which reduces the risk of colon cancer from 1 case in 5000 to 1 case in 10,000 over one year. The relative risk reduction is 0.5 (50%), while the absolute risk reduction is 0.0001 (0.01%). The absolute risk reduction reflects the low probability of getting colon cancer in the first place, while reporting only relative risk reduction, would run into risk of readers exaggerating the effectiveness of the drug.[4]
Authors such as Ben Goldacre believe that the risk difference is best presented as a natural number - drug reduces 2 cases of colon cancer to 1 case if you treat 10,000 people. Natural numbers, which are used in the number needed to treat approach, are easily understood by non-experts.[5]
Inference
Risk difference can be estimated from a 2x2 contingency table:
Group
Experimental (E) Control (C)
Events (E) EE CE
Non-events (N) EN CN
The point estimate of the risk difference is
$RD={\frac {EE}{EE+EN}}-{\frac {CE}{CE+CN}}.$
The sampling distribution of RD is approximately normal, with standard error
$SE(RD)={\sqrt {{\frac {EE\cdot EN}{(EE+EN)^{3}}}+{\frac {CE\cdot CN}{(CE+CN)^{3}}}}}.$
The $1-\alpha $ confidence interval for the RD is then
$CI_{1-\alpha }(RD)=RD\pm SE(RD)\cdot z_{\alpha },$
where $z_{\alpha }$ is the standard score for the chosen level of significance[2]
Numerical examples
Risk reduction
Example of risk reduction
Quantity Experimental group (E) Control group (C) Total
Events (E) EE = 15 CE = 100 115
Non-events (N) EN = 135 CN = 150 285
Total subjects (S) ES = EE + EN = 150 CS = CE + CN = 250 400
Event rate (ER) EER = EE / ES = 0.1, or 10% CER = CE / CS = 0.4, or 40% —
Variable Abbr. Formula Value
Absolute risk reduction ARR CER − EER 0.3, or 30%
Number needed to treat NNT 1 / (CER − EER) 3.33
Relative risk (risk ratio) RR EER / CER 0.25
Relative risk reduction RRR (CER − EER) / CER, or 1 − RR 0.75, or 75%
Preventable fraction among the unexposed PFu (CER − EER) / CER 0.75
Odds ratio OR (EE / EN) / (CE / CN) 0.167
Risk increase
Example of risk increase
Quantity Experimental group (E) Control group (C) Total
Events (E) EE = 75 CE = 100 175
Non-events (N) EN = 75 CN = 150 225
Total subjects (S) ES = EE + EN = 150 CS = CE + CN = 250 400
Event rate (ER) EER = EE / ES = 0.5, or 50% CER = CE / CS = 0.4, or 40% —
Variable Abbr. Formula Value
Absolute risk increase ARI EER − CER 0.1, or 10%
Number needed to harm NNH 1 / (EER − CER) 10
Relative risk (risk ratio) RR EER / CER 1.25
Relative risk increase RRI (EER − CER) / CER, or RR − 1 0.25, or 25%
Attributable fraction among the exposed AFe (EER − CER) / EER 0.2
Odds ratio OR (EE / EN) / (CE / CN) 1.5
See also
• Population Impact Measures
• Relative risk reduction
References
1. Porta, Miquel, ed. (2014). "Dictionary of Epidemiology - Oxford Reference". Oxford University Press. doi:10.1093/acref/9780199976720.001.0001. ISBN 9780199976720. Retrieved 2018-05-09.
2. J., Rothman, Kenneth (2012). Epidemiology : an introduction (2nd ed.). New York, NY: Oxford University Press. pp. 66, 160, 167. ISBN 9780199754557. OCLC 750986180.{{cite book}}: CS1 maint: multiple names: authors list (link)
3. Moher D, Hopewell S, Schulz KF, Montori V, Gøtzsche PC, Devereaux PJ, Elbourne D, Egger M, Altman DG (March 2010). "CONSORT 2010 explanation and elaboration: updated guidelines for reporting parallel group randomised trials". BMJ. 340: c869. doi:10.1136/bmj.c869. PMC 2844943. PMID 20332511.
4. Stegenga, Jacob (2015). "Measuring Effectiveness". Studies in History and Philosophy of Biological and Biomedical Sciences. 54: 62–71. doi:10.1016/j.shpsc.2015.06.003. PMID 26199055.
5. Ben Goldacre (2008). Bad Science. New York: Fourth Estate. pp. 239–260. ISBN 978-0-00-724019-7.
Clinical research and experimental design
Overview
• Clinical trial
• Trial protocols
• Adaptive clinical trial
• Academic clinical trials
• Clinical study design
Controlled study
(EBM I to II-1)
• Randomized controlled trial
• Scientific experiment
• Blind experiment
• Open-label trial
• Adaptive clinical trial
• Platform trial
Observational study
(EBM II-2 to II-3)
• Cross-sectional study vs. Longitudinal study, Ecological study
• Cohort study
• Retrospective
• Prospective
• Case–control study (Nested case–control study)
• Case series
• Case study
• Case report
Measures
Occurrence
Incidence, Cumulative incidence, Prevalence, Point prevalence, Period prevalence
Association
Risk difference, Number needed to treat, Number needed to harm, Risk ratio, Relative risk reduction, Odds ratio, Hazard ratio
Population impact
Attributable fraction among the exposed, Attributable fraction for the population, Preventable fraction among the unexposed, Preventable fraction for the population
Other
Clinical endpoint, Virulence, Infectivity, Mortality rate, Morbidity, Case fatality rate, Specificity and sensitivity, Likelihood-ratios, Pre- and post-test probability
Trial/test types
• In vitro
• In vivo
• Animal testing
• Animal testing on non-human primates
• First-in-man study
• Multicenter trial
• Seeding trial
• Vaccine trial
Analysis of clinical trials
• Risk–benefit ratio
• Systematic review
• Replication
• Meta-analysis
• Intention-to-treat analysis
Interpretation of results
• Selection bias
• Survivorship bias
• Correlation does not imply causation
• Null result
• Sex as a biological variable
• Category
• Glossary
• List of topics
|
Wikipedia
|
Risk inclination formula
The risk inclination formula uses the principle of moments, or Varignon's theorem,[1][2] to calculate the first factorial moment of probability in order to define this center point of balance among all confidence weights (i.e., the point of risk equilibration).
The formal derivation of the RIF is divided into three separate calculations: (1) calculation of 1st factorial moment, (2) calculation of inclination, and (3) calculation of the risk inclination score.
The RIF [3] is a component of the risk inclination model.
References
1. Coxeter, H. S. M. (1967). Quadrangles: Varignon's Theorem. Geometry Revisited. Washington, D.C.: The Mathematical Association of America. pp. 51–55.
2. Sharma, D. P. (2010). Engineering Mechanics. New Delhi, India: IDK Kindersley. pp. 8–9.
3. Jack, B.M.; Hung, K.M.; Liu, C.J.; Chiu, H.L. Utilitarian Model of Confidence Testing for Knowledge-based Societies. ERIC. ED519174.,
|
Wikipedia
|
Risk measure
In financial mathematics, a risk measure is used to determine the amount of an asset or set of assets (traditionally currency) to be kept in reserve. The purpose of this reserve is to make the risks taken by financial institutions, such as banks and insurance companies, acceptable to the regulator. In recent years attention has turned towards convex and coherent risk measurement.
Not to be confused with deviation risk measures, e.g. standard deviation.
Mathematically
A risk measure is defined as a mapping from a set of random variables to the real numbers. This set of random variables represents portfolio returns. The common notation for a risk measure associated with a random variable $X$ is $\rho (X)$. A risk measure $\rho :{\mathcal {L}}\to \mathbb {R} \cup \{+\infty \}$ :{\mathcal {L}}\to \mathbb {R} \cup \{+\infty \}} should have certain properties:[1]
Normalized
$\rho (0)=0$
Translative
$\mathrm {If} \;a\in \mathbb {R} \;\mathrm {and} \;Z\in {\mathcal {L}},\;\mathrm {then} \;\rho (Z+a)=\rho (Z)-a$
Monotone
$\mathrm {If} \;Z_{1},Z_{2}\in {\mathcal {L}}\;\mathrm {and} \;Z_{1}\leq Z_{2},\;\mathrm {then} \;\rho (Z_{2})\leq \rho (Z_{1})$
Set-valued
In a situation with $\mathbb {R} ^{d}$-valued portfolios such that risk can be measured in $m\leq d$ of the assets, then a set of portfolios is the proper way to depict risk. Set-valued risk measures are useful for markets with transaction costs.[2]
Mathematically
A set-valued risk measure is a function $R:L_{d}^{p}\rightarrow \mathbb {F} _{M}$, where $L_{d}^{p}$ is a $d$-dimensional Lp space, $\mathbb {F} _{M}=\{D\subseteq M:D=cl(D+K_{M})\}$, and $K_{M}=K\cap M$ where $K$ is a constant solvency cone and $M$ is the set of portfolios of the $m$ reference assets. $R$ must have the following properties:[3]
Normalized
$K_{M}\subseteq R(0)\;\mathrm {and} \;R(0)\cap -\mathrm {int} K_{M}=\emptyset $
Translative in M
$\forall X\in L_{d}^{p},\forall u\in M:R(X+u1)=R(X)-u$
Monotone
$\forall X_{2}-X_{1}\in L_{d}^{p}(K)\Rightarrow R(X_{2})\supseteq R(X_{1})$
Examples
• Value at risk
• Expected shortfall
• Superposed risk measures[4]
• Entropic value at risk
• Drawdown
• Tail conditional expectation
• Entropic risk measure
• Superhedging price
• Expectile
Variance
Variance (or standard deviation) is not a risk measure in the above sense. This can be seen since it has neither the translation property nor monotonicity. That is, $Var(X+a)=Var(X)\neq Var(X)-a$ for all $a\in \mathbb {R} $, and a simple counterexample for monotonicity can be found. The standard deviation is a deviation risk measure. To avoid any confusion, note that deviation risk measures, such as variance and standard deviation are sometimes called risk measures in different fields.
Relation to acceptance set
There is a one-to-one correspondence between an acceptance set and a corresponding risk measure. As defined below it can be shown that $R_{A_{R}}(X)=R(X)$ and $A_{R_{A}}=A$.[5]
Risk measure to acceptance set
• If $\rho $ is a (scalar) risk measure then $A_{\rho }=\{X\in L^{p}:\rho (X)\leq 0\}$ is an acceptance set.
• If $R$ is a set-valued risk measure then $A_{R}=\{X\in L_{d}^{p}:0\in R(X)\}$ is an acceptance set.
Acceptance set to risk measure
• If $A$ is an acceptance set (in 1-d) then $\rho _{A}(X)=\inf\{u\in \mathbb {R} :X+u1\in A\}$ defines a (scalar) risk measure.
• If $A$ is an acceptance set then $R_{A}(X)=\{u\in M:X+u1\in A\}$ is a set-valued risk measure.
Relation with deviation risk measure
There is a one-to-one relationship between a deviation risk measure D and an expectation-bounded risk measure $\rho $ where for any $X\in {\mathcal {L}}^{2}$
• $D(X)=\rho (X-\mathbb {E} [X])$
• $\rho (X)=D(X)-\mathbb {E} [X]$.
$\rho $ is called expectation bounded if it satisfies $\rho (X)>\mathbb {E} [-X]$ for any nonconstant X and $\rho (X)=\mathbb {E} [-X]$ for any constant X.[6]
See also
• Coherent risk measure
• Conditional value-at-risk
• Distortion risk measure
• Dynamic risk measure
• Entropic value at risk
• Managerial risk accounting
• Risk management
• Risk metric - the abstract concept that a risk measure quantifies
• Risk return ratio
• RiskMetrics - a model for risk management
• Spectral risk measure
• Value at risk
• Worst-case risk measure
References
1. Artzner, Philippe; Delbaen, Freddy; Eber, Jean-Marc; Heath, David (1999). "Coherent Measures of Risk" (PDF). Mathematical Finance. 9 (3): 203–228. doi:10.1111/1467-9965.00068. S2CID 6770585. Retrieved February 3, 2011.
2. Jouini, Elyes; Meddeb, Moncef; Touzi, Nizar (2004). "Vector–valued coherent risk measures". Finance and Stochastics. 8 (4): 531–552. CiteSeerX 10.1.1.721.6338. doi:10.1007/s00780-004-0127-6. S2CID 18237100.
3. Hamel, A. H.; Heyde, F. (2010). "Duality for Set-Valued Measures of Risk". SIAM Journal on Financial Mathematics. 1 (1): 66–95. CiteSeerX 10.1.1.514.8477. doi:10.1137/080743494.
4. Jokhadze, Valeriane; Schmidt, Wolfgang M. (2018). "Measuring model risk in financial risk management and pricing". SSRN. doi:10.2139/ssrn.3113139. S2CID 169594252. {{cite journal}}: Cite journal requires |journal= (help)
5. Andreas H. Hamel; Frank Heyde; Birgit Rudloff (2011). "Set-valued risk measures for conical market models". Mathematics and Financial Economics. 5 (1): 1–28. arXiv:1011.5986. doi:10.1007/s11579-011-0047-0. S2CID 154784949.
6. Rockafellar, Tyrrell; Uryasev, Stanislav; Zabarankin, Michael (2002). "Deviation Measures in Risk Analysis and Optimization" (PDF). Archived from the original (PDF) on September 16, 2011. Retrieved October 13, 2011. {{cite journal}}: Cite journal requires |journal= (help)
Further reading
• Crouhy, Michel; D. Galai; R. Mark (2001). Risk Management. McGraw-Hill. pp. 752 pages. ISBN 978-0-07-135731-9.
• Kevin, Dowd (2005). Measuring Market Risk (2nd ed.). John Wiley & Sons. pp. 410 pages. ISBN 978-0-470-01303-8.
• Foellmer, Hans; Schied, Alexander (2004). Stochastic Finance. de Gruyter Series in Mathematics. Vol. 27. Berlin: Walter de Gruyter. pp. xi+459. ISBN 978-311-0183467. MR 2169807.
• Shapiro, Alexander; Dentcheva, Darinka; Ruszczyński, Andrzej (2009). Lectures on stochastic programming. Modeling and theory. MPS/SIAM Series on Optimization. Vol. 9. Philadelphia: Society for Industrial and Applied Mathematics. pp. xvi+436. ISBN 978-0898716870. MR 2562798.
|
Wikipedia
|
Risk score
Risk score (or risk scoring) is the name given to a general practice in applied statistics, bio-statistics, econometrics and other related disciplines, of creating an easily calculated number (the score) that reflects the level of risk in the presence of some risk factors (e.g. risk of mortality or disease in the presence of symptoms or genetic profile, risk financial loss considering credit and financial history, etc.).
Risk scores are designed to be:
• Simple to calculate: In many cases all you need to calculate a score is a pen and a piece of paper (although some scores use rely on more sophisticated or less transparent calculations that require a computer program).
• Easily interpreted: The result of the calculation is a single number, and higher score usually means higher risk. Furthermore, many scoring methods enforce some form of monotonicity along the measured risk factors to allow a straight forward interpretation of the score (e.g. risk of mortality only increases with age, risk of payment default only increase with the amount of total debt the customer has, etc.).
• Actionable: Scores are designed around a set of possible actions that should be taken as a result of the calculated score. Effective score-based policies can be designed and executed by setting thresholds on the value of the score and associating them with escalating actions.
Formal definition
A typical scoring method is composed of 3 components:[1]
1. A set of consistent rules (or weights) that assign a numerical value ("points") to each risk factor that reflect our estimation of underlying risk.
2. A formula (typically a simple sum of all accumulated points) that calculates the score.
3. A set of thresholds that helps to translate the calculated score into a level of risk, or an equivalent formula or set of rules to translate the calculated score back into probabilities (leaving the nominal evaluation of severity to the practitioner).
Items 1 & 2 can be achieved by using some form of regression, that will provide both the risk estimation and the formula to calculate the score. Item 3 requires setting an arbitrary set of thresholds and will usually involve expert opinion.
Estimating risk with GLM
Risk score are designed to represent an underlying probability of an adverse event denoted $\lbrace Y=1\rbrace $ given a vector of $P$ explaining variables $\mathbf {X} $ containing measurements of the relevant risk factors. In order to establish the connection between the risk factors and the probability we estimate a set of weights $\beta $ is estimated using a generalized linear model:
${\begin{aligned}\operatorname {E} (\mathbf {Y} |\mathbf {X} )=\mathbf {P} (\mathbf {Y} =1|\mathbf {X} )=g^{-1}(\mathbf {X} \beta )\end{aligned}}$
Where $g^{-1}:\mathbb {R} \rightarrow [0,1]$ is a real-valued, monotonically increasing function that maps the values of the linear predictor $\mathbf {X} \beta $ to the interval $[0,1]$. GLM methods typically uses the logit or probit as the link function.
Estimating risk with other methods
While it's possible to estimate $\mathbf {P} (\mathbf {Y} =1|\mathbf {X} )$ using other statistical or machine learning methods, the requirements of simplicity and easy interpretation (and monotonicity per risk factor) make most of these methods difficult to use for scoring in this context:
• With more sophisticated methods it becomes difficult to attribute simple weights for each risk factor and to provide a simple formula for the calculation of the score. A notable exception are tree-based methods like CART, that can provide a simple set of decision rules and calculations, but cannot ensure the monotonicity of the scale across the different risk factors.
• The fact that we are estimating underlying risk across the population, and therefore cannot tag people in advance on an ordinal scale (we can't know in advance if a person belongs to a "high risk" group, we only see observed incidences) classification methods are only relevant if we want to classify people into 2 groups or 2 possible actions.
Constructing the score
When using GLM, the set of estimated weights $\beta $ can be used to assign different values (or "points") to different values of the risk factors in $\mathbf {X} $ (continuous or nominal as indicators). The score can then be expressed as a weighted sum:
${\begin{aligned}{\text{Score}}=\mathbf {X} \beta =\sum _{j=1}^{P}\mathbf {X} _{j}\beta _{j}\end{aligned}}$
• Some scoring methods will translate the score into probabilities by using $g^{-1}$ (e.g. SAPS II score[2] that gives an explicit function to calculate mortality from the score[3]) or a look-up table (e.g. ABCD² score[4][5] or the ISM7 (NI) Scorecard[6]). This practice makes the process of obtaining the score more complicated computationally but has the advantage of translating an arbitrary number to a more familiar scale of 0 to 1.
• The columns of $\mathbf {X} $ can represent complex transformations of the risk factors (including multiple interactions) and not just the risk factors themselves.
• The values of $\beta $ are sometimes scaled or rounded to allow working with integers instead of very small fractions (making the calculation simpler). While scaling has no impact ability of the score to estimate risk, rounding has the potential of disrupting the "optimality" of the GLM estimation.
Making score-based decisions
Let $\mathbf {A} =\lbrace \mathbf {a} _{1},...,\mathbf {a} _{m}\rbrace $ denote a set of $m\geq 2$ "escalating" actions available for the decision maker (e.g. for credit risk decisions: $\mathbf {a} _{1}$ = "approve automatically", $\mathbf {a} _{2}$ = "require more documentation and check manually", $\mathbf {a} _{3}$ = "decline automatically"). In order to define a decision rule, we want to define a map between different values of the score and the possible decisions in $\mathbf {A} $. Let $\tau =\lbrace \tau _{1},...\tau _{m-1}\rbrace $ be a partition of $\mathbb {R} $ into $m$ consecutive, non-overlapping intervals, such that $\tau _{1}<\tau _{2}<\ldots <\tau _{m-1}$.
The map is defined as follows:
${\begin{aligned}{\text{If Score}}\in [\tau _{j-1},\tau _{j})\rightarrow {\text{Take action }}\mathbf {a} _{j}\end{aligned}}$
• The values of $\tau $ are set based on expert opinion, the type and prevalence of the measured risk, consequences of miss-classification, etc. For example, a risk of 9 out of 10 will usually be considered as "high risk", but a risk of 7 out of 10 can be considered either "high risk" or "medium risk" depending on context.
• The definition of the intervals is on right open-ended intervals but can be equivalently defined using left open ended intervals $(\tau _{j-1},\tau _{j}]$.
• For scoring methods that are already translated the score into probabilities we either define the partition $\tau $ directly on the interval $[0,1]$ or translate the decision criteria into $[g^{-1}(\tau _{j-1}),g^{-1}(\tau _{j}))$, and the monotonicity of $g$ ensures a 1-to-1 translation.
Examples
Biostatistics
• Framingham Risk Score
• QRISK
• TIMI
• Rockall score
• ABCD² score
• CHA2DS2–VASc score
• SAPS II
(see more examples on the category page Category:Medical scoring system)
Financial industry
The primary use of scores in the financial sector is for Credit scorecards, or credit scores:
• In many countries (such as the US) credit score are calculated by commercial entities and therefore the exact method is not public knowledge (for example the Bankruptcy risk score, FICO score and others). Credit scores in Australia and UK are often calculated by using logistic regression to estimate probability of default, and are therefore a type of risk score.
• Other financial industries, such as the insurance industry also use scoring methods, but the exact implementation remains a trade secret, except for some rare cases[6]
Social Sciences
• COMPAS score for recidivism, as reverse-engineered by ProPublica[7] using logistic regression and Cox's proportional hazard model.
References
• Hastie, T. J.; Tibshirani, R. J. (1990). Generalized Additive Models. Chapman & Hall/CRC. ISBN 978-0-412-34390-2.
1. Toren, Yizhar (2011). "Ordinal Risk-Group Classification". arXiv:1012.5487 [stat.ML].
2. Le Gall, JR; Lemeshow, S; Saulnier, F (1993). "A new Simplified Acute Physiology Score (SAPS II) based on a European/North American multicenter study". JAMA. 270 (24): 2957–63. doi:10.1001/jama.1993.03510240069035. PMID 8254858.
3. "Simplified Acute Physiology Score (SAPS II) Calculator - ClinCalc.com". clincalc.com. Retrieved August 20, 2018.
4. Johnston SC; Rothwell PM; Nguyen-Huynh MN; Giles MF; Elkins JS; Bernstein AL; Sidney S. "Validation and refinement of scores to predict very early stroke risk after transient ischaemic attack" Lancet (2007): 369(9558):283-292
5. "ABCD² Score for TIA". www.mdcalc.com. Retrieved December 16, 2018.
6. "ISM7 (NI) Scorecard, Allstate Property & Casualty Company" (PDF). Retrieved December 16, 2018.
7. "How We Analyzed the COMPAS Recidivism Algorithm". Retrieved December 16, 2018.
|
Wikipedia
|
Ritabrata Munshi
Ritabrata Munshi (born 14 September 1976 in Calcutta (Kolkata), West Bengal) is an Indian mathematician specialising in number theory. He was awarded the Shanti Swarup Bhatnagar Prize for Science and Technology, the highest science award in India, for the year 2015 in mathematical science category.[1]
Ritabrata Munshi
Born14 September 1976 (age 46)
Alma mater
• Princeton University
Occupation
• Mathematician
Awards
• Shanti Swarup Bhatnagar Prize for Science and Technology (2015)
• ICTP Ramanujan Prize (2018)
• Infosys Prize (2017)
Academic career
Institutions
• Indian Statistical Institute
ThesisThe arithmetic of elliptic fibrations
Doctoral advisorAndrew Wiles
He is known for his contributions to the sub-convexity problem for automorphic L-functions. In a series of papers published in 2015[2][3][4] he introduced a new approach, based on the circle method, and was able to establish sub-convex bounds for genuine degree three L-functions. Later he extended his method to the case of GL(3)xGL(2) Rankin-Selberg convolutions.[5] These cases were earlier considered beyond the reach of known techniques.
He is affiliated to Tata Institute of Fundamental Research, Mumbai, and the Indian Statistical Institute, Kolkata. Munshi obtained PhD degree from Princeton University in 2006 under the guidance of Andrew John Wiles.[6] After that he was a Hill Assistant Professor at the Rutgers University and worked with Henryk Iwaniec. During 2009–2010 he was a member at the Institute for Advanced Study, Princeton.
Munshi was awarded the Swarna-Jayanti fellowship by the Department of Science and Technology, Government of India in 2012. He also received the B.M. Birla Science prize in 2013, and was elected a fellow of the Indian Academy of Sciences in 2016. For his outstanding contributions to analytic aspects of number theory, he was awarded the Infosys Prize 2017 in Mathematical Sciences.[7] On 8 November 2018 he was awarded the ICTP Ramanujan Prize in a ceremony held at the Budinich Lecture Hall, ICTP.[8] In 2018 he was an invited speaker at the International Congress of Mathematicians (ICM). He was elected a fellow of the Indian National Science Academy in 2020.
He serves in the editorial board of The Journal of the Ramanujan Mathematical Society and the Hardy-Ramanujan journal.
References
1. "Brief Profile of the Awardee". Shanti Swarup Bhatnagar Prize. CSIR Human Resource Development Group, New Delhi. Retrieved 4 November 2015.
2. Munshi, Ritabrata (2015). "The circle method and bounds for L-functions—II: Subconvexity for twists of GL(3) L-functions". American Journal of Mathematics. 137 (3): 791–812. doi:10.1353/ajm.2015.0018. ISSN 1080-6377. S2CID 14274076.
3. Munshi, Ritabrata (October 2015). "The circle method and bounds for 𝐿-functions—III: 𝑡-aspect subconvexity for 𝐺𝐿(3) 𝐿-functions". Journal of the American Mathematical Society. 28 (4): 913–938. arXiv:1301.1007. doi:10.1090/jams/843. ISSN 0894-0347. S2CID 124909878.
4. Munshi, Ritabrata (2015). "The circle method and bounds for L-functions - IV: Subconvexity for twists of GL(3) L-functions". Annals of Mathematics. 182 (2): 617–672. doi:10.4007/annals.2015.182.2.6. ISSN 0003-486X. JSTOR 24523345.
5. Munshi, Ritabrata (1 October 2018). "Subconvexity for $GL(3)\times GL(2)$ $L$-functions in $t$-aspect". arXiv:1810.00539 [math.NT].
6. Ritabrata Munshi at the Mathematics Genealogy Project
7. "Infosys Prize – Laureates 2017 – Prof. Ritabrata Munshi". infosys-science-foundation.com. Retrieved 1 December 2017.
8. "ICTP – Ramanujan Prize Winner 2018".
External links
• Munshi's home page
• Interview of Ritabrata Munshi
Recipients of Shanti Swarup Bhatnagar Prize for Science and Technology in Mathematical Science
1950s–70s
• K. S. Chandrasekharan & C. R. Rao (1959)
• K. G. Ramanathan (1965)
• A. S. Gupta & C. S. Seshadri (1972)
• P. C. Jain & M. S. Narasimhan (1975)
• K. R. Parthasarathy & S. K. Trehan (1976)
• M. S. Raghunathan (1977)
• E. M. V. Krishnamurthy (1978)
• S. Raghavan & S. Ramanan (1979)
1980s
• R. Sridharan (1980)
• J. K. Ghosh (1981)
• B. L. S. Prakasa Rao & J. B. Shukla (1982)
• I. B. S. Passi & Phoolan Prasad (1983)
• S. K. Malik & R. Parthasarathy (1985)
• T. Parthasarathy & U. B. Tewari (1986)
• Raman Parimala & T. N. Shorey (1987)
• M. B. Banerjee & K. B. Sinha (1988)
• Gopal Prasad (1989)
1990s
• R. Balasubramanian & S. G. Dani (1990)
• V. B. Mehta & A. Ramanathan (1991)
• Maithili Sharan (1992)
• Karmeshu & Navin M. Singhi (1993)
• N. Mohan Kumar (1994)
• Rajendra Bhatia (1995)
• V. S. Sunder (1996)
• Subhashis Nag & T. R. Ramadas (1998)
• Rajeeva Laxman Karandikar (1999)
2000s
• Rahul Mukerjee (2000)
• Gadadhar Misra & T. N. Venkataramana (2001)
• Dipendra Prasad & S. Thangavelu (2002)
• Manindra Agrawal & V. Srinivas (2003)
• Arup Bose & Sujatha Ramdorai (2004)
• Probal Chaudhuri & K. H. Paranjape (2005)
• Vikraman Balaji & Indranil Biswas (2006)
• B. V. Rajarama Bhat (2007)
• Rama Govindarajan (2007)
• Jaikumar Radhakrishnan (2008)
• Suresh Venapally (2009)
2010s
• Mahan Mitra & Palash Sarkar (2011)
• Siva Athreya & Debashish Goswami (2012)
• Eknath Prabhakar Ghate (2013)
• Kaushal Kumar Verma (2014)
• K Sandeep & Ritabrata Munshi (2015)
• Amalendu Krishna (2016)
• Naveen Garg (2016)
• (Not awarded) (2017)
• Amit Kumar & Nitin Saxena (2018)
• Neena Gupta & Dishant Mayurbhai Pancholi (2019)
2020s
• Rajat Subhra Hazra (2020)
• U. K. Anandavardhanan (2020)
• Anish Ghosh (2021)
• Saket Saurabh (2021)
Authority control
International
• ISNI
• VIAF
National
• United States
• Poland
Academics
• Google Scholar
• MathSciNet
• Mathematics Genealogy Project
• ORCID
• zbMATH
|
Wikipedia
|
Polynomial decomposition
In mathematics, a polynomial decomposition expresses a polynomial f as the functional composition $g\circ h$ of polynomials g and h, where g and h have degree greater than 1; it is an algebraic functional decomposition. Algorithms are known for decomposing univariate polynomials in polynomial time.
Polynomials which are decomposable in this way are composite polynomials; those which are not are indecomposable polynomials or sometimes prime polynomials[1] (not to be confused with irreducible polynomials, which cannot be factored into products of polynomials). The degree of a composite polynomial is always a composite number, the product of the degrees of the composed polynomials.
The rest of this article discusses only univariate polynomials; algorithms also exist for multivariate polynomials of arbitrary degree.[2]
Examples
In the simplest case, one of the polynomials is a monomial. For example,
$f=x^{6}-3x^{3}+1$
decomposes into
$g=x^{2}-3x+1{\text{ and }}h=x^{3}$
since
$f(x)=(g\circ h)(x)=g(h(x))=g(x^{3})=(x^{3})^{2}-3(x^{3})+1,$
using the ring operator symbol ∘ to denote function composition.
Less trivially,
${\begin{aligned}&x^{6}-6x^{5}+21x^{4}-44x^{3}+68x^{2}-64x+41\\={}&(x^{3}+9x^{2}+32x+41)\circ (x^{2}-2x).\end{aligned}}$
Uniqueness
A polynomial may have distinct decompositions into indecomposable polynomials where $f=g_{1}\circ g_{2}\circ \cdots \circ g_{m}=h_{1}\circ h_{2}\circ \cdots \circ h_{n}$ where $g_{i}\neq h_{i}$ for some $i$. The restriction in the definition to polynomials of degree greater than one excludes the infinitely many decompositions possible with linear polynomials.
Joseph Ritt proved that $m=n$, and the degrees of the components are the same up to linear transformations, but possibly in different order; this is Ritt's polynomial decomposition theorem.[1][3] For example, $x^{2}\circ x^{3}=x^{3}\circ x^{2}$.
Applications
A polynomial decomposition may enable more efficient evaluation of a polynomial. For example,
${\begin{aligned}&x^{8}+4x^{7}+10x^{6}+16x^{5}+19x^{4}+16x^{3}+10x^{2}+4x-1\\={}&\left(x^{2}-2\right)\circ \left(x^{2}\right)\circ \left(x^{2}+x+1\right)\end{aligned}}$
can be calculated with 3 multiplications and 3 additions using the decomposition, while Horner's method would require 7 multiplications and 8 additions.
A polynomial decomposition enables calculation of symbolic roots using radicals, even for some irreducible polynomials. This technique is used in many computer algebra systems.[4] For example, using the decomposition
${\begin{aligned}&x^{6}-6x^{5}+15x^{4}-20x^{3}+15x^{2}-6x-1\\={}&\left(x^{3}-2\right)\circ \left(x^{2}-2x+1\right),\end{aligned}}$
the roots of this irreducible polynomial can be calculated as[5]
$1\pm 2^{1/6},1\pm {\frac {\sqrt {-1\pm {\sqrt {3}}i}}{2^{1/3}}}.$
Even in the case of quartic polynomials, where there is an explicit formula for the roots, solving using the decomposition often gives a simpler form. For example, the decomposition
${\begin{aligned}&x^{4}-8x^{3}+18x^{2}-8x+2\\={}&(x^{2}+1)\circ (x^{2}-4x+1)\end{aligned}}$
gives the roots[5]
$2\pm {\sqrt {3\pm i}}$
but straightforward application of the quartic formula gives equivalent results but in a form that is difficult to simplify and difficult to understand; one of the four roots is:
$2-{\frac {\sqrt {{9\left({\frac {8{\sqrt {10}}i}{3^{3/2}}}+72\right)^{2/3}+36\left({\frac {8{\sqrt {10}}i}{3^{3/2}}}+72\right)^{1/3}+156} \over {\left({\frac {8{\sqrt {10}}i}{3^{3/2}}}+72\right)^{1/3}}}}{6}}-{{\sqrt {-\left({\frac {8{\sqrt {10}}i}{3^{3/2}}}+72\right)^{1/3}-{{52} \over {3\left({\frac {8{\sqrt {10}}i}{3^{3/2}}}+72\right)^{1/3}}}+8}} \over 2}.$
Algorithms
The first algorithm for polynomial decomposition was published in 1985,[6] though it had been discovered in 1976,[7] and implemented in the Macsyma/Maxima computer algebra system.[8] That algorithm takes exponential time in worst case, but works independently of the characteristic of the underlying field.
A 1989 algorithm runs in polynomial time but with restrictions on the characteristic.[9]
A 2014 algorithm calculates a decomposition in polynomial time and without restrictions on the characteristic.[10]
Notes
1. J.F. Ritt, "Prime and Composite Polynomials", Transactions of the American Mathematical Society 23:1:51–66 (January, 1922) doi:10.2307/1988911 JSTOR 1988911
2. Jean-Charles Faugère, Ludovic Perret, "An efficient algorithm for decomposing multivariate polynomials and its applications to cryptography", Journal of Symbolic Computation, 44:1676-1689 (2009), doi:10.1016/j.jsc.2008.02.005
3. Capi Corrales-Rodrigáñez, "A note on Ritt's theorem on decomposition of polynomials", Journal of Pure and Applied Algebra 68:3:293–296 (6 December 1990) doi:10.1016/0022-4049(90)90086-W
4. The examples below were calculated using Maxima.
5. Where each ± is taken independently.
6. David R. Barton, Richard Zippel (1985). "Polynomial Decomposition Algorithms". Journal of Symbolic Computation. 1 (2): 159–168. doi:10.1016/S0747-7171(85)80012-2.
7. Richard Zippel, Functional Decomposition, 1996.
8. See the polydecomp function.
9. Kozen, Dexter; Landau, Susan (1989). "Polynomial Decomposition Algorithms". Journal of Symbolic Computation. 7 (5): 445–456. doi:10.1016/S0747-7171(89)80027-6.
10. Raoul Blankertz (2014). "A polynomial time algorithm for computing all minimal decompositions of a polynomial" (PDF). ACM Communications in Computer Algebra. 48 (187): 1. Archived 2015-09-24 at the Wayback Machine
References
• Joel S. Cohen (2003). "Chapter 5. Polynomial Decomposition". Computer Algebra and Symbolic Computation. ISBN 1-56881-159-4.
|
Wikipedia
|
Exponential polynomial
In mathematics, exponential polynomials are functions on fields, rings, or abelian groups that take the form of polynomials in a variable and an exponential function.
This article is about polynomials in variables and exponential functions. For the polynomials involving Stirling numbers, see Touchard polynomials.
Definition
In fields
An exponential polynomial generally has both a variable x and some kind of exponential function E(x). In the complex numbers there is already a canonical exponential function, the function that maps x to ex. In this setting the term exponential polynomial is often used to mean polynomials of the form P(x, ex) where P ∈ C[x, y] is a polynomial in two variables.[1][2]
There is nothing particularly special about C here; exponential polynomials may also refer to such a polynomial on any exponential field or exponential ring with its exponential function taking the place of ex above.[3] Similarly, there is no reason to have one variable, and an exponential polynomial in n variables would be of the form P(x1, ..., xn, ex1, ..., exn), where P is a polynomial in 2n variables.
For formal exponential polynomials over a field K we proceed as follows.[4] Let W be a finitely generated Z-submodule of K and consider finite sums of the form
$\sum _{i=1}^{m}f_{i}(X)\exp(w_{i}X)\ ,$
where the fi are polynomials in K[X] and the exp(wi X) are formal symbols indexed by wi in W subject to exp(u + v) = exp(u) exp(v).
In abelian groups
A more general framework where the term 'exponential polynomial' may be found is that of exponential functions on abelian groups. Similarly to how exponential functions on exponential fields are defined, given a topological abelian group G a homomorphism from G to the additive group of the complex numbers is called an additive function, and a homomorphism to the multiplicative group of nonzero complex numbers is called an exponential function, or simply an exponential. A product of additive functions and exponentials is called an exponential monomial, and a linear combination of these is then an exponential polynomial on G.[5][6]
Properties
Ritt's theorem states that the analogues of unique factorization and the factor theorem hold for the ring of exponential polynomials.[4]
Applications
Exponential polynomials on R and C often appear in transcendental number theory, where they appear as auxiliary functions in proofs involving the exponential function. They also act as a link between model theory and analytic geometry. If one defines an exponential variety to be the set of points in Rn where some finite collection of exponential polynomials vanish, then results like Khovanskiǐ's theorem in differential geometry and Wilkie's theorem in model theory show that these varieties are well-behaved in the sense that the collection of such varieties is stable under the various set-theoretic operations as long as one allows the inclusion of the image under projections of higher-dimensional exponential varieties. Indeed, the two aforementioned theorems imply that the set of all exponential varieties forms an o-minimal structure over R.
Exponential polynomials appear in the characteristic equation associated with linear delay differential equations.
Notes
1. C. J. Moreno, The zeros of exponential polynomials, Compositio Mathematica 26 (1973), pp.69–78.
2. M. Waldschmidt, Diophantine approximation on linear algebraic groups, Springer, 2000.
3. Martin Bays, Jonathan Kirby, A.J. Wilkie, A Schanuel property for exponentially transcendental powers, (2008), arXiv:0810.4457v1
4. Everest, Graham; van der Poorten, Alf; Shparlinski, Igor; Ward, Thomas (2003). Recurrence sequences. Mathematical Surveys and Monographs. Vol. 104. Providence, RI: American Mathematical Society. p. 140. ISBN 0-8218-3387-1. Zbl 1033.11006.
5. László Székelyhidi, On the extension of exponential polynomials, Mathematica Bohemica 125 (2000), pp.365–370.
6. P. G. Laird, On characterizations of exponential polynomials, Pacific Journal of Mathematics 80 (1979), pp.503–507.
|
Wikipedia
|
Galerkin method
In mathematics, in the area of numerical analysis, Galerkin methods are named after the Soviet mathematician Boris Galerkin. They convert a continuous operator problem, such as a differential equation, commonly in a weak formulation, to a discrete problem by applying linear constraints determined by finite sets of basis functions.
Differential equations
Scope
Fields
• Natural sciences
• Engineering
• Astronomy
• Physics
• Chemistry
• Biology
• Geology
Applied mathematics
• Continuum mechanics
• Chaos theory
• Dynamical systems
Social sciences
• Economics
• Population dynamics
List of named differential equations
Classification
Types
• Ordinary
• Partial
• Differential-algebraic
• Integro-differential
• Fractional
• Linear
• Non-linear
By variable type
• Dependent and independent variables
• Autonomous
• Coupled / Decoupled
• Exact
• Homogeneous / Nonhomogeneous
Features
• Order
• Operator
• Notation
Relation to processes
• Difference (discrete analogue)
• Stochastic
• Stochastic partial
• Delay
Solution
Existence and uniqueness
• Picard–Lindelöf theorem
• Peano existence theorem
• Carathéodory's existence theorem
• Cauchy–Kowalevski theorem
General topics
• Initial conditions
• Boundary values
• Dirichlet
• Neumann
• Robin
• Cauchy problem
• Wronskian
• Phase portrait
• Lyapunov / Asymptotic / Exponential stability
• Rate of convergence
• Series / Integral solutions
• Numerical integration
• Dirac delta function
Solution methods
• Inspection
• Method of characteristics
• Euler
• Exponential response formula
• Finite difference (Crank–Nicolson)
• Finite element
• Infinite element
• Finite volume
• Galerkin
• Petrov–Galerkin
• Green's function
• Integrating factor
• Integral transforms
• Perturbation theory
• Runge–Kutta
• Separation of variables
• Undetermined coefficients
• Variation of parameters
People
List
• Isaac Newton
• Gottfried Leibniz
• Jacob Bernoulli
• Leonhard Euler
• Józef Maria Hoene-Wroński
• Joseph Fourier
• Augustin-Louis Cauchy
• George Green
• Carl David Tolmé Runge
• Martin Kutta
• Rudolf Lipschitz
• Ernst Lindelöf
• Émile Picard
• Phyllis Nicolson
• John Crank
Often when referring to a Galerkin method, one also gives the name along with typical assumptions and approximation methods used:
• Ritz–Galerkin method (after Walther Ritz) typically assumes symmetric and positive definite bilinear form in the weak formulation, where the differential equation for a physical system can be formulated via minimization of a quadratic function representing the system energy and the approximate solution is a linear combination of the given set of the basis functions.[1]
• Bubnov–Galerkin method (after Ivan Bubnov) does not require the bilinear form to be symmetric and substitutes the energy minimization with orthogonality constraints determined by the same basis functions that are used to approximate the solution. In an operator formulation of the differential equation, Bubnov–Galerkin method can be viewed as applying an orthogonal projection to the operator.
• Petrov–Galerkin method (after Georgii I. Petrov[2]) allows using basis functions for orthogonality constraints (called test basis functions) that are different from the basis functions used to approximate the solution. Petrov–Galerkin method can be viewed as an extension of Bubnov–Galerkin method, applying a projection that is not necessarily orthogonal in the operator formulation of the differential equation.
Examples of Galerkin methods are:
• the Galerkin method of weighted residuals, the most common method of calculating the global stiffness matrix in the finite element method,[3][4]
• the boundary element method for solving integral equations,
• Krylov subspace methods.[5]
Example: matrix linear system
We first introduce and illustrate the Galerkin method as being applied to a system of linear equations $A\mathbf {x} =\mathbf {b} $ with the following symmetric and positive definite matrix
$A={\begin{bmatrix}2&0&0\\0&2&1\\0&1&2\end{bmatrix}}$
and the solution and right-hand-side vectors
$\mathbf {x} ={\begin{bmatrix}1\\0\\0\end{bmatrix}},\quad \mathbf {b} ={\begin{bmatrix}2\\0\\0\end{bmatrix}}.$
Let us take
$V={\begin{bmatrix}0&0\\1&0\\0&1\end{bmatrix}},$
then the matrix of the Galerkin equation is
$V^{*}AV={\begin{bmatrix}2&1\\1&2\end{bmatrix}},$
the right-hand-side vector of the Galerkin equation is
$V^{*}\mathbf {b} ={\begin{bmatrix}0\\0\end{bmatrix}},$
so that we obtain the solution vector
$\mathbf {y} ={\begin{bmatrix}0\\0\end{bmatrix}}$
to the Galerkin equation $\left(V^{*}AV\right)\mathbf {y} =V^{*}\mathbf {b} $, which we finally uplift to determine the approximate solution to the original equation as
$V\mathbf {y} ={\begin{bmatrix}0\\0\\0\end{bmatrix}}.$
In this example, our original Hilbert space is actually the 3-dimensional Euclidean space $\mathbb {R} ^{3}$ equipped with the standard scalar product $(\mathbf {u} ,\mathbf {v} )=\mathbf {u} ^{T}\mathbf {v} $, our 3-by-3 matrix $A$ defines the bilinear form $a(\mathbf {u} ,\mathbf {v} )=\mathbf {u} ^{T}A\mathbf {v} $, and the right-hand-side vector $\mathbf {b} $ defines the bounded linear functional $f(\mathbf {v} )=\mathbf {b} ^{T}\mathbf {v} $. The columns
$\mathbf {e} _{1}={\begin{bmatrix}0\\1\\0\end{bmatrix}}\quad \mathbf {e} _{2}={\begin{bmatrix}0\\0\\1\end{bmatrix}},$
of the matrix $V$ form an orthonormal basis of the 2-dimensional subspace of the Galerkin projection. The entries of the 2-by-2 Galerkin matrix $V^{*}AV$ are $a(e_{j},e_{i}),\,i,j=1,2$, while the components of the right-hand-side vector $V^{*}\mathbf {b} $ of the Galerkin equation are $f(e_{i}),\,i=1,2$. Finally, the approximate solution $V\mathbf {y} $ is obtained from the components of the solution vector $\mathbf {y} $ of the Galerkin equation and the basis as $\sum _{j=1}^{2}y_{j}\mathbf {e} _{j}$.
Linear equation in a Hilbert space
Weak formulation of a linear equation
Let us introduce Galerkin's method with an abstract problem posed as a weak formulation on a Hilbert space $V$, namely,
find $u\in V$ such that for all $v\in V,a(u,v)=f(v)$.
Here, $a(\cdot ,\cdot )$ is a bilinear form (the exact requirements on $a(\cdot ,\cdot )$ will be specified later) and $f$ is a bounded linear functional on $V$.
Galerkin dimension reduction
Choose a subspace $V_{n}\subset V$ of dimension n and solve the projected problem:
Find $u_{n}\in V_{n}$ such that for all $v_{n}\in V_{n},a(u_{n},v_{n})=f(v_{n})$.
We call this the Galerkin equation. Notice that the equation has remained unchanged and only the spaces have changed. Reducing the problem to a finite-dimensional vector subspace allows us to numerically compute $u_{n}$ as a finite linear combination of the basis vectors in $V_{n}$.
Galerkin orthogonality
The key property of the Galerkin approach is that the error is orthogonal to the chosen subspaces. Since $V_{n}\subset V$, we can use $v_{n}$ as a test vector in the original equation. Subtracting the two, we get the Galerkin orthogonality relation for the error, $\epsilon _{n}=u-u_{n}$ which is the error between the solution of the original problem, $u$, and the solution of the Galerkin equation, $u_{n}$
$a(\epsilon _{n},v_{n})=a(u,v_{n})-a(u_{n},v_{n})=f(v_{n})-f(v_{n})=0.$
Matrix form of Galerkin's equation
Since the aim of Galerkin's method is the production of a linear system of equations, we build its matrix form, which can be used to compute the solution algorithmically.
Let $e_{1},e_{2},\ldots ,e_{n}$ be a basis for $V_{n}$. Then, it is sufficient to use these in turn for testing the Galerkin equation, i.e.: find $u_{n}\in V_{n}$ such that
$a(u_{n},e_{i})=f(e_{i})\quad i=1,\ldots ,n.$
We expand $u_{n}$ with respect to this basis, $u_{n}=\sum _{j=1}^{n}u_{j}e_{j}$ and insert it into the equation above, to obtain
$a\left(\sum _{j=1}^{n}u_{j}e_{j},e_{i}\right)=\sum _{j=1}^{n}u_{j}a(e_{j},e_{i})=f(e_{i})\quad i=1,\ldots ,n.$
This previous equation is actually a linear system of equations $Au=f$, where
$A_{ij}=a(e_{j},e_{i}),\quad f_{i}=f(e_{i}).$
Symmetry of the matrix
Due to the definition of the matrix entries, the matrix of the Galerkin equation is symmetric if and only if the bilinear form $a(\cdot ,\cdot )$ is symmetric.
Analysis of Galerkin methods
Here, we will restrict ourselves to symmetric bilinear forms, that is
$a(u,v)=a(v,u).$
While this is not really a restriction of Galerkin methods, the application of the standard theory becomes much simpler. Furthermore, a Petrov–Galerkin method may be required in the nonsymmetric case.
The analysis of these methods proceeds in two steps. First, we will show that the Galerkin equation is a well-posed problem in the sense of Hadamard and therefore admits a unique solution. In the second step, we study the quality of approximation of the Galerkin solution $u_{n}$.
The analysis will mostly rest on two properties of the bilinear form, namely
• Boundedness: for all $u,v\in V$ holds
$a(u,v)\leq C\|u\|\,\|v\|$ for some constant $C>0$
• Ellipticity: for all $u\in V$ holds
$a(u,u)\geq c\|u\|^{2}$ for some constant $c>0.$
By the Lax-Milgram theorem (see weak formulation), these two conditions imply well-posedness of the original problem in weak formulation. All norms in the following sections will be norms for which the above inequalities hold (these norms are often called an energy norm).
Well-posedness of the Galerkin equation
Since $V_{n}\subset V$, boundedness and ellipticity of the bilinear form apply to $V_{n}$. Therefore, the well-posedness of the Galerkin problem is actually inherited from the well-posedness of the original problem.
Quasi-best approximation (Céa's lemma)
Main article: Céa's lemma
The error $u-u_{n}$ between the original and the Galerkin solution admits the estimate
$\|u-u_{n}\|\leq {\frac {C}{c}}\inf _{v_{n}\in V_{n}}\|u-v_{n}\|.$
This means, that up to the constant $C/c$, the Galerkin solution $u_{n}$ is as close to the original solution $u$ as any other vector in $V_{n}$. In particular, it will be sufficient to study approximation by spaces $V_{n}$, completely forgetting about the equation being solved.
Proof
Since the proof is very simple and the basic principle behind all Galerkin methods, we include it here: by ellipticity and boundedness of the bilinear form (inequalities) and Galerkin orthogonality (equals sign in the middle), we have for arbitrary $v_{n}\in V_{n}$:
$c\|u-u_{n}\|^{2}\leq a(u-u_{n},u-u_{n})=a(u-u_{n},u-v_{n})\leq C\|u-u_{n}\|\,\|u-v_{n}\|.$
Dividing by $c\|u-u_{n}\|$ and taking the infimum over all possible $v_{n}$ yields the lemma.
Galerkin's best approximation property in the energy norm
For simplicity of presentation in the section above we have assumed that the bilinear form $a(u,v)$ is symmetric and positive definite, which implies that it is a scalar product and the expression $\|u\|_{a}={\sqrt {a(u,u)}}$ is actually a valid vector norm, called the energy norm. Under these assumptions one can easily prove in addition Galerkin's best approximation property in the energy norm.
Using Galerkin a-orthogonality and the Cauchy–Schwarz inequality for the energy norm, we obtain
$\|u-u_{n}\|_{a}^{2}=a(u-u_{n},u-u_{n})=a(u-u_{n},u-v_{n})\leq \|u-u_{n}\|_{a}\,\|u-v_{n}\|_{a}.$
Dividing by $\|u-u_{n}\|_{a}$ and taking the infimum over all possible $v_{n}\in V_{n}$ proves that the Galerkin approximation $u_{n}\in V_{n}$ is the best approximation in the energy norm within the subspace $V_{n}\subset V$, i.e. $u_{n}\in V_{n}$ is nothing but the orthogonal, with respect to the scalar product $a(u,v)$, projection of the solution $u$ to the subspace $V_{n}$.
Galerkin method for stepped Structures
I. Elishakof, M. Amato, A. Marzani, P.A. Arvan, and J.N. Reddy [6] [7] [8] [9] studied the application of the Galerkin method to stepped structures. They showed that the generalized function, namely unit-step function, Dirac’s delta function, and the doublet function are needed for obtaining accurate results.
History
The approach is usually credited to Boris Galerkin.[10][11] The method was explained to the Western reader by Hencky[12] and Duncan[13][14] among others. Its convergence was studied by Mikhlin[15] and Leipholz[16][17][18][19] Its coincidence with Fourier method was illustrated by Elishakoff et al.[20][21][22] Its equivalence to Ritz's method for conservative problems was shown by Singer.[23] Gander and Wanner[24] showed how Ritz and Galerkin methods led to the modern finite element method. One hundred years of method's development was discussed by Repin.[25] Elishakoff, Kaplunov and Kaplunov[26] show that the Galerkin’s method was not developed by Ritz, contrary to the Timoshenko’s statements.
See also
• Ritz method
References
1. A. Ern, J.L. Guermond, Theory and practice of finite elements, Springer, 2004, ISBN 0-387-20574-8
2. "Georgii Ivanovich Petrov (on his 100th birthday)", Fluid Dynamics, May 2012, Volume 47, Issue 3, pp 289-291, DOI 10.1134/S0015462812030015
3. S. Brenner, R. L. Scott, The Mathematical Theory of Finite Element Methods, 2nd edition, Springer, 2005, ISBN 0-387-95451-1
4. P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, 1978, ISBN 0-444-85028-7
5. Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd edition, SIAM, 2003, ISBN 0-89871-534-2
6. Elishakoff, I., Amato, M., Ankitha, A. P., & Marzani, A. (2021). Rigorous implementation of the Galerkin method for stepped structures needs generalized functions. Journal of Sound and Vibration, 490, 115708.
7. Elishakoff, I., Amato, M., & Marzani, A. (2021). Galerkin’s method revisited and corrected in the problem of Jaworsky and Dowell. Mechanical Systems and Signal Processing, 155, 107604.
8. Elishakoff, I., & Amato, M. (2021). Flutter of a beam in supersonic flow: truncated version of Timoshenko–Ehrenfest equation is sufficient. International Journal of Mechanics and Materials in Design, 1-17.
9. Amato, M., Elishakoff, I., & Reddy, J. N. (2021). Flutter of a Multicomponent Beam in a Supersonic Flow. AIAA Journal, 59(11), 4342-4353.
10. Galerkin, B.G.,1915, Rods and Plates, Series Occurring in Various Questions Concerning the Elastic Equilibrium of Rods and Plates, Vestnik Inzhenerov i Tekhnikov, (Engineers and Technologists Bulletin), Vol. 19, 897-908 (in Russian),(English Translation: 63-18925, Clearinghouse Fed. Sci. Tech. Info.1963).
11. "Le destin douloureux de Walther Ritz (1878-1909)", (Jean-Claude Pont, editor), Cahiers de Vallesia, 24, (2012), ISBN 978-2-9700636-5-0
12. Hencky H.,1927, Eine wichtige Vereinfachung der Methode von Ritz zur angennäherten Behandlung von Variationproblemen, ZAMM: Zeitschrift für angewandte Mathematik und Mechanik, Vol. 7, 80-81 (in German).
13. Duncan, W.J.,1937, Galerkin’s Method in Mechanics and Differential Equations, Aeronautical Research Committee Reports and Memoranda, No. 1798.
14. Duncan, W.J.,1938, The Principles of the Galerkin Method, Aeronautical Research Report and Memoranda, No. 1894.
15. S. G. Mikhlin, "Variational methods in Mathematical Physics", Pergamon Press, 1964
16. Leipholz H.H.E.,1976, Use of Galerkin’s Method for Vibration Problems, Shock and Vibration Digest, Vol. 8, 3-18
17. Leipholz H.H.E.,1967, Über die Wahl der Ansatzfunktionen bei der Durchführung des Verfahrens von Galerkin, Acta Mech., Vol. 3, 295-317 (in German).
18. Leipholz H.H.E., 1967, Über die Befreiung der Anzatzfunktionen des Ritzschen und Galerkinschen Verfahrens von den Randbedingungen, Ing. Arch., Vol. 36, 251-261 (in German).
19. Leipholz, H.H.E.,1976, Use of Galerkin’s Method for Vibration Problems, The Shock and Vibration Digest Vol. 8, 3-18, 1976.
20. Elishakoff, I., Lee,L.H.N.,1986, On Equivalence of the Galerkin and Fourier Series Methods for One Class of Problems, Journal of Sound and Vibration, Vol. 109, 174-177.
21. Elishakoff, I., Zingales, M.,2003, Coincidence of Bubnov-Galerkin and Exact Solution in an Applied Mechanics Problem, Journal of Applied Mechanics, Vol. 70, 777-779.
22. Elishakoff, I., Zingales M.,2004, Convergence of Bubnov-Galerkin Method Exemplified, AIAA Journal, Vol. 42(9), 1931-1933.
23. Singer J.,1962, On Equivalence of the Galerkin and Rayleigh-Ritz Methods, Journal of the Royal Aeronautical Society, Vol. 66, No. 621, p.592.
24. Gander, M.J, Wanner, G.,2012, From Euler, Ritz, and Galerkin to Modern Computing, SIAM Review, Vol. 54(4), 627-666.
25. ] Repin, S.,2017, One Hundred Years of the Galerkin Method, Computational Methods and Applied Mathematics, Vol.17(3), 351-357.
26. .Elishakoff, I., Julius Kaplunov, Elizabeth Kaplunov, 2020, “Galerkin’s method was not developed by Ritz, contrary to the Timoshenko’s statement”, in Nonlinear Dynamics of Discrete and Continuous Systems (A. Abramyan, I. Andrianov and V. Gaiko, eds.), pp. 63-82, Springer, Berlin.
External links
• "Galerkin method", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Galerkin Method from MathWorld
Numerical methods for partial differential equations
Finite difference
Parabolic
• Forward-time central-space (FTCS)
• Crank–Nicolson
Hyperbolic
• Lax–Friedrichs
• Lax–Wendroff
• MacCormack
• Upwind
• Method of characteristics
Others
• Alternating direction-implicit (ADI)
• Finite-difference time-domain (FDTD)
Finite volume
• Godunov
• High-resolution
• Monotonic upstream-centered (MUSCL)
• Advection upstream-splitting (AUSM)
• Riemann solver
• Essentially non-oscillatory (ENO)
• Weighted essentially non-oscillatory (WENO)
Finite element
• hp-FEM
• Extended (XFEM)
• Discontinuous Galerkin (DG)
• Spectral element (SEM)
• Mortar
• Gradient discretisation (GDM)
• Loubignac iteration
• Smoothed (S-FEM)
Meshless/Meshfree
• Smoothed-particle hydrodynamics (SPH)
• Peridynamics (PD)
• Moving particle semi-implicit method (MPS)
• Material point method (MPM)
• Particle-in-cell (PIC)
Domain decomposition
• Schur complement
• Fictitious domain
• Schwarz alternating
• additive
• abstract additive
• Neumann–Dirichlet
• Neumann–Neumann
• Poincaré–Steklov operator
• Balancing (BDD)
• Balancing by constraints (BDDC)
• Tearing and interconnect (FETI)
• FETI-DP
Others
• Spectral
• Pseudospectral (DVR)
• Method of lines
• Multigrid
• Collocation
• Level-set
• Boundary element
• Method of moments
• Immersed boundary
• Analytic element
• Isogeometric analysis
• Infinite difference method
• Infinite element method
• Galerkin method
• Petrov–Galerkin method
• Validated numerics
• Computer-assisted proof
• Integrable algorithm
• Method of fundamental solutions
Authority control: National
• Germany
• Israel
• United States
• Czech Republic
|
Wikipedia
|
Rayleigh–Ritz method
The Rayleigh–Ritz method is a direct numerical method of approximating eigenvalues, originated in the context of solving physical boundary value problems and named after Lord Rayleigh and Walther Ritz.
It is used in all applications that involve approximating eigenvalues and eigenvectors, often under different names. In quantum mechanics, where a system of particles is described using a Hamiltonian, the Ritz method uses trial wave functions to approximate the ground state eigenfunction with the lowest energy. In the finite element method context, mathematically the same algorithm is commonly called the Ritz-Galerkin method. The Rayleigh–Ritz method or Ritz method terminology is typical in mechanical and structural engineering to approximate the eigenmodes and resonant frequencies of a structure.
Naming and attribution
The name Rayleigh–Ritz is being debated[1][2] vs. the Ritz method after Walther Ritz, since the numerical procedure has been published by Walther Ritz in 1908-1909. According to A. W. Leissa,[1] Lord Rayleigh wrote a paper congratulating Ritz on his work in 1911, but stating that he himself had used Ritz's method in many places in his book and in another publication. This statement, although later disputed, and the fact that the method in the trivial case of a single vector results in the Rayleigh quotient make the arguable misnomer persist. According to S. Ilanko,[2] citing Richard Courant, both Lord Rayleigh and Walther Ritz independently conceived the idea of utilizing the equivalence between boundary value problems of partial differential equations on the one hand and problems of the calculus of variations on the other hand for numerical calculation of the solutions, by substituting for the variational problems simpler approximating extremum problems in which a finite number of parameters need to be determined; see the article Ritz method for details. Ironically for the debate, the modern justification of the algorithm drops the calculus of variations in favor of the simpler and more general approach of orthogonal projection as in Galerkin method named after Boris Galerkin, thus leading also to the Ritz-Galerkin method naming.
For matrix eigenvalue problems
In numerical linear algebra, the Rayleigh–Ritz method is commonly[3] applied to approximate an eigenvalue problem
$A\mathbf {x} =\lambda \mathbf {x} $
for the matrix $A\in \mathbb {C} ^{N\times N}$ of size $N$ using a projected matrix of a smaller size $m<N$, generated from a given matrix $V\in \mathbb {C} ^{N\times m}$ with orthonormal columns. The matrix version of the algorithm is the most simple:
1. Compute the $m\times m$ matrix $V^{*}AV$, where $V^{*}$ denotes the complex-conjugate transpose of $V$
2. Solve the eigenvalue problem $V^{*}AV\mathbf {y} _{i}=\mu _{i}\mathbf {y} _{i}$
3. Compute the Ritz vectors ${\tilde {\mathbf {x} }}_{i}=V\mathbf {y} _{i}$ and the Ritz value ${\tilde {\lambda }}_{i}=\mu _{i}$
4. Output approximations $({\tilde {\lambda }}_{i},{\tilde {\mathbf {x} }}_{i})$, called the Ritz pairs, to eigenvalues and eigenvectors of the original matrix $A$.
If the subspace with the orthonormal basis given by the columns of the matrix $V\in \mathbb {C} ^{N\times m}$ contains $k\leq m$ vectors that are close to eigenvectors of the matrix $A$, the Rayleigh–Ritz method above finds $k$ Ritz vectors that well approximate these eigenvectors. The easily computable quantity $\|A{\tilde {\mathbf {x} }}_{i}-{\tilde {\lambda }}_{i}{\tilde {\mathbf {x} }}_{i}\|$ determines the accuracy of such an approximation for every Ritz pair.
In the easiest case $m=1$, the $N\times m$ matrix $V$ turns into a unit column-vector $v$, the $m\times m$ matrix $V^{*}AV$ is a scalar that is equal to the Rayleigh quotient $\rho (v)=v^{*}Av/v^{*}v$, the only $i=1$ solution to the eigenvalue problem is $y_{i}=1$ and $\mu _{i}=\rho (v)$, and the only one Ritz vector is $v$ itself. Thus, the Rayleigh–Ritz method turns into computing of the Rayleigh quotient if $m=1$.
Another useful connection to the Rayleigh quotient is that $\mu _{i}=\rho (v_{i})$ for every Ritz pair $({\tilde {\lambda }}_{i},{\tilde {\mathbf {x} }}_{i})$, allowing to derive some properties of Ritz values $\mu _{i}$ from the corresponding theory for the Rayleigh quotient. For example, if $A$ is a Hermitian matrix, its Rayleigh quotient (and thus its every Ritz value) is real and takes values within the closed interval of the smallest and largest eigenvalues of $A$.
Example
The matrix
$A={\begin{bmatrix}2&0&0\\0&2&1\\0&1&2\end{bmatrix}}$
has eigenvalues $1,2,3$ and the corresponding eigenvectors
$\mathbf {x} _{\lambda =1}={\begin{bmatrix}0\\1\\-1\end{bmatrix}},\quad \mathbf {x} _{\lambda =2}={\begin{bmatrix}1\\0\\0\end{bmatrix}},\quad \mathbf {x} _{\lambda =3}={\begin{bmatrix}0\\1\\1\end{bmatrix}}.$
Let us take
$V={\begin{bmatrix}0&0\\1&0\\0&1\end{bmatrix}},$
then
$V^{*}AV={\begin{bmatrix}2&1\\1&2\end{bmatrix}}$
with eigenvalues $1,3$ and the corresponding eigenvectors
$\mathbf {y} _{\mu =1}={\begin{bmatrix}1\\-1\end{bmatrix}},\quad \mathbf {y} _{\mu =3}={\begin{bmatrix}1\\1\end{bmatrix}},$
so that the Ritz values are $1,3$ and the Ritz vectors are
$\mathbf {\tilde {x}} _{{\tilde {\lambda }}=1}={\begin{bmatrix}0\\1\\-1\end{bmatrix}},\quad \mathbf {\tilde {x}} _{{\tilde {\lambda }}=3}={\begin{bmatrix}0\\1\\1\end{bmatrix}}.$
We observe that each one of the Ritz vectors is exactly one of the eigenvectors of $A$ for the given $V$ as well as the Ritz values give exactly two of the three eigenvalues of $A$. A mathematical explanation for the exact approximation is based on the fact that the column space of the matrix $V$ happens to be exactly the same as the subspace spanned by the two eigenvectors $\mathbf {x} _{\lambda =1}$ and $\mathbf {x} _{\lambda =3}$ in this example.
For matrix singular value problems
Truncated singular value decomposition (SVD) in numerical linear algebra can also use the Rayleigh–Ritz method to find approximations to left and right singular vectors of the matrix $M\in \mathbb {C} ^{M\times N}$ of size $M\times N$ in given subspaces by turning the singular value problem into an eigenvalue problem.
Using the normal matrix
The definition of the singular value $\sigma $ and the corresponding left and right singular vectors is $Mv=\sigma u$ and $M^{*}u=\sigma v$. Having found one set (left of right) of approximate singular vectors and singular values by applying naively the Rayleigh–Ritz method to the Hermitian normal matrix $M^{*}M\in \mathbb {C} ^{N\times N}$ or $MM^{*}\in \mathbb {C} ^{M\times M}$, whichever one is smaller size, one could determine the other set of left of right singular vectors simply by dividing by the singular values, i.e., $u=Mv/\sigma $ and $v=M^{*}u/\sigma $. However, the division is unstable or fails for small or zero singular values.
An alternative approach, e.g., defining the normal matrix as $A=M^{*}M\in \mathbb {C} ^{N\times N}$ of size $N\times N$, takes advantage of the fact that for a given $N\times m$ matrix $W\in \mathbb {C} ^{N\times m}$ with orthonormal columns the eigenvalue problem of the Rayleigh–Ritz method for the $m\times m$ matrix
$W^{*}AW=W^{*}M^{*}MW=(MW)^{*}MW$
can be interpreted as a singular value problem for the $N\times m$ matrix $MW$. This interpretation allows simple simultaneous calculation of both left and right approximate singular vectors as follows.
1. Compute the $N\times m$ matrix $MW$.
2. Compute the thin, or economy-sized, SVD $MW=\mathbf {U} \Sigma \mathbf {V} _{h},$ with $N\times m$ matrix $\mathbf {U} $, $m\times m$ diagonal matrix $\Sigma $, and $m\times m$ matrix $\mathbf {V} _{h}$.
3. Compute the matrices of the Ritz left $U=\mathbf {U} $ and right $V_{h}=\mathbf {V} _{h}W^{*}$ singular vectors.
4. Output approximations $U,\Sigma ,V_{h}$, called the Ritz singular triplets, to selected singular values and the corresponding left and right singular vectors of the original matrix $M$ representing an approximate Truncated singular value decomposition (SVD) with left singular vectors restricted to the column-space of the matrix $W$.
The algorithm can be used as a post-processing step where the matrix $W$ is an output of an eigenvalue solver, e.g., such as LOBPCG, approximating numerically selected eigenvectors of the normal matrix $A=M^{*}M$.
Example
The matrix
$M={\begin{bmatrix}1&0&0&0\\0&2&0&0\\0&0&3&0\\0&0&0&4\\0&0&0&0\end{bmatrix}}$
has its normal matrix
$A=M^{*}M={\begin{bmatrix}1&0&0&0\\0&4&0&0\\0&0&9&0\\0&0&0&16\\\end{bmatrix}},$
singular values $1,2,3,4$ and the corresponding thin SVD
$A={\begin{bmatrix}0&0&0&1\\0&0&1&0\\0&1&0&0\\1&0&0&0\\0&0&0&0\end{bmatrix}}{\begin{bmatrix}4&0&0&0\\0&3&0&0\\0&0&2&0\\0&0&0&1\end{bmatrix}}{\begin{bmatrix}0&0&0&1\\0&0&1&0\\0&1&0&0\\1&0&0&0\end{bmatrix}},$
where the columns of the first multiplier from the complete set of the left singular vectors of the matrix $A$, the diagonal entries of the middle term are the singular values, and the columns of the last multiplier transposed (although the transposition does not change it)
${\begin{bmatrix}0&0&0&1\\0&0&1&0\\0&1&0&0\\1&0&0&0\end{bmatrix}}^{*}\quad =\quad {\begin{bmatrix}0&0&0&1\\0&0&1&0\\0&1&0&0\\1&0&0&0\end{bmatrix}}$
are the corresponding right singular vectors.
Let us take
$W={\begin{bmatrix}1/{\sqrt {2}}&1/{\sqrt {2}}\\1/{\sqrt {2}}&-1/{\sqrt {2}}\\0&0\\0&0\end{bmatrix}}$
with the column-space that is spanned by the two exact right singular vectors
${\begin{bmatrix}0&1\\1&0\\0&0\\0&0\end{bmatrix}}$
corresponding to the singular values 1 and 2.
Following the algorithm step 1, we compute
$MW={\begin{bmatrix}1/{\sqrt {2}}&1/{\sqrt {2}}\\{\sqrt {2}}&-{\sqrt {2}}\\0&0\\0&0\end{bmatrix}},$
and on step 2 its thin SVD $MW=\mathbf {U} {\Sigma }\mathbf {V} _{h}$ with
$\mathbf {U} ={\begin{bmatrix}0&1\\1&0\\0&0\\0&0\\0&0\end{bmatrix}},\quad \Sigma ={\begin{bmatrix}2&0\\0&1\end{bmatrix}},\quad \mathbf {V} _{h}={\begin{bmatrix}1/{\sqrt {2}}&-1/{\sqrt {2}}\\1/{\sqrt {2}}&1/{\sqrt {2}}\end{bmatrix}}.$
Thus we already obtain the singular values 2 and 1 from $\Sigma $ and from $\mathbf {U} $ the corresponding two left singular vectors $u$ as $[0,1,0,0,0]^{*}$ and $[1,0,0,0,0]^{*}$, which span the column-space of the matrix $W$, explaining why the approximations are exact for the given $W$.
Finally, step 3 computes the matrix $V_{h}=\mathbf {V} _{h}W^{*}$
$\mathbf {V} _{h}={\begin{bmatrix}1/{\sqrt {2}}&-1/{\sqrt {2}}\\1/{\sqrt {2}}&1/{\sqrt {2}}\end{bmatrix}}\,{\begin{bmatrix}1/{\sqrt {2}}&1/{\sqrt {2}}&0&0\\1/{\sqrt {2}}&-1/{\sqrt {2}}&0&0\end{bmatrix}}={\begin{bmatrix}0&1&0&0\\1&0&0&0\end{bmatrix}}$
recovering from its rows the two right singular vectors $v$ as $[0,1,0,0]^{*}$ and $[1,0,0,0]^{*}$. We validate the first vector: $Mv=\sigma u$
${\begin{bmatrix}1&0&0&0\\0&2&0&0\\0&0&3&0\\0&0&0&4\\0&0&0&0\end{bmatrix}}\,{\begin{bmatrix}0\\1\\0\\0\end{bmatrix}}=\,2\,{\begin{bmatrix}0\\1\\0\\0\\0\end{bmatrix}}$
and $M^{*}u=\sigma v$
${\begin{bmatrix}1&0&0&0&0\\0&2&0&0&0\\0&0&3&0&0\\0&0&0&4&0\end{bmatrix}}\,{\begin{bmatrix}0\\1\\0\\0\\0\end{bmatrix}}=\,2\,{\begin{bmatrix}0\\1\\0\\0\end{bmatrix}}.$
Thus, for the given matrix $W$ with its column-space that is spanned by two exact right singular vectors, we determine these right singular vectors, as well as the corresponding left singular vectors and the singular values, all exactly. For an arbitrary matrix $W$, we obtain approximate singular triplets which are optimal given $W$ in the sense of optimality of the Rayleigh–Ritz method.
See also
• Ritz method
• Rayleigh quotient
• Arnoldi iteration
Notes and references
1. Leissa, A.W. (2005). "The historical bases of the Rayleigh and Ritz methods". Journal of Sound and Vibration. 287 (4–5): 961–978. Bibcode:2005JSV...287..961L. doi:10.1016/j.jsv.2004.12.021.
2. Ilanko, Sinniah (2009). "Comments on the historical bases of the Rayleigh and Ritz methods". Journal of Sound and Vibration. 319 (1–2): 731–733. Bibcode:2009JSV...319..731I. doi:10.1016/j.jsv.2008.06.001.
3. Trefethen, Lloyd N.; Bau, III, David (1997). Numerical Linear Algebra. SIAM. p. 254. ISBN 978-0-89871-957-4.
External links
• Course on Calculus of Variations, has a section on Rayleigh–Ritz method.
|
Wikipedia
|
Sinuosity
Sinuosity, sinuosity index, or sinuosity coefficient of a continuously differentiable curve having at least one inflection point is the ratio of the curvilinear length (along the curve) and the Euclidean distance (straight line) between the end points of the curve. This dimensionless quantity can also be rephrased as the "actual path length" divided by the "shortest path length" of a curve. The value ranges from 1 (case of straight line) to infinity (case of a closed loop, where the shortest path length is zero or for an infinitely-long actual path[1]).
Interpretation
The curve must be continuous (no jump) between the two ends. The sinuosity value is really significant when the line is continuously differentiable (no angular point). The distance between both ends can also be evaluated by a plurality of segments according to a broken line passing through the successive inflection points (sinuosity of order 2).
The calculation of the sinuosity is valid in a 3-dimensional space (e.g. for the central axis of the small intestine), although it is often performed in a plane (with then a possible orthogonal projection of the curve in the selected plan; "classic" sinuosity on the horizontal plane, longitudinal profile sinuosity on the vertical plane).
The classification of a sinuosity (e.g. strong / weak) often depends on the cartographic scale of the curve (see the coastline paradox for further details) and of the object velocity which flowing therethrough (river, avalanche, car, bicycle, bobsleigh, skier, high speed train, etc.): the sinuosity of the same curved line could be considered very strong for a high speed train but low for a river. Nevertheless, it is possible to see a very strong sinuosity in the succession of few river bends, or of laces on some mountain roads.
Notable values
The sinuosity S of:
• 2 inverted continuous semicircles located in the same plane is $S={\tfrac {\pi }{2}}\approx 1.5708...$. It is independent of the circle radius;
• a sine function (over a whole number n of half-periods), which can be calculated by computing the sine curve's arclength on those periods, is $S=\textstyle {\tfrac {1}{n\pi }}\int _{0}^{n\pi }{\sqrt {1+(\cos x)^{2}}}dx\approx 1.216...$
With similar opposite arcs joints in the same plane, continuously differentiable:
Central angle Sinuosity
Degrees Radians Exact Decimal
30°${\frac {\pi }{6}}$ ${\frac {\pi }{3({\sqrt {6}}-{\sqrt {2}})}}$1.0115
60°${\frac {\pi }{3}}$ ${\frac {\pi }{3}}$1.0472
90°${\frac {\pi }{2}}$ ${\frac {\pi }{2{\sqrt {2}}}}$1.1107
120°${\frac {2\cdot \pi }{3}}$ ${\frac {2\cdot \pi }{3{\sqrt {3}}}}$1.2092
150°${\frac {5\cdot \pi }{6}}$ ${\frac {5\cdot \pi }{3({\sqrt {6}}+{\sqrt {2}})}}$1.3552
180°$\pi $ ${\frac {\pi }{2}}$1.5708
210°${\frac {7\cdot \pi }{6}}$ ${\frac {7\cdot \pi }{3({\sqrt {6}}+{\sqrt {2}})}}$1.8972
240°${\frac {4\cdot \pi }{3}}$ ${\frac {4\cdot \pi }{3{\sqrt {3}}}}$2.4184
270°${\frac {3\cdot \pi }{2}}$ ${\frac {3\cdot \pi }{2{\sqrt {2}}}}$3.3322
300°${\frac {5\cdot \pi }{3}}$ ${\frac {5\cdot \pi }{3}}$5.2360
330°${\frac {11\cdot \pi }{6}}$ ${\frac {11\cdot \pi }{3({\sqrt {6}}-{\sqrt {2}})}}$11.1267
Rivers
In studies of rivers, the sinuosity index is similar but not identical to the general form given above, being given by:
${\text{SI}}={\frac {\text{channel length}}{\text{downvalley length}}}$
The difference from the general form happens because the downvalley path is not perfectly straight. The sinuosity index can be explained, then, as the deviations from a path defined by the direction of maximum downslope. For this reason, bedrock streams that flow directly downslope have a sinuosity index of 1, and meandering streams have a sinuosity index that is greater than 1.[2]
It is also possible to distinguish the case where the stream flowing on the line could not physically travel the distance between the ends: in some hydraulic studies, this leads to assign a sinuosity value of 1 for a torrent flowing over rocky bedrock along a horizontal rectilinear projection, even if the slope angle varies.
For rivers, the conventional classes of sinuosity, SI, are:
• SI <1.05: almost straight
• 1.05 ≤ SI <1.25: winding
• 1.25 ≤ SI <1.50: twisty
• 1.50 ≤ SI: meandering
It has been claimed that river shapes are governed by a self-organizing system that causes their average sinuosity (measured in terms of the source-to-mouth distance, not channel length) to be π,[3] but this has not been borne out by later studies, which found an average value less than 2.[4]
See also
• Curvature
• Oxbow lake
References
1. Leopold, Luna B., Wolman, M.G., and Miller, J.P., 1964, Fluvial Processes in Geomorphology, San Francisco, W.H. Freeman and Co., 522p.
2. Mueller, Jerry (1968). "An Introduction to the Hydraulic and Topographic Sinuosity Indexes1". Annals of the Association of American Geographers. 58 (2): 371–385. doi:10.1111/j.1467-8306.1968.tb00650.x.
3. Stølum, Hans-Henrik (1996), "River Meandering as a Self-Organization Process", Science, 271 (5256): 1710–1713, Bibcode:1996Sci...271.1710S, doi:10.1126/science.271.5256.1710, S2CID 19219185.
4. Grime, James (March 14, 2015), "A meandering tale: the truth about pi and rivers", Alex Bellos's Adventures in Numberland, The Guardian.
River morphology
Large-scale features
• Alluvial plain
• Drainage basin
• Drainage system (geomorphology)
• Estuary
• Strahler number (stream order)
• River valley
• River delta
• River sinuosity
Alluvial rivers
• Anabranch
• Avulsion (river)
• Bar (river morphology)
• Braided river
• Channel pattern
• Cut bank
• Floodplain
• Meander
• Meander cutoff
• Mouth bar
• Oxbow lake
• Point bar
• Riffle
• Rapids
• Riparian zone
• River bifurcation
• River channel migration
• River mouth
• Slip-off slope
• Stream pool
• Thalweg
Bedrock river
• Canyon
• Knickpoint
• Plunge pool
Bedforms
• Ait
• Antidune
• Dune
• Current ripple
Regional processes
• Aggradation
• Base level
• Degradation (geology)
• Erosion and tectonics
• River rejuvenation
Mechanics
• Deposition (geology)
• Water erosion
• Exner equation
• Hack's law
• Helicoidal flow
• Playfair's law
• Sediment transport
• List of rivers that have reversed direction
• Category
• Portal
|
Wikipedia
|
Rng (algebra)
In mathematics, and more specifically in abstract algebra, a rng (or non-unital ring or pseudo-ring) is an algebraic structure satisfying the same properties as a ring, but without assuming the existence of a multiplicative identity. The term rng (IPA: /rʊŋ/) is meant to suggest that it is a ring without i, that is, without the requirement for an identity element.[1]
Algebraic structures
Group-like
• Group
• Semigroup / Monoid
• Rack and quandle
• Quasigroup and loop
• Abelian group
• Magma
• Lie group
Group theory
Ring-like
• Ring
• Rng
• Semiring
• Near-ring
• Commutative ring
• Domain
• Integral domain
• Field
• Division ring
• Lie ring
Ring theory
Lattice-like
• Lattice
• Semilattice
• Complemented lattice
• Total order
• Heyting algebra
• Boolean algebra
• Map of lattices
• Lattice theory
Module-like
• Module
• Group with operators
• Vector space
• Linear algebra
Algebra-like
• Algebra
• Associative
• Non-associative
• Composition algebra
• Lie algebra
• Graded
• Bialgebra
• Hopf algebra
There is no consensus in the community as to whether the existence of a multiplicative identity must be one of the ring axioms (see Ring (mathematics) § History). The term rng was coined to alleviate this ambiguity when people want to refer explicitly to a ring without the axiom of multiplicative identity.
A number of algebras of functions considered in analysis are not unital, for instance the algebra of functions decreasing to zero at infinity, especially those with compact support on some (non-compact) space.
Definition
Formally, a rng is a set R with two binary operations (+, ·) called addition and multiplication such that
• (R, +) is an abelian group,
• (R, ·) is a semigroup,
• Multiplication distributes over addition.
A rng homomorphism is a function f: R → S from one rng to another such that
• f(x + y) = f(x) + f(y)
• f(x · y) = f(x) · f(y)
for all x and y in R.
If R and S are rings, then a ring homomorphism R → S is the same as a rng homomorphism R → S that maps 1 to 1.
Examples
All rings are rngs. A simple example of a rng that is not a ring is given by the even integers with the ordinary addition and multiplication of integers. Another example is given by the set of all 3-by-3 real matrices whose bottom row is zero. Both of these examples are instances of the general fact that every (one- or two-sided) ideal is a rng.
Rngs often appear naturally in functional analysis when linear operators on infinite-dimensional vector spaces are considered. Take for instance any infinite-dimensional vector space V and consider the set of all linear operators f : V → V with finite rank (i.e. dim f(V) < ∞). Together with addition and composition of operators, this is a rng, but not a ring. Another example is the rng of all real sequences that converge to 0, with component-wise operations.
Also, many test function spaces occurring in the theory of distributions consist of functions decreasing to zero at infinity, like e.g. Schwartz space. Thus, the function everywhere equal to one, which would be the only possible identity element for pointwise multiplication, cannot exist in such spaces, which therefore are rngs (for pointwise addition and multiplication). In particular, the real-valued continuous functions with compact support defined on some topological space, together with pointwise addition and multiplication, form a rng; this is not a ring unless the underlying space is compact.
Example: even integers
The set 2Z of even integers is closed under addition and multiplication and has an additive identity, 0, so it is a rng, but it does not have a multiplicative identity, so it is not a ring.
In 2Z, the only multiplicative idempotent is 0, the only nilpotent is 0, and the only element with a reflexive inverse is 0.
Example: finite quinary sequences
The direct sum $ {\mathcal {T}}=\bigoplus _{i=1}^{\infty }\mathbf {Z} /5\mathbf {Z} $ equipped with coordinate-wise addition and multiplication is a rng with the following properties:
• Its idempotent elements form a lattice with no upper bound.
• Every element x has a reflexive inverse, namely an element y such that xyx = x and yxy = y.
• For every finite subset of ${\mathcal {T}}$, there exists an idempotent in ${\mathcal {T}}$ that acts as an identity for the entire subset: the sequence with a one at every position where a sequence in the subset has a non-zero element at that position, and zero in every other position.
Properties
• Ideals, quotient rings, and modules can be defined for rngs in the same manner as for rings.
• Working with rngs instead of rings complicates some related definitions, however. For example, in a ring R, the left ideal (f) generated by an element f, defined as the smallest left ideal containing f, is simply Rf, but if R is only a rng, then Rf might not contain f, so instead
$(f)=Rf+\mathbf {Z} f=\{af+nf:a\in R~{\text{and}}~n\in \mathbf {Z} \},$
where nf must be interpreted using repeated addition/subtraction since n need not represent an element of R. Similarly, the left ideal generated by elements f1, ..., fm of a rng R is
$(f_{1},\ldots ,f_{m})=\{a_{1}f_{1}+\cdots +a_{m}f_{m}+n_{1}f_{1}+\cdots n_{m}f_{m}:a_{i}\in R\;\mathrm {and} \;n_{i}\in \mathbf {Z} \},$
a formula that goes back to Emmy Noether.[2] Similar complications arise in the definition of submodule generated by a set of elements of a module.
• Some theorems for rings are false for rngs. For example, in a ring, every proper ideal is contained in a maximal ideal, so a nonzero ring always has at least one maximal ideal. Both these statements fail for rngs.
• A rng homomorphism f : R → S maps any idempotent element to an idempotent element.
• If f : R → S is a rng homomorphism from a ring to a rng, and the image of f contains a non-zero-divisor of S, then S is a ring, and f is a ring homomorphism.
Adjoining an identity element (Dorroh extension)
Every rng R can be enlarged to a ring R^ by adjoining an identity element. A general way in which to do this is to formally add an identity element 1 and let R^ consist of integral linear combinations of 1 and elements of R with the premise that none of its nonzero integral multiples coincide or are contained in R. That is, elements of R^ are of the form
n ⋅ 1 + r
where n is an integer and r ∈ R. Multiplication is defined by linearity:
(n1 + r1) ⋅ (n2 + r2) = n1n2 + n1r2 + n2r1 + r1r2.
More formally, we can take R^ to be the cartesian product Z × R and define addition and multiplication by
(n1, r1) + (n2, r2) = (n1 + n2, r1 + r2),
(n1, r1) · (n2, r2) = (n1n2, n1r2 + n2r1 + r1r2).
The multiplicative identity of R^ is then (1, 0). There is a natural rng homomorphism j : R → R^ defined by j(r) = (0, r). This map has the following universal property:
Given any ring S and any rng homomorphism f : R → S, there exists a unique ring homomorphism g : R^ → S such that f = gj.
The map g can be defined by g(n, r) = n · 1S + f(r).
There is a natural surjective ring homomorphism R^ → Z which sends (n, r) to n. The kernel of this homomorphism is the image of R in R^. Since j is injective, we see that R is embedded as a (two-sided) ideal in R^ with the quotient ring R^/R isomorphic to Z. It follows that
Every rng is an ideal in some ring, and every ideal of a ring is a rng.
Note that j is never surjective. So, even when R already has an identity element, the ring R^ will be a larger one with a different identity. The ring R^ is often called the Dorroh extension of R after the American mathematician Joe Lee Dorroh, who first constructed it.
The process of adjoining an identity element to a rng can be formulated in the language of category theory. If we denote the category of all rings and ring homomorphisms by Ring and the category of all rngs and rng homomorphisms by Rng, then Ring is a (nonfull) subcategory of Rng. The construction of R^ given above yields a left adjoint to the inclusion functor I : Ring → Rng. Notice that Ring is not a reflective subcategory of Rng because the inclusion functor is not full.
Properties weaker than having an identity
There are several properties that have been considered in the literature that are weaker than having an identity element, but not so general. For example:
• Rings with enough idempotents: A rng R is said to be a ring with enough idempotents when there exists a subset E of R given by orthogonal (i.e. ef = 0 for all e ≠ f in E) idempotents (i.e. e2 = e for all e in E) such that R = ⊕e∈E eR = ⊕e∈E Re.
• Rings with local units: A rng R is said to be a ring with local units in case for every finite set r1, r2, ..., rt in R we can find e in R such that e2 = e and eri = ri = rie for every i.
• s-unital rings: A rng R is said to be s-unital in case for every finite set r1, r2, ..., rt in R we can find s in R such that sri = ri = ris for every i.
• Firm rings: A rng R is said to be firm if the canonical homomorphism R ⊗R R → R given by r ⊗ s ↦ rs is an isomorphism.
• Idempotent rings: A rng R is said to be idempotent (or an irng) in case R2 = R, that is, for every element r of R we can find elements ri and si in R such that $ r=\sum _{i}r_{i}s_{i}$.
It is not hard to check that these properties are weaker than having an identity element and weaker than the previous one.
• Rings are rings with enough idempotents, using E = {1}. A ring with enough idempotents that has no identity is for example the ring of infinite matrices over a field with just a finite number of nonzero entries. The matrices that have just 1 over one element in the main diagonal and 0 otherwise are the orthogonal idempotents.
• Rings with enough idempotents are rings with local units just taking finite sums of the orthogonal idempotents to satisfy the definition.
• Rings with local units are in particular s-unital; s-unital rings are firm and firm rings are idempotent.
Rng of square zero
A rng of square zero is a rng R such that xy = 0 for all x and y in R.[3] Any abelian group can be made a rng of square zero by defining the multiplication so that xy = 0 for all x and y;[4] thus every abelian group is the additive group of some rng. The only rng of square zero with a multiplicative identity is the zero ring {0}.[4]
Any additive subgroup of a rng of square zero is an ideal. Thus a rng of square zero is simple if and only if its additive group is a simple abelian group, i.e., a cyclic group of prime order.[5]
Unital homomorphism
Given two unital algebras A and B, an algebra homomorphism
f : A → B
is unital if it maps the identity element of A to the identity element of B.
If the associative algebra A over the field K is not unital, one can adjoin an identity element as follows: take A × K as underlying K-vector space and define multiplication ∗ by
(x, r) ∗ (y, s) = (xy + sx + ry, rs)
for x, y in A and r, s in K. Then ∗ is an associative operation with identity element (0, 1). The old algebra A is contained in the new one, and in fact A × K is the "most general" unital algebra containing A, in the sense of universal constructions.
See also
• Semiring
Notes
1. Jacobson 1989, pp. 155–156.
2. Noether 1921, p. 30, §1.2
3. See Bourbaki (1998, p. 102), where it is called a pseudo-ring of square zero. Some other authors use the term "zero ring" to refer to any rng of square zero; see e.g. Szele (1949) and Kreinovich (1995).
4. Bourbaki 1998, p. 102
5. Zariski & Samuel 1958, p. 133
References
• Bourbaki, N. (1998). Algebra I, Chapters 1–3. Springer.
• Dummit, David S.; Foote, Richard M. (2003). Abstract Algebra (3rd ed.). Wiley. ISBN 978-0-471-43334-7.
• Dorroh, J. L. (1932). "Concerning Adjunctions to Algebras". Bull. Amer. Math. Soc. 38 (2): 85–88. doi:10.1090/S0002-9904-1932-05333-2.
• Jacobson, Nathan (1989). Basic algebra (2nd ed.). New York: W.H. Freeman. ISBN 0-7167-1480-9.
• Kreinovich, V. (1995). "If a polynomial identity guarantees that every partial order on a ring can be extended, then this identity is true only for a zero-ring". Algebra Universalis. 33 (2): 237–242. doi:10.1007/BF01190935. MR 1318988. S2CID 122388143.
• Herstein, I. N. (1996). Abstract Algebra (3rd ed.). Wiley. ISBN 978-0-471-36879-3.
• McCrimmon, Kevin (2004). A taste of Jordan algebras. Springer. ISBN 978-0-387-95447-9.
• Noether, Emmy (1921). "Idealtheorie in Ringbereichen" [Ideal theory in rings]. Mathematische Annalen (in German). 83 (1–2): 24–66. doi:10.1007/BF01464225. S2CID 121594471.
• Szele, Tibor (1949). "Zur Theorie der Zeroringe". Mathematische Annalen. 121: 242–246. doi:10.1007/bf01329628. MR 0033822. S2CID 122196446.
• Zariski, Oscar; Samuel, Pierre (1958). Commutative Algebra. Vol. 1. Van Nostrand.
|
Wikipedia
|
Mean line segment length
In geometry, the mean line segment length is the average length of a line segment connecting two points chosen uniformly at random in a given shape. In other words, it is the expected Euclidean distance between two random points, where each point in the shape is equally likely to be chosen.
Even for simple shapes such as a square or a triangle, solving for the exact value of their mean line segment lengths can be difficult because their closed-form expressions can get quite complicated. As an example, consider the following question:
What is the average distance between two randomly chosen points inside a square with side length 1?
While the question may seem simple, it has a fairly complicated answer; the exact value for this is ${\frac {2+{\sqrt {2}}+5\ln(1+{\sqrt {2}})}{15}}$.
Formal definition
The mean line segment length for an n-dimensional shape S may formally be defined as the expected Euclidean distance ||⋅|| between two random points x and y,[1]
$\mathbb {E} [\|x-y\|]={\frac {1}{\lambda (S)^{2}}}\int _{S}\int _{S}\|x-y\|\,d\lambda (x)\,d\lambda (y)$
where λ is the n-dimensional Lebesgue measure.
For the two-dimensional case, this is defined using the distance formula for two points (x1, y1) and (x2, y2)
${\frac {1}{\lambda (S)^{2}}}\iint _{S}\iint _{S}{\sqrt {(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}}}\,dx_{1}\,dy_{1}\,dx_{2}\,dy_{2}.$
Approximation methods
Since computing the mean line segment length involves calculating multidimensional integrals, various methods for numerical integration can be used to approximate this value for any shape.
One such method is the Monte Carlo method. To approximate the mean line segment length of a given shape, two points are randomly chosen in its interior and the distance is measured. After several repetitions of these steps, the average of these distances will eventually converge to the true value.
These methods can only give an approximation; they cannot be used to determine its exact value.
Formulas
Line segment
For a line segment of length d, the average distance between two points is 1/3d.[1]
Triangle
For a triangle with side lengths a, b, and c, the average distance between two points in its interior is given by the formula[2]
${\frac {4ss_{a}s_{b}s_{c}}{15}}\left[{\frac {1}{a^{3}}}\ln \left({\frac {s}{s_{a}}}\right)+{\frac {1}{b^{3}}}\ln \left({\frac {s}{s_{b}}}\right)+{\frac {1}{c^{3}}}\ln \left({\frac {s}{s_{c}}}\right)\right]+{\frac {a+b+c}{15}}+{\frac {(b+c)(b-c)^{2}}{30a^{2}}}+{\frac {(a+c)(a-c)^{2}}{30b^{2}}}+{\frac {(a+b)(a-b)^{2}}{30c^{2}}},$
where $s=(a+b+c)/2$ is the semiperimeter, and $s_{i}$ denotes $s-i$.
For an equilateral triangle with side length a, this is equal to
$\left({\frac {4+3\ln 3}{20}}\right)a\approx 0.364791843\ldots a.$
Square and rectangles
The average distance between two points inside a square with side length s is[3]
$\left({\frac {2+{\sqrt {2}}+5\ln(1+{\sqrt {2}})}{15}}\right)s\approx 0.521405433\ldots s.$
More generally, the mean line segment length of a rectangle with side lengths l and w is[1]
${\frac {1}{15}}\left[{\frac {l^{3}}{w^{2}}}+{\frac {w^{3}}{l^{2}}}+d\left(3-{\frac {l^{2}}{w^{2}}}-{\frac {w^{2}}{l^{2}}}\right)+{\frac {5}{2}}\left({\frac {w^{2}}{l}}\ln \left({\frac {l+d}{w}}\right)+{\frac {l^{2}}{w}}\ln \left({\frac {w+d}{l}}\right)\right)\right]$
where $d={\sqrt {l^{2}+w^{2}}}$ is the length of the rectangle's diagonal.
If the two points are instead chosen to be on different sides of the square, the average distance is given by[3][4]
$\left({\frac {2+{\sqrt {2}}+5\ln(1+{\sqrt {2}})}{9}}\right)s\approx 0.869009\ldots s.$
Cube and hypercubes
The average distance between points inside an n-dimensional unit hypercube is denoted as Δ(n), and is given as[5]
$\Delta (n)=\underbrace {\int _{0}^{1}\cdots \int _{0}^{1}} _{2n}{\sqrt {(x_{1}-y_{1})^{2}+(x_{2}-y_{2})^{2}+\cdots +(x_{n}-y_{n})^{2}}}\,dx_{1}\cdots \,dx_{n}\,dy_{1}\cdots \,dy_{n}$
The first two values, Δ(1) and Δ(2), refer to the unit line segment and unit square respectively.
For the three-dimensional case, the mean line segment length of a unit cube is also known as Robbins constant, named after David P. Robbins. This constant has a closed form,[6]
$\Delta (3)={\frac {4+17{\sqrt {2}}-6{\sqrt {3}}-7\pi }{105}}+{\frac {\ln(1+{\sqrt {2}})}{5}}+{\frac {2\ln(2+{\sqrt {3}})}{5}}.$
Its numerical value is approximately 0.661707182... (sequence A073012 in the OEIS)
Andersson et. al. (1976) showed that Δ(n) satisfies the bounds[7]
${\tfrac {1}{3}}n^{1/2}\leq \Delta (n)\leq ({\tfrac {1}{6}}n)^{1/2}{\sqrt {{\frac {1}{3}}\left[1+2\left(1-{\frac {3}{5n}}\right)^{1/2}\right]}}.$
Choosing points from two different faces of the unit cube also gives a result with a closed form, given by,[4]
${\frac {4+17{\sqrt {2}}-6{\sqrt {3}}-7\pi }{75}}+{\frac {7\ln {(1+{\sqrt {2}})}}{25}}+{\frac {14\ln {(2+{\sqrt {3}})}}{25}}.$
Circle and sphere
The average chord length between points on the circumference of a circle of radius r is[8]
${\frac {4}{\pi }}r\approx 1.273239544\ldots r$
And picking points on the surface of a sphere with radius r is [9]
${\frac {4}{3}}r$
Disks
The average distance between points inside a disk of radius r is[10]
${\frac {128}{45\pi }}r\approx 0.905414787\ldots r.$
The values for a half disk and quarter disk are also known.[11]
For a half disk of radius 1:
${\frac {64}{135}}{\frac {12\pi -23}{\pi ^{2}}}\approx 0.706053409\ldots $
For a quarter disk of radius 1:
${\frac {32}{135\pi ^{2}}}(6\ln {(2{\sqrt {2}}-2)}-94{\sqrt {2}}+48\pi +3)\approx 0.473877262\ldots $
Balls
For a three-dimensional ball, this is
${\frac {36}{35}}r\approx 1.028571428\ldots r.$
More generally, the mean line segment length of an n-ball is[1]
${\frac {2n}{2n+1}}\beta _{n}r$
where βn depends on the parity of n,
$\beta _{n}={\begin{cases}{\dfrac {2^{3n+1}\,(n/2)!^{2}\,n!}{(n+1)\,(2n)!\,\pi }}&({\text{for even }}n)\\{\dfrac {2^{n+1}\,n!^{3}}{(n+1)\,((n-1)/2)!^{2}\,(2n)!}}&({\text{for odd }}n)\end{cases}}$
General bounds
Burgstaller and Pillichshammer (2008) showed that for a compact subset of the n-dimensional Euclidean space with diameter 1, its mean line segment length L satisfies[1]
$L\leq {\sqrt {\frac {2n}{n+1}}}{\frac {2^{n-2}\Gamma (n/2)^{2}}{\Gamma (n-1/2){\sqrt {\pi }}}}$
where Γ denotes the gamma function. For n = 2, a stronger bound exists.
$L\leq {\frac {229}{800}}+{\frac {44}{75}}{\sqrt {2-{\sqrt {3}}}}+{\frac {19}{480}}{\sqrt {5}}=0.678442\ldots $
References
1. Burgstaller, Bernhard; Pillichshammer, Friedrich (2009). "The Average Distance Between Two Points". Bulletin of the Australian Mathematical Society. 80 (3): 353–359. doi:10.1017/S0004972709000707.
2. Weisstein, Eric W. "Triangle Line Picking". MathWorld.
3. Weisstein, Eric W. "Square Line Picking". MathWorld.
4. Bailey, David H.; Borwein, Jonathan M.; Kapoor, Vishaal; Weisstein, Eric W. (2006). "Ten Problems in Experimental Mathematics". The American Mathematical Monthly. 113 (6): 481–509. doi:10.2307/27641975. ISSN 0002-9890. JSTOR 27641975.
5. Weisstein, Eric W. "Hypercube Line Picking". MathWorld.
6. Robbins, David P.; Bolis, Theodore S. (1978), "Average distance between two points in a box (solution to elementary problem E2629)", American Mathematical Monthly, 85 (4): 277–278, doi:10.2307/2321177, JSTOR 2321177.
7. Anderssen, R. S.; Brent, R. P.; Daley, D. J.; Moran, P. A. P. (1976). "Concerning $\int _{0}^{1}\cdots \int _{0}^{1}(x_{1}^{2}+\cdots +x_{k}^{2})^{1/2}dx_{1}\cdots dx_{k}$ and a Taylor Series Method" (PDF). SIAM Journal on Applied Mathematics. 30 (1): 22–30. doi:10.1137/0130003.
8. Weisstein, Eric W. "Circle Line Picking". MathWorld.
9. Weisstein, Eric W. "Sphere Line Picking". MathWorld.
10. Weisstein, Eric W. "Disk Line Picking". MathWorld.
11. Weisstein, Eric W. "Circular Sector Line Picking". MathWorld.
External links
• Weisstein, Eric W. "Mean Line Segment Length". MathWorld.
|
Wikipedia
|
Robbins algebra
In abstract algebra, a Robbins algebra is an algebra containing a single binary operation, usually denoted by $\lor $, and a single unary operation usually denoted by $\neg $ satisfying the following axioms:
For all elements a, b, and c:
1. Associativity: $a\lor \left(b\lor c\right)=\left(a\lor b\right)\lor c$
2. Commutativity: $a\lor b=b\lor a$
3. Robbins equation: $\neg \left(\neg \left(a\lor b\right)\lor \neg \left(a\lor \neg b\right)\right)=a$
For many years, it was conjectured, but unproven, that all Robbins algebras are Boolean algebras. This was proved in 1996, so the term "Robbins algebra" is now simply a synonym for "Boolean algebra".
History
In 1933, Edward Huntington proposed a new set of axioms for Boolean algebras, consisting of (1) and (2) above, plus:
• Huntington's equation: $\neg (\neg a\lor b)\lor \neg (\neg a\lor \neg b)=a.$
From these axioms, Huntington derived the usual axioms of Boolean algebra.
Very soon thereafter, Herbert Robbins posed the Robbins conjecture, namely that the Huntington equation could be replaced with what came to be called the Robbins equation, and the result would still be Boolean algebra. $\lor $ would interpret Boolean join and $\neg $ Boolean complement. Boolean meet and the constants 0 and 1 are easily defined from the Robbins algebra primitives. Pending verification of the conjecture, the system of Robbins was called "Robbins algebra."
Verifying the Robbins conjecture required proving Huntington's equation, or some other axiomatization of a Boolean algebra, as theorems of a Robbins algebra. Huntington, Robbins, Alfred Tarski, and others worked on the problem, but failed to find a proof or counterexample.
William McCune proved the conjecture in 1996, using the automated theorem prover EQP. For a complete proof of the Robbins conjecture in one consistent notation and following McCune closely, see Mann (2003). Dahn (1998) simplified McCune's machine proof.
See also
• Algebraic structure
• Minimal axioms for Boolean algebra
References
• Dahn, B. I. (1998) Abstract to "Robbins Algebras Are Boolean: A Revision of McCune's Computer-Generated Solution of Robbins Problem," Journal of Algebra 208(2): 526–32.
• Mann, Allen (2003) "A Complete Proof of the Robbins Conjecture."
• William McCune, "Robbins Algebras Are Boolean," With links to proofs and other papers.
|
Wikipedia
|
Stochastic approximation
Stochastic approximation methods are a family of iterative methods typically used for root-finding problems or for optimization problems. The recursive update rules of stochastic approximation methods can be used, among other things, for solving linear systems when the collected data is corrupted by noise, or for approximating extreme values of functions which cannot be computed directly, but only estimated via noisy observations.
In a nutshell, stochastic approximation algorithms deal with a function of the form $ f(\theta )=\operatorname {E} _{\xi }[F(\theta ,\xi )]$ which is the expected value of a function depending on a random variable $ \xi $. The goal is to recover properties of such a function $ f$ without evaluating it directly. Instead, stochastic approximation algorithms use random samples of $ F(\theta ,\xi )$ to efficiently approximate properties of $ f$ such as zeros or extrema.
Recently, stochastic approximations have found extensive applications in the fields of statistics and machine learning, especially in settings with big data. These applications range from stochastic optimization methods and algorithms, to online forms of the EM algorithm, reinforcement learning via temporal differences, and deep learning, and others.[1] Stochastic approximation algorithms have also been used in the social sciences to describe collective dynamics: fictitious play in learning theory and consensus algorithms can be studied using their theory.[2]
The earliest, and prototypical, algorithms of this kind are the Robbins–Monro and Kiefer–Wolfowitz algorithms introduced respectively in 1951 and 1952.
Robbins–Monro algorithm
The Robbins–Monro algorithm, introduced in 1951 by Herbert Robbins and Sutton Monro,[3] presented a methodology for solving a root finding problem, where the function is represented as an expected value. Assume that we have a function $ M(\theta )$, and a constant $ \alpha $, such that the equation $ M(\theta )=\alpha $ has a unique root at $ \theta ^{*}$. It is assumed that while we cannot directly observe the function $ M(\theta )$, we can instead obtain measurements of the random variable $ N(\theta )$ where $ \operatorname {E} [N(\theta )]=M(\theta )$. The structure of the algorithm is to then generate iterates of the form:
$\theta _{n+1}=\theta _{n}-a_{n}(N(\theta _{n})-\alpha )$
Here, $a_{1},a_{2},\dots $ is a sequence of positive step sizes. Robbins and Monro proved[3], Theorem 2 that $\theta _{n}$ converges in $L^{2}$ (and hence also in probability) to $\theta ^{*}$, and Blum[4] later proved the convergence is actually with probability one, provided that:
• $ N(\theta )$ is uniformly bounded,
• $ M(\theta )$ is nondecreasing,
• $ M'(\theta ^{*})$ exists and is positive, and
• The sequence $ a_{n}$ satisfies the following requirements:
$\qquad \sum _{n=0}^{\infty }a_{n}=\infty \quad {\mbox{ and }}\quad \sum _{n=0}^{\infty }a_{n}^{2}<\infty \quad $
A particular sequence of steps which satisfy these conditions, and was suggested by Robbins–Monro, have the form: $ a_{n}=a/n$, for $ a>0$. Other series are possible but in order to average out the noise in $ N(\theta )$, the above condition must be met.
Complexity results
1. If $ f(\theta )$ is twice continuously differentiable, and strongly convex, and the minimizer of $ f(\theta )$ belongs to the interior of $ \Theta $, then the Robbins–Monro algorithm will achieve the asymptotically optimal convergence rate, with respect to the objective function, being $ \operatorname {E} [f(\theta _{n})-f^{*}]=O(1/n)$, where $ f^{*}$ is the minimal value of $ f(\theta )$ over $ \theta \in \Theta $.[5][6]
2. Conversely, in the general convex case, where we lack both the assumption of smoothness and strong convexity, Nemirovski and Yudin[7] have shown that the asymptotically optimal convergence rate, with respect to the objective function values, is $ O(1/{\sqrt {n}})$. They have also proven that this rate cannot be improved.
Subsequent developments and Polyak–Ruppert averaging
While the Robbins–Monro algorithm is theoretically able to achieve $ O(1/n)$ under the assumption of twice continuous differentiability and strong convexity, it can perform quite poorly upon implementation. This is primarily due to the fact that the algorithm is very sensitive to the choice of the step size sequence, and the supposed asymptotically optimal step size policy can be quite harmful in the beginning.[6][8]
Chung (1954)[9] and Fabian (1968)[10] showed that we would achieve optimal convergence rate $ O(1/{\sqrt {n}})$ with $ a_{n}=\bigtriangledown ^{2}f(\theta ^{*})^{-1}/n$ (or $ a_{n}={\frac {1}{(nM'(\theta ^{*}))}}$). Lai and Robbins[11][12] designed adaptive procedures to estimate $ M'(\theta ^{*})$ such that $ \theta _{n}$ has minimal asymptotic variance. However the application of such optimal methods requires much a priori information which is hard to obtain in most situations. To overcome this shortfall, Polyak (1991)[13] and Ruppert (1988)[14] independently developed a new optimal algorithm based on the idea of averaging the trajectories. Polyak and Juditsky[15] also presented a method of accelerating Robbins–Monro for linear and non-linear root-searching problems through the use of longer steps, and averaging of the iterates. The algorithm would have the following structure:
$\theta _{n+1}-\theta _{n}=a_{n}(\alpha -N(\theta _{n})),\qquad {\bar {\theta }}_{n}={\frac {1}{n}}\sum _{i=0}^{n-1}\theta _{i}$
The convergence of ${\bar {\theta }}_{n}$ to the unique root $\theta ^{*}$ relies on the condition that the step sequence $\{a_{n}\}$ decreases sufficiently slowly. That is A1)
$a_{n}\rightarrow 0,\qquad {\frac {a_{n}-a_{n+1}}{a_{n}}}=o(a_{n})$
Therefore, the sequence $ a_{n}=n^{-\alpha }$ with $ 0<\alpha <1$ satisfies this restriction, but $ \alpha =1$ does not, hence the longer steps. Under the assumptions outlined in the Robbins–Monro algorithm, the resulting modification will result in the same asymptotically optimal convergence rate $ O(1/{\sqrt {n}})$ yet with a more robust step size policy.[15] Prior to this, the idea of using longer steps and averaging the iterates had already been proposed by Nemirovski and Yudin[16] for the cases of solving the stochastic optimization problem with continuous convex objectives and for convex-concave saddle point problems. These algorithms were observed to attain the nonasymptotic rate $ O(1/{\sqrt {n}})$.
A more general result is given in Chapter 11 of Kushner and Yin[17] by defining interpolated time $ t_{n}=\sum _{i=0}^{n-1}a_{i}$, interpolated process $ \theta ^{n}(\cdot )$ and interpolated normalized process $ U^{n}(\cdot )$ as
$\theta ^{n}(t)=\theta _{n+i},\quad U^{n}(t)=(\theta _{n+i}-\theta ^{*})/{\sqrt {a_{n+i}}}\quad {\mbox{for}}\quad t\in [t_{n+i}-t_{n},t_{n+i+1}-t_{n}),i\geq 0$
Let the iterate average be $\Theta _{n}={\frac {a_{n}}{t}}\sum _{i=n}^{n+t/a_{n}-1}\theta _{i}$ and the associate normalized error to be ${\hat {U}}^{n}(t)={\frac {\sqrt {a_{n}}}{t}}\sum _{i=n}^{n+t/a_{n}-1}(\theta _{i}-\theta ^{*})$.
With assumption A1) and the following A2)
A2) There is a Hurwitz matrix $ A$ and a symmetric and positive-definite matrix $ \Sigma $ such that $ \{U^{n}(\cdot )\}$ converges weakly to $ U(\cdot )$, where $ U(\cdot )$ is the statisolution to
$dU=AU\,dt+\Sigma ^{1/2}\,dw$
where $ w(\cdot )$ is a standard Wiener process.
satisfied, and define $ {\bar {V}}=(A^{-1})'\Sigma (A')^{-1}$. Then for each $ t$,
${\hat {U}}^{n}(t){\stackrel {\mathcal {D}}{\longrightarrow }}{\mathcal {N}}(0,V_{t}),\quad {\text{where}}\quad V_{t}={\bar {V}}/t+O(1/t^{2}).$
The success of the averaging idea is because of the time scale separation of the original sequence $ \{\theta _{n}\}$ and the averaged sequence $ \{\Theta _{n}\}$, with the time scale of the former one being faster.
Application in stochastic optimization
Suppose we want to solve the following stochastic optimization problem
$g(\theta ^{*})=\min _{\theta \in \Theta }\operatorname {E} [Q(\theta ,X)],$
where $ g(\theta )=\operatorname {E} [Q(\theta ,X)]$ is differentiable and convex, then this problem is equivalent to find the root $\theta ^{*}$ of $\nabla g(\theta )=0$. Here $Q(\theta ,X)$ can be interpreted as some "observed" cost as a function of the chosen $\theta $ and random effects $X$. In practice, it might be hard to get an analytical form of $\nabla g(\theta )$, Robbins–Monro method manages to generate a sequence $(\theta _{n})_{n\geq 0}$ to approximate $\theta ^{*}$ if one can generate $(X_{n})_{n\geq 0}$ , in which the conditional expectation of $X_{n}$ given $\theta _{n}$ is exactly $\nabla g(\theta _{n})$, i.e. $X_{n}$ is simulated from a conditional distribution defined by
$\operatorname {E} [H(\theta ,X)|\theta =\theta _{n}]=\nabla g(\theta _{n}).$
Here $H(\theta ,X)$ is an unbiased estimator of $\nabla g(\theta )$. If $X$ depends on $\theta $, there is in general no natural way of generating a random outcome $H(\theta ,X)$ that is an unbiased estimator of the gradient. In some special cases when either IPA or likelihood ratio methods are applicable, then one is able to obtain an unbiased gradient estimator $H(\theta ,X)$. If $X$ is viewed as some "fundamental" underlying random process that is generated independently of $\theta $, and under some regularization conditions for derivative-integral interchange operations so that $\operatorname {E} {\Big [}{\frac {\partial }{\partial \theta }}Q(\theta ,X){\Big ]}=\nabla g(\theta )$, then $H(\theta ,X)={\frac {\partial }{\partial \theta }}Q(\theta ,X)$ gives the fundamental gradient unbiased estimate. However, for some applications we have to use finite-difference methods in which $H(\theta ,X)$ has a conditional expectation close to $\nabla g(\theta )$ but not exactly equal to it.
We then define a recursion analogously to Newton's Method in the deterministic algorithm:
$\theta _{n+1}=\theta _{n}-\varepsilon _{n}H(\theta _{n},X_{n+1}).$
Convergence of the algorithm
The following result gives sufficient conditions on $\theta _{n}$ for the algorithm to converge:[18]
C1) $\varepsilon _{n}\geq 0,\forall \;n\geq 0.$
C2) $\sum _{n=0}^{\infty }\varepsilon _{n}=\infty $
C3) $\sum _{n=0}^{\infty }\varepsilon _{n}^{2}<\infty $
C4) $|X_{n}|\leq B,{\text{ for a fixed bound }}B.$
C5) $g(\theta ){\text{ is strictly convex, i.e.}}$
$\inf _{\delta \leq |\theta -\theta ^{*}|\leq 1/\delta }\langle \theta -\theta ^{*},\nabla g(\theta )\rangle >0,{\text{ for every }}0<\delta <1.$
Then $\theta _{n}$ converges to $\theta ^{*}$ almost surely.
Here are some intuitive explanations about these conditions. Suppose $H(\theta _{n},X_{n+1})$ is a uniformly bounded random variables. If C2) is not satisfied, i.e. $\sum _{n=0}^{\infty }\varepsilon _{n}<\infty $ , then
$\theta _{n}-\theta _{0}=-\sum _{i=0}^{n-1}\varepsilon _{i}H(\theta _{i},X_{i+1})$
is a bounded sequence, so the iteration cannot converge to $\theta ^{*}$ if the initial guess $\theta _{0}$ is too far away from $\theta ^{*}$. As for C3) note that if $\theta _{n}$ converges to $\theta ^{*}$ then
$\theta _{n+1}-\theta _{n}=-\varepsilon _{n}H(\theta _{n},X_{n+1})\rightarrow 0,{\text{ as }}n\rightarrow \infty .$
so we must have $\varepsilon _{n}\downarrow 0$ ，and the condition C3) ensures it. A natural choice would be $\varepsilon _{n}=1/n$. Condition C5) is a fairly stringent condition on the shape of $g(\theta )$; it gives the search direction of the algorithm.
Example (where the stochastic gradient method is appropriate)[8]
Suppose $Q(\theta ,X)=f(\theta )+\theta ^{T}X$, where $f$ is differentiable and $X\in \mathbb {R} ^{p}$ is a random variable independent of $\theta $. Then $g(\theta )=\operatorname {E} [Q(\theta ,X)]=f(\theta )+\theta ^{T}\operatorname {E} X$ depends on the mean of $X$, and the stochastic gradient method would be appropriate in this problem. We can choose $H(\theta ,X)={\frac {\partial }{\partial \theta }}Q(\theta ,X)={\frac {\partial }{\partial \theta }}f(\theta )+X.$
Kiefer–Wolfowitz algorithm
The Kiefer–Wolfowitz algorithm was introduced in 1952 by Jacob Wolfowitz and Jack Kiefer,[19] and was motivated by the publication of the Robbins–Monro algorithm. However, the algorithm was presented as a method which would stochastically estimate the maximum of a function. Let $M(x)$ be a function which has a maximum at the point $\theta $. It is assumed that $M(x)$ is unknown; however, certain observations $N(x)$, where $\operatorname {E} [N(x)]=M(x)$, can be made at any point $x$. The structure of the algorithm follows a gradient-like method, with the iterates being generated as follows:
$x_{n+1}=x_{n}+a_{n}{\bigg (}{\frac {N(x_{n}+c_{n})-N(x_{n}-c_{n})}{2c_{n}}}{\bigg )}$
where $N(x_{n}+c_{n})$ and $N(x_{n}-c_{n})$ are independent, and the gradient of $M(x)$ is approximated using finite differences. The sequence $\{c_{n}\}$ specifies the sequence of finite difference widths used for the gradient approximation, while the sequence $\{a_{n}\}$ specifies a sequence of positive step sizes taken along that direction. Kiefer and Wolfowitz proved that, if $M(x)$ satisfied certain regularity conditions, then $x_{n}$ will converge to $\theta $ in probability as $n\to \infty $, and later Blum[4] in 1954 showed $x_{n}$ converges to $\theta $ almost surely, provided that:
• $\operatorname {Var} (N(x))\leq S<\infty $ for all $x$.
• The function $M(x)$ has a unique point of maximum (minimum) and is strong concave (convex)
• The algorithm was first presented with the requirement that the function $M(\cdot )$ maintains strong global convexity (concavity) over the entire feasible space. Given this condition is too restrictive to impose over the entire domain, Kiefer and Wolfowitz proposed that it is sufficient to impose the condition over a compact set $C_{0}\subset \mathbb {R} ^{d}$ which is known to include the optimal solution.
• The function $M(x)$ satisfies the regularity conditions as follows:
• There exists $\beta >0$ and $B>0$ such that
$|x'-\theta |+|x''-\theta |<\beta \quad \Longrightarrow \quad |M(x')-M(x'')|<B|x'-x''|$
• There exists $\rho >0$ and $R>0$ such that
$|x'-x''|<\rho \quad \Longrightarrow \quad |M(x')-M(x'')|<R$
• For every $\delta >0$, there exists some $\pi (\delta )>0$ such that
$|z-\theta |>\delta \quad \Longrightarrow \quad \inf _{\delta /2>\varepsilon >0}{\frac {|M(z+\varepsilon )-M(z-\varepsilon )|}{\varepsilon }}>\pi (\delta )$
• The selected sequences $\{a_{n}\}$ and $\{c_{n}\}$ must be infinite sequences of positive numbers such that
• $\quad c_{n}\rightarrow 0\quad {\text{as}}\quad n\to \infty $
• $\sum _{n=0}^{\infty }a_{n}=\infty $
• $\sum _{n=0}^{\infty }a_{n}c_{n}<\infty $
• $\sum _{n=0}^{\infty }a_{n}^{2}c_{n}^{-2}<\infty $
A suitable choice of sequences, as recommended by Kiefer and Wolfowitz, would be $a_{n}=1/n$ and $c_{n}=n^{-1/3}$.
Subsequent developments and important issues
1. The Kiefer Wolfowitz algorithm requires that for each gradient computation, at least $d+1$ different parameter values must be simulated for every iteration of the algorithm, where $d$ is the dimension of the search space. This means that when $d$ is large, the Kiefer–Wolfowitz algorithm will require substantial computational effort per iteration, leading to slow convergence.
1. To address this problem, Spall proposed the use of simultaneous perturbations to estimate the gradient. This method would require only two simulations per iteration, regardless of the dimension $d$.[20]
2. In the conditions required for convergence, the ability to specify a predetermined compact set that fulfills strong convexity (or concavity) and contains the unique solution can be difficult to find. With respect to real world applications, if the domain is quite large, these assumptions can be fairly restrictive and highly unrealistic.
Further developments
An extensive theoretical literature has grown up around these algorithms, concerning conditions for convergence, rates of convergence, multivariate and other generalizations, proper choice of step size, possible noise models, and so on.[21][22] These methods are also applied in control theory, in which case the unknown function which we wish to optimize or find the zero of may vary in time. In this case, the step size $a_{n}$ should not converge to zero but should be chosen so as to track the function.[21], 2nd ed., chapter 3
C. Johan Masreliez and R. Douglas Martin were the first to apply stochastic approximation to robust estimation.[23]
The main tool for analyzing stochastic approximations algorithms (including the Robbins–Monro and the Kiefer–Wolfowitz algorithms) is a theorem by Aryeh Dvoretzky published in the proceedings of the third Berkeley symposium on mathematical statistics and probability, 1956.[24]
See also
• Stochastic gradient descent
• Stochastic variance reduction
References
1. Toulis, Panos; Airoldi, Edoardo (2015). "Scalable estimation strategies based on stochastic approximations: classical results and new insights". Statistics and Computing. 25 (4): 781–795. doi:10.1007/s11222-015-9560-y. PMC 4484776. PMID 26139959.
2. Le Ny, Jerome. "Introduction to Stochastic Approximation Algorithms" (PDF). Polytechnique Montreal. Teaching Notes. Retrieved 16 November 2016.
3. Robbins, H.; Monro, S. (1951). "A Stochastic Approximation Method". The Annals of Mathematical Statistics. 22 (3): 400. doi:10.1214/aoms/1177729586.
4. Blum, Julius R. (1954-06-01). "Approximation Methods which Converge with Probability one". The Annals of Mathematical Statistics. 25 (2): 382–386. doi:10.1214/aoms/1177728794. ISSN 0003-4851.
5. Sacks, J. (1958). "Asymptotic Distribution of Stochastic Approximation Procedures". The Annals of Mathematical Statistics. 29 (2): 373–405. doi:10.1214/aoms/1177706619. JSTOR 2237335.
6. Nemirovski, A.; Juditsky, A.; Lan, G.; Shapiro, A. (2009). "Robust Stochastic Approximation Approach to Stochastic Programming". SIAM Journal on Optimization. 19 (4): 1574. doi:10.1137/070704277.
7. Problem Complexity and Method Efficiency in Optimization, A. Nemirovski and D. Yudin, Wiley -Intersci. Ser. Discrete Math 15 John Wiley New York (1983) .
8. Introduction to Stochastic Search and Optimization: Estimation, Simulation and Control, J.C. Spall, John Wiley Hoboken, NJ, (2003).
9. Chung, K. L. (1954-09-01). "On a Stochastic Approximation Method". The Annals of Mathematical Statistics. 25 (3): 463–483. doi:10.1214/aoms/1177728716. ISSN 0003-4851.
10. Fabian, Vaclav (1968-08-01). "On Asymptotic Normality in Stochastic Approximation". The Annals of Mathematical Statistics. 39 (4): 1327–1332. doi:10.1214/aoms/1177698258. ISSN 0003-4851.
11. Lai, T. L.; Robbins, Herbert (1979-11-01). "Adaptive Design and Stochastic Approximation". The Annals of Statistics. 7 (6): 1196–1221. doi:10.1214/aos/1176344840. ISSN 0090-5364.
12. Lai, Tze Leung; Robbins, Herbert (1981-09-01). "Consistency and asymptotic efficiency of slope estimates in stochastic approximation schemes". Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete. 56 (3): 329–360. doi:10.1007/BF00536178. ISSN 0044-3719. S2CID 122109044.
13. Polyak, B T (1990-01-01). "New stochastic approximation type procedures. (In Russian.)". 7 (7). {{cite journal}}: Cite journal requires |journal= (help)
14. Ruppert, D. "Efficient estimators from a slowly converging robbins-monro process". {{cite journal}}: Cite journal requires |journal= (help)
15. Polyak, B. T.; Juditsky, A. B. (1992). "Acceleration of Stochastic Approximation by Averaging". SIAM Journal on Control and Optimization. 30 (4): 838. doi:10.1137/0330046.
16. On Cezari's convergence of the steepest descent method for approximating saddle points of convex-concave functions, A. Nemirovski and D. Yudin, Dokl. Akad. Nauk SSR 2939, (1978 (Russian)), Soviet Math. Dokl. 19 (1978 (English)).
17. Kushner, Harold; George Yin, G. (2003-07-17). Stochastic Approximation and Recursive Algorithms and | Harold Kushner | Springer. ISBN 9780387008943. Retrieved 2016-05-16. {{cite book}}: |website= ignored (help)
18. Bouleau, N.; Lepingle, D. (1994). Numerical Methods for stochastic Processes. New York: John Wiley. ISBN 9780471546412.
19. Kiefer, J.; Wolfowitz, J. (1952). "Stochastic Estimation of the Maximum of a Regression Function". The Annals of Mathematical Statistics. 23 (3): 462. doi:10.1214/aoms/1177729392.
20. Spall, J. C. (2000). "Adaptive stochastic approximation by the simultaneous perturbation method". IEEE Transactions on Automatic Control. 45 (10): 1839–1853. doi:10.1109/TAC.2000.880982.
21. Kushner, H. J.; Yin, G. G. (1997). Stochastic Approximation Algorithms and Applications. doi:10.1007/978-1-4899-2696-8. ISBN 978-1-4899-2698-2.
22. Stochastic Approximation and Recursive Estimation, Mikhail Borisovich Nevel'son and Rafail Zalmanovich Has'minskiĭ, translated by Israel Program for Scientific Translations and B. Silver, Providence, RI: American Mathematical Society, 1973, 1976. ISBN 0-8218-1597-0.
23. Martin, R.; Masreliez, C. (1975). "Robust estimation via stochastic approximation". IEEE Transactions on Information Theory. 21 (3): 263. doi:10.1109/TIT.1975.1055386.
24. Dvoretzky, Aryeh (1956-01-01). "On Stochastic Approximation". The Regents of the University of California. {{cite journal}}: Cite journal requires |journal= (help)
Statistics
• Outline
• Index
Descriptive statistics
Continuous data
Center
• Mean
• Arithmetic
• Arithmetic-Geometric
• Cubic
• Generalized/power
• Geometric
• Harmonic
• Heronian
• Heinz
• Lehmer
• Median
• Mode
Dispersion
• Average absolute deviation
• Coefficient of variation
• Interquartile range
• Percentile
• Range
• Standard deviation
• Variance
Shape
• Central limit theorem
• Moments
• Kurtosis
• L-moments
• Skewness
Count data
• Index of dispersion
Summary tables
• Contingency table
• Frequency distribution
• Grouped data
Dependence
• Partial correlation
• Pearson product-moment correlation
• Rank correlation
• Kendall's τ
• Spearman's ρ
• Scatter plot
Graphics
• Bar chart
• Biplot
• Box plot
• Control chart
• Correlogram
• Fan chart
• Forest plot
• Histogram
• Pie chart
• Q–Q plot
• Radar chart
• Run chart
• Scatter plot
• Stem-and-leaf display
• Violin plot
Data collection
Study design
• Effect size
• Missing data
• Optimal design
• Population
• Replication
• Sample size determination
• Statistic
• Statistical power
Survey methodology
• Sampling
• Cluster
• Stratified
• Opinion poll
• Questionnaire
• Standard error
Controlled experiments
• Blocking
• Factorial experiment
• Interaction
• Random assignment
• Randomized controlled trial
• Randomized experiment
• Scientific control
Adaptive designs
• Adaptive clinical trial
• Stochastic approximation
• Up-and-down designs
Observational studies
• Cohort study
• Cross-sectional study
• Natural experiment
• Quasi-experiment
Statistical inference
Statistical theory
• Population
• Statistic
• Probability distribution
• Sampling distribution
• Order statistic
• Empirical distribution
• Density estimation
• Statistical model
• Model specification
• Lp space
• Parameter
• location
• scale
• shape
• Parametric family
• Likelihood (monotone)
• Location–scale family
• Exponential family
• Completeness
• Sufficiency
• Statistical functional
• Bootstrap
• U
• V
• Optimal decision
• loss function
• Efficiency
• Statistical distance
• divergence
• Asymptotics
• Robustness
Frequentist inference
Point estimation
• Estimating equations
• Maximum likelihood
• Method of moments
• M-estimator
• Minimum distance
• Unbiased estimators
• Mean-unbiased minimum-variance
• Rao–Blackwellization
• Lehmann–Scheffé theorem
• Median unbiased
• Plug-in
Interval estimation
• Confidence interval
• Pivot
• Likelihood interval
• Prediction interval
• Tolerance interval
• Resampling
• Bootstrap
• Jackknife
Testing hypotheses
• 1- & 2-tails
• Power
• Uniformly most powerful test
• Permutation test
• Randomization test
• Multiple comparisons
Parametric tests
• Likelihood-ratio
• Score/Lagrange multiplier
• Wald
Specific tests
• Z-test (normal)
• Student's t-test
• F-test
Goodness of fit
• Chi-squared
• G-test
• Kolmogorov–Smirnov
• Anderson–Darling
• Lilliefors
• Jarque–Bera
• Normality (Shapiro–Wilk)
• Likelihood-ratio test
• Model selection
• Cross validation
• AIC
• BIC
Rank statistics
• Sign
• Sample median
• Signed rank (Wilcoxon)
• Hodges–Lehmann estimator
• Rank sum (Mann–Whitney)
• Nonparametric anova
• 1-way (Kruskal–Wallis)
• 2-way (Friedman)
• Ordered alternative (Jonckheere–Terpstra)
• Van der Waerden test
Bayesian inference
• Bayesian probability
• prior
• posterior
• Credible interval
• Bayes factor
• Bayesian estimator
• Maximum posterior estimator
• Correlation
• Regression analysis
Correlation
• Pearson product-moment
• Partial correlation
• Confounding variable
• Coefficient of determination
Regression analysis
• Errors and residuals
• Regression validation
• Mixed effects models
• Simultaneous equations models
• Multivariate adaptive regression splines (MARS)
Linear regression
• Simple linear regression
• Ordinary least squares
• General linear model
• Bayesian regression
Non-standard predictors
• Nonlinear regression
• Nonparametric
• Semiparametric
• Isotonic
• Robust
• Heteroscedasticity
• Homoscedasticity
Generalized linear model
• Exponential families
• Logistic (Bernoulli) / Binomial / Poisson regressions
Partition of variance
• Analysis of variance (ANOVA, anova)
• Analysis of covariance
• Multivariate ANOVA
• Degrees of freedom
Categorical / Multivariate / Time-series / Survival analysis
Categorical
• Cohen's kappa
• Contingency table
• Graphical model
• Log-linear model
• McNemar's test
• Cochran–Mantel–Haenszel statistics
Multivariate
• Regression
• Manova
• Principal components
• Canonical correlation
• Discriminant analysis
• Cluster analysis
• Classification
• Structural equation model
• Factor analysis
• Multivariate distributions
• Elliptical distributions
• Normal
Time-series
General
• Decomposition
• Trend
• Stationarity
• Seasonal adjustment
• Exponential smoothing
• Cointegration
• Structural break
• Granger causality
Specific tests
• Dickey–Fuller
• Johansen
• Q-statistic (Ljung–Box)
• Durbin–Watson
• Breusch–Godfrey
Time domain
• Autocorrelation (ACF)
• partial (PACF)
• Cross-correlation (XCF)
• ARMA model
• ARIMA model (Box–Jenkins)
• Autoregressive conditional heteroskedasticity (ARCH)
• Vector autoregression (VAR)
Frequency domain
• Spectral density estimation
• Fourier analysis
• Least-squares spectral analysis
• Wavelet
• Whittle likelihood
Survival
Survival function
• Kaplan–Meier estimator (product limit)
• Proportional hazards models
• Accelerated failure time (AFT) model
• First hitting time
Hazard function
• Nelson–Aalen estimator
Test
• Log-rank test
Applications
Biostatistics
• Bioinformatics
• Clinical trials / studies
• Epidemiology
• Medical statistics
Engineering statistics
• Chemometrics
• Methods engineering
• Probabilistic design
• Process / quality control
• Reliability
• System identification
Social statistics
• Actuarial science
• Census
• Crime statistics
• Demography
• Econometrics
• Jurimetrics
• National accounts
• Official statistics
• Population statistics
• Psychometrics
Spatial statistics
• Cartography
• Environmental statistics
• Geographic information system
• Geostatistics
• Kriging
• Category
• Mathematics portal
• Commons
• WikiProject
|
Wikipedia
|
Robbins pentagon
In geometry, a Robbins pentagon is a cyclic pentagon whose side lengths and area are all rational numbers.
Unsolved problem in mathematics:
Can a Robbins pentagon have irrational diagonals?
(more unsolved problems in mathematics)
History
Robbins pentagons were named by Buchholz & MacDougall (2008) after David P. Robbins, who had previously given a formula for the area of a cyclic pentagon as a function of its edge lengths. Buchholz and MacDougall chose this name by analogy with the naming of Heron triangles after Hero of Alexandria, the discoverer of Heron's formula for the area of a triangle as a function of its edge lengths.
Area and perimeter
Every Robbins pentagon may be scaled so that its sides and area are integers. More strongly, Buchholz and MacDougall showed that if the side lengths are all integers and the area is rational, then the area is necessarily also an integer, and the perimeter is necessarily an even number.
Diagonals
Buchholz and MacDougall also showed that, in every Robbins pentagon, either all five of the internal diagonals are rational numbers or none of them are. If the five diagonals are rational (the case called a Brahmagupta pentagon by Sastry (2005)), then the radius of its circumscribed circle must also be rational, and the pentagon may be partitioned into three Heron triangles by cutting it along any two non-crossing diagonals, or into five Heron triangles by cutting it along the five radii from the circle center to its vertices.
Buchholz and MacDougall performed computational searches for Robbins pentagons with irrational diagonals but were unable to find any. On the basis of this negative result they suggested that Robbins pentagons with irrational diagonals may not exist.
References
• Buchholz, Ralph H.; MacDougall, James A. (2008), "Cyclic polygons with rational sides and area", Journal of Number Theory, 128 (1): 17–48, doi:10.1016/j.jnt.2007.05.005, MR 2382768.
• Robbins, David P. (1994), "Areas of polygons inscribed in a circle", Discrete and Computational Geometry, 12 (2): 223–236, doi:10.1007/BF02574377, MR 1283889
• Robbins, David P. (1995), "Areas of polygons inscribed in a circle", The American Mathematical Monthly, 102 (6): 523–530, doi:10.2307/2974766, JSTOR 2974766, MR 1336638.
• Sastry, K. R. S. (2005), "Construction of Brahmagupta n-gons" (PDF), Forum Geometricorum, 5: 119–126, MR 2195739.
|
Wikipedia
|
Robbins' theorem
In graph theory, Robbins' theorem, named after Herbert Robbins (1939), states that the graphs that have strong orientations are exactly the 2-edge-connected graphs. That is, it is possible to choose a direction for each edge of an undirected graph G, turning it into a directed graph that has a path from every vertex to every other vertex, if and only if G is connected and has no bridge.
This article is about Robbins' theorem in graph theory. For Robin's theorem in number theory, see divisor function.
Orientable graphs
Robbins' characterization of the graphs with strong orientations may be proven using ear decomposition, a tool introduced by Robbins for this task.
If a graph has a bridge, then it cannot be strongly orientable, for no matter which orientation is chosen for the bridge there will be no path from one of the two endpoints of the bridge to the other.
In the other direction, it is necessary to show that every connected bridgeless graph can be strongly oriented. As Robbins proved, every such graph has a partition into a sequence of subgraphs called "ears", in which the first subgraph in the sequence is a cycle and each subsequent subgraph is a path, with the two path endpoints both belonging to earlier ears in the sequence. (The two path endpoints may be equal, in which case the subgraph is a cycle.) Orienting the edges within each ear so that it forms a directed cycle or a directed path leads to a strongly connected orientation of the overall graph.[1]
Related results
An extension of Robbins' theorem to mixed graphs by Boesch & Tindell (1980) shows that, if G is a graph in which some edges may be directed and others undirected, and G contains a path respecting the edge orientations from every vertex to every other vertex, then any undirected edge of G that is not a bridge may be made directed without changing the connectivity of G. In particular, a bridgeless undirected graph may be made into a strongly connected directed graph by a greedy algorithm that directs edges one at a time while preserving the existence of paths between every pair of vertices; it is impossible for such an algorithm to get stuck in a situation in which no additional orientation decisions can be made.
Algorithms and complexity
A strong orientation of a given bridgeless undirected graph may be found in linear time by performing a depth-first search of the graph, orienting all edges in the depth-first search tree away from the tree root, and orienting all the remaining edges (which must necessarily connect an ancestor and a descendant in the depth-first search tree) from the descendant to the ancestor.[2] Although this algorithm is not suitable for parallel computers, due to the difficulty of performing depth-first search on them, alternative algorithms are available that solve the problem efficiently in the parallel model.[3] Parallel algorithms are also known for finding strongly connected orientations of mixed graphs.[4]
Applications
Robbins originally motivated his work by an application to the design of one-way streets in cities. Another application arises in structural rigidity, in the theory of grid bracing. This theory concerns the problem of making a square grid, constructed from rigid rods attached at flexible joints, rigid by adding more rods or wires as cross bracing on the diagonals of the grid. A set of added rods makes the grid rigid if an associated undirected graph is connected, and is doubly braced (remaining rigid if any edge is removed) if in addition it is bridgeless. Analogously, a set of added wires (which can bend to reduce the distance between the points they connect, but cannot expand) makes the grid rigid if an associated directed graph is strongly connected.[5] Therefore, reinterpreting Robbins' theorem for this application, the doubly braced structures are exactly the structures whose rods can be replaced by wires while remaining rigid.
Notes
1. Gross & Yellen (2006).
2. Vishkin (1985) credits this observation to Atallah (1984), and Balakrishnan (1996) credits it to Roberts (1978). But as Clark & Holton (1991) point out, the same algorithm is already included (with the assumption of 2-vertex-connectivity rather than 2-edge-connectivity) in the seminal earlier work of Hopcroft & Tarjan (1973) on depth-first search.
3. Vishkin (1985).
4. Soroker (1988).
5. Baglivo & Graver (1983).
References
• Atallah, Mikhail J. (1984), "Parallel strong orientation of an undirected graph", Information Processing Letters, 18 (1): 37–39, doi:10.1016/0020-0190(84)90072-3, MR 0742079.
• Baglivo, Jenny A.; Graver, Jack E. (1983), "3.10 Bracing structures", Incidence and Symmetry in Design and Architecture, Cambridge Urban and Architectural Studies, Cambridge, UK: Cambridge University Press, pp. 76–87, ISBN 9780521297844
• Balakrishnan, V. K. (1996), "4.6 Strong Orientation of Graphs", Introductory Discrete Mathematics, Mineola, NY: Dover Publications Inc., p. 135, ISBN 978-0-486-69115-2, MR 1402469.
• Boesch, Frank; Tindell, Ralph (1980), "Robbins's theorem for mixed multigraphs", The American Mathematical Monthly, 87 (9): 716–719, doi:10.2307/2321858, JSTOR 2321858, MR 0602828.
• Clark, John; Holton, Derek Allan (1991), "7.4 Traffic Flow", A first look at graph theory, Teaneck, NJ: World Scientific Publishing Co. Inc., pp. 254–260, ISBN 978-981-02-0489-1, MR 1119781.
• Gross, Jonathan L.; Yellen, Jay (2006), "Characterization of strongly orientable graphs", Graph Theory and its Applications, Discrete Mathematics and its Applications (2nd ed.), Boca Raton, FL: Chapman & Hall/CRC, pp. 498–499, ISBN 978-1-58488-505-4, MR 2181153.
• Hopcroft, John; Tarjan, Robert (1973), "Algorithm 447: efficient algorithms for graph manipulation", Communications of the ACM, 16 (6): 372–378, doi:10.1145/362248.362272, S2CID 14772567.
• Robbins, H. E. (1939), "A theorem on graphs, with an application to a problem on traffic control", American Mathematical Monthly, 46 (5): 281–283, doi:10.2307/2303897, JSTOR 2303897.
• Roberts, Fred S. (1978), "Chapter 2. The One-Way Street Problem", Graph Theory and its Applications to Problems of Society, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 29, Philadelphia, Pa.: Society for Industrial and Applied Mathematics (SIAM), pp. 7–14, ISBN 9780898710267, MR 0508050.
• Soroker, Danny (1988), "Fast parallel strong orientation of mixed graphs and related augmentation problems", Journal of Algorithms, 9 (2): 205–223, doi:10.1016/0196-6774(88)90038-7, MR 0936106.
• Vishkin, Uzi (1985), "On efficient parallel strong orientation", Information Processing Letters, 20 (5): 235–240, doi:10.1016/0020-0190(85)90025-0, MR 0801988.
|
Wikipedia
|
Robert Adrain
Robert Adrain (30 September 1775 – 10 August 1843) was an Irish political exile who won renown as a mathematician in the United States. He left Ireland after leading republican insurgents in the Rebellion of 1798, and settled in New Jersey and Pennsylvania. With Nathaniel Bowditch, he shares the distinction of being the first scholar to publish original mathematical research in America. This included his formulation of the method of least squares while working on a surveying problem (in two proofs of the exponential law of error published independently of Carl Friedrich Gauss) for which he is chiefly remembered.[1][2][3] His fields of applied mathematical interest included physics, astronomy and geodesy. Many of his mathematical investigations focussed on the shape of the Earth.[4]
Robert Adrain
portrait by Charles C. Ingham
Born30 September 1775
Carrickfergus, County Antrim, Ireland
Died10 August 1843(1843-08-10) (aged 67)
New Brunswick, New Jersey, US
Known forLeast squares method
Scientific career
FieldsDiophantine algebra
Statistics
InstitutionsQueen's College/Rutgers
Columbia College
University of Pennsylvania
Biography
Adrian was born in Carrickfergus, County Antrim, Ireland. His father, of French Huguenot descent, was a school teacher and maker of mathematical instruments, and he apparently received a good education until he was fifteen when both his parents died. He then supported himself and his four siblings by assuming his father's position as a teacher and as a private tutor.[3][4] In the cause of democratic reform and national independence, on 7 June 1798 he led a contingent of United Irishmen in the rebel army commanded by Henry Joy McCracken at the Battle of Antrim.[5] In the confrontation with British Crown forces, he was near fatally wounded by one of his own men. After being nursed back to health, with a bounty on his head he, his wife and infant child escaped to America.[3]
Although he was himself largely self-taught in mathematics, he secured a teaching position at the academy at Princeton, New Jersey. In 1801 he became president of the York County Academy in York, Pennsylvania. He wrote for the Mathematical Correspondent (edited by George Baron), the first mathematical journal in the United States, contributing the first article published in America on diophantine algebra. Later he twice attempted, in 1808 and 1814, to found his own journal, The Analyst, or, Mathematical Museum. While he failed to attract sufficient subscribers,[6] the first volume of the Analyst has been considered "the best collection of mathematical work produced in the United States up to that time". As well as from Adrian, it included contributions from Nathaniel Bowditch, Robert Patterson, John Gummere and Ferdinand Rudolph Hassler.[3] Recognition followed.
In 1809 Adrian was called to a professorship at Rutgers (then Queen's) College which, in 1810, awarded him an honorary M.A.. In 1812 he was elected a Fellow of the American Philosophical Society[7] and the next year, when he took a position at Columbia, of the American Academy of Arts and Sciences.[8] From 1827 he was Professor of Mathematics in the University of Pennsylvania
In 1825, he founded a somewhat more successful publication targeting a wider readership, The Mathematical Diary, which was published through 1832.[6] It was fashioned after the Correspondent, but at a higher level of mathematical involvement in problems solving and exposition.[4]
In 1834, Adrian was asked to resign from the University of Pennsylvania on grounds of class ill-discipline (instances of students overturning benches and throwing eggs). He returned to New Brunswick where he rented a school room and offered private tutoring until 1836. He then returned to New York and taught at the Columbia College Grammar School before retiring to New Brunswick in 1840 where three years later he died.[3]
It is suggested that Adrian, today, consistent with his conviction that "the last and highest department of mathematical science consists in its application to the laws and phenomenon of the natural world", would be considered an applied mathematician. Among his broad interests in physics, astronomy and geography, his paramount concern was dynamic geodesy, specifically the mathematical investigation of the shape of the earth.[4]
Adrain was the father of Congressman Garnett B. Adrain.[9] He is commemorated by a blue plaque, unveiled at Carrickfergus by the Ulster History Circle.
References
1. Dutka, Jacques (1990). "Robert Adrain and the method of least squares". Archive for History of Exact Sciences. Springer Science and Business Media LLC. 41 (2): 171–184. doi:10.1007/bf00411864. ISSN 0003-9519. S2CID 123167968.
2. Who Was Who in America, Historical Volume, 1607–1896. Marquis Who's Who. 1967.
3. Hogan, Edward R (1 May 1977). "Robert Adrian: American mathematician". Historia Mathematica. 4 (2): 157–172. doi:10.1016/0315-0860(77)90109-4. ISSN 0315-0860.
4. Swetz, Frank J. (2008). "The Mystery of Robert Adrain". Mathematics Magazine. 81 (5): (332–344) 333–336. doi:10.1080/0025570X.2008.11953574. ISSN 0025-570X. JSTOR 27643138. S2CID 126151584.
5. McConnell, Charles (1998), "The Aftermath of the Rising of 1798", in Archie Reid ed. The Liberty Tree:The story of the United Irishmen in and around the Borough of Newtownabbey, Newtownabbery Borough Council Bi-Centenary Publication, ISBN 0953337308, (65–68), 65
6. Parshall, Karen Hunger; David E. Rowe (1994). The Emergence of the American Mathematical Research Community, 1876–1900. American Mathematical Society. pp. 43–44. ISBN 0-8218-9004-2.
7. "APS Member History". search.amphilsoc.org. Retrieved 2 April 2021.
8. "Book of Members, 1780–2010: Chapter A" (PDF). American Academy of Arts and Sciences. Retrieved 6 April 2011.
9. Wilson, J. G.; Fiske, J., eds. (1900). "Adrain, Robert" . Appletons' Cyclopædia of American Biography. New York: D. Appleton.
Sources
• Webb, Alfred (1878). "Adrian, Robert" . A Compendium of Irish Biography. Dublin: M. H. Gill & son.
• Struik, D.J (1970). "Robert Adrain". Dictionary of Scientific Biography. Vol. 1. New York: Charles Scribner's Sons. pp. 65–66. ISBN 0-684-10114-9.
• Stigler, Stephen M. "Adrain, Robert". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/172. (Subscription or UK public library membership required.)
Attribution
• This article incorporates text from a publication now in the public domain: "Adrain, Robert". Dictionary of National Biography. London: Smith, Elder & Co. 1885–1900.
Further reading
• Robert Adrain. "Research concerning the probabilities of the errors which happen in making observations, &c". The Analyst, or Mathematical Museum. Vol. I, Article XIV, pp 93–109. Philadelphia: William P. Farrand and Co., 1808.
• Brian Hayes. "Science on the Farther Shore". American Scientist, 90(6):499, 2002. JSTOR 27857746
• Thomas Preveraud. « Vers des mathématiques américaines. Enseignements et éditions: de Robert Adrain à la genèse nationale d’une discipline (1800–1843). », université de Nantes, Centre François Viète.
• Stephen M. Stigler. "Mathematical statistics in the early States". Annals of Statistics, 6:239–265, 1978. doi:10.1214/aos/1176344123
External links
Wikimedia Commons has media related to Robert Adrain.
• MacTutor biography
• http://www.libraryireland.com/biography/RobertAdrain.php
Authority control
International
• FAST
• VIAF
National
• Germany
• United States
Academics
• MathSciNet
• zbMATH
People
• Ireland
Other
• SNAC
• IdRef
|
Wikipedia
|
Robert Alexander Rankin
Robert Alexander Rankin FRSE FRSAMD (27 October 1915 – 27 January 2001) was a Scottish mathematician who worked in analytic number theory.
Robert Rankin
Born(1915-10-27)27 October 1915
Garlieston, Scotland
Died27 January 2001(2001-01-27) (aged 85)
Glasgow, Scotland
Alma materClare College, Cambridge
AwardsSenior Whitehead Prize (1987)
De Morgan Medal (1998)
Scientific career
InstitutionsUniversity of Cambridge
University of Birmingham
University of Glasgow
Doctoral advisorG. H. Hardy and Albert Ingham
Doctoral studentsMichael P. Drazin
Life
Rankin was born in Garlieston in Wigtownshire the son of Rev Oliver Rankin (1885–1954), minister of Sorbie[1] and his wife, Olivia Theresa Shaw. His father took the name Oliver Shaw Rankin on marriage and became Professor of Old Testament Language, Literature and Theology in the University of Edinburgh.[2]
Rankin was educated at Fettes College then studied mathematics at Clare College, Cambridge, graduating in 1937. At Cambridge he was particularly influenced by J.E. Littlewood and A.E. Ingham.[1]
Rankin was elected a Fellow of Clare College in 1939, but his career was interrupted by the Second World War, during which he worked first for the Ministry of Supply then on rocketry research at Fort Halstead. In 1945 he returned to Cambridge as an assistant lecturer, and then moved to the University of Birmingham in 1951 as Mason professor of mathematics. In 1954 he became Professor of Mathematics, Glasgow University, retiring in 1982.[1]
In 1954 he was elected a Fellow of the Royal Society of Edinburgh. His proposers were William M. Smart, Robert Garry, James Norman Davidson and Robert Pollock Gillespie. He served as Vice President 1960 to 1963 and won the Society's Keith Prize for the period 1961–63.[2]
Rankin had a continuing interest in Srinivasa Ramanujan, working initially with G.H. Hardy on Ramanujan's unpublished notes. His research interests lay in the distribution of prime numbers and in modular forms. In 1939 he developed what is now known as the Rankin–Selberg method. In 1977 Cambridge University Press published Rankin's Modular Forms and Functions. In his review, Marvin Knopp wrote:
For, as much as any recent exposition of modular functions, this book succeeds in getting near the research frontier, and in some instances even reaches it – no small feat in this theory. Only someone of Rankin's stature as a research mathematician and experience in the classroom could aspire to such an accomplishment in a self-contained work – beginning with first principles.[3]
In 1987 Rankin received the Senior Whitehead Prize from the London Mathematical Society.[4]
Rankin died in Glasgow on 27 January 2001.[1]
Family
In 1942 he married Mary Ferrier Llewellyn.[1]
See also
• Rankin–Cohen bracket
Books
• An introduction to mathematical analysis, Pergamon Press 1963; Dover 2007.
• The modular group and its subgroups, Madras, Ramanujan Institute, 1969.
• Modular forms and functions, Cambridge University Press 1977
References
1. "Robert Rankin - Biography".
2. Biographical Index of Former Fellows of the Royal Society of Edinburgh 1783–2002 (PDF). The Royal Society of Edinburgh. July 2006. ISBN 0-902-198-84-X. Archived from the original (PDF) on 4 March 2016. Retrieved 3 February 2018.
3. Knopp, Marvin I. (1979). "Review: Modular forms and functions, by Robert A. Rankin; Modular functions and Dirichlet series in number theory, by Tom M. Apostol". Bull. Amer. Math. Soc. 1 (6): 935–943. doi:10.1090/S0273-0979-1979-14696-2.
4. List of Prizewinners from the London Mathematical Society
External links
• Robert Alexander Rankin at the Mathematics Genealogy Project
• O'Connor, John J.; Robertson, Edmund F., "Robert Alexander Rankin", MacTutor History of Mathematics Archive, University of St Andrews
De Morgan Medallists
• Arthur Cayley (1884)
• James Joseph Sylvester (1887)
• Lord Rayleigh (1890)
• Felix Klein (1893)
• S. Roberts (1896)
• William Burnside (1899)
• A. G. Greenhill (1902)
• H. F. Baker (1905)
• J. W. L. Glaisher (1908)
• Horace Lamb (1911)
• J. Larmor (1914)
• W. H. Young (1917)
• E. W. Hobson (1920)
• P. A. MacMahon (1923)
• A. E. H. Love (1926)
• Godfrey Harold Hardy (1929)
• Bertrand Russell (1932)
• E. T. Whittaker (1935)
• J. E. Littlewood (1938)
• Louis Mordell (1941)
• Sydney Chapman (1944)
• George Neville Watson (1947)
• A. S. Besicovitch (1950)
• E. C. Titchmarsh (1953)
• G. I. Taylor (1956)
• W. V. D. Hodge (1959)
• Max Newman (1962)
• Philip Hall (1965)
• Mary Cartwright (1968)
• Kurt Mahler (1971)
• Graham Higman (1974)
• C. Ambrose Rogers (1977)
• Michael Atiyah (1980)
• K. F. Roth (1983)
• J. W. S. Cassels (1986)
• D. G. Kendall (1989)
• Albrecht Fröhlich (1992)
• W. K. Hayman (1995)
• R. A. Rankin (1998)
• J. A. Green (2001)
• Roger Penrose (2004)
• Bryan John Birch (2007)
• Keith William Morton (2010)
• John Griggs Thompson (2013)
• Timothy Gowers (2016)
• Andrew Wiles (2019)
Authority control
International
• ISNI
• VIAF
National
• France
• BnF data
• Germany
• Israel
• United States
• Czech Republic
• Netherlands
• Poland
• Vatican
Academics
• DBLP
• MathSciNet
• Mathematics Genealogy Project
• zbMATH
People
• Deutsche Biographie
Other
• IdRef
|
Wikipedia
|
Robert B. Davis
Robert B. Davis (June 23, 1926 – December 21, 1997) was an American mathematician and mathematics educator.[1]
Davis was born in Fall River, Massachusetts.[2] He graduated from MIT with a B.S, M.S, and Ph.D. (1951) in mathematics. He was a professor and researcher at the University of New Hampshire, Syracuse University, the University of Illinois[3] and Rutgers University, where he was named New Jersey Professor of Mathematics Education in 1988.[4][5] He was one of the founders of the Madison Project, a study of mathematics education which spanned 15 years. The project is named for Madison Junior High School in Syracuse, where it began.[6] The project moved to Webster College near St, Louis, Missouri in 1961.[7]
Davis was the founding editor of The Journal of Mathematical Behavior (originally The Journal of Children's Mathematical Behavior), with Herbert Ginsburg in 1971.[1][6][4]
Davis was given the Ross Taylor/Glenn Gilbert National Leadership Award posthumously by the National Council of Supervisors of Mathematics in 1998.[8]
Selected publications
• Davis, R. B. (1964). Discovery In Mathematics: A Text For Teachers. Reading, Massachusetts: Addison-Wesley.[9]
• Davis, R. B. (1984). Learning Mathematics: The Cognitive Science Approach to Mathematics Education. Norwood, New Jersey: Ablex Publishing.[10]
• Davis, R. B.; Vinner, S. (1986). "The notion of limit: Some seemingly unavoidable misconception stages". The Journal of Mathematical Behavior. 5 (3): 281–303.
• Davis, R. B.; Maher, Carolyn A.; Noddings, Nel, eds. (1990). Constructivist Views on the Teaching and Learning of Mathematics. Journal for Research in Mathematics Education Monographs. Vol. 4. Reston, Virginia: National Council of Teachers of Mathematics.[11]
References
Archives at
IdentifiersR-MC 059
Sourcedigital description
How to use archival material
1. Kaput, James J. (March 1998). "Remembering Bob Davis" (PDF). Focus. Vol. 18, no. 3. Mathematical Association of America. p. 5.
2. Jacques Cattell Press; Dael L. Wolfle, eds. (1976). American Men and Women of Science. Vol. 2 (13 ed.). Bowker. p. 958. ISBN 0835208729.
3. Carolyn A. Maher; Robert Speiser (1998). "Robert Davis: In Memoriam" (PDF). Humanistic Mathematics Network Journal (17).
4. Ranzan, David (August 2006). "Guide to the Robert B. Davis Papers, 1957-1997". Special Collections and University Archives, Rutgers University Libraries. Rutgers University. Retrieved 2021-01-29.
5. Quinn, Laura (September 24, 1988). "Trying to transform the teaching of math". The Philadelphia Inquirer. p. 15.
6. Church, Phil (June 4, 2002). "Robert B. Davis Bio". Syracuse University Department of Mathematics. Retrieved March 11, 2017.
7. Angela Lynn Evans Walmsley (2003). A History of the "new Mathematics" Movement and Its Relationship with Current Mathematical Reform. University Press of America. p. 51. ISBN 9780761825128.
8. "Ross Taylor / Glenn Gilbert Gallery of Awardees". National Council of Supervisors of Mathematics. Retrieved 2021-01-29.
9. Reviews of Discovery In Mathematics:
• Storer, W. O. (February 1968). The Mathematical Gazette. 52 (379): 71–72. doi:10.2307/3614490. JSTOR 3614490. S2CID 125686468.{{cite journal}}: CS1 maint: untitled periodical (link)
• Wallace, Martha (November 1981). American Mathematical Monthly. 88 (9): 715. JSTOR 2320692.{{cite journal}}: CS1 maint: untitled periodical (link)
10. Reviews of Learning Mathematics:
• Brown, Margaret (March 1986). The Mathematical Gazette. 70 (451): 54–55. doi:10.2307/3615836. JSTOR 3615836.{{cite journal}}: CS1 maint: untitled periodical (link)
• Desforges, Charles (1985). British Educational Research Journal. 11 (3): 313–314. JSTOR 1500567.{{cite journal}}: CS1 maint: untitled periodical (link)
• Johnson, Jerry (March 1985). The Mathematics Teacher. 78 (3): 225–226. JSTOR 27964462.{{cite journal}}: CS1 maint: untitled periodical (link)
• Kaput, James J. (March 1985). "Minds, machines, mathematics, and metaphors". Journal for Research in Mathematics Education. 16 (2): 146–153. doi:10.2307/748372. JSTOR 748372.
• Kaput, James J. (September 1985). The College Mathematics Journal. 16 (4): 319–322. doi:10.2307/2686171. JSTOR 2686171.{{cite journal}}: CS1 maint: untitled periodical (link)
• Little, John (November 1, 1984). "New ways to learn old tricks". New Scientist: 53.
• Mason, John (March 1985). Instructional Science. 13 (4): 370–371. JSTOR 23369023.{{cite journal}}: CS1 maint: untitled periodical (link)
• Ormell, Chris (October 1985). British Journal of Educational Studies. 33 (3): 313–314. doi:10.2307/3121248. JSTOR 3121248.{{cite journal}}: CS1 maint: untitled periodical (link)
• Sowder, Larry (April 1985). The Arithmetic Teacher. 32 (8): 49. JSTOR 41192646.{{cite journal}}: CS1 maint: untitled periodical (link)
11. Reviews of Constructivist Views on the Teaching and Learning of Mathematics:
• Dawson, A. J. (Sandy) (October 1991). Educational Studies in Mathematics. 22 (5): 491–501. JSTOR 3482468.{{cite journal}}: CS1 maint: untitled periodical (link)
• Nattrass, George (September 1991). The Arithmetic Teacher. 39 (1): 49. ProQuest 208771288.{{cite journal}}: CS1 maint: untitled periodical (link)
Authority control
International
• FAST
• ISNI
• VIAF
National
• Germany
• Israel
• United States
• Netherlands
Academics
• MathSciNet
• Mathematics Genealogy Project
• zbMATH
|
Wikipedia
|
Robert Baldwin Hayward
Robert Baldwin Hayward (7 March 1829 – 2 February 1903) was an English educator and mathematician.
Life
Born on 7 March 1829, at Bocking, Essex, he was son of Robert Hayward by his wife Ann Baldwin; his father, from an old Quaker family, withdrew from the Quaker community on his marriage. Educated at University College, London, entered St John's College, Cambridge, in 1846, graduating as fourth wrangler in 1850. He was fellow from 30 March 1852 till 27 March 1860, and from 1852 till 1855 assistant tutor.[1][2]
From 1855 Baldwin was mathematical tutor and reader in natural philosophy at Durham University, leaving in 1859 to become a mathematical master at Harrow School. Hayward remained at Harrow till 1893, a period of 35 years. He reformed mathematics teaching there. He was president (1878–89) of the Association for the Improvement of Geometrical Teaching (afterwards the Mathematical Association).[1]
Hayward was a mountain climber and an original member of the Alpine Club from its foundation in 1858, withdrawing in 1865. He died at Shanklin, Isle of Wight, on 2 February 1903.[1]
Works
Baldwin published in 1895 a pamphlet, Hints on teaching Arithmetic. He was author of a text-book on Elementary Solid Geometry (1890) and The Algebra of Coplanar Vectors and Trigonometry (1899). In pure mathematics he published papers in the Transactions of the Cambridge Philosophical Society and the Quarterly Journal of Mathematics. He was elected Fellow of the Royal Society on 1 June 1876, in recognition of his work on the method of moving axes. He is known also for an article "Proportional Representation" in the Nineteenth century (February 1884).[1][3]
Family
Hayward married in 1860 Marianne, daughter of Henry Rowe, of Cambridge; his wife's sister married Henry William Watson. He had issue two sons and four daughters, including Sir Maurice Henry Weston Hayward, K.C.S.I., colonial administrator in India.[1][4]
Notes
1. Lee, Sidney, ed. (1912). "Hayward, Robert Baldwin" . Dictionary of National Biography (2nd supplement). Vol. 2. London: Smith, Elder & Co.
2. "Hayward, Robert Baldwin (HWRT846RB)". A Cambridge Alumni Database. University of Cambridge.
3. Price, Michael H. "Hayward, Robert Baldwin". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/33779. (Subscription or UK public library membership required.)
4. "Hayward, Maurice Henry Weston (HWRT886MH)". A Cambridge Alumni Database. University of Cambridge.
Attribution
This article incorporates text from a publication now in the public domain: Lee, Sidney, ed. (1912). "Hayward, Robert Baldwin". Dictionary of National Biography (2nd supplement). Vol. 2. London: Smith, Elder & Co.
Authority control
International
• ISNI
• VIAF
National
• Israel
• United States
Academics
• zbMATH
|
Wikipedia
|
Robert Berger (mathematician)
Robert Berger (born 1938) is an applied mathematician, known for discovering the first aperiodic tiling[1] using a set of 20,426 distinct tile shapes.
Contributions to tiling theory
The unexpected existence of aperiodic tilings, although not Berger's explicit construction of them, follows from another result proved by Berger: that the so-called domino problem is undecidable, disproving a conjecture of Hao Wang, Berger's advisor. The result is analogous to a 1962 construction used by Kahr, Moore, and Wang, to show that a more constrained version of the domino problem was undecidable.[2]
Education and career
Berger did his undergraduate studies at Rensselaer Polytechnic Institute, and studied applied physics at Harvard, earning a master's degree, before shifting to applied mathematics for his doctorate. Along with Hao Wang, Berger's other two doctoral committee members were Patrick Carl Fischer and Marvin Minsky. Later, he has worked in the Digital Integrated Circuits Group of the Lincoln Laboratory.[3]
Publications
Berger's work on tiling was published as "The Undecidability of the Domino Problem" in the Memoirs of the AMS in 1966.[4] This paper is essentially a reprint of Berger's 1964 dissertation at Harvard University.[5]
In 2009, a paper by Berger and other Lincoln Laboratories researchers, "Wafer-scale 3D integration of InGaAs image sensors with Si readout circuits", won the best paper award at the IEEE International 3D System Integration Conference (3DIC).[6] In 2010, a CMOS infrared imaging device with an analog-to-digital converter in each pixel, coinvented by Berger, was one of R&D Magazine's R&D 100 Award recipients.[7]
References
1. Darling, David J. (2004). The universal book of mathematics: from Abracadabra to Zeno's paradoxes. John Wiley and Sons. pp. 18–. ISBN 978-0-471-27047-8. Retrieved 29 September 2011.
2. Büchi, J. R. "The undecidability of the domino problem". Mathematical Reviews. 36 (49). MR 0216954.
3. Author biography from Raffel, J. I.; Mann, J. R.; Berger, R.; Soares, A. M.; Gilbert, S. (1989), "A generic architecture for wafer-scale neuromorphic systems" (PDF), The Lincoln Laboratory Journal, 2 (1): 63–76.
4. Berger, Robert (1966), "The Undecidability of the Domino Problem", Memoirs of the American Mathematical Society, 66 (66): 72 pp, doi:10.1090/memo/0066.
5. Robert Berger at the Mathematics Genealogy Project.
6. Awards and Recognition, Lincoln Laboratory Annual Report 2010, p. 50, retrieved 2011-09-30.
7. MIT Lincoln Laboratory receives five R&D 100 Awards, Lincoln Laboratory, retrieved 2011-09-30.
Authority control
International
• ISNI
• VIAF
National
• Catalonia
• Israel
• United States
• Netherlands
Academics
• MathSciNet
• Mathematics Genealogy Project
• zbMATH
Other
• IdRef
|
Wikipedia
|
Robert Bryant (mathematician)
Robert Leamon Bryant (born August 30, 1953, Kipling) is an American mathematician. He works at Duke University and specializes in differential geometry.[2]
Robert L. Bryant
Bryant at Oberwolfach in 2007
Born
Robert Leamon Bryant
(1953-08-30) August 30, 1953
Kipling, North Carolina, U.S.
NationalityAmerican
Alma materNorth Carolina State University at Raleigh
University of North Carolina at Chapel Hill
Known forBryant surface
Bryant soliton
AwardsSloan Research Fellowship, 1982
Scientific career
FieldsMathematics
InstitutionsDuke University
University of California, Berkeley
Rice University
Mathematical Sciences Research Institute
ThesisSome Aspects of the Local and Global Theory of Pfaffian Systems (1979)
Doctoral advisorRobert Brown Gardner
Doctoral studentsJeanne N. Clelland
Websitefds.duke.edu/db/aas/math/bryant
Education and career
Bryant grew up in a farming family in Harnett County and was a first-generation college student.[3] He obtained a bachelor's degree at North Caroline State University at Raleigh in 1974 and a PhD at University of North Carolina at Chapel Hill in 1979. His thesis was entitled "Some Aspects of the Local and Global Theory of Pfaffian Systems" and was written under the supervision of Robert Gardner.[4]
He worked at Rice University for seven years, as assistant professor (1979–1981), associate professor (1981–1982) and full professor (1982–1986). He then moved to Duke University, where he worked for twenty years as J. M. Kreps Professor.
Between 2007 and 2013 he worked as full professor at University of California, Berkeley, where he served as the director of the Mathematical Sciences Research Institute (MSRI).[5] In 2013 he returned to Duke University as Phillip Griffiths Professor of Mathematics.
Bryant was awarded in 1982 a Sloan Research Fellowship.[6] In 1986 he was invited speaker at the International Congress of Mathematicians in Berkeley.[7]
He was elected in 2002 a fellow of the American Academy of Arts and Sciences,[8] in 2007 a member of the National Academy of Sciences,[9] in 2013 a fellow of the American Mathematical Society[10] and in 2022 a fellow of the American Association for the Advancement of Science.[11][12] He is also a member of the Association for Women in Mathematics, the National Association of Mathematicians and the Mathematical Association of America.[13]
He served as the president of the American Mathematical Society for the 2-years term 2015–2016,[14][3] for which he was the first openly gay president.[3][15]
Bryant is on the board of directors of EDGE, a transition program for women entering graduate studies in the mathematical sciences.[16] He is also a board member of Spectra, an association for LGBT mathematicians that he helped to create.[17][18]
Research
Bryant's research has been influenced by Élie Cartan, Shiing-Shen Chern, and Phillip Griffiths.[3] His research interests cover many areas in Riemannian geometry, geometry of PDEs, Finsler geometry and mathematical physics.[19]
In 1987 he proved several properties of surfaces of unit constant mean curvature in hyperbolic space, which are now called Bryant surfaces in his honour.[20] In 2001 he contributed many advancements to the theory of Bochner-Kähler metrics, the class of Kähler metrics whose Bochner curvature vanishes.[21]
In 1987 he produced the first examples of Riemannian metrics with exceptional holonomy (i.e. whose holonomy groups are G2 or Spin(7)); this showed that every group in Marcel Berger's classification can arise as a holonomy group.[22] Later, he also contributed to the classification of exotic holonomy groups of arbitrary (i.e. non-Riemannian) torsion-free affine connections.[23][24]
Together with Phillip Griffiths and others co-authors, Bryant developed the modern theory of Exterior Differential Systems, writing two influential monographs, which have become the standard reference in the topic.[25][26] He also worked on their cohomology[27][28] and applications to PDEs.[29][30]
He is author of more than 60 papers,[31][32] and he has supervised 26 PhD students.[4]
Books
• A sampler of Riemann-Finsler Geometry, Cambridge University Press 2004 (editor with David Bao, S. S. Chern, Zhongmin Shen)
• Exterior Differential Systems, MSRI Publ. 18, Springer Verlag 1991, ISBN 0-226-07794-2 (with Robert Brown Gardner, S. S. Chern, H. L. Goldschmidt and Phillip Griffiths)
• Exterior Differential Systems and Euler-Lagrange Partial Differential Equations, Chicago Lectures in Mathematics, University of Chicago Press 2003, ISBN 0226077934 (with Phillip Griffiths and Dan Grossman)[33]
• Integral Geometry, Contemporary Mathematics 63, AMS 1987 (editor with Victor Guillemin, Sigurdur Helgason, R. O. Wells)
• An introduction to Lie groups and symplectic geometry, in Geometry and quantum field theory, IAS/Park City Math. Series 1, American Mathematical Society 1995, pp. 5–181
• Toward a Geometry of Differential Equations, in: Geometry, Topology & Physics, Conf. Proc. Lecture Notes Geom. Topology, VI, International Press, Cambridge, MA, 1995, pp. 1–76 (with Lucas Hsu and Phillip Griffiths)
Bryant and David Morrison are the editors of vol. 4 of the Selected Works of Phillip Griffiths.
References
1. Kusner, Rob (1987). "Conformal geometry and complete minimal surfaces". Bulletin of the American Mathematical Society. 17 (2): 291–295. doi:10.1090/S0273-0979-1987-15564-9. ISSN 0273-0979.
2. "Robert Bryant, Phillip Griffiths Professor of Mathematics and Professor of Computer Science and Chair".
3. "AMS Presidents: Robert Bryant" (PDF). American Mathematical Society. Retrieved August 6, 2021.
4. "Robert Bryant – The Mathematics Genealogy Project". www.mathgenealogy.org. Retrieved August 6, 2021.
5. "Biography: Robert Bryant". MSRI. 2008. Archived from the original on September 17, 2009.
6. "Past Fellows | Alfred P. Sloan Foundation". sloan.org. Archived from the original on March 14, 2018. Retrieved August 6, 2021.
7. Gleason, Andrew, ed. (1987). Proceedings of the International Congress of Mathematician 1986 (PDF). Berkley. p. 505.{{cite book}}: CS1 maint: location missing publisher (link)
8. "Robert L. Bryant". American Academy of Arts & Sciences. Retrieved August 6, 2021.
9. "Robert L. Bryant". www.nasonline.org. Retrieved August 6, 2021.
10. List of Fellows of the American Mathematical Society, retrieved 2012-11-10.
11. "Robert Bryant Named AAAS Fellow". Department of Mathematics. Retrieved January 30, 2022.
12. "Five Duke Faculty Named AAAS Fellows for 2021". today.duke.edu. Retrieved January 30, 2022.
13. "Members | Mathematical Association of America". www.maa.org. Retrieved January 30, 2022.
14. "Bryant Begins Term as AMS President". American Mathematical Society, Homepage. February 3, 2015.
15. Adriana Salerno (June 28, 2017). "Love simeq love : A celebration of LGBT+ Mathematicians". Retrieved October 25, 2021.
16. "Board of Directors". EDGE Foundation. Retrieved August 6, 2021.{{cite web}}: CS1 maint: url-status (link)
17. "Spectra". Retrieved September 30, 2019.
18. Bryant, Buckmire, Khadjavi, and Lind. ""The Origins of Spectra, an Organization for LGBT Mathematicians"" (PDF).{{cite news}}: CS1 maint: uses authors parameter (link)
19. "Robert Bryant – Simons Collaboration on Special Holonomy in Geometry, Analysis, and Physics". Retrieved January 30, 2022.
20. Bryant, Robert (1987). "Surfaces of mean curvature one in hyperbolic space". Astérisque. 154–155: 27. Zbl 0635.53047.
21. Bryant, Robert (2001). "Bochner-Kähler Metrics". Journal of the American Mathematical Society. 14 (3): 623–715. arXiv:math/0003099. doi:10.1090/S0894-0347-01-00366-6. JSTOR 827103. S2CID 119625517.
22. Bryant, Robert L. (1987). "Metrics with Exceptional Holonomy". Annals of Mathematics. 126 (3): 525–576. doi:10.2307/1971360. ISSN 0003-486X. JSTOR 1971360.
23. Bryant, Robert L. (1991), "Two exotic holonomies in dimension four, path geometries, and twistor theory", Complex Geometry and Lie Theory, Proceedings of Symposia in Pure Mathematics, vol. 53, Providence, Rhode Island: American Mathematical Society, pp. 33–88, doi:10.1090/pspum/053/1141197, ISBN 978-0-8218-1492-5, retrieved August 8, 2021
24. Bryant, Robert L. (2000). "Recent Advances in the Theory of Holonomy". Astérisque, Séminaire Bourbaki. 266: 351–374. arXiv:math/9910059.
25. Bryant, Robert L.; Chern, S. S.; Gardner, Robert B.; Goldschmidt, Hubert L.; Griffiths, P. A. (1991). Exterior Differential Systems. Mathematical Sciences Research Institute Publications. Vol. 18. New York, NY: Springer New York. doi:10.1007/978-1-4613-9714-4. ISBN 978-1-4613-9716-8.
26. Bryant, Robert L. (2003). Exterior differential systems and Euler-Lagrange partial differential equations. Phillip Griffiths, Daniel Andrew Grossman. Chicago: University of Chicago Press. ISBN 0-226-07793-4. OCLC 51804819.
27. Bryant, Robert L.; Griffiths, Phillip A. (1995). "Characteristic Cohomology of Differential Systems (I): General Theory". Journal of the American Mathematical Society. 8 (3): 507–596. doi:10.2307/2152923. ISSN 0894-0347. JSTOR 2152923.
28. Bryant, Robert L.; Griffiths, Phillip A. (June 1, 1995). "Characteristic cohomology of differential systems II: Conservation laws for a class of parabolic equations". Duke Mathematical Journal. 78 (3). doi:10.1215/S0012-7094-95-07824-7. ISSN 0012-7094.
29. Bryant, Robert; Griffiths, Phillip; Hsu, Lucas (March 1, 1995). "Hyperbolic exterior differential systems and their conservation laws, part I". Selecta Mathematica. 1 (1): 21–112. doi:10.1007/BF01614073. ISSN 1420-9020. S2CID 195271133.
30. Bryant, R.; Griffiths, P.; Hsu, L. (September 1, 1995). "Hyperbolic exterior differential systems and their conservation laws, part II". Selecta Mathematica. 1 (2): 265–323. doi:10.1007/BF01671567. ISSN 1420-9020. S2CID 15812302.
31. "MR: Bryant, Robert L. - 42675". mathscinet.ams.org. Retrieved August 6, 2021.
32. "Publications of Robert L. Bryant". www.msri.org. Retrieved August 6, 2021.
33. Olver, Peter J. (2005). "Review: Exterior differential systems and Euler-Lagrange partial differential equations, by R. L. Bryant, P. A Griffiths, and D. A. Grossman" (PDF). Bull. Amer. Math. Soc. (N.S.). 42 (3): 407–412. doi:10.1090/s0273-0979-05-01062-1.
External links
• Homepage at MSRI
• Homepage at Duke University
Presidents of the American Mathematical Society
1888–1900
• John Howard Van Amringe (1888–1890)
• Emory McClintock (1891–1894)
• George William Hill (1895–1896)
• Simon Newcomb (1897–1898)
• Robert Simpson Woodward (1899–1900)
1901–1924
• E. H. Moore (1901–1902)
• Thomas Fiske (1903–1904)
• William Fogg Osgood (1905–1906)
• Henry Seely White (1907–1908)
• Maxime Bôcher (1909–1910)
• Henry Burchard Fine (1911–1912)
• Edward Burr Van Vleck (1913–1914)
• Ernest William Brown (1915–1916)
• Leonard Eugene Dickson (1917–1918)
• Frank Morley (1919–1920)
• Gilbert Ames Bliss (1921–1922)
• Oswald Veblen (1923–1924)
1925–1950
• George David Birkhoff (1925–1926)
• Virgil Snyder (1927–1928)
• Earle Raymond Hedrick (1929–1930)
• Luther P. Eisenhart (1931–1932)
• Arthur Byron Coble (1933–1934)
• Solomon Lefschetz (1935–1936)
• Robert Lee Moore (1937–1938)
• Griffith C. Evans (1939–1940)
• Marston Morse (1941–1942)
• Marshall H. Stone (1943–1944)
• Theophil Henry Hildebrandt (1945–1946)
• Einar Hille (1947–1948)
• Joseph L. Walsh (1949–1950)
1951–1974
• John von Neumann (1951–1952)
• Gordon Thomas Whyburn (1953–1954)
• Raymond Louis Wilder (1955–1956)
• Richard Brauer (1957–1958)
• Edward J. McShane (1959–1960)
• Deane Montgomery (1961–1962)
• Joseph L. Doob (1963–1964)
• Abraham Adrian Albert (1965–1966)
• Charles B. Morrey Jr. (1967–1968)
• Oscar Zariski (1969–1970)
• Nathan Jacobson (1971–1972)
• Saunders Mac Lane (1973–1974)
1975–2000
• Lipman Bers (1975–1976)
• R. H. Bing (1977–1978)
• Peter Lax (1979–1980)
• Andrew M. Gleason (1981–1982)
• Julia Robinson (1983–1984)
• Irving Kaplansky (1985–1986)
• George Mostow (1987–1988)
• William Browder (1989–1990)
• Michael Artin (1991–1992)
• Ronald Graham (1993–1994)
• Cathleen Synge Morawetz (1995–1996)
• Arthur Jaffe (1997–1998)
• Felix Browder (1999–2000)
2001–2024
• Hyman Bass (2001–2002)
• David Eisenbud (2003–2004)
• James Arthur (2005–2006)
• James Glimm (2007–2008)
• George Andrews (2009–2010)
• Eric Friedlander (2011–2012)
• David Vogan (2013–2014)
• Robert Bryant (2015–2016)
• Ken Ribet (2017–2018)
• Jill Pipher (2019–2020)
• Ruth Charney (2021–2022)
• Bryna Kra (2023–2024)
Authority control
International
• ISNI
• VIAF
National
• Norway
• France
• BnF data
• Germany
• Israel
• United States
Academics
• DBLP
• MathSciNet
• Mathematics Genealogy Project
• zbMATH
Other
• SNAC
• IdRef
|
Wikipedia
|
Robert C. James
Robert Clarke James (1918[1] – September 25, 2004)[2] was an American mathematician who worked in functional analysis.
Biography
James attended UCLA as an undergraduate, where his father was a professor. As a devout Quaker, he was a conscientious objector during World War II.[2]
He obtained his PhD at Caltech in 1946 under the direction of Aristotle Demetrius Michal.[3] He spent a year as a Benjamin Peirce Fellow at Harvard and joined the faculty at UC Berkeley. In 1950, during the loyalty oath controversy, James refused to sign the oath and moved to Haverford College. Later he was the founding math department chair both at Harvey Mudd College and at the Claremont Graduate University.[2]
James constructed a number of counterexamples in the theory of Banach spaces, including James' space and James' tree space. He also characterized reflexivity for Banach spaces with an unconditional Schauder basis and proved an eponymous compactness criterion.
With his father Glenn James (a professor at UCLA and long-time editor of Mathematics Magazine) he published a mathematical dictionary which went through several editions.[4]
James was made a Fellow of the American Association for the Advancement of Science in 1978.[5]
Selected works
• James, Robert C.; James, Glenn (1992). Mathematics Dictionary (5th ed.). New York: Chapman & Hall. ISBN 0412990415.
• James, Robert C. (March 1951). "A non-reflexive Banach space isometric with its second conjugate space". Proceedings of the National Academy of Sciences. 37 (3): 174–177. Bibcode:1951PNAS...37..174J. doi:10.1073/pnas.37.3.174. PMC 1063327. PMID 16588998. MR44024
• James, Robert C. (1981). "Structure of Banach spaces: Radon-Nikodým and other properties". General topology and modern analysis (Proceedings of a conference at the University of California, Riverside, Calif., 1980). New York: Academic Press. pp. 347–363. ISBN 0-12-481820-X.
• James, Robert C. (November 1982). "Bases in Banach spaces". The American Mathematical Monthly. 89 (9): 625–640. doi:10.2307/2975644. JSTOR 2975644. MR678808
References
1. James, Robert C.; James, Glenn (1992). Mathematics Dictionary (5th ed.). New York: Chapman & Hall. ISBN 0412990415. On copyright page.
2. Platt, Joseph; Ives, Robin and Lori (Spring 2005). "Robert James, Founding Mathematics Chair" (PDF). MuddMath. Vol. 4, no. 1.
3. Robert C. James at the Mathematics Genealogy Project
4. "University of California: In Memoriam, April 1962. Glenn James, Mathematics: Los Angeles". California Digital Library. Retrieved 19 September 2022.
5. "Historic Fellows". www.aaas.org. American Association for the Advancement of Science. Retrieved 19 September 2022.
Authority control
International
• ISNI
• VIAF
National
• France
• BnF data
• Germany
• Israel
• Belgium
• United States
• Czech Republic
• Netherlands
Academics
• MathSciNet
• Mathematics Genealogy Project
• zbMATH
People
• Deutsche Biographie
Other
• IdRef
|
Wikipedia
|
Bob Vaughan
Robert Charles "Bob" Vaughan FRS (born 24 March 1945) is a British mathematician, working in the field of analytic number theory.
Robert Charles Vaughan
R. C. Vaughan in 2008
Born (1945-03-24) March 24, 1945
Alma materUniversity of London
Known forAnalytic number theory
Exponential sums
Hardy–Littlewood circle method
AwardsBerwick Prize (1979)
Fellow of the Royal Society
Scientific career
FieldsMathematician
InstitutionsPenn State
Imperial College
Doctoral advisorTheodor Estermann
Doctoral studentsTrevor Wooley
Life
Since 1999 he has been Professor at Pennsylvania State University, and since 1990 Fellow of the Royal Society. He did his PhD at the University of London under supervision of Theodor Estermann.[1] He supervised Trevor Wooley's PhD.[1]
Awards
In 2012, he became a fellow of the American Mathematical Society.[2]
See also
• Vaughan's identity
Writings
• The Hardy–Littlewood Method. Cambridge Tracts in Mathematics. Vol. 125 (2nd ed.). Cambridge University Press. 1997. ISBN 978-0-521-57347-4.
• Hugh L. Montgomery; Robert C. Vaughan (2007). Multiplicative number theory I. Classical theory. Cambridge tracts in advanced mathematics. Vol. 97. ISBN 978-0-521-84903-6.
References
1. Bob Vaughan at the Mathematics Genealogy Project
2. List of Fellows of the American Mathematical Society, retrieved 2013-08-28.
External links
• Robert C. Vaughan's Home page (includes CV and list of publications)
Fellows of the Royal Society elected in 1990
Fellows
• Roger Angel
• Michael Ashburner
• David Bohm
• David Brown
• Malcolm H. Chisholm
• Robin Clark
• Peter Clarricoats
• John G. Collier
• Simon Conway Morris
• Andrew Crawford
• Leslie Dutton
• Robert Fettiplace
• Erwin Gabathuler
• Nicholas C. Handy
• Allen Hill
• Jonathan Hodgkin
• Eric Jakeman
• George Jellicoe
• Louise Johnson
• Vaughan Jones
• Carole Jordan
• John Knott
• Harry Kroto
• Steven V. Ley
• Lew Mander
• Michael E. McIntyre
• Derek W. Moore
• Colin James Pennycuick
• John Albert Raven
• David Read
• Man Mohan Sharma
• Allan Snyder
• George Stark
• Azim Surani
• Bob Vaughan
• Herman Waldmann
• William Lionel Wilkinson
• Robert Hughes Williams
• Alan Williams
• Gregory Winter
• Semir Zeki
Foreign
• Edward Norton Lorenz
• Yasutomi Nishizuka
• Christiane Nüsslein-Volhard
• E. O. Wilson
• Bengt I. Samuelsson
• Lyman Spitzer
Authority control
International
• ISNI
• VIAF
National
• France
• BnF data
• Germany
• Israel
• United States
• Greece
• Netherlands
• Poland
Academics
• DBLP
• MathSciNet
• Mathematics Genealogy Project
• Scopus
• zbMATH
|
Wikipedia
|
Robert Coveyou
Robert R. Coveyou (February 9, 1915 – February 19, 1996) was an American research mathematician who worked at the Oak Ridge National Laboratory.[1] He also taught mathematics part-time for several years at Knoxville College and worked at the International Atomic Energy Agency in Vienna, Austria, while on leave from the Oak Ridge National Laboratory from 1968 until 1971.
Robert R Coveyou
BornFebruary 9, 1915
Petoskey, Michigan, US
Died(1996-02-19)February 19, 1996
Oak Ridge, Tennessee, US
NationalityAmerican
Alma materUniversity of Chicago
University of Tennessee
Known forPseudo-Random Number Generators, Medical Physics
Scientific career
FieldsMathematics
Computer science
InstitutionsOak Ridge National Laboratory
An expert on pseudo-random number generators, today he is probably best known for the title of an article published around 1970: "Random Number Generation is too Important to be Left to Chance".[2][3]
Coveyou was an original member of the small group of radiation protection specialists at the University of Chicago assembled under the leadership of Ernest O. Wollan in 1942/43 and moved to Oak Ridge, Tennessee as part of the Manhattan Project.[1]
After the end of World War II he returned to Chicago to finish his undergraduate degree in Mathematics, and in the following year he received his master's degree from the University of Tennessee, both while employed at the Oak Ridge National Laboratory. He then returned to the laboratory for the remainder of his career, retiring in 1976.
In the early 1950s, Coveyou was one of the scientists and engineers involved in the early introduction of computers to the Oak Ridge National Laboratory and has been credited with naming the first computer housed at the laboratory: the ORACLE (Oak Ridge Automatic Computer and Logical Engine). In preparation for working on the computer in Oak Ridge, he spent two stretches of several weeks each at Remington Rand Corporation in New York City working with their staff to learn how they used the new UNIVAC computer.
Coveyou was a tournament chess player, and was Tennessee State Champion eight times. He is a member of the Tennessee Chess Hall of Fame, having been inducted with the inaugural class in 1990. He also mentored many young Oak Ridge and Tennessee chess players, with an unusual and effective approach to tutoring young players, emphasizing the mastery of simple end games before tackling more complex aspects of the game, including openings.
One of Coveyou's memorable chess experiences was hosting 13-year-old Bobby Fischer at his hotel room in Cleveland, Ohio, after Fischer had just won the 1957 U.S. Open. Coveyou, Fischer, and Edmar Mednis, a chess master from New York and friend of Fischer's, played informal games of chess for hours after the conclusion of the tournament, lasting into the early morning hours of the next day.
Bob Coveyou was also active politically and in the civil rights movement. He helped lead an effort to establish Scarboro High School in the African-American neighborhood of Oak Ridge. Prior to the school's opening, African American children there had had to bus to Knoxville, 30 miles away, to attend Austin High School.[4] The school operated from 1950 until Oak Ridge High School was desegregated in the fall of 1955.
Notes
1. "Robert Coveyou Mathematician and Health Physicist, X-10 Graphite Reactor". Retrieved 22 September 2020.
2. Coveyou, R.R. (1969). "Random Number Generation is too Important to be Left to Chance". In Agins, B.R.; Kalos, M.H. (eds.). A collection of papers presented by invitation at the Symposia on Applied Probability and Monte Carlo Methods and Modern Aspects of Dynamics sponsored by the Air Force Office of Scientific Research at the 1967 National Meeting of SIAM in Washington, D.C. Studies in Applied Mathematics. Vol. 3. SIAM. pp. 70–111.
3. Peterson, Ivars. The Jungles of Randomness: A Mathematical Safari. Wiley, NY, 1998. (pp. 178) ISBN 0-471-16449-6
4. Smith, D. Ray. "Education in Oak Ridge – Pre - Oak Ridge and Early - Oak Ridge Schools, part 2" (PDF). Retrieved 2 January 2014.
External links
• HISTORY OF THE ENGINEERING PHYSICS AND MATHEMATICS DIVISION 1955–1993
• Smithsonian Institution Archives, Accession 90-105, Science Service Records, Image No. SIA2008-0295
|
Wikipedia
|
Robert D. Hough
Robert D. Hough is an American born mathematician specializing in number theory, probability and discrete mathematics. He is an associate professor of mathematics at Stony Brook University.
Robert D. Hough
Alma materStanford University
Awards
• David P. Robbins Prize (2017)
• Sloan Research Fellowship (2020)
• Stony Brook Trustees Faculty Award (2020)
Scientific career
FieldsMathematics
InstitutionsStony Brook University
ThesisDistribution problems in number theory (2012)
Doctoral advisorKannan Soundararajan
Websitemath.stonybrook.edu/~rdhough/
Early life and education
Hough holds BS in Math, MS in CS and PhD in Math degrees from Stanford University. He completed his PhD under Kannan Soundararajan in 2012. Hough was a post-doctoral researcher at Cambridge University and Oxford University in the United Kingdom working with Ben Green from 2013 to 2015, and was a post-doctoral member of the Institute for Advanced Study, Princeton, New Jersey from 2015 to 2016.
Achievements
Hough won the Mathematical Association of America's David P. Robbins Prize at the Joint Math Meetings in 2017.[1] The prize was given for finding the solution of a problem imposed by Paul Erdős.[2]
In February 2020, Hough won the Sloan Research Fellowship.[3] He has also won a Trustees Faculty Award from Stony Brook University.[4]
Career
Since 2016 Hough has been on the faculty of Stony Brook University.
References
1. "David P. Robbins Prize". Mathematical Association of America.
2. Montgomery, Hugh (1994). Ten lectures on the interface of analytic number theory and harmonic analysis. American Mathematical Society. ISBN 978-0821807378.
3. "Mathematics Professor Robert Hough Awarded Sloan Research Fellowship". Stony Brook News. February 24, 2020. Retrieved February 24, 2020.
4. "Dow alum Hough receives fellowship". Midland Daily News. August 8, 2020. Retrieved August 8, 2020.
Authority control: Academics
• MathSciNet
• Mathematics Genealogy Project
|
Wikipedia
|
Robert Edmund O'Malley
Robert Edmund O'Malley Jr. (born 1939) is an American mathematician.
Robert E. O'Malley
O'Malley in 2011
NationalityAmerican
Alma materUniversity of New Hampshire Stanford University
Known forSingular Perturbation Theory, Asymptotic Methods
Scientific career
FieldsMathematician
InstitutionsUniversity of Washington
Doctoral advisorGordon Eric Latta
O'Malley studied electrical engineering and mathematics at the University of New Hampshire, where he received his baccalaureate degree in 1960 and his master's in 1961. He then studied differential equations and singular perturbations at Stanford University, where he received his doctorate in mathematics in 1966. After brief appointments at the University of North Carolina (Chapel Hill), Bell Telephone Laboratories, the Courant Institute (New York University), and the Mathematics Research Center (the University of Wisconsin, Madison), O'Malley returned to New York University in 1968. He remained there, doing research on asymptotic methods and singular perturbations with Joseph Keller and a number of other stimulating colleagues and students. O'Malley spent a year at the University of Edinburgh, where his lecture notes formed the basis of his book, Introduction to Singular Perturbations (Academic Press, 1974). In 1973, he moved to the University of Arizona (Tucson) where he later organized a successful interdisciplinary program in applied mathematics, and where he applied singular perturbation ideas in control theory. After a sabbatical at Stanford University, O'Malley moved to Rensselaer Polytechnic Institute (Troy, New York) in 1981. At Rensselaer, he headed a mathematical sciences department which emphasized applied mathematics and computer science. There, he was active in campus affairs and served as the chairman of the faculty and the Ford Foundation Professor. Soon after a sabbatical at the Technical University of Vienna, where O'Malley studied asymptotic methods in semiconductor modeling, he moved to the University of Washington, Seattle.
O'Malley is currently at the University of Washington Department of Applied Mathematics as an emeritus faculty member. He served as the president of the Society for Industrial and Applied Mathematics (SIAM) [1] (1991–1992). In 2009 he became a SIAM Fellow.[2] In 2012 he became a fellow of the American Mathematical Society.[3]
Work
O'Malley's current research emphasizes the relationship between singular perturbation theory and various regularization methods for differential-algebraic systems, geometric approaches to understanding the limiting solutions to singularly perturbed boundary value problems, the motion of shock layers and other interfaces, the interplay between asymptotic and numerical methods, and tough problems of asymptotic matching. He continues to collaborate with an international collection of interesting characters, and receives support for his scholarly work from the National Science Foundation.
O'Malley is known for several pioneering contributions to singular perturbation theory and applications.[4]
O'Malley has been especially active as a member of SIAM, the Society for Industrial and Applied Mathematics. He was president of SIAM in 1991 and 1992, and has been vice-president in charge of their publication program which encompasses ten journals and several book series. Among his other positions with SIAM has been that of program chair for several meetings including ICIAM `91, the second international industrial and applied mathematics conference, which drew 2,200 participants.
O'Malley serves on a number of boards, including advisory committees for the National Institute of Standards and Technology, the Electric Power Research Institute, and Argonne National Laboratory, and more generally as a spokesperson for applied mathematics[5]
Selected publications
• On the motion of viscous shocks and the supersensitivity of their steady-state limits, Methods Applics. Analysis, 1, (1994), (with J.G.L. Laforgue).
• Shock layer movement for Burger's equation, SIAM J. Appl. Math. 54: (1994) (with J.G.L. Laforgue).
• On regularizing differential-algebraic equations, Trends and Developments in Ordinary Differential Equations, Y. Alavi and P.F. Hsieh, editors, World Scientific, 1994 (with L.V. Kalachev).
• The regularization of nonlinear differential-algebraic equations. SIAM J. Math Anal., 25: pp. 615–629, 1994 (with L.V. Kalachev).
• Super sensitive boundary value problems, "Asymptotic and Numerical Methods for Partial Differential Equations with Critical Parameters", H.G. Kaper and M. Garbey, editors, Kluwer, 1993, pp. 215–223 (with J.G.L. Laforgue).
• Singular Perturbation Methods for Ordinary Differential Equations, Springer-Verlag, New York, 1991.
References
1. SIAM Presidents http://www.siam.org/about/more/presidents.php
2. SIAM Fellows http://fellows.siam.org/index.php?sort=last#G
3. List of Fellows of the American Mathematical Society, retrieved 2013-03-20.
4. "SIAM: A Turning Point for Bob O'Malley". www.siam.org. Archived from the original on 2007-07-15.
5. "Robert O'Malley | Department of Applied Mathematics | University of Washington".
External links
• Home Page at University of Washington
• Robert Edmund O'Malley at the Mathematics Genealogy Project
• http://www.siam.org/news/news.php?id=768 Archived 2011-08-09 at the Wayback Machine
Authority control
International
• ISNI
• VIAF
National
• Norway
• France
• BnF data
• Germany
• Israel
• United States
• Netherlands
Academics
• CiNii
• MathSciNet
• Mathematics Genealogy Project
• zbMATH
People
• Deutsche Biographie
Other
• IdRef
|
Wikipedia
|
Robert Evert Stong
Robert Evert Stong (August 23, 1936, Oklahoma City – April 10, 2008, Charlottesville, Virginia) was a mathematician at the University of Virginia who proved the Hattori–Stong theorem.
Early life and education
Stong received a B.A. and M.A. in mathematics at the University of Oklahoma. He received a Ph.D. in mathematics from the University of Chicago in 1962.[1] His Ph.D. dissertation, Some relations among characteristic classes and numbers, was written under the supervision of Richard Lashof.[1] He served on active duty with the United States Army Reserves from 1962 to 1965 and was stationed at Fort Benjamin Harrison and the Pentagon. He worked on computer development and rose to the rank of captain.
Career
After serving with the Army, he went to the University of Oxford as a post-doctoral fellow (1964–66), and then to Princeton University as faculty (1966–68). In 1968 he became Professor of Mathematics at the University of Virginia, where he taught until his retirement in 2007.
His doctoral students included Nelson Saiers.[1]
Selected publications
• Stong, Robert E. (1966). "Finite topological spaces". Transactions of the American Mathematical Society. 123 (2): 325–340. doi:10.1090/s0002-9947-1966-0195042-2. MR 0195042.
• Stong, Robert E. (1968). Notes on cobordism theory (PDF). Mathematical notes. Princeton, NJ: Princeton University Press. MR 0248858. Archived from the original (PDF) on 2010-06-23.
• Kosniowski, Czes; Stong, Robert E. (1978). "Involutions and characteristic numbers". Topology. 17 (4): 309–330. doi:10.1016/0040-9383(78)90001-0. MR 0516213.
References
1. Robert Evert Stong at the Mathematics Genealogy Project
Sources
• "Robert E. Stong — Reminiscences". Archived from the original on 2016-03-04. Retrieved 2018-04-02.
Authority control
International
• ISNI
• VIAF
National
• Israel
• United States
• Netherlands
Academics
• MathSciNet
• Mathematics Genealogy Project
• zbMATH
Other
• IdRef
|
Wikipedia
|
Robert F. Coleman
Robert Frederick Coleman (November 22 1954 – March 24, 2014) was an American mathematician, and professor at the University of California, Berkeley.[1]
Robert F. Coleman
Robert Coleman at Oberwolfach in 1983
Born(1954-11-22)November 22, 1954
DiedMarch 24, 2014(2014-03-24) (aged 59)
NationalityAmerican
Alma mater
• Harvard University
• Princeton University
Known for
• p-adic integration
• Method of Coleman and Chabauty
• Coleman-Mazur eigencurve
• overconvergent p-adic modular forms
Awards
• MacArthur Fellow (1987)
• NSF-GRFP (1976)
• Intel STS (1972)
Scientific career
FieldsMathematics
InstitutionsUniversity of California, Berkeley
Doctoral advisorKenkichi Iwasawa
Biography
After graduating from Nova High School, he completed his bachelor's degree at Harvard University in 1976 and subsequently attended Cambridge University for Part III of the mathematical tripos. While there John H. Coates provided him with a problem for his doctoral thesis ("Division Values in Local Fields"), which he completed at Princeton University in 1979 under the advising of Kenkichi Iwasawa. He then had a one-year postdoctoral appointment at the Institute for Advanced Study and then taught at Harvard University for three years. In 1983, he moved to University of California, Berkeley. In 1985, he was struck with a severe case of multiple sclerosis, in which he lost the use of his legs. Despite this, he remained an active faculty member until his retirement in 2013. He was awarded a MacArthur fellowship in 1987.[2]
Coleman died on March 24, 2014.[3]
Research
He worked primarily in number theory, with specific interests in p-adic analysis and arithmetic geometry. In particular, he developed a theory of p-adic integration analogous to the classical complex theory of abelian integrals. Applications of Coleman integration include an effective version of Chabauty's theorem concerning rational points on curves and a new proof of the Manin-Mumford conjecture, originally proved by Raynaud. Coleman is also known for introducing p-adic Banach spaces into the study of modular forms and discovering important classicality criteria for overconvergent p-adic modular forms. With Barry Mazur, he introduced the eigencurve and established some of its fundamental properties. In 1990, Coleman found a gap in Manin's proof of the Mordell conjecture over function fields and managed to fill it in. With José Felipe Voloch, Coleman established an important unchecked compatibility in Benedict Gross's theory of companion forms.
Selected works
• Coleman, Robert F. (1979), "Division values in local fields.", Invent. Math., 53 (2): 91–116, Bibcode:1979InMat..53...91C, doi:10.1007/BF01390028, MR 0560409, S2CID 122569381 PhD thesis
• Coleman, Robert F. (1985), "Torsion points on curves and p-adic abelian integrals", Ann. of Math., 121 (1): 111–168, doi:10.2307/1971194, JSTOR 1971194, MR 0782557
• Coleman, Robert F. (1985), "Effective Chabauty", Duke Math. J., 52 (3): 765–770, doi:10.1215/s0012-7094-85-05240-8, MR 0808103
• Coleman, Robert F. (1987), "Ramified torsion points on curves", Duke Math. J., 54 (2): 615–640, doi:10.1215/s0012-7094-87-05425-1, MR 0899407
• Coleman, Robert F.; de Shalit, Ehud (1988), "p-adic regulators on curves and special values of p-adic L-functions", Invent. Math., 93 (2): 239–266, Bibcode:1988InMat..93..239C, doi:10.1007/bf01394332, MR 0948100, S2CID 122242212
• Coleman, Robert F. (1990), "Manin's proof of the Mordell conjecture over function fields", L'Enseignement Mathématique, 2e Série, 36 (3): 393–427, ISSN 0013-8584, MR 1096426, archived from the original on 2011-10-02
• Coleman, Robert F.; Voloch, José Felipe (1992), "Companion forms and Kodaira-Spencer theory", Invent. Math., 110: 263–281, Bibcode:1992InMat.110..263C, doi:10.1007/bf01231333, MR 1185584, S2CID 121416817
• Coleman, Robert F. (1996), "Classical and Overconvergent Modular Forms", Invent. Math., 124 (1–3): 215–241, Bibcode:1996InMat.124..215C, doi:10.1007/s002220050051, MR 1369416, S2CID 7995580
• Coleman, Robert F. (1997), "p-adic Banach spaces and families of modular forms.", Invent. Math., 127 (3): 417–479, Bibcode:1997InMat.127..417C, CiteSeerX 10.1.1.467.377, doi:10.1007/s002220050127, MR 1431135, S2CID 1677427
• Coleman, R.; Mazur, B. (1998), "The eigencurve" (PDF), Galois representations in arithmetic algebraic geometry (Durham, 1996), London Math. Soc. Lecture Note Ser., vol. 254, Cambridge: Cambridge Univ. Press, pp. 1–113, doi:10.1017/CBO9780511662010.003, ISBN 9780511662010, MR 1696469, archived from the original (PDF) on 2011-06-07
• Coleman, Robert F. (2003), "Stable maps of curves", Doc. Math., Extra Volume for Kazuya Kato's fiftieth birthday: 217–225, MR 2046600
References
1. "Robert F. Coleman | Department of Mathematics at University of California Berkeley". Math.berkeley.edu. Retrieved 2014-03-27.
2. (Freistadt 1987)
3. Baker, Matt (March 25, 2014). "Robert F. Coleman 1954-2014". Matt Baker's Math Blog. WordPress. Retrieved March 27, 2014. {{cite web}}: External link in |work= (help)
• Freistadt, Margo (1987), "MS victim counts on helping others", Spokane Chronicle (July 23): 1
External links
• Robert Coleman's Home Page
• Robert F. Coleman at the Mathematics Genealogy Project
• Robert F. Coleman's facebook page
• Matt Baker's blog: Robert F. Coleman 1954-2014
Authority control
International
• ISNI
• VIAF
National
• United States
Academics
• MathSciNet
• Mathematics Genealogy Project
• zbMATH
Other
• SNAC
|
Wikipedia
|
Robert Finn (mathematician)
Robert Samuel Finn (August 8, 1922 – August 16, 2022) was an American mathematician.
Robert Finn
BornAugust 8, 1922
Buffalo, New York, U.S.
DiedAugust 16, 2022 (aged 100)
Palo Alto, California, U.S.
Academic background
EducationRensselaer Polytechnic Institute (BS)
Syracuse University (PhD)
ThesisOn some properties of the solution of a class of non-linear partial differential equations
Doctoral advisorAbe Gelbart
Academic work
DisciplineMathematics
Sub-disciplineMinimal surfaces
Quasiconformal mapping
InstitutionsUniversity of Southern California
California Institute of Technology
Stanford University
Early life and education
Finn was born in Buffalo, New York. He earned a Bachelor of Science degree in physics from Rensselaer Polytechnic Institute and a PhD in mathematics from Syracuse University. Studying under Abe Gelbart, Finn completed a thesis titled On some properties of the solution of a class of non-linear partial differential equations.[1]
Career
He completed post-doctoral research at the Institute for Advanced Study in 1953 and at the Institute for Hydrodynamics of the University of Maryland from 1953 to 1954. In 1954, he became an assistant professor at the University of Southern California and in 1956 an associate professor at California Institute of Technology. Beginning in 1959, he was a professor at Stanford University.[2]
At the beginning of his career, Finn did research on minimal surfaces and quasiconformal mappings and later in his career on mathematical problems of hydrodynamics, such as mathematically rigorous treatments of capillary action. He was a visiting professor at the University of Bonn and several other universities. He was an exchange scientist in 1978 at the Soviet Academy of Sciences and in 1987 at the German Academy of Sciences at Berlin. In 1994 he received an honorary doctorate from the Leipzig University. For the academic years 1958–1959 and 1965–1966, he held Guggenheim Fellowships.[3] From 1979, he was an editor of the Pacific Journal of Mathematics.
Personal life
Finn turned 100 on August 8, 2022. He died eight days later, in Palo Alto, California, on August 16, 2022.[4]
Selected works
• Finn, R. (March 1953). "A Property of Minimal Surfaces". Proceedings of the National Academy of Sciences of the United States of America. 39 (3): 197–201. Bibcode:1953PNAS...39..197F. doi:10.1073/pnas.39.3.197. PMC 1063753. PMID 16589247.
• Finn, R. (October 1954). "On the Flow of a Perfect Fluid through a Polygonal Nozzle. I". Proc Natl Acad Sci U S A. 40 (10): 983–985. Bibcode:1954PNAS...40..983F. doi:10.1073/pnas.40.10.983. PMC 534204. PMID 16589589.
• Finn, R. (October 1954). "On the Flow of a Perfect Fluid through a Polygonal Nozzle. II". Proc Natl Acad Sci U S A. 40 (10): 985–987. Bibcode:1954PNAS...40..985F. doi:10.1073/pnas.40.10.985. PMC 534205. PMID 16589590.
• with Paul Concus: Concus, P.; Finn, R. (June 1969). "On the Behavior of a Capillary Surface in a Wedge". Proc Natl Acad Sci U S A. 63 (2): 292–299. Bibcode:1969PNAS...63..292C. doi:10.1073/pnas.63.2.292. PMC 223563. PMID 16591761.
• Equilibrium capillary surfaces. Grundlehren der mathematischen Wissenschaften. Springer Verlag. 1986.
References
1. Robert Finn at the Mathematics Genealogy Project
2. biographical information from American Men and Women of Science, Thomson Gale 2004
3. "Robert Finn". John Simon Guggenheim Memorial Foundation.
4. "Robert S. Finn". Trident Society. Retrieved August 20, 2022.
External links
• "Robert Finn". Stanford University.
Authority control
International
• ISNI
• VIAF
National
• France
• BnF data
• Germany
• Belgium
• United States
• Czech Republic
• Netherlands
Academics
• MathSciNet
• Mathematics Genealogy Project
• zbMATH
Other
• IdRef
|
Wikipedia
|
Robert Forsyth Scott
Sir Robert Forsyth Scott (28 July 1849 – 18 November 1933) was a mathematician, barrister and Master of St John's College, Cambridge
Life
Scott was born in Leith, near Edinburgh, the eldest son of Reverend George Scott, a Minister in the church at Dairsie and Mary Forsyth, daughter of the Edinburgh advocate Robert Forsyth.[1]
Scott was educated at the High School, Edinburgh, then in Stuttgart before becoming a student at University College, London. In 1870, while a student at University College, London, he was awarded a Whitworth Exhibition. He went on to read mathematics at St John's College, where he was fourth wrangler in the Tripos in 1875 and was elected to a fellowship in 1877.[2][3]
After publishing The Theory of Determinants and Their Applications in 1880, Scott turned his attention to the law, become a barrister in 1883, and to institutional history, including histories of St. John's College, Cambridge, published between 1882 and 1907.[1][3] In 1908 he was appointed as the Master of St John's College, a position he held until his death in Cambridge in 1933, and from 1910 to 1912 he served as Vice-chancellor of the University. On his death he left the library of St John's one of the largest collection of Burmese manuscripts in Europe.[3]
He was the elder brother of Sir James George Scott.
Publications
• History of St John's College, Cambridge
• The theory of determinants and their applications (2nd ed., revised by George Ballard Mathews), Cambridge University Press, 1904.
References
1. Biography of Robert Forsyth Scott
2. "Scott, Robert Forsyth (SCT871RF)". A Cambridge Alumni Database. University of Cambridge.
3. Digitization of History: Centre for History and Economics
External links
• Works by Robert Forsyth Scott at Project Gutenberg
• Works by or about Robert Forsyth Scott at Internet Archive
• O'Connor, John J.; Robertson, Edmund F., "Robert Forsyth Scott", MacTutor History of Mathematics Archive, University of St Andrews
Authority control
International
• ISNI
• VIAF
National
• Germany
• United States
• Greece
• Netherlands
Academics
• zbMATH
People
• Deutsche Biographie
• Trove
Other
• SNAC
• IdRef
|
Wikipedia
|
Robert G. Bartle
Robert Gardner Bartle (November 20, 1927 – September 18, 2003) was an American mathematician specializing in real analysis. He is known for writing the popular textbooks The Elements of Real Analysis (1964), The Elements of Integration (1966), and Introduction to Real Analysis (2011) with Donald R. Sherbert, published by John Wiley & Sons.
Robert G. Bartle
Born(1927-11-20)November 20, 1927
Kansas City, Missouri
DiedSeptember 18, 2003(2003-09-18) (aged 75)
Ann Arbor, Michigan
Alma materUniversity of Chicago
Known forReal Analysis
Scientific career
FieldsMathematics
InstitutionsUniversity of Illinois Urbana-Champaign,
Eastern Michigan University
Doctoral advisorLawrence Graves
Doctoral studentsMoedomo Soedigdomarto
Bartle was born in Kansas City, Missouri, and was the son of Glenn G. Bartle and Wanda M. Bartle. He was married to Doris Sponenberg Bartle (born 1927) from 1952 to 1982 and they had two sons, James A. Bartle (born 1955) and John R. Bartle (born 1958). He was on the faculty of the Department of Mathematics at the University of Illinois from 1955 to 1990.
Bartle was Executive Editor of Mathematical Reviews from 1976 to 1978 and from 1986 to 1990. From 1990 to 1999 he taught at Eastern Michigan University. In 1997, he earned a writing award from the Mathematical Association of America for his paper "Return to the Riemann Integral".[1]
References
1. Bartle, Robert G. (1996). "Return to the Riemann Integral". The American Mathematical Monthly. Mathematical Association of America. 103 (8): 625–632. doi:10.2307/2974874. hdl:10338.dmlcz/127740. JSTOR 2974874.
• Robert G. Bartle (1990) "A brief history of the mathematical literature".
• Jane E. Kister & Donald R. Sherbert (2004) "Robert G. Bartle (1927 — 2003)". Notices of the American Mathematical Society 51(2):239–40.
• O'Connor, Anahad (2003-11-03). "Robert G. Bartle, 75, Mathematician and Author". The New York Times. Retrieved 2019-05-17.
External links
• "In Memoriam ROBERT GARDNER BARTLE (1927 - 2003)" (PDF). ClassicalRealAnalysis.info. Retrieved 2019-05-17.
• Robert G. Bartle at the Mathematics Genealogy Project
Authority control
International
• ISNI
• VIAF
National
• Norway
• 2
• France
• BnF data
• Germany
• Israel
• United States
• Sweden
• Czech Republic
• Australia
• Korea
• Croatia
• Netherlands
Academics
• CiNii
• MathSciNet
• Mathematics Genealogy Project
• zbMATH
People
• Deutsche Biographie
• Trove
Other
• IdRef
|
Wikipedia
|
Robert Ghrist
Robert W. Ghrist (born 1969) is an American mathematician, known for his work on topological methods in applied mathematics.
Robert Ghrist
Born1969 (age 53–54)
Euclid, Ohio, U.S.
EducationUniversity of Toledo (BS)
Cornell University (MS), (PhD)
AwardsPresidential Early Career Award (2002)
Chauvenet Prize (2013)
Gauss Lectureship (2014)
Scientific career
FieldsMathematics & Engineering
InstitutionsUniversity of Pennsylvania
Life and Work
Ghrist received his bachelor's degree in mechanical engineering from the University of Toledo in 1991, and in 1994 his master's degree and in 1995 his PhD from Cornell University under Philip Holmes with thesis The link of periodic orbits of a flow.[1] From 1996 to 1998, he was R. H. Bing Instructor at the University of Texas and from 1998 an assistant professor and then from 2002 an associate professor at the Georgia Institute of Technology. In 2002 he became an associate professor and in 2004 a professor at the University of Illinois at Urbana-Champaign. From 2007, he was at the Information Trust Institute. In 2008, he was appointed Andrea Mitchell Penn Integrating Knowledge University Professor in Mathematics and Electrical/Systems Engineering at the University of Pennsylvania.
Ghrist was a visiting scientist in 1995 at the Institute for Advanced Study and in 2000 at the Isaac Newton Institute in Cambridge. He works on the application of topological methods to dynamical systems, robots, hydrodynamics, and information systems, such as sensor networks.[2]
Honors and awards
In 2002, Ghrist received a Presidential Early Career Award. In 2013, he received the Chauvenet Prize for Barcodes: The Persistent Topology of Data[3] and in 2014 the Gauss Lectureship of the German Mathematical Society.
Selected works
• Ghrist, Robert W.; Holmes, Philip; Sullivan, Michael (1997), Knots and Links in Three-dimensional Flows, Lecture Notes in Mathematics 1654, Springer Verlag, pp. x+208, ISBN 978-3540626282
• M. Farber; R. Ghrist; M. Burger; D. Koditschek, eds. (2007), Topology and Robotics, Contemporary Mathematics, American Mathematical Society, ISBN 978-0-8218-4246-1
• Ghrist, Robert W. (2014), Elementary Applied Topology, Createspace, ISBN 978-1502880857
• Ghrist, Robert (2008), "Barcodes: the persistent topology of data" (PDF), Bull. Amer. Math. Soc., 45: 61–75, doi:10.1090/s0273-0979-07-01191-3
See also
• Topological data analysis
References
1. Robert Ghrist at the Mathematics Genealogy Project
2. Vin de Silva, Robert Ghrist Homological Sensor Networks, Notices AMS, Januar 2007
3. Bulletin of the American Mathematical Society. vol. 45, 2008, pp. 61–75, Online
External links
• Homepage
• Applied topology and Dante: an interview with Robert Ghrist; The Endeavour, John D. Cook
• Ghrist's publications listed on the DBLP Bibliography Server
Chauvenet Prize recipients
• 1925 G. A. Bliss
• 1929 T. H. Hildebrandt
• 1932 G. H. Hardy
• 1935 Dunham Jackson
• 1938 G. T. Whyburn
• 1941 Saunders Mac Lane
• 1944 R. H. Cameron
• 1947 Paul Halmos
• 1950 Mark Kac
• 1953 E. J. McShane
• 1956 Richard H. Bruck
• 1960 Cornelius Lanczos
• 1963 Philip J. Davis
• 1964 Leon Henkin
• 1965 Jack K. Hale and Joseph P. LaSalle
• 1967 Guido Weiss
• 1968 Mark Kac
• 1970 Shiing-Shen Chern
• 1971 Norman Levinson
• 1972 François Trèves
• 1973 Carl D. Olds
• 1974 Peter D. Lax
• 1975 Martin Davis and Reuben Hersh
• 1976 Lawrence Zalcman
• 1977 W. Gilbert Strang
• 1978 Shreeram S. Abhyankar
• 1979 Neil J. A. Sloane
• 1980 Heinz Bauer
• 1981 Kenneth I. Gross
• 1982 No award given.
• 1983 No award given.
• 1984 R. Arthur Knoebel
• 1985 Carl Pomerance
• 1986 George Miel
• 1987 James H. Wilkinson
• 1988 Stephen Smale
• 1989 Jacob Korevaar
• 1990 David Allen Hoffman
• 1991 W. B. Raymond Lickorish and Kenneth C. Millett
• 1992 Steven G. Krantz
• 1993 David H. Bailey, Jonathan M. Borwein and Peter B. Borwein
• 1994 Barry Mazur
• 1995 Donald G. Saari
• 1996 Joan Birman
• 1997 Tom Hawkins
• 1998 Alan Edelman and Eric Kostlan
• 1999 Michael I. Rosen
• 2000 Don Zagier
• 2001 Carolyn S. Gordon and David L. Webb
• 2002 Ellen Gethner, Stan Wagon, and Brian Wick
• 2003 Thomas C. Hales
• 2004 Edward B. Burger
• 2005 John Stillwell
• 2006 Florian Pfender & Günter M. Ziegler
• 2007 Andrew J. Simoson
• 2008 Andrew Granville
• 2009 Harold P. Boas
• 2010 Brian J. McCartin
• 2011 Bjorn Poonen
• 2012 Dennis DeTurck, Herman Gluck, Daniel Pomerleano & David Shea Vela-Vick
• 2013 Robert Ghrist
• 2014 Ravi Vakil
• 2015 Dana Mackenzie
• 2016 Susan H. Marshall & Donald R. Smith
• 2017 Mark Schilling
• 2018 Daniel J. Velleman
• 2019 Tom Leinster
• 2020 Vladimir Pozdnyakov & J. Michael Steele
• 2021 Travis Kowalski
• 2022 William Dunham, Ezra Brown & Matthew Crawford
Authority control
International
• ISNI
• VIAF
National
• Norway
• Germany
• Israel
• Belgium
• United States
• Netherlands
Academics
• Google Scholar
• MathSciNet
• Mathematics Genealogy Project
• zbMATH
Other
• IdRef
|
Wikipedia
|
Robert Giffen
Sir Robert Giffen KCB FRS (22 July 1837 – 12 April 1910) was a Scottish statistician and economist.[1]
Robert Giffen
Born(1837-07-22)22 July 1837
Strathaven, Lanarkshire, Scotland
Died12 April 1910(1910-04-12) (aged 72)
Fort Augustus, Scotland
Alma materUniversity of Glasgow
Occupation(s)Economist, statistician
Life
Giffen was born at Strathaven, Lanarkshire. He entered a solicitor's office in Glasgow, and while in that city attended courses at the university. He drifted into journalism, and after working for the Stirling Journal he went to London in 1862 and joined the staff of the Globe. He also assisted John Morley, when the latter edited the Fortnightly Review. In 1868 he became Walter Bagehot's assistant-editor on The Economist; and his services were also secured in 1873 as city editor of the Daily News, and later of The Times.
His reputation as a financial journalist and statistician, gained in these years, led to his appointment in 1876 as head of the statistical department in the Board of Trade, and subsequently he became assistant secretary (1882) and finally controller-general (1892), retiring in 1897. As chief statistical adviser to the government, he drew up reports, gave evidence before commissions of inquiry, and acted as a government auditor.
Giffen was president of the Statistical Society (1882–1884); He was made a Companion of the Order of the Bath in 1891. In 1892 he was elected a Fellow of the Royal Society, and in 1894 he received the Guy Medal (gold) from the RSS. He was elected a member of the Royal Swedish Academy of Sciences in 1897.
Robert Giffen continued in later years to take a leading part in all public controversies connected with finance and taxation, and his high authority and practical experience were universally recognised. He was awarded a Knight Commander of the Order of the Bath in 1895.[2] He died somewhat suddenly in Fort Augustus, Scotland on 12 April 1910.
Works
Giffen published essays on financial subjects. His major publications were:
• American Railways as Investments (1873);
• Essays on Finance (1879 and 1884);
• The Progress of the Working Classes (1884);
• The Growth of Capital (1890);
• The Case against Bimetallism (1892); and
• Economic Inquiries and Studies (1904).[3]
The concept of a Giffen good is named after him. Alfred Marshall wrote in the third (1895) edition of his Principles of Economics:
As Mr. Giffen has pointed out, a rise in the price of bread makes so large a drain on the resources of the poorer labouring families and raises so much the marginal utility of money to them, that they are forced to curtail their consumption of meat and the more expensive farinaceous foods: and, bread being still the cheapest food which they can get and will take, they consume more, and not less of it.
Marshall's attribution identified no corresponding passage in Giffen's writings.
On 25 March 1908, Giffen spoke at the Royal United Services Institution in London, where he predicted that a major war would shock the world credit system, which in turn would virtually halt international trade.[4] This inspired the British Admiralty's plans for economic warfare at the outbreak of the First World War.[5]
References
• R. S. Mason Robert Giffen and the Giffen Paradox, Philip Allan (1989)
• A. E. Bateman, "Sir Robert Giffen", Journal of the Royal Statistical Society, 73, (1910) pp. 529–533. (includes photograph)
• F. Y. Edgeworth, "Sir Robert Griffen", Economic Journal, 20, (1910) pp. 318–321.
• "Sir Robert Giffen - Dead" (PDF). The New York Times. 13 April 1910.
Notes
1. "Giffen, Sir Robert". Who's Who: 738. 1910.
2. "Sir Robert Giffen". The Times. London. 13 April 1910. p. 13.
3. "Review of Economic Inquiries and Studies by Sir Robert Giffen, 2 vols". The Oxford Magazine. The Proprietors. 23: 419. 21 June 1905.
4. See:
• Giffen, Robert (1908). "The necessity of a war chest in this country, or a greatly increased gold reserve". Royal United Services Institution. Journal. 52 (368): 1329–1353. doi:10.1080/03071840809418923.
• "Our gold reserve: Necessity for a war chest," The Morning Post (London), 1908 March 26; reprinted in: James Carmichael Smith, Money and Profit-sharing: Or, The Double Standard Money System (London, England: Kegan Paul, Trench, Trübner & Co., 1908), pp. 196–200.
• Lambert, Nicholas A. (2012). Planning Armageddon. Cambridge, Massachusetts, USA: Harvard University Press. p. 115. ISBN 9780674063068.
5. (Lambert, 2012), p. 111.
External links
Wikisource has the text of the 1911 Encyclopædia Britannica article "Giffen, Sir Robert".
Works by or about Robert Giffen at Wikisource
• Royal Society citation
• Giffen correspondence
Attribution
The main part of this entry is taken from the 1911 Encyclopædia Britannica.
Guy Medallists
Gold Medallists
• Charles Booth (1892)
• Robert Giffen (1894)
• Jervoise Athelstane Baines (1900)
• Francis Ysidro Edgeworth (1907)
• Patrick G. Craigie (1908)
• G. Udny Yule (1911)
• T. H. C. Stevenson (1920)
• A. William Flux (1930)
• A. L. Bowley (1935)
• Major Greenwood (1945)
• R. A. Fisher (1946)
• A. Bradford Hill (1953)
• E. S. Pearson (1955)
• Frank Yates (1960)
• Harold Jeffreys (1962)
• Jerzy Neyman (1966)
• M. G. Kendall (1968)
• M. S. Bartlett (1969)
• Harald Cramér (1972)
• David Cox (1973)
• G. A. Barnard (1975)
• Roy Allen (1978)
• D. G. Kendall (1981)
• Henry Daniels (1984)
• Bernard Benjamin (1986)
• Robin Plackett (1987)
• Peter Armitage (1990)
• George E. P. Box (1993)
• Peter Whittle (1996)
• Michael Healy (1999)
• Dennis Lindley (2002)
• John Nelder (2005)
• James Durbin (2008)
• C. R. Rao (2011)
• John Kingman (2013)
• Bradley Efron (2014)
• Adrian Smith (2016)
• Stephen Buckland (2019)
• David Spiegelhalter (2020)
• Nancy Reid (2022)
Silver Medallists
• John Glover (1893)
• Augustus Sauerbeck (1894)
• A. L. Bowley (1895)
• F. J. Atkinson (1897)
• C. S. Loch (1899)
• Richard Crawford (1900)
• Thomas A. Welton (1901)
• R. H. Hooker (1902)
• Yves Guyot (1903)
• D. A. Thomas (1904)
• R. H. Rew (1905)
• W. H. Shaw (1906)
• N. A. Humphreys (1907)
• Edward Brabrook (1909)
• G. H. Wood (1910)
• R. Dudfield (1913)
• S. Rowson (1914)
• S. J. Chapman (1915)
• J. S. Nicholson (1918)
• J. C. Stamp (1919)
• A. William Flux (1921)
• H. W. Macrosty (1927)
• Ethel Newbold (1928)
• H. E. Soper (1930)
• J. H. Jones (1934)
• Ernest Charles Snow (1935)
• R. G. Hawtrey (1936)
• E. C. Ramsbottom (1938)
• L. Isserlis (1939)
• H. Leak (1940)
• M. G. Kendall (1945)
• Harry Campion (1950)
• F. A. A. Menzler (1951)
• M. S. Bartlett (1952)
• J. O. Irwin (1953)
• L. H. C. Tippett (1954)
• D. G. Kendall (1955)
• Henry Daniels (1957)
• G. A. Barnard (1958)
• E. C. Fieller (1960)
• D. R. Cox (1961)
• P. V. Sukhatme (1962)
• George E. P. Box (1964)
• C. R. Rao (1965)
• Peter Whittle (1966)
• Dennis Lindley (1968)
• Robin Plackett (1973)
• James Durbin (1976)
• John Nelder (1977)
• Peter Armitage (1978)
• Michael Healy (1979)
• M. Stone (1980)
• John Kingman (1981)
• Henry Wynn (1982)
• Julian Besag (1983)
• J. C. Gittins (1984)
• A. Bissell (1985)
• W. Pridmore (1985)
• Richard Peto (1986)
• John Copas (1987)
• John Aitchison (1988)
• F. P. Kelly (1989)
• David Clayton (1990)
• R. L. Smith (1991)
• Robert Nicholas Curnow (1992)
• A. F. M. Smith (1993)
• David Spiegelhalter (1994)
• B. W. Silverman (1995)
• Steffen Lauritzen (1996)
• Peter Diggle (1997)
• Harvey Goldstein (1998)
• Peter Green (1999)
• Walter Gilks (2000)
• Philip Dawid (2001)
• David Hand (2002)
• Kanti Mardia (2003)
• Peter Donnelly (2004)
• Peter McCullagh (2005)
• Michael Titterington (2006)
• Howell Tong (2007)
• Gareth Roberts (2008)
• Sylvia Richardson (2009)
• Iain M. Johnstone (2010)
• P. G. Hall (2011)
• David Firth (2012)
• Brian Ripley (2013)
• Jianqing Fan (2014)
• Anthony Davison (2015)
• Nancy Reid (2016)
• Neil Shephard (2017)
• Peter Bühlmann (2018)
• Susan Murphy (2019)
• Arnaud Doucet (2020)
• Håvard Rue (2021)
• Paul Fearnhead (2022)
Bronze Medallists
• William Gemmell Cochran (1936)
• R. F. George (1938)
• W. J. Jennett (1949)
• Peter Armitage (1962)
• James Durbin (1966)
• F. Downton (1967)
• Robin Plackett (1968)
• M. C. Pike (1969)
• P. G. Moore (1970)
• D. J. Bartholomew (1971)
• G. N. Wilkinson (1974)
• A. F. Bissell (1975)
• P. L. Goldsmith (1976)
• A. F. M. Smith (1977)
• Philip Dawid (1978)
• T. M. F. Smith (1979)
• A. J. Fox (1980)
• S. J. Pocock (1982)
• Peter McCullagh (1983)
• Bernard Silverman (1984)
• David Spiegelhalter (1985)
• D. F. Hendry (1986)
• Peter Green (1987)
• S. C. Darby (1988)
• S. M. Gore (1989)
• Valerie Isham (1990)
• M. G. Kenward (1991)
• C. Jennison (1992)
• Jonathan Tawn (1993)
• R. F. A. Poultney (1994)
• Iain M. Johnstone (1995)
• J. N. S. Matthews (1996)
• Gareth Roberts (1997)
• D. Firth (1998)
• P. W. F. Smith
• J. Forster (1999)
• J. Wakefield (2000)
• Guy Nason (2001)
• Geert Molenberghs (2002)
• Peter Lynn (2003)
• Nicola Best (2004)
• Steve Brooks (2005)
• Matthew Stephens (2006)
• Paul Fearnhead (2007)
• Fiona Steele (2008)
• Chris Holmes (2009)
• Omiros Papaspiliopoulos (2010)
• Nicolai Meinshausen (2011)
• Richard Samworth (2012)
• Piotr Fryzlewicz (2013)
• Ming Yuan (2014)
• Jinchi Lv (2015)
• Yingying Fan (2017)
• Peng Ding (2018)
• Jonas Peters (2019)
• Rachel McCrea (2020)
• Pierre E. Jacob (2021)
• Rajan Shah (2022)
Presidents of the Royal Statistical Society
19th century
• 1834–1836 The Marquess of Lansdowne
• 1836–1838 Sir Charles Lemon, Bt
• 1838–1840 The Earl FitzWilliam
• 1840–1842 Viscount Sandon
• 1842–1843 The Marquess of Lansdowne
• 1843–1845 Lord Ashley
• 1845–1847 The Lord Monteagle of Brandon
• 1847–1849 The Earl FitzWilliam
• 1849–1851 The Earl of Harrowby
• 1851–1853 The Lord Overstone
• 1853–1855 The Earl FitzWilliam
• 1855–1857 The Earl of Harrowby
• 1857–1859 Lord Stanley
• 1859–1861 Lord John Russell
• 1861–1863 Sir John Pakington, Bt
• 1863–1865 William Henry Sykes
• 1865–1867 The Lord Houghton
• 1867–1869 William Ewart Gladstone
• 1869–1871 William Newmarch
• 1871–1873 William Farr
• 1873–1875 William Guy
• 1875–1877 James Heywood
• 1877–1879 George Shaw-Lefevre
• 1879–1880 Thomas Brassey
• 1880–1882 James Caird
• 1882–1884 Robert Giffen
• 1884–1886 Rawson W. Rawson
• 1886–1888 George Goschen
• 1888–1890 Thomas Graham Balfour
• 1890–1892 Frederic J. Mouat
• 1892–1894 Charles Booth
• 1894–1896 The Lord Farrer
• 1896–1897 John Biddulph Martin
• 1897 Alfred Edmund Bateman
• 1897–1899 Leonard Courtney
• 1899–1900 Henry Fowler
• 1900–1902 The Lord Avebury
20th century
• 1902–1904 Patrick George Craigie
• 1904–1905 Sir Francis Powell, Bt
• 1905–1906 The Earl of Onslow
• 1906–1907 Richard Martin
• 1907–1909 Sir Charles Dilke, Bt
• 1909–1910 Jervoise Athelstane Baines
• 1910–1912 Lord George Hamilton
• 1912–1914 Francis Ysidro Edgeworth
• 1914–1915 The Lord Welby
• 1915–1916 Lord George Hamilton
• 1916–1918 Bernard Mallet, Registrar General
• 1918–1920 Herbert Samuel
• 1920–1922 R. Henry Rew
• 1922–1924 The Lord Emmott
• 1924–1926 Udny Yule
• 1926–1928 The Viscount D'Abernon
• 1928–1930 A. William Flux
• 1930–1932 Sir Josiah Stamp
• 1932–1934 The Lord Meston
• 1934–1936 Major Greenwood
• 1936–1938 The Lord Kennet
• 1938–1940 Arthur Lyon Bowley
• 1940–1941 Henry William Macrosty
• 1941 Hector Leak
• 1941–1943 William Beveridge
• 1943–1945 Ernest Charles Snow
• 1945–1947 The Lord Woolton
• 1947–1949 David Heron
• 1949–1950 Sir Geoffrey Heyworth
• 1950–1952 Austin Bradford Hill
• 1952–1954 Ronald Fisher
• 1954–1955 The Lord Piercy
• 1955–1957 Egon Pearson
• 1957–1959 Harry Campion
• 1959–1960 Hugh Beaver
• 1960–1962 Maurice Kendall
• 1962–1964 Joseph Oscar Irwin
• 1964–1965 Sir Paul Chambers
• 1965–1966 L. H. C. Tippett
• 1966–1967 M. S. Bartlett
• 1967–1968 Frank Yates
• 1968–1969 Arthur Cockfield
• 1969–1970 R. G. D. Allen
• 1970–1971 Bernard Benjamin
• 1971–1972 George Alfred Barnard
• 1972–1973 Harold Wilson
• 1973–1974 D. J. Finney
• 1974–1975 Henry Daniels
• 1975–1977 Stella Cunliffe
• 1977–1978 Henry Wynn
• 1978–1980 Sir Claus Moser
• 1980–1982 David Cox
• 1982–1984 Peter Armitage
• 1984–1985 Walter Bodmer
• 1985–1986 John Nelder
• 1986–1987 James Durbin
• 1987–1989 John Kingman
• 1989–1991 Peter G. Moore
• 1991–1993 T. M. F. Smith
• 1993–1995 D. J. Bartholomew
• 1995–1997 Adrian Smith
• 1997–1999 Robert Nicholas Curnow
• 1999–2001 Denise Lievesley
21st century
• 2001–2003 Peter Green
• 2003–2005 Andy Grieve
• 2005–2007 Tim Holt
• 2008–2009 David Hand
• 2010–2010 Bernard Silverman (resigned Feb 2010; replaced pro tem by David Hand)
• 2011–2012 Valerie Isham
• 2013–2014 John Pullinger
• 2014–2016 Peter Diggle
• 2017–2018 David Spiegelhalter
• 2019– Deborah Ashby
• Category
• List
Authority control
International
• FAST
• ISNI
• VIAF
National
• Norway
• Spain
• France
• BnF data
• Germany
• Israel
• United States
• Japan
• Australia
• Netherlands
• Portugal
Academics
• CiNii
People
• Trove
Other
• SNAC
• IdRef
|
Wikipedia
|
Robert Goldblatt
Robert Ian Goldblatt (born 1949) is a mathematical logician who is Emeritus Professor in the School of Mathematics and Statistics at Victoria University, Wellington, New Zealand. His doctoral advisor was Max Cresswell.[1] His most popular books are Logics of Time and Computation and Topoi: the Categorial Analysis of Logic. He has also written a graduate level textbook on hyperreal numbers which is an introduction to nonstandard analysis.
He has been Coordinating Editor of The Journal of Symbolic Logic and a Managing Editor of Studia Logica. He was elected Fellow and Councillor of the Royal Society of New Zealand, President of the New Zealand Mathematical Society, and represented New Zealand to the International Mathematical Union. In 2012 he was awarded the Jones Medal for lifetime achievement in mathematics.
Books and handbook chapters
• 1979: Topoi: The Categorial Analysis of Logic, North-Holland. Revised edition 1984. Dover Publications edition 2006. Internet edition, Project Euclid.
Benjamin C. Pierce recommends it as an "excellent beginner book", praising it for the use of simple set-theoretic examples and motivating intuitions, but noted that it "is sometimes criticized by category theorists for being misleading on some aspects of the subject, and for presenting long and difficult proofs where simple ones are available."[2] But the preface of the Dover edition observes (p. xv) that "This is a book about logic, rather than category theory per se. It aims to explain, in an introductory way, how certain logical ideas are illuminated by a category-theoretic perspective."
• 1982: Axiomatising the Logic of Computer Programming, Lecture Notes in Computer Science 130, Springer-Verlag.
• 1987: Orthogonality and Spacetime Geometry, Universitext Springer-Verlag ISBN 0-387-96519-X MR0888161
• 1987: Logics of Time and Computation. CSLI Lecture Notes, 7. Stanford University, Center for the Study of Language and Information MR1191162. Second edition 1992.
• 1993: Mathematics of Modality, CSLI Publications, ISBN 978-1-881526-24-7 MR1317099
• 1998: Lectures on the Hyperreals: An Introduction to Nonstandard Analysis. Graduate Texts in Mathematics, 188. Springer-Verlag.
Reviewer Perry Smith for MathSciNet wrote: "The author's ideas on how to achieve both intelligibility and rigor, explained in the preface, will be useful reading for anyone intending to teach nonstandard analysis."
• 2006: "Mathematical Modal Logic: a View of its Evolution" in Modalities in the Twentieth Century, Volume 7 of the Handbook of the History of Logic, edited by Dov M. Gabbay and John Woods, Elsevier, pp. 1–98.
• 2011: Quantifiers, Propositions and Identity: Admissible Semantics for Quantified Modal and Substructural Logics, Cambridge University Press and the Association for Symbolic Logic.
See also
• Influence of non-standard analysis
References
1. "Maxwell Cresswell - The Mathematics Genealogy Project". mathgenealogy.org. Retrieved 2023-04-09.
2. Benjamin C. Pierce (1991). Basic category theory for computer scientists. MIT Press. p. 73. ISBN 978-0-262-66071-6.
External links
• Home page
• Robert Goldblatt at the Mathematics Genealogy Project
Authority control
International
• ISNI
• VIAF
National
• France
• BnF data
• Catalonia
• Germany
• Israel
• United States
• Czech Republic
• Netherlands
• Poland
Academics
• DBLP
• MathSciNet
• Mathematics Genealogy Project
• zbMATH
Other
• IdRef
|
Wikipedia
|
Robert Goodell Brown
Robert Goodell Brown (1923–2013) was renowned in field of forecasting, specifically with major contributions of work regarding exponential smoothing. He was an International Institute of Forecasters member, and a past director. He contributed to the field of forecasting with practical books.[1]
References
1. http://forecasters.org/blog/2013/10/09/robert-g-brown-1923%E2%88%922013/
External links
• Obituary
• International Institute of Forecasters
Authority control
International
• ISNI
• VIAF
National
• Germany
• Israel
• United States
• Czech Republic
• Netherlands
Other
• IdRef
|
Wikipedia
|
Robert Guralnick
Robert Michael Guralnick (born 10 July 1950) is an American mathematician known for his work in group theory. He works as a Professor of Mathematics at the University of Southern California.[1]
Robert M. Guralnick
Guralnick at Oberwolfach, 2007
Born (1950-07-10) 10 July 1950
Los Angeles
NationalityAmerican
Alma materUCLA
AwardsCole Prize in Algebra (2018)
Scientific career
FieldsMathematics
InstitutionsUniversity of Southern California
Doctoral advisorBasil Gordon
Guralnick was named a Fellow of the American Mathematical Society in 2012, was an invited lecturer at the International Congress of Mathematicians in 2014, and was awarded the Cole Prize in 2018.[2] He is currently managing editor of Forum of Mathematics.
References
1. "Faculty Profile > USC Dana and David Dornsife College of Letters, Arts and Sciences". Dornsife.usc.edu. Retrieved 13 July 2018.
2. "2018 Frank Nelson Cole Prize in Algebra" (PDF). Ams.org. Retrieved 13 July 2018.
External links
• Robert Guralnick at the Mathematics Genealogy Project
Authority control
International
• ISNI
• VIAF
National
• Norway
• Germany
• Israel
• United States
• Netherlands
Academics
• DBLP
• Google Scholar
• MathSciNet
• Mathematics Genealogy Project
• ORCID
• Scopus
• zbMATH
People
• Deutsche Biographie
Other
• IdRef
|
Wikipedia
|
Robert Haldane (mathematician)
Robert Haldane FRSE (27 January 1772 in Perthshire – 9 March 1854 in St Andrews) was a British mathematician and minister of the Church of Scotland.
Robert Haldane
Born(1772-01-27)27 January 1772
Perthshire, Scotland
Died9 March 1854(1854-03-09) (aged 82)
St Andrews, Scotland
NationalityBritish
Alma materGlasgow University
Scientific career
FieldsMathematics
InstitutionsUniversity of St Andrews
Life
He was the son of a farmer at Overtown, Lecropt, on the borders of Perthshire and Stirlingshire; and was named after Robert Haldane, then proprietor of Airthrey Castle. He was educated at the school in Dunblane, and then at Glasgow University.[1]
Haldane became a private tutor, first in the family at Leddriegreen, Strathblane, and later with Col. Charles Moray of Abercairnie. On 5 December 1797, he was licensed as a preacher by the presbytery of Auchterarder, but he did not obtain a charge quickly. In August 1806, he was presented to the church of Drummelzier, in the presbytery of Peebles, and was ordained on 19 March 1807.[1]
When the chair of mathematics became vacant in the University of St. Andrews in 1807, Haldane was appointed to the professorship, and resigned his charge at Drummelzier on 2 October 1809. He remained in the post till 1820, when he was promoted by the crown to the pastoral charge of St. Andrews parish, vacant by the death of Principal George Hill, D.D. His predecessor had held the principalship of St. Mary's College in St. Andrews in conjunction with his ministerial office, and the same arrangement was followed in the case of Haldane, who was admitted on 28 September 1820. As principal he was ex officio primarius professor of divinity.[1]
On 17 May 1827 Haldane was elected moderator of the general assembly of the church of Scotland. At the time of the disruption of 1843 Haldane was called to the chair ad interim. He was elected a Fellow of the Royal Society of Edinburgh in 1820, his proposers being George Dunbar, Robert Jameson, Alexander Brunton and Patrick Neill.[2] In 1828 his role as Moderator was succeeded by Rev Stevenson McGill.
He died at St. Mary's College, St. Andrews, on 9 March 1854, in his eighty-third year, and was buried in the cathedral cemetery there. The grave lies on the north wall just left of the distinctive white military memorial to Lt Col Sir Hugh Lyon Playfair. Haldane's marble inscription is badly eroded.
His portrait was in the hall of the university library at St. Andrews. He was succeeded by John Tulloch.[1]
Haldane's only publication was a small work relating to the condition of the poor in St. Andrews (Cupar, 1841).[1]
References
• O'Connor, John J.; Robertson, Edmund F., "Robert Haldane (mathematician)", MacTutor History of Mathematics Archive, University of St Andrews
Notes
1. Stephen, Leslie; Lee, Sidney, eds. (1890). "Haldane, Robert (1772-1854)" . Dictionary of National Biography. Vol. 24. London: Smith, Elder & Co.
2. C D Waterston; A Macmillan Shearer (July 2006). Former Fellows of The Royal Society of Edinburgh, 1783–2002: Part 1 (A–J) (PDF). ISBN 090219884X. Archived from the original (PDF) on 24 January 2013. Retrieved 18 September 2015. {{cite book}}: |website= ignored (help)
Attribution
This article incorporates text from a publication now in the public domain: Stephen, Leslie; Lee, Sidney, eds. (1890). "Haldane, Robert (1772-1854)". Dictionary of National Biography. Vol. 24. London: Smith, Elder & Co.
Authority control
International
• VIAF
National
• Germany
|
Wikipedia
|
Robert Harley (mathematician)
Robert Harley (23 January 1828 – 26 July 1910) was an English Congregational minister and mathematician.
Life
Born in Liverpool on 23 January 1828, he was third son of Robert Harley by his wife Mary, daughter of William Stevenson, and niece of General Stevenson of Ayr.. The father, after a career as a merchant, became a minister of the Wesleyan Methodist Association. Harley's mathematical aptitude developed at school in Blackburn under William Hoole, and age 16 he was appointed to a mathematical mastership at Seacombe, near Liverpool, returning later to teach at Blackburn.[1][2]
In 1854 Harley entered the Congregational ministry, and was at Brighouse, Yorkshire, until 1868, after a time also filling the chair of mathematics and logic at Airedale College from 1864 to 1868.[3] In 1863 he was admitted Fellow of the Royal Society. He acted as secretary of the A section of the British Association at meetings at Norwich (1868) and Edinburgh (1871); and was a vice-president of the meetings at Bradford (1873), Bath (1888), and Cardiff (1891).[1]
From 1868 to 1872 he was pastor of the oldest Congregational church in Leicester, and from 1872 to 1881 was vice-principal of Mill Hill School, where he officiated in the chapel. At Mill Hill he was instrumental in erecting a public lecture hall for total abstinence talks as well as popular entertainment and instruction. From 1882 to 1885 he was principal of Huddersfield College, and from 1886 to 1890 minister of the Congregational church in Oxford, where he was made hon. M.A. in 1886.[1]
Having taken a ministerial appointment in Australia, Harley was pastor of Heath Church, Halifax, from 1892 until 1895, when he retired and settled at Forest Hill, near London. He continued to preach in London and the provinces, and as a temperance advocate.[1]
Harley died at Rosslyn, Westbourne Road, Forest Hill, on 26 July 1910, and was buried in Ladywell cemetery.[1]
Works
Throughout his career mathematics remained a major interest for Harley. He spent time on the theory of equations, especially the theory of the general equation of the fifth degree. His conclusions, which were published in Memoirs of the Manchester Lit. and Phil. Soc 1860, xv. 172–219, were independently reached at the same time by Sir James Cockle. Harley's two further papers on the Theory of Quintics[4] and an exposition of Cockle's method of symmetric products in Phil. Trans. (1860) attracted the attention of Arthur Cayley, who carried the research further.[1]
Harley failed to complete the treatise on quintics which he had begun, but continued to publish. A sketch of the life and work of George Boole appeared in the British Quarterly Review (July 1866), and a memoir of his friend, Sir James Cockle, is in the Proceedings of the Royal Society, vol. lix.[1]
Family
In 1854 Harley married Sara, daughter of James Stroyan of Wigan; she died in 1905.[1] Upon his death he was survived by two sons and a daughter.[3]
Their son Harold Harley (1860–1937), using the pen name Mark Ambient, was a dramatist and librettist of musical comedies.[5][6]
Notes
1. Lee, Sidney, ed. (1912). "Harley, Robert" . Dictionary of National Biography (2nd supplement). Vol. 2. London: Smith, Elder & Co.
2. Panteki, Maria. "Harley, Robert". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/33715. (Subscription or UK public library membership required.)
3. "Harley, Rev. Robert". Who's Who. 1910. p. 856.
4. Quarterly Journal of Mathematics 1860–2, iii. 343–59; v. 248–60
5. Alumni Cantabrigienses: A Biographical List of All Known Students, Graduates and Holders of Office at the University of Cambridge, from the Earliest Times to 1900 Volume 2, page 246.
6. "Ambient, Mark". Who's Who in the Theatre (4th ed.). 1922. p. 11.
Attribution
This article incorporates text from a publication now in the public domain: Lee, Sidney, ed. (1912). "Harley, Robert". Dictionary of National Biography (2nd supplement). Vol. 2. London: Smith, Elder & Co.
Authority control
International
• VIAF
National
• Germany
• United States
Academics
• zbMATH
|
Wikipedia
|
Robert Heath (mathematician)
Robert Heath (died 1779) was an English army officer, mathematician, and periodical editor.
Life
Heath was a captain in the British Army, and was described late in life as a "half-pay captain of invalids". For a time he served with his regiment in the Scilly Isles. He tried to establish a Palladium Society, under his own control.[1]
The Ladies' Diary
Heath is best known as a contributor to The Ladies' Diary, from 1737. He was taken onto the staff, and proposed the prize essays for 1739, 1740, 1742, 1746, and 1748. When Henry Beighton, editor of the Diary, died in October 1743, the proprietors, the Stationers' Company, allowed Beighton's widow to run it with Heath as her deputy. In that capacity Heath exercised full editorial control from 1744 to 1753, and continued to write under his own and assumed names.[1]
A personal quarrel with Thomas Simpson led Heath to denigrate in print Simpson's Doctrine of Ultimators (1750) and Doctrine of Fluxions (1751), while praising related works on the same subject by William Emerson. John Turner,[2] who like Emerson was a contributor to the Diary, inserted in his Mathematical Exercises (1750–3) a defence of Simpson against Heath, signed "Honestus". In 1753 the Stationers' Company dismissed Heath and installed Simpson in the editorial chair.[1]
Works
John Holmes and his Greek grammar were attacked by Heath and Robert Hankinson in a controversy from the period 1738–40. Holmes wrote in his own defence, and had support from Thomas Simpson.[3] Heath also published on the longitude problem.[4] He had a reputation for indecent writing and innuendo in the Town and Country Magazine and The Rambler's Magazine.[5]
While editor of the Ladies' Diary, Heath started in 1749 a journal on similar lines of his own account, The Palladium, which then ran for nearly 30 years, to 1778, under changing titles.[6] It was said that he poached for it better contributions sent to him as editor of the Diary.[1]
Heath wrote A History of the Islands of Scilly, with a Tradition of the Land called Lioness, and a General Account of Cornwall. The book, published in London in 1750, and dedicated to the Duke of Cumberland, included a new map of the isles, drawn by Heath from a survey made in 1744; it was reprinted in 1808 in John Pinkerton's Voyages and Travels, ii. 729–784. His other works included:[1]
• The Practical Arithmetician, 1750.
• Truth Triumphant: or Fluxions for the Ladies, 1752; part of the controversy with Simpson and Turner.[7]
• The Ladies' Chronologer, No. I. 1754 (amalgamated with the Palladium of 1755).
• The Ladies' Philosopher, No. I. 1752, II. 1753, III. 1754.
• Astronomia Accurata; or the Royal Astronomer and Navigator, 1760.
• General and Particular Account of the Annular Eclipse of the Sun which happened on Sunday, April 1, 1764.
Notes
1. Stephen, Leslie; Lee, Sidney, eds. (1891). "Heath, Robert (d.1779)" . Dictionary of National Biography. Vol. 25. London: Smith, Elder & Co.
2. Wallis, Ruth. "Heath, Robert". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/12844. (Subscription or UK public library membership required.)
3. Stoker, David. "Holmes, John". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/65275. (Subscription or UK public library membership required.)
4. Sir Henry Ellis (1843). Original Letters of Eminent Literary Men: Of the Sixteenth, Seventeenth, and Eighteenth Centuries. Camden Society. p. 304.
5. Joseph Clinton Robertson, ed. (1849). Mechanics' Magazine. Knight and Lacey. p. 468.
6. It was renamed The Gentleman and Lady's Palladium, 1750, The Gentleman's and Lady's Palladium and Chronologer, 1754, The Gentleman's and Lady's Military Palladium, 1759, The Palladium Extraordinary, 1763, The Palladium Enlarged, 1764, The Palladium of Fame, 1765, and The British Palladium, 1768.
7. Florian Cajori, A History of the Conceptions of Limits and Fluxions in Great Britain, from Newton to Woodhouse (1919), p. 212; archive.org.
Attribution
This article incorporates text from a publication now in the public domain: Stephen, Leslie; Lee, Sidney, eds. (1891). "Heath, Robert (d.1779)". Dictionary of National Biography. Vol. 25. London: Smith, Elder & Co.
Authority control
International
• VIAF
• WorldCat
National
• United States
• Netherlands
|
Wikipedia
|
Robert Henry Risch
Robert Henry Risch (born 1939) is an American mathematician who worked on computer algebra and is known for his work on symbolic integration, specifically the Risch algorithm.[1] This result was quoted as a milestone in the development of mathematics:
Calculus students worldwide depend on the algorithm, whenever they appeal to Wolfram Alpha to do their homework.[2]
He is also known for results on algebraic properties of elementary functions.[3] He received his PhD from University of California, Berkeley in 1968[4] under the supervision of Maxwell A. Rosenlicht.[5] After his PhD, he worked at the Thomas J. Watson Research Center Mathematics of AI group[6] and, between 1970 and 1972, the Institute for Advanced Study.[7]
References
1. Risch, Robert H. (1969). "The Problem of Integration in Finite Terms" (PDF). Transactions of the American Mathematical Society. 139: 67–189. doi:10.1090/S0002-9947-1969-0237477-8. Retrieved 8 January 2020.
2. Garcia, Stephan Ramon; Miller, Steven J. (2019). 100 years of math milestones : the Pi Mu Epsilon centennial collection. p. 302. ISBN 978-1-4704-3652-0. Retrieved 9 January 2020.
3. Risch, Robert H. (1979). "Algebraic Properties of the Elementary Functions of Analysis". American Journal of Mathematics. 101 (4): 743–759. doi:10.2307/2373917. JSTOR 2373917.
4. "Robert Henry Risch record at UC Berkeley". Robert Henry Risch. Retrieved 8 January 2020.
5. "Robert Henry Risch at Mathematics Genealogy Project". Retrieved 9 January 2020.
6. "Mathematics of AI". Retrieved 9 January 2020.
7. "Past Members: Robert Henry Risch". Institute of Advanced Studies. 9 December 2019.
Authority control: Academics
• MathSciNet
• Mathematics Genealogy Project
• zbMATH
|
Wikipedia
|
Robert Horton Cameron
Robert Horton Cameron (May 17, 1908 – July 17, 1989) was an American mathematician, who worked on analysis and probability theory. He is known for the Cameron–Martin theorem.
Robert Horton Cameron
Born(1908-05-17)May 17, 1908
Brooklyn, New York, USA
DiedJuly 17, 1989(1989-07-17) (aged 81)
Minneapolis, Minnesota, USA
NationalityAmerican
EducationCornell University
Known forCameron–Martin theorem
AwardsChauvenet Prize (1944)
Scientific career
Fieldsmathematician
InstitutionsMIT
University of Minnesota
ThesisAlmost Periodic Transformations[1] (1932)
Doctoral advisorW. A. Hurwitz
Doctoral studentsElizabeth Cuthill
Monroe D. Donsker
Education and career
Cameron received his Ph.D. in 1932 from Cornell University under the direction of W. A. Hurwitz.[2][3] He studied under a National Research Council postdoc at the Institute for Advanced Study in Princeton from 1933 to 1935.[4] Cameron was a faculty member at MIT from 1935 to 1945. He was then a faculty member at the University of Minnesota until his retirement. He spent the academic year 1953–1954 on sabbatical leave at the Institute for Advanced Study.[4] His doctoral students include Monroe D. Donsker and Elizabeth Cuthill. He had a total of 35 Ph.D. students at the University of Minnesota — his first two graduated in 1946 and his last one in 1977. Cameron published a total of 72 papers — his first in 1934 and his last, posthumously, in 1990.[5]
At MIT, he did some work with Norbert Wiener. During the 1940s Cameron and W. T. Martin, who was from 1943 to 1946 the chair of the mathematics department at Syracuse University, engaged in an ambitious program of extending Norbert Wiener's early work on mathematical models of Brownian motion.[6] In 1944, Cameron was awarded the Chauvenet Prize for '"Some Introductory Exercises in the Manipulation of Fourier Transforms", which appeared in National Mathematics Magazine, 1941, vol. 15, pages 331–356.
References
1. "Selected Graduate Students 1868--1968 | Department of Mathematics". math.cornell.edu. Retrieved 2022-11-20.
2. Robert Horton Cameron at the Mathematics Genealogy Project
3. Cameron, Robert Horton 1908– (WorldCat Identities) Cameron's thesis Almost periodic transformations was published in 3 different editions from 1932 to 1934. The copy in the U. S. Library of Congress is a 1934 edition. A 1932 edition published by Cornell U. is 170 pages long.
4. Cameron, Robert H., Community of Scholars Profile, IAS
5. Information provided by Prof. Emeritus David Skoug, U. of Nebraska, Feb. 2013
6. Kac, Mark (1985). Enigmas of Chance. New York: Harper & Row. p. 113. ISBN 0520059867.
Chauvenet Prize recipients
• 1925 G. A. Bliss
• 1929 T. H. Hildebrandt
• 1932 G. H. Hardy
• 1935 Dunham Jackson
• 1938 G. T. Whyburn
• 1941 Saunders Mac Lane
• 1944 R. H. Cameron
• 1947 Paul Halmos
• 1950 Mark Kac
• 1953 E. J. McShane
• 1956 Richard H. Bruck
• 1960 Cornelius Lanczos
• 1963 Philip J. Davis
• 1964 Leon Henkin
• 1965 Jack K. Hale and Joseph P. LaSalle
• 1967 Guido Weiss
• 1968 Mark Kac
• 1970 Shiing-Shen Chern
• 1971 Norman Levinson
• 1972 François Trèves
• 1973 Carl D. Olds
• 1974 Peter D. Lax
• 1975 Martin Davis and Reuben Hersh
• 1976 Lawrence Zalcman
• 1977 W. Gilbert Strang
• 1978 Shreeram S. Abhyankar
• 1979 Neil J. A. Sloane
• 1980 Heinz Bauer
• 1981 Kenneth I. Gross
• 1982 No award given.
• 1983 No award given.
• 1984 R. Arthur Knoebel
• 1985 Carl Pomerance
• 1986 George Miel
• 1987 James H. Wilkinson
• 1988 Stephen Smale
• 1989 Jacob Korevaar
• 1990 David Allen Hoffman
• 1991 W. B. Raymond Lickorish and Kenneth C. Millett
• 1992 Steven G. Krantz
• 1993 David H. Bailey, Jonathan M. Borwein and Peter B. Borwein
• 1994 Barry Mazur
• 1995 Donald G. Saari
• 1996 Joan Birman
• 1997 Tom Hawkins
• 1998 Alan Edelman and Eric Kostlan
• 1999 Michael I. Rosen
• 2000 Don Zagier
• 2001 Carolyn S. Gordon and David L. Webb
• 2002 Ellen Gethner, Stan Wagon, and Brian Wick
• 2003 Thomas C. Hales
• 2004 Edward B. Burger
• 2005 John Stillwell
• 2006 Florian Pfender & Günter M. Ziegler
• 2007 Andrew J. Simoson
• 2008 Andrew Granville
• 2009 Harold P. Boas
• 2010 Brian J. McCartin
• 2011 Bjorn Poonen
• 2012 Dennis DeTurck, Herman Gluck, Daniel Pomerleano & David Shea Vela-Vick
• 2013 Robert Ghrist
• 2014 Ravi Vakil
• 2015 Dana Mackenzie
• 2016 Susan H. Marshall & Donald R. Smith
• 2017 Mark Schilling
• 2018 Daniel J. Velleman
• 2019 Tom Leinster
• 2020 Vladimir Pozdnyakov & J. Michael Steele
• 2021 Travis Kowalski
• 2022 William Dunham, Ezra Brown & Matthew Crawford
Authority control
International
• FAST
• ISNI
• VIAF
National
• Germany
• Israel
• United States
• Netherlands
Academics
• MathSciNet
• Mathematics Genealogy Project
• zbMATH
Other
• IdRef
|
Wikipedia
|
Robert Hues
Robert Hues (1553 – 24 May 1632) was an English mathematician and geographer. He attended St. Mary Hall at Oxford, and graduated in 1578. Hues became interested in geography and mathematics, and studied navigation at a school set up by Walter Raleigh. During a trip to Newfoundland, he made observations which caused him to doubt the accepted published values for variations of the compass. Between 1586 and 1588, Hues travelled with Thomas Cavendish on a circumnavigation of the globe, performing astronomical observations and taking the latitudes of places they visited. Beginning in August 1591, Hues and Cavendish again set out on another circumnavigation of the globe. During the voyage, Hues made astronomical observations in the South Atlantic, and continued his observations of the variation of the compass at various latitudes and at the Equator. Cavendish died on the journey in 1592, and Hues returned to England the following year.
Robert Hues
The title page of a 1634 version of Hues' Tractatus de globis in the collection of the Biblioteca Nacional de Portugal
Born1553
Little Hereford, Herefordshire, England
Died24 May 1632 (aged 78–79)
Oxford, Oxfordshire, England
Alma materSt Mary Hall, Oxford (BA, 1578)
Known forpublishing Tractatus de globis et eorum usu (Treatise on Globes and their Use, 1594)
Scientific career
FieldsMathematics, geography
In 1594, Hues published his discoveries in the Latin work Tractatus de globis et eorum usu (Treatise on Globes and Their Use) which was written to explain the use of the terrestrial and celestial globes that had been made and published by Emery Molyneux in late 1592 or early 1593, and to encourage English sailors to use practical astronomical navigation. Hues' work subsequently went into at least 12 other printings in Dutch, English, French and Latin.
Hues continued to have dealings with Raleigh in the 1590s, and later became a servant of Thomas Grey, 15th Baron Grey de Wilton. While Grey was imprisoned in the Tower of London for participating in the Bye Plot, Hues stayed with him. Following Grey's death in 1614, Hues attended upon Henry Percy, the 9th Earl of Northumberland, when he was confined in the Tower; one source states that Hues, Thomas Harriot and Walter Warner were Northumberland's constant companions and known as his "Three Magi", although this is disputed. Hues tutored Northumberland's son Algernon Percy (who was to become the 10th Earl of Northumberland) at Oxford, and subsequently (in 1622–1623) Algernon's younger brother Henry. In later years, Hues lived in Oxford where he was a fellow of the University, and discussed mathematics and related subjects with like-minded friends. He died on 24 May 1632 in the city and was buried in Christ Church Cathedral.
Early years and education
Robert Hues was born in 1553 at Little Hereford in Herefordshire, England. In 1571, at the age of 18 years, he entered Brasenose College, University of Oxford.[1][2] English antiquarian Anthony à Wood (1632–1695) wrote that when Hues arrived at Oxford he was "only a poor scholar or servitor ... he continued for some time a very sober and serious servant ... but being sensible of the loss of time which he sustained there by constant attendance, he transferred himself to St Mary's Hall".[3] Hues graduated with a Bachelor of Arts (B.A.) degree on 12 July 1578,[4] having shown marked skill in Greek. He later gave advice to the dramatist and poet George Chapman for his 1616 English translation of Homer,[5] and Chapman referred to him as his "learned and valuable friend".[6] According to the Oxford Dictionary of National Biography, there is unsubstantiated evidence that after completing his degree Hues was held in the Tower of London, though no reason is given for this, then went abroad after his release.[1] It is possible he travelled to Continental Europe.[7]
Hues was a friend of the geographer Richard Hakluyt, who was then regent master of Christ Church. In the 1580s, Hakluyt introduced him to Walter Raleigh and explorers and navigators whom Raleigh knew. In addition, it is likely that Hues came to know astronomer and mathematician Thomas Harriot and Walter Warner at Thomas Allen's lectures in mathematics. The four men were later associated with Henry Percy, the 9th Earl of Northumberland,[1][8] who was known as the "Wizard Earl" for his interest in scientific and alchemical experiments and his library.[9]
Career
Hues became interested in geography and mathematics – an undated source indicates that he disputed accepted values of variations of the compass after making observations off the Newfoundland coast. He either went there on a fishing trip, or may have joined a 1585 voyage to Virginia arranged by Raleigh and led by Richard Grenville, which passed Newfoundland on the return journey to England. Hues perhaps become acquainted with the sailor Thomas Cavendish at this time, as both of them were taught by Harriot at Raleigh's school of navigation. An anonymous 17th-century manuscript states that Hues circumnavigated the world with Cavendish between 1586 and 1588 "purposely for taking the true Latitude of places";[11] he may have been the "NH" who wrote a brief account of the voyage that was published by Hakluyt in his 1589 work The Principall Navigations, Voiages, and Discoveries of the English Nation.[12] In the year that book appeared, Hues was with Edward Wright on the Earl of Cumberland's raiding expedition to the Azores to capture Spanish galleons.[1]
Beginning in August 1591, Hues joined Cavendish on another attempt to circumnavigate the globe. Sailing on the Leicester, they were accompanied by the explorer John Davis on the Desire. Cavendish and Davis agreed that they would part company once they had cleared the Strait of Magellan between Chile and Isla Grande de Tierra del Fuego, as Davis intended to sail to America to search for the Northwest Passage. The expedition was ultimately unsuccessful, although Davis did discover the Falkland Islands.[13] In the meantime, delayed in small harbours in the Strait with crew members dying from the cold, illness and starvation, Cavendish turned back eastwards to return to England. He was plagued by mutinous crewmen, and also by natives and Portuguese who attacked his sailors seeking food and water on shore. Increasingly depressed, Cavendish died in 1592 somewhere in the Atlantic Ocean, possibly a suicide.[7][14]
During the voyage, Hues made astronomical observations of the Southern Cross and other stars of the Southern Hemisphere while in the South Atlantic, and also observed the variation of the compass there and at the Equator. He returned to England after Cavendish died,[15] and published his discoveries in the work Tractatus de globis et eorum usu (Treatise on Globes and Their Use, 1594),[16] which he dedicated to Raleigh.[1] The book was written to explain the use of the terrestrial and celestial globes that had been made and published by Emery Molyneux in late 1592 or early 1593.[17] Apparently, the book was also intended to encourage English sailors to use practical astronomical navigation,[1] although Lesley Cormack has observed that the fact it was written in Latin suggests that it was aimed at scholarly readers on the Continent.[18] In 1595, William Sanderson, a London merchant who had largely financed the globes' construction, presented a small globe together with Hues' "Latin booke that teacheth the use of my great globes"[19] to Robert Cecil, a statesman who was spymaster and minister to Elizabeth I and James I. Hues' work subsequently went into at least 12 other printings in Dutch (1597, 1613 and 1622), English (1638 and 1659), French (1618) and Latin (1611, 1613, 1617, 1627, 1659 and 1663).[20] In his book An Accidence or The Path-way to Experience: Necessary for all Young Sea-men (1626),[21] John Smith, who founded the first permanent English settlement in North America at Jamestown, Virginia, listed Hues' book among the works that a young seaman should study.[22]
Tractatus de globis begins with a letter by Hues dedicated to Raleigh that recalled geographical discoveries made by Englishmen during Elizabeth I's reign. However, he felt that his countrymen would have surpassed the Spaniards and Portuguese if they had a complete knowledge of astronomy and geometry, which were essential to successful navigation.[23] In the preface of the book, Hues rehearsed arguments that proved the earth is a sphere, and refuted opposing theories.[24] The treatise was divided into five parts. The first part described elements common to Molyneux's terrestrial and celestial globes, including the circles and lines inscribed on them, zones and climates, and the use of each globe's wooden horizon circle and brass meridian.[25] The second part described planets, fixed stars and constellations; while the third part described the lands and seas shown on the terrestrial globe, and discussed the length of the circumference of the earth and of a degree of a great circle.[26] Part 4, which Hues considered the most important part of the work, explained how the globes enabled seamen to determine the sun's position, latitude, course and distance, amplitudes and azimuths, and time and declination.[27] The final part of the work contained a treatise inspired by Harriot on rhumb lines.[26][28] In the work, Hues also published for the first time the six fundamental navigational propositions involved in solving what was later termed the "nautical triangle" used for plane sailing. Difference of latitude and departure (or longitude) are two sides of the triangle forming a right angle, the distance travelled is the hypotenuse, and the angle between difference of latitude and distance is the course. If any two elements are known, the other two can be determined by plotting or calculation using tables of sines, tangents and secants.[29]
In the 1590s, Hues continued to have dealings with Raleigh – he was one of the executors of Raleigh's will[15] – and he may have been the "Hewes" who dined with Northumberland regularly in 1591. He later became a servant of Thomas Grey, the 15th and last Baron Grey de Wilton (1575–1614). For participating in the Bye Plot, a conspiracy by Roman Catholic priest William Watson to kidnap James I and force him to repeal anti-Catholic legislation, Grey was attainted and forfeited his title in 1603. The following year, he was imprisoned in the Tower of London. Grey was given consent for Hues to stay in the Tower with him.[1] Between 1605 and 1621, Northumberland was also confined in the Tower; he was suspected of involvement in the Gunpowder Plot of 1605 because his relative Thomas Percy was among the conspirators.
In 1616, following Grey's death, Hues began to be "attendant upon th'aforesaid Earle of Northumberland for matters of learning",[30] and was paid a yearly sum of £40 to support his research until Northumberland's death in 1632.[1][15] Wood stated that Harriot, Hues and Warner were Northumberland's "constant companions, and were usually called the Earl of Northumberland's Three Magi. They had a table at the Earl's charge, and the Earl himself did constantly converse with them, and with Sir Walter Raleigh, then in the Tower".[31] Together with the scientist Nathanael Toporley and the mathematician Thomas Allen, the men kept abreast of developments in astronomy, mathematics, physiology and the physical sciences, and made important contributions in these areas.[32] According to the letter writer John Chamberlain, Northumberland refused a pardon offered to him in 1617, preferring to remain with Harriot, Hues and Warner.[33] However, the fact that these companions of Northumberland were his "Three Magi" studying with him in the Tower of London has been regarded as a romanticisation by the antiquarian John Aubrey and disputed for lack of evidence.[1][34] Hues was tutor to Northumberland's sons: first Algernon Percy, who subsequently became the 10th Earl of Northumberland, at Oxford where he matriculated at Christ Church in 1617; and later Algernon's younger brother Henry in 1622–1623. Hues lived at Christ Church at this time, but may have occasionally attended upon Northumberland at Petworth House in Petworth, West Sussex, and at Syon House in London after the latter's release from the Tower in 1622.[32] Hues sometimes met Walter Warner in London, and they are known to have discussed the reflection of bodies.[1]
Later life
In later years, Hues lived in Oxford where he discussed mathematics and allied subjects with like-minded friends.[1][35] Cormack states he was a fellow at the University.[18] Under the terms of the will of Thomas Harriot, who died on 2 July 1621, Hues and Warner were given the responsibility of helping Harriot's executor Nathaniel Torporley to prepare Harriot's mathematical papers for publication. Hues was also required to help price Harriot's books and other possessions for sale to the Bodleian Library.[1]
Hues, who did not marry, died on 24 May 1632 in Stone House, St. Aldate's (opposite the Blue Boar in central Oxford).[36] This was the house of John Smith, M.A., the son of a cook at Christ Church named J. Smith.[15] In his will, Hues made many small bequests to his friends, including a sum of £20 to his "kinswoman" Mary Holly (of whom nothing is known), and 20 nobles to each of her three sisters. He was buried in Christ Church Cathedral, and a monumental brass to him was placed in Christ Church with the following inscription:[1]
Depositum viri literatissimi, morum ac religionis integerrimi, Roberti Husia, ob eruditionem omnigenem [sic: omnigenam?], Theologicam tum Historicam, tum Scholasticam, Philologicam, Philosophiam, præsertim vero Mathematicam (cujus insigne monumentum in typis reliquit) Primum Thomæ Candishio conjunctissimi, cujus in consortio, explorabundis [sic: explorabundus?] velis ambivit orbem: deinde Domino Baroni Gray; cui solator accessit in arca Londinensi. Quo defuncto, ad studia henrici Comitis Northumbriensis ibidem vocatus est, cujus filio instruendo cum aliquot annorum operam in hac Ecclesia dedisset et Academiae confinium locum valetudinariae senectuti commodum censuisset; in ædibus Johannis Smith, corpore exhaustus, sed animo vividus, expiravit die Maii 24, anno reparatae salutis 1632, aetatis suæ 79.[37] [Here lies a highly lettered man, of the highest moral and religious integrity, Robert Hues, on account of his erudition in all subjects, both Theology and History, and Rhetoric, Philology, and Philosophy, but especially Mathematics (of which a notable volume [i.e., his book] remains in print). He was most closely associated with Thomas Cavendish, in whose company he explored the world by sail; then with Lord Baron Gray, for whom he came as consoler in the Tower of London. When Gray died, he was summoned to study in the same place with Henry Earl of Northumberland, to teach his son, and when he had worked for some years in this Church [i.e., Christ Church Cathedral], and had decided that the place next to the School [i.e., Christ Church, Oxford] was suitable for his health in his old age, he breathed his last at the house of John Smith, his body exhausted, but with a lively spirit, on 24 May, in the year of our salvation 1632, at the age of 79.]
Works
• Hues, Robert (1594), Tractatus de globis et eorum usu: accommodatus iis qui Londini editi sunt anno 1593, sumptibus Gulielmi Sandersoni civis Londinensis, conscriptus à Roberto Hues [Treatise on Globes and their Use: Adapted to those which have been Published in London in the Year 1593, at the Expense of William Sanderson, a London Resident, Written by Robert Hues], London: In ædibus Thomæ Dawson [in the house of Thomas Dawson], OCLC 55576175 (in Latin), octavo. The following reprints are referred to by Clements Markham in his introduction to the Hakluyt Society's 1889 reprint of the English version of Tractatus de globis at pp. xxxviii–xl:
• 2nd printing: Hues, Robert (1597), Tractaet Ofte Hendelinge van het gebruijck der Hemelscher ende Aertscher Globe. Gheaccommodeert naer die Bollen, die eerst ghesneden zijn in Enghelandt door Io. Hondium, Anno 1693 [sic: 1593] ende nu gants door den selven vernieut, met alle de nieuwe ontdeckinghen van Landen, tot den daghe van heden geschiet, ende daerenboven van voorgaende fauten verbetert. In't Latijn beschreven, door Robertum Hues, Mathematicum, nu in Nederduijtsch overgheset, ende met diveersche nieuwe verclaringhe ende figueren vermeerdert en verciert. Door I. Hondium [Treatise or Essays on the Use of the Celestial and Terrestrial Globes. Tailored for the Globes which were First Made in England by J. Hondius, in the Year 1693 [sic: 1593], and which have now been Completely Revised by Him, with All New Discoveries of Countries up to the Present Day, and furthermore with Previous Errors Corrected. Described in Latin by Robert Hues, Mathematician, and now Translated into Dutch, and Enhanced and Ornamented with Several New Explanations and Figures, by J. Hondius], translated by Hondium, Iudocum, Amsterdam: Cornelis Claesz, OCLC 42811612 (in Dutch), quarto.[38]
• 3rd printing: Hues, Robert (1611), Tractatus de globis coelesti et terrestri ac eorum usu, conscriptus a Roberto Hues, denuo auctior & emendatior editus [Treatise on Globes Celestial and Terrestrial and their Use, written by Robert Hues, Second Enlarged and Corrected Edition], Amsterdam: Jodocus Hondius, OCLC 187141964 (in Latin), octavo. A reprint of the first edition of 1594.
• 4th printing: Hues, Robert (1613), Tractaut of te handebingen van het gebruych der hemelsike ende aertscher globe [Treatise or Essays on the Use of the Celestial and Terrestrial Globes], Amsterdam: [s.n.] (in Dutch), quarto.
• 5th printing: Hues, Robert (1613), Tractatvs de globis, coelesti et terrestri, ac eorvm vsu [Treatise on Globes, Celestial and Terrestrial, and their Use], Heidelberg: Typis [Printed by] Gotthardi Voegelini, OCLC 46414822 (in Latin). Contains the Index Geographicus. DeGolyer Collection in the History of Science and Technology (now History of Science Collections), University of Oklahoma.
• 6th printing: Hues, Robert (1617), Tractatvs de globis, coelesti et terrestri eorvmqve vsv. Primum conscriptus & editus a Roberto Hues. Anglo semelque atque iterum a Iudoco Hondio excusus, & nunc elegantibus iconibus & figuris locupletatus: ac de novo recognitus multisque observationibus oportunè illustratus as passim auctus opera ac studio Iohannis Isacii Pontani ... [Treatise on Globes, Celestial and Terrestrial, and their Use. First Written and Published by Robert Hues, Englishman, and in the First and Second Editions Drawn by Jodocus Hondius, and now Enlarged by Elegant Pictures and Drawings, and again Revised and Fittingly Illustrated by Many Observations, and throughout Enlarged by the Work and Effort of John Isaac Pontanus ...], Amsterdam: Excudebat [printed by] H[enricus] Hondius (in Latin), quarto.
• 7th printing: Hues, Robert (1618), Traicté des globes, et de leur usage, traduit du Latin de Robert Hues, et augmente de plusieurs nottes et operations du compas de proportion par D Henrion, mathematicien [A Treatise on Globes and their Use, Translated from the Latin version by Robert Hues, and Augmented with Several Notes and Operations of the Compass of Proportion by D Henrion, Mathematician], translated by Henrion, Denis, Paris: Chez Abraham Pacard, ruë sainct Iacques, au sacrifice d'Abraham [At Abraham Pacard, St. Jacques Street, with the sacrifice of Abraham], OCLC 37802904 (in French), octavo.[39]
• 8th printing: Hues, Robert (1622), Tractaet ofte handelinge van het gebruyck der hemelscher ende aertscher globe [Treatise or Essays on the Use of the Celestial and Terrestrial Globes], Amsterdam: Michiel Colijn, boeck-vercooper, woonende op't water, in't Huys-boeck, by de Oude Brugghe [Michiel Colijn, bookseller, who lives at the water's edge, in Huys-boeck, near the old bridge], OCLC 79659147 (in Dutch), quarto.[40]
• 9th printing: Hues, Robert (1627), Tractatvs de globis, coelesti et terrestri, ac eorvm vsv [Treatise on Globes, Celestial and Terrestrial, and their Use], Francofvrti ad Moenvm [Frankfurt am Main, Germany]: Typis & sumptibus VVechelianorum, apud Danielem & Dauidem Aubrios & Clementem Schleichium [Printed and paid for by the Wechelians, by Daniel and David Aubrios and Clement Schleich], OCLC 23625532 (in Latin), duodecimo.
• 10th printing: Hues, Robert (1638), A Learned Treatise of Globes, both Cœlestiall and Terrestriall: With their Severall Uses. Written first in Latine, by Mr Robert Hues: And by him so Published. Afterward Illustrated with Notes, by Io. Isa. Pontanus. And now Lastly made English, for the Benefit of the Unlearned by John Chilmead MrA of Christ-Church in Oxon, London: Printed by the assigne of T[homas] P[urfoot] for P[hilemon] Stephens and C[hristopher] Meredith, and are to be sold at their shop at the Golden Lion in Pauls-Church-yard, OCLC 165905181.[41]
• 11th printing: A Latin version by Jodocus Hondius and John Isaac Pontanus appeared in London in 1659. Octavo.[42]
• 12th printing: Hues, Robert; John Isaac Pontanus (1659), A Learned Treatise of Globes, both Cœlestiall and Terrestriall with their Several Uses .., London: Printed by J.S. for Andrew Kemb, and are to be sold at his shop ..., OCLC 11947725, octavo. Collection of Yale University Library.
• 13th printing: Hues, Robert (1663), Tractatus de globis coelesti et terrestri eorumque usu ac de novo recognitus multisq[ue] observationibus opportunè illustratus ac passim auctus, opera et studio Johannis Isacii Pontani ...; adjicitur Breviarium totius orbis terrarum Petri Bertii ... [Treatise on Globes Celestial and Terrestrial and their Use, Collected Anew and Suitably Illustrated with Many Observations and Enlarged Throughout, by the Effort and Devotion of John Isaac Pontanus ... A Brief Account of the Whole Globe is Added by Peter Bertius ...], Oxford: Excudebat [printed by] W.H., impensis [at the expense of] Ed. Forrest, OCLC 13197923 (in Latin).
The Hakluyt Society's reprint of the English version was itself published as:
• Hues, Robert (1889), Markham, Clements R. (ed.), Tractatus de globis et eorum usu: A Treatise Descriptive of the Globes Constructed by Emery Molyneux and Published in 1592 [Hakluyt Society, 1st ser., pt. II, no. 79a], London: Hakluyt Society, ISBN 978-0-8337-1759-7, OCLC 149869781.
The following works also are, or appear to be, versions of Tractatus de globis et eorum usu, though they are not mentioned by Markham:
• Hues, Robert (1623), Tractaet ofte Handelinge van het gebruyck der Hemelscher ende Aertscher Globe: In't Latyn eerst beschreven door Robertvm Hves, Mathematicum / en nu in Nederduytsch over-geset en met diversche nieuwe Verklaringen en Figuren vermeerdert en verciert / oock vele disputable questien gesolveert, door Iohannem Isacivm Pontanvm, Medicyn, en Professor der Philosophie inde vermaerde Schole te Harderwyck [Treatise or Essays on the Use of the Heavenly and Earthly Globe: First Described in Latin by Robert Hues, Mathematician / and now Translated into Dutch, and Expanded and Decorated with New Clarifications and Figures / also many Disputable Questions Solved, by John Isaac Pontanus, Physician and Professor of Philosophy of the renowned School in Harderwijk], Amsterdam: Iudocus Hondius, woonende op den Dam [living on the Dam ], OCLC 51084257.[43]
• Hues, Robert (1624), Tractatvs de globis, coelesti et terrestri eorvmqve vsv [Treatise on Globes, Celestial and Terrestrial, and their Use], Amsterdam: Excudebat [Printed by] H[enricus] Hondius, OCLC 8909075 (in Latin).
• Hues, Robert (1627), Tractatus duo mathematici: Quorum primus de globis coelesti et terrestri, eorum usu [Two Mathematical Treatises: Of which the First One is about the Celestial and Terrestial Globes, and their Use], Frankfurt: Bryana, OCLC 179907636.
• Hues, Robert; Nottnagel, Christoph (1627), Tractatus duo quorum primus de globis coelesti et terrestri, eorum usu, à Roberto Hues, Anglo, conscriptus. Alter breviarium totius orbis Terrarum, Petri Bertii. Nunc primum luci commißi [Two Treatises of which the First One is about the Celestial and Terrestial Globes, and their Use, signed by Robert Hues, Englishman. The Other One is an Anthology of Countries of the Whole World, by Peter Bertius. Now for the first time here gathered.] (3rd ed.), Wittenberg: [s.n.], OCLC 257661113.
• Hues, Robert (1634), Tractatvs de Globis Coelesti et Terrestri eorvmqve vsv: Primum conscriptus & editus à Roberto Hues Anglo semelque atque iteram à Iudoco Hondio excusus, & nunc elegantibus iconibus & figuris locupletatus: ac de novo recognitus multisque observationibus oportunè illustratus ac passim auctus opera ac studio. Iohannis Isacii Pontani Medici & Philosophiæ Professoris in Gymnasio Gelrico Hardervici [Treatise on Globes, Celestial and Terrestrial, and their Use. First Written and Published by Robert Hues, Englishman, and in the First and Second Editions Drawn by Jodocus Hondius, and now Enlarged by Elegant Pictures and Drawings, and again Revised and Fittingly Illustrated by Many Observations, and throughout Enlarged by the Work and Effort of John Isaac Pontanus, Physician and Professor of Philosophy of the School in Harderwijk], Amsterdam: Excudebat Henricus Hondius, sub signo Canis Vigilantis in Platea Vitulina prope Senatorium [Printed by Henricus Hondius, under the sign of the Watchful Dog in Calf Street [Kalverstraat] near the council hall].[44] Collection of the Biblioteca Nacional de Portugal.
• Hues, Robert (1651), Tractatus duo mathematici. Quorum primus de globis coelesti et terrestri, eorum usu, a Roberto Hues ... conscriptus. Alter breviarium totius orbis terrarum, Petri Bertii ... Editio prioribus auctior & emendatior [Two Mathematical Treatises. Of which the First One is about the Celestial and Terrestial Globes, and their Use, signed ... Robert Hues. The Other One an Anthology of Countries of the Whole World, of Peter Bertius ... First enlarged & improved edition], Oxford: Excudebat [Printed by] L. Lichfield, impensis [at the expense of] Ed. Forrest, OCLC 14913709, two pts. Collection of the Bodleian Library.
Notes
1. Susan M. Maxwell; Harrison, B. (January 2008). "Hues, Robert (1553–1632)". Oxford Dictionary of National Biography. Oxford Dictionary of National Biography (Online ed.). Oxford: Oxford University Press. doi:10.1093/ref:odnb/14045. (Subscription or UK public library membership required.)
2. Charles Buller Heberden, ed. (1909), Brasenose College Register, 1509–1909 [Oxford Historical Series; no. 55], Oxford: Blackwell for the Oxford Historical Society, OCLC 222963720
3. Anthony à Wood; Philip Bliss (1967), Athenae Oxonienses, an Exact History of All the Writers and Bishops who have had their Education in the University of Oxford; to which are Added the Fasti; or, Annals of the said University, vol. 2 (New ed.), New York, N.Y.: Johnson Reprint Corp., p. 534, OCLC 430234. At Oxford, a servitor was an undergraduate student who worked as a servant for fellows of the University in exchange for free accommodation and some meals, and exemption from paying fees for lectures.
4. University of Oxford (1968) [1891], Joseph Foster (ed.), Alumni Oxonienses: The Members of the University of Oxford, 1500–1714 ... Being the Matriculation Register of the University, Alphabetically Arranged, Revised and Annotated, Nendeln, Liechtenstein: Kraus Reprint, OCLC 5574505, vols. 1–2. Hues is listed under the name "Hughes".
5. Homer; George Chapman (c. 1616), The Whole Works of Homer; Prince of Poetts, in his Iliads, and Odysses. Translated according to the Greeke, by Geo. Chapman, London: [By Richard Field and William Jaggard] for Nathaniel Butter, OCLC 216610936
6. Thomas Warton (1871), The History of English Poetry, from the Close of the Eleventh to the Commencement of the Eighteenth Century, vol. 3, London: Reeves and Turner, p. 442, OCLC 8048047: see Markham, "Introduction", Tratatus de globis, p. xxxv. According to another source, Chapman called Hues "another right learned, honest, and entirely loved friend of mine": see Henry Stevens (1900), Thomas Hariot, the Mathematician, the Philosopher and the Scholar, London: [Privately printed at the Chiswick Press] (reproduced on Freeonlinebooks.org), OCLC 82784574, retrieved 19 April 2012. See also Jessica Wolfe (2004), "Homer in a nutshell: George Chapman and the mechanics of perspicuity [ch. 5]", Humanism, Machinery, and Renaissance Literature, Cambridge: Cambridge University Press, pp. 161–202, ISBN 978-0-521-83187-1
7. Markham, "Introduction", Tractatus de globis, p. xxxv.
8. According to Kargon, "[i]t was probably through Percy (although the reverse is possible)" that Harriot came to know Hues: Robert Hugh Kargon (1966), "Thomas Hariot and the Atomic View of Nature", Atomism in England from Hariot to Newton, Oxford: Clarendon Press, pp. 18–30 at 19, OCLC 531838
9. David Singmaster (28 February 2003), BSHM Gazetteer: Petworth, West Sussex, British Society for the History of Mathematics, archived from the original on 25 May 2009, retrieved 7 February 2008. See also David Singmaster (28 February 2003), BSHM Gazetteer: Thomas Harriot, British Society for the History of Mathematics, archived from the original on 25 May 2009, retrieved 7 February 2008
10. Henry Holland (1620), Herōologia Anglica, hoc est clarissimorvm et doctissimorvm aliqovt [sic: aliqvot] Anglorvm qvi florvervnt ab anno Cristi M.D. vsq' ad presentem annvm M.D.C.XX viuae effigies vitae et elogia [List of English Heroes, that is, Lifelike Images of the Lives and Epitaphs of the most Famous and Educated of the English who Flourished from the Year of Christ 1500 until the Present Year 1620], [Arnhem]: Impensis C. Passaei calcographus [sic] et Iansonij bibliopolae Arnhemiensis [Printed by Jan Jansson at the expense of Crispijn van de Passe and Jan Jansson], OCLC 6672789
11. MS Rawl. B 158, Bodleian Library, Oxford.
12. Richard Hakluyt (1589), The Principall Navigations, Voiages, and Discoveries of the English Nation: Made by Sea or Over Land to the Most Remote and Farthest Distant Quarters of the Earth at Any Time within the Compasse of These 1500 Years: Divided into Three Several Parts According to the Positions of the Regions Whereunto They Were Directed; the First Containing the Personall Travels of the English unto Indæa, Syria, Arabia ... the Second, Comprehending the Worthy Discoveries of the English Towards the North and Northeast by Sea, as of Lapland ... the Third and Last, Including the English Valiant Attempts in Searching Almost all the Corners of the Vaste and New World of America ... Whereunto is Added the Last Most Renowned English Navigation Round About the Whole Globe of the Earth, London: Imprinted by George Bishop and Ralph Newberie, deputies to Christopher Barker, printer to the Queen's Most Excellent Majestie, OCLC 77435498
13. Margaret Montgomery Larnder (2000), "Davis (Davys), John", Dictionary of Canadian Biography Online, archived from the original on 8 June 2009, retrieved 9 June 2009
14. David Judkins (2003), "Cavendish, Thomas (1560–1592)", in Jennifer Speake (ed.), Literature of Travel and Exploration: An Encyclopedia, vol. 1, New York, N.Y.: Fitzroy Dearborn, pp. 202–204 at 203, ISBN 978-1-57958-425-2
15. Markham, "Introduction", Tractatus de globis, p. xxxvi.
16. Robert Hues (1594), Tractatus de globis et eorum usu: accommodatus iis qui Londini editi sunt anno 1593, sumptibus Gulielmi Sandersoni civis Londinensis, conscriptus à Roberto Hues [Treatise on Globes and their Use: Adapted to those which have been Published in London in the Year 1593, at the Expense of William Sanderson, a London Resident, Written by Robert Hues], London: In ædibus Thomæ Dawson [in the house of Thomas Dawson], OCLC 55576175 (in Latin).
17. Helen M. Wallis (1951), "The first English globe: A recent discovery", The Geographical Journal, 117 (3): 280, doi:10.2307/1791852, JSTOR 1791852
18. Lesley B. Cormack (March 2006), "The Commerce of Utility: Teaching Mathematical Geography in Early Modern England", Science & Education, 15 (2–4): 305–322 at 311, Bibcode:2006Sc&Ed..15..305C, doi:10.1007/s11191-004-7690-2, S2CID 145588853
19. R. A. Skelton; John Summerson (1971), A Description of Maps and Architectural Drawings in the Collection made by William Cecil, First Baron Burghley, Now at Hatfield House, Oxford: Printed for presentation to the members of the Roxburghe Club, OCLC 181678336; Susan M. Maxwell; Harrison, B. (September 2004). "Molyneux, Emery (d. 1598)". Oxford Dictionary of National Biography. Oxford Dictionary of National Biography (Online ed.). Oxford: Oxford University Press. doi:10.1093/ref:odnb/50911. (Subscription or UK public library membership required.)
20. Markham, "Introduction", Tractatus de globis, pp. xxxviii–xl.
21. John Smith (1626), An Accidence or The Path-way to Experience. Necessary for all Young Sea-men, or those that are Desirous to Goe to Sea, briefly shewing the Phrases, Offices, and Words of Command, belonging to the Building, Ridging, and Sayling, a Man of Warre; and how to Manage a Fight at Sea. Together with the Charge and Duty of every Officer, and their Shares: Also the Names, VVeight, Charge, Shot, and Powder, of all Sorts of Great Ordnance. With the Vse of the Petty Tally. Written by Captaine Iohn Smith sometimes Governour of Virginia, and Admirall of New England, London: Printed [by Nicholas Okes] for Ionas Man, and Benjamin Fisher, and are to be sold at the signe of the Talbot, in Aldersgate streete, OCLC 55198107. Original in the Henry E. Huntington Library and Art Gallery (now The Huntington Library) in San Marino, California. Accidence is the branch of grammar that deals with the accidents or inflections of words. The term came to mean a book about the rudiments of grammar, and was extended to the rudiments or first principles of any subject: see "accidence2", OED Online (2nd ed.), Oxford: Oxford University Press, 1989, retrieved 23 July 2016
22. Susan Rose (2004), "Mathematics and the Art of Navigation: The Advance of Scientific Seamanship in Elizabethan England", Transactions of the Royal Historical Society, 6 (14): 175–184 at 177, doi:10.1017/S0080440104000192, S2CID 178295770
23. Markham, "Introduction", Tractatus de globis, p. xli.
24. Markham, "Introduction", Tractatus de globis, pp. xli–xlii.
25. Markham, "Introduction", Tractatus de globis, pp. xlii–xliii.
26. Markham, "Introduction", Tractatus de globis, p. xlii.
27. Markham, "Introduction", Tractatus de globis, pp. xlii and xlvi.
28. D[avid] B. Quinn; J[ohn] W[illiam] Shirley (1969), "A Contemporary List of Hariot References", Renaissance Quarterly, 22 (1): 9–26 at 13–14, doi:10.2307/2858975, JSTOR 2858975, S2CID 164132753
29. Derek Howse (2003), "Astronomical Navigation [pt. 8.18]", in I[vor] Grattan-Guinness (ed.), Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, Baltimore, Md.: Johns Hopkins University Press, pp. 1128, 1129, 1132, ISBN 978-0-8018-7397-3
30. John W[illiam] Shirley (1983), Thomas Harriot: A Biography, Oxford: Clarendon Press, p. 577, ISBN 978-0-19-822901-8
31. Anthony à Wood (1691–1692), Athenae Oxoniensis. An Exact History of All the Writers and Bishops who have had their Education in the Most Ancient and Famous University of Oxford, from the Fifteenth Year of King Henry the Seventh, Dom. 1500, to the End of the Year 1690. Representing the Birth, Fortune, Preferment, and Death of all those Authors and Prelates, the Great Accidents of their Lives, and the Fate and Character of their Writings. To which are Added, the Fasti or Annals, of the said University, for the Same Time, vol. 1, London: Printed for Tho. Bennett, p. 485, OCLC 70434257: see Arthur Collins (1709), The Peerage of England, or, An Historical and Genealogical Account of the Present Nobility: Containing, the Descent, Original Creations, and most Remarkable Actions, of their Respective Ancestors: Also, the Chief Titles of Honour and Preferment they now Enjoy, with their Marriages and Issue, Continu'd Down to this Present Year, 1709, and the Paternal Coats of Arms of each Family, in Blazon. Collected as well from our best Historians, Publick Records, and other Sufficient Authorities, as from the Personal Informations of most of the Nobility. To which is Prefix'd, an Introduction of the Present Royal Family of Great-Britain, Trac'd thro' its Several Branches down to this Time, and Terminating with the Protestant Succession, as Settled by Act of Parliament, London: Printed by G.J. for Abel Roper and Arthur Collins, ISBN 978-1-4021-7423-0, OCLC 224499127. See also Charles Hutton (1795–1796), "Harriot (Thomas)", A Mathematical and Philosophical Dictionary: Containing an Explanation of the Terms, and an Account of the Several Subjects, Comprized under the Heads of Mathematics, Astronomy, and Philosophy both Natural and Experimental: With an Historical Account of the Rise, Progress, and Present State of these Sciences: Also Memoirs of the Lives and Writings of the Most Eminent Authors, both Ancient and Modern, who by their Discoveries or Improvements have Contributed to the Advancement of them. In Two Volumes, with Many Cuts and Copper-plates, London: Printed by J. Davis, for J. Johnson, in St. Paul's Church-yard; and G.G. and J. Robinson, in Paternoster-Row (reproduced on the website of the Archimedes Project, Max Planck Institute for the History of Science), p. 584, OCLC 8166998, archived from the original on 16 July 2011, retrieved 10 November 2008
32. Kargon, "The Wizard Earl and the New Science" in Atomism in England, pp. 5–17 at 16.
33. John Chamberlain (1939), Norman Egbert McClure (ed.), The Letters [Memoirs of the American Philosophical Society, vol. 12, pts. 1–2], vol. 1, Philadelphia, Penn.: American Philosophical Society, p. 566, OCLC 221477966: see Kargon, "The Wizard Earl and the New Science" in Atomism in England, pp. 5–17 at 16.
34. John Aubrey (1975), Oliver Lawson Dick (ed.), Aubrey's Brief Lives, London: Secker and Warburg, p. 123, OCLC 1981442, critiqued by Shirley, Thomas Harriot, pp. 364–365. See Allan Chapman (1995), "The Astronomical Work of Thomas Harriot (1560–1621)", Quarterly Journal of the Royal Astronomical Society, 36: 97–107 at 99, Bibcode:1995QJRAS..36...97C
35. Feingold says that Hues became "a type of private tutor to Oxford men": Mordechai Feingold (1984), The Mathematicians' Apprenticeship: Science, Universities and Society in England, 1560–1640, Cambridge: Cambridge University Press, p. 84, ISBN 978-0-521-25133-4
36. In Historia et antiquitates universitatis Oxoniensis (History and Antiquities of the University of Oxford, 1674), vol. 2, p. 361, notice of Hues' death was given under St. Mary Hall as follows: "Oxonii in parochiâ Sancti Aldati, inque Domicilio speciatim lapides [sic: lapideo?], e regione insignis Afri [sic: Apri?] cærulei, fatis concessit, et in ecclesiâ Ædis Christi Cathedrali humatus fuit an: dom: CIƆDXXXII [sic: CIƆDCXXXII]" (He yielded to the Fates at Oxford, in the parish of St. Aldate, specifically in the Stone House, in the neighbourhood of the Blue Boar sign, and was buried in the church of Christ Church Cathedral in the year of our Lord 1532 [sic: 1632]). Historia et antiquitates universitatis Oxoniensis duobus voluminibus comprehensæ [History and Antiquities of the University of Oxford Taken together in Two Volumes], Oxford: E Theatro Sheldoniano [ Sheldonian Theatre, University of Oxford ], 1674, OCLC 13439733, was a Latin translation by Richard Peers and Richard Reeve under the direction of Dr. John Fell of an English manuscript by Anthony à Wood which the University purchased in 1670. The manuscript itself was later published as Anthony à Wood; John Gutch (1786–1790), The History & Antiquities of the Colleges and Halls in the University of Oxford ... Now First Published in English, from the Original Manuscript in the Bodleian Library ... with a Continuation to the Present Time, by the Editor, John Gutch. (Appendix to the History and Antiquities of the Colleges and Halls in the University of Oxford ... containing Fasti Oxonienses, or a Commentary on the Supreme Magistrates of the University: by Anthony à Wood, M.A. Now First Published in English ... with a Continuation to the Present Time, also Additions and Corrections ... and Indexes to the Whole, by the Editor, John Gutch.), Oxford: Clarendon Press, OCLC 84810015. See Markham, "Introduction", Tractatus de globis, p. xxxvii, n. 1.
37. Historia et antiquitates universitatis Oxoniensis, vol. 2, p. 534. The brass is also referred to at p. 288: "In laminâ œneâ, eidem pariati [sic: parieti?] impactâ talem cernis inscriptionem" (On the copper plate, driven to the same wall, one sees such an inscription). See Markham, "Introduction", Tractatus de globis, p. xxxvii, n. 1.
38. The title is from Helen M. Wallis (1955), "Further light on the Molyneux globes", The Geographical Journal, Blackwell Publishing, 121 (3): 304–311, doi:10.2307/1790894, JSTOR 1790894, and the imprint information from WorldCat (OCLC 42811612). According to Markham, "Introduction", Tractatus de globis, pp. xxxvii–xxxviii, the title of this version is Tractaut of te handebingen van het gebruych der hemel siker ende aertscher globe, and it was printed in Antwerp.
39. J.J. O'Connor; E.F. Robertson (August 2006), Pierre Hérigone, The MacTutor History of Mathematics archive, School of Mathematics and Statistics, University of St Andrews, archived from the original on 31 October 2007, retrieved 7 November 2008
40. According to Markham, "Introduction", Tractatus de globis, at p. xxxviii, this version was published by Jodocus Hondius in 1624. However, WorldCat (OCLC 8909075) suggests that the 1624 version was in Latin, not Dutch.
41. WorldCat (OCLC 61335670) suggests that printings of this work were also made in 1639; see also A learned treatise of globes both coelestiall and terrestriall, with their severall uses, by Robert Hues, Open Library, Internet Archive, OL 7015375M, retrieved 10 November 2008. According to Markham, "Introduction", Tractatus de globis, p. xxxix, although the title page of the work states that the translator was "John Chilmead", this is generally believed to be an error as no such person was known to have lived at the time. Instead, the translator is believed to be Edmund Chilmead (1610–1653), a translator, man of letters and music teacher who graduated in 1628 and was a chaplain of Christ Church, Oxford.
42. Markham, "Introduction", Tractatus de globis, pp. xxxix–xl.
43. See Figure 22: Title-page of the Dutch edition of Hues's account of the globes, illustrating a celestial globe by Hondius, The Measurers: A Flemish Image of Mathematics in the Sixteenth Century, Museum of the History of Science, Oxford, 7 August 1995, archived from the original on 2 May 2008, retrieved 11 November 2008
44. HUES, Robert, 1553–1632. Tractatvs de globis coelesti et terrestri eorvmqve vsv, Biblioteca Nacional de Portugal, 2002, archived from the original on 9 June 2011, retrieved 11 November 2008
References
• Kargon, Robert Hugh (1966), Atomism in England from Hariot to Newton, Oxford: Clarendon Press, OCLC 531838, chs. 2–4.
• Markham, Clements R., "Introduction", in Hues, Robert (1889), Markham, Clements R. (ed.), Tractatus de globis et eorum usu: A Treatise Descriptive of the Globes Constructed by Emery Molyneux and Published in 1592 [Hakluyt Society, 1st ser., pt. II, no. 79a], London: Hakluyt Society, ISBN 978-0-8337-1759-7.
• Maxwell, Susan M.; Harrison, B. (January 2008). "Hues, Robert (1553–1632)". Oxford Dictionary of National Biography. Oxford Dictionary of National Biography (Online ed.). Oxford: Oxford University Press. doi:10.1093/ref:odnb/14045. (Subscription or UK public library membership required.).
• Shirley, John W[illiam] (1983), "Thomas Harriot: A Biography", Journal for the History of Astronomy, Oxford: Clarendon Press, 17: 71, Bibcode:1986JHA....17...71D, doi:10.1177/002182868601700113, ISBN 978-0-19-822901-8, S2CID 125341236.
Further reading
Wikimedia Commons has media related to Robert Hues.
Articles
• J.O.M. (1851), "Robert Hues on the Use of Globes", Notes and Queries, s1-IV: 384, archived from the original on 15 April 2013.
Books
• Hutchinson, John (1890), Herefordshire Biographies, being a Record of such of Natives of the County as have Attained to more than Local Celebrity, with Notices of their Lives and Bibliographical References, together with an Appendix containing Notices of some other Celebrities, Intimately Connected with the County but not Natives of it, Hereford: Jakeman & Carver, OCLC 62357054.
Authority control
International
• ISNI
• VIAF
National
• Norway
• France
• BnF data
• Catalonia
• Germany
• Israel
• Belgium
• United States
• Sweden
• Netherlands
• Portugal
• Vatican
Academics
• zbMATH
People
• Netherlands
• Trove
Other
• SNAC
• IdRef
|
Wikipedia
|
Robert J. Elliott
Robert James Elliott (born 1940) is a British-Canadian mathematician, known for his contributions to control theory, game theory, stochastic processes and mathematical finance.
He was schooled at Swanwick Hall Grammar School in Swanwick, Derbyshire and studied mathematics in which he earn a B.A. (1961) and M.A. (1965) at the University of Oxford, as well as a Ph.D (thesis Some results in spectral synthesis advised by John Hunter Williamson, 1965)[1] and Sc.D. (1983) from the University of Cambridge.[2]
He taught and conducted research at University of Newcastle (1964), Yale University (1965–66), University of Oxford (1966–68), University of Warwick (1969–73), Northwestern University (1972–73), University of Hull (1973–86), University of Alberta (1985-2001), University of Calgary (2001-2009) and University of Adelaide (2009-2013).
He is the cousin of physicist Roger James Elliott.
Books
• Stochastic Processes, Finance and Control A Festschrift in Honor of Robert J Elliott (World Scientific Publishing, 2012)
• with Nigel Kalton, The Existence of Value for Differential Games (American Mathematical Society, 1972)
• Stochastic Calculus and Applications (Springer-Verlag, 1982)
• Viscosity Solutions and Optimal Control (Longman, 1987)
• Stokasticheski Analiz i evo Prilozeniya (M.I.R. Publications Moscow, 1986)
• with Lakhdar Aggoun and John B. Moore, Hidden Markov Models: Estimation and Control (Springer-Verlag, 1994)
• with P. Ekkehard Kopp, Mathematics of Financial Markets (Springer Verlag, 1999, in Hungarian 2000).
• with J. van der Hoek, Binomial Models in Finance (Springer Verlag, 2005)
• with Rogemar S. Mamon, Hidden Markov Models in Finance (Springer, 2007)
• with Samuel N. Cohen, Stochastic Calculus and Applications (Springer, 2015)
References
1. entry at Mathematics Genealogy Project
2. homepage
Authority control
International
• ISNI
• VIAF
National
• France
• BnF data
• Catalonia
• Germany
• Israel
• Belgium
• United States
• Czech Republic
• Netherlands
Academics
• CiNii
• DBLP
• MathSciNet
• Mathematics Genealogy Project
• zbMATH
Other
• IdRef
|
Wikipedia
|
Bethany Rose Marsh
Bethany Rose Marsh is a mathematician working in the areas of cluster algebras, representation theory of finite-dimensional algebras, homological algebra, tilting theory, quantum groups, algebraic groups, Lie algebras and Coxeter groups.[1] Marsh currently works at the University of Leeds as a Professor of pure mathematics. She was a EPSRC Leadership Fellow from 2008 to 2014.[1] In addition to her duties at the University of Leeds, Marsh was an editor of the Glasgow Mathematical Journal from 2008 to 2013 and served on the London Mathematical Society editorial board from 2014 to 2018.[2]
Bethany Rose Marsh
Alma materUniversity of Oxford (BA)
University of Warwick (MSc & PhD)
AwardsWhitehead Prize (2009)
Scientific career
FieldsMathematics
InstitutionsUniversity of Leeds
University of Leicester
University of Glasgow
University of Bielefeld
Doctoral advisorRoger Carter
Awards
In July 2009, Marsh was awarded the Whitehead Prize by the London Mathematical Society for her work on representation theory, and especially for her research on cluster categories and cluster algebras.[3]
Publications
• "MathSciNet".
• "ArXiv".
References
1. Bethany Marsh. "Bethany Marsh's Homepage".
2. Glasgow Mathematical Journal. "Editorial Board".
3. London Mathematical Society. "Prize Winners 2009". Archived from the original on 23 October 2009.
External links
• Bethany Rose Marsh at the Mathematics Genealogy Project
Authority control
International
• ISNI
• VIAF
National
• Germany
• Israel
• United States
Academics
• DBLP
• MathSciNet
• Mathematics Genealogy Project
• ORCID
• Publons
• ResearcherID
• Scopus
• zbMATH
Other
• IdRef
|
Wikipedia
|
Robert Joshua Rubin
Robert Joshua Rubin (/ˈruːbɪn/; August 17, 1926 – January 18, 2008)[1] was an American mathematician whose work involved modelling complex physical systems.[2] He worked principally at the National Bureau of Standards, and was a Fellow of the American Association for the Advancement of Science and also a Fellow of the American Physical Society.[2]
Robert Joseph Rubin
Born(1926-08-17)August 17, 1926
New York, New York
DiedJanuary 18, 2008(2008-01-18) (aged 81)
Washington, District of Columbia
NationalityAmerican
Alma materCornell University
Scientific career
FieldsMathematics
InstitutionsJohns Hopkins Applied Physics Laboratory, University of Illinois, Los Alamos National Laboratory, National Bureau of Standards, National Institutes of Health
Education and career
He received his bachelor's degree from Cornell University, and his doctorate also from Cornell, in 1951; his doctoral advisor was Peter Debye.[2]
He worked first at the Johns Hopkins Advanced Physics Laboratory, and then at the Bureau of Standards.
Personal life
From 1948 until his death in 2008, he was married to Vera Rubin. His children include Judith Young (astronomer) and Karl Rubin (mathematician).[2]
References
1. "Robert Joshua Rubin (1926 - 2008) - Genealogy".
2. Sullivan, Patricia (February 5, 2008). "Robert J. Rubin, 81; Scientist Whose Work Combined Disciplines". The Washington Post. Retrieved July 3, 2022.
|
Wikipedia
|
Robert Lazarsfeld
Robert Kendall Lazarsfeld (born April 15, 1953) is an American mathematician, currently a professor at Stony Brook University.[1] He was previously the Raymond L. Wilder Collegiate Professor of Mathematics at the University of Michigan.[2] He is the son of two sociologists, Paul Lazarsfeld and Patricia Kendall. His research focuses on algebraic geometry.
Robert Lazarsfeld
Born (1953-04-15) April 15, 1953
New York City
NationalityAmerican
Alma materBrown University Harvard University
Scientific career
FieldsMathematics
InstitutionsStony Brook University
University of Michigan
University of California, Los Angeles
Doctoral advisorWilliam Fulton
Doctoral studentsChristopher Hacon
Mihnea Popa
During 2002–2009, Lazarsfeld was an editor at the Journal of the American Mathematical Society (Managing Editor, 2007–2009).[3] In 2012–2013, he served as the Managing Editor of the Michigan Mathematical Journal.[4]
Lazarsfeld went to Harvard for undergraduate studies and earned his doctorate from Brown University in 1980 under supervision of William Fulton.[5]
In 2006 Lazarsfeld was elected a Fellow of the American Academy of Arts and Sciences.[6] In 2012 he became a fellow of the American Mathematical Society.[7] In 2015 he was awarded the AMS Leroy P. Steele Prize for Mathematical Exposition.[8]
Selected works
• Lazarsfeld, Robert (2004). Positivity in algebraic geometry, Vol. I. Berlin: Springer. ISBN 3-540-22533-1.; Lazarsfeld, R. K. (2004). Positivity in algebraic geometry, Vol. II. ISBN 3-540-22534-X.[9]
• Lazarsfeld, Robert; Van de Ven, Antonius (2012) [1984]. Topics in the geometry of projective space: Recent work of FL Zak. DMV Seminar Vol. 4. Birkhäuser. ISBN 9783034893480; with an addendum by Fyodor Zak{{cite book}}: CS1 maint: postscript (link)
References
1. Robert Lazarsfeld's CV
2. Univ. of Michigan Mathematics Dept. Faculty Listing, retrieved October 27, 2010.
3. Past editors of the Journal of the AMS
4. University of Michigan Michigan Math Journal Site
5. Robert Lazarsfeld at the Mathematics Genealogy Project
6. Three U-M faculty elected to American Academy of Arts and Sciences. The Record, May 8, 2006, University of Michigan. Accessed May 31, 2011
7. List of Fellows of the American Mathematical Society, retrieved January 27, 2013.
8. 2015 AMS Leroy P. Steele Prize for Mathematical Exposition
9. Kollár, János (2006). "Review: Positivity in algebraic geometry. I–II. by Robert Lazarsfeld" (PDF). Bull. Amer. Math. Soc. (N.S.). 43 (2): 279–280. doi:10.1090/s0273-0979-06-01087-1.
External links
• Website at Stony Brook University
• Robert Lazarsfeld publications indexed by Google Scholar
Authority control
International
• ISNI
• VIAF
• WorldCat
National
• Norway
• 2
• France
• BnF data
• Germany
• Israel
• United States
• Netherlands
Academics
• CiNii
• Google Scholar
• MathSciNet
• Mathematics Genealogy Project
• Scopus
• zbMATH
Other
• IdRef
|
Wikipedia
|
Robert Kozma (professor)
Robert Kozma is First Tennessee University Professor of Mathematics at the University of Memphis.
Biography
Kozma received his MS in Power Engineering from the Moscow Power Engineering Institute in 1982, his MS in mathematics from the Eötvös Loránd University in 1988, and his PhD in Applied Physics from Delft University of Technology in 1992[1]
Kozma has been associate professor at the Department of Quantum Science and Engineering, Tohoku University, Sendai, Japan since 1993. He became Assistant Professor/Lecturer at the Department of Information Sciences, Otago University, Dunedin, New Zealand in 1996. In 1998 in the USA he became a joint appointment at the Division of Neurobiology and the EECS Department, University of California, Berkeley. In 2009 he was appointed Professor of Computer Science, University of Memphis, Memphis, Tennessee, and since 2009 he is Professor of Mathematical Sciences, University of Memphis, Memphis, Tennessee. Since 2001 he is also director of Computational Neurodynamics Laboratory, presently CLION, FedEx Institute of Technology of the University of Memphis, Memphis.
He serves on the AdCom of IEEE Computational Intelligence Society CIS (2009–2012) and on the Governing Board of the International Neural Network Society INNS (2004–2012). He is Chair of the Distinguished Lecturer Program, IEEE CIS. He has been Technical Committee Member of IEEE Computational Intelligence Society since 1996, and IEEE Senior Member. He also served in leading positions at over 20 international conferences, including General Chair of IEEE/INNS International Joint Conference on Neural Networks IJCNN09 in Atlanta; Program Co-chair of International Joint Conference on Neural Networks IJCNN08/WCCI08 in Hong Kong; Program Co-chair of IJCNN04, Budapest, Hungary; chair for Finances of IEEE WCCI06, Vancouver, Canada.
He is Associate Editor of ‘Neural Networks (Elsevier),’ ‘IEEE Transactions on Neural Networks,’ ‘Neurocomputing’ (Elsevier), ‘Journal of Cognitive Neurodynamics’ (Springer), Area Editor of ‘New Mathematics and Natural Computation’ (World Scientific), and ‘Cognitive Systems Research.’
Work
[2]
Kozma's current research interests include spatio-temporal dynamics of neural processes, random graph approaches to large-scale networks, such as neural networks, computational intelligence methods for knowledge acquisition and autonomous decision making in biological and artificial systems.
Publications
He has published 100+ papers and 3+ books[3][4] in the several fields including signal processing; and design, analysis, and control of intelligent systems.
References
1. Robert Kozma. "VITA – ROBERT KOZMA" (PDF). Retrieved 25 May 2010.
2. Neuropercolation Article on Scholarpedia
3. "author:"Robert Kozma" – Google Scholar". Retrieved 25 May 2010.
4. author:"Robert Kozma" – Google Books. Retrieved 25 May 2010.
External links
• Kozma's homepage
• CLION
• Robert Kozma; Modeling Cortical Phase Transitions (2007)
• September 2009 Newsletter for University of Memphis
Authority control
Academics
• DBLP
• ORCID
• zbMATH
Artists
• Museum of Modern Art
• Photographers' Identities
|
Wikipedia
|
Robert Mackenzie Johnston
Robert Mackenzie Johnston F.L.S., (27 November 1843 – 20 April 1918)[1] was a Scottish-Australian statistician and scientist.
Early life
Johnston was born at Petty[2] near Inverness, Scotland, the son of Lachlan Johnstone,[2] a crofter, and his wife Mary, née Mackenzie.[1] Johnston was educated at the village school where his ability was quickly recognized. Johnson was influenced by the life of Hugh Miller, a stonemason and geologist,[1] whose books were lent to him. Johnston obtained work on the railways, read widely, and studied botany, geology, and chemistry at the Andersonian University under Professors Kennedy, Crosskey, and Penny.[2] Glasgow.
Career in Australia
Emigrating to Australia in 1870 he was given a position in the accountant's branch of the Launceston and Western District railway. He transferred to the government service in 1872, authoring "Field Memoranda for Tasmanian Botanists" (Launceston, 1874). In 1880 he became chief clerk in the Audit Department, his former railway colleagues presented him with a watch inscribed:
Presented to ROBERT MACKENZIE JOHNSTON by personal friends, on the occasion of his going from amongst them, in recognition not merely of his scientific attainments, but also of his social worth, and as a token of the high esteem and great regard in which he has been ever hold. Launceston, Tasmania, August, 1880. Invitum sequitur honos.
[3]
In 1882 Johnston was appointed registrar-general and government statistician. Johnston was appointed a royal commissioner to report on the fisheries of Tasmania, being the author of "Descriptive Catalogue of Tasmanian Fishes" (Hobart, 1882). Johnston also did much geological work, and in 1888 the government published his Systematic Account of the Geology of Tasmania. He was president of the economic and social science and statistics section at the meeting of the Australasian Association for the Advancement of Science held at Melbourne in 1890, and with the coming of federation he was able to influence very much the special problems of finance that were raised. He originated the scheme of per-capita payments by the Commonwealth to the states that was eventually adopted. Johnston was offered and declined the position of government statist for New South Wales, and declined to be a candidate for the position of Commonwealth statist.
Legacy
Johnston was also interested in all branches of science, in music, and in education. Johnston died at Hobart on 20 April 1918 of heart disease.[1] Johnston received the Imperial Service Order in 1903 and was fellow of the Linnean Society of London and the Royal Geographical Society of Australasia and honorary fellow of the Royal Statistical Society of London. A list of 103 of his papers is given in the Papers and Proceedings of the Royal Society of Tasmania for 1918, of which over 50 are on geological subjects. In 1903 The R. M. Johnston Memorial Volume, being a selection from his more important papers, was published by the Tasmanian government.
Species named in honor of Robert Mackenzie Johnston include the fish: Johnston's weedfish, Heteroclinus johnstoni, and the Tasmanian yellow gum Eucalyptus johnstonii.
References
Wikisource has original works by or about:
Robert Mackenzie Johnston
1. R. L. Wettenhall, 'Johnston, Robert Mackenzie (1843 - 1918)', Australian Dictionary of Biography, Volume 9, MUP, 1983, pp 501-503. Retrieved 8 November 2012
2. Mennell, Philip (1892). "Johnston, Robert Mackenzie" . The Dictionary of Australasian Biography. London: Hutchinson & Co – via Wikisource.
3. "PRESENTATION TO R. M. JOHNSTON ESQ". Launceston Examiner. Trove, National Library of Australia. 3 September 1880. Retrieved 8 November 2012.
4. International Plant Names Index. R.M.Johnst.
• Serle, Percival (1949). "Johnson, Robert Mackenzie". Dictionary of Australian Biography. Sydney: Angus & Robertson. Retrieved 18 August 2009.
Authority control
International
• ISNI
• VIAF
National
• United States
• Australia
Academics
• International Plant Names Index
People
• Australia
Other
• SNAC
• IdRef
|
Wikipedia
|
Robert MacPherson (mathematician)
Robert Duncan MacPherson (born May 25, 1944) is an American mathematician at the Institute for Advanced Study and Princeton University. He is best known for the invention of intersection homology with Mark Goresky, whose thesis he directed at Brown University, and who became his life partner. MacPherson previously taught at Brown University, the University of Paris, and the Massachusetts Institute of Technology. In 1983 he gave a plenary address at the International Congress of Mathematicians in Warsaw.
Robert MacPherson
MacPherson at Oberwolfach in 2008
Born (1944-05-25) May 25, 1944
Lakewood, Ohio
NationalityAmerican
Alma materSwarthmore College
Harvard University
AwardsNAS Award in Mathematics (1992)
Leroy P. Steele Prize (2002)
Heinz Hopf Prize (2009)
Scientific career
FieldsMathematics
InstitutionsMassachusetts Institute of Technology
Brown University
Princeton University
Doctoral advisorRaoul Bott
Doctoral studentsMark Goresky
Julianna Tymoczko
Kari Vilonen
Zhiwei Yun
Education and career
Educated at Swarthmore College and Harvard University, MacPherson received his PhD from Harvard in 1970. His thesis, written under the direction of Raoul Bott, was entitled Singularities of Maps and Characteristic Classes. Among his many PhD students are Kari Vilonen and Mark Goresky.
Honors and awards
In 1992, MacPherson was awarded the NAS Award in Mathematics from the National Academy of Sciences.[1] In 2002 he and Goresky were awarded the Leroy P. Steele Prize for Seminal Contribution to Research by the American Mathematical Society.[2][3] In 2009 he received the Heinz Hopf Prize from ETH Zurich. In 2012 he became a fellow of the American Mathematical Society.[4]
Personal
MacPherson's PhD advisee, Mark Goresky, later became his life partner. Their discovery of intersection homology made "both of them famous."[5] After the collapse of the Soviet Union, they were instrumental in channeling aid to Russian mathematicians, especially many who had to hide their sexuality.[5]
Selected publications
• Goresky, Mark; MacPherson, Robert, La dualité de Poincaré pour les espaces singuliers, C. R. Acad. Sci. Paris Sér. A-B 284 (1977), no. 24, A1549–A1551. MR0440533
• Goresky, Mark; MacPherson, Robert, Intersection homology theory, Topology 19 (1980), no. 2, 135–162. doi:10.1016/0040-9383(80)90003-8 MR0572580
• Goresky, Mark; MacPherson, Robert, Intersection homology. II, Inventiones Mathematicae 72 (1983), no. 1, 77–129. doi:10.1007/BF01389130 MR0696691
References
1. "NAS Award in Mathematics". National Academy of Sciences. Retrieved February 13, 2011.
2. "Notices of the AMS 2002 p. 466" (PDF). ams.org.
3. "List of Steele Prizes Seminal Contribution to Research". www.ams.org.
4. List of Fellows of the American Mathematical Society, retrieved February 2, 2013.
5. "Robert D. MacPherson". Simons Foundation.
External links
Wikimedia Commons has media related to Robert MacPherson (mathematician).
• Robert MacPherson at the Mathematics Genealogy Project
• IAS page, including links to publications and CV
• The work of Robert MacPherson
Authority control
International
• ISNI
• VIAF
National
• France
• BnF data
• Germany
• Israel
• United States
• Czech Republic
• Netherlands
Academics
• CiNii
• DBLP
• MathSciNet
• Mathematics Genealogy Project
• zbMATH
People
• Trove
Other
• IdRef
|
Wikipedia
|
Robert McCann (mathematician)
Robert John McCann FRSC is a Canadian mathematician, known for his work in transportation theory. He has worked as a professor at the University of Toronto since 1998, and as Canada Research Chair in Mathematics, Economics, and Physics since 2020.
Robert J. McCann
NationalityCanadian
Alma materPrinceton University
AwardsAMS Centennial Fellowship (1996)
Monroe Martin Prize (2000)
Coxeter–James Prize (2005)
Jeffery–Williams Prize (2017)
W.T. and Idalia Reid Prize (2023)
Scientific career
FieldsApplied mathematics
InstitutionsUniversity of Toronto
ThesisA Convexity Theory For Interacting Gases And Equilibrium Crystals (1994)
Doctoral advisorElliott H. Lieb
Life and work
McCann was raised in Windsor, Ontario. He studied engineering and physics at Queen's University before graduating with a degree in math, and earned a PhD in mathematics from Princeton University in 1994.[1] McCann was a Tamarkin Assistant Professor at Brown University from 1994, before joining the University of Toronto Department of Mathematics in the fall of 1998.[2] He served as editor-in-chief of the Canadian Journal of Mathematics from 2007 to 2016.[3] He was an invited speaker at the International Congress of Mathematicians in Seoul in 2014.[4] He was elected a Fellow of the American Mathematical Society in 2012, of the Royal Society of Canada in 2014, of the Fields Institute in 2015 and of the Canadian Mathematical Society in 2020.
He invented the displacement interpolation between probability measures and studied the convexity of various entropies and energies along it, later linking these to Ricci curvature and eventually to the Einstein equations of general relativity. He has pioneered applications of optimal transport to economic problems such as hedonic matching, investment to match, and multidimensional screening.
References
1. "Scientific Advisory Panel". The Fields Institute for Research in Mathematical Sciences. 24 December 2014. Retrieved 3 June 2019.
2. "New Faculty Appointments" (PDF). Mathematics. Department of Mathematics, University of Toronto: 4–5. July 2000.
3. "Professor Robert McCann to receive the 2017 CMS Jeffery-Williams Prize". Ottawa: Canadian Mathematical Society. 23 February 2017. Retrieved 3 June 2019.
4. McCann, Robert J. (2014). "Academic wages, singularities, phase transitions and pyramid schemes". Proceedings of the International Congress of Mathematicians (Seoul 2014). Vol. 3. pp. 835–849.
External links
• Robert McCann at the Mathematics Genealogy Project
• Robert McCann publications indexed by Google Scholar
Authority control
International
• VIAF
National
• Germany
• Israel
• United States
Academics
• DBLP
• Google Scholar
• MathSciNet
• Mathematics Genealogy Project
• ORCID
• zbMATH
Other
• IdRef
|
Wikipedia
|
Robert McLachlan (mathematician)
Robert Iain McLachlan FRSNZ (born 1964) is a New Zealand mathematician and Distinguished Professor in the School of Fundamental Sciences, Massey University, New Zealand.[1] His research in geometric integration encompasses both pure and applied mathematics, modelling the structure of systems such as liquids, climate cycles, and quantum mechanics. He is also writes for the public on the subject of climate change policy.
Robert McLachlan
Born1964 (age 58–59)
Christchurch, New Zealand
Known forGeometric integration
Academic background
Alma materCaltech
Thesis
• Separated Viscous Flows via Multigrid (1990)
Doctoral advisorHerbert Keller
Academic work
InstitutionsMassey University
Academic career
McLachlan was born in Christchurch, New Zealand in 1964,[2] and studied mathematics at the University of Canterbury, graduating with a BSc (Hons) First Class in 1984.[3] One formative experience was in his last year of high school, where he had free rein to experiment with assembly language programming on the school PDP-11/10.[3]
McLachlan went on to graduate work in numerical analysis in 1986. He received a PhD from Caltech (the California Institute of Technology) in 1990, supervised by Herbert Keller in computational fluid dynamics, with a thesis titled "Separated Viscous Flows via Multigrid".[3] He then worked as a postdoctoral fellow at the University of Colorado Boulder in what was then the new field of symplectic geometry. After meeting Jürgen Moser, who was visiting Boulder at the time, McLachlan spent six months on a postdoctoral fellowship in Switzerland, at the Swiss Federal Institute of Technology in Zurich.[3]
McLachlan joined Massey University in Palmerston North in 1994, and began a collaboration with Reinout Quispel at La Trobe University that resulted in over 26 publications on geometric integration.[3] In 2002 he became Professor of Applied Mathematics at Massey, and spent a year's sabbatical at the University of Geneva, working with Gerhard Wanner and Ernst Hairer, and the Norwegian Academy of Sciences in Oslo.[4] In 2007 he won the prestigious Germund Dahlquist Prize, the first mathematician from the southern hemisphere to do so.[5] From 2008 to 2012, along with Stephen Marsland and Matt Perlmutter, he worked on a Marsden grant project "Geodesics in diffeomorphism groups: geometry and applications", designing efficient numerical integrators that preserved the geometric properties of systems.[6] In 2013, McLachlan was the LMS-NZMS Aitken Lecturer, delivering talks on geometric numerical integration to six UK universities.[7][8]
McLachlan is a Fellow of the New Zealand Mathematical Society, and in 1998 organised the first of the annual Manawatu-Wellington Applied Mathematics Conferences. He was president of the NZMS in 2008–2009 and vice president in 2010, and edited the New Zealand Journal of Mathematics for six years.[3] Since 2016 he has been a Distinguished Professor in Massey's School of Fundamental Sciences.[9]
Research
McLachlan is a world leader in the field of geometric integration, a technique for the reliable simulation of large-scale complex systems, and in particular the use of symplectic techniques in the numerical analysis of differential equations.[3][10] This field, which McLachlan helped found in the 1990s, builds into its approach the underlying geometric structure of data sets.[9] Because it allows the simulation of large systems, it has the potential for solving practical problems in fields as disparate as the structure of liquids, climate cycles, the motion of the solar system, particles in circular accelerators, chaos in dynamical systems, and weather forecasting.[3][11][12] For example, during Hurricane Sandy in 2012, the European Centre for Medium-Range Weather Forecasts, using geometric integration models, correctly predicted the hurricane would suddenly turn 90 degrees towards New York six days in advance.[11] McLachlan's methods have been used in computational science to examine a possible celestial origin of the ice ages, biological models, and the dynamics of flexible structures.[11] His research contributed to a simulation of the solar system simulation that revised the dates of geophysical epochs by millions of years.[9] Although his work in geometric numerical integration has a wide range of real-world applications, he considers himself a pure mathematician.[11]
Awards and fellowships
• Fellow of the New Zealand Mathematical Society (2001)[1]
• Fellow of the Royal Society of New Zealand (2002)[13]
• NZ Association of Scientists Research Medal (2003)[10]
• NZMS Research Award (2005)[3]
• NZIMA Maclaurin Fellowship (2005)[3]
• SIAM Germund Dahlquist Prize (2007)[12]
• James Cook Research Fellow (2012)[14]
• Research fellow at the Isaac Newton Institute, Cambridge, UK[5]
• Research fellow at the Mathematical Sciences Research Institute, Berkeley, USA[9]
• Research fellow at the Centre for Advanced Study at the Norwegian Academy of Science and Letters, Oslo[9]
• Visiting fellow, Mathematical Research Institute of Oberwolfach, Germany[11]
Science communication
As an advocate for action on climate change McLachlan writes frequently for the public, in media like Scientific American's blog,[15] the blog Planetary Ecology and the Royal Society Te Apārangi's Sciblogs. His writing focuses on consequences of climate change,[15] the benefits of wind turbines,[16] electric cars,[17] and climate policy.[18][19]
Selected research
• McLachlan, Robert I.; Offen, Christian (2019). "Symplectic integration of boundary value problems". Numerical Algorithms. 81 (4): 1219–1233. arXiv:1804.09042. doi:10.1007/s11075-018-0599-7. S2CID 52297714.
• McLachlan, Robert I.; Modin, Klas; Verdier, Olivier (2014). "Symplectic integrators for spin systems". Physical Review E. 89 (6): 061301. arXiv:1402.4114. Bibcode:2014PhRvE..89f1301M. doi:10.1103/PhysRevE.89.061301. PMID 25019718. S2CID 16838116.
• McLachlan, Robert I.; Quispel, G. Reinout W. (2006). "Geometric integrators for ODEs". Journal of Physics A: Mathematical and General. 39 (19): 5251–5285. Bibcode:2006JPhA...39.5251M. doi:10.1088/0305-4470/39/19/S01.
• McLachlan, Robert I.; Quispel, G. Reinout W. (2002). "Splitting methods". Acta Numerica. 11: 341–434. doi:10.1017/S0962492902000053. S2CID 229168188.
• McLachlan, Robert I. (1995). "On the Numerical Integration of Ordinary Differential Equations by Symmetric Composition Methods". SIAM Journal on Scientific Computing. 16 (1): 151–168. Bibcode:1995SJSC...16..151M. doi:10.1137/0916010.
• Braunstein, Samuel L.; McLachlan, Robert I. (1987). "Generalized squeezing". Physical Review A. 35 (4): 1659–1667. Bibcode:1987PhRvA..35.1659B. doi:10.1103/PhysRevA.35.1659. PMID 9898327. S2CID 42118827.
References
1. McLachlan, Robert. "The emerging science of geometric integration". www.massey.ac.nz. Retrieved 16 April 2020.
2. "McLachlan, Robert I. (Robert Iain), 1964–". LOC. Retrieved 20 April 2020.
3. van Brunt, Bruce (2013). "Robert McLachlan" (PDF). New Zealand Mathematical Society Newsletter. 118: 18–19.
4. Vlieg-Hulstman, Marijcke (April 2002). "Massey University". Newsletter of the New Zealand Mathematical Society. 84: 10.
5. "Maths Prize for Massey Prof". NZEDGE. 13 August 2007. Retrieved 16 April 2020.
6. Tuffley, Christopher (December 2009). "Massey University". Newsletter of the New Zealand Mathematical Society. 107: 17.
7. Fisher, Elizabeth (25 July 2019). "LMS-NZMS Forder and Aitken Lectureships". London Mathematical Society.
8. "LMS – NZMS Aitken UK Lecture Tour 2013" (PDF). London Mathematical Society. 2013.
9. "Massey's eminent researchers". Rangahau | Research at Massey University. 2018. Retrieved 16 April 2020.
10. "New Zealand Association of Scientists – Hill Tinsley Medal". scientists.org.nz. Retrieved 16 April 2020.
11. Dickson, Anna (2018). "Mathematics in a geometric universe". Rangahau: Research at Massey University. Vol. 2. Auckland, New Zealand: Massey University Press. pp. 45–47.
12. Stephenson, Jessica (13 August 2007). "Robert McLachlan awarded Germund Dahlquist Prize". EurekAlert!. AAAS. Retrieved 16 April 2020.
13. "All Fellows: M-O". Royal Society Te Apārangi. Retrieved 16 April 2020.
14. "List of Recipients of James Cook Research Fellowships". Royal Society Te Apāirangi. Retrieved 16 April 2020.
15. McLachlan, Robert (4 June 2019). "Climate Change Is a Fourfold Tragedy". Scientific American Blog Network. Retrieved 16 April 2020.
16. McLachlan, Robert (19 December 2019). "Blow, wind of fruitfulness". Planetary Ecology. Retrieved 16 April 2020.
17. McLachlan, Robert (28 December 2019). "Should I ditch my fossil-fueled car?". Planetary Ecology. Retrieved 16 April 2020.
18. McLachlan, Robert (11 November 2019). "Zero Carbon: it's not just a good idea, it's the law". Planetary Ecology. Retrieved 16 April 2020.
19. McLachlan, Robert (18 December 2019). "Which countries are likely to meet their Paris Agreement targets". Sciblogs. Retrieved 16 April 2020.
External links
• Planetary Ecology blog
• Massey University staff web page
• Google Scholar profile
• Robert McLachlan on SciBlogs
• McLachlan's web page of geometric integration images
Authority control
International
• ISNI
• VIAF
National
• Germany
• United States
Academics
• DBLP
• Google Scholar
• ORCID
• Scopus
|
Wikipedia
|
Robert Megginson
Robert Eugene Megginson is an American mathematician, the Arthur F. Thurnau Professor of Mathematics at the University of Michigan.[1] His research concerns functional analysis and Banach spaces;[2] he is the author of the textbook An Introduction to Banach Space Theory (GTM 183, Springer, 1998).[3]
Megginson was born in 1948 in Washington, Illinois, of Oglala Sioux heritage on his mother's side,[2] and grew up in Sheldon, Illinois, where his father was mayor.[4] He earned a degree in physics from the University of Illinois at Urbana–Champaign in 1969, and became a software specialist for the Roper Corporation until 1977, when he returned to graduate school.[2] He earned a master's degree in statistics in 1983,[5] He completed his Ph.D. in 1984 at the University of Illinois, with a thesis on normed vector spaces supervised by Mahlon M. Day.[6] This accomplishment made him one of only approximately 12 Native Americans to hold a doctorate in mathematics, and he has taken great interest in underrepresented minorities in mathematics.[2]
Because his wife was employed nearby in Decatur, Illinois,[4] Megginson took a teaching position in 1983, joining the faculty of Eastern Illinois University as an assistant professor, rather than doing postdoctoral research. He moved to the University of Michigan in 1992, was on leave as the deputy director of the Mathematical Sciences Research Institute in Berkeley, California from 2002 to 2004, and became the Thurnau Professor at Michigan in 2008.[5]
Megginson won the U.S. Presidential Award for Excellence in Science, Mathematics, and Engineering Mentoring in 1997.[7] The American Indian Science and Engineering Society gave him their Ely S. Parker Award for lifetime service to the Native American community in 1999.[5] The American Association for the Advancement of Science elected him as a fellow in 2009,[8] and in the same year the Mathematical Association of America gave him their Yueh-Gin Gung and Dr. Charles Y. Hu Award for Distinguished Service, for his work on underrepresented minorities.[9] In 2012, Megginson became one of the inaugural fellows of the American Mathematical Society.[10]
References
1. Faculty detail, University of Michigan mathematics, retrieved 2016-06-13.
2. "Robert Eugene Megginson", Strengthening Underrepresented Minority Mathematics Achievement (SUMMA), Mathematical Association of America, retrieved 2016-06-13.
3. Review of An Introduction to Banach Space Theory by Ehrhard Behrends (1999), MR1650235.
4. Ross, Kenneth A. (January 6, 2011), Interview with Bob Megginson (PDF), Mathematical Association of America, retrieved 2016-06-13.
5. Curriculum vitae, retrieved 2016-06-13.
6. Robert Megginson at the Mathematics Genealogy Project
7. "Megginson, Robert", Presidential Awards for Excellence in Science, Mathematics and Engineering Mentoring, National Science Foundation, retrieved 2016-06-13.
8. Bailey, Laura (January 11, 2010), "Eleven university scientists named AAAS Fellows", University Record Online, University of Michigan.
9. "MAA Prizes Presented in Washington, DC" (PDF), Notices of the AMS, 56 (5): 633–635, May 2009.
10. List of Fellows of the American Mathematical Society, retrieved 2016-06-13.
Authority control
International
• ISNI
• VIAF
National
• Germany
• Israel
• United States
• Netherlands
Academics
• DBLP
• MathSciNet
• Mathematics Genealogy Project
• zbMATH
Other
• SNAC
• IdRef
|
Wikipedia
|
Robert Miller Hardt
Robert Miller Hardt (born 24 June 1945, Pittsburgh)[1] is an American mathematician. His research deals with geometric measure theory, partial differential equations, and continuum mechanics. He is particularly known for his work with Leon Simon proving the boundary regularity of volume minimizing hypersurfaces.
Hardt received in 1967 his bachelor's degree from Massachusetts Institute of Technology and in 1971 his Ph.D. from Brown University under Herbert Federer with thesis Slicing and Intersection Theory for Chains Associated with Real Analytic Varieties.[2] In 1971 he became an instructor and later a professor at the University of Minnesota. In 1988 he became a professor at Rice University, where he is W. L. Moody Professor. His doctoral students include Fang-Hua Lin.
Hardt was a visiting scholar at the Institute for Advanced Study in 1976 and at IHES in 1978 and 1981. He was a visiting professor at the University of Melbourne in 1979, at Stanford University and at the University of Wuppertal.
In 1986 Hardt was an Invited Speaker at the ICM in Berkeley, California. In 2015 he was elected a Fellow of the American Mathematical Society.
Selected publications
Articles
• Stratification of real analytic mappings and images, Inventiones Mathematicae, vol. 28, 1975, pp. 193–208
• with Leon Simon: Boundary regularity and embedded solutions of the oriented Plateau problem, Annals of Mathematics, vol. 110, 1979, pp. 439–486. doi:10.1090/S0273-0979-1979-14581-6
• with Fang‐Hua Lin: Mappings minimizing the Lp norm of the gradient, Communications on Pure and Applied Mathematics, vol. 40, no. 5, 1987, pp. 555–588. doi:10.1002/cpa.3160400503
• Singularities of Harmonic Maps, Bull. Amer. Math. Soc. vol. 34, 1997, pp. 15–34 doi:10.1090/S0273-0979-97-00692-7
Books
• with Leon Simon: Seminar on geometric measure theory. DMV Seminar, Vol. 7, Birkhauser, 1986.
• as editor with Michael Wolf: Nonlinear partial differential equations in differential geometry, AMS, Institute for Advanced Study 1996
• as editor: Six Themes on Variation, Student Mathematical Library, Vol. 26, AMS, 2004[3]
• with Thierry De Pauw and W.F. Pfeffer: Homology of normal chains and cohomology of charges, Providence, Rhode Island : American Mathematical Society, 2017.
References
1. biographical information from American Men and Women of Science, Thomson Gale 2004
2. Robert Miller Hardt at the Mathematics Genealogy Project
3. Sandifer, Ed (16 June 2005). "Review of Six Themes on Variation, ed. Robert Hardt". MAA Reviews, Mathematical Association of America.
External links
• Robert Hardt's home page, math.rice.edu
Authority control
International
• ISNI
• VIAF
National
• Norway
• France
• BnF data
• Catalonia
• Germany
• Israel
• United States
• Netherlands
Academics
• MathSciNet
• Mathematics Genealogy Project
• zbMATH
Other
• IdRef
|
Wikipedia
|
Robert Ellis (mathematician)
Robert Mortimer Ellis (1926–2013) was an American mathematician, specializing in topological dynamics.[2]
Robert Mortimer Ellis
Born(1926-09-16)September 16, 1926
Cleveland, Ohio, US
DiedDecember 6, 2013(2013-12-06) (aged 87)
NationalityAmerican
Known forEllis semigroup of a dynamical system;[1]
Ellis action of a dynamical system[1]
Scientific career
FieldsMathematics
Doctoral advisorWalter Gottschalk
Ellis grew up in Philadelphia, served briefly in the U.S. Army, and then studied at the University of Pennsylvania, where he received his Ph.D. in 1953.[3] He was a postdoc at the University of Chicago from 1953 to 1955. He was at Pennsylvania State University from 1955 to 1957 an assistant professor and from 1957 to 1963 an associate professor and at Wesleyan University from 1963 to 1967 a full professor. At the University of Minnesota he was a full professor from 1967 to 1995, when he retired as professor emeritus.[2]
He developed an algebraic approach to topological dynamics, leading to a strengthening with an alternate proof of the Furstenberg structure theorem.[4] He was the author or coauthor of about 40 research publications. In the year of his retirement, a conference was held in his honor at the University of Minnesota on April 5 and 6 1995; the conference proceedings were published in 1998 by the American Mathematical Society (AMS).[2][5] He was elected a Fellow of the AMS in 2012.
Ellis was predeceased by his wife. Upon his death he was survived by a grandchild, a daughter, and his son David, a professor of mathematics at Beloit College and a long-time collaborator with his father.[2]
References
1. Akin, Ethan (1997). Recurrence in Topological Dynamics: Furstenberg Families and Ellis Actions. Springer. pp. 133–134. ISBN 9780306455506.
2. "In Memoriam: Robert Ellis". School of Mathematics, University of Minnesota.
3. Robert Mortimer Ellis at the Mathematics Genealogy Project
4. Ellis, Robert (1978). "The Furstenberg structure theorem". Pacific J. Math. 76 (2): 345–348. doi:10.2140/pjm.1978.76.345.
5. Ellis, Robert, Mahesh G. Nerurkar, Douglas Dokken, and David Ellis. Topological Dynamics and Applications: A Volume in Honor of Robert Ellis: Proceedings of a Conference in Honor of the Retirement of Robert Ellis, April 5–6, 1995, University of Minnesota. Vol. 215. American Mathematical Soc., 1998.
Authority control
International
• ISNI
• VIAF
National
• Germany
• Israel
• United States
• Czech Republic
• Netherlands
Academics
• MathSciNet
• Mathematics Genealogy Project
• zbMATH
Other
• IdRef
|
Wikipedia
|
Robert Langlands
Robert Phelan Langlands, CC FRS FRSC (/ˈlæŋləndz/; born October 6, 1936) is a Canadian mathematician.[1][2] He is best known as the founder of the Langlands program, a vast web of conjectures and results connecting representation theory and automorphic forms to the study of Galois groups in number theory,[3][4] for which he received the 2018 Abel Prize. He was an emeritus professor and occupied Albert Einstein's office at the Institute for Advanced Study in Princeton, until 2020 when he retired.[5]
Robert Langlands
CC FRS FRSC
Born (1936-10-06) October 6, 1936
New Westminster, British Columbia, Canada
NationalityCanadian/American
Alma materUniversity of British Columbia,
Yale University
Known forLanglands program
AwardsJeffery–Williams Prize (1980)
Cole Prize (1982)
Wolf Prize (1995–96)
Steele Prize (2005)
Nemmers Prize (2006)
Shaw Prize (2007)
Abel Prize (2018)
Order of Canada (2019)
Scientific career
FieldsMathematics
InstitutionsPrinceton University,
Yale University,
Institute for Advanced Study
ThesisSemi-Groups and Representations of Lie Groups (1960)
Doctoral advisorCassius Ionescu-Tulcea
Doctoral studentsJames Arthur
Thomas Callister Hales
Diana Shelstad
Career
Langlands was born in New Westminster, British Columbia, Canada, in 1936 to Robert Langlands and Kathleen J Phelan. He has two younger sisters (Mary b 1938; Sally b 1941). In 1945, his family moved to White Rock, near the US border, where his parents had a building supply and construction business.[6][3][1]
He graduated from Semiahmoo Secondary School and started enrolling at the University of British Columbia at the age of 16, receiving his undergraduate degree in Mathematics in 1957;[7] he continued at UBC to receive an M. Sc. in 1958. He then went to Yale University where he received a PhD in 1960.[8]
His first academic position was at Princeton University from 1960 to 1967, where he worked as an associate professor.[3] He spent a year in Turkey at METU during 1967–68 in an office next to Cahit Arf's.[9] He was a Miller Research Fellow at the University of California, Berkeley from 1964 to 1965, then was a professor at Yale University from 1967 to 1972. He was appointed Hermann Weyl Professor at the Institute for Advanced Study in 1972, and became professor emeritus in January 2007.[5]
Research
Langlands' Ph.D. thesis was on the analytical theory of Lie semigroups,[10] but he soon moved into representation theory, adapting the methods of Harish-Chandra to the theory of automorphic forms. His first accomplishment in this field was a formula for the dimension of certain spaces of automorphic forms, in which particular types of Harish-Chandra's discrete series appeared.[11][12]
He next constructed an analytical theory of Eisenstein series for reductive groups of rank greater than one, thus extending work of Hans Maass, Walter Roelcke, and Atle Selberg from the early 1950s for rank one groups such as $\mathrm {SL} (2)$. This amounted to describing in general terms the continuous spectra of arithmetic quotients, and showing that all automorphic forms arise in terms of cusp forms and the residues of Eisenstein series induced from cusp forms on smaller subgroups. As a first application, he proved the Weil conjecture on Tamagawa numbers for the large class of arbitrary simply connected Chevalley groups defined over the rational numbers. Previously this had been known only in a few isolated cases and for certain classical groups where it could be shown by induction.[13]
As a second application of this work, he was able to show meromorphic continuation for a large class of $L$-functions arising in the theory of automorphic forms, not previously known to have them. These occurred in the constant terms of Eisenstein series, and meromorphicity as well as a weak functional equation were a consequence of functional equations for Eisenstein series. This work led in turn, in the winter of 1966–67, to the now well known conjectures[14] making up what is often called the Langlands program. Very roughly speaking, they propose a huge generalization of previously known examples of reciprocity, including (a) classical class field theory, in which characters of local and arithmetic abelian Galois groups are identified with characters of local multiplicative groups and the idele quotient group, respectively; (b) earlier results of Martin Eichler and Goro Shimura in which the Hasse–Weil zeta functions of arithmetic quotients of the upper half plane are identified with $L$-functions occurring in Hecke's theory of holomorphic automorphic forms. These conjectures were first posed in relatively complete form in a famous letter to Weil,[14] written in January 1967. It was in this letter that he introduced what has since become known as the $L$-group and along with it, the notion of functoriality.
The book by Hervé Jacquet and Langlands on $\mathrm {GL} (2)$ presented a theory of automorphic forms for the general linear group $\mathrm {GL} (2)$, establishing among other things the Jacquet–Langlands correspondence showing that functoriality was capable of explaining very precisely how automorphic forms for $\mathrm {GL} (2)$ related to those for quaternion algebras. This book applied the adelic trace formula for $\mathrm {GL} (2)$ and quaternion algebras to do this. Subsequently, James Arthur, a student of Langlands while he was at Yale, successfully developed the trace formula for groups of higher rank. This has become a major tool in attacking functoriality in general, and in particular has been applied to demonstrating that the Hasse–Weil zeta functions of certain Shimura varieties are among the $L$-functions arising from automorphic forms.[15]
The functoriality conjecture is far from proven, but a special case (the octahedral Artin conjecture, proved by Langlands[16] and Tunnell[17]) was the starting point of Andrew Wiles' attack on the Taniyama–Shimura conjecture and Fermat's Last Theorem.
In the mid-1980s Langlands turned his attention[18] to physics, particularly the problems of percolation and conformal invariance. In 1995, Langlands started a collaboration with Bill Casselman at the University of British Columbia with the aim of posting nearly all of his writings—including publications, preprints, as well as selected correspondence—on the Internet. The correspondence includes a copy of the original letter to Weil that introduced the $L$-group. In recent years he has turned his attention back to automorphic forms, working in particular on a theme he calls "beyond endoscopy".[19]
Awards and honors
Langlands has received the 1996 Wolf Prize (which he shared with Andrew Wiles),[20] the 2005 AMS Steele Prize, the 1980 Jeffery–Williams Prize, the 1988 NAS Award in Mathematics from the National Academy of Sciences,[21] the 2006 Nemmers Prize in Mathematics, the 2007 Shaw Prize in Mathematical Sciences (with Richard Taylor) for his work on automorphic forms. In 2018, Langlands was awarded the Abel Prize for "his visionary program connecting representation theory to number theory".[22]
He was elected a Fellow of the Royal Society of Canada in 1972 and a Fellow of the Royal Society in 1981.[23][24] In 2012, he became a fellow of the American Mathematical Society.[25] Langlands was elected as a member of the American Academy of Arts and Sciences in 1990.[26] He was elected as a member of the National Academy of Sciences in 1993[27] and a member of the American Philosophical Society 2004.[28]
Among other honorary degrees, in 2003, Langlands received a doctorate honoris causa from Université Laval.[29]
In 2019, Langlands was appointed a Companion of the Order of Canada.[30][31]
On January 10, 2020, Langlands was honoured at Semiahmoo Secondary, which installed a mural to celebrate his contributions to mathematics.
Personal life
Langlands has been married to Charlotte Lorraine Cheverie (b 1935) since 1957. They have four children (2 daughters and 2 sons).[3] He holds Canadian and American citizenships.
Langlands spent a year in Turkey in 1967–68, where his office at the Middle East Technical University was next to that of Cahit Arf.[32][33] In addition to his mathematical studies, Langlands likes to learn foreign languages, both for better understanding of foreign publications on his topic and just as a hobby. He speaks English, French, Turkish and German, and reads (but does not speak) Russian.[33]
Publications
• Euler Products, New Haven: Yale University Press, 1967, ISBN 0-300-01395-7
• On the Functional Equations Satisfied by Eisenstein Series, Berlin: Springer, 1976, ISBN 3-540-07872-X
• Base Change for GL(2), Princeton: Princeton University Press, 1980, ISBN 0-691-08272-3
• Automorphic Representations, Shimura Varieties, and Motives. Ein Märchen (PDF), Chelsea Publishing Company, 1979
See also
• Automorphic L-function
• Endoscopic group
• Geometric Langlands correspondence
• Jacquet–Langlands correspondence
• Langlands classification
• Langlands decomposition
• Langlands–Deligne local constant
• Langlands dual
• Langlands group
• Langlands–Shahidi method
• Local Langlands conjectures
• Standard L-function
• Taniyama group
References
1. Alex Bellos (March 20, 2018). "Abel Prize 2018: Robert Langlands wins for 'unified theory of maths'". The Guardian. Retrieved March 26, 2018.
2. "Robert Phelan Langlands". NAS. Retrieved March 26, 2018.
3. Contento, Sandro (March 27, 2015), "The Canadian Who Reinvented Mathematics", Toronto Star
4. D Mackenzie (2000) Fermat's Last Theorem's First Cousin, Science 287(5454), 792–793.
5. Edward Frenkel (2013). "preface". Love and Math: The Heart of Hidden Reality. Basic Books. ISBN 978-0-465-05074-1. Robert Langlands, the mathematician who currently occupies Albert Einstein's office at the Institute for Advanced Study in Princeton
6. "UBC Newsletter: Robert Langlands Interview" (PDF). 2010. Archived from the original (PDF) on April 7, 2014. Retrieved June 22, 2018.
7. Kenneth, Chang (March 20, 2018). "Robert P. Langlands Is Awarded the Abel Prize, a Top Math Honor". The New York Times. Retrieved March 20, 2018.
8. "Canadian mathematician Robert Langlands wins Abel Prize for 2018". The New Indian Express. March 21, 2018. Retrieved March 26, 2018.
9. "Robert Langlands wins Abel Prize 2018 for 'unified theory of maths' | Mathematics Department". math.metu.edu.tr. Retrieved July 26, 2021.
10. For context, see the note by Derek Robinson at the IAS site
11. "IAS publication paper 14". IAS. Retrieved March 26, 2018.
12. R. P. Langlands (January 1963). "The dimension of spaces of automorphic forms". American Journal of Mathematics. 85 (1): 99–125. doi:10.2307/2373189. JSTOR 2373189. MR 0156362.
13. Langlands, Robert P. (1966), "The volume of the fundamental domain for some arithmetical subgroups of Chevalley groups", Algebraic Groups and Discontinuous Subgroups, Proc. Sympos. Pure Math., Providence, R.I.: Amer. Math. Soc., pp. 143–148, MR 0213362
14. "IAS paper 43". IAS. Retrieved March 26, 2018.
15. "IAS paper 60". Institute of Advanced Studies. Retrieved March 26, 2018.
16. Langlands, Robert P, Base change for GL(2). Annals of Mathematics Studies, 96. Princeton University Press, Princeton, N.J.; ISBN 0-691-08263-4; MR 574808
17. Tunnell, Jerrold, Artin's conjecture for representations of octahedral type, Bulletin of the American Mathematical Society (N.S.) 5 (1981), no. 2, 173–175.
18. "IAS publication". Retrieved March 26, 2018.
19. "IAS paper 25". IAS. Retrieved March 26, 2018.
20. "AMS Notices" (PDF).
21. "NAS Award in Mathematics". National Academy of Sciences. Retrieved February 13, 2011.
22. "2018: Robert P. Langlands". The Abel Prize. Retrieved July 22, 2022.
23. "Search Fellows". Royal Society of Canada. Retrieved April 3, 2018.
24. "Robert Langlands". Royal Society. Retrieved April 3, 2018.
25. List of Fellows of the American Mathematical Society, retrieved January 27, 2013.
26. "Robert Phelan Langlands". American Academy of Arts & Sciences. Retrieved March 22, 2021.
27. "Robert Langlands". www.nasonline.org. Retrieved March 22, 2021.
28. "APS Member History". search.amphilsoc.org. Retrieved June 14, 2021.
29. "Robert Langlands, Université Laval". Archived from the original on June 29, 2016. Retrieved March 1, 2017.
30. Office of the Secretary to the Governor General (June 20, 2019). "Governor General Announces 83 New Appointments to the Order of Canada". The Governor General of Canada. Archived from the original on June 28, 2019. Retrieved June 27, 2019.
31. Dunlevy, T'Cha (June 27, 2019). "Alanis Obomsawin, 15 other Quebecers to receive Order of Canada". Montreal Gazette. Archived from the original on July 4, 2019. Retrieved July 4, 2019.
32. The work of Robert Langlands – Miscellaneous items, Digital Mathematics Archive, UBC SunSITE, last accessed December 10, 2013.
33. Interview with Robert Langlands, UBC Dept. of Math., 2010; last accessed April 5, 2014.
External links
Wikiquote has quotations related to Robert Langlands.
• O'Connor, John J.; Robertson, Edmund F., "Robert Langlands", MacTutor History of Mathematics Archive, University of St Andrews
• Robert Langlands at the Mathematics Genealogy Project
• The work of Robert Langlands (a nearly complete archive)
• Faculty page at IAS
• The Abel Prize Interview 2018 with Robert Langlands
• Contenta, Sandro. "The Canadian who reinvented mathematics". Toronto Star. Retrieved March 28, 2015.
• Julia Mueller, On the genesis of Robert P. Langlands' conjectures and his letter to André Weil, Bull. Amer. Math. Soc., January 25, 2018
Laureates of the Wolf Prize in Mathematics
1970s
• Israel Gelfand / Carl L. Siegel (1978)
• Jean Leray / André Weil (1979)
1980s
• Henri Cartan / Andrey Kolmogorov (1980)
• Lars Ahlfors / Oscar Zariski (1981)
• Hassler Whitney / Mark Krein (1982)
• Shiing-Shen Chern / Paul Erdős (1983/84)
• Kunihiko Kodaira / Hans Lewy (1984/85)
• Samuel Eilenberg / Atle Selberg (1986)
• Kiyosi Itô / Peter Lax (1987)
• Friedrich Hirzebruch / Lars Hörmander (1988)
• Alberto Calderón / John Milnor (1989)
1990s
• Ennio de Giorgi / Ilya Piatetski-Shapiro (1990)
• Lennart Carleson / John G. Thompson (1992)
• Mikhail Gromov / Jacques Tits (1993)
• Jürgen Moser (1994/95)
• Robert Langlands / Andrew Wiles (1995/96)
• Joseph Keller / Yakov G. Sinai (1996/97)
• László Lovász / Elias M. Stein (1999)
2000s
• Raoul Bott / Jean-Pierre Serre (2000)
• Vladimir Arnold / Saharon Shelah (2001)
• Mikio Sato / John Tate (2002/03)
• Grigory Margulis / Sergei Novikov (2005)
• Stephen Smale / Hillel Furstenberg (2006/07)
• Pierre Deligne / Phillip A. Griffiths / David B. Mumford (2008)
2010s
• Dennis Sullivan / Shing-Tung Yau (2010)
• Michael Aschbacher / Luis Caffarelli (2012)
• George Mostow / Michael Artin (2013)
• Peter Sarnak (2014)
• James G. Arthur (2015)
• Richard Schoen / Charles Fefferman (2017)
• Alexander Beilinson / Vladimir Drinfeld (2018)
• Jean-François Le Gall / Gregory Lawler (2019)
2020s
• Simon K. Donaldson / Yakov Eliashberg (2020)
• George Lusztig (2022)
• Ingrid Daubechies (2023)
Mathematics portal
Shaw Prize laureates
Astronomy
• Jim Peebles (2004)
• Geoffrey Marcy and Michel Mayor (2005)
• Saul Perlmutter, Adam Riess and Brian Schmidt (2006)
• Peter Goldreich (2007)
• Reinhard Genzel (2008)
• Frank Shu (2009)
• Charles Bennett, Lyman Page and David Spergel (2010)
• Enrico Costa and Gerald Fishman (2011)
• David C. Jewitt and Jane Luu (2012)
• Steven Balbus and John F. Hawley (2013)
• Daniel Eisenstein, Shaun Cole and John A. Peacock (2014)
• William J. Borucki (2015)
• Ronald Drever, Kip Thorne and Rainer Weiss (2016)
• Simon White (2017)
• Jean-Loup Puget (2018)
• Edward C. Stone (2019)
• Roger Blandford (2020)
• Victoria Kaspi and Chryssa Kouveliotou (2021)
• Lennart Lindegren and Michael Perryman (2022)
• Matthew Bailes, Duncan Lorimer and Maura McLaughlin (2023)
Life science
and medicine
• Stanley Norman Cohen, Herbert Boyer, Yuet-Wai Kan and Richard Doll (2004)
• Michael Berridge (2005)
• Xiaodong Wang (2006)
• Robert Lefkowitz (2007)
• Ian Wilmut, Keith H. S. Campbell and Shinya Yamanaka (2008)
• Douglas Coleman and Jeffrey Friedman (2009)
• David Julius (2010)
• Jules Hoffmann, Ruslan Medzhitov and Bruce Beutler (2011)
• Franz-Ulrich Hartl and Arthur L. Horwich (2012)
• Jeffrey C. Hall, Michael Rosbash and Michael W. Young (2013)
• Kazutoshi Mori and Peter Walter (2014)
• Bonnie Bassler and Everett Peter Greenberg (2015)
• Adrian Bird and Huda Zoghbi (2016)
• Ian R. Gibbons and Ronald Vale (2017)
• Mary-Claire King (2018)
• Maria Jasin (2019)
• Gero Miesenböck, Peter Hegemann and Georg Nagel (2020)
• Scott D. Emr (2021)
• Paul A. Negulescu and Michael J. Welsh (2022)
• Patrick Cramer and Eva Nogales (2023)
Mathematical
science
• Shiing-Shen Chern (2004)
• Andrew Wiles (2005)
• David Mumford and Wentsun Wu (2006)
• Robert Langlands and Richard Taylor (2007)
• Vladimir Arnold and Ludwig Faddeev (2008)
• Simon Donaldson and Clifford Taubes (2009)
• Jean Bourgain (2010)
• Demetrios Christodoulou and Richard S. Hamilton (2011)
• Maxim Kontsevich (2012)
• David Donoho (2013)
• George Lusztig (2014)
• Gerd Faltings and Henryk Iwaniec (2015)
• Nigel Hitchin (2016)
• János Kollár and Claire Voisin (2017)
• Luis Caffarelli (2018)
• Michel Talagrand (2019)
• Alexander Beilinson and David Kazhdan (2020)
• Jean-Michel Bismut and Jeff Cheeger (2021)
• Noga Alon and Ehud Hrushovski (2022)
• Vladimir Drinfeld and Shing-Tung Yau (2023)
Abel Prize laureates
• 2003 Jean-Pierre Serre
• 2004 Michael Atiyah
• Isadore Singer
• 2005 Peter Lax
• 2006 Lennart Carleson
• 2007 S. R. Srinivasa Varadhan
• 2008 John G. Thompson
• Jacques Tits
• 2009 Mikhail Gromov
• 2010 John Tate
• 2011 John Milnor
• 2012 Endre Szemerédi
• 2013 Pierre Deligne
• 2014 Yakov Sinai
• 2015 John Forbes Nash Jr.
• Louis Nirenberg
• 2016 Andrew Wiles
• 2017 Yves Meyer
• 2018 Robert Langlands
• 2019 Karen Uhlenbeck
• 2020 Hillel Furstenberg
• Grigory Margulis
• 2021 László Lovász
• Avi Wigderson
• 2022 Dennis Sullivan
• 2023 Luis Caffarelli
Fellows of the Royal Society elected in 1981
Fellows
• John Barker
• Eric Barnard
• Fraser John Bergersen
• Peter Bradshaw
• David Brink
• Fred Brown
• Ian Butterworth
• Brandon Carter
• Malcolm Clarke
• Robert Clayton
• John Conway
• Rex Dawson
• Dennis Evans
• Frank Farmer
• Ned Franklin
• Christopher Graham
• Norman Green
• Herbert Gutfreund
• Allan Hay
• John Hill
• Kenneth Holmes
• Robert Honeycombe
• Michael Horne
• Yuet Wai Kan
• Robert Langlands
• Simon Maddrell
• Ian McGregor
• John Nelder
• Mike O'Hara
• Autar Singh Paintal
• John Phillips
• Edward Roy Pike
• Walter Plowright
• Ken Pounds
• Dai Rees
• Colin Reese
• Edward Reich
• Wal Sargent
• John Taylor
• David Wheeler
Foreign
• Britton Chance
• Igor Shafarevich
• James Watson
• Steven Weinberg
Authority control
International
• FAST
• ISNI
• VIAF
National
• Norway
• France
• BnF data
• Germany
• Israel
• United States
• Latvia
• Czech Republic
• Australia
• Netherlands
• Poland
Academics
• DBLP
• MathSciNet
• Mathematics Genealogy Project
• Scopus
• zbMATH
People
• Deutsche Biographie
• Trove
Other
• SNAC
• IdRef
|
Wikipedia
|
Robert Phelps
Robert Ralph Phelps (March 22, 1926 – January 4, 2013) was an American mathematician who was known for his contributions to analysis, particularly to functional analysis and measure theory. He was a professor of mathematics at the University of Washington from 1962 until his death.
Robert R. Phelps
Born(1926-03-22)March 22, 1926
California
DiedJanuary 4, 2013(2013-01-04) (aged 86)
Washington state[1]
NationalityAmerican
Alma materUniversity of Washington
Known for
• Bishop–Phelps theorem
• Banach spaces & differentiability
• Choquet theory
SpouseElaine Phelps[2]
Scientific career
Fields
• Functional analysis
• measure theory
InstitutionsUniversity of Washington
Doctoral advisorVictor L. Klee[3]
Influenced
• John Rainwater
• John R. Giles
Biography
Phelps wrote his dissertation on subreflexive Banach spaces under the supervision of Victor Klee in 1958 at the University of Washington.[3] Phelps was appointed to a position at Washington in 1962.[4]
In 2012 he became a fellow of the American Mathematical Society.[5]
He was a convinced atheist.[6]
Research
With Errett Bishop, Phelps proved the Bishop–Phelps theorem, one of the most important results in functional analysis, with applications to operator theory, to harmonic analysis, to Choquet theory, and to variational analysis. In one field of its application, optimization theory, Ivar Ekeland began his survey of variational principles with this tribute:
The central result. The grandfather of it all is the celebrated 1961 theorem of Bishop and Phelps ... that the set of continuous linear functionals on a Banach space E which attain their maximum on a prescribed closed convex bounded subset X⊂E is norm-dense in E*. The crux of the proof lies in introducing a certain convex cone in E, associating with it a partial ordering, and applying to the latter a transfinite induction argument (Zorn's lemma).[7]
Phelps has written several advanced monographs, which have been republished. His 1966 Lectures on Choquet theory was the first book to explain the theory of integral representations.[8] In these "instant classic" lectures, which were translated into Russian and other languages, and in his original research, Phelps helped to lead the development of Choquet theory and its applications, including probability, harmonic analysis, and approximation theory.[9][10][11] A revised and expanded version of his Lectures on Choquet theory was republished as Phelps (2002).[11]
Phelps has also contributed to nonlinear analysis, in particular writing notes and a monograph on differentiability and Banach-space theory. In its preface, Phelps advised readers of the prerequisite "background in functional analysis": "the main rule is the separation theorem (a.k.a. [also known as] the Hahn–Banach theorem): Like the standard advice given in mountaineering classes (concerning the all-important bowline for tying oneself into the end of the climbing rope), you should be able to employ it using only one hand while standing blindfolded in a cold shower."[12] Phelps has been an avid rock-climber and mountaineer. Following the trailblazing research of Asplund and Rockafellar, Phelps hammered into place the pitons, linked the carabiners, and threaded the top rope by which novices have ascended from the frozen tundras of topological vector spaces to the Shangri-La of Banach space theory. His University College, London (UCL) lectures on the Differentiability of convex functions on Banach spaces (1977–1978) were "widely distributed". Some of Phelps's results and exposition were developed in two books,[13] Bourgin's Geometric aspects of convex sets with the Radon-Nikodým property (1983) and Giles's Convex analysis with application in the differentiation of convex functions (1982).[10][14] Phelps avoided repeating the results previously reported in Bourgin and Giles when he published his own Convex functions, monotone operators and differentiability (1989), which reported new results and streamlined proofs of earlier results.[13] Now, the study of differentiability is a central concern in nonlinear functional analysis.[15][16] Phelps has published articles under the pseudonym of John Rainwater.[17]
Selected publications
• Bishop, Errett; Phelps, R. R. (1961). "A proof that every Banach space is subreflexive". Bulletin of the American Mathematical Society. 67: 97–98. doi:10.1090/s0002-9904-1961-10514-4. MR 0123174.
• Phelps, Robert R. (1993) [1989]. Convex functions, monotone operators and differentiability. Lecture Notes in Mathematics. Vol. 1364 (2nd ed.). Berlin: Springer-Verlag. pp. xii+117. ISBN 3-540-56715-1. MR 1238715.
• Phelps, Robert R. (2001). Phelps, Robert R (ed.). Lectures on Choquet's theorem. Lecture Notes in Mathematics. Vol. 1757 (Second edition of 1966 ed.). Berlin: Springer-Verlag. pp. viii+124. doi:10.1007/b76887. ISBN 3-540-41834-2. MR 1835574.
• Namioka, I.; Phelps, R. R. (1975). "Banach spaces which are Asplund spaces". Duke Math. J. 42 (4): 735–750. doi:10.1215/s0012-7094-75-04261-1. hdl:10338.dmlcz/127336. ISSN 0012-7094.
Notes
1. Robert R. "Bob" Phelps Obituary
2. Page 21: Gritzmann, Peter; Sturmfels, Bernd (April 2008). "Victor L. Klee 1925–2007" (PDF). Notices of the American Mathematical Society. Providence, RI: American Mathematical Society. 55 (4): 467–473. ISSN 0002-9920.
3. Robert Phelps at the Mathematics Genealogy Project
4. University of Washington description of Phelps
5. List of Fellows of the American Mathematical Society, retrieved 2013-05-05.
6. "In Memoriam: Robert R. Phelps (1926-2013) « Math Drudge".
7. Ekeland (1979, p. 443)
8. Lacey, H. E. "Review of Gustave Choquet's (1969) Lectures on analysis, Volume III: Infinite dimensional measures and problem solutions". Mathematical Reviews. MR 0250013.
9. Asimow, L.; Ellis, A. J. (1980). Convexity theory and its applications in functional analysis. London Mathematical Society Monographs. Vol. 16. London-New York: Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers]. pp. x+266. ISBN 0-12-065340-0. MR 0623459.
10. Bourgin, Richard D. (1983). Geometric aspects of convex sets with the Radon-Nikodým property. Lecture Notes in Mathematics. Vol. 993. Berlin: Springer-Verlag. pp. xii+474. doi:10.1007/BFb0069321. ISBN 3-540-12296-6. MR 0704815.
11. Rao (2002)
12. Page iii of the first (1989) edition of Phelps (1993).
13. Nashed (1990)
14. Giles, John R. (1982). Convex analysis with application in the differentiation of convex functions. Research Notes in Mathematics. Vol. 58. Boston, Mass.-London: Pitman (Advanced Publishing Program). pp. x+278. ISBN 0-273-08537-9. MR 0650456.
15. Lindenstrauss, Joram and Benyamini, Yoav. Geometric nonlinear functional analysis Colloquium publications, 48. American Mathematical Society.
16. Mordukhovich, Boris S. (2006). Variational analysis and generalized differentiation I and II. Grundlehren Series (Fundamental Principles of Mathematical Sciences). Vol. 331. Springer. MR 2191745.
17. Phelps, Robert R. (2002). Melvin Henriksen (ed.). "Biography of John Rainwater". Topological Commentary. 7 (2). arXiv:math/0312462. Bibcode:2003math.....12462P.
References
• Ekeland, Ivar (1979). "Nonconvex minimization problems". Bulletin of the American Mathematical Society. New Series. 1 (3): 443–474. doi:10.1090/S0273-0979-1979-14595-6. MR 0526967.
• Nashed, M. Z. (1990). "Review of 1989 first edition of Phelps's Convex functions, monotone operators and differentiability". Mathematical Reviews. Lecture Notes in Mathematics. 1364. doi:10.1007/BFb0089089. ISBN 978-3-540-50735-2. MR 0984602. Review of first edition of Phelps (1993).
• Rao, T. S. S. R. K. (2002). Phelps, Robert R (ed.). "Review of Phelps (2002)". Mathematical Reviews. Lecture Notes in Mathematics. 1757. doi:10.1007/b76887. ISBN 978-3-540-41834-4. MR 1835574. Review of Phelps (2001).
External resources
• Professor Phelp's homepage at the University of Washington
• "Robert Phelps". University of Washington. Archived from the original on March 16, 2012.
• Mathematical Reviews. "Robert R. Phelps". Retrieved 2011-04-02.
• Robert Phelps at the Mathematics Genealogy Project
Authority control
International
• ISNI
• VIAF
National
• Norway
• France
• BnF data
• Germany
• Israel
• Belgium
• United States
• Latvia
• Czech Republic
• Netherlands
Academics
• MathSciNet
• Mathematics Genealogy Project
• zbMATH
People
• Deutsche Biographie
Other
• IdRef
|
Wikipedia
|
Robert Pollock Gillespie
Robert Pollock Gillespie FRSE (1903–1977) was a Scottish mathematician. He was twice President of the Edinburgh Mathematical Society (1946–7 and 1968–9). He published several important books on mathematics.
Life
He was born on 21 November 1903 in Johnstone, Renfrewshire the son of Thomas Gillespie a butcher and his wife, Jane Pollock. He was raised at Ashcot on Kilbarchan Road in Johnstone. He was educated locally then at Paisley Grammar School where he was dux. He then won a bursary to study Mathematics and Natural Philosophy (Physics) at Glasgow University graduating MA BSc in 1924. He did further postgraduate studies under E. W. Hobson at Cambridge University from 1924 to 1927 under a William Bryce Scholarship, gaining his doctorate (PhD) in 1932 due to a delay in submitting his thesis.[1]
He began lecturing in Mathematics at Glasgow University in 1929 under Prof Thomas Murray MacRobert and alongside Dr T S Graham.[2] In 1933 he was elected a Fellow of the Royal Society of Edinburgh due to his numerous publications on mathematics. His proposers were Thomas Murray MacRobert, Neil M'Arthur, Richard Alexander Robb and William Arthur.[3]
In the Second World War he served first in the Clyde River Patrol and in 1941 moved to the Air Traffic Control division of the RAF based at Prestwick. He served on the University Air Squadron after the war.[4]
In 1948 he was promoted to Senior Lecturer at Glasgow and remained there until retiral in 1969. On retiral he moved to Edzell and indulged his love as an amateur artist.
He died in Edzell on 1 January 1977.
Family
He was married to Maisie Bowman, daughter of Prof A. A. Bowman. Their son Alistair Gillespie was also a mathematician.
Publications
• Integration (1939)
• Partial Differentiation (1951)
• Solving Problems in Advanced Calculus (1972)
References
1. "Gillespie biography". Archived from the original on 9 August 2016. Retrieved 10 July 2016.
2. "RSE Obituary". Archived from the original on 9 August 2016. Retrieved 10 July 2016.
3. Biographical Index of Former Fellows of the Royal Society of Edinburgh 1783–2002 (PDF). The Royal Society of Edinburgh. July 2006. ISBN 0-902198-84-X.
4. "RSE Obituary". Archived from the original on 9 August 2016. Retrieved 10 July 2016.
Authority control
International
• ISNI
• VIAF
National
• Germany
• Israel
• United States
• Netherlands
Academics
• MathSciNet
• Mathematics Genealogy Project
• zbMATH
Other
• IdRef
|
Wikipedia
|
Robert Recorde
Robert Recorde (c. 1512 – 1558) was a Welsh[1][2] physician and mathematician. He invented the equals sign (=) and also introduced the pre-existing plus (+) and minus (−) signs to English speakers in 1557.
Robert Recorde
Robert Recorde (c.1512–1558)
Bornc. 1512
Tenby, Pembrokeshire, Wales
Died1558 (1559)
London, England
NationalityWelsh
Alma materUniversity of Oxford
University of Cambridge
Known forInventing the equals sign (=)
Scientific career
FieldsPhysician and mathematician
InstitutionsUniversity of Oxford
Royal Mint
Biography
Born around 1512, Robert Recorde was the second and last son of Thomas and Rose Recorde[3] of Tenby, Pembrokeshire, in Wales.[4]
Recorde entered the University of Oxford about 1525, and was elected a Fellow of All Souls College there in 1531. Having adopted medicine as a profession, he went to the University of Cambridge to take the degree of M.D. in 1545. He afterwards returned to Oxford, where he publicly taught mathematics, as he had done prior to going to Cambridge. He invented the "equals" sign, which consists of two horizontal parallel lines, stating that no two things can be more equal. It appears that he afterwards went to London, and acted as physician to King Edward VI and to Queen Mary, to whom some of his books are dedicated. He was also controller of the Royal Mint and served as Comptroller of Mines and Monies in Ireland.[5] After being sued for defamation by a political enemy, he was arrested for debt and died in the King's Bench Prison, Southwark, by the middle of June 1558.
Publications
Recorde published several works upon mathematical and medical subjects, chiefly in the form of dialogue between master and scholar, such as the following:
• The Grounde of Artes, teachings the Worke and Practise, of Arithmeticke, both in whole numbers and fractions (1543),[4] the first English language book on algebra.
• The Pathway to Knowledge, containing the First Principles of Geometry ... bothe for the use of Instrumentes Geometricall and Astronomicall, and also for Projection of Plattes (London, 1551)
• The Castle of Knowledge, containing the Explication of the Sphere both Celestiall and Materiall, etc. (London, 1556) A book explaining Ptolemaic astronomy while mentioning the Copernican heliocentric model in passing.
• The Whetstone of Witte, whiche is the seconde parte of Arithmeteke: containing thextraction of rootes; the cossike practise, with the rule of equation; and the workes of Surde Nombers (London, 1557). This was the book in which the equals sign was introduced within a printed edition.[6] With the publication of this book Recorde is credited with introducing algebra into the Island of Britain with a systematic notation.[7][8]
• A medical work, The Urinal of Physick (1548), frequently reprinted.[9]
Most of those works were written in the form of a catechism.[6] Several books whose authors are unknown have been attributed to him: Cosmographiae isagoge, De Arte faciendi Horologium and De Usu Globorum et de Statu temporum.[10]
See also
• Equality
• Equals sign
• Equation
• History of mathematical notation
• St. Mary's Church, Tenby
• The Ground of Arts
• Welsh mathematicians
• Zenzizenzizenzic – a word to describe a number to the eighth power coined by Robert Recorde
Notes
1. Mazur, Joseph (21 May 2014). "Notation, notation, notation: a brief history of mathematical symbols". The Guardian. ISSN 0261-3077. Retrieved 5 May 2023.
2. Western Mail, Saturday 24 March 1928 - https://www.britishnewspaperarchive.co.uk/viewer/bl/0000104/19280324/188/0006
3. "Robert Recorde: the Welshman who invented equality". The National Wales. Archived from the original on 6 February 2022. Retrieved 6 February 2022.
4. Johnston, Stephen (2004). "Recorde, Robert (c. 1512–1558)". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/23241. Retrieved 26 January 2012. (Subscription or UK public library membership required.)
5. Newman, James R. (1956). The World of Mathematics.
6. Smith, David Eugene (1 July 1917). "Medicine and Mathematics in the Sixteenth Century". Ann. Med. Hist. 1 (2): 125–140. OCLC 12650954. PMC 7927718. PMID 33943138. (here cited p. 131).
7. Jourdain, Philip E. B. (1913). The Nature of Mathematics.
8. Robert Recorde, The Whetstone of Witte (London, England: John Kyngstone, 1557), p. 236 (although the pages of this book are not numbered). From the chapter titled "The rule of equation, commonly called Algebers Rule" (p. 236): "Howbeit, for easie alteration of equations. I will propounde a fewe examples, bicause the extraction of their rootes, maie the more aptly bee wroughte. And to avoide the tediouse repetition of these woordes: is equalle to: I will sette as I doe often in worke use, a paire of paralleles, or Gemowe [twin, from gemew, from the French gemeau (twin / twins), from the Latin gemellus (little twin)] lines of one lengthe, thus: = , bicause noe .2. thynges, can be moare equalle." (However, for easy manipulation of equations, I will present a few examples in order that the extraction of roots may be more readily done. And to avoid the tedious repetition of these words "is equal to", I will substitute, as I often do when working, a pair of parallels or twin lines of the same length, thus: = , because no two things can be more equal.)
9. The Urinal of Physick, by Robert Recorde, 1548; at Google Books
10. John Hall, "An Historiall Expostulation", p. 60. In Early English Poetry, Ballads, and Popular Literature of the Middle Ages, v. XI. London: T. Richards, 1844
References
• This article incorporates text from a publication now in the public domain: Chisholm, Hugh, ed. (1911). "Recorde, Robert". Encyclopædia Britannica. Vol. 22 (11th ed.). Cambridge University Press. p. 966.
• James R. Newman (1956). The World of Mathematics Vol. 1 Commentary on Robert Recorde
• Philip E. B. Jourdain (1913). The Nature of Mathematics
• Gareth Roberts and Fenny Smith, editors (2012). Robert Recorde: The Life and Times of a Tudor Mathematician (University of Wales Press, distributed by University of Chicago Press) 232 pages
• Jack Williams (2011). Robert Recorde: Tudor Polymath, Expositor and Practitioner of Computation (Heidelberg, Springer) (History of Computing).
• J. W. S. Cassels (1976). Is This a Recorde?, The Mathematical Gazette Vol. 60 No. 411 March 1976 p 59-61
• Gordon Roberts (2016). Robert Recorde: Tudor Scholar and Mathematician (University of Wales Press).
• Frank J. Swetz and Victor J. Katz (2011). "Mathematical Treasures - Robert Recorde's Whetstone of Witte," Convergence (January 2011)
External links
Wikimedia Commons has media related to Robert Recorde.
• St. Andrew's University Maths History biography
• Robert Recorde: the Welshman who invented equality Archived 6 February 2022 at the Wayback Machine
• Robert Recorde and other Welsh Mathematicians
• 100 Welsh Heroes – Robert Recorde
• Earliest Uses of Symbols of Relation
• Earliest Known Uses of Some of the Words of Mathematics This contains numerous quotations from Recorde.
• RECORDE (Robert) in Charles Hutton's Mathematical and Philosophical Dictionary
• Robert Recorde's life and works on h2g2
• Current publisher of Robert Recorde's books in the form of original reproductions
• Works by Robert Record at Project Gutenberg
• Works by or about Robert Recorde at Internet Archive
Authority control
International
• FAST
• ISNI
• VIAF
National
• Norway
• Germany
• Israel
• Belgium
• United States
• Netherlands
• Portugal
• Vatican
Academics
• CiNii
• MathSciNet
• zbMATH
Other
• IdRef
|
Wikipedia
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.