text: 14 S.C. POWER AND B. SCHULZE N valency ﬁgure abc comment N 3 regular triangle hcb graphene Nhex Z 4 square sql 2D grid N 2 6 hexagon hxl triangle grid N tri 4 rectangle kgm 2D kagome Nkag 3 reg. triangle srs SrSi N K4 2 4 square nbo NbO NScaﬀ 4 reg. tetrahedron dia diamond NDia Z 6 reg. octahedron pcu primitive cubic N 3 8 cube bcu body centred cubic NBcu 12 cuboctahedron fcu face-centred cubic NFcu 6 hexagon hxg - Hxg R R 4. Regular meshes in 2 and 3 In this section we classify regular line meshes. As we have already observed the line segment meshes which are regular are necessarily linear. The proof makes use of the following theorem which is a consequence of Theorem 3.1. However there is also a more N N direct proof since there is no need to consider the nonlinear discrete nets and . K4 Dia Theorem 4.1. Up to conformal isomorphism there are 3 linear regular string-node nets N N N in three dimensions, namely , and Z . Scaﬀ Bcu 3 N √ Consider the standard realisation of the regular linear net = (N, S) with a node at tri the origin and period vectors a = (1, 0), a = (1/2, 3/2)). Let F be a countable additive R ∈ 1 N ⊗2 subgroup of with 1 F and deﬁne F to be the linear mesh whose strings are the tri translates ∈ ∩ ∈ λ a + λ a + l, λ , λ F [0, 1), l S. 1 1 2 2 1 2 N ⊗ Then it follows that F is a regular linear mesh whose vertex ﬁgure is a hexagon. Ntri R More generally, let be a regular linear net in d, for d = 2 or 3, which is in standard position. Recall that such nets by our deﬁnition are discrete and we may assume that the internodal distance are equal to 1. By a standard position for a linear 3D mesh we mean that the origin is a node, the x-axis is a string, and that there is a string through the origin in the x, y-plane which has minimal angle with the x-axis. Then we may deﬁne N ⊗ F in the same way as above but in terms of a triple of period vectors. N In two dimensions there is only one other possibility for a regular linear net , namely N N ⊗ M Z , and one can check that Z F = (F, F ), a regular linear mesh with vertex 2 2 grid N ⊗ ﬁgure equal to a square. Also it follows that for the regular linear meshes F and N ⊗ tri Z F the scaling group is the group of order preserving abelian group automorphisms 2 ∩ R of F and hence that the scaling group coincides with F∗ = F if and only if F is a + ﬁeld. N ⊗ N ⊗ In three dimensions one obtains similar conclusions for the meshes F and Z F . N Bcu 3 However, for the scaﬀolding net and certain choices of F we may obtain the identity N ⊗ N ⊗ Scaﬀ F = Z F . The reason for this, roughly speaking, is that if the additive Scaﬀ 3 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: S.C.

text: POWER

text: AND

text: B.

text: SCHULZE

text: N

text: valency

text: ﬁgure

text: abc

text: comment

text: N

text: regular

text: triangle

text: hcb

text: graphene

text: Nhex

text: Z

text: square

text: sql

text: grid

text: N

text: hexagon

text: hxl

text: triangle

text: grid

text: N

text: tri

text: rectangle

text: kgm

text: kagome

text: Nkag

text: reg.

text: triangle

text: srs

text: SrSi

text: N

text: square

text: nbo

text: NbO

text: NScaﬀ

text: reg.

text: tetrahedron

text: dia

text: diamond

text: NDia

text: Z

text: reg.

text: octahedron

text: pcu

text: primitive

text: cubic

text: N

text: cube

text: bcu

text: body

text: centred

text: cubic

text: NBcu

text: cuboctahedron

text: fcu

text: face-centred

text: cubic

text: NFcu

text: hexagon

text: hxg

text: Hxg

text: R

text: R

text: Regular

text: meshes

text: in

text: and

text: In

text: this

text: section

text: we

text: classify

text: regular

text: line

text: meshes.

text: As

text: we

text: have

text: already

text: observed

text: the

text: line

text: segment

text: meshes

text: which

text: are

text: regular

text: are

text: necessarily

text: linear.

text: The

text: proof

text: makes

text: use

text: of

text: the

text: following

text: theorem

text: which

text: is

text: a

text: consequence

text: of

text: Theorem

text: However

text: there

text: is

text: also

text: a

text: more

text: N

text: N

text: direct

text: proof

text: since

text: there

text: is

text: no

text: need

text: to

text: consider

text: the

text: nonlinear

text: discrete

text: nets

text: and

text: Dia

text: Theorem

text: Up

text: to

text: conformal

text: isomorphism

text: there

text: are

text: linear

text: regular

text: string-node

text: nets

text: N

text: N

text: N

text: in

text: three

text: dimensions,

text: namely

text: and

text: Z

text: Scaﬀ

text: Bcu

text: N

text: Consider

text: the

text: standard

text: realisation

text: of

text: the

text: regular

text: linear

text: net

text: (N,

text: S)

text: with

text: a

text: node

text: at

text: tri

text: the

text: origin

text: and

text: period

text: vectors

text: a

text: a

text: Let

text: F

text: be

text: a

text: countable

text: additive

text: R

text: N

text: ⊗2

text: subgroup

text: of

text: with

text: F

text: and

text: deﬁne

text: F

text: to

text: be

text: the

text: linear

text: mesh

text: whose

text: strings

text: are

text: the

text: tri

text: translates

text: λ

text: a

text: λ

text: a

text: l,

text: λ

text: λ

text: F

text: l

text: S.

text: N

text: Then

text: it

text: follows

text: that

text: F

text: is

text: a

text: regular

text: linear

text: mesh

text: whose

text: vertex

text: ﬁgure

text: is

text: a

text: hexagon.

text: Ntri

text: R

text: More

text: generally,

text: let

text: be

text: a

text: regular

text: linear

text: net

text: in

text: d,

text: for

text: d

text: or

text: which

text: is

text: in

text: standard

text: position.

text: Recall

text: that

text: such

text: nets

text: by

text: our

text: deﬁnition

text: are

text: discrete

text: and

text: we

text: may

text: assume

text: that

text: the

text: internodal

text: distance

text: are

text: equal

text: to

text: By

text: a

text: standard

text: position

text: for

text: a

text: linear

text: mesh

text: we

text: mean

text: that

text: the

text: origin

text: is

text: a

text: node,

text: the

text: x-axis

text: is

text: a

text: string,

text: and

text: that

text: there

text: is

text: a

text: string

text: through

text: the

text: origin

text: in

text: the

text: x,

text: y-plane

text: which

text: has

text: minimal

text: angle

text: with

text: the

text: x-axis.

text: Then

text: we

text: may

text: deﬁne

text: N

text: F

text: in

text: the

text: same

text: way

text: as

text: above

text: but

text: in

text: terms

text: of

text: a

text: triple

text: of

text: period

text: vectors.

text: N

text: In

text: two

text: dimensions

text: there

text: is

text: only

text: one

text: other

text: possibility

text: for

text: a

text: regular

text: linear

text: net

text: namely

text: N

text: N

text: M

text: Z

text: and

text: one

text: can

text: check

text: that

text: Z

text: F

text: (F,

text: F

text: a

text: regular

text: linear

text: mesh

text: with

text: vertex

text: grid

text: N

text: ﬁgure

text: equal

text: to

text: a

text: square.

text: Also

text: it

text: follows

text: that

text: for

text: the

text: regular

text: linear

text: meshes

text: F

text: and

text: N

text: tri

text: Z

text: F

text: the

text: scaling

text: group

text: is

text: the

text: group

text: of

text: order

text: preserving

text: abelian

text: group

text: automorphisms

text: R

text: of

text: F

text: and

text: hence

text: that

text: the

text: scaling

text: group

text: coincides

text: with

text: F∗

text: F

text: if

text: and

text: only

text: if

text: F

text: is

text: a

text: ﬁeld.

text: N

text: N

text: In

text: three

text: dimensions

text: one

text: obtains

text: similar

text: conclusions

text: for

text: the

text: meshes

text: F

text: and

text: Z

text: F

text: N

text: Bcu

text: However,

text: for

text: the

text: scaﬀolding

text: net

text: and

text: certain

text: choices

text: of

text: F

text: we

text: may

text: obtain

text: the

text: identity

text: N

text: N

text: Scaﬀ

text: F

text: Z

text: F

text: The

text: reason

text: for

text: this,

text: roughly

text: speaking,

text: is

text: that

text: if

text: the

text: additive

text: Scaﬀ

text: ##LTLine##

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text: 16 S.C. POWER AND B. SCHULZE In a similar manner one may deﬁne directly meshes based on the kagome net, namely M (cid:48) N ⊗ (cid:48) (F ) = F kag kag (cid:48) R where F is an odd countable additive subgroup of . These are precisely the nets which satisfy the conditions (i), (iii), and (iv), and have the property that the vertex ﬁgures is N equal to the nonregular rectangular vertex ﬁgure of . kag Q We now look more closely at regular meshes for which F is a unital subgroup of . Deﬁne the supernatural number, or generalised integer, n as the formal inﬁnite product n = 2r13r25r3 . . . , or, equivalently, as the sequence of exponents r , r , ..., with r a nonneg- ∞ 1 2 k ative integer or the symbol for each k. Two such numbers n, m are ﬁnitely equivalent if there are positive integers a, b such that there is an equality of the formal products for an and bm, in which case we write an = bm to indicate this. For the supernatural number n, as above, let n , k = 1, 2, . . . , be any increasing sequence k of natural numbers for which n divides n , for all k, and the multiplicity r(j, k) of the k →k+∞1 M prime divisor p of n tends to r as k . Deﬁne the proper linear mesh (n) as j k j tri the mesh (N, S) with body equal to ∪∞ 1 |N | . k=1 n tri k This is well-deﬁned in the sense that the set of strings, and hence nodes, is independent M N ⊗ Q Q of the choice of the sequence (n ) for n. In fact (n) = (n) where (n) is the k tri tri Q abelian group of positive rationals c/d with d a divisor of n. Note that (n) is a ﬁeld Q ∞ if and only if F = (corresponding to r = for all k), and that the scaling group is k trivial if and only if r is ﬁnite for all k k The proofs of the following results are now straightforward. Theorem 4.3. For d = 2 and also for d = 3 there are 2 strongly regular meshes, up to Q conformal aﬃne isomorphism, with dilation group contained in , namely N ⊗ Q N ⊗ Q , Z tri 2 and N ⊗ Q N ⊗ Q , Z Bcu 3 N N ⊗ Q N ⊗ Q Theorem 4.4. Let be a regular string-node net and let (n) and (m) be the regular meshes associated with the supernatural numbers n, m. Then these meshes are conformally isomorphic if and only if n and m are ﬁnitely equivalent. R Finally, let us deﬁne a pure mesh in d to be a strongly regular mesh such that every M(cid:48) M M regular submesh in is conformally isomorphic to . Theorem 4.5. The following statements are equivalent. M (i) is a pure mesh in dimension 2 or 3. M (ii) is conformally aﬃnely isomorphic one of the regular meshes of Theorem 4.2 (cid:48) Q ∞ where F (resp F ) is equal to (n), where n = p for some prime p (resp. odd prime p). ##LTLine##

label: paragraph

text: S.C.

text: POWER

text: AND

text: B.

text: SCHULZE

text: In

text: a

text: similar

text: manner

text: one

text: may

text: deﬁne

text: directly

text: meshes

text: based

text: on

text: the

text: kagome

text: net,

text: namely

text: M

text: N

text: (F

text: F

text: kag

text: kag

text: R

text: where

text: F

text: is

text: an

text: odd

text: countable

text: additive

text: subgroup

text: of

text: These

text: are

text: precisely

text: the

text: nets

text: which

text: satisfy

text: the

text: conditions

text: (i),

text: (iii),

text: and

text: (iv),

text: and

text: have

text: the

text: property

text: that

text: the

text: vertex

text: ﬁgures

text: is

text: N

text: equal

text: to

text: the

text: nonregular

text: rectangular

text: vertex

text: ﬁgure

text: of

text: kag

text: Q

text: We

text: now

text: look

text: more

text: closely

text: at

text: regular

text: meshes

text: for

text: which

text: F

text: is

text: a

text: unital

text: subgroup

text: of

text: Deﬁne

text: the

text: supernatural

text: number,

text: or

text: generalised

text: integer,

text: n

text: as

text: the

text: formal

text: inﬁnite

text: product

text: n

text: or,

text: equivalently,

text: as

text: the

text: sequence

text: of

text: exponents

text: r

text: r

text: with

text: r

text: a

text: nonneg-

text: k

text: ative

text: integer

text: or

text: the

text: symbol

text: for

text: each

text: k.

text: Two

text: such

text: numbers

text: n,

text: m

text: are

text: ﬁnitely

text: equivalent

text: if

text: there

text: are

text: positive

text: integers

text: a,

text: b

text: such

text: that

text: there

text: is

text: an

text: equality

text: of

text: the

text: formal

text: products

text: for

text: an

text: and

text: bm,

text: in

text: which

text: case

text: we

text: write

text: an

text: bm

text: to

text: indicate

text: this.

text: For

text: the

text: supernatural

text: number

text: n,

text: as

text: above,

text: let

text: n

text: k

text: be

text: any

text: increasing

text: sequence

text: k

text: of

text: natural

text: numbers

text: for

text: which

text: n

text: divides

text: n

text: for

text: all

text: k,

text: and

text: the

text: multiplicity

text: r(j,

text: k)

text: of

text: the

text: k

text: M

text: prime

text: divisor

text: p

text: of

text: n

text: tends

text: to

text: r

text: as

text: k

text: Deﬁne

text: the

text: proper

text: linear

text: mesh

text: (n)

text: as

text: j

text: k

text: j

text: tri

text: the

text: mesh

text: (N,

text: S)

text: with

text: body

text: equal

text: to

text: |N

text: n

text: tri

text: k

text: This

text: is

text: well-deﬁned

text: in

text: the

text: sense

text: that

text: the

text: set

text: of

text: strings,

text: and

text: hence

text: nodes,

text: is

text: independent

text: M

text: N

text: Q

text: Q

text: of

text: the

text: choice

text: of

text: the

text: sequence

text: (n

text: for

text: n.

text: In

text: fact

text: (n)

text: (n)

text: where

text: (n)

text: is

text: the

text: k

text: tri

text: tri

text: Q

text: abelian

text: group

text: of

text: positive

text: rationals

text: c/d

text: with

text: d

text: a

text: divisor

text: of

text: n.

text: Note

text: that

text: (n)

text: is

text: a

text: ﬁeld

text: Q

text: if

text: and

text: only

text: if

text: F

text: (corresponding

text: to

text: r

text: for

text: all

text: k),

text: and

text: that

text: the

text: scaling

text: group

text: is

text: k

text: trivial

text: if

text: and

text: only

text: if

text: r

text: is

text: ﬁnite

text: for

text: all

text: k

text: k

text: The

text: proofs

text: of

text: the

text: following

text: results

text: are

text: now

text: straightforward.

text: Theorem

text: For

text: d

text: and

text: also

text: for

text: d

text: there

text: are

text: strongly

text: regular

text: meshes,

text: up

text: to

text: Q

text: conformal

text: aﬃne

text: isomorphism,

text: with

text: dilation

text: group

text: contained

text: in

text: namely

text: N

text: Q

text: N

text: Q

text: Z

text: tri

text: and

text: N

text: Q

text: N

text: Q

text: Z

text: Bcu

text: N

text: N

text: Q

text: N

text: Q

text: Theorem

text: Let

text: be

text: a

text: regular

text: string-node

text: net

text: and

text: let

text: (n)

text: and

text: (m)

text: be

text: the

text: regular

text: meshes

text: associated

text: with

text: the

text: supernatural

text: numbers

text: n,

text: m.

text: Then

text: these

text: meshes

text: are

text: conformally

text: isomorphic

text: if

text: and

text: only

text: if

text: n

text: and

text: m

text: are

text: ﬁnitely

text: equivalent.

text: R

text: Finally,

text: let

text: us

text: deﬁne

text: a

text: pure

text: mesh

text: in

text: d

text: to

text: be

text: a

text: strongly

text: regular

text: mesh

text: such

text: that

text: every

text: M

text: M

text: regular

text: submesh

text: in

text: is

text: conformally

text: isomorphic

text: to

text: Theorem

text: The

text: following

text: statements

text: are

text: equivalent.

text: M

text: (i)

text: is

text: a

text: pure

text: mesh

text: in

text: dimension

text: or

text: M

text: (ii)

text: is

text: conformally

text: aﬃnely

text: isomorphic

text: one

text: of

text: the

text: regular

text: meshes

text: of

text: Theorem

text: Q

text: where

text: F

text: (resp

text: F

text: is

text: equal

text: to

text: (n),

text: where

text: n

text: p

text: for

text: some

text: prime

text: p

text: (resp.

text: odd

text: prime

text: p).

text: ##LTLine##


text: NETS AND MESHES 17 We now give a construction scheme to obtain a minimal enlargement of a given string- N N node net, say, to a string-node net whose dilation group contains a countable R × U subgroup U of ( , ). + N (cid:48) (cid:48) (cid:48) R ⊆ (cid:48) |N | ⊆ |N (cid:48)| We refer to any net = (N , S ) in d with the properties N N and as N R an extension of . This general construction, with U a dense group in , provides one + way of generating meshes from discrete nets. N The following lemma plays a role in deﬁning the strings of the net . U I R Lemma 4.6. Let be a countable family of nondegenerate closed intervals in . Then C R there is a family of disjoint closed intervals in with the following properties. C I I C (i) is a cover of in the sense that every interval of lies in an interval of . C C(cid:48) I (ii) is a minimal disjoint cover in the sense that if is a cover of by a family of ⊆ (cid:48) ∈ C(cid:48) ⊆ (cid:48) C disjoint closed intervals, and I C then C C where C is the interval of with ⊆ I C. I Proof. Note that the intervals of a ﬁnite or countable sequence of intervals in with consecutively pairwise nonvoid intersections, necessarily lie in the same interval of any disjoint cover. However, the closures of the unions of maximal sets of intervals which are intersection-connected in this manner may fail to be disjoint. The following coarser C equivalence relation leads to the existence of . R Let sep(A, B) denote the separation distance between two sets in given by the inﬁmum | − | ∈ ∈ ∼ R of distances a b with a A, b B. Deﬁne the equivalence relation I on by the ∼ requirement that x I y if for every (cid:15) > 0 there exists a ﬁnite set of intervals I , . . . , I I 1 N from so that the separation distances { } { } sep( x , I ), sep(I , I ), . . . , sep(I − , I ), sep(I , y ) 1 1 2 N 1 N N I are all no greater than (cid:15). We call such a ﬁnite set of intervals in an (cid:15)-chain. ∼ → → ∞ ∼ Note that if x I y and x x as k then x I y and it follows that the k k equivalence classes [x] are closed sets. Also each equivalence class [x] is convex. For if this were not the case then there exist y, z in [x] with y < z and an intermediate point w which is not in [x] and so, by the closedness of [x], w is contained in a ﬁnite closed interval J ∼ disjoint from [x], of width δ > 0 say. On the other hand since x I y for each (cid:15) = 1/m, ∈ Z m∩ (cid:54) ∅ with m and 1/m < δ, there exists a set J in an (cid:15)-chain for x, y with J J = . m m ∈ It follows from the compactness of J that there is a subsequence J and a point w J → → ∞ ∈ mk such that sep(w, J ) 0 as k . Thus w [x], a contradiction. The lemma follows C mk (cid:3) by taking to be the set of equivalence class [x] which are not singleton sets. N R ⊆ R Theorem 4.7. Let = (N, S) be a line segment net in d and let U be a countable R N+ N scaling group for d. Then there exists a uniquely determined extension of so that N N U (i) U is a subgroup of the scaling group D( ) of . N (cid:48) N U U (ii) If is an extension of the net whose scaling group contains U as a subgroup, N (cid:48) N then is an extension of . U

text: NETS

text: AND

text: MESHES

text: We

text: now

text: give

text: a

text: construction

text: scheme

text: to

text: obtain

text: a

text: minimal

text: enlargement

text: of

text: a

text: given

text: string-

text: N

text: N

text: node

text: net,

text: say,

text: to

text: a

text: string-node

text: net

text: whose

text: dilation

text: group

text: contains

text: a

text: countable

text: R

text: U

text: subgroup

text: U

text: of

text: N

text: R

text: |N

text: |N

text: We

text: refer

text: to

text: any

text: net

text: (N

text: S

text: in

text: d

text: with

text: the

text: properties

text: N

text: N

text: and

text: as

text: N

text: R

text: an

text: extension

text: of

text: This

text: general

text: construction,

text: with

text: U

text: a

text: dense

text: group

text: in

text: provides

text: one

text: way

text: of

text: generating

text: meshes

text: from

text: discrete

text: nets.

text: N

text: The

text: following

text: lemma

text: plays

text: a

text: role

text: in

text: deﬁning

text: the

text: strings

text: of

text: the

text: net

text: U

text: I

text: R

text: Lemma

text: Let

text: be

text: a

text: countable

text: family

text: of

text: nondegenerate

text: closed

text: intervals

text: in

text: Then

text: C

text: R

text: there

text: is

text: a

text: family

text: of

text: disjoint

text: closed

text: intervals

text: in

text: with

text: the

text: following

text: properties.

text: C

text: I

text: I

text: C

text: (i)

text: is

text: a

text: cover

text: of

text: in

text: the

text: sense

text: that

text: every

text: interval

text: of

text: lies

text: in

text: an

text: interval

text: of

text: C

text: I

text: (ii)

text: is

text: a

text: minimal

text: disjoint

text: cover

text: in

text: the

text: sense

text: that

text: if

text: is

text: a

text: cover

text: of

text: by

text: a

text: family

text: of

text: C

text: disjoint

text: closed

text: intervals,

text: and

text: I

text: C

text: then

text: C

text: C

text: where

text: C

text: is

text: the

text: interval

text: of

text: with

text: I

text: C.

text: I

text: Proof.

text: Note

text: that

text: the

text: intervals

text: of

text: a

text: ﬁnite

text: or

text: countable

text: sequence

text: of

text: intervals

text: in

text: with

text: consecutively

text: pairwise

text: nonvoid

text: intersections,

text: necessarily

text: lie

text: in

text: the

text: same

text: interval

text: of

text: any

text: disjoint

text: cover.

text: However,

text: the

text: closures

text: of

text: the

text: unions

text: of

text: maximal

text: sets

text: of

text: intervals

text: which

text: are

text: intersection-connected

text: in

text: this

text: manner

text: may

text: fail

text: to

text: be

text: disjoint.

text: The

text: following

text: coarser

text: C

text: equivalence

text: relation

text: leads

text: to

text: the

text: existence

text: of

text: R

text: Let

text: sep(A,

text: B)

text: denote

text: the

text: separation

text: distance

text: between

text: two

text: sets

text: in

text: given

text: by

text: the

text: inﬁmum

text: R

text: of

text: distances

text: a

text: b

text: with

text: a

text: A,

text: b

text: B.

text: Deﬁne

text: the

text: equivalence

text: relation

text: I

text: on

text: by

text: the

text: requirement

text: that

text: x

text: I

text: y

text: if

text: for

text: every

text: there

text: exists

text: a

text: ﬁnite

text: set

text: of

text: intervals

text: I

text: I

text: I

text: N

text: from

text: so

text: that

text: the

text: separation

text: distances

text: sep(

text: x

text: I

text: sep(I

text: I

text: sep(I

text: I

text: sep(I

text: y

text: N

text: N

text: N

text: I

text: are

text: all

text: no

text: greater

text: than

text: We

text: call

text: such

text: a

text: ﬁnite

text: set

text: of

text: intervals

text: in

text: an

text: Note

text: that

text: if

text: x

text: I

text: y

text: and

text: x

text: x

text: as

text: k

text: then

text: x

text: I

text: y

text: and

text: it

text: follows

text: that

text: the

text: k

text: k

text: equivalence

text: classes

text: [x]

text: are

text: closed

text: sets.

text: Also

text: each

text: equivalence

text: class

text: [x]

text: is

text: convex.

text: For

text: if

text: this

text: were

text: not

text: the

text: case

text: then

text: there

text: exist

text: y,

text: z

text: in

text: [x]

text: with

text: y

text: z

text: and

text: an

text: intermediate

text: point

text: w

text: which

text: is

text: not

text: in

text: [x]

text: and

text: so,

text: by

text: the

text: closedness

text: of

text: [x],

text: w

text: is

text: contained

text: in

text: a

text: ﬁnite

text: closed

text: interval

text: J

text: disjoint

text: from

text: [x],

text: of

text: width

text: δ

text: say.

text: On

text: the

text: other

text: hand

text: since

text: x

text: I

text: y

text: for

text: each

text: Z

text: m∩

text: with

text: m

text: and

text: δ,

text: there

text: exists

text: a

text: set

text: J

text: in

text: an

text: for

text: x,

text: y

text: with

text: J

text: J

text: m

text: m

text: It

text: follows

text: from

text: the

text: compactness

text: of

text: J

text: that

text: there

text: is

text: a

text: subsequence

text: J

text: and

text: a

text: point

text: w

text: J

text: mk

text: such

text: that

text: sep(w,

text: J

text: as

text: k

text: Thus

text: w

text: [x],

text: a

text: contradiction.

text: The

text: lemma

text: follows

text: C

text: mk

text: by

text: taking

text: to

text: be

text: the

text: set

text: of

text: equivalence

text: class

text: [x]

text: which

text: are

text: not

text: singleton

text: sets.

text: N

text: R

text: R

text: Theorem

text: Let

text: (N,

text: S)

text: be

text: a

text: line

text: segment

text: net

text: in

text: d

text: and

text: let

text: U

text: be

text: a

text: countable

text: R

text: N+

text: N

text: scaling

text: group

text: for

text: d.

text: Then

text: there

text: exists

text: a

text: uniquely

text: determined

text: extension

text: of

text: so

text: that

text: N

text: N

text: U

text: (i)

text: U

text: is

text: a

text: subgroup

text: of

text: the

text: scaling

text: group

text: D(

text: of

text: N

text: N

text: U

text: U

text: (ii)

text: If

text: is

text: an

text: extension

text: of

text: the

text: net

text: whose

text: scaling

text: group

text: contains

text: U

text: as

text: a

text: subgroup,

text: N

text: N

text: then

text: is

text: an

text: extension

text: of

text: U


text: NETS AND MESHES 19 G structure graph of or, in reticular chemistry, as the underlying topology of G. Similarly, N a discrete string-node net = (N, S) has an evident underlying structure graph. On the other hand there is no discrete structure graph which underlies a string-node M mesh . The appropriate associated ”pre-metric” structure can be considered to be the M |M| topological space X( ) = ( , τ) where the topology τ has a base of open sets provided R by the sets γ([a, b]) in d arising from parametrisations of the strings and parameter values a < b. In fact, although, for simplicity, we have chosen to deﬁne strings as nonoverlapping subsets, one could equally well (from the perspective of rigidity and ﬂexibility, for example) allow strings to overlap, and indeed, admit all sets γ([a, b]) between nodes as strings. M M With this perspective the topological space X( ) underlying the mesh supports a M Borel measure, deﬁned by the ”placement” , namely the string length measure µM, determined by the strings in the canonical way. M |M| We note that the mesh = (N, S) also gives rise naturally to a metric space ( , d). The metric d is given by · · · d(x, y) = inf(d (x , x ) + d (x , x ) + + d (x − , x )) γ1 1 2 γ2 2 3 γr r 1 r where the inﬁmum is taken over all sets of points x = x , x , . . . , x = y, where consecutive 1 2 r · · points lie on the same strings, L , . . . , L − respectively, and where d ( , ) is the string 1 r 1 γi |M| distance separation for points on the same string L . The metric space ( , d), the |M| i |M| topological space ( , τ) and the Borel measure space ( , µM) can be regarded as ”internal to the mesh” in the sense that their various isomorphism classes do not change under the deformation motions we consider below. 5. Flexibility and rigidity We now consider dynamical aspects of meshes, in the sense of their continuous motions and smooth motions which preserve all the internodal distances as measured along each string. Since the nodes of a mesh that lie on a particular string l are dense in l this means that the position (or placement) of a string at a time value t is given by a length preserving R map from l to d. The terms ﬂex and motion are used interchangeably. Also there will be no need in the following discussion to consider inﬁnitesimal ﬂexes or general velocity ﬁelds. M R Deﬁnition 5.1. Let = (N, S) be a line segment mesh in d. M |M| → R (i) A placement (resp. C -smooth placement) of is an injective map p : d k such that the restriction of p to each string is continuous (resp. C -smooth) and is length k preserving. M → ∈ (ii) A continuous ﬂex of is a pointwise continuous path t p , t [0, 1], of place- M t ments of . M → ∈ (iii) A C -smooth ﬂex of is a pointwise continuous path t p , t [0, 1], of C - k M t k smooth placements of . M → ∈ (iv) A uniformly C -smooth ﬂex of is a C -smooth ﬂex t p , t [0, 1], such that k |M| × R k t the associated map from [0, 1] to d is a k-fold continuously diﬀerentiable function |M|− × when restricted to the set ( )o [0, 1].

text: NETS

text: AND

text: MESHES

text: G

text: structure

text: graph

text: of

text: or,

text: in

text: reticular

text: chemistry,

text: as

text: the

text: underlying

text: topology

text: of

text: G.

text: Similarly,

text: N

text: a

text: discrete

text: string-node

text: net

text: (N,

text: S)

text: has

text: an

text: evident

text: underlying

text: structure

text: graph.

text: On

text: the

text: other

text: hand

text: there

text: is

text: no

text: discrete

text: structure

text: graph

text: which

text: underlies

text: a

text: string-node

text: M

text: mesh

text: The

text: appropriate

text: associated

text: ”pre-metric”

text: structure

text: can

text: be

text: considered

text: to

text: be

text: the

text: M

text: |M|

text: topological

text: space

text: X(

text: τ)

text: where

text: the

text: topology

text: τ

text: has

text: a

text: base

text: of

text: open

text: sets

text: provided

text: R

text: by

text: the

text: sets

text: γ([a,

text: b])

text: in

text: d

text: arising

text: from

text: parametrisations

text: of

text: the

text: strings

text: and

text: parameter

text: values

text: a

text: b.

text: In

text: fact,

text: although,

text: for

text: simplicity,

text: we

text: have

text: chosen

text: to

text: deﬁne

text: strings

text: as

text: nonoverlapping

text: subsets,

text: one

text: could

text: equally

text: well

text: (from

text: the

text: perspective

text: of

text: rigidity

text: and

text: ﬂexibility,

text: for

text: example)

text: allow

text: strings

text: to

text: overlap,

text: and

text: indeed,

text: admit

text: all

text: sets

text: γ([a,

text: b])

text: between

text: nodes

text: as

text: strings.

text: M

text: M

text: With

text: this

text: perspective

text: the

text: topological

text: space

text: X(

text: underlying

text: the

text: mesh

text: supports

text: a

text: M

text: Borel

text: measure,

text: deﬁned

text: by

text: the

text: ”placement”

text: namely

text: the

text: string

text: length

text: measure

text: µM,

text: determined

text: by

text: the

text: strings

text: in

text: the

text: canonical

text: way.

text: M

text: |M|

text: We

text: note

text: that

text: the

text: mesh

text: (N,

text: S)

text: also

text: gives

text: rise

text: naturally

text: to

text: a

text: metric

text: space

text: d).

text: The

text: metric

text: d

text: is

text: given

text: by

text: d(x,

text: y)

text: inf(d

text: (x

text: x

text: d

text: (x

text: x

text: d

text: (x

text: x

text: γ1

text: γ2

text: γr

text: r

text: r

text: where

text: the

text: inﬁmum

text: is

text: taken

text: over

text: all

text: sets

text: of

text: points

text: x

text: x

text: x

text: x

text: y,

text: where

text: consecutive

text: r

text: points

text: lie

text: on

text: the

text: same

text: strings,

text: L

text: L

text: respectively,

text: and

text: where

text: d

text: is

text: the

text: string

text: r

text: γi

text: |M|

text: distance

text: separation

text: for

text: points

text: on

text: the

text: same

text: string

text: L

text: The

text: metric

text: space

text: d),

text: the

text: |M|

text: i

text: |M|

text: topological

text: space

text: τ)

text: and

text: the

text: Borel

text: measure

text: space

text: µM)

text: can

text: be

text: regarded

text: as

text: ”internal

text: to

text: the

text: mesh”

text: in

text: the

text: sense

text: that

text: their

text: various

text: isomorphism

text: classes

text: do

text: not

text: change

text: under

text: the

text: deformation

text: motions

text: we

text: consider

text: below.

text: Flexibility

text: and

text: rigidity

text: We

text: now

text: consider

text: dynamical

text: aspects

text: of

text: meshes,

text: in

text: the

text: sense

text: of

text: their

text: continuous

text: motions

text: and

text: smooth

text: motions

text: which

text: preserve

text: all

text: the

text: internodal

text: distances

text: as

text: measured

text: along

text: each

text: string.

text: Since

text: the

text: nodes

text: of

text: a

text: mesh

text: that

text: lie

text: on

text: a

text: particular

text: string

text: l

text: are

text: dense

text: in

text: l

text: this

text: means

text: that

text: the

text: position

text: (or

text: placement)

text: of

text: a

text: string

text: at

text: a

text: time

text: value

text: t

text: is

text: given

text: by

text: a

text: length

text: preserving

text: R

text: map

text: from

text: l

text: to

text: d.

text: The

text: terms

text: ﬂex

text: and

text: motion

text: are

text: used

text: interchangeably.

text: Also

text: there

text: will

text: be

text: no

text: need

text: in

text: the

text: following

text: discussion

text: to

text: consider

text: inﬁnitesimal

text: ﬂexes

text: or

text: general

text: velocity

text: ﬁelds.

text: M

text: R

text: Deﬁnition

text: Let

text: (N,

text: S)

text: be

text: a

text: line

text: segment

text: mesh

text: in

text: d.

text: M

text: |M|

text: R

text: (i)

text: A

text: placement

text: (resp.

text: C

text: -smooth

text: placement)

text: of

text: is

text: an

text: injective

text: map

text: p

text: d

text: k

text: such

text: that

text: the

text: restriction

text: of

text: p

text: to

text: each

text: string

text: is

text: continuous

text: (resp.

text: C

text: -smooth)

text: and

text: is

text: length

text: k

text: preserving.

text: M

text: (ii)

text: A

text: continuous

text: ﬂex

text: of

text: is

text: a

text: pointwise

text: continuous

text: path

text: t

text: p

text: t

text: of

text: place-

text: M

text: t

text: ments

text: of

text: M

text: (iii)

text: A

text: C

text: -smooth

text: ﬂex

text: of

text: is

text: a

text: pointwise

text: continuous

text: path

text: t

text: p

text: t

text: of

text: C

text: k

text: M

text: t

text: k

text: smooth

text: placements

text: of

text: M

text: (iv)

text: A

text: uniformly

text: C

text: -smooth

text: ﬂex

text: of

text: is

text: a

text: C

text: -smooth

text: ﬂex

text: t

text: p

text: t

text: such

text: that

text: k

text: |M|

text: R

text: k

text: t

text: the

text: associated

text: map

text: from

text: to

text: d

text: is

text: a

text: k-fold

text: continuously

text: diﬀerentiable

text: function

text: |M|−

text: when

text: restricted

text: to

text: the

text: set

text: )o


text: 20 S.C. POWER AND B. SCHULZE Note that we assume that placements are deﬁned by injective maps and in particular these are noncrossing placements in the sense that the placed strings cannot intersect at nonnodal points. Also, unlike the usual freedom adopted for bar-joint frameworks, a continuous ﬂex in the sense above admits no ”collisions”. Thus a continuous ﬂex of a string-node net or mesh may be viewed as a homotopy in the ambient space, as in the category of knots. We have deﬁned continuous ﬂexes and smooth ﬂexes of a line segment mesh without direct reference to smooth meshes. One may also regard such a smooth ﬂex as a smooth path in the space of smooth meshes. However we need not consider this level of abstraction. N ⊗ Q ∩ Consider ﬁrst the planar bounded grid mesh ( Z ) [0, 1]2, which we now denote M 2→ R more simply as . We say that a map f : [0, 1]2 2 is a laminar map if it has the q form f(s, t) = α (s) + α (t) 1 2 → R where the sum is a vector sum for the maps α , α : [0, 1] 2. A laminar placement of M |M | → R 1 2 is a placement p : 2 which is the restriction of a laminar map. q q M Note that a laminar map is string-length preserving for if the maps α , α are length q 1 2 preserving. Simple examples show that a laminar map need not be injective and so need not deﬁne a placement. However, we have the following suﬃcient condition. Lemma 5.2. Let α , α be continuously diﬀerentiable and length preserving and suppose 1 (cid:48) 2 (cid:48) that the unit vectors α (s), α (t) satisfy the angle constraints 1 2 √ √ (cid:48) · (cid:48) · ∈ α (s) (1, 0) > 1/ 2, α (t) (0, 1) > 1/ 2, s, t (0, 1). 1 2 → R Then the associate laminar map f : [0, 1]2 2 determines a C -smooth placement of M 1 . q Proof. Let α (t) = (α (t), α (t)) for i = 1, 2. By the hypotheses we have i i,x i,y (cid:48) √1 (cid:48) α (s) > > α (s) 1,x 1,y 2 ∈ ≤ and so α (s) > α (s) for s (0, 1). It follows that for any 0 s < s the vector − 1,x 1,y | | 1 2 α (s ) α (s ) lies in the open cone y > x . 1 2 1 1 − It follows, similarly, that for 0 < t < t the vector α (t ) α (t ) and its negative lie | | | | 1 2 2 2 2 1 in the open cone y < x . ≤ (cid:54) − Suppose now that f(s , t ) = f(s , t ) with s s and t = t . Then α (s ) α (s ) = − − 1 1 2 2 1 2 1 2 1 2 1 1 (cid:3) (α (t ) α (t )) which is a contradiction. 2 2 2 1 M Deﬁnition 5.3. A continuous ﬂex γ = (γ ) of the grid mesh is a laminar ﬂex if each t q placement map γ is laminar. t ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: S.C.

text: POWER

text: AND

text: B.

text: SCHULZE

text: Note

text: that

text: we

text: assume

text: that

text: placements

text: are

text: deﬁned

text: by

text: injective

text: maps

text: and

text: in

text: particular

text: these

text: are

text: noncrossing

text: placements

text: in

text: the

text: sense

text: that

text: the

text: placed

text: strings

text: cannot

text: intersect

text: at

text: nonnodal

text: points.

text: Also,

text: unlike

text: the

text: usual

text: freedom

text: adopted

text: for

text: bar-joint

text: frameworks,

text: a

text: continuous

text: ﬂex

text: in

text: the

text: sense

text: above

text: admits

text: no

text: ”collisions”.

text: Thus

text: a

text: continuous

text: ﬂex

text: of

text: a

text: string-node

text: net

text: or

text: mesh

text: may

text: be

text: viewed

text: as

text: a

text: homotopy

text: in

text: the

text: ambient

text: space,

text: as

text: in

text: the

text: category

text: of

text: knots.

text: We

text: have

text: deﬁned

text: continuous

text: ﬂexes

text: and

text: smooth

text: ﬂexes

text: of

text: a

text: line

text: segment

text: mesh

text: without

text: direct

text: reference

text: to

text: smooth

text: meshes.

text: One

text: may

text: also

text: regard

text: such

text: a

text: smooth

text: ﬂex

text: as

text: a

text: smooth

text: path

text: in

text: the

text: space

text: of

text: smooth

text: meshes.

text: However

text: we

text: need

text: not

text: consider

text: this

text: level

text: of

text: abstraction.

text: N

text: Q

text: Consider

text: ﬁrst

text: the

text: planar

text: bounded

text: grid

text: mesh

text: Z

text: which

text: we

text: now

text: denote

text: M

text: 2→

text: R

text: more

text: simply

text: as

text: We

text: say

text: that

text: a

text: map

text: f

text: is

text: a

text: laminar

text: map

text: if

text: it

text: has

text: the

text: q

text: form

text: f(s,

text: t)

text: α

text: (s)

text: α

text: (t)

text: R

text: where

text: the

text: sum

text: is

text: a

text: vector

text: sum

text: for

text: the

text: maps

text: α

text: α

text: A

text: laminar

text: placement

text: of

text: M

text: |M

text: R

text: is

text: a

text: placement

text: p

text: which

text: is

text: the

text: restriction

text: of

text: a

text: laminar

text: map.

text: q

text: q

text: M

text: Note

text: that

text: a

text: laminar

text: map

text: is

text: string-length

text: preserving

text: for

text: if

text: the

text: maps

text: α

text: α

text: are

text: length

text: q

text: preserving.

text: Simple

text: examples

text: show

text: that

text: a

text: laminar

text: map

text: need

text: not

text: be

text: injective

text: and

text: so

text: need

text: not

text: deﬁne

text: a

text: placement.

text: However,

text: we

text: have

text: the

text: following

text: suﬃcient

text: condition.

text: Lemma

text: Let

text: α

text: α

text: be

text: continuously

text: diﬀerentiable

text: and

text: length

text: preserving

text: and

text: suppose

text: that

text: the

text: unit

text: vectors

text: α

text: (s),

text: α

text: (t)

text: satisfy

text: the

text: angle

text: constraints

text: α

text: (s)

text: α

text: (t)

text: s,

text: t

text: R

text: Then

text: the

text: associate

text: laminar

text: map

text: f

text: determines

text: a

text: C

text: -smooth

text: placement

text: of

text: M

text: q

text: Proof.

text: Let

text: α

text: (t)

text: (α

text: (t),

text: α

text: (t))

text: for

text: i

text: By

text: the

text: hypotheses

text: we

text: have

text: i

text: i,x

text: i,y

text: √1

text: α

text: (s)

text: α

text: (s)

text: and

text: so

text: α

text: (s)

text: α

text: (s)

text: for

text: s

text: It

text: follows

text: that

text: for

text: any

text: s

text: s

text: the

text: vector

text: α

text: (s

text: α

text: (s

text: lies

text: in

text: the

text: open

text: cone

text: y

text: x

text: It

text: follows,

text: similarly,

text: that

text: for

text: t

text: t

text: the

text: vector

text: α

text: (t

text: α

text: (t

text: and

text: its

text: negative

text: lie

text: in

text: the

text: open

text: cone

text: y

text: x

text: Suppose

text: now

text: that

text: f(s

text: t

text: f(s

text: t

text: with

text: s

text: s

text: and

text: t

text: t

text: Then

text: α

text: (s

text: α

text: (s

text: (α

text: (t

text: α

text: (t

text: which

text: is

text: a

text: contradiction.

text: M

text: Deﬁnition

text: A

text: continuous

text: ﬂex

text: γ

text: (γ

text: of

text: the

text: grid

text: mesh

text: is

text: a

text: laminar

text: ﬂex

text: if

text: each

text: t

text: q

text: placement

text: map

text: γ

text: is

text: laminar.

text: t

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 22 S.C. POWER AND B. SCHULZE We consider next the triadic kagome mesh. Recall that the inﬁnite periodic kagome bar-joint framework admits diverse continuous motions which are periodic in some sense. See [14], [18] for example. We now show that in contrast to the rectangular case above, this ﬂexibility does not carry over in any way to the triadic kagome mesh. Theorem 5.6. The triadic kagome mesh is rigid with respect to uniformly C -smooth 1 ﬂexes. M Proof. We ﬁrst show that any C -smooth placement of is congruent to the identity M 1 kag placement of . Note that such a placement, p say, has a unique continuous extension R → R kag t of p : 2 2. Moreover, a restriction of this extension deﬁnes a string-length preserving t M placement of the containing triadic triangular mesh, . It suﬃces then to show that M M tri the C -smooth mesh = p ( ), deﬁned by the range of this placement, is congruent M1 t tri to . tri M Let 0 < t < 1 and let q be a node of the mesh . By the C -smoothness of the strings M 0 1 of there are 3 three unit tangent vectors to the three strings through q , say a , a , a . − − − 0 1 2 3 We may choose these vectors so that the vectors a , a , a , a , a , a correspond to 1 2 3 1 2 3 a consecutive sequence of points on the unit circle, with the angles between a , a and 1 2 between a , a acute angles. We claim that all the consecutive angles are equal to π/3. 2 3 To see this it is suﬃcient to note that no consecutive angle is greater than π/3. We now show that this follows from the string-length preservation property applied to cross-strings connecting nodes on adjacent strings through q . (cid:48) M 0 (cid:48) (cid:48) Let q be the node in which is the preimage of q under p , and let q and q be nodes 0 tri (cid:48) 0 t 1 2 on adjacent strings through q which lie on a common cross string. The internodal string 0 (cid:107) (cid:48) − (cid:48) (cid:107) (cid:107) (cid:48) − (cid:48) (cid:107) (cid:107) (cid:48) − (cid:48) (cid:107) lengths agree with the Euclidean distances, q q , q q and q q respectively, 1 2 1 (cid:48) (cid:48)0 2 0 and these lengths are equal. Let q , q be the images of q , q under p . Since p preserves 1 2 1 2 t t string lengths we have d = d = d where d is the string distance between q and q 01 M12 02 ij i j for the (unique) string l of through q , q . ij i j Suppose now, by way of contradiction, that angle between the tangent vectors at q for 0 the strings l and l is greater than π/3. Then 01 02 (cid:107) − (cid:107) (cid:107) − (cid:107) (i) the ratio q q / q q has a ﬁnite limit greater than 1 as q tends to q . 1 2 1 0 1 0 Also, by string-length preservation, (cid:107) − (cid:107) (ii) the ratio d / q q has limit 1, as q tends to q , 01 1 0 1 0 and, by the triangle inequality, (cid:107) − (cid:107) (iii) d is not less than q q . 12 1 2 Combining the above it follows that if q is suﬃciently close to q then 1 0 (cid:107) − (cid:107) (cid:107) − (cid:107) d d q q q q 12 = (cid:107) −12 (cid:107) (cid:107) 1 − 2(cid:107) 1 0 > 1 d q q q q d 01 1 2 1 0 01 which is a contradiction. M M To see that is rigid for uniformly smooth ﬂexes we may note that contains triM tri the 4-valent mesh whose strings are either parallel to the x-axis or at angle π/3 to ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: S.C.

text: POWER

text: AND

text: B.

text: SCHULZE

text: We

text: consider

text: next

text: the

text: triadic

text: kagome

text: mesh.

text: Recall

text: that

text: the

text: inﬁnite

text: periodic

text: kagome

text: bar-joint

text: framework

text: admits

text: diverse

text: continuous

text: motions

text: which

text: are

text: periodic

text: in

text: some

text: sense.

text: See

text: for

text: example.

text: We

text: now

text: show

text: that

text: in

text: contrast

text: to

text: the

text: rectangular

text: case

text: above,

text: this

text: ﬂexibility

text: does

text: not

text: carry

text: over

text: in

text: any

text: way

text: to

text: the

text: triadic

text: kagome

text: mesh.

text: Theorem

text: The

text: triadic

text: kagome

text: mesh

text: is

text: rigid

text: with

text: respect

text: to

text: uniformly

text: C

text: -smooth

text: ﬂexes.

text: M

text: Proof.

text: We

text: ﬁrst

text: show

text: that

text: any

text: C

text: -smooth

text: placement

text: of

text: is

text: congruent

text: to

text: the

text: identity

text: M

text: kag

text: placement

text: of

text: Note

text: that

text: such

text: a

text: placement,

text: p

text: say,

text: has

text: a

text: unique

text: continuous

text: extension

text: R

text: R

text: kag

text: t

text: of

text: p

text: Moreover,

text: a

text: restriction

text: of

text: this

text: extension

text: deﬁnes

text: a

text: string-length

text: preserving

text: t

text: M

text: placement

text: of

text: the

text: containing

text: triadic

text: triangular

text: mesh,

text: It

text: suﬃces

text: then

text: to

text: show

text: that

text: M

text: M

text: tri

text: the

text: C

text: -smooth

text: mesh

text: p

text: deﬁned

text: by

text: the

text: range

text: of

text: this

text: placement,

text: is

text: congruent

text: t

text: tri

text: to

text: tri

text: M

text: Let

text: t

text: and

text: let

text: q

text: be

text: a

text: node

text: of

text: the

text: mesh

text: By

text: the

text: C

text: -smoothness

text: of

text: the

text: strings

text: M

text: of

text: there

text: are

text: three

text: unit

text: tangent

text: vectors

text: to

text: the

text: three

text: strings

text: through

text: q

text: say

text: a

text: a

text: a

text: We

text: may

text: choose

text: these

text: vectors

text: so

text: that

text: the

text: vectors

text: a

text: a

text: a

text: a

text: a

text: a

text: correspond

text: to

text: a

text: consecutive

text: sequence

text: of

text: points

text: on

text: the

text: unit

text: circle,

text: with

text: the

text: angles

text: between

text: a

text: a

text: and

text: between

text: a

text: a

text: acute

text: angles.

text: We

text: claim

text: that

text: all

text: the

text: consecutive

text: angles

text: are

text: equal

text: to

text: To

text: see

text: this

text: it

text: is

text: suﬃcient

text: to

text: note

text: that

text: no

text: consecutive

text: angle

text: is

text: greater

text: than

text: We

text: now

text: show

text: that

text: this

text: follows

text: from

text: the

text: string-length

text: preservation

text: property

text: applied

text: to

text: cross-strings

text: connecting

text: nodes

text: on

text: adjacent

text: strings

text: through

text: q

text: M

text: Let

text: q

text: be

text: the

text: node

text: in

text: which

text: is

text: the

text: preimage

text: of

text: q

text: under

text: p

text: and

text: let

text: q

text: and

text: q

text: be

text: nodes

text: tri

text: t

text: on

text: adjacent

text: strings

text: through

text: q

text: which

text: lie

text: on

text: a

text: common

text: cross

text: string.

text: The

text: internodal

text: string

text: lengths

text: agree

text: with

text: the

text: Euclidean

text: distances,

text: q

text: q

text: q

text: q

text: and

text: q

text: q

text: respectively,

text: and

text: these

text: lengths

text: are

text: equal.

text: Let

text: q

text: q

text: be

text: the

text: images

text: of

text: q

text: q

text: under

text: p

text: Since

text: p

text: preserves

text: t

text: t

text: string

text: lengths

text: we

text: have

text: d

text: d

text: d

text: where

text: d

text: is

text: the

text: string

text: distance

text: between

text: q

text: and

text: q

text: ij

text: i

text: j

text: for

text: the

text: (unique)

text: string

text: l

text: of

text: through

text: q

text: q

text: ij

text: i

text: j

text: Suppose

text: now,

text: by

text: way

text: of

text: contradiction,

text: that

text: angle

text: between

text: the

text: tangent

text: vectors

text: at

text: q

text: for

text: the

text: strings

text: l

text: and

text: l

text: is

text: greater

text: than

text: Then

text: (i)

text: the

text: ratio

text: q

text: q

text: q

text: q

text: has

text: a

text: ﬁnite

text: limit

text: greater

text: than

text: as

text: q

text: tends

text: to

text: q

text: Also,

text: by

text: string-length

text: preservation,

text: (ii)

text: the

text: ratio

text: d

text: q

text: q

text: has

text: limit

text: as

text: q

text: tends

text: to

text: q

text: and,

text: by

text: the

text: triangle

text: inequality,

text: (iii)

text: d

text: is

text: not

text: less

text: than

text: q

text: q

text: Combining

text: the

text: above

text: it

text: follows

text: that

text: if

text: q

text: is

text: suﬃciently

text: close

text: to

text: q

text: then

text: d

text: d

text: q

text: q

text: q

text: q

text: d

text: q

text: q

text: q

text: q

text: d

text: which

text: is

text: a

text: contradiction.

text: M

text: M

text: To

text: see

text: that

text: is

text: rigid

text: for

text: uniformly

text: smooth

text: ﬂexes

text: we

text: may

text: note

text: that

text: contains

text: triM

text: tri

text: the

text: mesh

text: whose

text: strings

text: are

text: either

text: parallel

text: to

text: the

text: x-axis

text: or

text: at

text: angle

text: to

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 2 35 16 phonon state. The parameter ω0 represents the value of LCCO A B the maximum phonon frequency. 30 14 Next, the Hamiltonian H can be rewritten in the mo- mentum representation. The canonical transformation 25 12 eliminating the phonon degrees of freedom allows to ob- tain the fundamental thermodynamic equ(cid:16)ation [47]: (cid:17) 20 10 (cid:18) (cid:19) (cid:90) √ 15 8 1 = V + U |∆ (T) |2 ω0 dερ (ε) tanh √β2 ε2 + E2 , 10 6 6 (cid:0) −ω0 (cid:1) 2 ε2 + E2 5 4 (6) Krockenberger et al. 0 Skinta et al. ≡ | | | | 2 where: E V + U ∆ (T) 2 ∆ (T) . The parame- 6 0.04 0.09 0.14 0.19 0.24 0.08 0.10 0.12 0.14 ters V and U denote the electron-phonon potential (EPh) x x and the electron-electron-phonon potential (EEPh), re- (cid:80) ≡ ≡ − spectively: V V + V and U V U . FIG. 1: The inﬂuence of the cerium concentration on the ε0 t0 U0 0 ≡ The order parameter is deﬁned by the formula: ∆ value of (A) the critical temperature and (B) the energy gap (cid:104) (cid:105) ω0 c−↓c−↑ , while c denotes the annihilation opera- for the superconductor LCCO [17], [67]. σ tor for the electron state with the momentum k. The (cid:104) (cid:105) symbol ... represents the thermodynamic average [59], TABLE I: The thermodynamic parameters of the LCCO [60]. The summation over k should be performed taking ≡ ≤ | | superconductor. Additionally, it has been deﬁned: U(0) under consideration the condition: ε ω , where ε is | | 0 U ∆ (0) 2. the electron band energy. The inverse temperature has 6 ≡ been deﬁned with the help of the expression: β 1/kBT, x T G(0) V U(0) T(cid:63)(cid:63) C where k is the Boltzmann constant. B (K) (meV) (meV) (meV) (K) Let us notice that the band energy for the two- dimensional square lattice can be written as: ε = − 0.087 28.7 15.1 436.6 123.9 28.9 2t0 [cos (kx) + cos (ky)]. Thus, the|elect|ron density o≡f − states has the form: ρ (ε) = b ln ε/b , where b 0.107 28.9 13.7 438.0 95.6 − − ≡ 1 2 1 − 0.04687t 1 and b 21.17796t [61], [62], [63], [64], 0.112 29.3 11.9 440.7 56.7 0 2 0 − − [65]. 0.135 21.7 4.7 387.8 50.8 The presented model can explain (at the qualitative level) the essential properties of the high-temperature su- perconducting state [47]. Namely, the high critical tem- equal to 380 meV [68], [69], [70], [71]. perature is related to the existence of the van Hove singu- II. RESULTS larity in the electron density of states. The van Hove sin- (cid:0) (cid:1) gularity is also responsible for the relatively low isotope In f1 (A)-(B), the dependence of the critical temper- coeﬃcient. The anomalously high value of the energy ≡ | | | | ature and the low-temperature value of the energy gap gap (G (0), where G (0) 2 V + U ∆ (0) 2 ∆ (0) ) in 6 (T = 1.6 K) on the concentration of cerium has been respect to the critical temperature, very often observed presented. The symbols have been prepared on the basis for δ < δ , can be explained by the high ratio U/V . It max of the experimental data [17], [67] (see also t1). should be noted that in the considered case, the function In the case of the critical temperature, it may be noted G (T) weakly depends on the temperature, which is also that the function T (x) has the typical course observed in agreement with the experimental data [66]. C in the high-temperature superconductors, corresponding The very important feature of the considered approach approximately to the inverse parabola. The dependence is related to the existence of the non-zero value of the en- of the energy gap on x is quite diﬀerent - the value of ergy gap above the critical temperature. The performed G (0) quickly decreases with the increase of the cerium numerical calculations show that G (T) vanishes at the concentration. Nernst temperature (T(cid:63)(cid:63)) [47]. Next, based on the data from the paper [17] and with In the present paper, basing on knowledge of the ex- the help of r1, the full dependence of the energy gap on perimental values of the critical temperature and the the temperature has been calculated. Additionally, it has energy gap close to zero Kelvin, the full form of the been assumed that the characteristic phonon frequency function G (T) has been determined. In particular, the is equal to 29 meV [72]. The obtained results have been following electron-doped superconductors, in the form plotted in f2. It can be seen that for the lowest value for of thin layers, have been taken into account: LCCO, the concentration of cerium, the function G (T) sharply Pr − Ce CuO (PCCO), and Nd − Ce CuO (NCCO). diﬀers from the BCS dependence of the energy gap on 2 x x 4 2 x x 4 Additionally, it has been assumed that ε = 0 and t is the temperature [34], [35]. In particular, G is much less 0 0 )K( T C )Vem( )0(G ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

font: default

text: phonon

text: state.

text: The

text: parameter

text: ω0

text: represents

text: the

text: value

text: of

text: LCCO

text: A

text: B

text: the

text: maximum

text: phonon

text: frequency.

text: Next,

text: the

text: Hamiltonian

text: H

text: can

text: be

text: rewritten

text: in

text: the

text: mo-

text: mentum

text: representation.

text: The

text: canonical

text: transformation

text: eliminating

text: the

text: phonon

text: degrees

text: of

text: freedom

text: allows

text: to

text: ob-

text: tain

text: the

text: fundamental

text: thermodynamic

label: equation

text: V

text: U

text: (T)

text: ω0

text: dερ

text: (ε)

text: tanh

text: √β2

text: ε2

text: −ω0

text: ε2

text: Krockenberger

text: et

text: al.

text: Skinta

text: et

text: al.

text: where:

text: E

text: V

text: U

text: (T)

text: (T)

text: The

text: parame-

text: ters

text: V

text: and

text: U

text: denote

text: the

text: electron-phonon

text: potential

text: (EPh)

text: x

text: x

text: and

text: the

text: electron-electron-phonon

text: potential

text: (EEPh),

text: re-

text: spectively:

text: V

text: V

text: V

text: and

text: U

text: V

text: U

text: FIG.

text: The

text: inﬂuence

text: of

text: the

text: cerium

text: concentration

text: on

text: the

text: ε0

text: The

text: order

text: parameter

text: is

text: deﬁned

text: by

text: the

text: formula:

text: value

text: of

text: (A)

text: the

text: critical

text: temperature

text: and

text: (B)

text: the

text: energy

text: gap

text: ω0

text: c−↓c−↑

text: while

text: c

text: denotes

text: the

text: annihilation

text: opera-

text: for

text: the

text: superconductor

text: LCCO

text: σ

text: tor

text: for

text: the

text: electron

text: state

text: with

text: the

text: momentum

text: k.

text: The

text: symbol

text: represents

text: the

text: thermodynamic

text: average

text: TABLE

text: I:

text: The

text: thermodynamic

text: parameters

text: of

text: the

text: LCCO

text: The

text: summation

text: over

text: k

text: should

text: be

text: performed

text: taking

text: superconductor.

text: Additionally,

text: it

text: has

text: been

text: deﬁned:

text: under

text: consideration

text: the

text: condition:

text: ε

text: ω

text: where

text: ε

text: is

text: U

text: the

text: electron

text: band

text: energy.

text: The

text: inverse

text: temperature

text: has

text: been

text: deﬁned

text: with

text: the

text: help

text: of

text: the

text: expression:

text: β

text: x

text: T

text: V

text: C

text: where

text: k

text: is

text: the

text: Boltzmann

text: constant.

text: B

text: (K)

text: (meV)

text: (meV)

text: (meV)

text: (K)

text: Let

text: us

text: notice

text: that

text: the

text: band

text: energy

text: for

text: the

text: two-

text: dimensional

text: square

text: lattice

text: can

text: be

text: written

text: as:

text: ε

text: [cos

text: (kx)

text: cos

text: (ky)].

text: Thus,

text: the|elect|ron

text: density

text: o≡f

text: states

text: has

text: the

text: form:

text: ρ

text: (ε)

text: b

text: ln

text: ε/b

text: where

text: b

text: and

text: b

text: The

text: presented

text: model

text: can

text: explain

text: (at

text: the

text: qualitative

text: level)

text: the

text: essential

text: properties

text: of

text: the

text: high-temperature

text: su-

text: perconducting

text: state

text: Namely,

text: the

text: high

text: critical

text: tem-

text: equal

text: to

text: meV

text: perature

text: is

text: related

text: to

text: the

text: existence

text: of

text: the

text: van

text: Hove

text: singu-

text: II.

text: RESULTS

text: larity

text: in

text: the

text: electron

text: density

text: of

text: states.

text: The

text: van

text: Hove

text: sin-

text: gularity

text: is

text: also

text: responsible

text: for

text: the

text: relatively

text: low

text: isotope

text: In

text: (A)-(B),

text: the

text: dependence

text: of

text: the

text: critical

text: temper-

text: coeﬃcient.

text: The

text: anomalously

text: high

text: value

text: of

text: the

text: energy

text: ature

text: and

text: the

text: low-temperature

text: value

text: of

text: the

text: energy

text: gap

text: gap

text: (G

text: where

text: G

text: V

text: U

text: in

text: (T

text: K)

text: on

text: the

text: concentration

text: of

text: cerium

text: has

text: been

text: respect

text: to

text: the

text: critical

text: temperature,

text: very

text: often

text: observed

text: presented.

text: The

text: symbols

text: have

text: been

text: prepared

text: on

text: the

text: basis

text: for

text: δ

text: δ

text: can

text: be

text: explained

text: by

text: the

text: high

text: ratio

text: U/V

text: It

text: max

text: of

text: the

text: experimental

text: data

text: (see

text: also

text: should

text: be

text: noted

text: that

text: in

text: the

text: considered

text: case,

text: the

text: function

text: In

text: the

text: case

text: of

text: the

text: critical

text: temperature,

text: it

text: may

text: be

text: noted

text: G

text: (T)

text: weakly

text: depends

text: on

text: the

text: temperature,

text: which

text: is

text: also

text: that

text: the

text: function

text: T

text: (x)

text: has

text: the

text: typical

text: course

text: observed

text: in

text: agreement

text: with

text: the

text: experimental

text: data

text: C

text: in

text: the

text: high-temperature

text: superconductors,

text: corresponding

text: The

text: very

text: important

text: feature

text: of

text: the

text: considered

text: approach

text: approximately

text: to

text: the

text: inverse

text: parabola.

text: The

text: dependence

text: is

text: related

text: to

text: the

text: existence

text: of

text: the

text: non-zero

text: value

text: of

text: the

text: en-

text: of

text: the

text: energy

text: gap

text: on

text: x

text: is

text: quite

text: diﬀerent

text: the

text: value

text: of

text: ergy

text: gap

text: above

text: the

text: critical

text: temperature.

text: The

text: performed

text: G

text: quickly

text: decreases

text: with

text: the

text: increase

text: of

text: the

text: cerium

text: numerical

text: calculations

text: show

text: that

text: G

text: (T)

text: vanishes

text: at

text: the

text: concentration.

text: Nernst

text: temperature

text: Next,

text: based

text: on

text: the

text: data

text: from

text: the

text: paper

text: and

text: with

text: In

text: the

text: present

text: paper,

text: basing

text: on

text: knowledge

text: of

text: the

text: ex-

text: the

text: help

text: of

text: the

text: full

text: dependence

text: of

text: the

text: energy

text: gap

text: on

text: perimental

text: values

text: of

text: the

text: critical

text: temperature

text: and

text: the

text: the

text: temperature

text: has

text: been

text: calculated.

text: Additionally,

text: it

text: has

text: energy

text: gap

text: close

text: to

text: zero

text: Kelvin,

text: the

text: full

text: form

text: of

text: the

text: been

text: assumed

text: that

text: the

text: characteristic

text: phonon

text: frequency

text: function

text: G

text: (T)

text: has

text: been

text: determined.

text: In

text: particular,

text: the

text: is

text: equal

text: to

text: meV

text: The

text: obtained

text: results

text: have

text: been

text: following

text: electron-doped

text: superconductors,

text: in

text: the

text: form

text: plotted

text: in

text: It

text: can

text: be

text: seen

text: that

text: for

text: the

text: lowest

text: value

text: for

text: of

text: thin

text: layers,

text: have

text: been

text: taken

text: into

text: account:

text: LCCO,

text: the

text: concentration

text: of

text: cerium,

text: the

text: function

text: G

text: (T)

text: sharply

text: Pr

text: Ce

text: CuO

text: (PCCO),

text: and

text: Nd

text: Ce

text: CuO

text: (NCCO).

text: diﬀers

text: from

text: the

text: BCS

text: dependence

text: of

text: the

text: energy

text: gap

text: on

text: x

text: x

text: x

text: x

text: Additionally,

text: it

text: has

text: been

text: assumed

text: that

text: ε

text: and

text: t

text: is

text: the

text: temperature

text: In

text: particular,

text: G

text: is

text: much

text: less

text: )K(

text: T

text: C

text: )Vem(

text: ##LTFigure##

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text: ∗ Valentin V. Karasiev, L´azaro Calder´ın, and S.B. Trickey Quantum Theory Project, Dept. of Physics and Dept. of Chemistry, University of Florida, Gainesville FL 32611-8435 (Dated: 13 May 2016) Eﬀects of explicit temperature dependence in the exchange-correlation (XC) free-energy functional upon calculated properties of matter in the warm dense regime are investigated. The comparison is between the KSDT ﬁnite-temperature local density approximation (TLDA) XC functional [Phys. Rev. Lett. 112, 076403 (2014)] parametrized from restricted path integral Monte Carlo data on the homogeneous electron gas (HEG) and the conventional Monte Carlo parametrization ground- state LDA XC functional (Perdew-Zunger, “PZ”) evaluated with T-dependent densities. Both Kohn-Sham (KS) and orbital-free density functional theory (OFDFT) are used, depending upon computational resource demands. Compared to the PZ functional, the KSDT functional generally lowers the direct-current (DC) electrical conductivity of low density Al, yielding improved agreement with experiment. The greatest lowering is about 15% for T= 15 kK. Correspondingly, the KS band structure of low-density fcc Al from KSDT exhibits a clear increase in inter-band separation above the Fermi level compared to the PZ bands. In some density-temperature regimes, the Deuterium equations of state obtained from the two XC functionals exhibit pressure diﬀerences as large as 4% and a 6% range of diﬀerences. However, the Hydrogen principal Hugoniot is insensitive to explicit XC T-dependence because of cancellation between the energy and pressure-volume work diﬀerence terms in the Rankine-Hugoniot equation. Finally, the temperature at which the HEG becomes ≥ unstable is T 7200 K for T-dependent XC, a result that the ground-state XC underestimates by about 1000 K. PACS numbers: 51.30.+i, 05.30.-d, 71.15.Mb, 52.25.Fi I. INTRODUCTION Warm dense matter (WDM), characterized by elevated 100000 -0.5 temperatures and wide compression ranges, plays an im- portant role in planetary-interior physics and materials 10000 -1 under extreme conditions, including the path to inertial conﬁnement fusion, heavy ion beam experiments, and Z- 1000 -1.5 pinch compression experiments [1–6]. Development of computational and theoretical methods to treat WDM 100 -2 applications is important both for interpreting experi- mental results and for gaining insight about thermody- 10 -2.5 namic regions that are diﬃcult to access experimentally. 1 -3 Current practice is to treat the WDM electronic de- grees of freedom via ﬁnite-temperature density functional 0.1 -3.5 theory (DFT) [7–9]. That necessitates use of an approxi- mate exchange-correlation (XC) free-energy density func- F 0.01 -4 tional, xc[n(T), T]. A common approximation [10–12] 0.1 1 10 100 is to use a ground-state XFC functional≈evaluated with the rs (bohr) ﬁnite-T density, that is [n(T), T] E [n(T)]. This xc xc FIG. 1: Map in (r , T) plane which shows the rela- s is the “ground state approximation” or GSA. Ref. 13 pre- tive importance of explicit T-dependence in the exchange- sented a rationale for why the GSA might be expected to correlation free-energy functional for the HEG measured as | − | | | | | work well. The essence of that argument is that GSA au- log ( f (r , T) e (r ) /[ f (r , T) + e (r ) ]). 10 xc s xc s s s xc s tomatically fulﬁlls certain constraints. The present study gives clear demonstrations of GSA deﬁciencies for speciﬁc systems in certain thermodynamic conditions and phys- self, and what are the magnitudes of the eﬀects? ical properties. The study involves systematic investiga- For compactness in what follows we use the phrase tions of three essential questions. What properties are F “XC thermal eﬀects” as a shortened expression for “ef- aﬀected by the explicit T-dependence of , over what xc fects of the explicit T-dependence in the XC free energy”. thermodynamic regime does the dependence manifest it- Thus, XC thermal eﬀects are those not included in the GSA. As orientation to the issue, Fig. 1 shows the rel- ative importance of XC thermal eﬀects as a function of ∗ r (the Wigner-Seitz radius, r = (3/4πn)1/3) and T as Electronic address: \markcomm{footercolor}{vkarasev@qtp. s | − | | s | | | ufl.edu} log10( fxc(rs, T) exc(rs) /[ fs(rs, T) + exc(rs) ]) for the )Kk( T ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Valentin

label: author

text: V.

text: Karasiev,

text: L´azaro

text: Calder´ın,

text: and

text: S.B.

text: Trickey

text: Quantum

text: Theory

text: Project,

text: Dept.

text: of

text: Physics

text: and

text: Dept.

text: of

text: Chemistry,

text: University

text: of

text: Florida,

text: Gainesville

text: FL

text: (Dated:

label: date

text: May

text: Eﬀects

label: abstract

text: of

text: explicit

text: temperature

text: dependence

text: in

text: the

text: exchange-correlation

text: (XC)

text: free-energy

text: functional

text: upon

text: calculated

text: properties

text: of

text: matter

text: in

text: the

text: warm

text: dense

text: regime

text: are

text: investigated.

text: The

text: comparison

text: is

text: between

text: the

text: KSDT

text: ﬁnite-temperature

text: local

text: density

text: approximation

text: (TLDA)

text: XC

text: functional

text: [Phys.

text: Rev.

text: Lett.

text: parametrized

text: from

text: restricted

text: path

text: integral

text: Monte

text: Carlo

text: data

text: on

text: the

text: homogeneous

text: electron

text: gas

text: (HEG)

text: and

text: the

text: conventional

text: Monte

text: Carlo

text: parametrization

text: ground-

text: state

text: LDA

text: XC

text: functional

text: (Perdew-Zunger,

text: “PZ”)

text: evaluated

text: with

text: T-dependent

text: densities.

text: Both

text: Kohn-Sham

text: (KS)

text: and

text: orbital-free

text: density

text: functional

text: theory

text: (OFDFT)

text: are

text: used,

text: depending

text: upon

text: computational

text: resource

text: demands.

text: Compared

text: to

text: the

text: PZ

text: functional,

text: the

text: KSDT

text: functional

text: generally

text: lowers

text: the

text: direct-current

text: (DC)

text: electrical

text: conductivity

text: of

text: low

text: density

text: Al,

text: yielding

text: improved

text: agreement

text: with

text: experiment.

text: The

text: greatest

text: lowering

text: is

text: about

text: for

text: T=

text: kK.

text: Correspondingly,

text: the

text: KS

text: band

text: structure

text: of

text: low-density

text: fcc

text: Al

text: from

text: KSDT

text: exhibits

text: a

text: clear

text: increase

text: in

text: inter-band

text: separation

text: above

text: the

text: Fermi

text: level

text: compared

text: to

text: the

text: PZ

text: bands.

text: In

text: some

text: density-temperature

text: regimes,

text: the

text: Deuterium

text: equations

text: of

text: state

text: obtained

text: from

text: the

text: two

text: XC

text: functionals

text: exhibit

text: pressure

text: diﬀerences

text: as

text: large

text: as

text: and

text: a

text: range

text: of

text: diﬀerences.

text: However,

text: the

text: Hydrogen

text: principal

text: Hugoniot

text: is

text: insensitive

text: to

text: explicit

text: XC

text: T-dependence

text: because

text: of

text: cancellation

text: between

text: the

text: energy

text: and

text: pressure-volume

text: work

text: diﬀerence

text: terms

text: in

text: the

text: Rankine-Hugoniot

text: equation.

text: Finally,

text: the

text: temperature

text: at

text: which

text: the

text: HEG

text: becomes

text: unstable

text: is

text: T

text: K

text: for

text: T-dependent

text: XC,

text: a

text: result

text: that

text: the

text: ground-state

text: XC

text: underestimates

text: by

text: about

text: K.

text: PACS

text: numbers:

text: I.

text: INTRODUCTION

text: Warm

text: dense

text: matter

text: (WDM),

text: characterized

text: by

text: elevated

text: temperatures

text: and

text: wide

text: compression

text: ranges,

text: plays

text: an

text: im-

text: portant

text: role

text: in

text: planetary-interior

text: physics

text: and

text: materials

text: under

text: extreme

text: conditions,

text: including

text: the

text: path

text: to

text: inertial

text: conﬁnement

text: fusion,

text: heavy

text: ion

text: beam

text: experiments,

text: and

text: Z-

text: pinch

text: compression

text: experiments

text: Development

text: of

text: computational

text: and

text: theoretical

text: methods

text: to

text: treat

text: WDM

text: applications

text: is

text: important

text: both

text: for

text: interpreting

text: experi-

text: mental

text: results

text: and

text: for

text: gaining

text: insight

text: about

text: thermody-

text: namic

text: regions

text: that

text: are

text: diﬃcult

text: to

text: access

text: experimentally.

text: Current

text: practice

text: is

text: to

text: treat

text: the

text: WDM

text: electronic

text: de-

text: grees

text: of

text: freedom

text: via

text: ﬁnite-temperature

text: density

text: functional

text: theory

text: (DFT)

text: That

text: necessitates

text: use

text: of

text: an

text: approxi-

text: mate

text: exchange-correlation

text: (XC)

text: free-energy

text: density

text: func-

text: F

text: tional,

text: xc[n(T),

text: T].

text: A

text: common

text: approximation

text: is

text: to

text: use

text: a

text: ground-state

text: XFC

text: functional≈evaluated

text: with

text: the

text: rs

text: (bohr)

text: ﬁnite-T

text: density,

text: that

text: is

text: [n(T),

text: T]

text: E

text: [n(T)].

text: This

text: xc

text: xc

text: FIG.

text: Map

text: in

text: (r

text: T)

text: plane

text: which

text: shows

text: the

text: rela-

text: s

text: is

text: the

text: “ground

text: state

text: approximation”

text: or

text: GSA.

text: Ref.

text: pre-

text: tive

text: importance

text: of

text: explicit

text: T-dependence

text: in

text: the

text: exchange-

text: sented

text: a

text: rationale

text: for

text: why

text: the

text: GSA

text: might

text: be

text: expected

text: to

text: correlation

text: free-energy

text: functional

text: for

text: the

text: HEG

text: measured

text: as

text: work

text: well.

text: The

text: essence

text: of

text: that

text: argument

text: is

text: that

text: GSA

text: au-

text: log

text: f

text: (r

text: T)

text: e

text: (r

text: f

text: (r

text: T)

text: e

text: (r

text: xc

text: s

text: xc

text: s

text: s

text: s

text: xc

text: s

text: tomatically

text: fulﬁlls

text: certain

text: constraints.

text: The

text: present

text: study

text: gives

text: clear

text: demonstrations

text: of

text: GSA

text: deﬁciencies

text: for

text: speciﬁc

text: systems

text: in

text: certain

text: thermodynamic

text: conditions

text: and

text: phys-

text: self,

text: and

text: what

text: are

text: the

text: magnitudes

text: of

text: the

text: eﬀects?

text: ical

text: properties.

text: The

text: study

text: involves

text: systematic

text: investiga-

text: For

text: compactness

text: in

text: what

text: follows

text: we

text: use

text: the

text: phrase

text: tions

text: of

text: three

text: essential

text: questions.

text: What

text: properties

text: are

text: F

text: “XC

text: thermal

text: eﬀects”

text: as

text: a

text: shortened

text: expression

text: for

text: “ef-

text: aﬀected

text: by

text: the

text: explicit

text: T-dependence

text: of

text: over

text: what

text: xc

text: fects

text: of

text: the

text: explicit

text: T-dependence

text: in

text: the

text: XC

text: free

text: energy”.

text: thermodynamic

text: regime

text: does

text: the

text: dependence

text: manifest

text: it-

text: Thus,

text: XC

text: thermal

text: eﬀects

text: are

text: those

text: not

text: included

text: in

text: the

text: GSA.

text: As

text: orientation

text: to

text: the

text: issue,

text: Fig.

text: shows

text: the

text: rel-

text: ative

text: importance

text: of

text: XC

text: thermal

text: eﬀects

text: as

text: a

text: function

text: of

text: r

text: (the

text: Wigner-Seitz

text: radius,

text: r

text: and

text: T

text: as

text: Electronic

text: address:

text: \markcomm{footercolor}{vkarasev@qtp.

text: s

text: s

text: ufl.edu}

text: fxc(rs,

text: T)

text: exc(rs)

text: fs(rs,

text: T)

text: exc(rs)

text: for

text: the

text: )Kk(

text: T

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##