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Trace

Trace may refer to

• Trace on a field extension
• Trace of a square matrix
• Trace on a C*-algebra
• Trace of a trace-class operator
• An element of a trace monoid
• Reduced trace on a central simple algebra

Trace monoid

Let $A$ be an alphabet with an irreflexive symmetric relation $I$ called independence. The complementary relation $I = A \times A \setminus I$ is the "dependence" relation. Such an alphabet is a concurrence or dependency alphabet. The free monoid on $A$ modulo the relations $ab=ba$ when $a,b \in I$ is the trace monoid on $(A,D)$. The elements of a trace monoid are "traces" and the subsets are the "trace languages".

Trace monoids are used to model concurrency in computer languages.

References

• Diekert, Volker; Rozenberg, Grzegorz (edd) "The Book Of Traces" (World Scientific, 1995) ISBN 981-02-2058-8

Trace-class operator

An operator $T$ on a Hilbert space $H$ with complete orthonormal set $(e_n)$ for which the sum $\sum_n \langle Tx_n , x_n \rangle$ is finite. For such operators, the trace is defined to be the value of this sum. The set of trace-class operators on $H$ coincides with the set of squares of Hilbert-Schmidt operators. The trace-class operators are precisely the Schatten class for $p=1$.

References

• Retherford, J. R. "Hilbert space: Compact operators and the trace theorem" London Mathematical Society Student Texts 27. (Cambridge University Press, 1993) ISBN 0-521-42933-1. Zbl 0783.47031

Schatten class

Schatten ideal

A class of operators on a Hilbert space. Let $T$ be an operator with singular values $\sigma_n$. For $1 \le p < \infty$ we say that $T$ is in the Schatten $p$-class if the sequence $(\sigma_n)$ is in $\ell_p$: that is, if $\sum_n |\sigma_n|^p$ converges, and then the $p$-root of the value is the Schatten $p$-norm of $T$. The Schatten classes form ideals of the operator algebra.

The Schatten $2$-class is precisely the Hilbert–Schmidt operators. The Schatten $1$-class is the trace-class operators.

References

• Retherford, J. R. "Hilbert space: Compact operators and the trace theorem" London Mathematical Society Student Texts 27. (Cambridge University Press, 1993) ISBN 0-521-42933-1. Zbl 0783.47031
• Schatten, Robert. "Norm ideals of completely continuous operators" Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue Folge, 27 . (Springer-Verlag, 1960) ISBN 0090.09402

Singular value

The singular values of a complex matrix $A$ are the eigenvalues of $A^*A$, or equivalently of $AA^*$. The singular value decomposition of $A$ is the expression $A=U\Sigma V$, with $U$ a unitary $(m\times n)$-matrix, $V$ a unitary $(n\times n)$-matrix and $\Sigma$ of the form $$ \Sigma = \begin{pmatrix} {\mathcal D} & 0\\ 0 & 0\end{pmatrix}, $$ where ${\mathcal D}$ is diagonal with entries the singular values $s_1,\dots,s_k$ of $A$ and $k$ the rank of $A$.

In the case of a closed operator $A$ on a Hilbert space, then $A^*$ is a positive operator and the singular values of $A$ are the spectrum of $A^*A$.

Span

Span may refer to

• Linear hull, also called linear span or span
• Span (category theory)

Span (category theory)

A diagram in a category of the form $$ \begin{array}{ccccc} & & C & & \\ & f \swarrow & & \searrow g & \\ A & & & & B \end{array} $$

Two spans with arrows $(f,g)$ and $(f',g')$ are equivalent if for all $D,p,q$ the diagrams $$ \begin{array}{ccccc} & & C & & \\ & f \swarrow & & \searrow g & \\ A & & & & B \\ & p \searrow & & \swarrow q \\ & & D & & \\ \end{array} \ \ \text{and}\ \ \begin{array}{ccccc} & & C & & \\ & f' \swarrow & & \searrow g' & \\ A & & & & B \\ & p \searrow & & \swarrow q \\ & & D & & \\ \end{array} $$ either both commute or both do not commute.

A pushout is the colimit of a span.

References

[1]	S. MacLane, "Categories for the working mathematician" , Springer (1971). ISBN 0-387-98403-8

Standard construction

A concept in category theory. Other names are triple, monad and functor-algebra.

Let $\mathfrak{S}$ be a category. A standard construction is a functor $T:\mathfrak{S} \rightarrow \mathfrak{S}$ equipped with natural transformations $\eta:1\rightarrow T$ and $\mu:T^2\rightarrow T$ such that the following diagrams commute: $$ \begin{array}{ccc} T^3 Y & \stackrel{T\mu_Y}{\rightarrow} & T^2 Y \\ \mu_{TY}\downarrow& & \downarrow_Y \\ T^2 & \stackrel{T_y}{\rightarrow} & Y \end{array} $$ $$ \begin{array}{ccccc} TY & \stackrel{TY}{\rightarrow} & T^2Y & \stackrel{T_{\eta Y}}{\leftarrow} & TY \\ & 1\searrow & \downarrow\mu Y & \swarrow1 & \\ & & Y & & \\ \end{array} $$

The basic use of standard constructions in topology is in the construction of various classifying spaces and their algebraic analogues, the so-called bar-constructions.

References

[1]	J.M. Boardman, R.M. Vogt, "Homotopy invariant algebraic structures on topological spaces" , Springer (1973)
[2]	J.F. Adams, "Infinite loop spaces" , Princeton Univ. Press (1978)
[3]	J.P. May, "The geometry of iterated loop spaces" , Lect. notes in math. , 271 , Springer (1972)
[4]	S. MacLane, "Categories for the working mathematician" , Springer (1971)

Comments

The term "standard construction" was introduced by R. Godement [a1] for want of a better name for this concept. It is now entirely obsolete, having been generally superseded by "monad" (although a minority of authors still use the term "triple" ). Monads have many other uses besides the one mentioned above, for example in the categorical approach to universal algebra (see [a2], [a3]).

References

[a1]	R. Godement, "Théorie des faisceaux" , Hermann (1958)
[a2]	E.G. Manes, "Algebraic theories" , Springer (1976)
[a3]	M. Barr, C. Wells, "Toposes, triples and theories" , Springer (1985)

Ostrowski representation

MSC 11A67

Let $[a_1,a_2,\ldots]$ be the partial quotients of an infinite continued fraction and $(c_n)$ the corresponding continuants, $c_0 = 1$, $c_1 = a_1$ and $c_{n+1} = a_{n+1} c_n + c_{n-1}$. An Ostrowski representation of $N$ is $$ N = \sum_{k=0}^n x_{k+1} c_k $$ where $0 \le x_k \le a_k$ and if $x_k = a_k$ then $x_{k-1} = 0$. Every positive integer $M$ has a unique Ostrowski representation.

When the $a_n$ are all equal to $1$, the $c_n$ are the Fibonacci numbers, and the Ostrowski representation is just the Zeckendorf representation. The addition of two $n$-digit numbers in Ostrowski representation based on a given continued fraction can be computed by three linear passes over the input.

References

• Philipp Hieronymi; Alonza Terry jun. "Ostrowski numeration systems, addition, and finite automata" Notre Dame J. Formal Logic 59 (2018) 215-232 Zbl 06870290

Dirac comb

A sum of Dirac delta-functions supported on a locally finite point set.

References

• Michael Baake, Uwe Gromm; "Aperiodic order", vol.1, Encyclopedia of Mathematics and its Applications 149 (Cambridge, 2013) ISBN 978-0-521-86991-1 Zbl 1295.37001
• Marjorie Senechal; "Quasicrystals and geometry" (Cambridge, 1995) ISBN 0-521-57541-9 Zbl 0828.52007

Delone set

Delaunay set, $(r,R)$-set

A subset $D$ of $\mathbf{R}^n$ which is both discrete: there exists $r>0$ such that the balls of radius $r$ centred on points of $D$ are disjoint; and relatively dense: there exists $R$ such that the balls of radius $R$ centred on points of $D$ cover $\mathbf{R}^n$.

See also Covering and packing.

The spectrum of a Delone set $D$ is defined as the Fourier transform $$ \hat\gamma(s) = \lim_{T\to\infty} \frac{1}{(2T)^n} \sum_{d \in D_T} \exp(-2\pi i\, d\cdot s) $$ where $D_T = D \cap [-T,T]^n$. The spectrum $\hat\gamma$ is a positive measure and has a Lebesgue decomposition into a sum of discrete and continuous measures. If the discrete measure is supported on a countably infinite set $S$, then $D$ is said to satisfy the diffraction condition.

References

• Marjorie Senechal; "Quasicrystals and geometry" (Cambridge, 1995) ISBN 0-521-57541-9 Zbl 0828.52007

SIS model

A simple model in mathematical epidemiology which reduces to the logistic equation. Assume that the population falls into two subgroups, "susceptible" ($S$) and "infected" ($I$), with susceptible members being infected at a rate proportional to the number of infected, and infected members recovering and returning to the susceptible subgroup at a constant rate. We therefore have $$ S' = - \beta S \cdot I + \alpha I $$ and $$ I' = \beta S \cdot I - \alpha I $$ Since $S+I = N$ is constant, we have $I' = r I (1- k^{-1} I)$ where $r = \beta N - \alpha$ is the growth rate and $k = r / \beta$. The basic reproduction number $R_0 = \beta N / \alpha$. If $R_0 < 1$, so that $r<0$, then $I$ decreases to zero. Otherwise we have the explicit solution $$ I(t) = \frac{ k B e^{rt} }{ 1 + B e^{rt} } $$ where $B= I(0) / (k - I(0))$ and $I(t)$ tends to $k$ as $t \to \infty$

Reference

• Maia Martcheva, "An Introduction to Mathematical Epdidemiology" Texts in Applied Mathematics 41 (Springer, 2015) ISBN 978-1-4899-7611-6 Zbl 1333.92006

Pregroup

A pregroup generalises the notion of a free group with amalgamation. A pregroup is a partially ordered monoid $(M,{\cdot},1,{\ge})$ with left and right adjoint maps $L$ and $R$ satisfying $$ x^L \cdot x \ge 1 \ge x \cdot x^L $$ $$ x \cdot x^R \ge 1 \ge x^R \cdot x $$ It follows that $x^{LR} = x = x^{RL}$

A partially ordered group is a pregroup, with both adjoint maps being group inversion.

References

• J. Stallings, "Group theory and three-dimensional manifolds" Yale Univ. Monogr. 4 (1971) Zbl 0241.57001