OpenLLM-France/wikimedia huggingface dataset:
# Linear algebra (Osnabrück 2024-2025)/Skew lines/R^3/Distance/Linear system/Example/Examplereferencenumber

Example 35.13
# Linear algebra (Osnabrück 2024-2025)/Euclidean space/Affine subspace/Perpendicular/Fact/Factreferencenumber

Lemma 35.12
# MDLD/triangles

Start=triangles See=triangle MDLD/triangle Target=[[MDLD/triangle]]
# MDLD/triangle

Start=triangle Target=Triangle geometry/Introduction/Section
# MDLD/nondegenerate triangle

Start=nondegenerate triangle Target=Triangle geometry/Introduction/Section
# MDLD/circumcenter

Start=circumcenter Target=Triangle/Perpendicular bisector/Circumcenter/Definition
# MDLD/altitudes (triangle)

Start=altitudes (triangle) See=altitude (triangle) MDLD/altitude (triangle) Target=Triangle/Altitude/Definition
# Hard properties of Boolean functions

Properties of Boolean functions :
| hard | soft |
| binary | binary |
| integer | integer |
ShortcutsBoolf-hardBoolf-EC :

## simple

* valency: number of relevant arguments (i.e. the number of circles needed to draw an Euler diagram, especially for a blightless BF)
* adicity: Atoms are numbered from 0. The adicity of a BF is its biggest atom plus 1. The period length of its truth table $2^{adicity}$.
* weight: quotient of true places and all places of the truth table (0 for the contradiction, 1 for the tautology)

## subsets

Belonging to a subset can also be seen as a property.
* linear: Walsh functions and their complements
* seal: conjunctions of linears
* dense: no gaps before or between the atoms, i.e. valency = adicity
* balanced: same number of true and false places, i.e. weight = 0.5
* monotonic: no true place under false place (when places are arranged in a Hasse diagram) (counted by Dedekind numbers)

## equivalence classes based on similarity

see also:
* sequences
* examples of extended families and clans
* smallest Zhegalkin index of families, factions and clans

### basic (negation and permutation)
#### family (negation)

Functions can be turned into each other by negating inputs.
The 10 (exactly) 2-ary Boolean functions form three families, whose representative functions are AND, OR and XOR.  (See this table of 2-ary functions, where equivalents are in the same row.)
The lexicographically smallest truth tables are encoded in A227722 = 0, 1, 3, 5, 6, 7, 15, 17, 18...

#### faction (permutation)

Functions can be turned into each other by permuting inputs. While families and clans can be self-complementary, factions can not. (Therefore the number of factions is always even.)

#### clan (negation, permutation)

Functions can be turned into each other by negating and permuting inputs.
The lexicographically smallest truth tables are encoded in A227723 = 0, 1, 3, 6, 7, 15...

### super (extension with complement)
## parity and depravity

quadrants :
| even evil (0) | even odious (2) |
| odd evil (1) | odd odious (3) |
The parity and depravity of a Boolean function are based on the first and last place of its truth table. It is odd (even), iff the first place is true (false). Its Zhegalkin index is also odd (even). It is odious (evil), iff the last place is true (false). Its Zhegalkin index has odd (even) binary weight. It is ugly, iff parity and depravity are different (i.e. iff the quadrant is 1 or 2). The quadrant is an integer 0...3, and calculated as $parity+2\cdot depravity$.

## prefect

The prefect is a way to assign each BF to a linear BF. A linear is assigned to itself. (3-ary images)

## gender and honesty

These binary properties seem to be related. Apparently there are no dishonest male BF.

### gender

Gender is based on parity. For any positive arity there are slightly more males than females. See Gender of Boolean functions.

### honesty

The XOR of all members of a family is either the tautology or the contradiction. Where it is the tautology, the BF is honest. Most BF are.
# Linear algebra (Osnabrück 2024-2025)/Part I

This is the English version of the material (lecture notes, exercises) of a mathematical course taught at the University of Osnabrück. So this is not an online course, only the material for a real university course. The material is organized in a modular way (e.g., facts, proofs, examples, and exercises are in separate files) so that it can also be used in other courses. The mathematical material is quite standard for a university beginners class, and the course has been taught several times. The English version is a direct translation. Suggestions are welcome.
Linear algebra is a fundamental course in mathematics. It deals with "linear structures“: linear systems of equations, vectors and their representation, bases and base change, linear mappings and matrices, determinants, eigenvalues and eigenvectors, optimal descriptions of linear mappings, affine mappings, normed vector spaces, lengths and angles, analytic geometry, isometries and symmetries, bilinear forms, more general algebraic structures like equivalence relations, groups, residue class construction, and multilinear algebra.
Part II
* List of definitions
* Important theorems
# MDLD/group homomorphisms

Start=group homomorphisms See=group homomorphism MDLD/group homomorphism Target=Group homomorphism/Definition
# Linear algebra (Osnabrück 2024-2025)/Group homomorphism/Factorization/Fact/Factreferencenumber

Theorem 47.8
# MDLD/equivalence relation

Start=equivalence relation Target=Set/Equivalence relation/Definition
# Linear algebra (Osnabrück 2024-2025)/Group theory/Finite/Order/Different/Fact/Factreferencenumber

Lemma 46.6
# Linear algebra (Osnabrück 2024-2025)/Group theory (algebra)/Lagrange's theorem/Fact/Factreferencenumber

Theorem 46.7
# Group theory/Finite/Order/Different/Fact

Let $G$ be a finite group.
$g\in G$ has finite
The powers
    $g^{0}=e_{G},\,g^{1}=g,\,g^{2},\ldots ,g^{\operatorname {ord} \,(g)-1}$
are all different.
# Linear algebra (Osnabrück 2024-2025)/Normal subgroup/Characterization/Fact/Factreferencenumber

Lemma 46.11
# MDLD/normal subgroup

Start=normal subgroup Target=Subgroup/Normal subgroup/Definition
# Linear algebra (Osnabrück 2024-2025)/Subgroups of Z/One generator/Fact/Factreferencenumber

Theorem 44.3
# MDLD/Linear mappings

Start=Linear mappings See=linear mapping MDLD/linear mapping Target=Linear mapping/Field/Definition
# Linear algebra (Osnabrück 2024-2025)/Group homomorphism/Inverse to inverse/Fact/Factreferencenumber

Lemma 44.6
# Linear algebra (Osnabrück 2024-2025)/Group homomorphism/Z to group/Fact/Factreferencenumber

Lemma 44.12
# MDLD/group isomorphism

Start=group isomorphism Target=Group isomorphism/Definition
# Linear algebra (Osnabrück 2024-2025)/Group homomorphism/Kernel/Subgroup/Fact/Factreferencenumber

Lemma 44.21
# Linear algebra (Osnabrück 2024-2025)/Group homomorphism/Injectivity and kernel/Fact/Factreferencenumber

Lemma 44.22
# MDLD/kernel (group)

Start=kernel (group) Target=Group homomorphism/Kernel/Definition
# MDLD/relation

Start=relation Target=Sets/Relation/Definition
# MDLD/relation (set)

Start=relation (set) Target=Sets/Relation/Definition
# Linear algebra (Osnabrück 2024-2025)/Equivalence relation/Fiber of a mapping/Example/Examplereferencenumber

Example 45.6
# Linear algebra (Osnabrück 2024-2025)/Equivalence relation/Symmetric accessibility relation/Example/Examplereferencenumber

Example 45.12
# MDLD/equivalence class

Start=equivalence class Target=Equivalence relation/Equivalence class/Definition
# MDLD/equivalence classes

Start=equivalence classes See=equivalence class MDLD/equivalence class Target=Equivalence relation/Equivalence class/Definition
# Equivalence classes/Partition/Quotient set/Properties/Fact

Let $M$ be a set, and let $\sim$ denote an equivalence relation on $M$, together with the equivalence classes $[x]$ and the quotient set $M/\sim$. Then the following statements hold.
* $x\sim y$ if and only if $[x]=[y]$, and this holds if and only if $[x]\cap [y]\neq \emptyset$.
* $M=\bigcup _{[x]\in M/\sim }[x]$ is a disjoint union.
* The canonical projection is surjective.
    $q\colon M\longrightarrow M/\sim ,x\longmapsto [x],$
* We have $q^{-1}([x])=[x]$.
# Linear algebra (Osnabrück 2024-2025)/Integer numbers/Construction from natural numbers/Example/Examplereferencenumber

Example 45.19
# Sequence Chrysanthemum

| | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 0-padded triangular numbers (~A000217) | 0 | 0 | 1 | 3 | 6 | 10 | 15 | 21 |
| Chrysanthemum (A006125) | 1 | 1 | 2 | 8 | 64 | 1024 | 32768 | 2097152 |
This is the diagonal in triangles [[|Rhododendron$_{(Drop)}$]].
# MDLD/bilinear form

Start=bilinear form Target=Vector space/Bilinear form/Definition
# Bilinear form/Linear forms/Nondegenerate/Fact/Proof

(1) follows immediately from bilinearity. (2). Let $u_{1},u_{2}\in V$ and $a_{1},a_{2}\in K$. Then, for every vector $v\in V$, we have
    $\left\langle a_{1}u_{1}+a_{2}u_{2},v\right\rangle =a_{1}\left\langle u_{1},v\right\rangle +a_{2}\left\langle u_{2},v\right\rangle \,,$
and this means the linearity of the assignment. (3). Since the assignment is linear by part (2), we have to show that its kernel is not trivial. So let $u\in V$ be such that $\left\langle u,-\right\rangle$ is the zero mapping. This means that $\left\langle u,v\right\rangle =0$ for all $v\in V$. This implies, by the definition of nondegenerate, that $u=0$. If $V$ has finite dimension, then we have an injective linear mapping between vector spaces of the same dimension. Such a mapping is also bijective, by fact.
# Euclidean space/Linear form/Corresponding vector/Fact

Let $(V,\left\langle -,-\right\rangle )$ be a Euclidean space, and let
    $f\colon V\longrightarrow \mathbb {R}$
denote a linear form.
$w\in V$ such that
    $f(v)=\left\langle w,v\right\rangle \,.$
If $u_{1},\ldots ,u_{n}$ is an orthonormal basis of $V$, and $f(u_{i})=a_{i}$, then this vector equals $w=\sum _{i=1}^{n}a_{i}u_{i}$.
# MDLD/symmetric (bilinear)

Start=symmetric (bilinear) Target=Bilinear form/Symmetric/Definition
# MDLD/antilinear

Start=antilinear Target=Vector spaces over C/Mapping/Antilinear/Definition
# MDLD/sesquilinear form

Start=sesquilinear form Target=Vector space over C/Sesquilinear form/Definition
# MDLD/Gram matrices (sesquilinear)

Start=Gram matrices (sesquilinear) See=Gram matrix (sesquilinear) MDLD/Gram matrix (sesquilinear) Target=[[MDLD/Gram matrix (sesquilinear)]]
# MDLD/Gram matrix (sesquilinear)

Start=Gram matrix (sesquilinear) Target=Sesquilinear form/Gram matrix/Definition
# MDLD/type (bilinear)

Start=type (bilinear) Target=Bilinear form/Symmetric/Type/Definition
# MDLD/negative semidefinite

Start=negative semidefinite Target=Bilinear form/Symmetric/Definiteness/Definition
# MDLD/positive definite

Start=positive definite Target=Bilinear form/Symmetric/Definiteness/Definition
# Bilinear form/Symmetric/Type/Minor criterion/Fact

Let $\left\langle -,-\right\rangle$ be a symmetric bilinear form on a finite-dimensional real vector space $V$. Let $v_{1},\ldots ,v_{n}$ denote a basis of $V$. Let $G$ be the Gram matrix of $\left\langle -,-\right\rangle$ with respect to this basis. Suppose that the determinant $D_{k}$ of the square submatrices
    $M_{k}={\left(\left\langle v_{i},v_{j}\right\rangle \right)}_{1\leq i,j\leq k}\,$
are not $0$ for $k=1,\ldots ,n$. Let $a$ be the number of changes of sign in the sequence
    $D_{0}=1,\,D_{1}=\det M_{1},\,D_{2}=\det M_{2},\ldots ,D_{n}=\det M_{n}=\det G.$
type
$\left\langle -,-\right\rangle$
$(n-a,a)$
# MDLD/Gram matrices

Start=Gram matrices See=Gram matrix MDLD/Gram matrix Target=[[MDLD/Gram matrix]]
# MDLD/Gram matrix

Start=Gram matrix Target=Bilinear form/Gram matrix/Definition
# Bilinear form/Symmetric/Eigenvalue criterion/Fact

Let $\left\langle -,-\right\rangle$ be a symmetric bilinear form on a finite-dimensional real vector space, and let $v_{1},\ldots ,v_{n}$ be a basis of $V$. Let $G$ denote the Gram matrix of $\left\langle -,-\right\rangle$ with respect to this basis.
type $(p,q)$ of the form has the following interpretation: $p$ is the sum of the dimensions of the eigenspaces of $G$ for positive eigenvalues,
$q$
$G$
# MDLD/negative definite

Start=negative definite Target=Bilinear form/Symmetric/Definiteness/Definition
# MDLD/symmetric bilinear form

Start=symmetric bilinear form Target=Bilinear form/Symmetric/Definition
# MDLD/nondegenerate

Start=nondegenerate Target=Bilinear form/Nondegenerate/Definition
# MDLD/Minkowski-space

Start=Minkowski-space Target=Minkowski-space/Type/Definition
# Minkowski-space/Observer vector/Definition

Let $V$ be a Minkowski-space, endowed with a Minkowski-form $\left\langle -,-\right\rangle$. A vector $v\in V$ fulfilling
    $\left\langle v,v\right\rangle =-1\,$
is called Observer vector or four-velocity of an observer.
# MDLD/observer vector

Start=observer vector Target=Minkowski-space/Observer vector/Definition
# Minkowski-space/Observer vectors/Fact

Let $V$ be a Minkowski-space, endowed with a Minkowski-form $\left\langle -,-\right\rangle$. Then the following statements hold.
* For every observer vector $v\in V$, we have a direct sum decomposition where the restriction of the Minkowski-form to $\mathbb {R} v$ is negative definite, and where the restriction of the Minkowski-form to $V_{v}=(\mathbb {R} v)^{\perp }$ is positive definite. Here, $V_{v}$ consists of spacelike vectors.
    $V=\mathbb {R} v\oplus (\mathbb {R} v)^{\perp }\,,$
* For two observer vectors $v,w\in V$ from the same half cone, we have
    $\left\langle v,w\right\rangle <0\,.$
* For timelike vectors $v,w\in V$, we have
    $\left\langle v,w\right\rangle ^{2}\geq \left\langle v,v\right\rangle \cdot \left\langle w,w\right\rangle \,.$
# MDLD/adjoint endomorphism

Start=adjoint endomorphism Target=Inner product/Adjoint endomorphism/Definition
# MDLD/adjoint mapping

Start=adjoint mapping Target=Inner product/Adjoint endomorphism/Definition
# MDLD/standard inner product

Start=standard inner product Target=R^n/Standard inner product/Example
# MDLD/normal endomorphism

Start=normal endomorphism Target=Vector space/K/Finite-dimensional/Normal endomorphism/Definition
# MDLD/normal (endomorphism)

Start=normal (endomorphism) Target=Vector space/K/Finite-dimensional/Normal endomorphism/Definition
# Principal axis transformation/Hermitian/Orthonormal system/Fact

Let $V$ be a finite-dimensional ${\mathbb {K} }$-vector space, endowed with an inner product. Let $\Psi$ denote an Hermitian form on $V$.
orthonormal basis of $V$ (with respect to the inner product), which is an orthogonal basis
$\Psi$
# MDLD/monomials

Start=monomials See=monomial MDLD/monomial Target=[[MDLD/monomial]]
# MDLD/monomial

Start=monomial Target=Monomial/Definition
# MDLD/polynomials (field n)

Start=polynomials (field n) See=polynomial (field n) MDLD/polynomial (field n) Target=[[MDLD/polynomial (field n)]]
# MDLD/polynomial (field n)

Start=polynomial (field n) Target=Polynomial/Field/n variables/Directly/Definition
# MDLD/polynomial (n)

Start=polynomial (n) Target=Polynomial/Field/n variables/Directly/Definition
# Sequence Ranunculus
## Sequences Ranunculus$_{(Drop)}$

| | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 🌊 Ranunculus (A028361) | 1 | 2 | 6 | 30 | 270 | 4590 | 151470 | 9845550 |
| 💧 RanunculusDrop (A126391) | 1 | 1 | 4 | 24 | 240 | 4320 | 146880 | 9694080 |
| 2$^{n−1}$ | | 1 | 2 | 4 | 8 | 16 | 32 | 64 |
${\text{for}}~n>0{\text{:}}~~~{\text{RanunculusDrop}}(n)=2^{n-1}\cdot {\text{Ranunculus}}(n-1)$
row sums of Rhododendron$_{(Drop)}$ :
🌊    triangle Rhododendron    row sums Ranunculus (A028361) :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | sums |
| 0 | 1 · 1 1 | | | | | | | | 1 |
| 1 | 1 · 1 1 | 1 · 1 1 | | | | | | | 2 |
| 2 | 1 · 1 1 | 3 · 1 3 | 1 · 2 2 | | | | | | 6 |
| 3 | 1 · 1 1 | 7 · 1 7 | 7 · 2 14 | 1 · 8 8 | | | | | 30 |
| 4 | 1 · 1 1 | 15 · 1 15 | 35 · 2 70 | 15 · 8 120 | 1 · 64 64 | | | | 270 |
| 5 | 1 · 1 1 | 31 · 1 31 | 155 · 2 310 | 155 · 8 1240 | 31 · 64 1984 | 1 · 1024 1024 | | | 4590 |
| 6 | 1 · 1 1 | 63 · 1 63 | 651 · 2 1302 | 1395 · 8 11160 | 651 · 64 41664 | 63 · 1024 64512 | 1 · 32768 32768 | | 151470 |
| 7 | 1 · 1 1 | 127 · 1 127 | 2667 · 2 5334 | 11811 · 8 94488 | 11811 · 64 755904 | 2667 · 1024 2731008 | 127 · 32768 4161536 | 1 · 2097152 2097152 | 9845550 |
💧    triangle RhododendronDrop    row sums RanunculusDrop :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | sums |
| 0 | 1 * 1 1 | | | | | | | | 1 |
| 1 | 0 | 1 * 1 1 | | | | | | | 1 |
| 2 | 0 | 2 * 1 2 | 1 * 2 2 | | | | | | 4 |
| 3 | 0 | 4 * 1 4 | 6 * 2 12 | 1 * 8 8 | | | | | 24 |
| 4 | 0 | 8 * 1 8 | 28 * 2 56 | 14 * 8 112 | 1 * 64 64 | | | | 240 |
| 5 | 0 | 16 * 1 16 | 120 * 2 240 | 140 * 8 1120 | 30 * 64 1920 | 1 * 1024 1024 | | | 4320 |
| 6 | 0 | 32 * 1 32 | 496 * 2 992 | 1240 * 8 9920 | 620 * 64 39680 | 62 * 1024 63488 | 1 * 32768 32768 | | 146880 |
| 7 | 0 | 64 * 1 64 | 2016 * 2 4032 | 10416 * 8 83328 | 11160 * 64 714240 | 2604 * 1024 2666496 | 126 * 32768 4128768 | 1 * 2097152 2097152 | 9694080 |

## Sequence TornRanunculus

Half of Ranunculus for positive keys.
| | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 🌊 TornRanunculus (A028362) | 1 | 1 | 3 | 15 | 135 | 2295 | 75735 | 4922775 |
| 💧 TornRanunculusDrop | 1 | 0 | 2 | 12 | 120 | 2160 | 73440 | 4847040 |
| A353196 | | | 1 | 6 | 60 | 1080 | 36720 | 2423520 |
# Triangle Cypress

turned Rhododendron with columns multiplied by SmallGrass
triangle Cypress :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | sums |
| 0 | 1 * 1 1 | | | | | | | | 1 |
| 1 | 1 * 1 1 | 1 * 3 3 | | | | | | | 4 |
| 2 | 2 * 1 2 | 3 * 3 9 | 1 * 15 15 | | | | | | 26 |
| 3 | 8 * 1 8 | 14 * 3 42 | 7 * 15 105 | 1 * 255 255 | | | | | 410 |
| 4 | 64 * 1 64 | 120 * 3 360 | 70 * 15 1050 | 15 * 255 3825 | 1 * 65535 65535 | | | | 70834 |
| 5 | 1024 * 1 1024 | 1984 * 3 5952 | 1240 * 15 18600 | 310 * 255 79050 | 31 * 65535 2031585 | 1 * 4294967295 4294967295 | | | 4297112728 |
| 6 | 32768 * 1 32768 | 64512 * 3 193536 | 41664 * 15 624960 | 11160 * 255 2845800 | 1302 * 65535 85326570 | 63 * 4294967295 270582939585 | 1 * 18446744073709551615 18446744073709551615 | | 18446744344381514834 |
| 7 | 2097152 * 1 2097152 | 4161536 * 3 12484608 | 2731008 * 15 40965120 | 755904 * 255 192755520 | 94488 * 65535 6192271080 | 5334 * 4294967295 22909355551530 | 127 * 18446744073709551615 2342736497361113055105 | 1 * 340282366920938463463374607431768211455 340282366920938463463374607431768211455 | 340282366920938465806111127708677391570 |

## Triangle SignedCypress

SignedCypress    row sums SmallLily (A305737) :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | sums |
| 0 | 1 * 1 1 | | | | | | | | 1 |
| 1 | 1 * 1 −1 | 1 * 3 3 | | | | | | | 2 |
| 2 | 2 * 1 2 | 3 * 3 −9 | 1 * 15 15 | | | | | | 8 |
| 3 | 8 * 1 −8 | 14 * 3 42 | 7 * 15 −105 | 1 * 255 255 | | | | | 184 |
| 4 | 64 * 1 64 | 120 * 3 −360 | 70 * 15 1050 | 15 * 255 −3825 | 1 * 65535 65535 | | | | 62464 |
| 5 | 1024 * 1 −1024 | 1984 * 3 5952 | 1240 * 15 −18600 | 310 * 255 79050 | 31 * 65535 −2031585 | 1 * 4294967295 4294967295 | | | 4293001088 |
| 6 | 32768 * 1 32768 | 64512 * 3 −193536 | 41664 * 15 624960 | 11160 * 255 −2845800 | 1302 * 65535 85326570 | 63 * 4294967295 −270582939585 | 1 * 18446744073709551615 18446744073709551615 | | 18446743803209556992 |
| 7 | 2097152 * 1 −2097152 | 4161536 * 3 12484608 | 2731008 * 15 −40965120 | 755904 * 255 192755520 | 94488 * 65535 −6192271080 | 5334 * 4294967295 22909355551530 | 127 * 18446744073709551615 −2342736497361113055105 | 1 * 340282366920938463463374607431768211455 340282366920938463463374607431768211455 | 340282366920938461120638132973980614656 |
# Triangle Rhododendron

* Triangle Oak with multiplied columns

## Triangles Rhododendron$_{(Drop)}$

🌊    triangle Rhododendron    row sums Ranunculus (A028361) :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | sums |
| 0 | 1 · 1 1 | | | | | | | | 1 |
| 1 | 1 · 1 1 | 1 · 1 1 | | | | | | | 2 |
| 2 | 1 · 1 1 | 3 · 1 3 | 1 · 2 2 | | | | | | 6 |
| 3 | 1 · 1 1 | 7 · 1 7 | 7 · 2 14 | 1 · 8 8 | | | | | 30 |
| 4 | 1 · 1 1 | 15 · 1 15 | 35 · 2 70 | 15 · 8 120 | 1 · 64 64 | | | | 270 |
| 5 | 1 · 1 1 | 31 · 1 31 | 155 · 2 310 | 155 · 8 1240 | 31 · 64 1984 | 1 · 1024 1024 | | | 4590 |
| 6 | 1 · 1 1 | 63 · 1 63 | 651 · 2 1302 | 1395 · 8 11160 | 651 · 64 41664 | 63 · 1024 64512 | 1 · 32768 32768 | | 151470 |
| 7 | 1 · 1 1 | 127 · 1 127 | 2667 · 2 5334 | 11811 · 8 94488 | 11811 · 64 755904 | 2667 · 1024 2731008 | 127 · 32768 4161536 | 1 · 2097152 2097152 | 9845550 |
💧    triangle RhododendronDrop    row sums RanunculusDrop :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | sums |
| 0 | 1 * 1 1 | | | | | | | | 1 |
| 1 | 0 | 1 * 1 1 | | | | | | | 1 |
| 2 | 0 | 2 * 1 2 | 1 * 2 2 | | | | | | 4 |
| 3 | 0 | 4 * 1 4 | 6 * 2 12 | 1 * 8 8 | | | | | 24 |
| 4 | 0 | 8 * 1 8 | 28 * 2 56 | 14 * 8 112 | 1 * 64 64 | | | | 240 |
| 5 | 0 | 16 * 1 16 | 120 * 2 240 | 140 * 8 1120 | 30 * 64 1920 | 1 * 1024 1024 | | | 4320 |
| 6 | 0 | 32 * 1 32 | 496 * 2 992 | 1240 * 8 9920 | 620 * 64 39680 | 62 * 1024 63488 | 1 * 32768 32768 | | 146880 |
| 7 | 0 | 64 * 1 64 | 2016 * 2 4032 | 10416 * 8 83328 | 11160 * 64 714240 | 2604 * 1024 2666496 | 126 * 32768 4128768 | 1 * 2097152 2097152 | 9694080 |
The main part of RhododendronDrop is Rhododendron with rows multiplied by powers of two.
This triangle is the basis of Cypress, whose signed version has row sums SmallLily.

## Triangle SignedRhododendron

The row sums of positive entries are TornRanunculus.
triangle SignedRhododendron (A158474) :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | sums |
| 0 | 1 · 1 1 | | | | | | | | 1 |
| 1 | 1 · 1 1 | 1 · 1 −1 | | | | | | | 0 |
| 2 | 1 · 1 1 | 3 · 1 −3 | 1 · 2 2 | | | | | | 0 |
| 3 | 1 · 1 1 | 7 · 1 −7 | 7 · 2 14 | 1 · 8 −8 | | | | | 0 |
| 4 | 1 · 1 1 | 15 · 1 −15 | 35 · 2 70 | 15 · 8 −120 | 1 · 64 64 | | | | 0 |
| 5 | 1 · 1 1 | 31 · 1 −31 | 155 · 2 310 | 155 · 8 −1240 | 31 · 64 1984 | 1 · 1024 −1024 | | | 0 |
| 6 | 1 · 1 1 | 63 · 1 −63 | 651 · 2 1302 | 1395 · 8 −11160 | 651 · 64 41664 | 63 · 1024 −64512 | 1 · 32768 32768 | | 0 |
| 7 | 1 · 1 1 | 127 · 1 −127 | 2667 · 2 5334 | 11811 · 8 −94488 | 11811 · 64 755904 | 2667 · 1024 −2731008 | 127 · 32768 4161536 | 1 · 2097152 −2097152 | 0 |
# History of Sri Lanka/Chandrika Kumaratunga

Chandrika Kumaratunga (Sinhala: චන්ද්‍රිකා බන්ඩාරනායක කුමාරතුංග, Tamil: சந்திரிகா பண்டாரநாயக்க குமாரதுங்க; born on June 29, 1945) was the fifth President of Sri Lanka, who served from 12 November 1994 to 19 November 2005. The country's only female president to date, she is the daughter of two former prime ministers and also the leader of the Sri Lanka Freedom Party (SLFP) until the end of 2005.
She is the longest-serving president in Sri Lankan history. She was elected in 1994, defeating UNP candidate Srima Dissanayake in the largest landslide victory in Sri Lankan history, securing 62% of the votes. Kumaratunga was reelected in 1999, defeating UNP candidate and Leader of the Opposition, Ranil Wickremesinghe.

## External Links

* ↑ "Profile: Chandrika Kumaratunga". BBC Profile: Chandrika Kumaratunga. 2005-08-26. Retrieved 2025-03-13.