User:Richard Pinch/sandbox-8

Erdos–Kac theorem

One of the main theorems in this area, and a far-reaching generalization of the Hardy–Ramanujan theorem, is the Erdös–Kac theorem, which states that the functions $f=\omega$ and $f=\Omega$ (and, in fact, a large class of additive arithmetic functions $f$ satisfy $$ \lim_{x\rightarrow\infty}\frac{1}{x} \sharp\left\lbrace{ n \le x : f(n) \le \log\log n + t \sqrt{\log\log n} }\right\rbrace = \frac{1}{2\pi} \int_{-\infty}^t e^{-x^2/2} dx\ \ \ (\,t \in \mathbf{R}\,) \ . $$

This shows that the values of $\omega(n)$ and $\Omega(n)$ are, in a sense, normally distributed with mean $\log\log n$ and standard deviation $\sqrt{\log\log n}$ (cf. also Normal distribution).

The Erdös–Kac theorem gives some more precise information on $\omega(n)$. Denote by $N(x;a,b)$ the number of integers in the interval $[3,x]$ for which the inequalities $$ a \le \frac{ \omega(n) - \log\log n }{ \sqrt{\log\log n} } \le b $$ hold. Then, as $x \rightarrow \infty$, $$ N(x;a,b) = (x+o(x)) \frac{1}{\sqrt{2\pi}} \int_a^b e^{-t^2/2} dt \ . $$

See [a1].

References

[a1]	W. Narkiewicz, "Number theory" , World Sci. (1983) pp. 251–259 (Translated from Polish) Zbl 0528.10001

Mediate cardinal

A cardinal which is larger than any finite set (not an inductive cardinal) but cannot be put into one-to-one correspondence with a proper subset of itself (not a reflexive cardinal). The terminology is that of Russell and Whitehead.

The non-existence of mediate cardinals follows from the Axiom of Choice.

References

• Hartley Rogers jr, "Theory of recursive functions and effective computability", MIT Press (1987) ISBN 0-262-68052-1
• Bertrand Russell, "Introduction to mathematical philosophy", Allen & Unwin (1920) Zbl 47.0036.12

Reflexive cardinal

The cardinality of a reflexive set: a set which can be placed into one-to-one correspondence with a proper subset of itself.

References

• Rudolf Carnap, "Introduction to symbolic logic and its applications" Dover (1958) Zbl 0083.00104
• Bertrand Russell, "Introduction to mathematical philosophy", Allen & Unwin (1920) Zbl 47.0036.12

Inductive cardinal

A cardinal which is either zero or derived from zero by a finite number of inductive steps of addition cardinal one.

References

• Rudolf Carnap, "Introduction to symbolic logic and its applications" Dover (1958) Zbl 0083.00104
• Bertrand Russell, "Introduction to mathematical philosophy", Allen & Unwin (1920) Zbl 47.0036.12

Comparability of cardinals

The proposition that any two cardinal numbers $\mathfrak{a}, \mathfrak{b}$ are comparable, so that one of $\mathfrak{a} < \mathfrak{b}$, $\mathfrak{a} = \mathfrak{b}$, $\mathfrak{a} > \mathfrak{b}$ holds. This asserts that of any two sets, one may be put into one-to-one correspondence with a subset of the other. Comparability follows from the well-ordering theorem, which implies that infinite cardinal numbers are all alephs: conversely, the comparability of sets in general implies the well-ordering theorem. Hence this is an equivalent of the Axiom of Choice.

References

• Abraham A. Fraenkel, "Abstract set theory", North-Holland (1953) Zbl 0050.04903

Dedekind finiteness

A set is said to be infinite in the sense of Dedekind if it can be put into one-to-one correspondence with a proper subset of itself, and finite otherwise. Russell and Whithead call such sets "reflexive".

References

• Bertrand Russell, "Introduction to mathematical philosophy", Allen & Unwin (1920) Zbl 47.0036.12

Adjugate matrix

adjoint matrix

The signed transposed matrix of cofactors for a given square matrix $A$. The $(i,j)$ entry of $\mathrm{adj}\,A$ is $$ \mathrm{adj}\, A_{ij} = (-1)^{i+j} \det A(j,i) $$ where $A(j,i)$ is the minor formed by deleting the row and column through the matrix entry $A_{ji}$.

The expansion of the determinant in cofactors is expressed by the identity $$ A . \mathrm{adj}\, A = \mathrm{adj}\, A . A = (\det A) I $$ where $I$ is the identity matrix.

Some texts do not include the transposition in their definition.

References

• A.C. Aitken, "Determinants and matrices", Oliver and Boyd (1939) Zbl 65.1111.05 Zbl 0022.10005

Rank metric

A metric on the space of matrices of given size over a field, given by the rank of their difference.

References

• È.M. Gabidulin, "Theory of codes with maximum rank distance" Probl. Inf. Transm. 21 (1985) 1-12; tr. of Probl. Peredachi Inf. 21 (1985) 3-16 Zbl 0585.94013

T-group

A group $G$ in which normality is transitive: if $N$ is a normal subgroup of $G$, and $H$ is a normal subgroup of $N$, then $H$ is a normal subgroup of $G$. Equivalently, every subnormal subgroup of $G$ is normal in $G$. Every completely reducible group is a T-group; the dihedral group of order 8 is the smallest finite group that is not a T-group. A group is a T-group if and only if it is equal to its own Wielandt subgroup.

References

• Derek Robinson, "A Course in the Theory of Groups", Graduate Texts in Mathematics 80 Springer (1996) ISBN 0-387-94461-3 Zbl 0483.20001

Completely reducible group

A group which is a direct product of simple groups.

References

• Derek Robinson, "A Course in the Theory of Groups", Graduate Texts in Mathematics 80 Springer (1996) ISBN 0-387-94461-3 Zbl 0483.20001

Ado theorem

Every finite-dimensional Lie algebra over a field of characteristic zero is isomorphic to a subalgebra of the Lie algebra of endomorphisms of a finite-dimensional vector space.

The corresponding assertion in finite characteristic is Iwasawa's Theorem.

References

• William Fulton, Joe Harris, "Representation Theory: A First Course", Graduate Texts in Mathematics 129 Springer (1991) ISBN 0-387-97527-6 Zbl 0744.22001
• Terence Tao, "Hilbert's Fifth Problem and Related Topics", Graduate Studies in Mathematics 153 American Mathematical Society (2014) ISBN 147041564X Zbl 1298.22001

Fibonacci expansion

Every positive integer can be expressed uniquely as a sum of Fibonacci numbers $F_k$ with the condition that consecutive Fibonacci numbers $F_{k+1}, F_k$ are not used. This expansion may be computed using a greedy algorithm: given $N$, subtract the largest $F_k$ with $F_k \le N$ and repeat with $N' = N - F_k$.

Tribonacci sequence

A linear recursive sequence, an extension of the sequence of Fibonacci numbers having the recurrence relation $$ t_{n+3}=t_{n+2}+t_{n+1}+t_n $$ for $n\ge 0$ and initial conditions $$ t_0=a\ ,t_1=b,\ t_2=c, $$ (with $a$, $b$, $c$ given constants).

The concept was introduced by the fourteen-year-old student M. Feinberg in 1963 in [a1] for the case: $a=b=1$, $c=2$: the term 'Tribonacci numbers is used for this specific case. The basic properties are introduced in [a5], [a3], [a4], [a2].

The formula for the $n$-th number is given by A. Shannon in [a3]: $$ T_n = \sum_{m=0}^{[n/2]} \sum_{r=0}^{[n/3]} \binom{ n-m-2r }{ m+r }\binom{ m+r }{ r } $$

Binet's formula for the $n$-th number is given by W. Spickerman in [b2]: $$ T_n = \frac{\rho^{n+2}}{ (\rho-\sigma)(\rho-\bar\sigma) } + \frac{\sigma^{n+2}}{ (\sigma-\rho)(\sigma-\bar\sigma) } + \frac{\bar\sigma^{n+2}}{ (\bar\sigma-\rho)(\bar\sigma-\sigma) } $$ where $$ \rho = \frac{1}{3}\left({ (19+3\sqrt{33})^{1/3} + (19-3\sqrt{33})^{1/3} +1 }\right)\,, $$ and $$ \sigma = \frac{1}{6}\left({ 2-(19+3\sqrt{33})^{1/3} - (19-3\sqrt{33})^{1/3} }\right) + \frac{\sqrt3 i}{6}\left({ (19+3\sqrt{33})^{1/3} - (19-3\sqrt{33})^{1/3} }\right) \ . $$ Here $\rho,\sigma,\bar\sigma$ are the roots of the companion polynomial $z^3 = z^2 + z + 1$.

The Tribonacci numbers count the binary sequences with no occurrences of three consecutive digits $\mathbf{1}$. [Ref?]

The Tribonacci sequence was generalized in [a6], [a7] to the form of two sequences:

$$a_{n+3}=u_{n+2}+w_{n+1}+y_n,$$

$$b_{n+3}=v_{n+2}+x_{n+1}+z_n,$$

where $u,v,w,x,y,z\in\{a,b\}$ and each of the tuples $(u,v)$, $(w,x)$, $(y,z)$ contains the two symbols $a$ and $b$. There are eight different such schemes. An open problem (as of 2000) is the construction of an explicit formula for each of them.

References

[a1]	M. Feinberg, "Fibonacci–Tribonacci" The Fibonacci Quart. , 1 : 3 (1963) pp. 71–74
[a2]	C. Valavigi, "Properties of Tribonacci numbers" The Fibonacci Quart. , 10 : 3 (1972) pp. 231–246
[a3]	A. Scott, T. Delaney, V. Hoggatt Jr., "The Tribonacci sequence" The Fibonacci Quart. , 15 : 3 (1977) pp. 193–200
[a4]	A. Shannon, "Tribonacci numbers and Pascal's pyramid" The Fibonacci Quart. , 15 : 3 (1977) pp. 268–275 Zbl 0385.05006
[a5]	I. Bruce, "A modified Tribonacci sequence" The Fibonacci Quart. , 22 : 3 (1984) pp. 244–246
[a6]	K. Atanassov, J. Hlebarova, S. Mihov, "Recurrent formulas of the generalized Fibonacci and Tribonacci sequences" The Fibonacci Quart. , 30 : 1 (1992) pp. 77–79
[a7]	J.-Z. Lee, J.-S. Lee, "Some properties of the generalization of the Fibonacci sequence" The Fibonacci Quart. , 25 : 2 (1987) pp. 111–117
[b2]	W. Spickerman, "Binet's formula for the Tribonacci sequence" The Fibonacci Quart. , 20 (1981) pp. 118–120 Zbl 0486.10011

Density of a sequence

A concept in general additive number theory, in which one studies addition laws for sequences of general form. The density of a sequence is a measure of what part of the sequence of all natural numbers belongs to a given sequence $A=\{a_k\}$ of integers $a_0=0<1\leq a_1<\ldots<a_k$. Let $$ A(x) = \sum_{a_i \le x \\ a_i \in A}\,1 $$ be the counting function of $A$.

Schnirelmann density. $$ \sigma(A) = \inf_n \frac{A(n)}{n} $$.

Let $W = \sum_n w_n$ be a divergent series of positive real numbers and let $W(x) = \sum_{n \le x} w_n$ be its summatory function. We can define a density relative to $W$ $$ d_W(X) = \lim_{x \rightarrow \infty} \frac{A(x)}{W(x)} $$ if this limit exists, and upper and lower densities to be the corresponding upper and lower limits if these exist.

Asymptotic density, a particular case of this being the natural density. This is density with respect to the "natural" series $w_n = 1$.

Logarithmic density. This is density with respect to the series $w_n = 1/n$. If a sequence has natural density then it has logarithmic density and the two are equal, but the converse does not hold.

Let $P(\sigma)$ denote a probability distribution on natural numbers, that is, a sequence $p_n = p_n(\sigma)$ such that $\p_n \ge 0$ and $\sum_n p_n$ converges to 1; we suppose that $P(\sigma)$ depends on a parameter $\sigma$.

The concept of density is also extended to numerical sequences differing from the natural sequence, for example to the sequence of integers in algebraic number fields. As a result it is possible to examine bases in algebraic fields.

References

[1]	A.O. Gel'fond, Yu.V. Linnik, "Elementary methods in the analytic theory of numbers" , M.I.T. (1966) (Translated from Russian)
[2]	H.H. Ostmann, "Additive Zahlentheorie" , Springer (1956)
[3]	Gérald Tenenbaum, "Introduction to analytic and probabilistic number theory" , (tr. C.B.Thomas) Cambridge University Press (1995) ISBN 0-521-41261-7

Schnirelmann Density

A definition of the density of a sequence, introduced in 1930 by L.G. Shnirel'man.

By the density of a sequence $A$ one means the density $d(A)$ of $A$, namely

$$d(A)=\inf\frac{A(n)}{n},$$

where

$$A(n)=\sum_{1\leq a_k\leq n}1.$$

The density $d(A)=1$ if and only if $A$ coincides with the set $\mathbf N_0$ of all non-negative integers. Let $A+B$ be the arithmetic sum of two sequences $A=\{a_k\}$ and $B=\{b_t\}$, i.e. the set $A+B=\{a_k+b_t\}$ where the numbers $a_k+b_t$ are taken without repetition. If $A=B$, one puts $2A=A+A$, and similarly $3A=A+A+A$, etc. If $hA=\mathbf N_0$, then $A$ is called a basis of order $h$. On examining the structures of sets obtained by summing sequences determined only by their densities, one uses the following theorems on the density of the sum of two sequences:

$$d(A+B)\geq d(A)+d(B)-d(A)d(B)$$

(Shnirel'man's inequality) and the stronger inequality

$$d(A+B)\geq\min(d(A)+d(B),1)$$

(the Mann–Dyson inequality).

Shnirel'man's inequality implies that any sequence of positive density is a basis of finite order. This can be used in additive problems, in which one frequently sums sequences of zero density by the preliminary construction of new sequences with positive density from the given ones. For example, it has been shown by sieve methods that the sequence $\{p\}+\{p\}$, where $p$ runs through the prime numbers, has positive density. Shnirel'man's theorem follows from this: There exists an integer $c_0>0$ such that any natural number is the sum of at most $c_0$ prime numbers. This theorem gives a solution to the so-called weak Goldbach problem (see also Additive number theory).

A variant of this concept of density is that of asymptotic density, a particular case of this being the natural density. The concept of density is also extended to numerical sequences differing from the natural sequence, for example to the sequence of integers in algebraic number fields. As a result it is possible to examine bases in algebraic fields.

References

[1]	A.O. Gel'fond, Yu.V. Linnik, "Elementary methods in the analytic theory of numbers" , M.I.T. (1966) (Translated from Russian)
[2]	H.H. Ostmann, "Additive Zahlentheorie" , Springer (1956)