Pacific Journal ofMathematics

A CONDITIONAL ENTROPY FOR THE SPACE OFPSEUDO-MENGER MAPS

ALAN SALESKI

Vol. 59, No. 2 June 1975

PACIFIC JOURNAL OF MATHEMATICSVol. 59, No. 2, 1975

A CONDITIONAL ENTROPY FOR THE SPACEOF PSEUDO-MENGER MAPS

ALAN SALESKI

Let X be a set and T: X ~» X be a bi jection. Considerthe space ^ of pseudo-Menger maps on X which induce acompact topology on X for which T is a homeomorphism.The lattice properties of Λί are investigated and a bivariatenonnegative function of -^ is defined which possesses certainproperties analogous to those of the usual conditional entropyfunction defined on the space of measurable partitions of aprobability space.

l Introduction* A pseudo-Menger map on a set X is, roughlyspeaking, an assignment of a distribution function to every pair ofpoints in X in a manner consistent with the axioms of a pseudo-metric space. Each such map induces a topology on X as definedby Schweizer and Sklar [9]. Let T be a bi jection of X onto itself.Let ^£ denote the space of all pseudo-Menger maps which induce acompact topology on X for which T is a homeomorphism. Jf θ e Λflet h(T, θ) denote the topological entropy of T with respect to thetopology on X induced by θ. In [7] it was shown that h(T, •) is left-continuous in the sense that if θ, θne^f, θn^>θ and θn(p, q)—+θ(p, q)in distribution for all p, q e X then h(T, θn)~+h(T, θ). In an effort toextend this result one is led to ask the question whether ^£ is closedunder the operations of Max and Min and, if so, what can one sayabout the entropy of T acting on the topology engendered by themaps obtained as a result of such operations. We now proceed toprovide precise definitions and notation.

2* Preliminaries* Let / denote the closed unit interval, R thereal numbers and Z+ the positive integers. Let <2ί be the set of allleft-continuous monotone increasing functions F: R~+1 satisfyingF(0) = 0 and sup,, F(x) = 1. Endowed with the Levy metric &r is acomplete metric space. If Fn, Fe& then F%^F will denote con-vergence with respect to the Levy topology. It is well-known thatFn -i F if and only if Fn(x) —> F(x) for each x e R at which F is con-tinuous. If F, Ge 3ί then F ^ G will mean F(x) :> G(x) for all x e R.Let He & be the function defined by: H(t) = 0 for t ^ 0 and H(t) = 1for t > 0.

Let X be a fixed set. Let ^(X) denote the collection of allfunctions θ: X x X—> £gr. For convenience we shall often write θpq

in place of θ(p, q). A statistical pseudo-metric space is an orderedpair (X, θ) where θ e ̂ satisfies:

525

526 ALAN SALESKI

(SM 1) θpq = θqp for all p, q e X

(SM 2) θpq(a + b) = 1 whenever θpr{a) = θrg(b) = 1 for some r e X

(SM 3) θpp = H for all p e l .

If, in addition, θ satisfies:

(SM 4) θpq = if only if p = q

then (X, θ) is a statistical metric space. Let ^ ( X ) denote the collec-tion of all θ for which (X, θ) is a statistical pseudo-metric space.

A triangular norm is a function Δ: I x I—>/which is associative,commutative, non-decreasing in each variable and satisfies Δ(y, 1) = yfor each y e I. A continuous Menger space [pseudo-Menger space] isa statistical metric space [statistical pseudo-metric space] (X, θ) forwhich there exists a continuous triangular norm Δ satisfying:

(SM 5) θpr(a + b) ^ Δ{βpq(a), θqr{b)) for all p, g, r 6 X and α, 6 ε i2 .

Let ^€(X) denote the set of all θ for which (X, θ) is a continuouspseudo-Menger space. If θn, θ e Λ€(X) and θn(p, q) -^ <̂ (p, g) for allp, qeX then we will write θn -̂> 61. Similarly, if θ, Γe ^f(X) and^ g ^ Γpq for all p,qeX then we write 0 ^ Γ. Let Ξ e ^ f (X) bedefined by Ξpq = Jϊ for all p, qe X.

If θ e ^(X) let X be endowed with the topology, denoted τ(θ),generated by all sets of the form N(p, e, λ, θ) = {g G X: 0pg(e) > 1 — λ}where p e l , e > 0, λ > 0. Let T:X—>X be a bijection. Let^^(X, Γ) = {θ e ^t(X): T is a self-homeomorphism of (X, τ(0)) andτ(0) is compact}.

If #e^T(X, T) we will let h(T, θ) denote the topological entropyof T with respect to the τ{θ) topology. We will follow the notationand definitions of topological entropy developed in [1]. The only excep-tion is the understanding that if T is a self-homeomorphism of (X, τ)where τ is not a compact topology and ^ c r is a cover of X whichpossesses a finite subcover then h(T, %S) = l i m ^ (l/k)£Γ(VJzί Γ ^ ) .We let ^ ( X , Γ) - {0 G ̂ f (X, Γ): h(T, θ)< oo}.

Let |0 be a pseudo-metric on a set y and let % be an opencover of (Yf p). Then |θ-diam <%S will mean the sup {p-diam A: Ae %f}.Let τ^) denote the topology on Y determined by p. In addition, ife > 0 and ae Y then let B(Y, a, p, ε) = {qe Y: ρ(q, a) < e}. If D isanother pseudo-metric on Y then p ^ D will mean p(ay b) ̂ Z)(α, 6)for all a,beY. Finally, if X is a pseudo-metric space then X* willdenote the (unique up to uniform isomorphism) completion for whichX* ~ X is Hausdorff. Such an X* will be called the pseudo-metricspace completion of X.

A CONDITIONAL ENTROPY FOR THE SPACE OF PSEUDO-MENGER MAPS 527

3 . L a t t i c e o p e r a t i o n s . I f θ, Ψe^(X,T) d e f i n e ΘVΨ =

Min (0, Ψ) and θ A Ψ = Max (θ, Ψ). It is easy to construct examplesin which θ Λ Ψ fails even to belong to S^(X). However, we willshow that θ V Ψ admits a canonical extension to a map belonging to^T(X*, Tη w h e r e x* i s t h e completion of (X, τ(θ V ¥)) and T* isthe extension of Γ to P .

P R O P O S I T I O N 1 . If θ,Ψe ^t(X) then βvΨe ^//{X).

Proof. Let Δt and Δ2 be continuous triangular norms for θ andΨ respectively which satisfy:

0pq{a + b) ̂ ^ ( M α )

and

y pq(a + 6 ) ̂ Δ 2 ( Ψ p r ( a ) , Ψrq(b)) f o r a l l α , 6 G J B a n d a l l p,q,reX.

It is easy to check that Δz = Min (J l f J2) is a continuous triangularnorm. Using the monotonicity of Δ1 and Δ2 we verify the triangleinequality for g V Ψ with respect to Δ%\

(θ V n , r ( α + 6) = Min (^r(α + 6), F,r(α + 6))^ Min(Λ(^(α), θqr(b)), ΔlΨvq(a\ Ψqr{b)))^ Min (Δlf J2)(Min (ίp f f(α), y M (α)) , Min (θqr(b), Ψqr(b)))

= J8((ff V f) f,(α) f (» V f

LEMMA 1. Assume θ,We^έf(X) determine compact topologiesτ(θ) and τ(Ψ) respectively. Let dλ and d2 be pseudo-metrics on Xwhich generate the topologies τ(θ) and τ(Ψ) respectively. Then thepseudo-metric D = dx + d2 determines the topology τ(θ V Ψ).

Proof. As a consequence of Lemma 4 of [7] and the above pro-position we know that τ{θ\/Ψ) => τ{θ) U τ{Ψ). Thus it suffices to showthat τ(ΘVΨ) is generated by {Af]C:Aeτ(θ) and Ceτ(Ψ)}. LetpeX, ε > 0 and λ > 0 be given. Let q e N(p, ε, λ, θ V Ψ). For eachneZ+ choose An = N(q91/n, 1/n, θ) and Cn = i%, 1/w, 1/w, ?Γ). Sup-pose for each n there exists yne Anf] Cn such that ynί iV(p, ε,X,ΘVΨ).Then θqyjl/n) > 1 - (1/n) and ΨgyJl/n) > 1 - (1M) from which(θVΨ)qyn-+ H. Since (θ \J Ψ)pq is left-continuous and (0 V y)M(ε) >1 - λ, there exists a δ > 0 for which (0 V y)pff(e - δ) > 1 - λ. Now(0 V Γ )w#(e) ^ Δ((θ V ? U e - ί), (θ V f )„.(*)) - ( « V Γ U e - δ) > 1 - λfrom which one draws the contradiction that /̂̂ e N(p, ε,X, θ V Ψ)for large n.

PROPOSITION 2. Lei 0, Ψ e ̂ £(X) and suppose τ(θ) and τ{Ψ) areeach compact. Then τ(θ V Ψ) is totally bounded.

528 ALAN SALESKI

Proof. Let dlf d2 be pseudo-metrics on X which determine τ(θ)and τ(Ψ) respectively. Then D — dt + d2 is a pseudo-metric forτ(β V Ψ). Let ε > 0 be given. Let pif qόeX, l^i^N, 1 ̂ j ^ Mf

be chosen such that Uf=i -B(^, P., ̂ , e/2) = Uf=i # ( ^ , ίi, d* β/2) = X.Then it is easy to verify that, for each i and j, B(X, piy du e/2) nB(X, qh d2, e/2) c £(X, *„, A e) for any ziS e B(X, pi9 du e/2) Πi?(X, qh d2, e/2) provided this intersection is nonempty.

THEOREM 1. Let θ,We ̂ ( X , T) and let (X*, τ*) denote thepseudo-metric space completion of (X, τ{θ V ¥)). Then:

1. T admits a unique extension to a self-homeomorphism T*of (X*, τ ).

2. θ V Ψ admits a unique extension to a map (θ V W)*"*X* x

3. (0 V Ψ)* e ̂ TίJΓ11, Γ*)4. τ* = r((0 V ?Γ)*)

Proof. Let D* denote the pseudo-metric on (X*, r*) which ex-tends the pseudo-metric D on (X, τ(0 V f)). Since θ\/W:XxX->^ris a uniformly continuous map [8] it can be extended (Cor. 6.2, Ch.14 of [3]) to a continuous map (θ V ?0*: X* x X* - > ^ . The work ofSherwood [11, 12] implies that {θ V 3F)* € .^f (X) and τ((θ V ?Γ)*) = τ*.Since Γ is uniformly continuous on (X, D) there exists an extensionT* which is a self-homeomorphism of (X*, r*). Now since r* is com-pact, (0 V 30* € ̂ T(X*, Γ*).

4* Entropy• We begin by investigating the relation amongh{T,θ\ί W), h{T, θ) and h(T, Ψ). Several lemmas are required.

LEMMA 2. Let (X, p) be a compact pseudo-metric space andT:X—+Xbe a homeomorphism. Let {^n:neZ+} be a sequence ofopen covers of X satisfying p-diam ^ n —> 0 as n —• ©o. Then

Proof. Let {^nj: j e Z+} be a subsequence of the {^J. Usingthe Lebesgue covering lemma one can select a subsequence {mά} ofthe {%} such that <2fm. < %fmj+1 for j ^ 1. Now applying the Corol-lary on page 314 of [1] the desired result is obtained.

LEMMA 3. Suppose D and d are pseudo-metrics on X satisfyingD Ξ> d, τ(d) is compact and τ(D) is totally bounded. Assume T isa self-homeomorphism of (X, d) and of (X, D). Let { Ψl: n e Z+} bea sequence of τ(D)-open covers of X such that D-diam % ^ 0 as% —-• oo and each Ψl possesses a finite subcover of X. Then sup{h(T, ??): <& c τ(d)} ^ ϊίm.^0 h(T, Tn).

A CONDITIONAL ENTROPY FOR THE SPACE OF PSEUDO-MENGER MAPS 529

Proof. Let *Wn be a sequence of r(ώ)-open covers of X suchthat cZ-diam "Wl —> 0. Then for each n > 0 there exists an m :> wsuch that <Wn< Tm. Thus limΛλ(Γ, 3^) ̂ Ίimwλ(!Γ,sup {Λ(Γ, ̂ ) : ̂ c τ(d)} = lim, h(T,

LEMMA 4. Let θ, Ψ e ̂ €{X, T) and let Dbe a pseudo-metric on Xfor which τ(D) = τ(θVΨ). For each ε>0 let ^ε=^{B(Xt p, D, ε): p e X}.Then h{T*, {θ V ¥)*) = supe

Proo/. Let ^ e * = {S(X*, p, J9*, ε): p 6 X}. Then ^ £ = ( A Π X:^e*} and h(T*, (θ V y)*) - sups Λ(Γ*f ^ * ) . It is easy to verify

that NWfT^ft^NO/fTVf.) for all K^O. Consequentlyh(T*, %S*) = A(Γ, ^ . ) and the lemma is proven.

THEOREM 2. Let θ,Ψe ̂ f(X, T). Then:

Max (h(T, θ\ h{T, ¥)) S h(T*, (θ V ¥)*) ^ h(T, θ) + h(T, ¥) .

Proof. Let dx and ώ2 be pseudo-metrics on X which generateτ(θ) and τ(¥) respectively. Then D = dx + d2 is a pseudo-metric forr(0 V ?Γ). Let eH be a sequence of positive numbers such that εn —> 0.Let 5*ς-{J?(X,p, D , e , ) : p 6 l } and TS = {^(X*, p, D*, e j : p e X}.Applying lemma 3, we have λ(Γ*, (θ V f)*) = lim^^ h(T*, 7T) ^ίίΐήn_ Λ(Γ, ^ς) ̂ sup {λ(Γ, ^ ) : ^ c r^)} - λ(Γ, θ).

Let ^ - {S(X, pf dlf 1/Λ): p e X}, ^ . = {£(X, p, d2,1/n): p e X}and ^ = {β(X, p, D, 1/ )̂: p e l } . Since ^ w •< ^4n V K̂ we haveh(T, &n) ^ λ(Γ, ^n V ώl ) ̂ λ(Γ, ^U) + λ(Γ, ώ^). Lemma 4 yieldsA(Γ*, (β V 50*)= lim^oo fc(Γ, ̂ ) ^ lim.^o h(T, ^n)A(Γ, 0) + MΓ, ?").

EXAMPLE 1. Let Y = {0, 1, 2} and X = Γ z . Define the shiftT:X—>X by Γ({?/J) = {i/<+i}. Let dx and d2 be pseudo-metrics on Xgiven by:

where

_ JO if a = 2

(1 if a = 0, 1

and

00 I /v(oι \ r

530 ALAN SALESKI

where

α(α)=f° i f α = °11 if α = l , 2

for all {Ui} and {2J e X.Define 0, y e ̂ (X, T) by:

Me) = #(* ~ din, z))

and

?Γtt,(e) = H(e - d2(u, z)) for all ε > 0 and all κ , ^ e l .

Then it follows that h(T*, (θ V ¥)*) = ϊrc 3, Max (Λ(Γ, 0), Λ(Γ, ?0) = In2, and Λ(Γ, 0) + h(T, Ψ) = in 4.

DEFINITION. If ί, f* e ̂ €F(X, T) let λΓ(* I V) = M?7*, (0 V

PROPOSITION 3. Assume Θ,W, Γe ^£'F(X, T). Then:( a ) 0£hτ(θ\Ψ)£h(T,θ)( b) hτ{θ I θ) = 0( c ) /^(0 V ?Γ I Γ) = M * | y v f ) + ΛΓ(y | Γ) provided ΘV

ΨVΓe^tF(X, T)

(d) M « | S )

Proof. Statement (a) is a corollary of Theorem 2. Statements(b), (c) and (d) follow quickly from the definitions.

PROPOSITION 4. Let θ, Γe^?F(X, T). Suppose ΘVΓe^TF(X, T)and that (X, Γ) is a Menger space. Then hτ(θ \ Γ) - 0.

Proof. This follows from the fact that any two compact metri-zable topologies on X, each of which renders T a homeomorphism,yield the same topological entropy for T.

PROPOSITION 5. Let θ,Ψe ^fF(X, T). Assume that θ v Ψ e, T) and that θpq = H if and only if Ψpq = H. Then hτ(θ\Ψ) = 0.

Proof. For x,yeX, define x ~ y if and only if θxy = if. Thisequivalence relation on X induces a self-homeomorphism T of X/~.It is easy to verify that h(T) = MΓ). One can then apply Proposi-tion 4.

PROPOSITION 6. Let θ, Ψ, Γ e ̂ €F(X, T) and suppose Ψ ̂ Γ.Then hτ(Ψ \ θ) ̂ hτ(Γ \ θ).

A CONDITIONAL ENTROPY FOR THE SPACE OF PSEUDO-MENGER MAPS 531

Proof. Lemma 4 of [7] yields τ{Γ) c τ(Ψ V θ). Since τ(θ) cτ(W V θ) we have τ(Γ V θ) c r ( r V 0). Let (X?, rf) and (X?, r?)denote the completions of (X, r(0 V Γ)) and (X, r(0 V ¥)) respectively,and let T* and T* denote the extensions of T to X? and X? re-spectively. One may assume that XfcX? and that Tt extends T*.Then the relative topology on X* induced |!by τ* contains r*. LetA and D2 denote pseudo-metrics on X* which generate τ* and r? |x*(the topology induced on X? by r?) respectively. By replacing D2

with A + DZ9 if necessary, we may assume that Dι ^ D2. Let £)?denote the extension of D2 to (X2*, r2*). Let ^ς* = {B(X?, p, D?, 1/n):peXΐ} and Tn = {An Xΐ:Ae Tn*}. Applying Lemma 3 togetherwith the fact that fc(2?, Tn) = M27*, 3̂ *)» we have:

rs)

EXAMPLE 2. Two examples are given to show that, in general,if θ, Ψ, Γ e ^ ( X , T) and f ^ Γ then ΛΓ(β | Ψ) may be less than orgreater than hτ(θ \ Γ).

First, note that Ψ S Ξ and hτ{θ \ Ξ) = λ(Γ, ί). Thus, if h(T, θ) > 0,then 0 = hτ(θ \ θ) < hτ{θ \ Ξ).

We now provide an example in which Ψ ^ Γ and hτ(θ | Ψ) >hτ(θ\Γ). Let Y = {1, 2, 3, 4, 5} and X = Yz. Define three pseudo-metrics on X as follows:

where

c(α)= 1 i f α = 5

0 otherwise

where

β(α) =

'2 if α = 1 or 3

1 if a = 2 or 4

0 if α = 5

532 ALAN SALESKI

where

1 if a = 1 or 2

(0 otherwise

for all {Vt}f{zt}eX.Define θ, Ψ, Γ e ^£(X) as follows:

θpq(e) = JΓ(e - p(p, g))

Γ,g(ε) - H(ε - J5(p, g))

Ψpg(ε) = ff(e - <Z(p, g))

for all p, qe X.Let T:X~*X be the shift given by T({yt}) = {yi+ι}. Then

0, Γ, f e ^ ( J , Γ), Γ ^ Γ, hτ(θ I ?r) = λ(Γ, 0 V F) - λ(Γf ?f) = ̂ 5 -^ 3 = In 5/3, and M# I Π = fc(Γ, θ V Γ) - h{T9 Γ) = in 3 - in 2 =to 3/2.

THEOREM 3. Lei #„, y., ^,iFe ̂ 5 (Jf, T). Suppose θn^θ, Ψn

θn^θ,Ψn^Ψ and θyψe ^?{X, T). Then hτ(θn \ Ψn) -> hτ{θ \ Ψ) asn—> oo.

The proof of this theorem is based upon the following lemma.

LEMMA 5. Assume Σe^{X,T), β e ^ ( I ) , τ{Ω) is totallybounded, Ω ̂ Σ and T is a self-homeomorphism of (X, τ(Ω)). Then

, Γ).

Proof. We must show τ{Ω) is compact. Let X* be the com-pletion of (X, τ(Ω)) and Γ* be the extension of T to X*. So 42* e^"(X*, Γ*) Since X* is compact, Lemma 3 of [7] is applicable.Thus, given g e l * , {N(q, ε, λ, £?*): ε, λ > 0} is a local basis for τ(fl*)at g. If pe Xobserve that iSΓ(p, ε, λ, β * ) f l l = JV(p, ε, λ, 13). Hence,for p e l , {iV(2>, ε, λ, β): ε, λ > 0} is a local basis for τ(£?) at p. Nowan argument analogous to that used in lemma 4 of [7] yields τ(Ω) cτ(Σ).

Proof of Theorem 3. Since θnV Ψn^θ V We ^ C ( X , T) and

θ%VΨne^(X) the preceding lemma yields 0W V Ψ%s^{X, T).Now, applying the main theorem of [7], h(T, θn V ΨJ->h(T, θ \J Ψ)and Λ(T, Ψn)~+h(T, Ψ). The desired result now follows.

We conclude with a brief consideration of a special class ofpseudo-Menger maps.

A CONDITIONAL ENTROPY FOR THE SPACE OF PSEUDO-MENGER MAPS 533

DEFINITION. The map Θ e ^f(X, T) is (X, T)-deterministie ifh(T, θ) = 0.

PROPOSITION 7. Let θ, Γ e ^ f (X, T) and suppose that Θ^Γ.If Γ is (X, T)-deterministie then so is θ.

Proof. This follows at once from Lemma 4 of [7] where it isshown that τ(θ) c τ{Γ).

The following two propositions are consequences of Theorem 2.

PROPOSITION 8. Let θ, Γe^f(X, T). Then θ and Γ are (X, 2>deterministic if and only if θ V Γ is (X*, ϊ7*)--deterministic.

PROPOSITION 9. // Γ e ^ # ( X , T) is {X, Tydeterministic andθ e ^TF(X, T) then hT{θ \ Γ) = h(T, θ).

ACKNOWLEDGMENT. The author is grateful to the referee foruncovering several errors in the original version of this paper.

REFERENCES

1. R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy, Trans.Amer. Math. Soc, 114 (1965), 309-319.2. P. Billingsley, Ergodic Theory and Information, Wiley and Sons, New York, 1965.3. J. Dugundji, Topology, Allyn and Bacon, Boston, 1967.4. M. Loeve, Probability Theory, 3rd Ed., Van Nostrand Co., Princeton, N. J., 1963.5. K. Menger, Statistical metrics, Proc. Nat. Acad. Sci., 2 8 (1942), 535-537.6. W. Parry, Entropy and Generators in Ergodic Theory, Benjamin, New York, 1969.7. A. Saleski, Entropy of self-homeomorphisms of statistical pseudo-metric spaces,Pacific J. Math., 51 (1974), 537-542.8. B. Schweizer, On the uniform continuity of the probabilistic distance, Z. Wahr.,5 (1966), 357-360.9. B. Schweizer and A. Sklar, Statistical metric spaces, Pacific J. Math., 10 (1960),313-334.10. B. Schweizer, A. Sklar and E. Thorp, The metrization of statistical metric spaces,Pacific J. Math., 10 (1960), 673-675.11. H. Sherwood, On the completion of probabilistic metric spaces, Z. Wahr., 6 (1966),62-64.

12. , Complete probabilistic metric spaces, Z. Wahr., 20 (1971), 117-128.

Received October 4, 1974 and in revised form April 21, 1975.

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Pacific Journal of MathematicsVol. 59, No. 2 June, 1975

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two-point boundary value problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327Howard Edwin Bell, Infinite subrings of infinite rings and near-rings . . . . . . . . . . . . . . 345Grahame Bennett, Victor Wayne Goodman and Charles Michael Newman, Norms of

random matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359Beverly L. Brechner, Almost periodic homeomorphisms of E2 are periodic . . . . . . . . . 367Beverly L. Brechner and R. Daniel Mauldin, Homeomorphisms of the plane . . . . . . . . 375Jia-Arng Chao, Lusin area functions on local fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383Frank Rimi DeMeyer, The Brauer group of polynomial rings . . . . . . . . . . . . . . . . . . . . . 391M. V. Deshpande, Collectively compact sets and the ergodic theory of

semi-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399Raymond Frank Dickman and Jack Ray Porter, θ -closed subsets of Hausdorff

spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407Charles P. Downey, Classification of singular integrals over a local field . . . . . . . . . . . 417Daniel Reuven Farkas, Miscellany on Bieberbach group algebras . . . . . . . . . . . . . . . . . 427Peter A. Fowler, Infimum and domination principles in vector lattices . . . . . . . . . . . . . 437Barry J. Gardner, Some aspects of T -nilpotence. II: Lifting properties over

T -nilpotent ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445Gary Fred Gruenhage and Phillip Lee Zenor, Metrization of spaces with countable

large basis dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455J. L. Hickman, Reducing series of ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461Hugh M. Hilden, Generators for two groups related to the braid group . . . . . . . . . . . . 475Tom (Roy Thomas Jr.) Jacob, Some matrix transformations on analytic sequence

spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487Elyahu Katz, Free products in the category of kw-groups . . . . . . . . . . . . . . . . . . . . . . . . . 493Tsang Hai Kuo, On conjugate Banach spaces with the Radon-Nikodým property . . . . 497Norman Eugene Liden, K -spaces, their antispaces and related mappings . . . . . . . . . . 505Clinton M. Petty, Radon partitions in real linear spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 515Alan Saleski, A conditional entropy for the space of pseudo-Menger maps . . . . . . . . . 525Michael Singer, Elementary solutions of differential equations . . . . . . . . . . . . . . . . . . . . 535Eugene Spiegel and Allan Trojan, On semi-simple group algebras. I . . . . . . . . . . . . . . . 549Charles Madison Stanton, Bounded analytic functions on a class of open Riemann

surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557Sherman K. Stein, Transversals of Latin squares and their generalizations . . . . . . . . . 567Ivan Ernest Stux, Distribution of squarefree integers in non-linear sequences . . . . . . . 577Lowell G. Sweet, On homogeneous algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585Lowell G. Sweet, On doubly homogeneous algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595Florian Vasilescu, The closed range modulus of operators . . . . . . . . . . . . . . . . . . . . . . . . 599Arthur Anthony Yanushka, A characterization of the symplectic groups PSp(2m, q)

as rank 3 permutation groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611James Juei-Chin Yeh, Inversion of conditional Wiener integrals . . . . . . . . . . . . . . . . . . . 623

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