Logarithmic pair

In algebraic geometry, a logarithmic pair consists of a variety, together with a divisor along which one allows mild logarithmic singularities. They were studied by .
Definition.
A boundary Q-divisor on a variety is a Q-divisor "D" of the form Σ"d""i""D""i" where the "D""i" are the distinct irreducible components of "D" and all coefficients are rational numbers with 0≤"d""i"≤1.
A logarithmic pair, or log pair for short, is a pair ("X","D") consisting of a normal variety "X" and a boundary Q-divisor "D".
The log canonical divisor of a log pair ("X","D") is "K"+"D" where "K" is the canonical divisor of "X".
A logarithmic 1-form on a log pair ("X","D") is allowed to have logarithmic singularities of the form
d log("z") = d"z"/"z" along components of the divisor given locally by "z"=0.


Discrete Hartley transform

A discrete Hartley transform (DHT) is a Fourier-related transform of discrete, periodic data similar to the discrete Fourier transform (DFT), with analogous applications in signal processing and related fields. Its main distinction from the DFT is that it transforms real inputs to real outputs, with no intrinsic involvement of complex numbers. Just as the DFT is the discrete analogue of the continuous Fourier transform (FT), the DHT is the discrete analogue of the continuous Hartley transform (HT), introduced by Ralph V. L. Hartley in 1942.
Because there are fast algorithms for the DHT analogous to the fast Fourier transform (FFT), the DHT was originally proposed by Ronald N. Bracewell in 1983 as a more efficient computational tool in the common case where the data are purely real. It was subsequently argued, however, that specialized FFT algorithms for real inputs or outputs can ordinarily be found with slightly fewer operations than any corresponding algorithm for the DHT.
Definition.
Formally, the discrete Hartley transform is a linear, invertible function "H": R"n" → R"n" (where R denotes the set of real numbers). The "N" real numbers "x"0, ..., "x""N"−1 are transformed into the "N" real numbers "H"0, ..., "H""N"−1 according to the formula
The combination formula_2 formula_3 is sometimes denoted , and should not be confused with , or which appears in the DFT definition (where "i" is the imaginary unit).
As with the DFT, the overall scale factor in front of the transform and the sign of the sine term are a matter of convention. Although these conventions occasionally vary between authors, they do not affect the essential properties of the transform.
Properties.
The transform can be interpreted as the multiplication of the vector ("x"0, ..., "x""N"−1) by an "N"-by-"N" matrix; therefore, the discrete Hartley transform is a linear operator. The matrix is invertible; the inverse transformation, which allows one to recover the "xn" from the "Hk", is simply the DHT of "Hk" multiplied by 1/"N". That is, the DHT is its own inverse (involutory), up to an overall scale factor.
The DHT can be used to compute the DFT, and vice versa. For real inputs "xn", the DFT output "X""k" has a real part ("H""k" + "H""N−k")/2 and an imaginary part ("H""N−k" − "Hk")/2. Conversely, the DHT is equivalent to computing the DFT of "xn" multiplied by 1 + "i", then taking the real part of the result.
As with the DFT, a cyclic convolution z = x∗y of two vectors x = ("xn") and y = ("yn") to produce a vector z = ("zn"), all of length "N", becomes a simple operation after the DHT. In particular, suppose that the vectors X, Y, and Z denote the DHT of x, y, and z respectively. Then the elements of Z are given by:
where we take all of the vectors to be periodic in "N" ("XN" = "X"0, et cetera). Thus, just as the DFT transforms a convolution into a pointwise multiplication of complex numbers ("pairs" of real and imaginary parts), the DHT transforms a convolution into a simple combination of "pairs" of real frequency components. The inverse DHT then yields the desired vector z. In this way, a fast algorithm for the DHT (see below) yields a fast algorithm for convolution. (This is slightly more expensive than the corresponding procedure for the DFT, not including the costs of the transforms below, because the pairwise operation above requires 8 real-arithmetic operations compared to the 6 of a complex multiplication. This count doesn't include the division by 2, which can be absorbed e.g. into the 1/"N" normalization of the inverse DHT.)
Fast algorithms.
Just as for the DFT, evaluating the DHT definition directly would require O("N"2) arithmetical operations (see Big O notation). There are fast algorithms similar to the FFT, however, that compute the same result in only O("N" log "N") operations. Nearly every FFT algorithm, from Cooley–Tukey to prime-factor to Winograd (1985) to Bruun's (1993), has a direct analogue for the discrete Hartley transform. (However, a few of the more exotic FFT algorithms, such as the QFT, have not yet been investigated in the context of the DHT.)
In particular, the DHT analogue of the Cooley–Tukey algorithm is commonly known as the fast Hartley transform (FHT) algorithm, and was first described by Bracewell in 1984. This FHT algorithm, at least when applied to power-of-two sizes "N", is the subject of the United States patent number 4,646,256, issued in 1987 to Stanford University. Stanford placed this patent in the public domain in 1994 (Bracewell, 1995).
As mentioned above, DHT algorithms are typically slightly less efficient (in terms of the number of floating-point operations) than the corresponding DFT algorithm (FFT) specialized for real inputs (or outputs). This was first argued by Sorensen et al. (1987) and Duhamel & Vetterli (1987). The latter authors obtained what appears to be the lowest published operation count for the DHT of power-of-two sizes, employing a split-radix algorithm (similar to the split-radix FFT) that breaks a DHT of length "N" into a DHT of length "N"/2 and two real-input DFTs ("not" DHTs) of length "N"/4. In this way, they argued that a DHT of power-of-two length can be computed with, at best, 2 more additions than the corresponding number of arithmetic operations for the real-input DFT.
On present-day computers, performance is determined more by cache and CPU pipeline considerations than by strict operation counts, and a slight difference in arithmetic cost is unlikely to be significant. Since FHT and real-input FFT algorithms have similar computational structures, neither appears to have a substantial "a priori" speed advantage ( and Šević, 1994). As a practical matter, highly optimized real-input FFT libraries are available from many sources (e.g. from CPU vendors such as Intel), whereas highly optimized DHT libraries are less common.
On the other hand, the redundant computations in FFTs due to real inputs are more difficult to eliminate for large prime "N", despite the existence of O("N" log "N") complex-data algorithms for such cases, because the redundancies are hidden behind intricate permutations and/or phase rotations in those algorithms. In contrast, a standard prime-size FFT algorithm, Rader's algorithm, can be directly applied to the DHT of real data for roughly a factor of two less computation than that of the equivalent complex FFT (Frigo and Johnson, 2005). On the other hand, a non-DHT-based adaptation of Rader's algorithm for real-input DFTs is also possible (Chu & Burrus, 1982).
Multi-Dimensional Discrete Hartley Transform (MD-DHT).
The rD-DHT (MD-DHT with "r" dimensions) is given by
formula_5
with formula_6 and where formula_7
Similar to the 1-D case, as a real and symmetric transform, the MD-DHT is simpler than the MD-DFT. For one, the inverse DHT is identical to the forward transform, with the addition of a scaling factor;
and second, since the kernel is real, it avoids the computational complexity of complex numbers. Additionally, the DFT is directly obtainable from the DHT by a simple additive operation (Bracewell, 1983).
The MD-DHT is widely used in areas like image and optical signal processing. Specific applications include computer vision, high-definition television, and teleconferencing, areas that process or analyze motion images (Zeng, 2000).
Fast algorithms for the MD-DHT.
As computing speed keeps increasing, bigger multidimensional problems become computationally feasible, requiring the need for fast multidimensional algorithms. Three such algorithms follow.
In pursuit of separability for efficiency, we consider the following transform (Bracewell, 1983),
formula_8
It was shown in Bortfeld (1995), that the two can be related by a few additions. For example, in 3-D,
formula_9
For formula_10, row-column algorithms can then be implemented. This technique is commonly used due to the simplicity of such R-C algorithms, but they are not optimized for general M-D spaces.
Other fast algorithms have been developed, such as radix-2, radix-4, and split radix. For example, Boussakta (2000) developed the 3-D vector radix,
formula_11
formula_12
formula_13
formula_14
It was also presented in Boussakta (2000) that this 3D-vector radix algorithm takes formula_15 multiplications and formula_16 additions compared to formula_17 multiplications and formula_18 additions from the row-column approach. The drawback is that the implementation of these radix-type of algorithms is hard to generalize for signals of arbitrary dimensions.
Number theoretic transforms have also been used for solving the MD-DHT, since they perform extremely fast convolutions. In Boussakta (1988), it was shown how to decompose the MD-DHT transform into a form consisting of convolutions:
For the 2-D case (the 3-D case is also covered in the stated reference),
formula_19 formula_20 , formula_21
can be decomposed into 1-D and 2-D circular convolutions as follows,
formula_22
where
formula_23 formula_24
formula_25 formula_26
formula_27
formula_28
formula_29
Developing formula_30 further,
formula_31
formula_32
At this point we present the Fermat number transform (FNT). The tth Fermat number is given by formula_33, with formula_34. The well known Fermat numbers are for formula_35 (formula_36 is prime for formula_37), (Boussakta, 1988). The Fermat number transform is given by
formula_38
with formula_39. formula_40 and formula_41 are roots of unity of order formula_42 and formula_43 respectively formula_44.
Going back to the decomposition, the last term for formula_45 will be denoted as formula_46, then
formula_47
formula_48
formula_29
If formula_50 and formula_51 are primitive roots of formula_42 and formula_43 (which are guaranteed to exist if formula_43 and formula_42 are prime) then formula_50 and formula_51 map formula_58 to formula_59 So, mapping formula_60 and formula_61 to formula_62 and formula_63, one gets the following,
formula_64
formula_65
formula_66.
Which is now a circular convolution. With formula_67, formula_68, and formula_69, one has
formula_70
formula_71
where formula_72 denotes term by term multiplication. It was also stated in (Boussakta, 1988) that this algorithm reduces the number of multiplications by a factor of 8–20 over other DHT algorithms at a cost of a slight increase in the number of shift and add operations, which are assumed to be simpler than multiplications. The drawback of this algorithm is the constraint that each dimension of the transform has a primitive root.


Smooth topology

In algebraic geometry, the smooth topology is a certain Grothendieck topology, which is finer than étale topology. Its main use is to define the cohomology of an algebraic stack with coefficients in, say, the étale sheaf formula_1.
To understand the problem that motivates the notion, consider the classifying stack formula_2 over formula_3. Then formula_4 in the étale topology; i.e., just a point. However, we expect the "correct" cohomology ring of formula_2 to be more like that of formula_6 as the ring should classify line bundles. Thus, the cohomology of formula_2 should be defined using smooth topology for formulae like Behrend's fixed point formula to hold.


Denis Osin

Denis Osin is a mathematician at Vanderbilt University working in geometric group theory and geometric topology. He was a speaker at the International Congress of Mathematicians in Rio de Janeiro in 2018.
Osin received a Ph.D. at Moscow State University in 1999 under the supervision of Aleksandr Olshansky. He worked at the Financial University under the Government of the Russian Federation, at the City College of CUNY, and joined Vanderbilt in 2008. He was promoted to a Full Professor in 2013. He is an editor at "Groups, Geometry, and Dynamics".
He was named to the 2021 class of fellows of the American Mathematical Society "for contributions in geometric group theory, specifically groups acting on hyperbolic spaces".


Definite matrix

In mathematics, a symmetric matrix formula_1 with real entries is positive-definite if the real number formula_2 is positive for every nonzero real column vector formula_3 where formula_4 is the transpose of More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is
positive-definite if the real number formula_5 is positive for every nonzero complex column vector formula_3 were formula_7 denotes the conjugate transpose of formula_8
Positive semi-definite matrices are defined similarly, except that the scalars formula_2 and formula_5 are required to be positive "or zero" (that is nonnegative). Negative-definite and negative semi-definite matrices are defined analogously. A matrix that is not positive semi-definite and not negative semi-definite is sometimes called indefinite.
A matrix is thus positive-definite if and only if it is the matrix of a positive-definite quadratic form or Hermitian form. In other words, a matrix is positive-definite if and only if it defines an inner product.
Positive-definite and positive-semidefinite matrices can be characterized in many ways, which may explain the importance of the concept in various parts of mathematics. A matrix is positive-definite (resp. positive-semidefinite) if and only if satisfies any of the following equivalent conditions.
Positive-definite and positive-semidefinite real matrices are at the basis of convex optimization, since, given a function of several real variables that is twice differentiable, then if its Hessian matrix (matrix of its second partial derivatives) is positive-definite at a point , then the function is convex near , and, conversely, if the function is convex near , then the Hessian matrix is positive-semidefinite at .
Some authors use more general definitions of definiteness, including some non-symmetric real matrices, or non-Hermitian complex ones.
Definitions.
In the following definitions, formula_14 is the transpose of formula_15, formula_16 is the conjugate transpose of formula_15 and formula_18 denotes the "n"-dimensional zero-vector.
Definitions for real matrices.
An formula_19 symmetric real matrix formula_1 is said to be positive-definite if formula_21 for all non-zero formula_22 in formula_23. Formally,
An formula_19 symmetric real matrix formula_1 is said to be positive semidefinite or non-negative-definite if formula_26 for all formula_22 in formula_23. Formally,
An formula_19 symmetric real matrix formula_1 is said to be negative-definite if formula_31 for all non-zero formula_22 in formula_23. Formally,
An formula_19 symmetric real matrix formula_1 is said to be negative-semidefinite or non-positive-definite if formula_36 for all formula_37 in formula_23. Formally,
An formula_19 symmetric real matrix which is neither positive semidefinite nor negative semidefinite is called indefinite.
Definitions for complex matrices.
The following definitions all involve the term formula_40. Notice that this is always a real number for any Hermitian square matrix formula_1.
An formula_19 Hermitian complex matrix formula_1 is said to be positive-definite if formula_44 for all non-zero formula_22 in formula_46. Formally,
An formula_19 Hermitian complex matrix formula_1 is said to be positive semi-definite or non-negative-definite if formula_49 for all formula_37 in formula_51. Formally,
An formula_19 Hermitian complex matrix formula_1 is said to be negative-definite if formula_54 for all non-zero formula_22 in formula_46. Formally,
An formula_19 Hermitian complex matrix formula_1 is said to be negative semi-definite or non-positive-definite if formula_59 for all formula_22 in formula_46. Formally,
An formula_19 Hermitian complex matrix which is neither positive semidefinite nor negative semidefinite is called indefinite.
Consistency between real and complex definitions.
Since every real matrix is also a complex matrix, the definitions of "definiteness" for the two classes must agree.
For complex matrices, the most common definition says that "formula_1 is positive-definite if and only if formula_64 is real and positive for all non-zero "complex" column vectors formula_65". This condition implies that formula_1 is Hermitian (i.e. its transpose is equal to its conjugate). To see this, consider the matrices formula_67 and formula_68, so that formula_69 and formula_70. The matrices formula_71 and formula_11 are Hermitian, therefore formula_73 and formula_74 are individually real. If formula_64 is real, then formula_74 must be zero for all formula_65. Then formula_11 is the zero matrix and formula_79, proving that formula_1 is Hermitian.
By this definition, a positive-definite "real" matrix formula_1 is Hermitian, hence symmetric; and formula_82 is positive for all non-zero "real" column vectors formula_83. However the last condition alone is not sufficient for formula_1 to be positive-definite. For example, if
then for any real vector formula_83 with entries formula_87 and formula_88 we have formula_89, which is always positive if formula_65 is not zero. However, if formula_65 is the complex vector with entries formula_92 and formula_93, one gets
which is not real. Therefore, formula_1 is not positive-definite.
On the other hand, for a "symmetric" real matrix formula_1, the condition "formula_97 for all nonzero real vectors formula_65" "does" imply that formula_1 is positive-definite in the complex sense.
Notation.
If a Hermitian matrix formula_1 is positive semi-definite, one sometimes writes formula_101 and if formula_1 is positive-definite one writes formula_103. To denote that formula_1 is negative semi-definite one writes formula_105 and to denote that formula_1 is negative-definite one writes formula_107.
The notion comes from functional analysis where positive semidefinite matrices define positive operators.
A common alternative notation is formula_108, formula_109, formula_110 and formula_111 for positive semi-definite and positive-definite, negative semi-definite and negative-definite matrices, respectively. This may be confusing, as sometimes nonnegative matrices (respectively, nonpositive matrices) are also denoted in this way.
Eigenvalues.
Let formula_1 be an formula_19 Hermitian matrix. This implies all its eigenvalues are real.
Let formula_119 be an eigendecomposition of formula_1, where formula_121 is a unitary complex matrix whose columns comprise an orthonormal basis of eigenvectors of formula_1, and formula_123 is a "real" diagonal matrix whose main diagonal contains the corresponding eigenvalues. The matrix formula_1 may be regarded as a diagonal matrix formula_123 that has been re-expressed in coordinates of the (eigenvectors) basis formula_121. Put differently, applying "M" to some vector z in our coordinates system ("M"z), is the same as changing the basis of our z to the eigenvector coordinate system using "P"−1 ("P"−1z), applying the stretching transformation "D" to it ("DP"−1z), and then changing the basis back to our system using "P" ("PDP"−1z).
With this in mind, the one-to-one change of variable formula_127 shows that formula_64 is real and positive for any complex vector formula_65 if and only if formula_130 is real and positive for any formula_131; in other words, if formula_123 is positive definite. For a diagonal matrix, this is true only if each element of the main diagonal—that is, every eigenvalue of formula_1—is positive. Since the spectral theorem guarantees all eigenvalues of a Hermitian matrix to be real, the positivity of eigenvalues can be checked using Descartes' rule of alternating signs when the characteristic polynomial of a real, symmetric matrix formula_1 is available.
Decomposition.
Let formula_1 be an formula_19 Hermitian matrix.
formula_1 is positive semidefinite if and only if it can be decomposed as a product
of a matrix formula_11 with its conjugate transpose.
When formula_1 is real, formula_11 can be real as well and the decomposition can be written as
formula_1 is positive definite if and only if such a decomposition exists with formula_11 invertible.
More generally, formula_1 is positive semidefinite with rank formula_146 if and only if a decomposition exists with a formula_147 matrix formula_11 of full row rank (i.e. of rank formula_146).
Moreover, for any decomposition formula_138, formula_151.
If formula_138, then formula_153, so formula_1 is positive semidefinite.
If moreover formula_11 is invertible then the inequality is strict for formula_156, so formula_1 is positive definite.
If formula_11 is formula_147 of rank formula_146, then formula_161.
In the other direction, suppose formula_1 is positive semidefinite.
Since formula_1 is Hermitian, it has an eigendecomposition formula_164 where formula_165 is unitary and formula_123 is a diagonal matrix whose entries are the eigenvalues of formula_1
Since formula_1 is positive semidefinite, the eigenvalues are non-negative real numbers, so one can define formula_169 as the diagonal matrix whose entries are non-negative square roots of eigenvalues.
Then formula_170 for formula_171.
If moreover formula_1 is positive definite, then the eigenvalues are (strictly) positive, so formula_169 is invertible, and hence formula_171 is invertible as well.
If formula_1 has rank formula_146, then it has exactly formula_146 positive eigenvalues and the others are zero, hence in formula_171 all but formula_146 rows are all zeroed.
Cutting the zero rows gives a formula_180 matrix formula_181 such that formula_182.
The columns formula_183 of formula_11 can be seen as vectors in the complex or real vector space formula_185, respectively.
Then the entries of formula_1 are inner products (that is dot products, in the real case) of these vectors
In other words, a Hermitian matrix formula_1 is positive semidefinite if and only if it is the Gram matrix of some vectors formula_183.
It is positive definite if and only if it is the Gram matrix of some linearly independent vectors.
In general, the rank of the Gram matrix of vectors formula_183 equals the dimension of the space spanned by these vectors.
Uniqueness up to unitary transformations.
The decomposition is not unique:
if formula_138 for some formula_147 matrix formula_11 and if formula_165 is any unitary formula_195 matrix (meaning formula_196),
then formula_197 for formula_198.
However, this is the only way in which two decompositions can differ: the decomposition is unique up to unitary transformations.
More formally, if formula_71 is a formula_147 matrix and formula_11 is a formula_202 matrix such that formula_203,
then there is a formula_204 matrix formula_165 with orthonormal columns (meaning formula_206) such that formula_207.
When formula_208 this means formula_165 is unitary.
This statement has an intuitive geometric interpretation in the real case:
let the columns of formula_71 and formula_11 be the vectors formula_212 and formula_183 in formula_185.
A real unitary matrix is an orthogonal matrix, which describes a rigid transformation (an isometry of Euclidean space formula_185) preserving the 0 point (i.e. rotations and reflections, without translations).
Therefore, the dot products formula_216 and formula_217 are equal if and only if some rigid transformation of formula_185 transforms the vectors formula_212 to formula_183 (and 0 to 0).
Square root.
A matrix formula_1 is positive semidefinite if and only if there is a positive semidefinite matrix formula_11 (in particular formula_11 is Hermitian, so formula_224) satisfying formula_225. This matrix formula_11 is unique, is called the "non-negative square root" of formula_1, and is denoted with formula_228.
When formula_1 is positive definite, so is formula_230, hence it is also called the "positive square root" of formula_1.
The non-negative square root should not be confused with other decompositions formula_232.
Some authors use the name "square root" and formula_230 for any such decomposition, or specifically for the Cholesky decomposition,
or any decomposition of the form formula_225;
other only use it for the non-negative square root.
If formula_235 then formula_236.
Cholesky decomposition.
A positive semidefinite matrix formula_1 can be written as formula_238, where formula_239 is lower triangular with non-negative diagonal (equivalently formula_232 where formula_241 is upper triangular); this is the Cholesky decomposition.
If formula_1 is positive definite, then the diagonal of formula_239 is positive and the Cholesky decomposition is unique.
The Cholesky decomposition is especially useful for efficient numerical calculations.
A closely related decomposition is the LDL decomposition, formula_244, where formula_123 is diagonal and formula_239 is lower unitriangular.
Other characterizations.
Let formula_1 be an formula_19 Hermitian matrix. The following properties are equivalent to formula_1 being positive definite:
A positive semidefinite matrix is positive definite if and only if it is invertible.
A matrix formula_1 is negative (semi)definite if and only if formula_273 is positive (semi)definite.
Quadratic forms.
The (purely) quadratic form associated with a real formula_19 matrix formula_1 is the function formula_276 such that formula_277 for all formula_37. formula_1 can be assumed symmetric by replacing it with formula_280.
A symmetric matrix formula_1 is positive definite if and only if its quadratic form is a strictly convex function.
More generally, any quadratic function from formula_282 to formula_283 can be written as formula_284 where formula_1 is a symmetric formula_19 matrix, formula_88 is a real formula_288-vector, and formula_289 a real constant. This quadratic function is strictly convex, and hence has a unique finite global minimum, if and only if formula_1 is positive definite. For this reason, positive definite matrices play an important role in optimization problems.
Simultaneous diagonalization.
A symmetric matrix and another symmetric and positive definite matrix can be simultaneously diagonalized, although not necessarily via a similarity transformation. This result does not extend to the case of three or more matrices. In this section we write for the real case. Extension to the complex case is immediate.
Let formula_1 be a symmetric and formula_292 a symmetric and positive definite matrix. Write the generalized eigenvalue equation as formula_293 where we impose that formula_37 be normalized, i.e. formula_295. Now we use Cholesky decomposition to write the inverse of formula_292 as formula_297. Multiplying by formula_165 and letting formula_299, we get formula_300, which can be rewritten as formula_301 where formula_302. Manipulation now yields formula_303 where formula_304 is a matrix having as columns the generalized eigenvectors and formula_305 is a diagonal matrix of the generalized eigenvalues. Now premultiplication with formula_306 gives the final result: formula_307 and formula_308, but note that this is no longer an orthogonal diagonalization with respect to the inner product where formula_302. In fact, we diagonalized formula_1 with respect to the inner product induced by formula_292.
Note that this result does not contradict what is said on simultaneous diagonalization in the article Diagonalizable matrix, which refers to simultaneous diagonalization by a similarity transformation. Our result here is more akin to a simultaneous diagonalization of two quadratic forms, and is useful for optimization of one form under conditions on the other.
Properties.
Induced partial ordering.
For arbitrary square matrices formula_1, formula_292 we write formula_314 if formula_315 i.e., formula_316 is positive semi-definite. This defines a partial ordering on the set of all square matrices. One can similarly define a strict partial ordering formula_317. The ordering is called the Loewner order.
Inverse of positive definite matrix.
Every positive definite matrix is invertible and its inverse is also positive definite. If formula_318 then formula_319. Moreover, by the min-max theorem, the "k"th largest eigenvalue of formula_1 is greater than the "k"th largest eigenvalue of formula_292.
Scaling.
If formula_1 is positive definite and formula_323 is a real number, then formula_324 is positive definite.
Submatrices.
Every principal submatrix of a positive definite matrix is positive definite.
Trace.
The diagonal entries formula_346 of a positive-semidefinite matrix are real and non-negative. As a consequence the trace, formula_347. Furthermore, since every principal sub-matrix (in particular, 2-by-2) is positive semidefinite,
and thus, when formula_349,
An formula_19 Hermitian matrix formula_1 is positive definite if it satisfies the following trace inequalities:
Another important result is that for any formula_1 and formula_292 positive-semidefinite matrices, formula_356
Hadamard product.
If formula_357, although formula_339 is not necessary positive semidefinite, the Hadamard product is, formula_359 (this result is often called the Schur product theorem).
Regarding the Hadamard product of two positive semidefinite matrices formula_360, formula_361, there are two notable inequalities:
Kronecker product.
If formula_357, although formula_339 is not necessary positive semidefinite, the Kronecker product formula_366.
Frobenius product.
If formula_357, although formula_339 is not necessary positive semidefinite, the Frobenius product formula_369 (Lancaster–Tismenetsky, "The Theory of Matrices", p. 218).
Convexity.
The set of positive semidefinite symmetric matrices is convex. That is, if formula_1 and formula_292 are positive semidefinite, then for any formula_372 between 0 and 1, formula_373 is also positive semidefinite. For any vector formula_15:
This property guarantees that semidefinite programming problems converge to a globally optimal solution.
Relation with cosine.
The positive-definiteness of a matrix formula_71 expresses that the angle formula_377 between any vector formula_15 and its image formula_379 is always formula_380:
Further properties.
A Hermitian matrix is positive semidefinite if and only if all of its principal minors are nonnegative. It is however not enough to consider the leading principal minors only, as is checked on the diagonal matrix with entries 0 and −1.
Block matrices.
A positive formula_402 matrix may also be defined by blocks:
where each block is formula_19. By applying the positivity condition, it immediately follows that formula_71 and formula_123 are hermitian, and formula_407.
We have that formula_408 for all complex formula_65, and in particular for formula_410. Then
A similar argument can be applied to formula_123, and thus we conclude that both formula_71 and formula_123 must be positive definite matrices, as well.
Converse results can be proved with stronger conditions on the blocks, for instance using the Schur complement.
Local extrema.
A general quadratic form formula_415 on formula_288 real variables formula_417 can always be written as formula_418 where formula_22 is the column vector with those variables, and formula_1 is a symmetric real matrix. Therefore, the matrix being positive definite means that formula_421 has a unique minimum (zero) when formula_22 is zero, and is strictly positive for any other formula_22.
More generally, a twice-differentiable real function formula_421 on formula_288 real variables has local minimum at arguments formula_417 if its gradient is zero and its Hessian (the matrix of all second derivatives) is positive semi-definite at that point. Similar statements can be made for negative definite and semi-definite matrices.
Covariance.
In statistics, the covariance matrix of a multivariate probability distribution is always positive semi-definite; and it is positive definite unless one variable is an exact linear function of the others. Conversely, every positive semi-definite matrix is the covariance matrix of some multivariate distribution.
Extension for non-Hermitian square matrices.
The definition of positive definite can be generalized by designating any complex matrix formula_1 (e.g. real non-symmetric) as positive definite if formula_428 for all non-zero complex vectors formula_65, where formula_430 denotes the real part of a complex number formula_289. Only the Hermitian part formula_432 determines whether the matrix is positive definite, and is assessed in the narrower sense above. Similarly, if formula_15 and formula_1 are real, we have formula_435 for all real nonzero vectors formula_15 if and only if the symmetric part formula_437 is positive definite in the narrower sense. It is immediately clear that formula_438is insensitive to transposition of "M".
Consequently, a non-symmetric real matrix with only positive eigenvalues does not need to be positive definite. For example, the matrix formula_439 has positive eigenvalues yet is not positive definite; in particular a negative value of formula_440 is obtained with the choice formula_441 (which is the eigenvector associated with the negative eigenvalue of the symmetric part of
In summary, the distinguishing feature between the real and complex case is that, a bounded positive operator on a complex Hilbert space is necessarily Hermitian, or self adjoint. The general claim can be argued using the polarization identity. That is no longer true in the real case.
Applications.
Heat conductivity matrix.
Fourier's law of heat conduction, giving heat flux formula_442 in terms of the temperature gradient formula_443 is written for anisotropic media as formula_444, in which formula_445 is the symmetric thermal conductivity matrix. The negative is inserted in Fourier's law to reflect the expectation that heat will always flow from hot to cold. In other words, since the temperature gradient formula_446 always points from cold to hot, the heat flux formula_442 is expected to have a negative inner product with formula_446 so that formula_449. Substituting Fourier's law then gives this expectation as formula_450, implying that the conductivity matrix should be positive definite.


Bruhat decomposition

In mathematics, the Bruhat decomposition (introduced by François Bruhat for classical groups and by Claude Chevalley in general) "G" = "BWB" of certain algebraic groups "G" into cells can be regarded as a general expression of the principle of Gauss–Jordan elimination, which generically writes a matrix as a product of an upper triangular and lower triangular matrices—but with exceptional cases. It is related to the Schubert cell decomposition of Grassmannians: see Weyl group for this.
More generally, any group with a ("B", "N") pair has a Bruhat decomposition.
Definitions.
The Bruhat decomposition of "G" is the decomposition
of "G" as a disjoint union of double cosets of "B" parameterized by the elements of the Weyl group "W". (Note that although "W" is not in general a subgroup of "G", the coset "wB" is still well defined because the maximal torus is contained in "B".)
Examples.
Let "G" be the general linear group GL"n" of invertible formula_2 matrices with entries in some algebraically closed field, which is a reductive group. Then the Weyl group "W" is isomorphic to the symmetric group "S""n" on "n" letters, with permutation matrices as representatives. In this case, we can take "B" to be the subgroup of upper triangular invertible matrices, so Bruhat decomposition says that one can write any invertible matrix "A" as a product "U"1"PU"2 where "U"1 and "U"2 are upper triangular, and "P" is a permutation matrix. Writing this as "P" = "U"1−1"AU"2−1, this says that any invertible matrix can be transformed into a permutation matrix via a series of row and column operations, where we are only allowed to add row "i" (resp. column "i") to row "j" (resp. column "j") if "i" > "j" (resp. "i" < "j"). The row operations correspond to "U"1−1, and the column operations correspond to "U"2−1.
The special linear group SL"n" of invertible formula_2 matrices with determinant 1 is a semisimple group, and hence reductive. In this case, "W" is still isomorphic to the symmetric group "S""n". However, the determinant of a permutation matrix is the sign of the permutation, so to represent an odd permutation in SL"n", we can take one of the nonzero elements to be −1 instead of 1. Here "B" is the subgroup of upper triangular matrices with determinant 1, so the interpretation of Bruhat decomposition in this case is similar to the case of GL"n".
Geometry.
The cells in the Bruhat decomposition correspond to the Schubert cell decomposition of Grassmannians. The dimension of the cells corresponds to the length of the word "w" in the Weyl group. Poincaré duality constrains the topology of the cell decomposition, and thus the algebra of the Weyl group; for instance, the top dimensional cell is unique (it represents the fundamental class), and corresponds to the longest element of a Coxeter group.
Computations.
The number of cells in a given dimension of the Bruhat decomposition are the coefficients of the "q"-polynomial of the associated Dynkin diagram.
Double Bruhat Cells.
With two opposite Borels one may intersect the Bruhat cells for each of them.


Iwasawa theory

In number theory, Iwasawa theory is the study of objects of arithmetic interest over infinite towers of number fields. It began as a Galois module theory of ideal class groups, initiated by (), as part of the theory of cyclotomic fields. In the early 1970s, Barry Mazur considered generalizations of Iwasawa theory to abelian varieties. More recently (early 1990s), Ralph Greenberg has proposed an Iwasawa theory for motives.
Formulation.
Iwasawa worked with so-called formula_1-extensions: infinite extensions of a number field formula_2 with Galois group formula_3 isomorphic to the additive group of p-adic integers for some prime "p". Every closed subgroup of formula_4 is of the form formula_5 so by Galois theory, a formula_1-extension formula_7 is the same thing as a tower of fields
such that formula_9 Iwasawa studied classical Galois modules over formula_10 by asking questions about the structure of modules over formula_11
More generally, Iwasawa theory asks questions about the structure of Galois modules over extensions with Galois group a p-adic Lie group.
Example.
Let formula_12 be a prime number and let formula_13 be the field generated over formula_14 by the formula_12th roots of unity. Iwasawa considered the following tower of number fields:
where formula_17 is the field generated by adjoining to formula_18 the "p""n"+1-st roots of unity and
The fact that formula_20 implies, by infinite Galois theory, that formula_21 In order to get an interesting Galois module, Iwasawa took the ideal class group of formula_17, and let formula_23 be its "p"-torsion part. There are norm maps formula_24 whenever formula_25, and this gives us the data of an inverse system. If we set
then it is not hard to see from the inverse limit construction that formula_27 is a module over formula_28 In fact, formula_29 is a module over the Iwasawa algebra formula_30. This is a 2-dimensional, regular local ring, and this makes it possible to describe modules over it. From this description it is possible to recover information about the "p"-part of the class group of formula_31
The motivation here is that the "p"-torsion in the ideal class group of formula_18 had already been identified by Kummer as the main obstruction to the direct proof of Fermat's Last Theorem.
Connections with p-adic analysis.
From this beginning in the 1950s, a substantial theory has been built up. A fundamental connection was noticed between the module theory, and the p-adic L-functions that were defined in the 1960s by Kubota and Leopoldt. The latter begin from the Bernoulli numbers, and use interpolation to define p-adic analogues of the Dirichlet L-functions. It became clear that the theory had prospects of moving ahead finally from Kummer's century-old results on regular primes.
Iwasawa formulated the main conjecture of Iwasawa theory as an assertion that two methods of defining p-adic L-functions (by module theory, by interpolation) should coincide, as far as that was well-defined. This was proved by for formula_14 and for all totally real number fields by . These proofs were modeled upon Ken Ribet's proof of the converse to Herbrand's theorem (the so-called Herbrand–Ribet theorem).
Karl Rubin found a more elementary proof of the Mazur-Wiles theorem by using Kolyvagin's Euler systems, described in and , and later proved other generalizations of the main conjecture for imaginary quadratic fields.
Generalizations.
The Galois group of the infinite tower, the starting field, and the sort of arithmetic module studied can all be varied. In each case, there is a "main conjecture" linking the tower to a "p"-adic L-function.
In 2002, Christopher Skinner and Eric Urban claimed a proof of a "main conjecture" for GL(2). In 2010, they posted a preprint .


Full and faithful functors

In category theory, a faithful functor (respectively a full functor) is a functor that is injective (respectively surjective) when restricted to each set of morphisms that have a given source and target.
Formal definitions.
Explicitly, let "C" and "D" be (locally small) categories and let "F" : "C" → "D" be a functor from "C" to "D". The functor "F" induces a function
for every pair of objects "X" and "Y" in "C". The functor "F" is said to be
for each "X" and "Y" in "C".
Properties.
A faithful functor need not be injective on objects or morphisms. That is, two objects "X" and "X"′ may map to the same object in "D" (which is why the range of a full and faithful functor is not necessarily isomorphic to "C"), and two morphisms "f" : "X" → "Y" and "f"′ : "X"′ → "Y"′ (with different domains/codomains) may map to the same morphism in "D". Likewise, a full functor need not be surjective on objects or morphisms. There may be objects in "D" not of the form "FX" for some "X" in "C". Morphisms between such objects clearly cannot come from morphisms in "C".
A full and faithful functor is necessarily injective on objects up to isomorphism. That is, if "F" : "C" → "D" is a full and faithful functor and formula_2 then formula_3.
Generalization to (∞, 1)-categories.
The notion of a functor being 'full' or 'faithful' does not translate to the notion of a (∞, 1)-category. In an (∞, 1)-category, the maps between any two objects are given by a space only up to homotopy. Since the notion of injection and surjection are not homotopy invariant notions (consider an interval embedding into the real numbers vs. an interval mapping to a point), we do not have the notion of a functor being "full" or "faithful." However, we can define a functor of quasi-categories to be "fully faithful" if for every "X" and "Y" in "C," the map formula_4is a weak equivalence.


Algebraically closed group

In group theory, a group formula_1 is algebraically closed if any finite set of equations and inequations that "make sense" in formula_1 have a solution in formula_1 without needing a group extension. This notion will be made precise later in the article in .
Informal discussion.
Suppose we wished to find an element formula_4 of a group formula_5 satisfying the conditions (equations and inequations):
Then it is easy to see that this is impossible because the first two equations imply formula_9. In this case we say the set of conditions are inconsistent with formula_5. (In fact this set of conditions are inconsistent with any group whatsoever.)
Now suppose formula_5 is the group with the multiplication table:
Then the conditions:
have a solution in formula_5, namely formula_15.
However the conditions:
Do not have a solution in formula_5, as can easily be checked.
However if we extend the group formula_19 to the group formula_20 with multiplication table:
Then the conditions have two solutions, namely formula_21 and formula_22.
Thus there are three possibilities regarding such conditions:
It is reasonable to ask whether there are any groups formula_29 such that whenever a set of conditions like these have a solution at all, they have a solution in formula_29 itself? The answer turns out to be "yes", and we call such groups algebraically closed groups.
Formal definition.
We first need some preliminary ideas.
If formula_5 is a group and formula_32 is the free group on countably many generators, then by a finite set of equations and inequations with coefficients in formula_5 we mean a pair of subsets formula_34 and formula_35 of formula_36 the free product of formula_32 and formula_5.
This formalizes the notion of a set of equations and inequations consisting of variables formula_39 and elements formula_40 of formula_5. The set formula_34 represents equations like:
The set formula_35 represents inequations like
By a solution in formula_5 to this finite set of equations and inequations, we mean a homomorphism formula_50, such that formula_51 for all formula_52 and formula_53 for all formula_54, where formula_55 is the unique homomorphism formula_56 that equals formula_57 on formula_32 and is the identity on formula_5.
This formalizes the idea of substituting elements of formula_5 for the variables to get true identities and inidentities. In the example the substitutions formula_61 and formula_62 yield:
We say the finite set of equations and inequations is consistent with formula_5 if we can solve them in a "bigger" group formula_69. More formally:
The equations and inequations are consistent with formula_5 if there is a groupformula_69 and an embedding formula_72 such that the finite set of equations and inequations formula_73 and formula_74 has a solution in formula_69, where formula_76 is the unique homomorphism formula_77 that equals formula_78 on formula_5 and is the identity on formula_32.
Now we formally define the group formula_1 to be algebraically closed if every finite set of equations and inequations that has coefficients in formula_1 and is consistent with formula_1 has a solution in formula_1.
Known Results.
It is difficult to give concrete examples of algebraically closed groups as the following results indicate:
The proofs of these results are in general very complex. However, a sketch of the proof that a countable group formula_85 can be embedded in an algebraically closed group follows.
First we embed formula_85 in a countable group formula_87 with the property that every finite set of equations with coefficients in formula_85 that is consistent in formula_87 has a solution in formula_87 as follows:
There are only countably many finite sets of equations and inequations with coefficients in formula_85. Fix an enumeration formula_92 of them. Define groups formula_93 inductively by:
Now let:
Now iterate this construction to get a sequence of groups formula_97 and let:
Then formula_1 is a countable group containing formula_85. It is algebraically closed because any finite set of equations and inequations that is consistent with formula_1 must have coefficients in some formula_102 and so must have a solution in formula_103.


Paul Cohn

Paul Moritz Cohn FRS (8 January 1924 – 20 April 2006) was Astor Professor of Mathematics at University College London, 1986–1989, and author of many textbooks on algebra. His work was mostly in the area of algebra, especially non-commutative rings.
Ancestry and early life.
He was the only child of Jewish parents, James (or Jakob) Cohn, owner of an import business, and Julia ("née" Cohen), a schoolteacher.
Both of his parents were born in Hamburg, as were three of his grandparents. His ancestors came from various parts of Germany. His father fought in the German army in World War I; he was wounded several times and awarded the Iron Cross. A street in Hamburg is named in memory of his mother.
When he was born, his parents were living with his mother's mother in Isestraße. After her death in October 1925, the family moved to a rented flat in a new building in Lattenkamp, in the Winterhude quarter. He attended a kindergarten then, in April 1930, moved to Alsterdorfer Straße School. After a while, he had a new teacher, a National Socialist, who picked on him and punished him without cause. Thus in 1931, he moved to the Meerweinstraße School where his mother taught.
Following the rise of the Nazis in 1933, his father's business was confiscated and his mother dismissed. He moved to the Talmud-Tora-Schule, a Jewish school. In mid-1937, the family moved to Klosterallee. This was nearer the school, the synagogue and other pupils, being in the Jewish area. His German teacher was Dr. Ernst Loewenberg, the son of the poet Jakob Loewenberg.
On the night of 9/10 November 1938 (Kristallnacht), his father was arrested and sent to Sachsenhausen concentration camp. He was released after four months but told to emigrate. Cohn went to Britain in May 1939 on the Kindertransport to work on a chicken farm, and never saw his parents again. He corresponded regularly with them until late 1941. At the end of the War, he learned that they were deported to Riga on 6 December 1941 and never returned. At the end of 1941, the farm closed. He trained as a precision engineer, acquired a work permit and worked in a factory for 4½ years. He passed the Cambridge Scholarship Examination, and won an exhibition to Trinity College, Cambridge.
Career.
He received a B.A in Mathematics from Cambridge University in 1948 and a Ph.D. (supervised by Philip Hall) in 1951. He then spent a year as a Chargé de Recherches at the University of Nancy. On his return, he became a lecturer in mathematics at Manchester University. He was a visiting professor at Yale University in 1961–1962, and for part of 1962 was at the University of California at Berkeley. On his return, he became Reader at Queen Mary College. He was a visiting professor at the University of Chicago in 1964 and at the State University of New York at Stony Brook in 1967. By then, he was regarded as one of the world's leading algebraists.
Also in 1967, he became head of the Department of Mathematics at Bedford College. He held several visiting professorships, in America, Paris, Delhi, Canada, Haifa and Bielefeld. He was awarded the Lester R. Ford Award from the Mathematical Association of America in 1972 and the Senior Berwick Prize of the London Mathematical Society in 1974.
In the early 1980s, funding cuts caused the closure of the small colleges of the University of London. Cohn moved to University College in 1984, together with the two other experts at Bedford on ring theory, Bill Stephenson and Warren Dicks. He became Astor Professor of Mathematics there in 1986. He continued to be a visiting professor, for example to the University of Alberta in 1986 and to Bar Ilan University in 1987. He retired in 1989, but remained active as Professor Emeritus and Honorary Research Fellow until his death.
He was President of the London Mathematical Society, 1982–1984, having been its secretary, 1965–1967 and a Council member in 1968–1971, 1972–1975 and 1979–1982. He was editor of the Society's Monographs in 1968–1977 and 1980–1993. He was elected a Fellow of the Royal Society in 1980 and was on its council, 1985–1987. He was a member of the Mathematical Committee of the Science Research Council, 1977–1980. He chaired the National Committee for Mathematics, 1988–1989.
Mathematical work.
In all, Cohn wrote nearly 200 mathematical papers. He worked in many areas of algebra, mainly in non-commutative ring theory. His first papers, covering many topics, were published in 1952. He generalised a theorem due to Wilhelm Magnus, and worked on the structure of tensor spaces. In 1953 he published a joint paper with Kurt Mahler on pseudo-valuations and in 1954 he published a work on Lie algebras.
Papers over the next few years covered areas such as group theory, field theory, Lie rings, semigroups, Abelian groups and ring theory. He published his first book, "Lie groups", in 1957. After that, he moved into the areas of Jordan algebras, Lie division rings, skew fields, free ideal rings and non-commutative unique factorisation domains. He published his second book, "Linear equations", in 1958 and his third, "Solid geometry", in 1961. "Universal algebra" appeared in 1965 (second edition 1981). After that, he concentrated on non-commutative ring theory and the theory of algebras.
His monograph "Free rings and their relations" appeared in 1971. It covered the work of Cohn and others on free associative algebras and related classes of rings, especially free ideal rings. He included all of his own published results on the embedding of rings into skew fields. The second, enlarged edition appeared in 1985.
Cohn also wrote undergraduate textbooks. "Algebra" volume I appeared in 1974 and volume II in 1977. The second edition, in three volumes, was published by Wiley between 1982 and 1991. These volumes were in line with the British (rather than American) curricula at the time and include both linear algebra and abstract algebra. Cohn wrote a subsequent revised iteration the first volume as "Classical Algebra" (Wiley, 2000) as a more "user friendly" version for undergraduates (according to its preface); this book also includes a few selected topics from volumes II and III of "Algebra". The final incarnation of Cohn's algebra textbooks appeared in 2003 as two Springer volumes "Basic Algebra" and "Further Algebra and Applications"; the material in "Basic Algebra" is (according to its preface) rather more concise and while corresponding roughly with volume I of "Algebra" assumes knowledge of linear algebra; the material on basic theories (groups, rings, fields) is pursued in more depth in "Basic Algebra" compared to volume I of "Algebra". "Further Algebra and Applications" roughly corresponds to volumes II and III of "Algebra", but reflects the shift of some material from these volumes to "Basic Algebra".
Private life.
His recreation was etymology and language in all its forms. He married Deirdre Sharon in 1958, and they had two daughters.


Lower p-series


John H. Walter

John Harris Walter (born 14 December 1927, Los Angeles) is an American mathematician known for proving the Walter theorem in the theory of finite groups.
Walter received from California Institute of Technology his bachelor's degree in 1951. He received from the University of Michigan his master's degree in 1953 and his Ph.D. in 1954 with thesis "Automorphisms of the Projective Unitary Groups" under the supervision of Leonard Tornheim. Walter was a visiting professor in 1960/61 and 1965/66 at the University of Chicago, 1967/68 at Harvard University, and 1972/73 at the University of Cambridge, UK. He is a professor emeritus of mathematics at the University of Illinois at Urbana–Champaign, where he became an associate professor in 1961 and a full professor in 1966. In 2012 he was elected a fellow of the American Mathematical Society.


Set of uniqueness

In mathematics, a set of uniqueness is a concept relevant to trigonometric expansions which are not necessarily Fourier series. Their study is a relatively pure branch of harmonic analysis.
Definition.
A subset "E" of the circle is called a set of uniqueness, or a "U"-set, if any trigonometric expansion
which converges to zero for formula_2 is identically zero; that is, such that
Otherwise "E" is a set of multiplicity (sometimes called an "M"-set or a Menshov set). Analogous definitions apply on the real line, and in higher dimensions. In the latter case one needs to specify the order of summation, e.g. "a set of uniqueness with respect to summing over balls".
To understand the importance of the definition it is important to get out of the Fourier mind-set. In Fourier analysis there is no question of uniqueness, since the coefficients "c"("n") are derived by integrating the function. Hence in Fourier analysis the order of actions is
In the theory of uniqueness the order is different:
In effect, it is usually sufficiently interesting (as in the definition above) to assume that the sum converges to zero and ask if that means that all the "c"("n") must be zero. As is usual in analysis, the most interesting questions arise when one discusses pointwise convergence. Hence the definition above, which arose when it became clear that neither "convergence everywhere" nor "convergence almost everywhere" give a satisfactory answer.
Early research.
The empty set is a set of uniqueness. This is just a fancy way to say that if a trigonometric series converges to zero "everywhere" then it is trivial. This was proved by Riemann, using a delicate technique of double formal integration; and showing that the resulting sum has some generalized kind of second derivative using Toeplitz operators. Later on, Cantor generalized Riemann's techniques to show that any countable, closed set is a set of uniqueness, a discovery which led him to the development of set theory. Paul Cohen, another great innovator in set theory, started his career with a thesis on sets of uniqueness.
As the theory of Lebesgue integration developed, it was assumed that any set of zero measure would be a set of uniqueness — in one dimension the locality principle for Fourier series shows that any set of positive measure is a set of multiplicity (in higher dimensions this is still an open question). This was disproved by D. E. Menshov who in 1916 constructed an example of a set of multiplicity which has measure zero.
Transformations.
A translation and dilation of a set of uniqueness is a set of uniqueness. A union of a countable family of "closed" sets of uniqueness is a set of uniqueness. There exists an example of two sets of uniqueness whose union is not a set of uniqueness, but the sets in this example are not Borel. It is an open problem whether the union of any two Borel sets of uniqueness is a set of uniqueness.
Singular distributions.
A closed set is a set of uniqueness if and only if there exists a distribution "S" supported on the set (so in particular it must be singular) such that
(formula_5 here are the Fourier coefficients). In all early examples of sets of uniqueness the distribution in question was in fact a measure. In 1954, though, Ilya Piatetski-Shapiro constructed an example of a set of uniqueness which does not support any measure with Fourier coefficients tending to zero. In other words, the generalization of distribution is necessary.
Complexity of structure.
The first evidence that sets of uniqueness have complex structure came from the study of Cantor-like sets. Salem and Zygmund showed that a Cantor-like set with dissection ratio ξ is a set of uniqueness if and only if 1/ξ is a Pisot number, that is an algebraic integer with the property that all its conjugates (if any) are smaller than 1. This was the first demonstration that the property of being a set of uniqueness has to do with "arithmetic" properties and not just some concept of size (Nina Bari had proved the case of ξ rational -- the Cantor-like set is a set of uniqueness if and only if 1/ξ is an integer -- a few years earlier).
Since the 50s, much work has gone into formalizing this complexity. The family of sets of uniqueness, considered as a set inside the space of compact sets (see Hausdorff distance), was located inside the analytical hierarchy. A crucial part in this research is played by the "index" of the set, which is an ordinal between 1 and ω1, first defined by Pyatetskii-Shapiro. Nowadays the research of sets of uniqueness is just as much a branch of descriptive set theory as it is of harmonic analysis. See the Kechris-Louveau book referenced below.


James Milne (mathematician)

James S. Milne (born 10 October 1942 in Invercargill, New Zealand) is a New Zealand mathematician working in arithmetic geometry.
Life.
Milne attended the High School in Invercargill in New Zealand until 1959, and then studied at the University of Otago in Dunedin (B.A. 1964) and Harvard University (Masters 1966, Ph.D. 1967 under John Tate). From then to 1969 he was a lecturer at University College London. After that he was at the University of Michigan, as Assistant Professor (1969-1972), Associate Professor (1972-1977), Professor (1977-2000), and Professor Emeritus (since 2000). He has also been a visiting professor at King's College in London, at the Institut des hautes études scientifiques in Paris (1975, 1978), at the Mathematical Sciences Research Institute in Berkeley, California (1986–87), and the Institute for Advanced Study in Princeton, New Jersey (1976–77, 1982, 1988).
In his dissertation, entitled "The conjectures of Birch and Swinnerton-Dyer for constant abelian varieties over function fields," he proved the conjecture of Birch and Swinnerton–Dyer for constant abelian varieties over function fields in one variable over a finite field. He also gave the first examples of nonzero abelian varieties with finite Tate–Shafarevich group. He went on to study Shimura varieties (certain hermitian symmetric spaces, low-dimensional examples being modular curves) and motives.
His students include Piotr Blass, Michael Bester, Matthew DeLong, Pierre Giguere, William Hawkins Jr, Matthias Pfau, Victor Scharaschkin, Stefan Treatman, Anthony Vazzana, and Wafa Wei.
Milne is also an avid mountain climber.


Joseph J. Rotman

Joseph J. Rotman (May 26, 1934 – October 16, 2016) was a Professor of Mathematics at the University of Illinois at Urbana–Champaign and also a published author of 10 textbooks.
Rotman was born in Chicago. He did his undergraduate and graduate work at the University of Chicago, where he received his doctorate in 1959 with a thesis in abelian groups written under the direction of Irving Kaplansky. In 1959 he moved to the University of Illinois at Urbana–Champaign, where he spent the rest of his mathematical career. Rotman retired from UIUC in 2004. His research interests lay in the area of algebra, involving abelian groups, modules, homological algebra, and combinatorics.
Rotman was the Managing Editor of the "Proceedings of the American Mathematical Society" in 1972–1973. In 1985 he was the Annual Visiting Lecturer of the South African Mathematical Society.
A partial list of Rotman's publications includes:


Modular subgroup

In mathematics, in the field of group theory, a modular subgroup is a subgroup that is a modular element in the lattice of subgroups, where the meet operation is defined by the intersection and the join operation is defined by the subgroup generated by the union of subgroups.
By the modular property of groups, every quasinormal subgroup (that is, a subgroup that permutes with all subgroups) is modular. In particular, every normal subgroup is modular.


Character table

In group theory, a branch of abstract algebra, a character table is a two-dimensional table whose rows correspond to irreducible representations, and whose columns correspond to conjugacy classes of group elements. The entries consist of characters, the traces of the matrices representing group elements of the column's class in the given row's group representation. In chemistry, crystallography, and spectroscopy, character tables of point groups are used to classify "e.g." molecular vibrations according to their symmetry, and to predict whether a transition between two states is forbidden for symmetry reasons. Many university level textbooks on physical chemistry, quantum chemistry, spectroscopy and inorganic chemistry devote a chapter to the use of symmetry group character tables.
Definition and example.
The irreducible complex characters of a finite group form a character table which encodes much useful information about the group "G" in a compact form. Each row is labelled by an irreducible character and the entries in the row are the values of that character on any representative of the respective conjugacy class of "G" (because characters are class functions). The columns are labelled by (representatives of) the conjugacy classes of "G". It is customary to label the first row by the character of the trivial representation, which is the trivial action of on a 1-dimensional vector space by formula_1 for all formula_2. Each entry in the first row is therefore 1. Similarly, it is customary to label the first column by the identity. The entries of the first column are the values of the irreducible characters at the identity, the degrees of the irreducible characters. Characters of degree "1" are known as linear characters.
Here is the character table of "C"3 = "", the cyclic group with three elements and generator "u":
where ω is a primitive third root of unity. The character table for general cyclic groups is (a scalar multiple of) the DFT matrix.
Another example is the character table of formula_3:
where (12) represents conjugacy class consisting of (12),(13),(23), and (123) represents conjugacy class consisting of (123),(132). To learn more about character table of symmetric groups, see .
The first row of the character table always consists of 1s, and corresponds to the trivial representation (the 1-dimensional representation consisting of 1×1 matrices containing the entry 1). Further, the character table is always square because (1) irreducible characters are pairwise orthogonal, and (2) no other non-trivial class function is orthogonal to every character. (A class function is one that is constant on conjugacy classes.) This is tied to the important fact that the irreducible representations of a finite group "G" are in bijection with its conjugacy classes. This bijection also follows by showing that the class sums form a basis for the center of the group algebra of "G", which has dimension equal to the number of irreducible representations of "G".
Orthogonality relations.
The space of complex-valued class functions of a finite group "G" has a natural inner-product:
where formula_5 means the complex conjugate of the value of formula_6 on formula_7. With respect to this inner product, the irreducible characters form an orthonormal basis
for the space of class-functions, and this yields the orthogonality relation for the rows of the character
table:
For formula_9 the orthogonality relation for columns is as follows:
where the sum is over all of the irreducible characters formula_11 of "G" and the symbol formula_12 denotes the order of the centralizer of formula_7.
For an arbitrary character formula_11, it is irreducible if and only if formula_15.
The orthogonality relations can aid many computations including:
If the irreducible representation "V" is non-trivial, then formula_18.
More specifically, consider the regular representation which is the permutation obtained from a finite group "G" acting on itself. The characters of this representation are formula_19 and formula_20 for formula_7 not the identity. Then given an irreducible representation formula_22,
Then decomposing the regular representations as a sum of irreducible representations of "G", we get formula_24. From which we conclude
over all irreducible representations formula_22. This sum can help narrow down the dimensions of the irreducible representations in a character table. For example, if the group has order 10 and 4 conjugacy classes (for instance, the dihedral group of order 10) then the only way to express the order of the group as a sum of four squares is formula_27, so we know the dimensions of all the irreducible representations.
Properties.
Complex conjugation acts on the character table: since the complex conjugate of a representation is again a representation, the same is true for characters, and thus a character that takes on non-trivial complex values has a conjugate character.
Certain properties of the group "G" can be deduced from its character table:
The character table does not in general determine the group up to isomorphism: for example, the quaternion group "Q" and the dihedral group of 8 elements ("D"4) have the same character table. Brauer asked whether the character table, together with the knowledge of how the powers of elements of its conjugacy classes are distributed, determines a finite group up to isomorphism. In 1964, this was answered in the negative by E. C. Dade.
The linear representations of are themselves a group under the tensor product, since the tensor product of 1-dimensional vector spaces is again 1-dimensional. That is, if formula_29 and formula_30 are linear representations, then formula_31 defines a new linear representation. This gives rise to a group of linear characters, called the character group under the operation formula_32. This group is connected to Dirichlet characters and Fourier analysis.
Outer automorphisms.
The outer automorphism group acts on the character table by permuting columns (conjugacy classes) and accordingly rows, which gives another symmetry to the table. For example, abelian groups have the outer automorphism formula_33, which is non-trivial except for elementary abelian 2-groups, and outer because abelian groups are precisely those for which conjugation (inner automorphisms) acts trivially. In the example of formula_34 above, this map sends formula_35 and accordingly switches formula_36 and formula_37 (switching their values of formula_38 and formula_39). Note that this particular automorphism (negative in abelian groups) agrees with complex conjugation.
Formally, if formula_40 is an automorphism of "G" and formula_41 is a representation, then formula_42 is a representation. If formula_43 is an inner automorphism (conjugation by some element "a"), then it acts trivially on representations, because representations are class functions (conjugation does not change their value). Thus a given class of outer automorphisms, it acts on the characters – because inner automorphisms act trivially, the action of the automorphism group Aut descends to the quotient Out.
This relation can be used both ways: given an outer automorphism, one can produce new representations (if the representation is not equal on conjugacy classes that are interchanged by the outer automorphism), and conversely, one can restrict possible outer automorphisms based on the character table.


Power iteration

In mathematics, power iteration (also known as the power method) is an eigenvalue algorithm: given a diagonalizable matrix formula_1, the algorithm will produce a number formula_2, which is the greatest (in absolute value) eigenvalue of formula_1, and a nonzero vector formula_4, which is a corresponding eigenvector of formula_2, that is, formula_6.
The algorithm is also known as the Von Mises iteration.
Power iteration is a very simple algorithm, but it may converge slowly. The most time-consuming operation of the algorithm is the multiplication of matrix formula_1 by a vector, so it is effective for a very large sparse matrix with appropriate implementation.
The method.
The power iteration algorithm starts with a vector formula_8, which may be an approximation to the dominant eigenvector or a random vector. The method is described by the recurrence relation
So, at every iteration, the vector formula_10 is multiplied by the matrix formula_1 and normalized.
If we assume formula_1 has an eigenvalue that is strictly greater in magnitude than its other eigenvalues and the starting vector formula_8 has a nonzero component in the direction of an eigenvector associated with the dominant eigenvalue, then a subsequence formula_14 converges to an eigenvector associated with the dominant eigenvalue.
Without the two assumptions above, the sequence formula_14 does not necessarily converge. In this sequence,
where formula_17 is an eigenvector associated with the dominant eigenvalue, and formula_18. The presence of the term formula_19 implies that formula_20 does not converge unless formula_21. Under the two assumptions listed above, the sequence formula_22 defined by
converges to the dominant eigenvalue (with Rayleigh quotient).
One may compute this with the following algorithm (shown in Python with NumPy):
import numpy as np
def power_iteration(A, num_simulations: int):
 # Ideally choose a random vector
 # To decrease the chance that our vector
 # Is orthogonal to the eigenvector
 b_k = np.random.rand(A.shape[1])
 for _ in range(num_simulations):
 # calculate the matrix-by-vector product Ab
 b_k1 = np.dot(A, b_k)
 # calculate the norm
 b_k1_norm = np.linalg.norm(b_k1)
 # re normalize the vector
 b_k = b_k1 / b_k1_norm
 return b_k
power_iteration(np.array(0.5, 0.5], [0.2, 0.8), 10)
The vector formula_10 to an associated eigenvector. Ideally, one should use the Rayleigh quotient in order to get the associated eigenvalue.
This algorithm is used to calculate the "Google PageRank".
The method can also be used to calculate the spectral radius (the eigenvalue with the largest magnitude, for a square matrix) by computing the Rayleigh quotient
Analysis.
Let formula_1 be decomposed into its Jordan canonical form: formula_27, where the first column of formula_28 is an eigenvector of formula_1 corresponding to the dominant eigenvalue formula_30. Since the dominant eigenvalue of formula_1 is unique, the first Jordan block of formula_32 is the formula_33 matrix formula_34 where formula_30 is the largest eigenvalue of "A" in magnitude. The starting vector formula_8 can be written as a linear combination of the columns of "V":
By assumption, formula_38 has a nonzero component in the direction of the dominant eigenvalue, so formula_39.
The computationally useful recurrence relation for formula_40 can be rewritten as:
where the expression: formula_42 is more amenable to the following analysis.
The expression above simplifies as formula_44
The limit follows from the fact that the eigenvalue of formula_46 is less than 1 in magnitude, so
It follows that:
Using this fact, formula_49 can be written in a form that emphasizes its relationship with formula_50 when "k" is large:
where formula_52 and formula_53 as formula_44
The sequence formula_55 is bounded, so it contains a convergent subsequence. Note that the eigenvector corresponding to the dominant eigenvalue is only unique up to a scalar, so although the sequence formula_56 may not converge,
formula_49 is nearly an eigenvector of "A" for large "k".
Alternatively, if "A" is diagonalizable, then the following proof yields the same result
Let λ1, λ2, ..., λ"m" be the m eigenvalues (counted with multiplicity) of A and let "v"1, "v"2, ..., "v""m" be the corresponding eigenvectors. Suppose that formula_58 is the dominant eigenvalue, so that formula_59 for formula_60.
The initial vector formula_8 can be written:
If formula_8 is chosen randomly (with uniform probability), then "c"1 ≠ 0 with probability 1. Now,
On the other hand:
Therefore, formula_10 converges to (a multiple of) the eigenvector formula_17. The convergence is geometric, with ratio
where formula_69 denotes the second dominant eigenvalue. Thus, the method converges slowly if there is an eigenvalue close in magnitude to the dominant eigenvalue.
Applications.
Although the power iteration method approximates only one eigenvalue of a matrix, it remains useful for certain computational problems. For instance, Google uses it to calculate the PageRank of documents in their search engine, and Twitter uses it to show users recommendations of whom to follow. The power iteration method is especially suitable for sparse matrices, such as the web matrix, or as the matrix-free method that does not require storing the coefficient matrix formula_1 explicitly, but can instead access a function evaluating matrix-vector products formula_71. For non-symmetric matrices that are well-conditioned the power iteration method can outperform more complex Arnoldi iteration. For symmetric matrices, the power iteration method is rarely used, since its convergence speed can be easily increased without sacrificing the small cost per iteration; see, e.g., Lanczos iteration and LOBPCG.
Some of the more advanced eigenvalue algorithms can be understood as variations of the power iteration. For instance, the inverse iteration method applies power iteration to the matrix formula_72. Other algorithms look at the whole subspace generated by the vectors formula_10. This subspace is known as the Krylov subspace. It can be computed by Arnoldi iteration or Lanczos iteration.


Commutative algebra

Commutative algebra is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings include polynomial rings; rings of algebraic integers, including the ordinary integers formula_1; and "p"-adic integers.
Commutative algebra is the main technical tool in the local study of schemes.
The study of rings that are not necessarily commutative is known as noncommutative algebra; it includes ring theory, representation theory, and the theory of Banach algebras.
Overview.
Commutative algebra is essentially the study of the rings occurring in algebraic number theory and algebraic geometry.
In algebraic number theory, the rings of algebraic integers are Dedekind rings, which constitute therefore an important class of commutative rings. Considerations related to modular arithmetic have led to the notion of a valuation ring. The restriction of algebraic field extensions to subrings has led to the notions of integral extensions and integrally closed domains as well as the notion of ramification of an extension of valuation rings.
The notion of localization of a ring (in particular the localization with respect to a prime ideal, the localization consisting in inverting a single element and the total quotient ring) is one of the main differences between commutative algebra and the theory of non-commutative rings. It leads to an important class of commutative rings, the local rings that have only one maximal ideal. The set of the prime ideals of a commutative ring is naturally equipped with a topology, the Zariski topology. All these notions are widely used in algebraic geometry and are the basic technical tools for the definition of scheme theory, a generalization of algebraic geometry introduced by Grothendieck.
Many other notions of commutative algebra are counterparts of geometrical notions occurring in algebraic geometry. This is the case of Krull dimension, primary decomposition, regular rings, Cohen–Macaulay rings, Gorenstein rings and many other notions.
History.
The subject, first known as ideal theory, began with Richard Dedekind's work on ideals, itself based on the earlier work of Ernst Kummer and Leopold Kronecker. Later, David Hilbert introduced the term "ring" to generalize the earlier term "number ring". Hilbert introduced a more abstract approach to replace the more concrete and computationally oriented methods grounded in such things as complex analysis and classical invariant theory. In turn, Hilbert strongly influenced Emmy Noether, who recast many earlier results in terms of an ascending chain condition, now known as the Noetherian condition. Another important milestone was the work of Hilbert's student Emanuel Lasker, who introduced primary ideals and proved the first version of the Lasker–Noether theorem.
The main figure responsible for the birth of commutative algebra as a mature subject was Wolfgang Krull, who introduced the fundamental notions of localization and completion of a ring, as well as that of regular local rings. He established the concept of the Krull dimension of a ring, first for Noetherian rings before moving on to expand his theory to cover general valuation rings and Krull rings. To this day, Krull's principal ideal theorem is widely considered the single most important foundational theorem in commutative algebra. These results paved the way for the introduction of commutative algebra into algebraic geometry, an idea which would revolutionize the latter subject.
Much of the modern development of commutative algebra emphasizes modules. Both ideals of a ring "R" and "R"-algebras are special cases of "R"-modules, so module theory encompasses both ideal theory and the theory of ring extensions. Though it was already incipient in Kronecker's work, the modern approach to commutative algebra using module theory is usually credited to Krull and Noether.
Main tools and results.
Noetherian rings.
In mathematics, more specifically in the area of modern algebra known as ring theory, a Noetherian ring, named after Emmy Noether, is a ring in which every non-empty set of ideals has a maximal element. Equivalently, a ring is Noetherian if it satisfies the ascending chain condition on ideals; that is, given any chain:
there exists an "n" such that:
For a commutative ring to be Noetherian it suffices that every prime ideal of the ring is finitely generated. (The result is due to I. S. Cohen.)
The notion of a Noetherian ring is of fundamental importance in both commutative and noncommutative ring theory, due to the role it plays in simplifying the ideal structure of a ring. For instance, the ring of integers and the polynomial ring over a field are both Noetherian rings, and consequently, such theorems as the Lasker–Noether theorem, the Krull intersection theorem, and the Hilbert's basis theorem hold for them. Furthermore, if a ring is Noetherian, then it satisfies the descending chain condition on "prime ideals". This property suggests a deep theory of dimension for Noetherian rings beginning with the notion of the Krull dimension.
Hilbert's basis theorem.
Theorem. If "R" is a left (resp. right) Noetherian ring, then the polynomial ring "R"["X"] is also a left (resp. right) Noetherian ring.
Hilbert's basis theorem has some immediate corollaries:
Primary decomposition.
An ideal "Q" of a ring is said to be "primary" if "Q" is proper and whenever "xy" ∈ "Q", either "x" ∈ "Q" or "yn" ∈ "Q" for some positive integer "n". In Z, the primary ideals are precisely the ideals of the form ("pe") where "p" is prime and "e" is a positive integer. Thus, a primary decomposition of ("n") corresponds to representing ("n") as the intersection of finitely many primary ideals.
The "Lasker–Noether theorem", given here, may be seen as a certain generalization of the fundamental theorem of arithmetic:
Lasker-Noether Theorem. Let "R" be a commutative Noetherian ring and let "I" be an ideal of "R". Then "I" may be written as the intersection of finitely many primary ideals with distinct radicals; that is:
with "Qi" primary for all "i" and Rad("Qi") ≠ Rad("Qj") for "i" ≠ "j". Furthermore, if:
is decomposition of "I" with Rad("Pi") ≠ Rad("Pj") for "i" ≠ "j", and both decompositions of "I" are "irredundant" (meaning that no proper subset of either {"Q"1, ..., "Qt"} or {"P"1, ..., "Pk"} yields an intersection equal to "I"), "t" = "k" and (after possibly renumbering the "Qi") Rad("Qi") = Rad("Pi") for all "i".
For any primary decomposition of "I", the set of all radicals, that is, the set {Rad("Q"1), ..., Rad("Qt")} remains the same by the Lasker–Noether theorem. In fact, it turns out that (for a Noetherian ring) the set is precisely the assassinator of the module "R"/"I"; that is, the set of all annihilators of "R"/"I" (viewed as a module over "R") that are prime.
Localization.
The localization is a formal way to introduce the "denominators" to a given ring or a module. That is, it introduces a new ring/module out of an existing one so that it consists of fractions
where the denominators "s" range in a given subset "S" of "R". The archetypal example is the construction of the ring Q of rational numbers from the ring Z of integers.
Completion.
A completion is any of several related functors on rings and modules that result in complete topological rings and modules. Completion is similar to localization, and together they are among the most basic tools in analysing commutative rings. Complete commutative rings have simpler structure than the general ones and Hensel's lemma applies to them.
Zariski topology on prime ideals.
The Zariski topology defines a topology on the spectrum of a ring (the set of prime ideals). In this formulation, the Zariski-closed sets are taken to be the sets
where "A" is a fixed commutative ring and "I" is an ideal. This is defined in analogy with the classical Zariski topology, where closed sets in affine space are those defined by polynomial equations . To see the connection with the classical picture, note that for any set "S" of polynomials (over an algebraically closed field), it follows from Hilbert's Nullstellensatz that the points of "V"("S") (in the old sense) are exactly the tuples ("a1", ..., "an") such that ("x1" - "a1", ..., "xn" - "an") contains "S"; moreover, these are maximal ideals and by the "weak" Nullstellensatz, an ideal of any affine coordinate ring is maximal if and only if it is of this form. Thus, "V"("S") is "the same as" the maximal ideals containing "S". Grothendieck's innovation in defining Spec was to replace maximal ideals with all prime ideals; in this formulation it is natural to simply generalize this observation to the definition of a closed set in the spectrum of a ring.
Examples.
The fundamental example in commutative algebra is the ring of integers formula_1. The existence of primes and
the unique factorization theorem laid the foundations for concepts such as Noetherian rings and the primary decomposition.
Other important examples are:
Connections with algebraic geometry.
Commutative algebra (in the form of polynomial rings and their quotients, used in the definition of algebraic varieties) has always been a part of algebraic geometry. However, in the late 1950s, algebraic varieties were subsumed into Alexander Grothendieck's concept of a scheme. Their local objects are affine schemes or prime spectra, which are locally ringed spaces, which form a category that is antiequivalent (dual) to the category of commutative unital rings, extending the duality between the category of affine algebraic varieties over a field "k", and the category of finitely generated reduced "k"-algebras. The gluing is along the Zariski topology; one can glue within the category of locally ringed spaces, but also, using the Yoneda embedding, within the more abstract category of presheaves of sets over the category of affine schemes. The Zariski topology in the set-theoretic sense is then replaced by a Zariski topology in the sense of Grothendieck topology. Grothendieck introduced Grothendieck topologies having in mind more exotic but geometrically finer and more sensitive examples than the crude Zariski topology, namely the étale topology, and the two flat Grothendieck topologies: fppf and fpqc. Nowadays some other examples have become prominent, including the Nisnevich topology. Sheaves can be furthermore generalized to stacks in the sense of Grothendieck, usually with some additional representability conditions, leading to Artin stacks and, even finer, Deligne–Mumford stacks, both often called algebraic stacks.


OpenBLAS

In scientific computing, OpenBLAS is an open-source implementation of the BLAS (Basic Linear Algebra Subprograms) and LAPACK APIs with many hand-crafted optimizations for specific processor types. It is developed at the Lab of Parallel Software and Computational Science, ISCAS.
OpenBLAS adds optimized implementations of linear algebra kernels for several processor architectures, including Intel Sandy Bridge
and Loongson. It claims to achieve performance comparable to the Intel MKL: this mostly holds true on the BLAS part, while the LAPACK part falls behind.
OpenBLAS is a fork of GotoBLAS2, which was created by Kazushige Goto at the Texas Advanced Computing Center.


Algebraic independence

In abstract algebra, a subset formula_1 of a field formula_2 is algebraically independent over a subfield formula_3 if the elements of formula_1 do not satisfy any non-trivial polynomial equation with coefficients in formula_3.
In particular, a one element set formula_6 is algebraically independent over formula_3 if and only if formula_8 is transcendental over formula_3. In general, all the elements of an algebraically independent set formula_1 over formula_3 are by necessity transcendental over formula_3, and over all of the field extensions over formula_3 generated by the remaining elements of formula_1.
Example.
The two real numbers formula_15 and formula_16 are each transcendental numbers: they are not the roots of any nontrivial polynomial whose coefficients are rational numbers. Thus, each of the two singleton sets formula_17 and formula_18 are algebraically independent over the field formula_19 of rational numbers.
However, the set formula_20 is "not" algebraically independent over the rational numbers, because the nontrivial polynomial
is zero when formula_22 and formula_23.
Algebraic independence of known constants.
Although both formula_24 and "e" are known to be transcendental,
it is not known whether the set of both of them is algebraically independent over formula_19. In fact, it is not even known if formula_26 is irrational.
Nesterenko proved in 1996 that:
Lindemann–Weierstrass theorem.
The Lindemann–Weierstrass theorem can often be used to prove that some sets are algebraically independent over formula_19. It states that whenever formula_38 are algebraic numbers that are linearly independent over formula_19, then formula_40 are also algebraically independent over formula_19.
Algebraic matroids.
Given a field extension formula_42 which is not algebraic, Zorn's lemma can be used to show that there always exists a maximal algebraically independent subset of formula_2 over formula_3. Further, all the maximal algebraically independent subsets have the same cardinality, known as the transcendence degree of the extension.
For every set formula_1 of elements of formula_2, the algebraically independent subsets of formula_1 satisfy the axioms that define the independent sets of a matroid. In this matroid, the rank of a set of elements is its transcendence degree, and the flat generated by a set formula_48 of elements is the intersection of formula_2 with the field formula_50. A matroid that can be generated in this way is called an algebraic matroid. No good characterization of algebraic matroids is known, but certain matroids are known to be non-algebraic; the smallest is the Vámos matroid.
Many finite matroids may be represented by a matrix over a field formula_3, in which the matroid elements correspond to matrix columns, and a set of elements is independent if the corresponding set of columns is linearly independent. Every matroid with a linear representation of this type may also be represented as an algebraic matroid, by choosing an indeterminate for each row of the matrix, and by using the matrix coefficients within each column to assign each matroid element a linear combination of these transcendentals. The converse is false: not every algebraic matroid has a linear representation.


Mark Ronan

Mark Andrew Ronan (born 1947) is Emeritus Professor of Mathematics at the University of Illinois at Chicago and Honorary Professor of Mathematics at University College London. He has lived and taught in: Germany (at the University of Braunschweig and the Free University of Berlin); in England, where from 1989 to 1992 he was Mason Professor of Mathematics at the University of Birmingham; and America at the University of Illinois at Chicago, where his teaching included courses on ancient literature from Mesopotamia, and on the history of the calendar, as well as mathematics.
In addition to his research papers, Ronan's popular account of the quest to discover and classify all the finite building blocks for symmetry (the finite simple groups) was published in 2006. In 2008 it formed the basis for a series of BBC radio broadcasts. In his research work he is an expert on the theory of buildings, with a standard text on the subject, now being reissued in paperback. Apart from his professional work, he has acted in many operas at the Lyric Opera of Chicago, danced in the Nutcracker, and maintains a blog on opera, ballet and theatre reviews.
Bibliography.
"Lectures on Buildings", original edition, Academic Press 1989; paperback edition, updated and revised, University of Chicago Press 2009.
"Symmetry and the Monster", Oxford University Press 2006.


Hom functor

In mathematics, specifically in category theory, hom-sets, i.e. sets of morphisms between objects, give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applications in category theory and other branches of mathematics.
Formal definition.
Let "C" be a locally small category (i.e. a category for which hom-classes are actually sets and not proper classes).
For all objects "A" and "B" in "C" we define two functors to the category of sets as follows:
The functor Hom(–,"B") is also called the "functor of points" of the object "B".
Note that fixing the first argument of Hom naturally gives rise to a covariant functor and fixing the second argument naturally gives a contravariant functor. This is an artifact of the way in which one must compose the morphisms.
The pair of functors Hom("A",–) and Hom(–,"B") are related in a natural manner. For any pair of morphisms "f" : "B" → "B"′ and "h" : "A"′ → A the following diagram commutes:
Both paths send "g" : "A" → "B" to "f"∘"g"∘"h" : "A"′ → "B"′.
The commutativity of the above diagram implies that Hom(–,–) is a bifunctor from "C" × "C" to Set which is contravariant in the first argument and covariant in the second. Equivalently, we may say that Hom(–,–) is a covariant bifunctor
where "C"op is the opposite category to "C". The notation HomC(–,–) is sometimes used for Hom(–,–) in order to emphasize the category forming the domain.
Yoneda's lemma.
Referring to the above commutative diagram, one observes that every morphism
gives rise to a natural transformation
and every morphism
gives rise to a natural transformation
Yoneda's lemma implies that "every" natural transformation between Hom functors is of this form. In other words, the Hom functors give rise to a full and faithful embedding of the category "C" into the functor category Set"C""op" (covariant or contravariant depending on which Hom functor is used).
Internal Hom functor.
Some categories may possess a functor that behaves like a Hom functor, but takes values in the category "C" itself, rather than Set. Such a functor is referred to as the internal Hom functor, and is often written as
to emphasize its product-like nature, or as
to emphasize its functorial nature, or sometimes merely in lower-case:
Categories that possess an internal Hom functor are referred to as closed categories. One has that
where "I" is the unit object of the closed category. For the case of a closed monoidal category, this extends to the notion of currying, namely, that
where formula_6 is a bifunctor, the internal product functor defining a monoidal category. The isomorphism is natural in both "X" and "Z". In other words, in a closed monoidal category, the internal Hom functor is an adjoint functor to the internal product functor. The object formula_7 is called the internal Hom. When formula_6 is the Cartesian product formula_9, the object formula_7 is called the exponential object, and is often written as formula_11.
Internal Homs, when chained together, form a language, called the internal language of the category. The most famous of these are simply typed lambda calculus, which is the internal language of Cartesian closed categories, and the linear type system, which is the internal language of closed symmetric monoidal categories.
Properties.
Note that a functor of the form
is a presheaf; likewise, Hom(A, –) is a copresheaf.
A functor "F" : "C" → Set that is naturally isomorphic to Hom(A, –) for some A in "C", is called a representable functor (or representable copresheaf); likewise, a contravariant functor equivalent to Hom(–, A) might be called corepresentable.
Note that Hom(–, –) : "C"op × "C" → Set is a profunctor, and, specifically, it is the identity profunctor formula_12.
The internal hom functor preserves limits; that is, formula_13 sends limits to limits, while formula_14 sends limits in formula_15, that is colimits formula_16, into limits. In a certain sense, this can be taken as the definition of a limit or colimit.
Other properties.
If A is an abelian category and "A" is an object of A, then HomA("A",–) is a covariant left-exact functor from A to the category Ab of abelian groups. It is exact if and only if "A" is projective.
Let "R" be a ring and "M" a left "R"-module. The functor HomR("M",–): Mod-"R" → Ab is right adjoint to the tensor product functor – formula_6R M: Ab → Mod-"R".


Edge and vertex spaces

In the mathematical discipline of graph theory, the edge space and vertex space of an undirected graph are vector spaces defined in terms of the edge and vertex sets, respectively. These vector spaces make it possible to use techniques of linear algebra in studying the graph.
Definition.
Let formula_1 be a finite undirected graph. The vertex space formula_2 of "G" is the vector space over the finite field of two elements
formula_3 of all functions formula_4. Every element of formula_2 naturally corresponds the subset of "V" which assigns a 1 to its vertices. Also every subset of "V" is uniquely represented in formula_2 by its characteristic function. The edge space formula_7 is the formula_8-vector space freely generated by the edge set "E". The dimension of the vertex space is thus the number of vertices of the graph, while the dimension of the edge space is the number of edges.
These definitions can be made more explicit. For example, we can describe the edge space as follows:
The singleton subsets of "E" form a basis for formula_7.
One can also think of formula_2 as the power set of "V" made into a vector space with similar vector addition and scalar multiplication as defined for formula_7.
Properties.
The incidence matrix formula_17 for a graph formula_18 defines one possible linear transformation
between the edge space and the vertex space of formula_18. The incidence matrix of formula_18, as a linear transformation, maps each edge to its two incident vertices. Let formula_22 be the edge between formula_23 and formula_24 then
The cycle space and the cut space are linear subspaces of the edge space.


Tracy-Singh product


Gould polynomials

In mathematics the Gould polynomials "G""n"("x"; "a","b") are polynomials introduced by H. W. Gould and named by Roman in 1984.
They are given by
where