REVERSE MATHEMATICS AND PROPERTIES OF FINITE CHARACTER
|
|
- Martha Baker
- 5 years ago
- Views:
Transcription
1 REVERSE MATHEMATICS AND PROPERTIES OF FINITE CHARACTER DAMIR D. DZHAFAROV AND CARL MUMMERT Abstract. We study the reverse mathematcs of the prncple statng that, for every property of fnte character, every set has a maxmal subset satsfyng the property. In the context of set theory, ths varant of Tukey s lemma s equvalent to the axom of choce. We study ts behavor n the context of second-order arthmetc, where t apples to sets of natural numbers only, and gve a full characterzaton of ts strength n terms of the quantfer structure of the formula defnng the property. We then study the nteracton between propertes of fnte character and fntary closure operators, and the nteracton between these propertes and a class of nondetermnstc closure operators. 1. Introducton A formula ϕ wth one free set varable s of fnte character, and has the fnte character property, f ϕ( ) holds and, for every set A, ϕ(a) holds f and only f ϕ(f ) holds for every fnte F A. In ths paper, we restrct our attenton to formulas of second-order arthmetc, and consder several varants and restrctons of the prncple FCP (Defnton 2.1) whch asserts that for every formula of fnte character, every subset of N has a maxmal subset satsfyng that formula. Because the empty set satsfes any formula of fnte character, the soundness of ths prncple n second-order arthmetc can be verfed n ZFC by straghtforward applcaton of Zorn s lemma. Detaled defntons of second-order arthmetc and the subsystems studed n ths paper are gven by Smpson [4]. The prncple CE (Defnton 3.3) asserts that gven sets A B N, a formula ϕ of fnte character and a fntary closure operator D, such that A s a D-closed set satsfyng the formula, there s a set X whch s maxmal wth respect to the condtons that A X B, ϕ(x) holds, and X s D- closed. In the thrd secton, we gve a full characterzaton of the strength of fragments of CE n terms of the complexty of the formulas of fnte character to whch they apply. The authors are grateful to Dens Hrschfeldt, Antono Montalbán, and Robert Soare for valuable comments and suggestons, and to an anonymous referee who suggested an mprovement that strengthened Proposton 4.4. The frst author was partally supported by an NSF Graduate Research Fellowshp and an NSF Postdoctoral Fellowshp. 1
2 2 DAMIR D. DZHAFAROV AND CARL MUMMERT We can further generalze CE by replacng the fntary closure operator wth a more general knd of operator whch we name a nondetermnstc closure operator. The correspondng prncple, NCE (Defnton 4.2), s studed n the fnal secton, where a full characterzaton of ts strength s obtaned. We were led to study the reverse mathematcs of FCP by our separate work [1] on the prncple FIP whch states that every countable famly of subsets of N has a maxmal subfamly wth the fnte ntersecton property. All the prncples studed there are consequences of approprate restrctons of FCP. Smlarly, Propostons 3.7 and 4.4 below demonstrate how CE and NCE can be used to prove facts about countable algebrac objects n second-order arthmetc. In lght of these applcatons, we fnd t worthwle to have a complete understandng of the reverse mathematcs strengths of these prncples. Consderng ths paper together wth our work on FIP gves a new example of two prncples, FCP and FIP, whch are each equvalent to the axom of choce when formalzed n set theory, but whch have drastcally dfferent strengths when formalzed n second-order arthmetc. The axom scheme for FCP s equvalent to full comprehenson n second-order arthmetc, whle FIP s weaker than ACA 0 and ncomparable wth WKL Propertes of fnte character We begn wth the study of varous forms of the followng prncple. Defnton 2.1. The followng scheme s defned n RCA 0. (FCP) For each L 2 formula ϕ of fnte character, whch may have arbtrary set parameters, every set A has a -maxmal subset B such that ϕ(b) holds. FCP s analogous to the set-theoretc prncple M 7 n the catalog of Rubn and Rubn [3], whch s equvalent to the axom of choce [3, p. 34 and Theorem 4.3]. In order to better gauge the reverse mathematcal strength of FCP, we consder restrctons of the formulas to whch t apples. As wth other such ramfcatons, we wll prmarly be nterested n restrctons to classes n the arthmetcal and analytcal herarches. In partcular, for each {0, 1} and n 0, we make the followng defntons: Σ n-fcp s the restrcton of FCP to Σ n formulas; Π n-fcp s the restrcton of FCP to Π n formulas; n-fcp s the scheme whch says that for every Σ n formula ϕ(x) and every Π n formula ψ(x), f ϕ(x) s of fnte character and ( X)[ϕ(X) ψ(x)], then every set A has a -maxmal set B such that ϕ(b) holds. We also defne QF-FCP to be the restrcton of FCP to the class of quantferfree formulas wthout parameters. The followng proposton demonstrates two monotoncty propertes of formulas of fnte character.
3 REVERSE MATHEMATICS AND PROPERTIES OF FINITE CHARACTER 3 Proposton 2.2. Let ϕ(x) be a formula of fnte character. The followng are provable n RCA 0 : (1) f A B and ϕ(b) holds then ϕ(a) holds; (2) f A 0 A 1 A 2 s a sequence of sets such that ϕ(a ) holds for each N, and N A exsts, then ϕ( N A ) holds. Proof. The proof of (1) s mmedate from the defntons. For (2), the key pont s to show that f F s a fnte subset of N A then there s some j N wth F A j. Ths follows from nducton on the Σ 0 1 formula ψ(n, F ) ( m)( < n)( F = A m ), n whch F s a set parameter. Our frst theorem n ths secton characterzes most of the above restrctons of FCP (see Corollary 2.5). We draw partcular attenton to part (2) of the theorem, where Σ 0 1 does not appear n the lst of classes of formulas. The reason behnd ths wll be made apparent by Theorem 2.6. Theorem 2.3. For {0, 1} and n 1, let Γ be any of Π n, Σ n, or n. (1) Γ-FCP s provable n Γ-CA 0 ; (2) If Γ s Π 0 n, Π 1 n, Σ 1 n, or 1 n, then Γ-FCP mples Γ-CA 0 over RCA 0. The proof of ths theorem wll make use of the followng techncal lemma, whch s needed only because there are no term-formng operatons for sets n the language L 2 of second-order arthmetc. For example, there s no term n L 2 that takes a set X and a number n and returns X D n where, as n the rest of ths paper, D n denotes the fnte set wth canoncal ndex n, or f n s not a canoncal ndex. The moral of the lemma s that such terms can be nterpreted nto L 2 n a natural way. The codng of fnte sets by ther canoncal ndces can be formalzed n RCA 0 n such a way that the predcate D n s defned by a formula ρ(, n) wth only bounded quantfers, and such that the set of canoncal ndces s also defnable by a bounded-quantfer formula [4, Theorem II.2.5]. Moreover, RCA 0 proves that every fnte set has a canoncal ndex. We use the notaton Y = D n to abbrevate the formula ( )[ Y ρ(, n)], along wth smlar notaton for subsets of fnte sets. Lemma 2.4. Let ϕ(x) be a formula wth one free set varable. There s a formula ϕ(x) wth one free number varable such that RCA 0 proves (2.4.1) ( A)( n)[a = D n = (ϕ(a) ϕ(n))]. Moreover, we may take ϕ to have the same complextes n the arthmetcal and analytc herarches as ϕ. Proof. Let ρ(, n) be the formula defnng the relaton D n, as dscussed above. We may assume ϕ s wrtten n prenex normal form. Form ϕ(n) by replacng each occurrence t X of ϕ, t a term, wth the formula ρ(t, n). Let ψ(x, Ȳ, m) be the quantfer-free matrx of ϕ, where Ȳ and m are sequences of varables that are quantfed n ϕ. Smlarly, let ψ(n, Ȳ, m) be the matrx of ϕ. Fx any model M of RCA 0 and fx n, A M such that
4 4 DAMIR D. DZHAFAROV AND CARL MUMMERT M = A = D n. proves that A straghtforward metanducton on the structure of ψ M = ( Ȳ )( m)[ψ(a, Ȳ, m) ψ(n, Ȳ, m)]. The key pont s that the atomc formulas n ψ(a, Ȳ, m) are the same as those n ψ(n, Ȳ, m), wth the excepton of formulas of the form t A, whch have been replaced wth the equvalent formulas of the form ρ(t, n). A second metanducton on the quantfer structure of ϕ shows that we may adjon quantfers to ψ and ψ untl we have obtaned ϕ and ϕ, whle mantanng logcal equvalence. Thus every model of RCA 0 satsfes (2.4.1). Because ρ has only bounded quantfers, the substtuton requred to pass from ϕ to ϕ does not change the complexty of the formula. We shall sometmes dentfy a fnte set wth ts canoncal ndex. Thus, f F s fnte and n s ts canoncal ndex, we may wrte ϕ(f ) for ϕ(n). Proof of Theorem 2.3. For (1), let ϕ(x) and A = {a : N} be an nstance of Γ-FCP. Defne g : 2 <N N {0, 1} by { 1 f ϕ({a j : τ(j) = 1} {a }) holds, g(τ, ) = 0 otherwse. where ϕ s as n the lemma. The functon g exsts by Γ comprehenson. By prmtve recurson, there exsts a functon h: N {0, 1} such that for all N, h() = 1 f and only f g(h, ) = 1. For each N, let B = {a j : j < h(j) = 1}. An nducton on ϕ shows that ϕ(b ) holds for every N. Let B = {a : h() = 1} = N B. Because Proposton 2.2 s provable n RCA 0 and hence n Γ-CA 0, t follows that ϕ(b) holds. By the same token, f ϕ(b {a k }) holds for some k then so must ϕ(b k {a k }), and therefore a k B k+1, whch means that a k B. Therefore B s -maxmal, and we have shown that Γ-CA 0 proves Γ-FCP. For (2), we assume Γ s one of Π 0 n, Π 1 n, or Σ 1 n; the proof for 1 n s smlar. We work n RCA 0 + Γ-FCP. Let ϕ(n) be a formula n Γ and let ψ(x) be the formula ( n)[n X = ϕ(n)]. It s easly seen that ψ s of fnte character, and t belongs to Γ because Γ s closed under unversal number quantfcaton. By Γ-FCP, N contans a -maxmal subset B such that ψ(b) holds. For any y, f y B then ϕ(y) holds. On the other hand, f ϕ(y) holds then so does ψ(b {y}), so y must belong to B by maxmalty. Therefore B = {y N : ϕ(y)}, and we have shown that Γ-FCP mples Γ-CA 0. The corollary below summarzes the theorem as t apples to the varous classes of formulas we are nterested n. Of specal note s part (5), whch says that FCP tself (that s, FCP for arbtrary L 2 -formulas) s as strong as any theorem of second-order arthmetc can be. Corollary 2.5. The followng are provable n RCA 0 :
5 REVERSE MATHEMATICS AND PROPERTIES OF FINITE CHARACTER 5 (1) 0 1 -FCP, Σ0 0-FCP, and QF-FCP; (2) for each n 1, ACA 0 s equvalent to Π 0 n-fcp; (3) for each n 1, 1 n-ca 0 s equvalent to 1 n-fcp; (4) for each n 1, Π 1 n-ca 0 s equvalent to Π 1 n-fcp and to Σ 1 n-fcp; (5) Z 2 s equvalent to FCP. The case of FCP for Σ 0 1 formulas s anomalous. The proof of part (2) of Theorem 2.3 does not go through for Σ 0 1 because ths class s not closed under unversal quantfcaton. As the next theorem shows, ths lmtaton s qute sgnfcant. Intutvely, the proof uses the fact that a Σ 0 1 formula ϕ s contnuous n the sense that f ϕ(x) holds then there s an N such that ϕ(y ) holds for any Y wth X {0,..., N} = Y {0,..., N}. Theorem 2.6. Σ 0 1 -FCP s provable n RCA 0. Proof. Let ϕ(x) be a Σ 0 1 formula of fnte character. We clam that there exsts some c ϕ N such that for every set A, f A {0,..., c ϕ } = then ϕ(a) holds. To show ths, put ϕ(x) n normal form, so that ϕ(x) ( m)ρ(x[m]) where ρ s Σ 0 0. As ϕ( ) holds, there s some c = c ϕ such that ρ( [c]) holds. Now let A be any set such that A {0,..., c} =. Then ρ(a[c]) holds, so ϕ(a) holds. Ths proves the clam. Now fx any set A. By the clam, we know that ϕ(a {0,..., c ϕ }) holds. We may use bounded Σ 0 1 comprehenson [4, Theorem II.3.9] to form the set I of m such that D m {0,..., c ϕ } and ϕ(d m (A {0,..., c ϕ })) holds. We may then choose m I such that D m has maxmal cardnalty among the sets wth ndces n I. It follows mmedately that D m (A {0,..., c ϕ }) s a maxmal subset of A satsfyng ϕ. The above proof contans an mplct non-unformty n choosng a fnte set of maxmal cardnalty. The next proposton shows that ths nonunformty s essental, by showng that a sequental form of Σ 0 1-FCP s a strctly stronger prncple.
6 6 DAMIR D. DZHAFAROV AND CARL MUMMERT Proposton 2.7. The followng are equvalent over RCA 0 : (1) ACA 0 ; (2) for every famly A = A : N of sets, and every Σ 0 1 formula ϕ(x, x) wth one free set varable and one free number varable such that for all N, the formula ϕ(x, ) s of fnte character, there exsts a famly B = B : N of sets such that for all, B s a -maxmal subset of A satsfyng ϕ(x, ). Proof. The forward mplcaton follows by a straghtforward modfcaton of the proof of Theorem 2.3. For the reversal, let a one-to-one functon f : N N be gven. For each N, let A = {}, and let ϕ(x, x) be the formula ( y)[x X = f(y) = x]. Then, for each, ϕ(x, ) has the fnte character property, and for every set S that contans, ϕ(s, ) holds f and only f range(f). Thus, f B = B : N s the subfamly obtaned by applyng part (2) to the famly A = A : N and the formula ϕ(x, x), then It follows that the range of f exsts. range(f) B = {} B. Remark 2.8. Proposton 2.7 would not hold wth the class of boundedquantfer formulas of fnte character n place of the class of Σ 0 1 such formulas, because n that case part (2) s provable n RCA 0. Thus, n spte of the smlarty between the two classes suggested by the proof of Theorem 2.6, they do not concde. 3. Fntary closure operators We can strengthen FCP by mposng addtonal requrements on the maxmal set beng constructed. In partcular, we now consder requrng the maxmal set to satsfy a fntary closure property as well as a property of fnte character. Defnton 3.1. A fntary closure operator s a set of pars F, n n whch F s (the canoncal ndex for) a fnte (possbly empty) subset of N and n N. A set A N s closed under a fntary closure operator D, or D-closed, f for every F, n D, f F A then n A. Ths defnton of a closure operator s not the standard set-theoretc defnton presented by Rubn and Rubn [3, Defnton 6.3]. However, t s easy to see that for each operator of the one knd there s an operator of the other such that the same sets are closed under both. Our defnton has the advantage of beng readly formalzable n RCA 0. The followng prncple expresses the monotoncty of fntary closure operators. The proof follows drectly from defntons.
7 REVERSE MATHEMATICS AND PROPERTIES OF FINITE CHARACTER 7 Proposton 3.2. It can be proved n RCA 0 that f D s a fntary closure operator and A 0 A 1 A 2 s a sequence of sets such that N A exsts and each A s D-closed, then N A s D-closed. The prncple n the next defnton s analogous to prncple AL 3 of Rubn and Rubn [3], whch s equvalent to the axom of choce n the context of set theory [3, p. 96, and Theorems 6.4 and 6.5]. Defnton 3.3. The followng scheme s defned n RCA 0. (CE) If D s a fntary closure operator, ϕ s an L 2 formula of fnte character, and A s any set, then every D-closed subset of A satsfyng ϕ s contaned n a maxmal such subset. In the termnology of Rubn and Rubn [3], ths s a prmed statement, meanng that t asserts the exstence not merely of a maxmal subset of a gven set, but the exstence of a maxmal extenson of any gven subset. Prmed versons of FCP and ts restrctons can be formed, and are equvalent to the unprmed versons over RCA 0. By contrast, CE has only a prmed form. Ths s because f A s a set, ϕ s a formula of fnte character, and D s a fntary closure operator, A need not have any D-closed subset of whch ϕ holds. For example, suppose ϕ holds only of, and D contans a par of the form, a for some a A. Ths leads to the observaton that the requrements n the CE scheme that the maxmal set must both be D-closed and satsfy a property of fnte character are, ntutvely, n opposton to each other. Satsfyng a fntary closure property s a postve requrement, n the sense that formng the closure of a set usually requres addng elements to the set. Satsfyng a property of fnte character can be seen as a negatve requrement n lght of part (1) of Proposton 2.2. We consder restrctons of CE as we dd restrctons of FCP above. By analogy, f Γ s a class of formulas, we use the notaton Γ-CE to denote the restrcton of CE to the formulas n Γ. We begn wth the followng analogue of part (1) of Theorem 2.3 from the prevous secton. Theorem 3.4. For {0, 1} and n 1, let Γ be Π n, Σ n, or 1 n. Then Γ-CE s provable n Γ-CA 0. Proof. Let ϕ be a formula of fnte character n Γ, whch may have parameters, and let D be a fntary closure operator. Let A be any set and let C be a D-closed subset of A such that ϕ(c) holds. For any X A, let cl D (X) denote the D-closure of X. That s, cl D (X) = N X, where X 0 = X and for each N, X +1 s the set of all n N such that ether n X or there s a fnte set F X such that F, n D. Because we take D to be a set, cl D (X) can be defned usng a Σ 0 1 formula wth parameter D. Defne a formula ψ(k, X) by ψ(k, X) ( n)[(d n cl D (X D k ) = ϕ(n)] cl D (X D k ) A,
8 8 DAMIR D. DZHAFAROV AND CARL MUMMERT where ϕ s as n Lemma 2.4. Note that ψ s arthmetcal f Γ s Π 0 n or Σ 0 n, and s n Γ otherwse. Defne a functon f : N {0, 1} nductvely such that f() = 1 f and only f ψ({j < : f(j) = 1} {}, C) holds. The characterzaton of the complexty of ψ ensures that ths f can be constructed usng Γ comprehenson, by frst formng the oracle {k : ψ(k, C)}. Now, for each N, let B = cl D (C {j < : f(j) = 1}), and let B = N B. The constructon of f ensures that ϕ(b ) mples ϕ(b +1 ) for all N, and we have assumed that ϕ holds of B 0 = cl D (C) = C. Therefore, an nstance of nducton shows that ϕ holds of B for all N, and thus also of B by Proposton 2.2. Ths also shows that B A. Smlarly, because each B s D-closed, the formalzed verson of Proposton 3.2 mples B s D-closed. Fnally, we check that B s maxmal. Suppose that H s a D-closed set such that B H A and ϕ(h) holds. Fxng H, because B B H and H s D-closed, we have cl D (B {}) H. Thus, ϕ(f ) holds for every fnte subset F of cl D (B {}), so by constructon f() = 1 and B +1 = cl D (B {}). Because B +1 B, we conclude that B. Thus B = H, as desred. It follows that for most standard syntactcal classes Γ, Γ-CE s equvalent to Γ-FCP. Indeed, for any class Γ we have that Γ-CE mples Γ-FCP, because any nstance of the latter can be regarded as an nstance of the former by addng an empty fntary closure operator. Conversely, f Γ s Π 0 n, Π 1 n, Σ 1 n, or 1 n, then Γ-FCP s equvalent to Γ-CA 0 by Theorem 2.3 (2), and hence equvalent to Γ-CE. Thus, n partcular, parts (2) (5) of Corollary 2.5 hold for CE n place of FCP, and the full scheme CE tself s equvalent to Z 2. The proof of the precedng theorem does not work for Γ = 0 1, because then Γ-CA 0 s just RCA 0, and we need at least ACA 0 to prove the exstence of the functon f defned there (the formula ψ(σ, X) beng arthmetcal at best). The next theorem shows that ths cannot be avoded, even for a class of consderably weaker formulas. Theorem 3.5. QF-CE mples ACA 0 over RCA 0. Proof. Assume a one-to-one functon f : N N s gven. Let ϕ(x) be the quantfer-free formula 0 / X, whch trvally has fnte character, and let p : N be an enumeraton of all prmes. Let D be the fntary closure operator consstng, for all, n N, of all pars of the form {p n+1 }, p n+2 ; {p n+2 }, p n+1 ; {p n+1 }, 0, f f(n) =. The set D exsts by 0 1 comprehenson relatve to f and our enumeraton of prmes.
9 REVERSE MATHEMATICS AND PROPERTIES OF FINITE CHARACTER 9 Note that s a D-closed subset of N and ϕ( ) holds. Thus, we may apply CE for quantfer-free formulas to obtan a maxmal D-closed subset B of N such that ϕ(b) holds. By defnton of D, for every N, B ether contans every postve power of p or no postve power. Now f f(n) = for some n, then no postve power of p can be n B, because otherwse p n+1 would necessarly be n B and hence so would 0. On the other hand, f f(n) for all n then B {p n+1 : n N} s D-closed and satsfes ϕ, so by maxmalty p n+1 must belong to B for every n. It follows that range(f) f and only f p B, so the range of f exsts. The next corollary can be contrasted wth 2.5 part (1) and Theorem 2.6 to llustrate a dfference between CE from FCP n terms of some of ther weakest restrctons. Corollary 3.6. The followng are equvalent over RCA 0 : (1) ACA 0 ; (2) Σ 0 1 -CE; (3) Σ 0 0 -CE; (4) QF-CE. We conclude ths secton wth one addtonal llustraton of how formulas of fnte character can be used n conjuncton wth fntary closure operators. Recall the followng concepts from order theory: a countable jon-semlattce s a countable poset L, L wth a maxmal element 1 L and a jon operaton L : L L L such that for all a, b L, a L b s the least upper bound of a and b; an deal on a countable jon-semlattce L s a subset I of L that s downward closed under L and closed under L. The prncple n the followng proposton s the countable analogue of a varant of AL 1 n Rubn and Rubn [3]; compare wth Proposton 4.4 below. For more on the computablty theory of deals on lattces, see Turlngton [5]. Proposton 3.7. Over RCA 0, QF-CE mples that every proper deal on a countable jon-semlattce extends to a maxmal proper deal. Proof. Let L be a countable jon-semlattce. Let ϕ be the formula 1 X, and let D be the fntary closure operator consstng of all pars of the form {a, b}, c where a, b L and c = a b; {a}, b, where b L a. Because we defne a jon-semlattce to come wth both the order relaton and the jon operaton, the set D s 0 0 wth parameters, so RCA 0 proves D exsts. It s mmedate that a set X s closed under D f and only f X s an deal n L. We have not been able to prove a reversal correspondng to the prevous proposton.
10 10 DAMIR D. DZHAFAROV AND CARL MUMMERT Queston 3.8. What s the strength of the prncple assertng that every proper deal on a countable jon-semlattce extends to a maxmal proper deal? Ths queston s further motvated by work of Turlngton [5, Theorem ] on the smlar problem of constructng prme deals on computable lattces. However, because a maxmal deal on a countable lattce need not be a prme deal, Turlngton s results do not drectly resolve our queston. 4. Nondetermnstc fntary closure operators It appears that the underlyng reason that the restrcton of CE to arthmetcal formulas s provable n ACA 0 (and more generally, why Γ-CE s provable n Γ-CA 0 f Γ s as n Theorem 3.4) s that our defnton of fntary closure operator s very constranng. Intutvely, f D s such an operator and ϕ s an arthmetcal formula, and we seek to extend some D-closed subset B satsfyng ϕ to a maxmal such subset, we can focus largely on ensurng that ϕ holds. Achevng closure under D s relatvely straghtforward, because at each stage we only need to search through all fnte subsets F of our current extenson, and then adjon all n such that F, n D. Ths closure process becomes far less trval f we are gven a choce of whch elements to adjon. We now consder the case when each fnte subset F can be assocated wth a possbly nfnte set of numbers from whch we must choose at least one to adjon. Intutvely, ths change adds an aspect of dependent choce when we wsh to form the closure of a set. We wll show that ths weaker noton of closure operator leads to a strctly stronger analogue of CE. Defnton 4.1. A nondetermnstc fntary closure operator s a sequence of sets of the form F, S where F s (the canoncal ndex for) a fnte (possbly empty) subset of N and S s a nonempty subset of N. A set A N s closed under a nondetermnstc fntary closure operator N, or N-closed, f for each F, S n N, f F A then A S. Note that f D s a determnstc fntary closure operator, that s, a fntary closure operator n the stronger sense of the prevous secton, then for any set A there s a unque -mnmal D-closed set extendng A. Ths s not true for nondetermnstc fntary closure operators. For example, let N be the operator such that, N N and, for each N and each j >, {}, {j} N. Then any N-closed set extendng wll be of the form { N : k} for some k, and any set of ths form s N-closed. Thus there s no -mnmal N-closed set. In ths secton we study the followng nondetermnstc verson of CE. Defnton 4.2. The followng scheme s defned n RCA 0. (NCE) If N s a nondetermnstc closure operator, ϕ s an L 2 formula of fnte character, and A s any set, then every N-closed subset of A satsfyng ϕ s contaned n a maxmal such subset.
11 REVERSE MATHEMATICS AND PROPERTIES OF FINITE CHARACTER 11 Because the unon of a chan of N-closed sets s agan N-closed, NCE can be proved n set theory usng Zorn s lemma. Restrctons of NCE to varous syntactcal classes of formulas are defned as for CE and FCP. Remark 4.3. We mght expect to be able to prove NCE from CE by sutably transformng a gven nondetermnstc fntary closure operator N nto a determnstc one. For nstance, we could go through the members of N one by one, and for each such member F, S add F, n to D for some n S (e.g., the least n). All D-closed sets would then ndeed be N-closed. The converse, however, would not necessarly be true, because a set could have F as a subset for some F, S N, yet t could contan a dfferent n S than the one chosen n defnng D. In partcular, a maxmal D-closed subset of a gven set mght not be maxmal among N-closed subsets. The results of ths secton demonstrate that t s mpossble, n general, to reduce nondetermnstc closure operators to determnstc ones n weak systems. Recall that an deal on a countable poset P, P s a subset I of P downward closed under P and such that for all p, q I there s an r I wth p P r and q P r. The next proposton s smlar to Proposton 3.7 above, whch dealt wth deals on countable jon-semlattces. In the proof of that proposton, we defned a determnstc fntary closure operator D n such a way that D-closed sets were closed under the jon operaton. For ths we reled on the fact that for every two elements n the semlattce there s a unque element that s ther jon. The reason we need nondetermnstc fntary closure operators below s that, for deals on countable posets, there are no longer unque elements wtnessng the relevant closure property. Proposton 4.4. Over RCA 0, QF-NCE mples that every deal on a countable poset can be extended to a maxmal deal. Proof. Let P, P be a countable poset; wthout loss of generalty we may assume P s nfnte. Form an extended poset P by adjonng a new element t to P and declarng q < P t for all q P. It follows mmedately that the deals on P correspond exactly to the deals of P that do not contan t, and each deal on P whch s maxmal among deals not contanng t corresponds to a maxmal deal on P. Fx an enumeraton {p : N} of P. We form a nondetermnstc closure operator N = N : N such that, for each N, f = 2 j, k and p j P p k then N = {p k }, {p j } ; f = 2 j, k, l + 1 and p j P p l and p k P p l then N = {p j, p k }, {p n : (p j P p n ) (p k P p n )} ; otherwse, N = {p }, {p }. Ths constructon gves a quantfer-free defnton of each N unformly n, so RCA 0 s able to construct N. Moreover, a subset of P s N-closed f and only f t s an deal.
12 12 DAMIR D. DZHAFAROV AND CARL MUMMERT Let ϕ(x) be the formula t X, whch s of fnte character. Fx an deal I P. Vewng I as a subset of P, we see that I s N-closed and ϕ(i) holds. Thus, by QF-NCE, there s a maxmal N-closed extenson J P satsfyng ϕ. Ths mmedately yelds a maxmal deal on P extendng I. Mummert [2, Theorem 2.4] showed that the proposton that every deal on a countable poset extends to a maxmal deal s equvalent to Π 1 1 -CA 0 over RCA 0, whch leads to the followng corollary. Ths contrasts sharply wth Theorem 3.4, whch showed that CE for arthmetcal formulas s provable n ACA 0. Corollary 4.5. QF-NCE mples Π 1 1 -CA 0 over RCA 0. We wll state the precse strength of QF-NCE n Corollary 4.7 below. We must frst prove the followng upper bound. The proof uses a technque nvolvng countable coded β-models, parallel to Lemma 2.4 of Mummert [2]. In ACA 0, a countable coded β-model s defned as a sequence M = M : N of subsets of N such that for every Σ 1 1 formula ϕ wth parameters from M, ϕ holds f and only f M = ϕ. Π 1 1 -CA 0 proves that every set s ncluded n some countable coded β-model. Complete nformaton on countable coded β-models s gven by Smpson [4, Secton VII.2]. Theorem 4.6. Σ 1 1 -NCE s provable n Π1 1 -CA 0. Proof. Let ϕ be a Σ 1 1 formula of fnte character (possbly wth parameters) and let N be a nondetermnstc closure operator. Let A be any set and let C be an N-closed subset of A such that ϕ(c) holds. Let M = M : N be a countable coded β-model contanng A, C, N, and any parameters of ϕ. Usng Π 1 1 comprehenson, we may form the set { : M = ϕ(m )}. Workng outsde M, we buld an ncreasng sequence B : N of N- closed extensons of C. Let B 0 = C. Gven, ask whether there s a j such that M j s an N-closed subset of A; B M j ; M j ; and ϕ(m j ) holds. If there s, choose the least such j and let B +1 = M j. Otherwse, let B +1 = B. Fnally, let B = N B. Because the nductve constructon only asks arthmetcal questons about M, t can be carred out n Π 1 1 -CA 0, and so Π 1 1 -CA 0 proves that B exsts. Clearly C B A. An arthmetcal nducton shows that for all N, ϕ(b ) holds and B s N-closed. Therefore, the formalzed verson of Proposton 2.2 shows that ϕ(b) holds, and the analogue of Proposton 3.2 for nondetermnstc fntary closure operators shows that B s N-closed.
13 REVERSE MATHEMATICS AND PROPERTIES OF FINITE CHARACTER 13 Now suppose that H s an N-closed set such that B H A and ϕ(h) holds. Fx H. Because ϕ s Σ 1 1, the property (4.6.1) ( X)[X s N-closed B X A X ϕ(x)] s expressble by a Σ 1 1 sentence wth parameters from M, and H wtnesses that t s true. Thus, because M s a β-model, ths sentence must be satsfed by M, whch means that some M j must also wtness t. The nductve constructon must therefore have selected such an M j to be B +1, whch means B +1 and hence B. It follows that B s maxmal. We can now characterze the strength of Σ 1 1-NCE and ts restrctons. Corollary 4.7. For each n 1, the followng are equvalent over RCA 0 : (1) Π 1 1 -CA 0; (2) Σ 1 1 -NCE; (3) Σ 0 n-nce; (4) QF-NCE. Proof. Theorem 4.6 shows that (1) mples (2), and t s obvous that (2) mples (3) and (3) mples (4). Corollary 4.5 shows that (4) mples (1). Our fnal results characterze the strength of NCE for formulas hgher n the analytcal herarchy. Theorem 4.8. For each n 1, (1) Σ 1 n-nce and Π 1 n-nce are provable n Π 1 n-ca 0 ; (2) 1 n-nce s provable n 1 n-ca 0. Proof. We prove part (1), the proof of part (2) beng smlar. Let ϕ(x) be a Σ 1 n formula of fnte character, respectvely a Π 1 n such formula. Let N be a nondetermnstc closure operator, let A be any set, and let C be an N-closed subset of A such that ϕ(c) holds. By Lemma 4.5, let ϕ be a Σ 1 n formula, respectvely a Π 1 n formula, such that ( X)( n)[x = D n = (ϕ(x) ϕ(n))]. We may use Π 1 n comprehenson to form the set W = {n : ϕ(n)}. Defne ψ(x) to be the arthmetcal formula ( n)[d n X = n W ]. We clam that for every set X, ψ(x) holds f and only f ϕ(x) holds. The defntons of W and ψ ensure that ψ(x) holds f and only f ϕ(d n ) holds for every fnte D n X, whch s true f and only f ϕ(x) holds because ϕ has fnte character. Ths establshes the clam. By the clam, ψ s a property of fnte character and ψ(c) holds. Usng Σ 1 1 -NCE, whch s provable n Π1 1 -CA 0 by Theorem 4.6 and thus s provable n Π 1 n-ca 0, there s a maxmal N-closed subset B of A extendng C wth property ψ. Agan by the clam, B s a maxmal N-closed subset of A extendng B wth property ϕ.
14 14 DAMIR D. DZHAFAROV AND CARL MUMMERT Corollary 4.9. The followng are provable n RCA 0 : (1) for each n 1, 1 n-ca 0 s equvalent to 1 n-nce; (2) for each n 1, Π 1 n-ca 0 s equvalent to Π 1 n-nce and to Σ 1 n-nce; (3) Z 2 s equvelent to NCE. Proof. The mplcatons from 1 n-ca 0, Π 1 n-ca 0, and Z 2 follow by Theorem 4.8. On the other hand, each restrcton of NCE trvally mples the correspondng restrcton of FCP, so the reversals follow by Corollary 2.5. Remark The characterzatons n ths secton shed lght on the role of the closure operator n the prncples CE and NCE. For n 1, we have shown that Σ 1 n-fcp, Σ 1 n-ce, and Σ 1 n-nce are all equvalent over RCA 0. However, QF-FCP s provable n RCA 0, QF-CE s equvalent to ACA 0 over RCA 0, and QF-NCE s equvalent to Π 1 1 -CA 0 over RCA 0. Thus the closure operators n the stronger prncples serve as a sort of replacement for arthmetcal quantfcaton n the case of CE, and for Σ 1 1 quantfcaton n the case of NCE. Ths allows these prncples to have greater strength than mght be suggested by the property of fnte character alone. At hgher levels of the analytcal herarchy, the prncples become equvalent because the complexty of the property of fnte character overtakes the complexty of the closure notons. References 1. Damr D. Dzhafarov and Carl Mummert, On the strength of the fnte ntersecton prncple, submtted, Carl Mummert, Reverse mathematcs of MF spaces, J. Math. Log. 6 (2006), no. 2, MR MR (2008d:03011) 3. Herman Rubn and Jean E. Rubn, Equvalents of the axom of choce. II, Studes n Logc and the Foundatons of Mathematcs, vol. 116, North-Holland Publshng Co., Amsterdam, MR MR (87c:04004) 4. Stephen G. Smpson, Subsystems of second order arthmetc, second ed., Perspectves n Logc, Cambrdge Unversty Press, Cambrdge, MR MR (2010e:03073) 5. Amy Turlngton, Computablty of Heytng algebras and dstrbutve lattces, Ph.D. dssertaton, Unversty of Connectcut, Department of Mathematcs, Unversty of Notre Dame, Department of Mathematcs, Unversty of Notre Dame, 255 Hurley Hall, Notre Dame, Indana USA E-mal address: ddzhafar@nd.edu Department of Mathematcs, Marshall Unversty, 1 John Marshall Drve, Huntngton, West Vrgna USA E-mal address: mummertc@marshall.edu
Affine transformations and convexity
Affne transformatons and convexty The purpose of ths document s to prove some basc propertes of affne transformatons nvolvng convex sets. Here are a few onlne references for background nformaton: http://math.ucr.edu/
More informationWeek 2. This week, we covered operations on sets and cardinality.
Week 2 Ths week, we covered operatons on sets and cardnalty. Defnton 0.1 (Correspondence). A correspondence between two sets A and B s a set S contaned n A B = {(a, b) a A, b B}. A correspondence from
More informationThe Order Relation and Trace Inequalities for. Hermitian Operators
Internatonal Mathematcal Forum, Vol 3, 08, no, 507-57 HIKARI Ltd, wwwm-hkarcom https://doorg/0988/mf088055 The Order Relaton and Trace Inequaltes for Hermtan Operators Y Huang School of Informaton Scence
More informationa b a In case b 0, a being divisible by b is the same as to say that
Secton 6.2 Dvsblty among the ntegers An nteger a ε s dvsble by b ε f there s an nteger c ε such that a = bc. Note that s dvsble by any nteger b, snce = b. On the other hand, a s dvsble by only f a = :
More informationLinear, affine, and convex sets and hulls In the sequel, unless otherwise specified, X will denote a real vector space.
Lnear, affne, and convex sets and hulls In the sequel, unless otherwse specfed, X wll denote a real vector space. Lnes and segments. Gven two ponts x, y X, we defne xy = {x + t(y x) : t R} = {(1 t)x +
More informationMaximizing the number of nonnegative subsets
Maxmzng the number of nonnegatve subsets Noga Alon Hao Huang December 1, 213 Abstract Gven a set of n real numbers, f the sum of elements of every subset of sze larger than k s negatve, what s the maxmum
More information20. Mon, Oct. 13 What we have done so far corresponds roughly to Chapters 2 & 3 of Lee. Now we turn to Chapter 4. The first idea is connectedness.
20. Mon, Oct. 13 What we have done so far corresponds roughly to Chapters 2 & 3 of Lee. Now we turn to Chapter 4. The frst dea s connectedness. Essentally, we want to say that a space cannot be decomposed
More informationFoundations of Arithmetic
Foundatons of Arthmetc Notaton We shall denote the sum and product of numbers n the usual notaton as a 2 + a 2 + a 3 + + a = a, a 1 a 2 a 3 a = a The notaton a b means a dvdes b,.e. ac = b where c s an
More informationFACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP
C O L L O Q U I U M M A T H E M A T I C U M VOL. 80 1999 NO. 1 FACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP BY FLORIAN K A I N R A T H (GRAZ) Abstract. Let H be a Krull monod wth nfnte class
More informationOn the smoothness and the totally strong properties for nearness frames
Int. Sc. Technol. J. Namba Vol 1, Issue 1, 2013 On the smoothness and the totally strong propertes for nearness frames Martn. M. Mugoch Department of Mathematcs, Unversty of Namba 340 Mandume Ndemufayo
More informationREAL ANALYSIS I HOMEWORK 1
REAL ANALYSIS I HOMEWORK CİHAN BAHRAN The questons are from Tao s text. Exercse 0.0.. If (x α ) α A s a collecton of numbers x α [0, + ] such that x α
More informationFirst day August 1, Problems and Solutions
FOURTH INTERNATIONAL COMPETITION FOR UNIVERSITY STUDENTS IN MATHEMATICS July 30 August 4, 997, Plovdv, BULGARIA Frst day August, 997 Problems and Solutons Problem. Let {ε n } n= be a sequence of postve
More information12 MATH 101A: ALGEBRA I, PART C: MULTILINEAR ALGEBRA. 4. Tensor product
12 MATH 101A: ALGEBRA I, PART C: MULTILINEAR ALGEBRA Here s an outlne of what I dd: (1) categorcal defnton (2) constructon (3) lst of basc propertes (4) dstrbutve property (5) rght exactness (6) localzaton
More informationExercise Solutions to Real Analysis
xercse Solutons to Real Analyss Note: References refer to H. L. Royden, Real Analyss xersze 1. Gven any set A any ɛ > 0, there s an open set O such that A O m O m A + ɛ. Soluton 1. If m A =, then there
More informationAPPENDIX A Some Linear Algebra
APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,
More informationDIFFERENTIAL FORMS BRIAN OSSERMAN
DIFFERENTIAL FORMS BRIAN OSSERMAN Dfferentals are an mportant topc n algebrac geometry, allowng the use of some classcal geometrc arguments n the context of varetes over any feld. We wll use them to defne
More informationMATH 241B FUNCTIONAL ANALYSIS - NOTES EXAMPLES OF C ALGEBRAS
MATH 241B FUNCTIONAL ANALYSIS - NOTES EXAMPLES OF C ALGEBRAS These are nformal notes whch cover some of the materal whch s not n the course book. The man purpose s to gve a number of nontrval examples
More informationMore metrics on cartesian products
More metrcs on cartesan products If (X, d ) are metrc spaces for 1 n, then n Secton II4 of the lecture notes we defned three metrcs on X whose underlyng topologes are the product topology The purpose of
More informationAppendix B. Criterion of Riemann-Stieltjes Integrability
Appendx B. Crteron of Remann-Steltes Integrablty Ths note s complementary to [R, Ch. 6] and [T, Sec. 3.5]. The man result of ths note s Theorem B.3, whch provdes the necessary and suffcent condtons for
More informationGraph Reconstruction by Permutations
Graph Reconstructon by Permutatons Perre Ille and Wllam Kocay* Insttut de Mathémathques de Lumny CNRS UMR 6206 163 avenue de Lumny, Case 907 13288 Marselle Cedex 9, France e-mal: lle@ml.unv-mrs.fr Computer
More informationprinceton univ. F 17 cos 521: Advanced Algorithm Design Lecture 7: LP Duality Lecturer: Matt Weinberg
prnceton unv. F 17 cos 521: Advanced Algorthm Desgn Lecture 7: LP Dualty Lecturer: Matt Wenberg Scrbe: LP Dualty s an extremely useful tool for analyzng structural propertes of lnear programs. Whle there
More informationFORCING COPIES OF ω 1 WITH PFA(S)
FORCING COPIES OF ω 1 WITH PFA(S) ALAN DOW Abstract. We work n the PFA(S)[S] model. We show that a non-compact sequentally compact space of small character wll contan a copy of ω 1. Droppng the character
More informationIntroductory Cardinality Theory Alan Kaylor Cline
Introductory Cardnalty Theory lan Kaylor Clne lthough by name the theory of set cardnalty may seem to be an offshoot of combnatorcs, the central nterest s actually nfnte sets. Combnatorcs deals wth fnte
More informationA combinatorial problem associated with nonograms
A combnatoral problem assocated wth nonograms Jessca Benton Ron Snow Nolan Wallach March 21, 2005 1 Introducton. Ths work was motvated by a queston posed by the second named author to the frst named author
More informationAnti-van der Waerden numbers of 3-term arithmetic progressions.
Ant-van der Waerden numbers of 3-term arthmetc progressons. Zhanar Berkkyzy, Alex Schulte, and Mchael Young Aprl 24, 2016 Abstract The ant-van der Waerden number, denoted by aw([n], k), s the smallest
More informationFINITELY-GENERATED MODULES OVER A PRINCIPAL IDEAL DOMAIN
FINITELY-GENERTED MODULES OVER PRINCIPL IDEL DOMIN EMMNUEL KOWLSKI Throughout ths note, s a prncpal deal doman. We recall the classfcaton theorem: Theorem 1. Let M be a fntely-generated -module. (1) There
More informationBOUNDEDNESS OF THE RIESZ TRANSFORM WITH MATRIX A 2 WEIGHTS
BOUNDEDNESS OF THE IESZ TANSFOM WITH MATIX A WEIGHTS Introducton Let L = L ( n, be the functon space wth norm (ˆ f L = f(x C dx d < For a d d matrx valued functon W : wth W (x postve sem-defnte for all
More informationVolume 18 Figure 1. Notation 1. Notation 2. Observation 1. Remark 1. Remark 2. Remark 3. Remark 4. Remark 5. Remark 6. Theorem A [2]. Theorem B [2].
Bulletn of Mathematcal Scences and Applcatons Submtted: 016-04-07 ISSN: 78-9634, Vol. 18, pp 1-10 Revsed: 016-09-08 do:10.1805/www.scpress.com/bmsa.18.1 Accepted: 016-10-13 017 ScPress Ltd., Swtzerland
More informationOn C 0 multi-contractions having a regular dilation
SUDIA MAHEMAICA 170 (3) (2005) On C 0 mult-contractons havng a regular dlaton by Dan Popovc (mşoara) Abstract. Commutng mult-contractons of class C 0 and havng a regular sometrc dlaton are studed. We prove
More informationSubset Topological Spaces and Kakutani s Theorem
MOD Natural Neutrosophc Subset Topologcal Spaces and Kakutan s Theorem W. B. Vasantha Kandasamy lanthenral K Florentn Smarandache 1 Copyrght 1 by EuropaNova ASBL and the Authors Ths book can be ordered
More informationA CHARACTERIZATION OF ADDITIVE DERIVATIONS ON VON NEUMANN ALGEBRAS
Journal of Mathematcal Scences: Advances and Applcatons Volume 25, 2014, Pages 1-12 A CHARACTERIZATION OF ADDITIVE DERIVATIONS ON VON NEUMANN ALGEBRAS JIA JI, WEN ZHANG and XIAOFEI QI Department of Mathematcs
More informationDifference Equations
Dfference Equatons c Jan Vrbk 1 Bascs Suppose a sequence of numbers, say a 0,a 1,a,a 3,... s defned by a certan general relatonshp between, say, three consecutve values of the sequence, e.g. a + +3a +1
More informationAssortment Optimization under MNL
Assortment Optmzaton under MNL Haotan Song Aprl 30, 2017 1 Introducton The assortment optmzaton problem ams to fnd the revenue-maxmzng assortment of products to offer when the prces of products are fxed.
More informationPolynomials. 1 More properties of polynomials
Polynomals 1 More propertes of polynomals Recall that, for R a commutatve rng wth unty (as wth all rngs n ths course unless otherwse noted), we defne R[x] to be the set of expressons n =0 a x, where a
More informationGeometry of Müntz Spaces
WDS'12 Proceedngs of Contrbuted Papers, Part I, 31 35, 212. ISBN 978-8-7378-224-5 MATFYZPRESS Geometry of Müntz Spaces P. Petráček Charles Unversty, Faculty of Mathematcs and Physcs, Prague, Czech Republc.
More informationOn the Operation A in Analysis Situs. by Kazimierz Kuratowski
v1.3 10/17 On the Operaton A n Analyss Stus by Kazmerz Kuratowsk Author s note. Ths paper s the frst part slghtly modfed of my thess presented May 12, 1920 at the Unversty of Warsaw for the degree of Doctor
More informationProblem Set 9 Solutions
Desgn and Analyss of Algorthms May 4, 2015 Massachusetts Insttute of Technology 6.046J/18.410J Profs. Erk Demane, Srn Devadas, and Nancy Lynch Problem Set 9 Solutons Problem Set 9 Solutons Ths problem
More informationPerfect Competition and the Nash Bargaining Solution
Perfect Competton and the Nash Barganng Soluton Renhard John Department of Economcs Unversty of Bonn Adenauerallee 24-42 53113 Bonn, Germany emal: rohn@un-bonn.de May 2005 Abstract For a lnear exchange
More informationEdge Isoperimetric Inequalities
November 7, 2005 Ross M. Rchardson Edge Isopermetrc Inequaltes 1 Four Questons Recall that n the last lecture we looked at the problem of sopermetrc nequaltes n the hypercube, Q n. Our noton of boundary
More informationREGULAR POSITIVE TERNARY QUADRATIC FORMS. 1. Introduction
REGULAR POSITIVE TERNARY QUADRATIC FORMS BYEONG-KWEON OH Abstract. A postve defnte quadratc form f s sad to be regular f t globally represents all ntegers that are represented by the genus of f. In 997
More informationA new construction of 3-separable matrices via an improved decoding of Macula s construction
Dscrete Optmzaton 5 008 700 704 Contents lsts avalable at ScenceDrect Dscrete Optmzaton journal homepage: wwwelsevercom/locate/dsopt A new constructon of 3-separable matrces va an mproved decodng of Macula
More informationRestricted divisor sums
ACTA ARITHMETICA 02 2002) Restrcted dvsor sums by Kevn A Broughan Hamlton) Introducton There s a body of work n the lterature on varous restrcted sums of the number of dvsors of an nteger functon ncludng
More informationLecture 7: Gluing prevarieties; products
Lecture 7: Glung prevaretes; products 1 The category of algebrac prevaretes Proposton 1. Let (f,ϕ) : (X,O X ) (Y,O Y ) be a morphsm of algebrac prevaretes. If U X and V Y are affne open subvaretes wth
More informationCommunication Complexity 16:198: February Lecture 4. x ij y ij
Communcaton Complexty 16:198:671 09 February 2010 Lecture 4 Lecturer: Troy Lee Scrbe: Rajat Mttal 1 Homework problem : Trbes We wll solve the thrd queston n the homework. The goal s to show that the nondetermnstc
More informationTHE CHINESE REMAINDER THEOREM. We should thank the Chinese for their wonderful remainder theorem. Glenn Stevens
THE CHINESE REMAINDER THEOREM KEITH CONRAD We should thank the Chnese for ther wonderful remander theorem. Glenn Stevens 1. Introducton The Chnese remander theorem says we can unquely solve any par of
More informationLectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix
Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could
More informationSupplement: Proofs and Technical Details for The Solution Path of the Generalized Lasso
Supplement: Proofs and Techncal Detals for The Soluton Path of the Generalzed Lasso Ryan J. Tbshran Jonathan Taylor In ths document we gve supplementary detals to the paper The Soluton Path of the Generalzed
More information2.3 Nilpotent endomorphisms
s a block dagonal matrx, wth A Mat dm U (C) In fact, we can assume that B = B 1 B k, wth B an ordered bass of U, and that A = [f U ] B, where f U : U U s the restrcton of f to U 40 23 Nlpotent endomorphsms
More informationInfinitely Split Nash Equilibrium Problems in Repeated Games
Infntely Splt ash Equlbrum Problems n Repeated Games Jnlu L Department of Mathematcs Shawnee State Unversty Portsmouth, Oho 4566 USA Abstract In ths paper, we ntroduce the concept of nfntely splt ash equlbrum
More informationarxiv: v1 [math.co] 1 Mar 2014
Unon-ntersectng set systems Gyula O.H. Katona and Dánel T. Nagy March 4, 014 arxv:1403.0088v1 [math.co] 1 Mar 014 Abstract Three ntersecton theorems are proved. Frst, we determne the sze of the largest
More informationNOTES FOR QUANTUM GROUPS, CRYSTAL BASES AND REALIZATION OF ŝl(n)-modules
NOTES FOR QUANTUM GROUPS, CRYSTAL BASES AND REALIZATION OF ŝl(n)-modules EVAN WILSON Quantum groups Consder the Le algebra sl(n), whch s the Le algebra over C of n n trace matrces together wth the commutator
More informationA CLASS OF RECURSIVE SETS. Florentin Smarandache University of New Mexico 200 College Road Gallup, NM 87301, USA
A CLASS OF RECURSIVE SETS Florentn Smarandache Unversty of New Mexco 200 College Road Gallup, NM 87301, USA E-mal: smarand@unmedu In ths artcle one bulds a class of recursve sets, one establshes propertes
More informationNP-Completeness : Proofs
NP-Completeness : Proofs Proof Methods A method to show a decson problem Π NP-complete s as follows. (1) Show Π NP. (2) Choose an NP-complete problem Π. (3) Show Π Π. A method to show an optmzaton problem
More informationThe Multiple Classical Linear Regression Model (CLRM): Specification and Assumptions. 1. Introduction
ECONOMICS 5* -- NOTE (Summary) ECON 5* -- NOTE The Multple Classcal Lnear Regresson Model (CLRM): Specfcaton and Assumptons. Introducton CLRM stands for the Classcal Lnear Regresson Model. The CLRM s also
More informationSTEINHAUS PROPERTY IN BANACH LATTICES
DEPARTMENT OF MATHEMATICS TECHNICAL REPORT STEINHAUS PROPERTY IN BANACH LATTICES DAMIAN KUBIAK AND DAVID TIDWELL SPRING 2015 No. 2015-1 TENNESSEE TECHNOLOGICAL UNIVERSITY Cookevlle, TN 38505 STEINHAUS
More informationRandom Walks on Digraphs
Random Walks on Dgraphs J. J. P. Veerman October 23, 27 Introducton Let V = {, n} be a vertex set and S a non-negatve row-stochastc matrx (.e. rows sum to ). V and S defne a dgraph G = G(V, S) and a drected
More informationGenericity of Critical Types
Genercty of Crtcal Types Y-Chun Chen Alfredo D Tllo Eduardo Fangold Syang Xong September 2008 Abstract Ely and Pesk 2008 offers an nsghtful characterzaton of crtcal types: a type s crtcal f and only f
More informationSL n (F ) Equals its Own Derived Group
Internatonal Journal of Algebra, Vol. 2, 2008, no. 12, 585-594 SL n (F ) Equals ts Own Derved Group Jorge Macel BMCC-The Cty Unversty of New York, CUNY 199 Chambers street, New York, NY 10007, USA macel@cms.nyu.edu
More informationn ). This is tight for all admissible values of t, k and n. k t + + n t
MAXIMIZING THE NUMBER OF NONNEGATIVE SUBSETS NOGA ALON, HAROUT AYDINIAN, AND HAO HUANG Abstract. Gven a set of n real numbers, f the sum of elements of every subset of sze larger than k s negatve, what
More informationwhere a is any ideal of R. Lemma 5.4. Let R be a ring. Then X = Spec R is a topological space Moreover the open sets
5. Schemes To defne schemes, just as wth algebrac varetes, the dea s to frst defne what an affne scheme s, and then realse an arbtrary scheme, as somethng whch s locally an affne scheme. The defnton of
More informationMARTIN S AXIOM AND SEPARATED MAD FAMILIES
MARTIN S AXIOM AND SEPARATED MAD FAMILIES ALAN DOW AND SAHARON SHELAH Abstract. Two famles A, B of subsets of ω are sad to be separated f there s a subset of ω whch mod fnte contans every member of A and
More informationAn Introduction to Morita Theory
An Introducton to Morta Theory Matt Booth October 2015 Nov. 2017: made a few revsons. Thanks to Nng Shan for catchng a typo. My man reference for these notes was Chapter II of Bass s book Algebrac K-Theory
More informationDirichlet s Theorem In Arithmetic Progressions
Drchlet s Theorem In Arthmetc Progressons Parsa Kavkan Hang Wang The Unversty of Adelade February 26, 205 Abstract The am of ths paper s to ntroduce and prove Drchlet s theorem n arthmetc progressons,
More informationk(k 1)(k 2)(p 2) 6(p d.
BLOCK-TRANSITIVE 3-DESIGNS WITH AFFINE AUTOMORPHISM GROUP Greg Gamble Let X = (Z p d where p s an odd prme and d N, and let B X, B = k. Then t was shown by Praeger that the set B = {B g g AGL d (p} s the
More informationOn the set of natural numbers
On the set of natural numbers by Jalton C. Ferrera Copyrght 2001 Jalton da Costa Ferrera Introducton The natural numbers have been understood as fnte numbers, ths wor tres to show that the natural numbers
More informationMAT 578 Functional Analysis
MAT 578 Functonal Analyss John Qugg Fall 2008 Locally convex spaces revsed September 6, 2008 Ths secton establshes the fundamental propertes of locally convex spaces. Acknowledgment: although I wrote these
More informationTHE WEIGHTED WEAK TYPE INEQUALITY FOR THE STRONG MAXIMAL FUNCTION
THE WEIGHTED WEAK TYPE INEQUALITY FO THE STONG MAXIMAL FUNCTION THEMIS MITSIS Abstract. We prove the natural Fefferman-Sten weak type nequalty for the strong maxmal functon n the plane, under the assumpton
More information3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X
Statstcs 1: Probablty Theory II 37 3 EPECTATION OF SEVERAL RANDOM VARIABLES As n Probablty Theory I, the nterest n most stuatons les not on the actual dstrbuton of a random vector, but rather on a number
More informationInner Product. Euclidean Space. Orthonormal Basis. Orthogonal
Inner Product Defnton 1 () A Eucldean space s a fnte-dmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,
More informationSalmon: Lectures on partial differential equations. Consider the general linear, second-order PDE in the form. ,x 2
Salmon: Lectures on partal dfferental equatons 5. Classfcaton of second-order equatons There are general methods for classfyng hgher-order partal dfferental equatons. One s very general (applyng even to
More informationBallot Paths Avoiding Depth Zero Patterns
Ballot Paths Avodng Depth Zero Patterns Henrch Nederhausen and Shaun Sullvan Florda Atlantc Unversty, Boca Raton, Florda nederha@fauedu, ssull21@fauedu 1 Introducton In a paper by Sapounaks, Tasoulas,
More informationTHE SUMMATION NOTATION Ʃ
Sngle Subscrpt otaton THE SUMMATIO OTATIO Ʃ Most of the calculatons we perform n statstcs are repettve operatons on lsts of numbers. For example, we compute the sum of a set of numbers, or the sum of the
More informationLECTURE V. 1. More on the Chinese Remainder Theorem We begin by recalling this theorem, proven in the preceeding lecture.
LECTURE V EDWIN SPARK 1. More on the Chnese Remander Theorem We begn by recallng ths theorem, proven n the preceedng lecture. Theorem 1.1 (Chnese Remander Theorem). Let R be a rng wth deals I 1, I 2,...,
More informationYong Joon Ryang. 1. Introduction Consider the multicommodity transportation problem with convex quadratic cost function. 1 2 (x x0 ) T Q(x x 0 )
Kangweon-Kyungk Math. Jour. 4 1996), No. 1, pp. 7 16 AN ITERATIVE ROW-ACTION METHOD FOR MULTICOMMODITY TRANSPORTATION PROBLEMS Yong Joon Ryang Abstract. The optmzaton problems wth quadratc constrants often
More informationExample: (13320, 22140) =? Solution #1: The divisors of are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 27, 30, 36, 41,
The greatest common dvsor of two ntegers a and b (not both zero) s the largest nteger whch s a common factor of both a and b. We denote ths number by gcd(a, b), or smply (a, b) when there s no confuson
More informationCanonical transformations
Canoncal transformatons November 23, 2014 Recall that we have defned a symplectc transformaton to be any lnear transformaton M A B leavng the symplectc form nvarant, Ω AB M A CM B DΩ CD Coordnate transformatons,
More information9 Characteristic classes
THEODORE VORONOV DIFFERENTIAL GEOMETRY. Sprng 2009 [under constructon] 9 Characterstc classes 9.1 The frst Chern class of a lne bundle Consder a complex vector bundle E B of rank p. We shall construct
More informationOnline Appendix. t=1 (p t w)q t. Then the first order condition shows that
Artcle forthcomng to ; manuscrpt no (Please, provde the manuscrpt number!) 1 Onlne Appendx Appendx E: Proofs Proof of Proposton 1 Frst we derve the equlbrum when the manufacturer does not vertcally ntegrate
More informationDOUBLE POINTS AND THE PROPER TRANSFORM IN SYMPLECTIC GEOMETRY
DOUBLE POINTS AND THE PROPER TRANSFORM IN SYMPLECTIC GEOMETRY JOHN D. MCCARTHY AND JON G. WOLFSON 0. Introducton In hs book, Partal Dfferental Relatons, Gromov ntroduced the symplectc analogue of the complex
More informationON THE EXTENDED HAAGERUP TENSOR PRODUCT IN OPERATOR SPACES. 1. Introduction
ON THE EXTENDED HAAGERUP TENSOR PRODUCT IN OPERATOR SPACES TAKASHI ITOH AND MASARU NAGISA Abstract We descrbe the Haagerup tensor product l h l and the extended Haagerup tensor product l eh l n terms of
More informationSelf-complementing permutations of k-uniform hypergraphs
Dscrete Mathematcs Theoretcal Computer Scence DMTCS vol. 11:1, 2009, 117 124 Self-complementng permutatons of k-unform hypergraphs Artur Szymańsk A. Paweł Wojda Faculty of Appled Mathematcs, AGH Unversty
More informationThe Minimum Universal Cost Flow in an Infeasible Flow Network
Journal of Scences, Islamc Republc of Iran 17(2): 175-180 (2006) Unversty of Tehran, ISSN 1016-1104 http://jscencesutacr The Mnmum Unversal Cost Flow n an Infeasble Flow Network H Saleh Fathabad * M Bagheran
More informationPartitions and compositions over finite fields
Parttons and compostons over fnte felds Muratovć-Rbć Department of Mathematcs Unversty of Saraevo Zmaa od Bosne 33-35, 71000 Saraevo, Bosna and Herzegovna amela@pmf.unsa.ba Qang Wang School of Mathematcs
More informationthe ordinal-least-upper-bound of any set of cardinals is a cardinal; for any set I, and cardinals λ is a cardinal.
11 Regular cardnals In what follows, κ, λ, µ, ν, ρ always denote cardnals. A cardnal κ s sad to be regular f κ s nfnte, and the unon of fewer than κ sets, each of whose cardnalty s less than κ, s of cardnalty
More informationComplete subgraphs in multipartite graphs
Complete subgraphs n multpartte graphs FLORIAN PFENDER Unverstät Rostock, Insttut für Mathematk D-18057 Rostock, Germany Floran.Pfender@un-rostock.de Abstract Turán s Theorem states that every graph G
More informationThe lower and upper bounds on Perron root of nonnegative irreducible matrices
Journal of Computatonal Appled Mathematcs 217 (2008) 259 267 wwwelsevercom/locate/cam The lower upper bounds on Perron root of nonnegatve rreducble matrces Guang-Xn Huang a,, Feng Yn b,keguo a a College
More informationDeterminants Containing Powers of Generalized Fibonacci Numbers
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol 19 (2016), Artcle 1671 Determnants Contanng Powers of Generalzed Fbonacc Numbers Aram Tangboonduangjt and Thotsaporn Thanatpanonda Mahdol Unversty Internatonal
More informationCOMPLEX NUMBERS AND QUADRATIC EQUATIONS
COMPLEX NUMBERS AND QUADRATIC EQUATIONS INTRODUCTION We know that x 0 for all x R e the square of a real number (whether postve, negatve or ero) s non-negatve Hence the equatons x, x, x + 7 0 etc are not
More informationRepresentation theory and quantum mechanics tutorial Representation theory and quantum conservation laws
Representaton theory and quantum mechancs tutoral Representaton theory and quantum conservaton laws Justn Campbell August 1, 2017 1 Generaltes on representaton theory 1.1 Let G GL m (R) be a real algebrac
More informationEvery planar graph is 4-colourable a proof without computer
Peter Dörre Department of Informatcs and Natural Scences Fachhochschule Südwestfalen (Unversty of Appled Scences) Frauenstuhlweg 31, D-58644 Iserlohn, Germany Emal: doerre(at)fh-swf.de Mathematcs Subject
More informationR n α. . The funny symbol indicates DISJOINT union. Define an equivalence relation on this disjoint union by declaring v α R n α, and v β R n β
Readng. Ch. 3 of Lee. Warner. M s an abstract manfold. We have defned the tangent space to M va curves. We are gong to gve two other defntons. All three are used n the subject and one freely swtches back
More informationRandić Energy and Randić Estrada Index of a Graph
EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 5, No., 202, 88-96 ISSN 307-5543 www.ejpam.com SPECIAL ISSUE FOR THE INTERNATIONAL CONFERENCE ON APPLIED ANALYSIS AND ALGEBRA 29 JUNE -02JULY 20, ISTANBUL
More informationwhere a is any ideal of R. Lemma Let R be a ring. Then X = Spec R is a topological space. Moreover the open sets
11. Schemes To defne schemes, just as wth algebrac varetes, the dea s to frst defne what an affne scheme s, and then realse an arbtrary scheme, as somethng whch s locally an affne scheme. The defnton of
More informationSolutions to the 71st William Lowell Putnam Mathematical Competition Saturday, December 4, 2010
Solutons to the 7st Wllam Lowell Putnam Mathematcal Competton Saturday, December 4, 2 Kran Kedlaya and Lenny Ng A The largest such k s n+ 2 n 2. For n even, ths value s acheved by the partton {,n},{2,n
More informationBézier curves. Michael S. Floater. September 10, These notes provide an introduction to Bézier curves. i=0
Bézer curves Mchael S. Floater September 1, 215 These notes provde an ntroducton to Bézer curves. 1 Bernsten polynomals Recall that a real polynomal of a real varable x R, wth degree n, s a functon of
More informationNUMERICAL DIFFERENTIATION
NUMERICAL DIFFERENTIATION 1 Introducton Dfferentaton s a method to compute the rate at whch a dependent output y changes wth respect to the change n the ndependent nput x. Ths rate of change s called the
More informationBasic Regular Expressions. Introduction. Introduction to Computability. Theory. Motivation. Lecture4: Regular Expressions
Introducton to Computablty Theory Lecture: egular Expressons Prof Amos Israel Motvaton If one wants to descrbe a regular language, La, she can use the a DFA, Dor an NFA N, such L ( D = La that that Ths
More informationEconomics 101. Lecture 4 - Equilibrium and Efficiency
Economcs 0 Lecture 4 - Equlbrum and Effcency Intro As dscussed n the prevous lecture, we wll now move from an envronment where we looed at consumers mang decsons n solaton to analyzng economes full of
More informationRandom Partitions of Samples
Random Parttons of Samples Klaus Th. Hess Insttut für Mathematsche Stochastk Technsche Unverstät Dresden Abstract In the present paper we construct a decomposton of a sample nto a fnte number of subsamples
More informationMath 217 Fall 2013 Homework 2 Solutions
Math 17 Fall 013 Homework Solutons Due Thursday Sept. 6, 013 5pm Ths homework conssts of 6 problems of 5 ponts each. The total s 30. You need to fully justfy your answer prove that your functon ndeed has
More information