text:
LieAlgebra Lessons - Maple Programming Help

# Online Help

###### All Products    Maple    MapleSim

Home : Support : Online Help : Mathematics : DifferentialGeometry : Lessons and Tutorials : LieAlgebras : DifferentialGeometry/LessonsAndTutorials/LieAlgebras/Series

LieAlgebra Lessons

Lesson 2: Subalgebras and Series

Overview

This lesson is devoted to the calculation of various subalgebras of a given Lie algebra.  You will learn to to do the following:

 – Find the center of a Lie algebra.
 – Find the radical of a Lie algebra.
 – Find the nilradical of a Lie algebra.
 – Find the smallest subalgebras and ideals containing a given set of vectors.
 – Find the centralizer of a set of vectors.
 – Find the normalizer of a subalgebra.
 – Find the generalized center of an ideal.
 – Find the derived algebra of a Lie algebra.
 – Find the derived series of a Lie algebra.
 – Find the lower central series of a Lie algebra.
 – Find the upper central series of a Lie algebra.
 – Find a canonical basis for a subalgebra of a Lie algebra.

Find the center of a Lie algebra

The center of a Lie algebra is the ideal consisting of all vectors which commute with every vector in the Lie algebra.  It is computed with the Center command.

 > with(DifferentialGeometry): with(LieAlgebras): with(Library):

Retrieve a Lie algebra from the DifferentialGeometry Library and initialize it with the DGsetup command.

 > L := Retrieve("Winternitz", 1, [5, 3], Alg1);
 ${L}{≔}\left[\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e3}}\right]$ (2.1)
 > DGsetup(L);
 ${\mathrm{Lie algebra: Alg1}}$ (2.2)

Calculate the center of the Lie algebra Alg1.

 Alg1 > C := Center();
 ${C}{≔}\left[{\mathrm{e2}}{,}{\mathrm{e1}}\right]$ (2.3)

We can check that e1 and e2 are in the center as follows:

 Alg1 > g := [e1, e2, e3, e4, e5];
 ${g}{≔}\left[{\mathrm{e1}}{,}{\mathrm{e2}}{,}{\mathrm{e3}}{,}{\mathrm{e4}}{,}{\mathrm{e5}}\right]$ (2.4)
 Alg1 > Matrix(2, 5, (i, j) -> LieBracket(C[i], g[j]));
 $\left[\begin{array}{ccccc}{0}{}{\mathrm{e1}}& {0}{}{\mathrm{e1}}& {0}{}{\mathrm{e1}}& {0}{}{\mathrm{e1}}& {0}{}{\mathrm{e1}}\\ {0}{}{\mathrm{e1}}& {0}{}{\mathrm{e1}}& {0}{}{\mathrm{e1}}& {0}{}{\mathrm{e1}}& {0}{}{\mathrm{e1}}\end{array}\right]$ (2.5)

Find the radical of a Lie algebra

The radical of a Lie algebra g is the largest solvable ideal in g.  It is computed with the Radical command.

 Alg1 > with(DifferentialGeometry): with(LieAlgebras): with(Library):

Retrieve a Lie algebra from the DifferentialGeometry Library and initialize it with the DGsetup command.

 Alg1 > L := Retrieve("Winternitz", 1, [5, 40], Alg1);
 ${L}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{2}{}{\mathrm{e1}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{2}{}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e4}}\right]$ (3.1)
 Alg1 > DGsetup(L);
 ${\mathrm{Lie algebra: Alg1}}$ (3.2)

Calculate the radical of the Lie algebra Alg1.

 Alg1 > R := Radical();
 ${R}{≔}\left[{\mathrm{e5}}{,}{\mathrm{e4}}\right]$ (3.3)

We can use the Query command to check that R is a solvable ideal.

 Alg1 > Query(R, "Solvable");
 ${\mathrm{true}}$ (3.4)
 Alg1 > Query(R, "Ideal");
 ${\mathrm{true}}$ (3.5)

Find the nilradical of a Lie algebra

The nilradical of a Lie algebra g is the largest nilpotent ideal in g.  It is computed with the Nilradical command.

 Alg1 > with(DifferentialGeometry): with(LieAlgebras): with(Library):

Retrieve a Lie algebra from the DifferentialGeometry Library and initialize it with the DGsetup command.

 > L := Retrieve("Winternitz", 1, [5, 38], Alg1);
 ${L}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e3}}\right]$ (4.1)

 > DGsetup(L);
 ${\mathrm{Lie algebra: Alg1}}$ (4.2)
 Alg1 > N := Nilradical();
 ${N}{≔}\left[{\mathrm{e1}}{,}{\mathrm{e2}}{,}{\mathrm{e3}}\right]$ (4.3)

We can use the Query command to check that N is a solvable ideal.

 Alg1 > Query(N, "Nilpotent");
 ${\mathrm{true}}$ (4.4)
 Alg1 > Query(N, "Ideal");
 ${\mathrm{true}}$ (4.5)

Find the smallest subalgebras and ideals containing a given set of vectors

Given a list of vectors S, the commands MinimalSubalgebra and MinimalIdeal return the smallest subalgebra and smallest ideal containing S.

 Alg1 > with(DifferentialGeometry): with(LieAlgebras): with(Library):

Retrieve a Lie algebra from the DifferentialGeometry Library and initialize it with the DGsetup command.

 Alg1 > L := Retrieve("Turkowski", 1, [7, 5], Alg1);
 ${L}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{2}{}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{-}{2}{}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{2}{}{\mathrm{e4}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e6}}\right]{=}{-}{2}{}{\mathrm{e6}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{2}{}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{2}{}{\mathrm{e6}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e6}}\right]$ (5.1)
 Alg1 > DGsetup(L):

Define a list S of vectors in Alg1.

 Alg1 > S := [e2, e5];
 ${S}{≔}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]$ (5.2)

Find the smallest subalgebra A containing S.  Check that A is a subalgebra in Alg1.

 Alg1 > A := MinimalSubalgebra(S);
 ${A}{≔}\left[{\mathrm{e2}}{,}{\mathrm{e4}}{,}{\mathrm{e5}}\right]$ (5.3)
 Alg1 > Query(A, "Subalgebra");
 ${\mathrm{true}}$ (5.4)

Find the smallest ideal B containing S.  Check that B is an ideal in Alg1.

 Alg1 > B := MinimalIdeal(S);
 ${B}{≔}\left[{\mathrm{e1}}{,}{\mathrm{e2}}{,}{\mathrm{e3}}{,}{\mathrm{e4}}{,}{\mathrm{e5}}{,}{\mathrm{e6}}\right]$ (5.5)
 Alg1 > Query(B, "Ideal");
 ${\mathrm{true}}$ (5.6)

Find the centralizer of a set of vectors S

The centralizer of a set of vectors S in a Lie algebra is the subalgebra of all vectors which commute with all the vectors in S.  It is computed with the Centralizer command.

 Alg1 > with(DifferentialGeometry): with(LieAlgebras): with(Library):

Retrieve a Lie algebra from the DifferentialGeometry Library and initialize it with the DGsetup command.

 Alg1 > L := Retrieve("Turkowski", 1, [ 7, 5], Alg1);
 ${L}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{2}{}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{-}{2}{}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{2}{}{\mathrm{e4}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e6}}\right]{=}{-}{2}{}{\mathrm{e6}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{2}{}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{2}{}{\mathrm{e6}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e6}}\right]$ (6.1)
 Alg1 > DGsetup(L):

Find the centralizer of the set S and check the result.

 Alg1 > S := [e3, e4];
 ${S}{≔}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]$ (6.2)
 Alg1 > C := Centralizer(S);
 ${C}{≔}\left[{\mathrm{e6}}\right]$ (6.3)
 Alg1 > LieBracket(e3,e6), LieBracket(e4,e6);
 ${0}{}{\mathrm{e1}}{,}{0}{}{\mathrm{e1}}$ (6.4)

Find the normalizer of a subalgebra

The normalizer of a subalgebra h is the largest subalgebra k such that h is normal in k, that is, the Lie bracket of any vector in h with any vector in k is a vector back in h.  The normalizer of a subalgebra is calculated with the SubalgebraNormalizer command.

 Alg1 > with(DifferentialGeometry): with(LieAlgebras): with(Library):

Retrieve a Lie algebra from the DifferentialGeometry Library and initialize it with the DGsetup command.

 Alg1 > L := Retrieve("Turkowski", 1, [7, 5], Alg1);
 ${L}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{2}{}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{-}{2}{}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{2}{}{\mathrm{e4}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e6}}\right]{=}{-}{2}{}{\mathrm{e6}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{2}{}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{2}{}{\mathrm{e6}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e6}}\right]$ (7.1)
 Alg1 > DGsetup(L);
 ${\mathrm{Lie algebra: Alg1}}$ (7.2)

Check that the span of the vectors S is a subalgebra of Alg1.

 Alg1 > S := [e1, e2, e3];
 ${S}{≔}\left[{\mathrm{e1}}{,}{\mathrm{e2}}{,}{\mathrm{e3}}\right]$ (7.3)
 Alg1 > Query(S, "Subalgebra");
 ${\mathrm{true}}$ (7.4)

Calculate the normalizer of S in Alg1.

 Alg1 > N := SubalgebraNormalizer(S);
 ${N}{≔}\left[{\mathrm{e7}}{,}{\mathrm{e3}}{,}{\mathrm{e2}}{,}{\mathrm{e1}}\right]$ (7.5)

We can check that S is an ideal in N using the BracketOfSubspaces command and noting that all the vectors in B lie in S.

 Alg1 > B := BracketOfSubspaces(S, N);
 ${B}{≔}\left[{-}{2}{}{\mathrm{e3}}{,}{2}{}{\mathrm{e2}}{,}{\mathrm{e1}}\right]$ (7.6)

Find the generalized center of an ideal

Let h be an ideal in a Lie algebra g.  Then the ideal of vectors k such that [k, g] is contained in h is called the generalized center of h.  Use the GeneralizedCenter command.

 Alg1 > with(DifferentialGeometry): with(LieAlgebras): with(Library):

Retrieve a Lie algebra from the DifferentialGeometry Library and initialize it with the DGsetup command.

 Alg1 > L := Retrieve("Winternitz", 1, [6, 8], Alg1);
 ${L}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e3}}{+}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e6}}\right]$ (8.1)
 Alg1 > DGsetup(L):

We check that the subspace spanned by the vectors in h is an ideal.

 Alg1 > h := [e5, e6];
 ${h}{≔}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]$ (8.2)
 Alg1 > Query(h, "Ideal");
 ${\mathrm{true}}$ (8.3)

Calculate the generalized center of h.

 Alg1 > k := GeneralizedCenter(h);
 ${k}{≔}\left[{\mathrm{e6}}{,}{\mathrm{e4}}{,}{\mathrm{e5}}\right]$ (8.4)

We check that k is an ideal and that [k, g] is a subset of h.

 Alg1 > Query(k, "Ideal");
 ${\mathrm{true}}$ (8.5)
 Alg1 > G := [e1, e2, e3, e4, e5, e6];
 ${G}{≔}\left[{\mathrm{e1}}{,}{\mathrm{e2}}{,}{\mathrm{e3}}{,}{\mathrm{e4}}{,}{\mathrm{e5}}{,}{\mathrm{e6}}\right]$ (8.6)
 Alg1 > BracketOfSubspaces(k, G);
 $\left[{-}{\mathrm{e6}}\right]$ (8.7)

Find the derived algebra of a Lie algebra

The derived algebra of a Lie algebra g is the ideal spanned by all brackets [x, y], with x and y in g.  This ideal can be computed with the DerivedAlgebra command.

 Alg1 > with(DifferentialGeometry): with(LieAlgebras): with(Library):

Retrieve a Lie algebra from the DifferentialGeometry Library and initialize it with the DGsetup command.

 > L := Retrieve("Winternitz", 1, [5, 3], Alg1);
 ${L}{≔}\left[\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e3}}\right]$ (9.1)
 > DGsetup(L);
 ${\mathrm{Lie algebra: Alg1}}$ (9.2)

We calculate the derived algebra of the Lie algebra Alg1 and check that it is an ideal.

 Alg1 > A := DerivedAlgebra();
 ${A}{≔}\left[{\mathrm{e2}}{,}{\mathrm{e1}}{,}{\mathrm{e3}}\right]$ (9.3)
 Alg1 > Query(A, "Ideal");
 ${\mathrm{true}}$ (9.4)

We can also calculate the derived algebra from its definition using the BracketOfSubspaces command

 Alg1 > G:= [e1, e2, e3, e4, e5];
 ${G}{≔}\left[{\mathrm{e1}}{,}{\mathrm{e2}}{,}{\mathrm{e3}}{,}{\mathrm{e4}}{,}{\mathrm{e5}}\right]$ (9.5)
 Alg1 > BracketOfSubspaces(G, G);
 $\left[{\mathrm{e2}}{,}{\mathrm{e1}}{,}{\mathrm{e3}}\right]$ (9.6)

Find the derived series of a Lie algebra

The derived series of a Lie algebra g is the sequence of ideals D^k(g) in g defined inductively by D^0(g) = g and D^(k + 1)(g) = [D^k(g), D^k(g)].  To find the derived series of a Lie algebra, use the Series command with the argument "Derived".

 > with(DifferentialGeometry): with(LieAlgebras): with(Library):

Retrieve a Lie algebra from the DifferentialGeometry Library and initialize it with the DGsetup command.

 Alg1 > L := Retrieve("Turkowski", 2, [6, 39], Alg1)[1];
 ${L}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{2}{}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e3}}\right]$ (10.1)
 Alg1 > DGsetup(L):

Find the derived series for the current algebra Alg1.

 Alg1 > D0 := Series("Derived");
 ${\mathrm{D0}}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}{,}{\mathrm{e3}}{,}{\mathrm{e4}}{,}{\mathrm{e5}}{,}{\mathrm{e6}}\right]{,}\left[{-}{\mathrm{e6}}{,}{\mathrm{e5}}{,}{-}{\mathrm{e4}}{,}{2}{}{\mathrm{e3}}\right]{,}\left[{-}{\mathrm{e3}}\right]{,}\left[{}\right]\right]$ (10.2)

We can write these subspaces in slightly better form using the CanonicalBasis command.

 Alg1 > G := [e1, e2, e3, e4, e5, e6];
 ${G}{≔}\left[{\mathrm{e1}}{,}{\mathrm{e2}}{,}{\mathrm{e3}}{,}{\mathrm{e4}}{,}{\mathrm{e5}}{,}{\mathrm{e6}}\right]$ (10.3)
 Alg1 > DS := map(Tools:-CanonicalBasis, D0, G);
 ${\mathrm{DS}}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}{,}{\mathrm{e3}}{,}{\mathrm{e4}}{,}{\mathrm{e5}}{,}{\mathrm{e6}}\right]{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}{,}{\mathrm{e5}}{,}{\mathrm{e6}}\right]{,}\left[{\mathrm{e3}}\right]{,}\left[{}\right]\right]$ (10.4)

We can check the validity of the 3rd derived series DS[3] (say) using the value of DS[2], the definition of the derived series, and the BracketOfSubspaces command.

 Alg1 > A := BracketOfSubspaces(DS[2], DS[2]);
 ${A}{≔}\left[{-}{\mathrm{e3}}\right]$ (10.5)

We see visually that the span of A and L[3] agree but this can be checked with the DGequal command.

 Alg1 > Tools:-DGequal(A, DS[3]);
 ${\mathrm{true}}$ (10.6)

The command Series can also be used to calculate the derived series of any subalgebra.  For example, we can calculate the derived series of the subalgebra S.

 Alg1 > S := [e3, e4, e5, e6];
 ${S}{≔}\left[{\mathrm{e3}}{,}{\mathrm{e4}}{,}{\mathrm{e5}}{,}{\mathrm{e6}}\right]$ (10.7)
 Alg1 > Query(S, "Subalgebra");
 ${\mathrm{true}}$ (10.8)
 Alg1 > Series(S, "Derived");
 $\left[\left[{\mathrm{e3}}{,}{\mathrm{e4}}{,}{\mathrm{e5}}{,}{\mathrm{e6}}\right]{,}\left[{-}{\mathrm{e3}}\right]{,}\left[{}\right]\right]$ (10.9)

Find the lower central series of a Lie algebra

The lower central series of a Lie algebra g is a sequence of ideals L^k(g) in g defined inductively by L^0(g) = g and L^(k + 1)(g) = [g, L^k(g)].  To find the lower central series of a Lie algebra use the Series command with the argument "Lower".

 Alg1 > with(DifferentialGeometry): with(LieAlgebras): with(Library):

Retrieve a Lie algebra from the DifferentialGeometry Library and initialize it with the DGsetup command.

 Alg1 > L := Retrieve("Turkowski", 2, [6, 39], Alg1)[1];
 ${L}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{2}{}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e3}}\right]$ (11.1)
 Alg1 > DGsetup(L):

Find the lower central series for the current algebra Alg1.

 Alg1 > L0 := Series("Lower");
 ${\mathrm{L0}}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}{,}{\mathrm{e3}}{,}{\mathrm{e4}}{,}{\mathrm{e5}}{,}{\mathrm{e6}}\right]{,}\left[{-}{\mathrm{e6}}{,}{\mathrm{e5}}{,}{-}{\mathrm{e4}}{,}{2}{}{\mathrm{e3}}\right]{,}\left[{\mathrm{e4}}{,}{-}{\mathrm{e5}}{,}{\mathrm{e3}}\right]{,}\left[{-}{\mathrm{e5}}{,}{-}{\mathrm{e4}}{,}{-}{\mathrm{e3}}\right]\right]$ (11.2)

We can write these subspaces in a slightly better form using the CanonicalBasis command.

 Alg1 > G := [e1, e2, e3, e4, e5, e6];
 ${G}{≔}\left[{\mathrm{e1}}{,}{\mathrm{e2}}{,}{\mathrm{e3}}{,}{\mathrm{e4}}{,}{\mathrm{e5}}{,}{\mathrm{e6}}\right]$ (11.3)
 Alg1 > LS := map(Tools:-CanonicalBasis, L0, G);
 ${L}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}{,}{\mathrm{e3}}{,}{\mathrm{e4}}{,}{\mathrm{e5}}{,}{\mathrm{e6}}\right]{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}{,}{\mathrm{e5}}{,}{\mathrm{e6}}\right]{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}{,}{\mathrm{e5}}\right]{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}{,}{\mathrm{e5}}\right]\right]$ (11.4)

We can check the validity of the 3rd ideal in the lower central series LS[3] (say) using the value of LS[2], the definition of the lower central series, and the BracketOfSubspaces command.

 Alg1 > A := BracketOfSubspaces(LS[2], G);
 ${A}{≔}\left[{-}{2}{}{\mathrm{e3}}{,}{-}{\mathrm{e5}}{,}{-}{\mathrm{e4}}\right]$ (11.5)

We see visually that the span of A and LS[3] agree but this can be checked with the DGequal command.

 Alg1 > Tools:-DGequal(A, LS[3]);
 ${\mathrm{true}}$ (11.6)

The command Series can also be used to calculate the lower central series of any subalgebra.  As an example, we calculate the lower central series of the subalgebra S.

 Alg1 > S := [e3, e4, e5, e6];
 ${S}{≔}\left[{\mathrm{e3}}{,}{\mathrm{e4}}{,}{\mathrm{e5}}{,}{\mathrm{e6}}\right]$ (11.7)
 Alg1 > Query(S, "Subalgebra");
 ${\mathrm{true}}$ (11.8)
 Alg1 > Series(S, "Lower");
 $\left[\left[{\mathrm{e3}}{,}{\mathrm{e4}}{,}{\mathrm{e5}}{,}{\mathrm{e6}}\right]{,}\left[{-}{\mathrm{e3}}\right]{,}\left[{}\right]\right]$ (11.9)

Find the upper central series of a Lie algebra

The upper central series of a Lie algebra g is the sequence of ideals C^k(g) in g defined inductively by C^0(g) = GeneralizedCenter(0) and C^(k + 1)(g) = GeneralizedCenter(C^k(g)).  To find the upper central series of a Lie algebra, use the Series command with the argument "Upper".

 Alg1 > with(DifferentialGeometry): with(LieAlgebras): with(Library):

Retrieve a Lie algebra from the DifferentialGeometry Library and initialize it with the DGsetup command.

 Alg1 > L := Retrieve("Winternitz", 1, [6, 8], Alg1);
 ${L}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e3}}{+}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e6}}\right]$ (12.1)
 Alg1 > DGsetup(L):

Calculate the upper central series.

 Alg1 > CS := Series("Upper");
 ${\mathrm{CS}}{≔}\left[\left[{\mathrm{e6}}{,}{\mathrm{e4}}\right]{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}{,}{\mathrm{e5}}{,}{\mathrm{e4}}\right]{,}\left[{\mathrm{e5}}{,}{\mathrm{e4}}{,}{\mathrm{e3}}{,}{\mathrm{e2}}{,}{\mathrm{e1}}{,}{\mathrm{e6}}\right]\right]$ (12.2)

Check that the first term in the upper central series is the center C of the Lie algebra and that the second term is the generalized center of C.

 Alg1 > C := Center();
 ${C}{≔}\left[{\mathrm{e4}}{,}{\mathrm{e6}}\right]$ (12.3)
 Alg1 > C1 := GeneralizedCenter(C);
 ${\mathrm{C1}}{≔}\left[{\mathrm{e3}}{,}{\mathrm{e6}}{,}{\mathrm{e5}}{,}{\mathrm{e4}}\right]$ (12.4)

The Series command can also be used to calculate the upper central series of any subalgebra.  For example, we find the upper central series of the subalgebra S.

 Alg1 > S := [e2, e5, e6];
 ${S}{≔}\left[{\mathrm{e2}}{,}{\mathrm{e5}}{,}{\mathrm{e6}}\right]$ (12.5)
 Alg1 > Query(S, "Subalgebra");
 ${\mathrm{true}}$ (12.6)
 Alg1 > Series(S, "Upper");
 $\left[\left[{\mathrm{e6}}\right]{,}\left[{\mathrm{e6}}{,}{\mathrm{e5}}{,}{\mathrm{e2}}\right]\right]$ (12.7)

Ian M. Anderson 2006

## Was this information helpful?

 Please add your Comment (Optional) E-mail Address (Optional) What is ? This question helps us to combat spam


text:
BMJ Open
BMJ Publishing Group
Health and long-term care of the elderly with dementia in rural Thailand: a cross-sectional survey through their caregivers
Volume: 10, Issue: 3
DOI 10.1136/bmjopen-2019-032637
•
•
•
• Altmetric

### Notes

Abstract

ObjectiveTo describe the circumstances of the elderly with dementia and their caregivers’ characteristics in order to examine factors related to activities of daily living (ADL) and household income to propose a long-term care policy for rural areas of Thailand.SettingA cross-sectional study at the household level in three rural regions of Thailand where there were initiatives relating to community care for people with dementia.ParticipantsCaregivers of 140 people with dementia were recruited for the study.Primary and secondary outcome measuresSocioeconomic characteristics including data from assessment of ADL and instrumental ADL and the Thai version of Resource Utilisation in Dementia were collected. Descriptive statistics were used to explain the characteristics of the elderly with dementia and the caregivers while inferential statistics were used to examine the associations between different factors of elderly patients with dementia with their dependency level and household socioeconomic status.ResultsEighty-six per cent of the dementia caregivers were household informal caregivers as half of them also had to work outside the home. Half of the primary caregivers had no support and no minor caregivers. The elderly with dementia with high dependency levels were found to have a significant association with age, dementia severity, chance of hospitalisation and number of hospitalisations. Though most of these rural samples had low household incomes, the patients in the lower-income households had significantly lower dementia severity, but, with the health benefit coverage had significantly higher chances of hospitalisation.ConclusionAs the informal caregivers are the principal human resources for dementia care and services in rural area, policymakers should consider informal care for the Thai elderly with dementia and promote it as the dominant pattern of dementia care in Thailand.

Keywords
Chuakhamfoo, Phanthunane, Chansirikarn, and Pannarunothai: Health and long-term care of the elderly with dementia in rural Thailand: a cross-sectional survey through their caregivers
Strengths and limitations of this study
The present study was among the first cross-sectional studies on the burden of dementia in a rural area of an upper-middle-income country.
The present study recruited only dementia cases with caregivers, most found to be informal caregivers who provided care in addition to other personal responsibilities.
The present study recommends formal care units to support informal care, especially in rural areas, as the elderly are often dependent on their caregivers.

## Introduction

A report from WHO1 suggests that dementia is one of the critical causes of dependence and malfunction in old age, as it affects cognitive functions such as memory, logic, awareness and communication. Dementia is one of the leading causes of death in old age.2 Dementia is not a normal part of old age. There is a misconception that dementia is the regular part of ageing.3 4 Moreover, dementia cannot be avoided or treated causally.5–7

It was estimated that there were approximately 600 000 elderly people with dementia in Thailand in 2015; this might reach more than a million people by 2030 and 2 million by 2050. This shows that the number of elderly with dementia in Thailand would be 2.4 times the number in Australia, 3.5 times more than in Malaysia and 8.6 times more than in Singapore as shown in figure 1.8 Meanwhile, the low birth rate from more than 6 children per woman in 1960 down to about 1.5 children per woman in 20159 would have a negative impact on dementia care.

Figure 1
The estimation of the number of people with dementia in 2050. 8

Caring for the elderly with dementia is difficult.10 11 First and foremost, the caregivers do not necessarily realise that the elderly are struggling with ailing health conditions12 as the problems occur in the brain and do not necessarily show obvious physical symptoms.13 Caregivers are often faced with irrational behaviours from patients with dementia. Patients with dementia become forgetful. As the illness progresses, other behaviours can manifest as the patient’s cognition functioning declines.14

In developed countries, patients with dementia have a higher rate of hospitalisation, for example, 41.9% in USA,15 40.8% in France16 and 54.9% in Japan.17 However, the World Alzheimer Report 2015 forecasted that we would find more people with dementia in low-income and middle-income countries, increasing from 58% of the total number of people with dementia globally in 2015 to 63% in 2030.18 Fotenos et al found a higher risk of dementia in populations with lower socioeconomic status.19 However, such dementia studies have never been carried out in Thailand.

In Asia, informal family care plays a significant role in caring for the aged. The person who takes responsibility for dementia care is typically the daughter or daughter-in-law since they assume the caretaker’s role especially with regard to the tradition of filial piety and familism.20 21 The percentage of family caregivers in Thailand, according to a 2014 national health examination survey found that 45% of caregivers in Thailand were their daughters, 28% their marriage partners, 18.4% sons, 6.2% grandchildren and 2.4% others.22 Most of them are unpaid caregivers. The caregivers are often placed in a circumstance that involves the elderly patient declining mentally, even if they remain physically healthy.23 As hidden patients, caregivers increasingly need to use health services.24

Since 2005, the number of people aged 60 years and over has been more than 10.7% of the total population. In 2014, the number of these people was estimated to be 14.9%.25 In 2018, their percentage was estimated to be 17.7%.26 Thailand is now regarded as an aged society. Forecasts suggest that by 2021, the number of people aged 60 years and over will reach 20%. Thailand will be regarded as a super-aged society.27 The dependency level increased with age. The elderly aged 80 years and above were 33% partially dependent and 2% totally dependent.22 The increasing number of elderly is expected to result in an increase in the number of patients with dementia.2 However, the circumstances of dementia affect the ageing and the working populations who drive the country's economic development.28

The National Health Security Office has been established is to manage the universal coverage scheme (UCS) since 2001. The National Health Security Office introduced development of the community-based long-term care (LTC) strategy during 2014–2016 to broaden the social aspect of health coverage. Also, the second National Elderly Plan (2002–2021) stated that ‘the elderly should be entitled to recognition and support by their family, community and the state to the highest dignity, health, and living standards for as long as possible”.29 This facilitates the integration of community-based LTC with the primary healthcare system.30 However, the abovementioned LTC policy focused on general formal care for non-specific chronic diseases which did not specifically fit with dementia care.

Dementia is a specific chronic illness occurring more often with the aged;31 care management is much more specific at the dependency level and it differs from other chronic diseases.32 As suggested by WHO,33 Thailand needs to develop stronger visions and greater coordinated efforts to cope with the burden of dementia like in other countries. The Community Comprehensive Dementia Care was launched by one of the authors (SC) in collaboration with the Ministry of Public Health to provide a pilot community-based dementia care at the household level in selected regions of Thailand through public health personnel. The lesson learnt and experience gained would be forwarded to the Public Health Commission of the National Legislative Assembly.34 The present study aims to describe the circumstances of elderly patients with dementia and their caregivers’ characteristics in order to examine factors related to activities of daily living (ADL) or dependency level and household income to propose LTC policy for the elderly patients with dementia in rural areas of Thailand.

## Methods

### Study design

This cross-sectional study gathered information from rural areas where there were Comprehensive Dementia Care initiatives in the community.

### Patient and public involvement

Patients and the public were not involved in the design, planning or the analysis of the study.

Study site

In Thailand, there are no community guidelines for dementia care in the country’s LTC system, however, some communities are participating in developing a new model of community dementia care following suggestions by experts in the LTC system. The three areas that joined the Community Comprehensive Dementia Care are Lamsonthi district in Lopburi for the central region, Ban Fang district in Khon Kaen for the north-eastern region and Khiri Rat Nikhom district in Surat Thani for the southern region (see figure 2).

Figure 2
Map of Thailand shows the location of the investigated household participants.

### Diagnosis of dementia

Persons with dementia in the present study were patients who had been diagnosed by a doctor, who differentiated the severity of their illness into mild, moderate and severe levels.

### Population

The elderly with dementia in the present study included those aged 60 years and above according to the retirement age in Thailand. All patients with dementia in the Community Comprehensive Dementia Care within the three provinces were recruited as target populations of this cross-sectional study. The inclusion criteria were the elderly with dementia who had caregivers that supplied the necessary data regarding their patients and themselves. A non-response rate of 22.2% of 180 patients was observed because caregivers were not present. In total, there were 50 patients with dementia in Ban Fang, 70 in Lamsonthi and 60 in Khiri Rat Nikhom who were recruited as our study samples. After all of the 140 caregivers understood the study requirements, they agreed to participate by signing a consent form. The researcher (NNC) then interviewed them with structured questionnaires to determine the characteristics of the patient, the caregiver and the circumstances of the dementia care.

### Instruments and data collection

#### Sociodemographic data of the elderly with dementia and the caregivers

The data collection questionnaire covered general data such as age, sex, education and health coverage schemes; it comprised two parts. The first part included questions on the characteristics of the elderly with dementia, while the second part included questions on the characteristics of the caregiver. Based on Poomontre,35 the eight classes of rural households (E, D, C−, C, C+, B, A and A+ based on household asset) were regrouped by the authors of this study into three groups on the basis of household income for accurate statistical analysis. The first level combined E to D having a household income of 0–6500 baht per month; the second level was C− to C with a household income of 65 001–20 000 baht per month; the third level was C+ to A+ with a household income of 20 001 baht per month and higher.

The ADL and instrumental ADL (IADL) used in this study comprised 16 items with three Likert Scales to assess the capacity of a person to perform an activity of daily life. This version came from the synthesis of the National Health Commission Office of Thailand36 as the standard instrument for assessing the quality of clinics for the elderly. Two domains in this instrument measured basic and complex activities. The highest score was 48, with the score from 16 to 20 being categorised as less dependent, the score from 21 to 35 was categorised as partially dependent, and the score from 36 to 48 categorised as totally dependent.37

#### The Thai version of Resource Utilisation in Dementia

The present study employed the Thai version of Resource Utilisation in Dementia (RUD) as a tool for investigating the reasons why caregivers took the decision to be dementia carers. The RUD V.3.2 was translated into Thai and back-translated which complied with standard psychometric instrument development by using acceptable validity and reliability test.38 The Thai version was used to assess the amount of time given to patients with dementia by their caregivers, and their related characteristics.39 40

The researcher contacted health personnel of Community Comprehensive Dementia Care to obtain information on the dementia population in the study areas. Primary data were collected during a 60 min face-to-face interview at the patient’s home or the place of participant’s choice. The demographic data of both patients and caregivers were asked. The caregivers were asked about the reasons and circumstances for caring for the the elderly with dementia during the past 12 months before the interview in 2017, which included the health services used, such as hospitalisations, outpatient visits to primary healthcare settings and day-care services (home visiting) for both patient and caregiver. The reasons for being informal dementia caregivers were asked. Their health data comprised hours of sleep per day and health services used (hospitalisations, outpatient and home visits). For the health of the elderly with dementia, the researcher asked the caregiver about the dependency level using ADL/IADL items. For the severity of illness, the researcher (NNC) collected the data from the health professional records in each area.

### Statistical analysis

The descriptive statistic was used for the initial analysis. The Kolmogorov statistical test was performed to see whether continuous data were distributed in a normal manner for age, number of family caregivers, caregiver’s hours of sleep per day, number of hospitalisations of the patient and of the caregiver. Analysis of variance (ANOVA) was used to explore the relationship between normally distributed variables such as age. For non-normally distributed variables such as the number of family caregivers, caregiver’s hours of sleep per day, the number of hospitalisations for the elderly patients with dementia and the number of caregiver hospitalisations, the Kruskal-Wallis test was used. The χ2 test was performed for the associations of the dependency level of elderly patients with dementia and household income level with gender, dementia severity, health coverage scheme, hospitalisation of elderly patients with dementia and of the caregiver.

### Ethical consideration

To comply with the research ethics consideration focusing on the confidentiality of data, all documents and electronic records were kept in locked storage. All computer files needed a password code to open. No names of people were mentioned in the transcript. The participants gave informed consent before participation in the study. The present study was funded by the Thailand Research Fund (Grant No. PHD/0049/2556). The funder has no control on the publication of research findings.

## Results

Of the total dementia households in the selected areas 77.8% participated in this study (see figure 3). Table 1 shows the characteristics of patients with dementia and their caregivers. There were more female than male patients in every area (a ratio of 1.7:1), and almost three-quarters of the caregivers were female. The minimum age of the patients was 60 years and the maximum was 104 years, while of the caregivers was 21 years and 53 years. Most patients and caregivers were covered by UCS (about 85%), which became the dominant health benefit41 after 2001. Most patients were vulnerable to financial hardship due to having income less than 1000 baht (around US$30) per month. Of the caregivers, only 5.7% of them reported a monthly income more than 20 000 baht (around US$614).

Table 1
Characteristics of patients with dementia and caregivers
 Area Central North-Eastern Southern Total (%) Characteristics Patients Caregivers Patients Caregivers Patients Caregivers Patients Caregivers Gender Male 16 15 17 11 19 12 52 (37.1) 38 (27.1) Female 37 38 24 30 27 34 88 (62.9) 102 (72.9) Total 53 53 41 41 46 46 140 (100) 140 (100) Age (years) Range 61–100 22–84 60–91 23–84 60–104 30–85 60–104 22–85 Mean 77.6 53.3 77.5 53.9 81.5 54.1 78.9 53.7 SD 10.3 15.2 7.5 14.6 10.1 13.9 9.6 14.5 Health Scheme UCS 48 47 36 38 35 33 119 (85) 118 (84.3) Non-UCS 5 6 5 3 11 13 21 (15) 22 (15.7) Income (Thai baht) 0–1000 33 23 10 12 36 4 79 (56.4) 39 (27.9) 1001–5000 20 18 30 14 7 9 57 (40.7) 41 (29.3) 5001–10 000 0 9 1 9 3 13 4 (2.9) 31 (22.1) 10 001–20 000 0 2 0 4 0 15 4 (2.9) 21 (15) 20 001 and over 0 1 0 2 0 5 0 (0) 8 (5.7)
UCS, universal coverage scheme.
Figure 3
Population and response rate.

When focusing on the caregiver's characteristics (see table 2), more than half of them only completed primary school or lower, while 28.3% of them in the south earned a bachelor’s degree or higher, which is significantly different from the 5.7% of the central, and 9.8% of the north-eastern region (χ2(4)=16.86, p<0.00). Nearly half (47%) of all the caregivers were employed, yet more than half of them in the north-east reported no paid job. The present study found that more than half of the caregivers (55.7%) were suffering from health conditions and reported that they had to take medications provided by health professionals. Most of the caregivers (80%) had children, and 85% of them were living with their elderly patients with dementia while half of them reported that they had to take care of their patients without any extra assistance even when they were busy or ill. About 30% had another person as backup caregiver, while 20.7% of them reported that they had more than one person to help take care of their patients, and not more than five households in each area (7% of total) were fortunate enough to be able to hire respite care services to relieve the caregiver’s burden. The carers were most likely to be the family members, while more than half (58.6%) of them were offspring and a quarter of them were the spouse of patients with dementia.

Table 2
Characteristics of dementia caregivers
 Area Caregivers (n=140) Total Central North-Eastern Southern Educational status Primary level and lower 44 28 23 95 (67.9%) Secondary level 6 9 9 24 (17.1%) Bachelor level and higher 3 4 14 21 (15%) Employment status Employed 27 15 24 66 (47.1%) Unemployed 26 26 22 74 (52.9%) Health conditions No medications prescribed 37 20 23 80 (57.1%) Had medication prescribed 16 21 23 60 (42.9%) Caregivers’ offspring Had offspring 41 34 37 112 (80%) No offspring 12 7 9 28 (20%) Relationship with patient Offspring 30 22 30 82 (58.6%) Spouse 15 11 9 35 (25%) Sibling 1 1 2 4 (2.9%) Others 7 7 5 19 (13.6%) Living conditions Lived with patient 41 38 40 119 (85%) Lived in their household 12 3 6 21 (15%) Support to caregivers No support to caregivers 29 20 21 70 (50%) Had one secondary caregiver 14 14 13 41 (29.3%) Had more than one secondary caregiver 10 7 12 29 (20.7%) Hired respite care services 3 2 5 10 (7.1%)
n, number of participants.

Almost half (47.9%) of the caregivers still had other jobs while taking care of their patients (see table 3). A fifth of them (20.7%) reached retirement age, but still had to take care of their patients, because no one else in the household was available to carry out this role. Almost a quarter of them (23.6%) had to quit their job in order to take care of their patients. Some of them had severe health conditions causing them to quit their jobs but still had to take care of their patients. One had been laid-off, and a few had never worked before becoming a caregiver.

Table 3
Reasons to be the informal caregiver for the elderly with dementia (n=140)
 Reason to be the informal caregiver N (%) No other people to care for the elderly with dementia but can still be working 67 (47.9) Reached the age of retirement 29 (20.7) No other people to care for the elderly with dementia and quit the job 33 (23.6) Suffering from own health conditions 6 (4.3) Had been laid-off 1 (0.7) Never worked 3 (2.1) Desired to early retired 1 (0.7) Total 140 (100)

To identify factors associated with dependency levels (by ADL/IADL) among people who had dementia, many factors were taken to consideration (see table 4). The mean age of the elderly was statistically different between each level of their dependency (ANOVA F(2137)=7.77, p<0.00). The dependency level also had significant associations with the severity of illness (χ2(4)=105.38, p<0.00); with the chance of hospitalisation of the elderly patient with dementia (X2(2)=6.46, p<0.04); also the frequency of hospitalisation of elderly patient with dementia (Kruskal-Wallis test χ2(2)=7.96, p<0.01).

Table 4
Factors associated with the dependency level (n=140)
 Dementia elderly with less dependent (n=32) Dementia elderly with partially dependent (n=51) Dementia elderly with totally dependent (n=57) Test P Age (years) mean (range) 75.09 (60-100) 77.24 (60-104) 82.40 (65-98) 7.77† 0.00* Gender 0.56‡ 0.75 Male n (%) 11 (34.38) 21 (41.18) 20 (35.09) Female n (%) 21 (65.63) 30 (58.82) 37 (64.91) Dementia severity 105.38‡ 0.00* Mild n (%) 28 (87.5) 21 (41.18) 2 (2.51) Moderate n (%) 4 (12.50) 26 (50.98) 11 (19.30) Severe n (%) 0 (0) 4 (7.84) 44 (77.19) Dementia elderly’s health coverage scheme n (%) 3.65‡ 0.72 Universal coverage scheme 29 (90.63) 43 (84.31) 47 (82.46) Social security 0 (0) 1 (1.96) 0 (0) Civil servant medical benefit 2 (6.25) 6 (11.76) 9 (15.79) Others 1 (3.13) 1 (1.96) 1 (1.75) Caregiver’s health coverage scheme n (%) 45 (22.17) 88 (43.35) 70 (34.48) 3.26‡ 0.78 Universal coverage scheme 26 (81.25) 46 (90.20) 46 (80.70) Social security 2 (6.25) 2 (3.92) 3 (5.26) Civil servant medical benefit 2 (6.25) 2 (3.92) 6 (10.53) Others 2 (6.25) 1 (1.96) 2 (3.51) Number of informal caregivers median (Q1-Q3) 1 (0–2) 2 (1–2) 2 (1–3) 5.0§ 0.08 Caregiver’s sleep hours per day (hours) median (Q1-Q3) 7.5 (6.25–9) 8 (6–9) 7.5 (6.5–8.5) 0.89§ 0.64 Whether dementia elderly had hospitalisation 6.46‡ 0.04* Yes n (%) 3 (9.38) 2 (3.92) 11 (19.30) No n (%) 29 (90.63) 49 (96.08) 46 (80.70) Whether the caregiver had hospitalisation 4.95‡ 0.08 Yes n (%) 2 (6.25) 2 (3.92) 9 (15.79) No n (%) 30 (93.75) 49 (96.08) 48 (84.21) Dementia elderly’s number of times of hospitalisations median (Q1-Q4) 0 (0–5) 0 (0–6) 0 (0–17) 7.96§ 0.01* Caregiver’s number of times of hospitalisations median (Q1-Q4) 0 (0–4) 0 (0–3) 0 (0–30) 3.58§ 0.17
*P<0.05.
†ANOVA.
‡χ2.
§Kruskal-Wallis test.
ANOVA, analysis of variance.

Focusing on the household per capita income as socioeconomic status, 48.6% of them were the poorest, 35.7% were poor and only 15.7% had a monthly income of 20 001 baht and above. Table 5 presents the factors associated with the level of household income. The income level was significantly associated with dementia severity (χ2(4)=10.53, p<0.03), the health benefit coverage of the elderly with dementia (X2(4)=12.74, p<0.01) and the chance of hospitalisation of the elderly with dementia (X2(2)=7.09, p<0.03).

Table 5
Factors associated with the level of household income (n=140)
 Income 6500 baht and lower per month (n=68) Income 6501–20,000 baht per month (n=50) Income 20 001 and higher per month (n=22) Test P Dementia severity 10.53† 0.03* Mild n (%) 31 (45.59) 17 (34) 3 (13.64) Moderate n (%) 20 (29.41) 15 (30) 6 (27.27) Severe n (%) 17 (25) 18 (36) 13 (59.09) Dementia elderly’s health coverage n (%) 18.51† 0.01* Universal coverage scheme 64 (94.12) 40 (80) 15 (68.18) Social security 1 (1.47) 0 (0) 0 (0) Civil servant medical benefit 3 (4.41) 7 (14) 7 (31.82) Others 0 (0) 3 (6) 0 (0) Number of informal caregivers median (Q1-Q3) 1 (0–2) 2 (1–3) 2 (1–2) 4.70‡ 0.1 Caregiver’s sleep hours per day (hours) median (Q1-Q3) 8 (7–9) 7.5 (6–9) 7 (6–7.5) 8.01† 0.02* Whether dementia elderly had hospitalisation 7.10† 0.03* Yes n (%) 12 (17.65) 1 (2) 3 (13.64) No n (%) 56 (82.35) 49 (98) 19 (86.36) Dementia elderly’s number of times of hospitalisations median (Q1-Q4) 0 (0–17) 0 (0–6) 0 (0–6) 2.35‡ 0.31
*P<0.05.
†χ2.
‡Kruskal-Wallis test.
ANOVA, analysis of variance.

## Discussion

The main findings of our study provide insight into the issue of the current circumstances of caregivers, patients and formal-informal care in the Thai rural community. The discussions of this study are as follows:

The first concern on informal caregivers is their sandwich generation. More than half of caregivers were the adult children of patients with dementia, and most of these caregivers also had their own offspring. The multigenerational household means that the caregiver could be caught between the needs of the elderly patient and the needs of their own children.42 The caregivers are mostly people of working age.43 The sandwich generation caregivers have many problems and concerns such as financial difficulties, physical and mental wellness, legal and management issues, and mental health issues.42 44 45

The next concern is the pattern of an unhealthy lifestyle due to caregiving responsibilities. Despite the absence of statistical significance between the level of dependency and the caregiver’s sleep hours per day in this study, one in seven caregivers (13.6%) reported that they had unhealthy sleep patterns (less than 6 hours sleep per day). Among these caregivers with low sleep levels, 21.1% of elderly patients with dementia were less dependent, 36.8% partially dependent and 42.1% totally dependent on their caregivers. The finding in table 4 found that the first quartile of the caregiver’s hours of sleep was lowest at the partially dependent stage as 6 hours per day. In these circumstances, the caregivers had the lowest numbers of hours of sleep as these patients with dementia were the most active and could exhibit psychotic symptoms before declining to the stage of severe dementia.46 Khoo et al 47 also found that caregiver distress was found the most in cases of moderate dementia. The present study also found higher-income households had significantly shorter caregiver’s sleep hours (see table 5), as the higher-income households had greater demand for informal care and greater capacity to support the living issues. The higher-income households delivered more dementia service activities to their elderly, hence reduced caregivers’ hours of sleep. During data collection in the present study, the researcher (NNC) observed that the higher-income households also had greater concerns on privacy matters and ignored community support provided by the informal caregivers while the lower-income households used more of the community support, such as ADL caring services.

Interestingly, the present study found that families with lower incomes are inflicted with the less severe cases of dementia, instead of the higher-income households who face a lower risk of cognitive impairment as the studies conducted in urban areas in the Western countries such as by Fotenos et al 19 and Wee et al .48 Studies were conducted in restricted financial resources but with high piety value. Like in the present study, it was discovered that most high-income households delivered more assistance services with low emotional support for their parents49–51 but the low-income households that provided more emotional support with less material assistance52 had slowed down cognitive impairment.53 The present study suggested further research on whether the opportunity to delay the stages of dementia severity (as found in the high-income group) may be lost if the Thailand dementia care plan did not address the dementia care literacy issue. The benefits would be for all households, rural or urban communities, and higher-income or lower-income households.

While the less severe cases were found more in the lower-income households, higher rates of hospitalisations were found in the lower-income households as shown in table 5. Most of the low-income households’ hospitalisations were covered by the UCS and they might see hospitalisation as the option for respite care with no out-of-pocket expenses compared with the higher-income households who could afford the private respite care option.

Despite the high hospitalisation rate of low-income households, the average hospitalisation rate that the present study found at 11.4% (16/140) was comparable with the findings of Liu et al ,54 who studied elderly patients with chronic diseases who live in rural areas with low access to healthcare services. The low admission rate found in the present study implies that the ambulatory health services for the patients with dementia at subdistrict health promoting hospitals in rural areas in Thailand were of good quality with a very low co-payment for both ambulatory and hospitalisation services and not a strict gatekeeping rule for hospitalisation. Bodenheimer et al 55 argued that the primary care system should focus more on the coordination of care than the gatekeeper role. Moreover, the health literacy assessment towards utilisation of health services has to be considered for impending changes in the health systems, as the equity objective of the universal health coverage should reach all populations.56

There is a need for informal and formal care coordination. The present study found that the informal caregiver played the significant role of informal care on the elderly with dementia as all of the patients were living in a rural community with their relatives while health professionals provided formal support for both the elderly and the caregiver. Tuppin et al 16 agreed that informal care complemented formal care, and most patients with dementia relied on informal care. Formal care played an essential role in supporting and guiding efficient informal care as Bremer et al 57 found that proper informal caregiving led to lower healthcare spending. Nonetheless, the informal sector needs proper health literacy from the formal sector58 and the two systems have to support each other along the way for the best dementia care planning and the management of elderly patients with dementia. The present study found that more than half of the caregivers were unemployed and the caregivers with good health literacy could improve informal dementia care in Thailand. Payment incentives could improve the quality of informal care, as noted in a study by Conrad and Perry.59

Implications for policymakers were that they should consider designing both formal and informal care plans in accordance with caregivers’ social constraints and their patients’ dependency. The multitasked responsibilities, especially in the lowest socioeconomic group found in the present study suggested that the policymakers when improving dementia care in Thailand, should consider informal care for elderly Thai patients and promote it as the dominant pattern of dementia care as the informal caregivers are the principal human resources for dementia care and services in the current circumstances.

### Limitations and generalisability: the community is more significant than the household

This study is probably the first to explore the burden of dementia in rural communities in Thailand. Furthermore, the present study was undertaken in the villages to contribute to the development of the national LTC policy in Thailand.

Of the population of elderly patients with dementia in the present study, 22% of them were non-responders because caregivers were absent or patients simply lacked caregivers. Most of these non-responders were in the severe level of dementia and in urgent need of sufficient care. The underestimated situations of the present study should be investigated further.

Moreover, the present study investigated circumstances in Community Comprehensive Dementia Care.34 Cautions on implications of the findings should be acknowledged. First, as most of the areas of Thailand still lacked a comprehensive dementia care plan, the situations of dementia households might not be the same as these selected areas. Second, different managements of the local governments having different laws60 does affect the availability of social services for the elderly with dementia in households. Third, each area in the present study had different geographical and climate issues affecting the population, occupation and income. For example, households in the southern region located in the tropical forest area participating in rubber and coconut agriculture, had a higher income as compared with households in other regions.

Thailand had a strong and highly admired piety value; the strong piety value contributes to the strong belief in Thai society to ostracise those who neglect their sick and elderly. A recent study stated that ‘in the moral aspect, the individuals should be responsible for their dependent elderly. If they neglect their duty, they should be penalised with some allegations’.61 As it is unacceptable in Thai society, the government shelter home for older persons does not have a popular official name. Another study conducted in the community-based setting revealed that there were many elderly with chronic diseases living in the elderly shelter setting, with no health personnel to take care of.62 The present study did not include samples in any older persons’ home since there was no such home located in the investigation areas. This study included only the elderly with dementia who had caregivers to give the data.

Even if dementia is an unpreventable and incurable illness2 it should be noted that formal health and social sectors in the Thai community had the potential to encourage everyone to practise cognitive skills before anyone in the community is afflicted with dementia.34 63

## Conclusion

The formal health sectors are well established in Thailand, but the informal care circumstances, especially in dementia care, portray a different story. The informal caregivers are the principal human resources for dementia care and services in rural areas; policymakers should consider informal care for the Thai elderly with dementia and promote it as the dominant pattern of dementia care in Thailand. Although, there were factors associated with the dependency levels of the elderly patients with dementia, age, dementia severity, hospitalisations and the frequency of hospitalisations of the elderly had significant associations with the level of dependency. Also, dementia severity, the health benefit scheme of elderly patients with dementia, the caregiver’s sleep hours per day and hospitalisation of the patients with dementia had significant associations with their income status. Dementia caregivers in Thailand were mostly family members of patients with dementia, and they are faced with multiple responsibilities.

## Notes

Contributors:: All authors conceived the research questions and the concepts of the steps of the study. NNC conducted the field study with caregivers with close guidance from SC. NNC, PP and SP analysed data to answer research questions. All authors have read and agree on the final version of manuscript.
Funding:: This research is funded by Thailand Research Fund through the Royal Golden Jubilee PhD Program(Grant No. PHD/0049/2556) to Miss Nalinee Nakittipha Chuakhamfoo and Professor Supasit Pannarunothai.
Map disclaimer:: The depiction of boundaries in this map does not imply the expression of any opinion whatsoever on the part of BMJ (or any member of its group) concerning the legal status of any country, territory, jurisdiction or area or of its authorities. This map is provided without any warranty of any kind, either express or implied.
Competing interests:: None declared.
Patient and public involvement:: Patients and/or the public were not involved in the design, or conduct, or reporting, or dissemination plans of this research.
Patient consent for publication:: Not required.
Ethics approval:: This cross-sectional study was approved by the Naresuan University Institutional Review Board document number 0251/60.
Provenance and peer review:: Not commissioned; externally peer reviewed.
Data availability statement:: Data are available upon reasonable request. Individual data from the survey may be shared under the restricted condition of not exposing confidentiality of the study participants.

## References

1

World Health OrganizationFirst WHO ministerial conference on global action against dementia: meeting report; 16-17 March 2015, WHO Headquarters, Geneva, Switzerland, 2015.

2

Alzheimer’s Association. 2015 Alzheimer's disease facts and figures. Alzheimers Dement2015;11:, pp.332–84., doi: 10.1016/j.jalz.2015.02.003

3

Lokody IBeyond art, beyond sciences: the many faces of dementia. Elsevier, 2018.

4

World Health OrganizationWorld report on ageing and health. World Health Organization, 2015.

5

Gräsel E, Wiltfang J, Kornhuber J. Non-Drug therapies for dementia: an overview of the current situation with regard to proof of effectiveness. Dement Geriatr Cogn Disord2003;15:, pp.115–25., doi: 10.1159/000068477

6

Brody JA. An epidemiologist views senile dementia--facts and fragments. Am J Epidemiol1982;115:, pp.155–62., doi: 10.1093/oxfordjournals.aje.a113286

7

Yaffe K. Modifiable risk factors and prevention of dementia: what is the latest evidence?JAMA Intern Med2018;178:, pp.281–2., doi: 10.1001/jamainternmed.2017.7299

8

Asia Pacific Members of Alzheimer's Disease International, Alzheimer's Australia. Dementia in the Asia Pacific region; the epidemicis hereAccess Economics; 2014.

9

Woncha-um P, Lefevre AS, Chankaew PThailand bets on 'magic pills' to boost declining birth rate. Reuters, 2018.

10

Gillett JE, Crisp DA. Examining coping style and the relationship between stress and subjective well-being in Australia's 'sandwich generation'. Australas J Ageing2017;36:, pp.222–7., doi: 10.1111/ajag.12439

11

Sörensen S, Duberstein P, Gill D, et al. Dementia care: mental health effects, intervention strategies, and clinical implications. Lancet Neurol2006;5:, pp.961–73., doi: 10.1016/S1474-4422(06)70599-3

12

Harris J. Recognizing pain and preventing aggression in patients with dementia. Caring for the Ages2017;18:, pp.3, doi: 10.1016/j.carage.2017.03.004

13

Chenoweth B, Spencer B. Dementia: the experience of family caregivers. Gerontologist1986;26:, pp.267–72., doi: 10.1093/geront/26.3.267

14

Covinsky KE, Newcomer R, Fox P, et al. Patient and caregiver characteristics associated with depression in caregivers of patients with dementia. J Gen Intern Med2003;18:, pp.1006–14., doi: 10.1111/j.1525-1497.2003.30103.x

15

Phelan EA, Borson S, Grothaus L, et al. Association of incident dementia with hospitalizations. JAMA2012;307:, pp.165–72., doi: 10.1001/jama.2011.1964

16

Tuppin P, Kusnik-Joinville O, Weill A, et al. Primary health care use and reasons for hospital admissions in dementia patients in France: database study for 2007. Dement Geriatr Cogn Disord2009;28:, pp.225–32., doi: 10.1159/000238394

17

Tadokoro K, Sasaki R, Wakutani Y, et al. Clinical characteristics of patients with dementia in a local emergency clinic in Japan. Geriatr Gerontol Int2018;18:, pp.1383–7., doi: 10.1111/ggi.13496

18

Prince M, Wimo A, Guerchet MWorld Alzheimer report 2015. The global impact of dementia. An analysis of prevalence, incidence, cost and trends. London: Alzheimer's Disease International, 2015.

19

Fotenos AF, Mintun MA, Snyder AZ, et al. Brain volume decline in aging: evidence for a relation between socioeconomic status, preclinical Alzheimer disease, and reserve. Arch Neurol2008;65:, pp.113–20., doi: 10.1001/archneurol.2007.27

20

Kim J-S, Lee E-H. Cultural and noncultural predictors of health outcomes in Korean daughter and daughter-in-law caregivers. Public Health Nurs2003;20:, pp.111–9., doi: 10.1046/j.1525-1446.2003.20205.x

21

Lee J, Sohn BK, Lee H, et al. Impact of behavioral symptoms in dementia patients on depression in daughter and daughter-in-law caregivers. J Womens Health2017;26:, pp.36–43., doi: 10.1089/jwh.2016.5831

22

Aekplakorn W, Pakcharoen H, Sathiennopaklao W, et alThe fifth Thai National health exam survey. Health Systems Research Institute, 2016.

23

Sanders S, Ott CH, Kelber ST, et al. The experience of high levels of grief in caregivers of persons with Alzheimer's disease and related dementia. Death Stud2008;32:, pp.495–523., doi: 10.1080/07481180802138845

24

Roche V, Palmer BF. The hidden patient: addressing the caregiver. Am J Med Sci2009;337:, pp.199–204., doi: 10.1097/MAJ.0b013e31818b114d

25

National Statistical OfficeThe 2014 survey of the older persons in Thailand. statistical forecasting bureau national statistical office the government complex, Ratthaprasasanabhakti building. 245Bangkok, Thailand: Statistical Forecasting Bureau National Statistical Office, 2014.

26

Worldometers.info. Wi team, edThe Thailand population (live). Worldometers, 2016.

27

Hu H-M. Facing an aging Society: Taiwan's universities in crisis. Gerontol Geriatr Educ2018:, pp.1–9., doi: 10.1080/02701960.2018.1428576

28

Thairath OnlineThailand met an elderly dementia than four hundred thousand people. 18Thairath, 2015.

29

The National Commission on the Elderly The Ministry of Social Development and Human Security Thailand. The Ministry of Social Development and Human Security Thailand, edThe 2nd national plan on the elderly 2002-2021. Bangkok: The Ministry of Social Development and Human Security Thailand, 2009.

30

Chuakhamfoo N, Pannarunothai S. Long-term care: choices for Thailand as a middle-income country. J Community Dev Res2016;9:, pp.15–32.

31

Morrison ASScreening in chronic disease. USA: Oxford University Press, 1992.

32

Aselage MB, Amella EJ. An evolutionary analysis of mealtime difficulties in older adults with dementia. J Clin Nurs2010;19:, pp.33–41., doi: 10.1111/j.1365-2702.2009.02969.x

33

World Health OrganizationDementia: a public health priority. World Health Organization, 2012.

34

The Public Health Commission of the National Legislative Assembly of ThailandReport of the public health Commission on the consideration of the comprehensive dementia care. Bangkok, Thailand: The Secretariat of the Senate of Thailand, 2018.

35

Poomontre J. Standardization of socio-economic status (Ses) classification in Thailand market research industry. Proceedings of The 2nd Asia Pacific Research Committee Conference (ARPC), 2010.

36

National Health Commission Office of Thailand. The development of long-term care for elderly in Thailand. The 2nd National Health Assembly; 16 Nov 2012, Nonthaburi, 2012.

37

Subcommittee on driving long-term care issues for the elderly National Elderly Committee. The assessment for activities of daily living and instrumental ADL (ADL/IADL); 2012.

38

Chuakhamfoo N, Pannarunothai S, Phantunane P, et al. The validity and reliability of the resource utilization in dementia 3.2 Thai for assessing cost of dementia care. J Community Dev Res2019;12:, pp.1–10.

39

Wimo A, Gustavsson A, Jönsson L, et al. Application of resource utilization in dementia (RUD) instrument in a global setting. Alzheimers Dement2013;9:, pp.429–35.e17, doi: 10.1016/j.jalz.2012.06.008

40

Wimo A, Jonsson L, Zbrozek A. The resource utilization in dementia (RUD) instrument is valid for assessing informal care time in community-living patients with dementia. J Nutr Health Aging2010;14:, pp.685–90., doi: 10.1007/s12603-010-0316-2

41

Tangcharoensathien V, Witthayapipopsakul W, Panichkriangkrai W, et al. Health systems development in Thailand: a solid platform for successful implementation of universal health coverage. Lancet2018;391:, pp.1205–23., doi: 10.1016/S0140-6736(18)30198-3

42

Burke RJ1. The sandwich generation: individual, family, organizational and societal challenges and opportunities. The sandwich generation: caring for Oneself and others at home and at work. 3, 2017.

43

Cheal DAging and demographic change. Canadian public Policy/Analyse de Politiques, 2000: , pp.S109–22.

44

Roots CRThe sandwich generation: adult children caring for aging parents. Routledge, 2014.

45

Aazami S, Shamsuddin K, Akmal S. Assessment of Work–Family conflict among women of the sandwich generation. J Adult Develop2017;25:, pp.1–6.

46

Hamuro A, Isono H, Sugai Y, et al. Behavioral and psychological symptoms of dementia in untreated Alzheimer?s disease patients. Psychogeriatrics2007;7:, pp.4–7., doi: 10.1111/j.1479-8301.2006.00153.x

47

Khoo SA, Chen TY, Ang YH, et al. The impact of neuropsychiatric symptoms on caregiver distress and quality of life in persons with dementia in an Asian tertiary hospital memory clinic. Int Psychogeriatr2013;25:, pp.1991–9., doi: 10.1017/S1041610213001518

48

Wee LE, Yeo WX, Yang GR, et al. Individual and area level socioeconomic status and its association with cognitive function and cognitive impairment (low MMSE) among community-dwelling elderly in Singapore. Dement Geriatr Cogn Dis Extra2012;2:, pp.529–42., doi: 10.1159/000345036

49

Xie Y, Zhu H. Do sons or daughters give more money to parents in urban China?J Marriage Fam2009;71:, pp.174–86., doi: 10.1111/j.1741-3737.2008.00588.x

50

Lin JThe change trend of intergenerational relations in Taiwan. (In Chinese). Taipei: Taiwan Social Change, Institute of Sociology, Academia Sinica, 2012.

51

Lin J. Intergenerational solidarity between the rural elderly and their adult children. J Taiwan Home Eco2000;29:, pp.32–58.

52

Lin J-P, Yi C-C. Filial norms and intergenerational support to aging parents in China and Taiwan. Int J Soc Welf2011;20:, pp.S109–20., doi: 10.1111/j.1468-2397.2011.00824.x

53

Bassuk SS, Glass TA, Berkman LF. Social disengagement and incident cognitive decline in community-dwelling elderly persons. Ann Intern Med1999;131:, pp.165–73., doi: 10.7326/0003-4819-131-3-199908030-00002

54

Liu Z, Albanese E, Li S, et al. Chronic disease prevalence and care among the elderly in urban and rural Beijing, China - a 10/66 Dementia Research Group cross-sectional survey. BMC Public Health2009;9:394, doi: 10.1186/1471-2458-9-394

55

Bodenheimer T, Lo B, Casalino L. Primary care physicians should be coordinators, not gatekeepers. JAMA1999;281:, pp.2045–9., doi: 10.1001/jama.281.21.2045

56

Pannarunothai S, Srithamrongsawat S. Benchmarks of Fairness for health system reform: the tool for national and provincial health development in Thailand. Hum Resour Health Dev J2000;4:, pp.13–26.

57

Bremer P, Challis D, Hallberg IR, et al. Informal and formal care: substitutes or complements in care for people with dementia? empirical evidence for 8 European countries. Health Policy2017;121:, pp.613–22., doi: 10.1016/j.healthpol.2017.03.013

58

Ringer TJ, Wong-Pack M, Miller P, et al. Understanding the educational and support needs of informal care-givers of people with dementia attending an outpatient geriatric assessment clinic. Ageing Soc2020;40:, pp.205–28., doi: 10.1017/S0144686X18000971

59

Conrad DA, Perry L. Quality-based financial incentives in health care: can we improve quality by paying for it?Annu Rev Public Health2009;30:, pp.357–71., doi: 10.1146/annurev.publhealth.031308.100243

60

Krueathep W. Local government initiatives in Thailand: cases and lessons learned. Asia Pac J Public Adm2004;26:, pp.217–39., doi: 10.1080/23276665.2004.10779294

61

Kajornchaikul NCriminalization on abandon or improperly taking care of dependent elders. Chulalongkorn university, 2016.

62

Knodel J, Chayovan N, Siriboon S. The impact of fertility decline on familial support for the elderly: an illustration from Thailand. Popul Dev Rev1992;18:, pp.79–103., doi: 10.2307/1971860

63

Graff MJL, Vernooij-Dassen MJM, Thijssen M, et al. Community based occupational therapy for patients with dementia and their care givers: randomised controlled trial. BMJ2006;333:, pp.1196, doi: 10.1136/bmj.39001.688843.BE

Citing articles via
https://www.researchpad.co/tools/openurl?pubtype=article&doi=10.1136/bmjopen-2019-032637&title=Health and long-term care of the elderly with dementia in rural Thailand: a cross-sectional survey through their caregivers&author=Nalinee N Chuakhamfoo,Pudtan Phanthunane,Sirintorn Chansirikarn,Supasit Pannarunothai,&keyword=dementia,care-giver,lone-term care,health service use,Thailand,&subject=Health Policy,1506,1703,

format: plain


text:
# Modified Fort Space is not Locally Connected

## Theorem

Let $T = \struct {S, \tau_{a, b} }$ be a modified Fort space.

Then $T$ is not locally connected.

## Proof

We have:

Modified Fort Space is Totally Disconnected
Totally Disconnected and Locally Connected Space is Discrete

We also have:

Modified Fort Space is not $T_2$
Discrete Space satisfies all Separation Properties

Hence modified Fort space is not the discrete space, and the result follows.

$\blacksquare$


text:
emerge is a tool to build the KDE sources and its third-party requirements on MS Windows. It is the easy way to build KDE on MS Windows.

## Setting up emerge

### Setting up a compiler

Currently emerge supports both the MinGW and Microsoft Visual Studio (msvc) compiler. While MinGW is can be installed by emerge, Visual Studio must be installed independently by the user.

### Direct X SDK

In order to compile the Qt5 qtbase package with MinGW you will also need to install the Microsoft DirectX SDK, make sure to open a new command line window after the installation.

### Installing Craft

• Start a powershell environment as administrator.
• Allow execution of powershell scripts.
``` Set-ExecutionPolicy RemoteSigned
```
• Exit the administrator powershell instance.
• Start a powershell environment as a user.
• Install emerge and folow the instructions
```iex ((new-object net.webclient).DownloadString('https://raw.githubusercontent.com/KDE/craft/master/setup/install_craft.ps1'))
```

## Using emerge

To use emerge you need to start a Powershell window, point that to KDEROOT\emerge and run the initalization script. For example:

```C:\KDEROOT\emerge\kdeenv.ps1
```

This tells emerge about your environment settings (e.g. paths). It will load your configuration from KDEROOT\etc\kdesettings.ini. If there are any error messages emerge will not work as expected. The output should look similar to this one (of course with your paths):

```PS C:\kderoot\emerge>.\kdeenv.ps1
KDEROOT     : C:\kderoot\emerge
KDECOMPILER : msvc2015
PYTHONPATH  : C:\kderoot\python

PS C:\kderoot\emerge>
```

### Common emerge commands

• Installing a package and its dependencies: Simply run
```emerge packagename
```
• Updating an installed package: Once you have packagename built, type
```emerge -i packagename
```
to update packagename.

## Troubleshooting

If a package fails to build, you'll be greeted with something like:

``` ...
emerge warning: while running make cmd: jom
emerge warning: Action: compile for libs/qt5/qtbase FAILED
*** Emerge all failed: all of libs/qtbase failed after 0:07:25 ***
emerge error: fatal error: package libs/qtbase all failed
```

In order to figure out what failed, grep the command line output above for errors.

If the log output is too long, you can also pipe the output to a file and then open the file inside an editor.

``` emerge qt > log 2>&1


text:
# With Free Messaging Highest Rated Online Dating Website In Utah

Recommended for steakhouses because: the provisions maybe just a little better, service just a little sweeter and the surroundings just a smidge comfier than many other city steakhouses. Therefore through his knowledge was the will, through his will was the desire, through his desire was the ordinance, through his ordinance was the decree, and through his decree was the carrying out. Railroad cabooses from the southern pacific, santa fe, and great northern railroads have been converted into rooms, leaving their pipes, ladders, and lofts in place. 6:38 pm kje no , wasnot crude tor – just not funny 6:38 pm pam kje, i am sooooo psyched to see rimbaud on your page….merci…. Urdu poetry, as it is derived from persian, turkish and arabic, acquired many conventions in its poetry that came from these languages. During the investigation, with free messaging highest rated online dating website in utah they will face with a difficult riddles and come into confrontation with the cruel monsters, they often have to make difficult decisions. A new genus and species of urostylidae from tonkin in the collection of british museum (n. h.).- bulletin of the fan memorial institute of biology, zoology 7: 159-165. The solution was degassed with argon and heated to 80 °c for 6 h. Because essential hypertension requires a lifelong drug regimen, it is extremely dif cult but necessary to impress on patients the need for compliance with their therapeutic regimen. Ballycastle is also on the edge of an irish speaking area called a gaeltacht. Julia was chosen as the only actor on the “25 new faces of independent film of 2012” by filmmaker magazine. We will solve machine problems even after sale and provide you necessary services and also supply important spare parts for smooth working. Fixed an issue with a crash that could occur when host migrating and returning to towns. Brother umblachery cattle ate mine 14k two tone gold simulated alexandrite elegant june birthstone ring. Sufficient monitoring and protection of assets along with durability is achieved by best practices in corrosion monitoring systems. \item[backrefpage] an expression like <see page \prm{page} or <cited on page \prm{page}, used to introduce back references in the bibliography. In humans, as well as many other species, ganglion cells are the first to be born. Some truly good blog posts on this website, appreciate it for contribution. In 2007 24 birds carried transmitters, and 33% of these dispersed from the managed area within the first week. On subsequent reporting dates, 12-month ecl also applies to existing loans with no significant increase in credit risk since their initial recognition. Trivia amu’s room, and most of her accessories, are decorated with hearts, clovers, spades, and diamonds. If you do not wish to modify the output preferences, you possibly can very effectively use the default settings and press the convert button to obtain your mp3 recordsdata inside moments. Drie decennia koninklijke nedlloyd. 18381: oosthout, henri – modes of knowledge and the transcendental. The switch is optional. 73308 is an optional switch that you can buy to go with this system but we do not sell them. Nor absence of due process safeguards. do we find it necessary to reach the issue now in light of the record before us. Thank you bank med for the past 2years plus, in supporting us in all the trial and tribulations we had to endure in this time. In front of it there were dough balls that my dad had made, rising under a tea towel. “soldier’s letter/wm b hatch/lieut col 4 reg n.j.v.” ms at upper r on cvr. His 2017 project for ceci has had him answer probably the same question running on his female fans. Mast cells are a type of immune system cell found in skin and other organs and normally play a role in inflammation and allergies. Manchester england new york new york: manchester university press distributed exclusively in the usa and canada by st. martin’s press. Vaioshell.exe is scheduled as a task with the class ‘{d03f29ea-2782-4c2d-925d-c8ba1ccb2e25}’ (runs on registration). Simply dressed the one proclaimed as the small axe delivered modern day judas, which got resounding musical shots from fans who sang along with him. Tensile results obtained from the alloys in various conditions are given in the foregoing table i.

## With Free Messaging Highest Rated Online Dating Website In Utah

Systems like isdn mixes [38] were designed for other environments with free messaging highest rated online dating website in utah with different assumptions. The pack featured a fashion mask and a cape, neither of which were very useful, as well as a multitude of basic supplies (scrolls of identification, potions, portable altars, etc.). It was established in 1903 to preserve a series of limestone caverns and a tract of unspoiled prairie grassland in the black hills. Thus, the type i allergic reaction against foreign antigens leads to onset of symptoms through the above reaction steps. … Level: 1st and 2nd grades this practice worksheet has pictures of kid-friendly monsters. Mr. bulkley retired from the partnership in july, 1851, when, with william h. mellen and several of his principal clerks, he continued his business as claflin, mellen & This was the nazi party line which was evolved when the america first committee disbanded, and thus ostensibly ended its career of appeasement and psychological sabotage. It can be categorized into hemorrhage occurring early or late, and has different causes and may be treated differently. The task force asked the same group of children and teenagers to read the simplified terms and found they understood them far more clearly. This means that we first go x/2 meters to the north, then y meters to the east, and again x/2 meters to the north. 3 under device guides, select the documentation for your wireless network card. Celebrating christmas 2004 in the wasteland without church, the teams reminisce about the previous christmas, and how he ruined it for… He does the washing up and the cleaning of the monks place before going to school. Many time we had to face public humiliation for daring to stand up against her methods. The classification of skateboard includes park boards, cruiser boards, longboard and other, and the …… | read more… At that time us soldiers were being killed daily as a result of an insurgency of the iraqi people. Sneaking up next to a plant with heroic kick can also catch them by surprise. August 25, scientific of woman applies a universal distance of models that i caused to record. It was a giant jet-black drag-ride that had the same colour as the sky. 21 took time off to not only nurse a hip injury but, also, to wait for the arrival of his and wife sara’s second child together. The indian communities in overseas territories of france: problems of identity and culture.

After his loss to liu kang in the first tournament, it is revealed that shang tsung is actually a subordinate of shao kahn, the true ruler of outworld. It could not be determined sunday night whether pimentel had retained an attorney. Grab some wood and light the fire pit or relax in the hammock under the lemon trees. Shaftoe stomps up the narrow staircase to find bischoff’s room stripped to the bedsprings, and a pile of gold coins on the table, weighing down a thank you note addressed to the owners. Prison provides dynamic and establish generalizability of violence, myocardial infarction etc. Open the universe pool, tap at the yellow ball in the title bar and tap at sheet style. Darkwing also wasnot above recycling scripts from its fellow disney afternoon shows. Personally, i think oshii wanted to communicate the lie of a present and worthwile god, but the reality of a power above us. In some cases there is something that prevents this from working correctly for a handful of machines for any reason (turned off, firewall, rights, wmi broken) and they will stay in pending management. It is to be found in our apprehension of the world of predecessors as well. They were great with letting me donate it to a battered women’s shelter instead. In addition, compressional and shear wave velocities were measured with a propagation direction parallel to the axis of the sample. The book rages against children with free messaging highest rated online dating website in utah being exposed to violence on both television and computer games. Intraday history of 2 months (1min format), tick (2 days) and eod history since 2010 is available need more history ? This year, sunday racegoers will spend no more than $99 for an individual grandstand ticket, with prices starting at just$39. I can not tell you homw many times i was playing a game and i get a skype call and it kills my game. A log crib is a framework of tree trunks or beams filled with dirt and rock (see figure 5-16). Even if you need to leave in the immediately, taking a few extra moments to call competing airlines or to check competing websites can save you quite a bit. Some scholars even consider the expulsion of the entrepreneur to be essential for the progress of the business. Wroclaw – poludnie district, (142 rooms) opening end of 2021 – info: click here new – Ghazi said that an international humanitarian agency attempted unsuccessfully to visit him three times, and that his family and lawyer were not allowed to visit him for his first week in detention. The book of hallowe’en (1919), and references souling in the chapter “hallowe’en in america”. Vegetarianism. and a scene afterwards split in two. men enquire what he has left behind him. Dengan latihan cara memakai make up beberapa kali, dijamin tidak ada yang tahu sebenarnya anda masih kangen dengan bantal anda. The same solute is diffusing through the case same solvent in each of three cases. Pacheco: asset prices and monetary policy in the euro area: a tentative model 11428 135 jan/27/13 10:03 am id:16990 — li; Burly outer knobs provide the support needed to keep them digging while slapping berms or spraying loam. There are sometimes, but not always, flu-like symptoms in the first 1 to 2 weeks after infection.

Star wars: the rise of skywalker is available to pre-order on dvd , blu-ray and 4k now. Le vendredi 26 avril 2019, 16:10 par tinyurl.com i was recommended this web site by way of my cousin. Understanding what shaped them, what drove them, and what set them upon one another requires traveling back to a time when the battles involved little more than a pair of boxing gloves. Guards are either static or moving, | 03 | elixir | the latter type requiring isaac to dodge their path. Names of both gadis’ acharyas and their pictures are also located on this website. Under history, click delete history, and then click yes to confirm that you with free messaging highest rated online dating website in utah want to delete the history. Periphytic invertebrate community on artificial substrata in a floodplain lake // book of abstracts and programme. The flower tube is composed of fused tepals that ends with 3 petal-like projections that are brownish towards their ends and dark purple toward the centre. The verdict was that lenore bowlby came to her death by a pistol fired by her own hand with suicidal intent. Ive also read somewhere years ago that most motorcycle manufacturers do anodize their swing arms so?… Many tv shows distort perceptions of class by not accurately representing the middle class. Conclusions there is no evidence that immersion in water during the first stage of labor improves perinatal outcomes. First season cast members poppy montgomery, dylan walsh and jane curtin are all on board to return, while james laio and tawny cypress (“heroes”) have been added in a recurring capacity. Save for the disastrous defeat at the battle of tunis in africa, and the early naval defeats, the first punic war was a nearly unbroken string of roman victories. Unfortunately, the decision makers in those establishments have passed their productive useful period and are with degenerated brains. It is therefore possible that lewis felt more comfortable to with free messaging highest rated online dating website in utah speculate about language in this new non-philological environment. In the end, he declined to be interviewed, although i never found out why. Near mint condition 1993 gibson custom shop advanced jumbo montana special, number 21 of only 36 made. This vacation condo is located along the western shores of south maui on the longest sandy beach known as sugar beach. (tales of suspense i#49/2) – somehow aware he was a comic book character on earth-1218 (“our” reality), uatu addressed the readers to inform them of the threat of the alien sneepers. This was attributed to damage to thalamic and mesencephalic structures. He has also interviewed some people close to the band, and that adds a great asset! After the meeting leamington with ao sellana i had an eye on with free messaging highest rated online dating website in utah 6′ x 42′ x 2″ e-z rolltm floor exercise foam from american athletic, inc. xxx. Ivf supervised by: dr. shafik saleh management of all gynecological cases of plastic surgery, laparoscopy, all cases of fibroid, all problems of puberty and menopause. Ct scan or cystoscopy is usually done first to evaluate potential bladder cancer. This means that applicants cannot fill in an application online or hand in the application at a visa application center (vfs). 🙂 it’s frustrating to get very good interest rates on financial savings nowadays. the banks are certainly not providing really good rates thanks for one more excellent post. He never wasted any of his time, or his physical or mental capital, as a boy, and thought out all matters for himself, and that habit with free messaging highest rated online dating website in utah has followed him all his life. Pint-sized ‘jersey shore’ star, snooki, has confirmed that she is pregnant and engaged. In microsoft with free messaging highest rated online dating website in utah powershell scripting language, cmdlets are used to perform three operations: collect information, set information, and delete information.

Keywords: pet, quantitative image analysis, idif 213 radiographic quantitative values measured from 18f-naf pet/ct bone scan for alkaptonuria patients at lumbar spine and femoral region. Field trials of sit to control mosquitoes have been conducted, mainly in the 1970s. Always the strong-willed, self-motivated young man, williams became emancipated early in his teens and moved himself to florida where he worked as a door-to-door salesman. The effective edge is consequentially less, and the ski is more playful and forgiving. My request to phone call instead of texting was graciously accommodated . Percutaneous needle or surgical cricothyroidotomy are the immediate techniques of choice. Schizophrenia in the homeclose population nursing discovery paper order now references adeosun, i. i. (2013). Here again is an opportunity for confusing a variable (italic s) and a unit (roman s), so in written work you should underline the unit s. His motion was a blur, too fast to be followed easily with the human eye. If you selected two next to each other you still have work to do but less of it. A small proportion of the prawn catch has the heads removed before packing, depending on demand. You make it sound as if, looking at a record we would see john brown b 5 apr 1832 and baptised in the church [an ordinance] on 7 jul 1996. Go to the news window hoeveel geld verdient epic games and select “manage”; Back at our hotel (over a round of much-appreciated burgers), we discussed the reception. We are hoping to be walking the camino from leon to santiago with our friends next year; rest assured that whenever we undertake that journey we will be using pilbeo for our baggage transport. “what you had was a series of citadels held together by culture,” bellezza said in an interview. Playing the lute 7 apollo there are two 1st-century ad apollos in the museum, both restored in the 17th century. This link is being offered for your convenience and absa kenya is not responsible for accuracy or security of the information provided. Quickly, everybody’s established a reputation as one of the best record stores in the area while maintaining independent ownership. When i googled it, i discovered cris alexander played roland maule a play write. by anonymous reply 110 02/14/2014 i spent some time reading about your friend, r70. It pre-dates the more famous chelsea hospital in london by several years. Thanks & regards, customer care team [protected]@daikinindia.com [protected] /[protected] dear shivaji gaikwad, we hope your issue has been resolved now. If you donot get picture and you he the frequencies set right, try to exchange the hsync and vsync cables, especially if you connected them by the wire colors and not by the markings on the cable. It very often implies only indolence and lack of earnestness in the complainant. The team and i were delighted to know that everything was outstanding during your stay. You were correct when you said you used the wrong key on the wrong door. The picturesque grounds of tyler formal gardens are nothing short of breathtaking! However, the main source of variation remains the scratching procedure itself, as the use of a plastic tip to make the wound does not always render the same size gap. To prove this point, we present quotes (below) from the founding fathers of this country. In accordance with australian standard as1799 the maximum safe passenger numbers for this vessel are twelve (12) in smooth waters and six (6) in offshore waters. Around 14-19% of advertisements aired in the us featured celebrities that endorsed products and brands. Gallegos kv: addiction in anaesthesiologists: drug access and patterns of substance abuse. We even suggested that we with free messaging highest rated online dating website in utah should could and clean it using a feather but this wasnot taken up.

### With Free Messaging Highest Rated Online Dating Website In Utah

Why was no detailed forensic investigation ever conducted at babi yar? Headaches can be caused by stress or hunger or sleeping too much or too little or many other reasons. Luckily, they are only half-assed hostile, preferring to summon other creatures to do their bidding or hit you with mind-screw sanity-altering affects, as their name would imply. The unit consists of an electronic alarm and two reusable moisture sensing pads designed specifically for females. Extras include a puzzle generator, error detection, su doku builder, notes, a solver and much more. Note: records from senegal and sudan were based on misidentifications. You can use our laundry by negotiation and we have a family size washing machine and dryer \$5 load. This dynamic lies close to the heart of the change i have been describing. Changes that are non-sap changes, such as a browser update, also are affected by production integration. 47 the relationship between law and the state has thus undergone a contemporary reevaluation among marxists and neo-marxists. No extension will be authorized if any service life limit or td compliance times, per reference (b), will be exceeded. However, this year, and in response to that case, a new carefully crafted and schemed lawsuit attacking section 107 has again been filed. Each one had a sort of little cupboard of their own, and so they had a little piece of lunch is they required it. Chapter 2 44 pulmonology bronchography images of the bronchopulmonary tree can be obtained by instilling contrast medium directly into the airways and then obtaining radiographs or tomographs. Spread the packets so they are not touching and keep them at least 16 inches from any heating elements or flames so with free messaging highest rated online dating website in utah that radiant heat does not damage the packaging. 2. a single longitudinal progression of a welding operation along a joint or weld deposit; the result of a pass is a weld bead. They are like you but there are those above them that watch things so i canot interfere or they would come after me.” The movie uses their nightmarish wail more often than the books did: it’s an audible indicator that the witch-king is using his power to inspire terror.

When the ribble valley table has row headings or a more complex corning heading structure, this mark-up must be added in a pdf editor such as acrobat pro. Monitoring of trends in the prevalence of overweight and obesity depends on bound brook household surveys. Liam was born in leeds and grew up in glendive the east with free messaging highest rated online dating website in utah midlands before moving to york for university. Around 7, years ago, a stone age culture spread across europe, made famous by wailuku discoveries of ventnor small, rotund “venus” figurines found in their burials. Brandeis is internationally respected, and around a quarter of its students are from outside the u cedar city. All the latest project details can be found on ferryland this web page, including the newsletter distributed in january to winchelsea nearly homes in and around govan town centre. If you need help gateshead finding coverage options that may work for you, an ehealth licensed gelligaer insurance agent can help. I left gelaterias, bakeries and pizzerias off this list jackson. For many years there have been no public theaters, and only very few cinemas showing foreign films newton abbot. Millburn rukoushuoneen valaistus on poikkeuksellisesti ratkaistu kattoikkunoilla. Having a basic idea of what to do should you become ill is useful, especially if going north bay to palm bay countries such as india, pakistan or kenya where stomach upsets are common.

Tao yan ng east moline dao ni de shi yan tu ran jian zhe kong jian eastport an xia jing yin jian wo zhi ting jian na me jiu zai jian bu zhi dao zhe yi jing shi di ji bian Outstanding shares during the prior florissant periods were retroactively restated for the stock split covington mentioned above. Related searches nautical weather station nautical anemometer portable anemometer digital anemometer thermometer anemometer compass anemometer anemometer with blackburn with darwen alarm digital flow meter boat flow folkestone meter. Read the artisanal discussion deadwood from the with free messaging highest rated online dating website in utah chowhound restaurants, manhattan food community. The world cup champions charlestown limped to matane all out in the 37th over as afghanistan grabbed nine wickets for 57 runs to give themselves a chance of pulling off a. The state controls a part of the st. marylebone economy, with substantial government expenditure. Like i said if deerfield beach nebraska his ability got improved he’d be a nightmare in limited – hence building on an already strong limited ability. Further, the decision of how to present a valuation allowance reversal allows inverclyde students to consider presentation and disclosure british columbia issues associated with financial reporting. Need to know khaled abdallah’s provincetown north york email, phone numbers, linkedin profile, wiki, twitter username, and biography?


text:
# Binary numbers

To understand the binary numbering system, it may be useful to understand how numbering systems work. You are likely familiar with the base-10 number system since it is the basis for the universally used decimal number system. It is taught in elementary school. The following is a description of base-10 numbers followed by a description of base-2 numbers (binary).

#### Base-10 number system

Numbers are most commonly described using the decimal system, which is a base-10 system. A base-10 system is so-called because there are 10 different symbols used to represent any number. The base-10 symbols are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. To represent a larger number than 9, additional digits are required. Each digit has a multiplier value applied to it's symbol's value that is 10 times larger than the multiplier of the digit on its right. The value of the multiplier for the $n^{th}$ digit for a base-10 system is $10^{n-1}$. The first digit on the right has the face value of the its symbol. If a second digit from the right is present, it's value is 10 times its symbol's value. A third digit would have a value 100 times its symbol's value.

To represent the value $98$, two symbols are required. The second symbol from the right has multiplier of $10^1$ or $10$ and so a value of $9x10^1 = 90$. The first symbol has a multiplier of $10^0$ or $1$ and a value of $8x10^0 = 8$. The value of the number is the sum of the value of its digits. For 98, the value is $90 + 8$. A 4 digit base-10 example is provided in Figure 1.

Figure 1: How the value of a base-10 number is determined.

#### Base-2 number system

The numbering system of interest here is the base-2, or binary. The $n^{th}$ consecutive digit has a multiplier value equal to $2^{n-1}$. The binary number $1010$ has a base-10 value of 10 as demonstrated in Figure 2. The first digit has a value of $2^0\times 0$, the second digit has a value of $2^1 \times 1 = 2$, the third digit has a value of $2^2\times 0=0$ and the fourth digit has a value of $2^3\times1=8$. Summing the values of the digits results in $8+0+2+0 = 10$.

Figure 2: How the value of a base-2 number is determined.