text: 9 (cid:89) for odd N↑ and N↓, and (N /2, N /2), (N /6, 5N /6), φ φ φ φ and (5N /6, N /6) for even N↑ and N↓. As a conse- N/2 φ φ Ψ = (z − z )3 (ω − ω )3 , (21) quence, the momentum sector at Qy = 0 (mod Nφ) has 330 i j i j the triple degeneracy. Also, due to the center-of-mass i<j shift, there are other similar triple degeneracies in the momentum sectors at Q = N and 2N (mod N ) with y φ which can be in turn obtained as the exact zero-energy N = N↑+N↓. It is worthwhile to mention that, along the ground state of the BQH Hamiltonian in Eq. (7) with x-direction, the momentum of all the above degenerate V intra set equal to a nonzero positive number and all (cid:48) 1 states is the same; Q = Q ↑ + Q ↓ = 0 (mod N ) with the other Haldane pseudopotentials to zero. Practically, (cid:48) x x x N = gcd(N, N ). φ we use the fact that Ψ is the direct product of two 330 For odd N↑ and N↓, two degenerate copies of the Laughlin states, each of which can be obtained in a much Halperin (330) state can be obtained from (Q ↑, Q ↓) = smaller Hilbert space. For example, for N = 12 in the y y (N /3, 2N /3) and (2N /3, N /3), both of which reduce torus geometry, the size of the Hilbert space for the 2/3 φ φ φ φ BQH system is around 3.6 × 1015, whereas that for the to a single identical state at Qy = 0 (mod Nφ) after 1/3 FQH system is around 2.9 × 106. Once Ψ is ob- antisymmetrization. Meanwhile, the antisymmetrized 330 projection of the Halperin (330) state obtained from tained, we apply the antisymmetrization operator to ob- (Q ↑, Q ↓) = (0, 0) remains distinct from this state. tain AΨ = Ψ and then the PH conjugation operator y y 330 Z4 Consequently, a double ground-state degeneracy is gen- to obtain C Ψ . PH Z4 erated at Q = 0. Since Q = 0 for all the above As mentioned in Sec. , there is an issue of the ground- y x states, this means that there is the double ground- state degeneracy mismatch between the antisymmetrized state degeneracy for the antisymmetrized Halperin (330) Halperin (330) and the Z parafermion states in the torus 4 state at Q = (Q , Q ) = (0, 0). Similarly, for even N↑ geometry. Fortunately, this ground-state degeneracy is- x y and N↓, two degenerate copies of the Halperin (330) sue is much less severe, compared to that between the an- state can be obtained from (Q ↑, Q ↓) = (N /6, 5N /6) tisymmetrized Halperin (551) and the fermionic Haﬀnian y y φ φ and (5N /6, N /6), both of which reduce to a sin- states discussed in the preceding section. Speciﬁcally, φ φ gle identical state at Q = 0 (mod N ) after anti- there is a certain momentum sector in the torus geome- y φ symmetrization. Also, a diﬀerent single state is ob- try, where all degenerate copies of the antisymmetrized tained by antisymmetrizing the Halperin (330) state at Halperin (330) state are exactly identical to the appro- (Q ↑, Q ↓) = (N /2, N /2), consequently generating the priate counterparts of the Z parafermion state. y y φ φ 4 double ground-state degeneracy at Q = (0, 0) similar to To understand what this momentum sector is, let us the case of odd N↑ and N↓. Eventually, there is alto- consider the root conﬁgurations of the Halperin (330) gether the six-fold ground-state degeneracy due to the and the Z parafermion states. First, the Halperin (330) 4 center-of-mass shift. state is constructed as the direct product between two Laughlin states in the top (↑) and bottom (↓) layers, It is important to note that, strictly speaking, anti- each of which has three degenerate copies via the center- symmetrization would annihilate any root conﬁgurations of-mass shift, leading to the nine-fold ground-state de- obtained in the momentum sector satisfying Qy↑ = Qy↓, generacy of the Halperin (330) state [81]. Speciﬁcally, i.e., where electrons with opposite layer indices are ex- the Laughlin state has the following three root conﬁg- actly on top of each other. Fortunately, there are extra urations: | • ◦ ◦ • ◦ ◦ • ◦ ◦ · · · (cid:105), | ◦ • ◦ ◦ • ◦ ◦ • ◦ · · · (cid:105), “ﬂuctuation conﬁgurations,” which are derived from the and | ◦ ◦ • ◦ ◦ • ◦ ◦ • · · · (cid:105), where • and ◦ indicate ﬁlled root conﬁgurations away from the thin-torus limit. These and empty sites, respectively. These three root conﬁgu- ﬂuctuation conﬁgurations can generally survive antisym- rations are related with each other via the center-of-mass metrization. shift. Evaluating the pseudomomentum of each root con- Now, let us switch gears and consider the ground-state ﬁguration, one can show that three degenerate copies of degeneracy of the Z parafermion state. There are al- 4 the Laughlin state occur at Q = 0, N /3, and 2N /3 together ﬁfteen diﬀerent root conﬁgurations for the Z y φ φ 4 for odd, and N /6, N /2, and 5N /6 for even particle parafermion state: (i) | • • • • ◦ ◦ · · · (cid:105) and its six other φ φ φ numbers, where N is the number of ﬂux quanta. We set center-of-mass-shifted copies, (ii) |•••◦•◦ · · · (cid:105) and its six φ N = 3N↑ = 3N↓ since we are interested in the BQH sys- other center-of-mass-shifted copies, and (iii) |••◦••◦ · · · (cid:105) φ tem with the top and bottom layers having the (equal) and its three other center-of-mass-shifted copies. Phys- particle number N↑ and N↓, respectively. Note that there ically, these root conﬁgurations can be understood as is no ﬂux shift in the torus geometry. all possible ways of distributing four electrons within The root conﬁgurations of the Halperin (330) state can the unit cell of six sites, followed by the center-of-mass be obtained by taking the direct product between those shift [82, 83]. Having absolutely zero amplitudes of ﬁve- of the two Laughlin states in the top and bottom lay- electron cluster, these root conﬁgurations generate the ers. Among the nine degenerate copies obtained from zero-energy ground states of the fermionic ﬁve-body δ- this product, the Halperin (330) state can occur partic- function interaction Hamiltonian, which imposes an en- ularly at Q = Q ↑ + Q ↓ = 0 (mod N ) by combining ergy cost only on ﬁve-electron cluster. Note that, by y y y φ (Q ↑, Q ↓) = (0, 0), (N /3, 2N /3), and (2N /3, N /3) deﬁnition, the Z parafermion state is the zero-energy y y φ φ φ φ k

label: equation

text: for

text: odd

text: N↑

text: and

text: N↓,

text: and

text: (N

text: N

text: (N

text: φ

text: φ

text: φ

text: φ

text: and

text: N

text: for

text: even

text: N↑

text: and

text: N↓.

text: As

text: a

text: conse-

text: φ

text: φ

text: Ψ

text: (z

text: z

text: (ω

text: ω

text: quence,

text: the

text: momentum

text: sector

text: at

text: Qy

text: (mod

text: Nφ)

text: has

text: i

text: j

text: i

text: j

text: the

text: triple

text: degeneracy.

text: Also,

text: due

text: to

text: the

text: center-of-mass

text: i<j

text: shift,

text: there

text: are

text: other

text: similar

text: triple

text: degeneracies

text: in

text: the

text: momentum

text: sectors

text: at

text: Q

text: N

text: and

text: (mod

text: N

text: with

text: y

text: φ

text: which

text: can

text: be

text: in

text: turn

text: obtained

text: as

text: the

text: exact

text: zero-energy

text: N

text: N↑+N↓.

text: It

text: is

text: worthwhile

text: to

text: mention

text: that,

text: along

text: the

text: ground

text: state

text: of

text: the

text: BQH

text: Hamiltonian

text: in

text: Eq.

text: with

text: x-direction,

text: the

text: momentum

text: of

text: all

text: the

text: above

text: degenerate

text: V

text: intra

text: set

text: equal

text: to

text: a

text: nonzero

text: positive

text: number

text: and

text: all

text: states

text: is

text: the

text: same;

text: Q

text: Q

text: Q

text: (mod

text: N

text: with

text: the

text: other

text: Haldane

text: pseudopotentials

text: to

text: zero.

text: Practically,

text: x

text: x

text: x

text: N

text: gcd(N,

text: N

text: φ

text: we

text: use

text: the

text: fact

text: that

text: Ψ

text: is

text: the

text: direct

text: product

text: of

text: two

text: For

text: odd

text: N↑

text: and

text: N↓,

text: two

text: degenerate

text: copies

text: of

text: the

text: Laughlin

text: states,

text: each

text: of

text: which

text: can

text: be

text: obtained

text: in

text: a

text: much

text: Halperin

text: state

text: can

text: be

text: obtained

text: from

text: (Q

text: Q

text: smaller

text: Hilbert

text: space.

text: For

text: example,

text: for

text: N

text: in

text: the

text: y

text: y

text: (N

text: and

text: N

text: both

text: of

text: which

text: reduce

text: torus

text: geometry,

text: the

text: size

text: of

text: the

text: Hilbert

text: space

text: for

text: the

text: φ

text: φ

text: φ

text: φ

text: BQH

text: system

text: is

text: around

text: whereas

text: that

text: for

text: the

text: to

text: a

text: single

text: identical

text: state

text: at

text: Qy

text: (mod

text: Nφ)

text: after

text: FQH

text: system

text: is

text: around

text: Once

text: Ψ

text: is

text: ob-

text: antisymmetrization.

text: Meanwhile,

text: the

text: antisymmetrized

text: projection

text: of

text: the

text: Halperin

text: state

text: obtained

text: from

text: tained,

text: we

text: apply

text: the

text: antisymmetrization

text: operator

text: to

text: ob-

text: (Q

text: Q

text: remains

text: distinct

text: from

text: this

text: state.

text: tain

text: AΨ

text: Ψ

text: and

text: then

text: the

text: PH

text: conjugation

text: operator

text: y

text: y

text: Consequently,

text: a

text: double

text: ground-state

text: degeneracy

text: is

text: gen-

text: to

text: obtain

text: C

text: Ψ

text: PH

text: erated

text: at

text: Q

text: Since

text: Q

text: for

text: all

text: the

text: above

text: As

text: mentioned

text: in

text: Sec.

text: there

text: is

text: an

text: issue

text: of

text: the

text: ground-

text: y

text: x

text: states,

text: this

text: means

text: that

text: there

text: is

text: the

text: double

text: ground-

text: state

text: degeneracy

text: mismatch

text: between

text: the

text: antisymmetrized

text: state

text: degeneracy

text: for

text: the

text: antisymmetrized

text: Halperin

text: Halperin

text: and

text: the

text: Z

text: parafermion

text: states

text: in

text: the

text: torus

text: state

text: at

text: Q

text: (Q

text: Q

text: Similarly,

text: for

text: even

text: N↑

text: geometry.

text: Fortunately,

text: this

text: ground-state

text: degeneracy

text: is-

text: x

text: y

text: and

text: N↓,

text: two

text: degenerate

text: copies

text: of

text: the

text: Halperin

text: sue

text: is

text: much

text: less

text: severe,

text: compared

text: to

text: that

text: between

text: the

text: an-

text: state

text: can

text: be

text: obtained

text: from

text: (Q

text: Q

text: (N

text: tisymmetrized

text: Halperin

text: and

text: the

text: fermionic

text: Haﬀnian

text: y

text: y

text: φ

text: φ

text: and

text: N

text: both

text: of

text: which

text: reduce

text: to

text: a

text: sin-

text: states

text: discussed

text: in

text: the

text: preceding

text: section.

text: Speciﬁcally,

text: φ

text: φ

text: gle

text: identical

text: state

text: at

text: Q

text: (mod

text: N

text: after

text: anti-

text: there

text: is

text: a

text: certain

text: momentum

text: sector

text: in

text: the

text: torus

text: geome-

text: y

text: φ

text: symmetrization.

text: Also,

text: a

text: diﬀerent

text: single

text: state

text: is

text: ob-

text: try,

text: where

text: all

text: degenerate

text: copies

text: of

text: the

text: antisymmetrized

text: tained

text: by

text: antisymmetrizing

text: the

text: Halperin

text: state

text: at

text: Halperin

text: state

text: are

text: exactly

text: identical

text: to

text: the

text: appro-

text: (Q

text: Q

text: (N

text: N

text: consequently

text: generating

text: the

text: priate

text: counterparts

text: of

text: the

text: Z

text: parafermion

text: state.

text: y

text: y

text: φ

text: φ

text: double

text: ground-state

text: degeneracy

text: at

text: Q

text: similar

text: to

text: To

text: understand

text: what

text: this

text: momentum

text: sector

text: is,

text: let

text: us

text: the

text: case

text: of

text: odd

text: N↑

text: and

text: N↓.

text: Eventually,

text: there

text: is

text: alto-

text: consider

text: the

text: root

text: conﬁgurations

text: of

text: the

text: Halperin

text: gether

text: the

text: six-fold

text: ground-state

text: degeneracy

text: due

text: to

text: the

text: and

text: the

text: Z

text: parafermion

text: states.

text: First,

text: the

text: Halperin

text: center-of-mass

text: shift.

text: state

text: is

text: constructed

text: as

text: the

text: direct

text: product

text: between

text: two

text: Laughlin

text: states

text: in

text: the

text: top

text: and

text: bottom

text: layers,

text: It

text: is

text: important

text: to

text: note

text: that,

text: strictly

text: speaking,

text: anti-

text: each

text: of

text: which

text: has

text: three

text: degenerate

text: copies

text: via

text: the

text: center-

text: symmetrization

text: would

text: annihilate

text: any

text: root

text: conﬁgurations

text: of-mass

text: shift,

text: leading

text: to

text: the

text: nine-fold

text: ground-state

text: de-

text: obtained

text: in

text: the

text: momentum

text: sector

text: satisfying

text: Qy↑

text: Qy↓,

text: generacy

text: of

text: the

text: Halperin

text: state

text: Speciﬁcally,

text: i.e.,

text: where

text: electrons

text: with

text: opposite

text: layer

text: indices

text: are

text: ex-

text: the

text: Laughlin

text: state

text: has

text: the

text: following

text: three

text: root

text: conﬁg-

text: actly

text: on

text: top

text: of

text: each

text: other.

text: Fortunately,

text: there

text: are

text: extra

text: urations:

text: “ﬂuctuation

text: conﬁgurations,”

text: which

text: are

text: derived

text: from

text: the

text: and

text: where

text: and

text: indicate

text: ﬁlled

text: root

text: conﬁgurations

text: away

text: from

text: the

text: thin-torus

text: limit.

text: These

text: and

text: empty

text: sites,

text: respectively.

text: These

text: three

text: root

text: conﬁgu-

text: ﬂuctuation

text: conﬁgurations

text: can

text: generally

text: survive

text: antisym-

text: rations

text: are

text: related

text: with

text: each

text: other

text: via

text: the

text: center-of-mass

text: metrization.

text: shift.

text: Evaluating

text: the

text: pseudomomentum

text: of

text: each

text: root

text: con-

text: Now,

text: let

text: us

text: switch

text: gears

text: and

text: consider

text: the

text: ground-state

text: ﬁguration,

text: one

text: can

text: show

text: that

text: three

text: degenerate

text: copies

text: of

text: degeneracy

text: of

text: the

text: Z

text: parafermion

text: state.

text: There

text: are

text: al-

text: the

text: Laughlin

text: state

text: occur

text: at

text: Q

text: N

text: and

text: together

text: ﬁfteen

text: diﬀerent

text: root

text: conﬁgurations

text: for

text: the

text: Z

text: y

text: φ

text: φ

text: for

text: odd,

text: and

text: N

text: N

text: and

text: for

text: even

text: particle

text: parafermion

text: state:

text: (i)

text: and

text: its

text: six

text: other

text: φ

text: φ

text: φ

text: numbers,

text: where

text: N

text: is

text: the

text: number

text: of

text: ﬂux

text: quanta.

text: We

text: set

text: center-of-mass-shifted

text: copies,

text: (ii)

text: and

text: its

text: six

text: φ

text: N

text: since

text: we

text: are

text: interested

text: in

text: the

text: BQH

text: sys-

text: other

text: center-of-mass-shifted

text: copies,

text: and

text: (iii)

text: φ

text: tem

text: with

text: the

text: top

text: and

text: bottom

text: layers

text: having

text: the

text: (equal)

text: and

text: its

text: three

text: other

text: center-of-mass-shifted

text: copies.

text: Phys-

text: particle

text: number

text: N↑

text: and

text: N↓,

text: respectively.

text: Note

text: that

text: there

text: ically,

text: these

text: root

text: conﬁgurations

text: can

text: be

text: understood

text: as

text: is

text: no

text: ﬂux

text: shift

text: in

text: the

text: torus

text: geometry.

text: all

text: possible

text: ways

text: of

text: distributing

text: four

text: electrons

text: within

text: The

text: root

text: conﬁgurations

text: of

text: the

text: Halperin

text: state

text: can

text: the

text: unit

text: cell

text: of

text: six

text: sites,

text: followed

text: by

text: the

text: center-of-mass

text: be

text: obtained

text: by

text: taking

text: the

text: direct

text: product

text: between

text: those

text: shift

text: Having

text: absolutely

text: zero

text: amplitudes

text: of

text: ﬁve-

text: of

text: the

text: two

text: Laughlin

text: states

text: in

text: the

text: top

text: and

text: bottom

text: lay-

text: electron

text: cluster,

text: these

text: root

text: conﬁgurations

text: generate

text: the

text: ers.

text: Among

text: the

text: nine

text: degenerate

text: copies

text: obtained

text: from

text: zero-energy

text: ground

text: states

text: of

text: the

text: fermionic

text: ﬁve-body

text: δ-

text: this

text: product,

text: the

text: Halperin

text: state

text: can

text: occur

text: partic-

text: function

text: interaction

text: Hamiltonian,

text: which

text: imposes

text: an

text: en-

text: ularly

text: at

text: Q

text: Q

text: Q

text: (mod

text: N

text: by

text: combining

text: ergy

text: cost

text: only

text: on

text: ﬁve-electron

text: cluster.

text: Note

text: that,

text: by

text: y

text: y

text: y

text: φ

text: (Q

text: Q

text: (N

text: and

text: N

text: deﬁnition,

text: the

text: Z

text: parafermion

text: state

text: is

text: the

text: zero-energy

text: y

text: y

text: φ

text: φ

text: φ

text: φ

text: k


text: 3 LOGICAL GAMES—REPRESENTING STRATEGIC GAMES 17 ≤ 5. Let n be a positive integer and m n. Then there is a characteristic formula for m in Ł . n n (cid:76) Proof. Claim 1 follows, e.g., from the theory of Farey–Schauder hats as developed in [8, Section 3]. To obtain claim 2 for any rational a = m (where m, n are relatively prime), it is suﬃcient to take n the formula n ξ , where ξ is as in claim 1. Claims 3 and 4 are direct consequences i=1 m,n m,n of functional representation of L(cid:32) ukasiewicz inﬁnite-valued logic (see Table 1). Finally, in order to obtain claim 5 it suﬃces to take the formula &ni=1 χm , where χm is a pseudo-characteristic formula for m in [0, 1]Ł (a more involved proof of this factnis provided nin [17, Lemma 3] and in [18, n Lemma 7.2]). LH Thus all logical Łn-games (including, e.g.G, the game Łm of Example 3.1(cid:23)0G) ⊆areQweakly ex- pressible and so are all logicalE[C0, 1]Ł-games such and only such that [0, 1]Ł . Observe that, consequently, the game Ł of Example 3.13 is not even weakly expressible (although it would become weakly expressible if restricted to rational payoﬀs). Furthermore, recall that the algebras Łc and Gc introduced in Example 2.2 contain truth n n constants for all elements of their domains; consequently, all Łc - and Gc -games are expressible NT n n (including, e.g., the game Łc4 of Example 3.2∈). Exam∩pQle 2.2 also introduced several standard algebras containing truth constants r¯ for all r [0, 1] : clearly for any such algebra A and G (cid:23)G ⊆ Q G VA(cid:52) any A-game , if A then is expressible. (Thus, e.g., the game Q of Example 3.11 is Ł expressible.) G H{ere w}e(cid:40)can als(cid:23)oGil⊆lusQtrate the usefulneGss of Proposition 3.4: Consider a [0, 1]Ł-game such that 0, 1 [0, 1]Ł . WeGknow that is weakly expressible, but not expressible. However, by Proposition 3.4, we can see as a [0, 1]QŁ-game which has the same pure and mixed equilibria and is clearly expressible. 3.4 Representing strategic games—general results In the next three propositions we investigate a particularly simple type of games: ﬁnite strategic games with at most two possible payoﬀ values. Note that all ﬁnite win/loose games fall into this category. It is easy to show that all such games (modulo certain encodings of strategies) can be presented as Boolean games (logical 2-games in our terminology). The only possible complication arises from the fact that the sets of strategies need not be limited to a binary choice. This implies that each player may have to control more than one variable, in general, so that all of her possible choices can be represented. This result is implicit already in [14]; however, we make it explicit and prove it, as it provides a convenient preparation for more complex cases elaborated later. Convention 3.17. By Lemma 2.16 one may identify without loss of generality any given set of strategies of a ﬁnite game with an initial segment of the set of natural numbers. For sake of conciseness we will do so in the rest of the paper. G (cid:104) { | ∈ } { | ∈ }(cid:105) Proposition 3.18. Let = N, S i N , f i N be a ﬁnite strategic game such that the { } i i ∈ union of ranges of all f is a, b , where a < b. For any i N, let n denote the least natural | | ≤i ∈ (cid:104) i(cid:105) ∈ number such that Si 2ni. Moreover, for any si Si, let s1i , . . . , sni i 2ni denote the binary representation of s . (Recall that by Convention 3.17 we assume that s is a natural number). i G i G (cid:104) { | ∈ Th}en{ th(cid:48) e| g∈ame} { is |aﬃ∈nely}(cid:105)re(cid:83)presented by (cid:104)an(cid:105)expressible logical 2-g∈ame ˆ = N, V, Vi i N , S i N , ϕ i N via g and c = c ∈ , where for each i N: i i i i N • V = {v1, . . . , vni} (and V = ∈ V ). i i i i N i • Si(cid:48) = {(cid:104)s1i(cid:95), . . . , sni i(cid:94)(cid:105) | s(cid:94)i ∈ Si}. (Recall that by our conventions, strategies can be identiﬁed with tuples of elements of the algebra; see beginning of Section .) • ϕ = δ (vj), i sj k ∈ ×···× ∈ ≤ k s S1 Sn k N j nk fi(s)=b ∈ { } ¬ where δ (v) is a characteristic formula of x 0, 1 in 2 (e.g., δ (v) = v and δ (v) = v). x 0 1 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: LOGICAL

text: GAMES—REPRESENTING

text: STRATEGIC

text: GAMES

label: list

text: Let

text: n

text: be

text: a

text: positive

text: integer

text: and

text: m

text: n.

text: Then

text: there

text: is

text: a

text: characteristic

text: formula

text: for

text: m

text: in

text: Ł

text: n

text: n

text: Proof.

text: Claim

text: follows,

text: e.g.,

text: from

text: the

text: theory

text: of

text: Farey–Schauder

text: hats

text: as

text: developed

text: in

text: Section

text: To

text: obtain

text: claim

text: for

text: any

text: rational

text: a

text: m

text: (where

text: m,

text: n

text: are

text: relatively

text: prime),

text: it

text: is

text: suﬃcient

text: to

text: take

text: n

text: the

text: formula

text: n

text: ξ

text: where

text: ξ

text: is

text: as

text: in

text: claim

text: Claims

text: and

text: are

text: direct

text: consequences

text: m,n

text: m,n

text: of

text: functional

text: representation

text: of

text: ukasiewicz

text: inﬁnite-valued

text: logic

text: (see

text: Table

text: Finally,

text: in

text: order

text: to

text: obtain

text: claim

text: it

text: suﬃces

text: to

text: take

text: the

text: formula

text: χm

text: where

text: χm

text: is

text: a

text: pseudo-characteristic

text: formula

text: for

text: m

text: in

text: (a

text: more

text: involved

text: proof

text: of

text: this

text: factnis

text: provided

text: nin

text: Lemma

text: and

text: in

text: n

text: Lemma

text: LH

text: Thus

text: all

text: logical

text: Łn-games

text: (including,

text: e.g.G,

text: the

text: game

text: Łm

text: of

text: Example

text: ⊆areQweakly

text: ex-

text: pressible

text: and

text: so

text: are

text: all

text: such

text: and

text: only

text: such

text: that

text: Observe

text: that,

text: consequently,

text: the

text: game

text: Ł

text: of

text: Example

text: is

text: not

text: even

text: weakly

text: expressible

text: (although

text: it

text: would

text: become

text: weakly

text: expressible

text: if

text: restricted

text: to

text: rational

text: payoﬀs).

text: Furthermore,

text: recall

text: that

text: the

text: algebras

text: Łc

text: and

text: Gc

text: introduced

text: in

text: Example

text: contain

text: truth

text: n

text: n

text: constants

text: for

text: all

text: elements

text: of

text: their

text: domains;

text: consequently,

text: all

text: Łc

text: and

text: Gc

text: -games

text: are

text: expressible

text: NT

text: n

text: n

text: (including,

text: e.g.,

text: the

text: game

text: of

text: Example

text: Exam∩pQle

text: also

text: introduced

text: several

text: standard

text: algebras

text: containing

text: truth

text: constants

text: r¯

text: for

text: all

text: r

text: clearly

text: for

text: any

text: such

text: algebra

text: A

text: and

text: G

text: Q

text: G

text: any

text: A-game

text: if

text: A

text: then

text: is

text: expressible.

text: (Thus,

text: e.g.,

text: the

text: game

text: Q

text: of

text: Example

text: is

text: Ł

text: expressible.)

text: G

text: H{ere

text: the

text: usefulneGss

text: of

text: Proposition

text: Consider

text: a

text: such

text: that

text: WeGknow

text: that

text: is

text: weakly

text: expressible,

text: but

text: not

text: expressible.

text: However,

text: by

text: Proposition

text: we

text: can

text: see

text: as

text: a

text: which

text: has

text: the

text: same

text: pure

text: and

text: mixed

text: equilibria

text: and

text: is

text: clearly

text: expressible.

text: Representing

text: strategic

text: games—general

text: results

text: In

text: the

text: next

text: three

text: propositions

text: we

text: investigate

text: a

text: particularly

text: simple

text: type

text: of

text: games:

text: ﬁnite

text: strategic

text: games

text: with

text: at

text: most

text: two

text: possible

text: payoﬀ

text: values.

text: Note

text: that

text: all

text: ﬁnite

text: win/loose

text: games

text: fall

text: into

text: this

text: category.

text: It

text: is

text: easy

text: to

text: show

text: that

text: all

text: such

text: games

text: (modulo

text: certain

text: encodings

text: of

text: strategies)

text: can

text: be

text: presented

text: as

text: Boolean

text: games

text: (logical

text: in

text: our

text: terminology).

text: The

text: only

text: possible

text: complication

text: arises

text: from

text: the

text: fact

text: that

text: the

text: sets

text: of

text: strategies

text: need

text: not

text: be

text: limited

text: to

text: a

text: binary

text: choice.

text: This

text: implies

text: that

text: each

text: player

text: may

text: have

text: to

text: control

text: more

text: than

text: one

text: variable,

text: in

text: general,

text: so

text: that

text: all

text: of

text: her

text: possible

text: choices

text: can

text: be

text: represented.

text: This

text: result

text: is

text: implicit

text: already

text: in

text: however,

text: we

text: make

text: it

text: explicit

text: and

text: prove

text: it,

text: as

text: it

text: provides

text: a

text: convenient

text: preparation

text: for

text: more

text: complex

text: cases

text: elaborated

text: later.

text: Convention

text: By

text: Lemma

text: one

text: may

text: identify

text: without

text: loss

text: of

text: generality

text: any

text: given

text: set

text: of

text: strategies

text: of

text: a

text: ﬁnite

text: game

text: with

text: an

text: initial

text: segment

text: of

text: the

text: set

text: of

text: natural

text: numbers.

text: For

text: sake

text: of

text: conciseness

text: we

text: will

text: do

text: so

text: in

text: the

text: rest

text: of

text: the

text: paper.

text: G

text: Proposition

text: Let

text: N,

text: S

text: i

text: N

text: f

text: i

text: N

text: be

text: a

text: ﬁnite

text: strategic

text: game

text: such

text: that

text: the

text: i

text: i

text: union

text: of

text: ranges

text: of

text: all

text: f

text: is

text: a,

text: b

text: where

text: a

text: b.

text: For

text: any

text: i

text: N,

text: let

text: n

text: denote

text: the

text: least

text: natural

text: ≤i

text: number

text: such

text: that

text: Si

text: Moreover,

text: for

text: any

text: si

text: Si,

text: let

text: sni

text: i

text: denote

text: the

text: binary

text: representation

text: of

text: s

text: (Recall

text: that

text: by

text: Convention

text: we

text: assume

text: that

text: s

text: is

text: a

text: natural

text: number).

text: i

text: G

text: i

text: G

text: Th}en{

text: e|

text: g∈ame}

text: is

text: by

text: logical

text: ˆ

text: N,

text: V,

text: Vi

text: i

text: N

text: S

text: i

text: N

text: ϕ

text: i

text: N

text: via

text: g

text: and

text: c

text: c

text: where

text: for

text: each

text: i

text: N:

text: i

text: i

text: i

text: i

text: N

text: V

text: vni}

text: (and

text: V

text: V

text: i

text: i

text: i

text: i

text: N

text: i

text: sni

text: Si}.

text: (Recall

text: that

text: by

text: our

text: conventions,

text: strategies

text: can

text: be

text: identiﬁed

text: with

text: tuples

text: of

text: elements

text: of

text: the

text: algebra;

text: see

text: beginning

text: of

text: Section

text: ϕ

text: δ

text: (vj),

text: i

text: sj

text: k

text: k

text: s

text: Sn

text: k

text: N

text: j

text: nk

text: fi(s)=b

text: where

text: δ

text: (v)

text: is

text: a

text: characteristic

text: formula

text: of

text: x

text: in

text: (e.g.,

text: δ

text: (v)

text: v

text: and

text: δ

text: (v)

text: v).

text: x

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: REFERENCES 26 by logical formulas. In particular they also considered operations on games, a form of relativized validity and satisﬁability motivated by their games, and a calculus for deriving winning strategies. It would be certainly interesting to see to what extent these and related topics can be developed also in the considerably more general many-valued setting presented here. References [1] Stefano Aguzzoli, Simone Bova, and Brunella Gerla. Free algebras and functional represen- tation for fuzzy logics. In Petr Cintula, Petr H´ajek, and Carles Noguera, editors, Handbook of Mathematical Fuzzy Logic - Volume 2, volume 38 of Studies in Logic, Mathematical Logic and Foundations, pages 713–791. College Publications, London, 2011. [2] Matthias Baaz, Christian G. Fermu¨ller, and Helmut Veith. An analytic calculus for quantiﬁed propositional G¨odel logic. In Proceedings of TABLEAUX 2000, volume 1847 of Lecture Notes in Computer Science, pages 112–126, Berlin, 2000. Springer. [3] Matthias Baaz and Norbert Preining. G¨odel–Dummett logics. In Petr Cintula, Petr H´ajek, and Carles Noguera, editors, Handbook of Mathematical Fuzzy Logic - Volume 2, volume 38 of Studies in Logic, Mathematical Logic and Foundations, pages 585–625. College Publications, London, 2011. [4] Libor Bˇehounek, Petr Cintula, and Petr H´ajek. Introduction to mathematical fuzzy logic. In Petr Cintula, Petr H´ajek, and Carles Noguera, editors, Handbook of Mathematical Fuzzy Logic - Volume 1, volume 37 of Studies in Logic, Mathematical Logic and Foundations, pages 1–101. College Publications, London, 2011. [5] Garrett Birkhoﬀ and Richard S. Pierce. Lattice-ordered rings. Anais da Academia Brasileira de Ciˆencias, 28:41–69, 1956. [6] F´elix Bou, Francesc Esteva, Llu´ıs Godo, and Ricardo O. Rodr´ıguez. On the minimum many-valued modal logic over a ﬁnite residuated lattice. Journal of Logic and Computa- tion, 21(5):739–790, 2011. [7] Valerio Capraro and Marco Scarsini. Existence of equilibria in countable games: An algebraic approach. Games and Economic Behavior, 79:163–180, 2013. [8] Roberto Cignoli, Itala M.L. D’Ottaviano, and Daniele Mundici. Algebraic Foundations of Many-Valued Reasoning, volume 7 of Trends in Logic. Kluwer, Dordrecht, 1999. [9] Petr Cintula and Carles Noguera. A general framework for mathematical fuzzy logic. In Petr Cintula, Petr H´ajek, and Carles Noguera, editors, Handbook of Mathematical Fuzzy Logic - Volume 1, volume 37 of Studies in Logic, Mathematical Logic and Foundations, pages 103–207. College Publications, London, 2011. [10] Melvin Dresher. Games of Strategy: Theory and Applications. Prentice Hall, 1961. [11] Paul E. Dunne and Wiebe van der Hoek. Representation and complexity in Boolean games. In Jos´e Ju´lio Alferes and Jo˜ao Alexandre Leite, editors, Logics in Artiﬁcial Intelligence, Proceedings of 9th European Conference, JELIA, volume 3229 of Lecture Notes in Computer Science, pages 347–359, Springer, 2004. [12] Francesc Esteva, Llu´ıs Godo, and Enrico Marchioni. Fuzzy logics with enriched language. In Petr Cintula, Petr H´ajek, and Carles Noguera, editors, Handbook of Mathematical Fuzzy Logic - Volume 2, volume 38 of Studies in Logic, Mathematical Logic and Foundations, pages 627–711. College Publications, London, 2011. [13] Petr H´ajek. Metamathematics of Fuzzy Logic, volume 4 of Trends in Logic. Kluwer, Dordrecht, 1998.

text: REFERENCES

text: by

text: logical

text: formulas.

text: In

text: particular

text: they

text: also

text: considered

text: operations

text: on

text: games,

text: a

text: form

text: of

text: relativized

text: validity

text: and

text: satisﬁability

text: motivated

text: by

text: their

text: games,

text: and

text: a

text: calculus

text: for

text: deriving

text: winning

text: strategies.

text: It

text: would

text: be

text: certainly

text: interesting

text: to

text: see

text: to

text: what

text: extent

text: these

text: and

text: related

text: topics

text: can

text: be

text: developed

text: also

text: in

text: the

text: considerably

text: more

text: general

text: many-valued

text: setting

text: presented

text: here.

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text: Stefano

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text: Fuzzy

text: Logic

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text: In

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text: Jo˜ao

text: Alexandre

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text: editors,

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text: Intelligence,

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text: Conference,

text: JELIA,

text: volume

text: of

text: Lecture

text: Notes

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text: Science,

text: pages

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text: Francesc

text: Esteva,

text: Llu´ıs

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text: and

text: Enrico

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text: Fuzzy

text: logics

text: with

text: enriched

text: language.

text: In

text: Petr

text: Cintula,

text: Petr

text: H´ajek,

text: and

text: Carles

text: Noguera,

text: editors,

text: Handbook

text: of

text: Mathematical

text: Fuzzy

text: Logic

text: Volume

text: volume

text: of

text: Studies

text: in

text: Logic,

text: Mathematical

text: Logic

text: and

text: Foundations,

text: pages

text: College

text: Publications,

text: London,

text: Petr

text: H´ajek.

text: Metamathematics

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text: Logic,

text: volume

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text: Trends

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text: Logic.

text: Kluwer,

text: Dordrecht,


text: 2 PRELIMINARY NOTIONS 5 Algebra Functions Domains Pieces [0, 1]Ł continuous linear linear functions with integer coeﬃcients (cid:52) [0, 1] all linear linear functions with integer coeﬃcients Ł [0, 1]QŁ continuous linear linear functions with integer coeﬃcients and a rational shift (cid:52) [0, 1]QŁ all linear linear functions with integer coeﬃcients and a rational shift (cid:52) [0, 1] all all polynomials with integer coeﬃcients PŁ (cid:52) [0, 1]QPŁ all all polynomials with rational coeﬃcients [0, 1] all all fractions of polynomials with integer coeﬃcients ŁΠ12 { } ⊆ { } [0, 1]ŁΠ all all functions f expressible in [0, 1]ŁΠ12 such that f[ 0, 1 n] 0, 1 Table 1: Characterization of functions expressible in prominent standard algebras. Let us recall several standard notions of the algebraic semantics of many-valued logics. (For a detailed modern exposition see, e.g., [9].) Deﬁnition 2.3. Let A be a standard algebra. An A-evaluation is a mapping e assigning an element of A to each propositional variable. Every A-evaluation can be uniquely extended to a ◦ ◦ mapping from FmL into A, by setting e( (ϕ , . . . , ϕ )) = A(e(ϕ ), . . . , e(ϕ )) for each n-ary ◦ ∈ L A 1 n 1 n connective and formulas ϕ , . . . , ϕ . L A 1 n An -formula ϕ is satisﬁed by an A-evaluation e if e(ϕ) = 1. A formula ϕ is A-satisﬁable if A it is satisﬁed by some A-evaluation. | The logic of A is identiﬁed with the consequence relation = , deﬁned as follows: A | ⊆ { } Γ = ϕ if and only if for each A-evaluation e: if e[Γ] 1 , then e(ϕ) = 1. A A trivial, but important observation is that the value of a formula ϕ in an A-evaluation depends only on the variables occurring in ϕ. Let v be a sequence v , . . . , v of pairwise diﬀerent variables; 1 n we shall write ϕ(v , . . . , v ), or just ϕ(v), to denote that all variables occurring in ϕ are among 1 n those in v. Given a formula ϕ(v , . . . , v ) and a sequence of formulas ψ , . . . , ψ , we shall write 1 n 1 n ϕ(ψ , . . . , ψ ) to denote the formula where each variable v is replaced by the formula ψ . 1 n i i → Deﬁnition 2.4. Given a formula ϕ(v , . . . , v ), we deﬁne the mapping ϕA : An A by setting: 1 n ϕA(a , . . . , a ) = e(ϕ), 1 n where e is any A-evaluation such that e(v ) = a . i i For any given standard algebra A, it is an interesting question how to describe the class of all functions ϕA. The classes of functions expressible in prominent standard algebras are described in Table 1. These delimitations are of a “piecewise” character, i.e., based on a decomposition of the corresponding hypercube [0, 1]n into domains; in particular, each row of the table speciﬁes: (i) whether the functions are all those which satisfy the other constraints listed on the row, or just the continuous ones; (ii) whether the possible domains are all Q-semialgebraic sets,1 or just the linear ones; and (iii) how the functions restricted to these domains are characterized.2 For further details see [20, 22] or [1, Sect. 4.1], where also the (more complicated) result for G¨odel algebras is presented. Let us furthermore remark that the case of [0, 1]PŁ is a long-standing open problem related to the so-called Pierce–Birkhoﬀ conjecture [5, 16]. ⊆ {(cid:104) (cid:105) ∈ 1A |set S [0,1]n is (li}near) Q-semialgebraic if it is a Boolean combination of sets of the form x1,...,xn [0,1]n P(x1,...,xn) > 0 for (linear) polynomials P with integer coeﬃcients. 2In the table, a shift means an absolute coeﬃcient of the polynomial. ##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: PRELIMINARY

text: NOTIONS

text: Algebra

label: table

text: Functions

text: Domains

text: Pieces

text: continuous

text: linear

text: linear

text: functions

text: with

text: integer

text: coeﬃcients

text: all

text: linear

text: linear

text: functions

text: with

text: integer

text: coeﬃcients

text: Ł

text: continuous

text: linear

text: linear

text: functions

text: with

text: integer

text: coeﬃcients

text: and

text: a

text: rational

text: shift

text: all

text: linear

text: linear

text: functions

text: with

text: integer

text: coeﬃcients

text: and

text: a

text: rational

text: shift

text: all

text: all

text: polynomials

text: with

text: integer

text: coeﬃcients

text: PŁ

text: all

text: all

text: polynomials

text: with

text: rational

text: coeﬃcients

text: all

text: all

text: fractions

text: of

text: polynomials

text: with

text: integer

text: coeﬃcients

text: ŁΠ12

text: 1]ŁΠ

text: all

text: all

text: functions

text: f

text: expressible

text: in

text: such

text: that

text: f[

text: n]

text: Table

text: Characterization

text: of

text: functions

text: expressible

text: in

text: prominent

text: standard

text: algebras.

text: Let

text: us

text: recall

text: several

text: standard

text: notions

text: of

text: the

text: algebraic

text: semantics

text: of

text: many-valued

text: logics.

text: (For

text: a

text: detailed

text: modern

text: exposition

text: see,

text: e.g.,

text: Deﬁnition

text: Let

text: A

text: be

text: a

text: standard

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text: An

text: A-evaluation

text: is

text: a

text: mapping

text: e

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text: an

text: element

text: of

text: A

text: to

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text: Every

text: A-evaluation

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text: a

text: mapping

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text: FmL

text: into

text: A,

text: by

text: setting

text: e(

text: (ϕ

text: ϕ

text: A(e(ϕ

text: e(ϕ

text: for

text: each

text: n-ary

text: L

text: A

text: n

text: n

text: connective

text: and

text: formulas

text: ϕ

text: ϕ

text: L

text: A

text: n

text: An

text: -formula

text: ϕ

text: is

text: satisﬁed

text: by

text: an

text: A-evaluation

text: e

text: if

text: e(ϕ)

text: A

text: formula

text: ϕ

text: is

text: A-satisﬁable

text: if

text: A

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text: some

text: A-evaluation.

text: The

text: logic

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text: A

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text: consequence

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text: A

text: Γ

text: ϕ

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text: only

text: if

text: for

text: each

text: A-evaluation

text: e:

text: if

text: e[Γ]

text: then

text: e(ϕ)

text: A

text: A

text: trivial,

text: but

text: important

text: observation

text: is

text: that

text: the

text: value

text: of

text: a

text: formula

text: ϕ

text: in

text: an

text: A-evaluation

text: depends

text: only

text: on

text: the

text: variables

text: occurring

text: in

text: ϕ.

text: Let

text: v

text: be

text: a

text: sequence

text: v

text: v

text: of

text: pairwise

text: diﬀerent

text: variables;

text: n

text: we

text: shall

text: write

text: ϕ(v

text: v

text: or

text: just

text: ϕ(v),

text: to

text: denote

text: that

text: all

text: variables

text: occurring

text: in

text: ϕ

text: are

text: among

text: n

text: those

text: in

text: v.

text: Given

text: a

text: formula

text: ϕ(v

text: v

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text: a

text: sequence

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text: formulas

text: ψ

text: ψ

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text: n

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text: ϕ(ψ

text: ψ

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text: denote

text: the

text: formula

text: where

text: each

text: variable

text: v

text: is

text: replaced

text: by

text: the

text: formula

text: ψ

text: n

text: i

text: i

text: Deﬁnition

text: Given

text: a

text: formula

text: ϕ(v

text: v

text: we

text: deﬁne

text: the

text: mapping

text: ϕA

text: An

text: A

text: by

text: setting:

text: n

text: ϕA(a

text: a

text: e(ϕ),

text: n

text: where

text: e

text: is

text: any

text: A-evaluation

text: such

text: that

text: e(v

text: a

text: i

text: i

text: For

text: any

text: given

text: standard

text: algebra

text: A,

text: it

text: is

text: an

text: interesting

text: question

text: how

text: to

text: describe

text: the

text: class

text: of

text: all

text: functions

text: ϕA.

text: The

text: classes

text: of

text: functions

text: expressible

text: in

text: prominent

text: standard

text: algebras

text: are

text: described

text: in

text: Table

text: These

text: delimitations

text: are

text: of

text: a

text: “piecewise”

text: character,

text: i.e.,

text: based

text: on

text: a

text: decomposition

text: of

text: the

text: corresponding

text: hypercube

text: into

text: domains;

text: in

text: particular,

text: each

text: row

text: of

text: the

text: table

text: speciﬁes:

text: (i)

text: whether

text: the

text: functions

text: are

text: all

text: those

text: which

text: satisfy

text: the

text: other

text: constraints

text: listed

text: on

text: the

text: row,

text: or

text: just

text: the

text: continuous

text: ones;

text: (ii)

text: whether

text: the

text: possible

text: domains

text: are

text: all

text: Q-semialgebraic

text: or

text: just

text: the

text: linear

text: ones;

text: and

text: (iii)

text: how

text: the

text: functions

text: restricted

text: to

text: these

text: domains

text: are

text: For

text: further

text: details

text: see

text: or

text: Sect.

text: where

text: also

text: the

text: (more

text: complicated)

text: result

text: for

text: G¨odel

text: algebras

text: is

text: presented.

text: Let

text: us

text: furthermore

text: remark

text: that

text: the

text: case

text: of

text: is

text: a

text: long-standing

text: open

text: problem

text: related

text: to

text: the

text: so-called

text: Pierce–Birkhoﬀ

text: conjecture

text: |set

text: S

text: is

text: (li}near)

text: Q-semialgebraic

text: if

text: it

text: is

text: a

text: Boolean

text: combination

text: of

text: sets

text: of

text: the

text: form

text: for

text: (linear)

text: polynomials

text: P

text: with

text: integer

text: coeﬃcients.

text: the

text: table,

text: a

text: shift

text: means

text: an

text: absolute

text: coeﬃcient

text: of

text: the

text: polynomial.

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text: 2 Cao et al. used CPL parametrization for dark energy equation of state (Linder 2003) and is equivalent to the ﬁrst order expansion in the scale factor a(t) which is the only gravitational de- 0.16 gree of freedom in Freedman - Robertson - Walker cosmol- ogy. These two parametrizations have been extensively used to investigate the properties of dark energy in the literature 0.14 (Cao et al. 2012; Cao & Zhu 2014, 2015). Assuming that the Etherington theorem is valid (which is quite a reasonable as- 0.12 sumption), the best-ﬁt value obtained with a given data set should be η = 1 and η = 0. Our results indicate that, the 0 P non-isothermal double-β model tends to be more compatible 0.1 with the reciprocity relation than the isothermal β model. This kind of result is an interesting example of how general 0.08 principles (like DDR) could be used to assess the validity of assumptions concerning local physical conditions. β model This paper is organized as follows. In Section we present 0.06 Double β model ΛCDM1 two samples of gas mass fraction data from 38 X-ray lumi- ΛCDM2 nous galaxy clusters and their corresponding mass density XCDM 0.04 proﬁles. Statistical method and constraint results on η(z) 0.1 0.3 0.5 0.7 0.9 parameters are shown in Section . Finally, we summarize z our main conclusions and make a discussion in Section . 2 GALAXY CLUSTER SAMPLES Figure 1. The cluster gas mass fraction data derived from a sample of 38 luminous X-ray clusters observed by Chandra X-ray The cluster gas mass fraction is deﬁned as a ratio of the Observatory (LaRoque et al. 2006). Black circles and red squares X-ray emitting gas mass to the total mass of a cluster, i.e., with the error bars correspond to the results obtained under the f = M /M . Gas mass M is derived from the X-ray assumption of isothermal β model and non-isothermal double-β gas gas tot gas surface brightness while the total mass M can be obtained model, respectively. The predictions of fgas(z) under the assump- tot tion of DDR are also overplotted, for the three reference cosmolo- by assuming that the gas is in hydrostatic equilibrium with gies (see Table 1) considered in this paper. the cluster NFW potential. In order to perform appropriate calculations, one needs the gas mass density proﬁle. There- fore, in our analysis aimed at constraining free parameters (η0, ηP) in general expression for η(z), we will use the gas non-parametric Wilcoxon signed rank test reveals that dif- mass fractions obtained with diﬀerent gas mass density pro- ference between these two datasets is signiﬁcant at the level ﬁles. of p = 0.0001. More speciﬁcally, the assumption of isother- In order to calculate η from the data (according to mality can potentially aﬀect the gas mass fraction measure- obs − Eq.(6) shown below) we use two sets of gas mass fraction ments by about ( 5%) through its eﬀects on both gas mass f both derived from a sample of 38 luminous X-ray clus- and total mass. Therefore, an appropriate systematic uncer- gas ters with temperatures T > 5 keV, observed by Chandra tainty will be assigned to the gas mass fractions derived from gas X-ray Observatory (LaRoque et al. 2006) located at red- the isothermal β model. In order to show the major sources shifts from z = 0.14 to z = 0.89. In fact, LaRoque et al. of uncertainty in the present analysis, we display the ratio (2006) presented results of their analysis using both X-ray between systematic and statistical uncertainties for these only data (for isothermal β-model) and a combination of data in Fig. 2. Chandra X-ray data and BIMA/OVRO interferometric ra- In the frequently used isothermal β model, the 3- dio SZE data for the double - β model. However, according to dimensional electron number density n can be written as e Holanda, Goncalves & Alcaniz (2012), the gas mass fraction (Cavaliere & Fusco-Femiano 1976, 1978; Grego et al. 2001; measurement via SZE depends only on the angular diame- Reese et al. 2002; Ettori et (cid:18)al. 2004)(cid:19) ter distance, so it is insensitive to the validity of the DDR. Therefore, only the high resolution Chandra X-ray data are − considered in our analysis. The above mentioned two sets of r2 3β/2 n (r) = n 1 + , (2) fgas were obtained within the most commonly used isother- e e0 rc2 mal β model and under assumption of the non-isothermal double - β model, respectively. where n is the central electron number density, r is the e0 Fig. 1 displays these data and their observational statis- radius from the center of the cluster, r is the core radius of c tical uncertainties. We will also consider possible systematic the intracluster medium (ICM), and β is a power law index. uncertainties inﬂuencing the derived gas mass fraction fol- Although it has been widely used as a useful assumption in lowing the discussion in (Bonamente et al. 2006) concerning galaxy cluster studies, the isothermal β model still encoun- their method of calculating angular diameter distances to ters some problems with describing the central emission ex- clusters. The eﬀect of these systematical uncertainties on cess seen in some clusters (Mohr et al. 1999; LaRoque et al. − the gas mass fractions is summarized in Table 3 of LaRoque 2006) and the X-ray surface brightness at r > (1 1.5) r , 2500 et al. (2006). Note that the systematic diﬀerence between where r is the radius at which the mean enclosed mass 2500 f derived from the isothermal β model and the double - density is equal to 2500ρ . Such features, non-compatible gas crit β model still exists. From a purely statistical point of view, with the isothermal β model have been observed in archival (cid:13) c RAS, MNRAS 000, 1–7 f sag ##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## 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text: Cao

text: et

text: al.

text: used

text: CPL

text: parametrization

text: for

text: dark

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text: and

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font: default

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text: ogy.

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text: two

text: parametrizations

text: have

text: been

text: extensively

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text: the

text: properties

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text: dark

text: energy

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text: the

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text: theorem

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text: the

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text: a

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text: η

text: and

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text: Our

text: results

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text: that,

text: the

text: P

text: non-isothermal

text: double-β

text: model

text: tends

text: to

text: be

text: more

text: compatible

text: with

text: the

text: reciprocity

text: relation

text: than

text: the

text: isothermal

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text: This

text: kind

text: of

text: result

text: is

text: an

text: interesting

text: example

text: of

text: how

text: general

text: principles

text: (like

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text: could

text: be

text: used

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text: the

text: validity

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text: assumptions

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text: local

text: physical

text: conditions.

text: β

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text: Double

text: β

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text: two

text: samples

text: of

text: gas

text: mass

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text: from

text: X-ray

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text: nous

text: galaxy

text: clusters

text: and

text: their

text: corresponding

text: mass

text: density

text: XCDM

text: proﬁles.

text: Statistical

text: method

text: and

text: constraint

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text: η(z)

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text: we

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text: Section

text: GALAXY

text: CLUSTER

text: SAMPLES

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text: The

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text: gas

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text: Observatory

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text: Black

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text: derived

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text: the

text: X-ray

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text: isothermal

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text: the

text: total

text: mass

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text: respectively.

text: The

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text: are

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text: for

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text: analysis

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text: constraining

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text: parameters

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text: in

text: general

text: expression

text: for

text: η(z),

text: we

text: will

text: use

text: the

text: gas

text: non-parametric

text: Wilcoxon

text: signed

text: rank

text: test

text: reveals

text: that

text: dif-

text: mass

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text: with

text: diﬀerent

text: gas

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text: data

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text: obs

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text: we

text: use

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text: derived

text: from

text: a

text: sample

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text: luminous

text: X-ray

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text: and

text: total

text: mass.

text: Therefore,

text: an

text: appropriate

text: systematic

text: uncer-

text: gas

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text: with

text: temperatures

text: T

text: keV,

text: observed

text: by

text: Chandra

text: tainty

text: will

text: be

text: assigned

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text: the

text: gas

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text: gas

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text: et

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text: located

text: at

text: red-

text: the

text: isothermal

text: β

text: model.

text: In

text: order

text: to

text: show

text: the

text: major

text: sources

text: shifts

text: from

text: z

text: to

text: z

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text: fact,

text: LaRoque

text: et

text: al.

text: of

text: uncertainty

text: in

text: the

text: present

text: analysis,

text: we

text: display

text: the

text: ratio

text: presented

text: results

text: of

text: their

text: analysis

text: using

text: both

text: X-ray

text: between

text: systematic

text: and

text: statistical

text: uncertainties

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text: these

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text: (for

text: isothermal

text: β-model)

text: and

text: a

text: combination

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text: data

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text: Fig.

text: Chandra

text: X-ray

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text: BIMA/OVRO

text: interferometric

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text: In

text: the

text: frequently

text: used

text: isothermal

text: β

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text: the

text: dio

text: SZE

text: data

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text: the

text: double

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text: However,

text: according

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text: the

text: gas

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text: measurement

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text: so

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text: the

text: validity

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text: DDR.

text: Therefore,

text: only

text: the

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text: X-ray

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text: are

text: considered

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text: our

text: analysis.

text: The

text: above

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text: n

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text: core

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text: We

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text: intracluster

text: medium

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text: and

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text: uncertainties

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text: derived

text: gas

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text: Although

text: it

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text: been

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text: discussion

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text: concerning

text: galaxy

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text: the

text: isothermal

text: β

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text: encoun-

text: their

text: method

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text: clusters.

text: The

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text: the

text: gas

text: mass

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text: summarized

text: in

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text: the

text: X-ray

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text: r

text: r

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text: Note

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text: systematic

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text: where

text: r

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text: the

text: radius

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text: which

text: the

text: mean

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text: mass

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text: the

text: isothermal

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text: the

text: double

text: density

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text: equal

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text: Such

text: features,

text: non-compatible

text: gas

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text: β

text: model

text: still

text: exists.

text: From

text: a

text: purely

text: statistical

text: point

text: of

text: view,

text: with

text: the

text: isothermal

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text: model

text: have

text: been

text: observed

text: in

text: archival

text: c

text: RAS,

text: MNRAS

text: f

text: sag

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text: Objective functionals that are motivated by failure probabilities of mechanical compo- nents that are subject to cyclic mechanical loading have been introduced in [5, 15, 21] to the ﬁeld of shape optimization [9, 16, 24]. Here, failure times are modelled by spatio-temporal Poisson Point Processes (PPP) and their ﬁrst occurrence times. In this paper, we take enhanced material damage into account that occurs at elevated temperature. This is due to thermal stresses and also due to reduced durability of materials at higher temperatures. The assumed design objective is to choose the shape of a component in a set of admissible shapes such that the failure probability after a given number of load cycles is minimal. The reliability assessment of cooled components, e.g. in gas turbines or vessels, leads to a set of multi-physical partial diﬀerential equations that is known as the the thermo- mechanical equation [17]. In this paper, we investigate a shape optimization problems with the thermo-mechanical system of PDEs as state equation. We consider fairly singular set of objective functionals that are motivated by the probability of failure under cyclic themo-mechanical loading, as it is the case for low cycle fatigue (LCF). This set of objective functionals introduce temperature dependence to the objective functionals in [15, 22, 23]. We show that, under suitable regularity assumptions on the admissible shapes and the boundary conditions, there exist of shapes with minimal failure probability. The boundary conditions to the thermal equation in this paper are of Robin type. This corresponds to convective heat transfer at the boundary of the component, which is most frequently used in engineering applications. For the coupling of thermal stress to the mechanical equation, we follow a partially coupled approach, see e.g. [17]. When writing the coupled system in strong form, the gradient of the temperature ﬁeld becomes a volume force density for the elasticity PDE and the temperature diﬀerence to a baseline temperature, where the component is in its original stress free state, becomes a surface load density in the elasticity equation. This re-deﬁnes the right hand side of the elasticity equation. In this situation, uniform regularity estimates for the temperature ﬁelds serve two purposes: On the one hand, the temperature ﬁeld itself is part of the solution of the state equation in the sense of shape optimization with PDE constraints. We thus need some compactness for this component of the state space. Secondly, regularity assumptions on volume forces and surface loads are crucial input for the regularity estimates for the elasticity equation. Here uniform bounds on C1-Hölder norms for the volume force densities and C2-Hölder for the surface load densities are required for the uniform Schauder estimates in [1, 2, 14, 15] . In this paper we prove that such estimates in fact hold true. Having established suitable uniform Schauder estimates on both components of the state space, we can now proceed to prove graph compactness in the sense of shape optimization using the Arzela-Ascoli theorem for Hölder spaces [3, Section 8]. This implies existence of optimal shapes in the class of admissible shapes for all objective functionals that are continuous with respect to the state space topology, which is a direct sum topology for the temperature and displacement ﬁeld as speciﬁed above. Proofs for the existence of optimal shapes are not new in shape optimization, see e.g. 2

text: Objective

text: functionals

text: that

text: are

text: motivated

text: by

text: failure

text: probabilities

text: of

text: mechanical

text: compo-

text: nents

text: that

text: are

text: subject

text: to

text: cyclic

text: mechanical

text: loading

text: have

text: been

text: introduced

text: in

text: to

text: the

text: ﬁeld

text: of

text: shape

text: optimization

text: Here,

text: failure

text: times

text: are

text: modelled

text: by

text: spatio-temporal

text: Poisson

text: Point

text: Processes

text: (PPP)

text: and

text: their

text: ﬁrst

text: occurrence

text: times.

text: In

text: this

text: paper,

text: we

text: take

text: enhanced

text: material

text: damage

text: into

text: account

text: that

text: occurs

text: at

text: elevated

text: temperature.

text: This

text: is

text: due

text: to

text: thermal

text: stresses

text: and

text: also

text: due

text: to

text: reduced

text: durability

text: of

text: materials

text: at

text: higher

text: temperatures.

text: The

text: assumed

text: design

text: objective

text: is

text: to

text: choose

text: the

text: shape

text: of

text: a

text: component

text: in

text: a

text: set

text: of

text: admissible

text: shapes

text: such

text: that

text: the

text: failure

text: probability

text: after

text: a

text: given

text: number

text: of

text: load

text: cycles

text: is

text: minimal.

text: The

text: reliability

text: assessment

text: of

text: cooled

text: components,

text: e.g.

text: in

text: gas

text: turbines

text: or

text: vessels,

text: leads

text: to

text: a

text: set

text: of

text: multi-physical

text: partial

text: diﬀerential

text: equations

text: that

text: is

text: known

text: as

text: the

text: the

text: thermo-

text: mechanical

text: equation

text: In

text: this

text: paper,

text: we

text: investigate

text: a

text: shape

text: optimization

text: problems

text: with

text: the

text: thermo-mechanical

text: system

text: of

text: PDEs

text: as

text: state

text: equation.

text: We

text: consider

text: fairly

text: singular

text: set

text: of

text: objective

text: functionals

text: that

text: are

text: motivated

text: by

text: the

text: probability

text: of

text: failure

text: under

text: cyclic

text: themo-mechanical

text: loading,

text: as

text: it

text: is

text: the

text: case

text: for

text: low

text: cycle

text: fatigue

text: (LCF).

text: This

text: set

text: of

text: objective

text: functionals

text: introduce

text: temperature

text: dependence

text: to

text: the

text: objective

text: functionals

text: in

text: We

text: show

text: that,

text: under

text: suitable

text: regularity

text: assumptions

text: on

text: the

text: admissible

text: shapes

text: and

text: the

text: boundary

text: conditions,

text: there

text: exist

text: of

text: shapes

text: with

text: minimal

text: failure

text: probability.

text: The

text: boundary

text: conditions

text: to

text: the

text: thermal

text: equation

text: in

text: this

text: paper

text: are

text: of

text: Robin

text: type.

text: This

text: corresponds

text: to

text: convective

text: heat

text: transfer

text: at

text: the

text: boundary

text: of

text: the

text: component,

text: which

text: is

text: most

text: frequently

text: used

text: in

text: engineering

text: applications.

text: For

text: the

text: coupling

text: of

text: thermal

text: stress

text: to

text: the

text: mechanical

text: equation,

text: we

text: follow

text: a

text: partially

text: coupled

text: approach,

text: see

text: e.g.

text: When

text: writing

text: the

text: coupled

text: system

text: in

text: strong

text: form,

text: the

text: gradient

text: of

text: the

text: temperature

text: ﬁeld

text: becomes

text: a

text: volume

text: force

text: density

text: for

text: the

text: elasticity

text: PDE

text: and

text: the

text: temperature

text: diﬀerence

text: to

text: a

text: baseline

text: temperature,

text: where

text: the

text: component

text: is

text: in

text: its

text: original

text: stress

text: free

text: state,

text: becomes

text: a

text: surface

text: load

text: density

text: in

text: the

text: elasticity

text: equation.

text: This

text: re-deﬁnes

text: the

text: right

text: hand

text: side

text: of

text: the

text: elasticity

text: equation.

text: In

text: this

text: situation,

text: uniform

text: regularity

text: estimates

text: for

text: the

text: temperature

text: ﬁelds

text: serve

text: two

text: purposes:

text: On

text: the

text: one

text: hand,

text: the

text: temperature

text: ﬁeld

text: itself

text: is

text: part

text: of

text: the

text: solution

text: of

text: the

text: state

text: equation

text: in

text: the

text: sense

text: of

text: shape

text: optimization

text: with

text: PDE

text: constraints.

text: We

text: thus

text: need

text: some

text: compactness

text: for

text: this

text: component

text: of

text: the

text: state

text: space.

text: Secondly,

text: regularity

text: assumptions

text: on

text: volume

text: forces

text: and

text: surface

text: loads

text: are

text: crucial

text: input

text: for

text: the

text: regularity

text: estimates

text: for

text: the

text: elasticity

text: equation.

text: Here

text: uniform

text: bounds

text: on

text: norms

text: for

text: the

text: volume

text: force

text: densities

text: and

text: for

text: the

text: surface

text: load

text: densities

text: are

text: required

text: for

text: the

text: uniform

text: Schauder

text: estimates

text: in

text: In

text: this

text: paper

text: we

text: prove

text: that

text: such

text: estimates

text: in

text: fact

text: hold

text: true.

text: Having

text: established

text: suitable

text: uniform

text: Schauder

text: estimates

text: on

text: both

text: components

text: of

text: the

text: state

text: space,

text: we

text: can

text: now

text: proceed

text: to

text: prove

text: graph

text: compactness

text: in

text: the

text: sense

text: of

text: shape

text: optimization

text: using

text: the

text: Arzela-Ascoli

text: theorem

text: for

text: Hölder

text: spaces

text: Section

text: This

text: implies

text: existence

text: of

text: optimal

text: shapes

text: in

text: the

text: class

text: of

text: admissible

text: shapes

text: for

text: all

text: objective

text: functionals

text: that

text: are

text: continuous

text: with

text: respect

text: to

text: the

text: state

text: space

text: topology,

text: which

text: is

text: a

text: direct

text: sum

text: topology

text: for

text: the

text: temperature

text: and

text: displacement

text: ﬁeld

text: as

text: speciﬁed

text: above.

text: Proofs

text: for

text: the

text: existence

text: of

text: optimal

text: shapes

text: are

text: not

text: new

text: in

text: shape

text: optimization,

text: see

text: e.g.