Based on Hrushovski, Palacín and Pillay’s example ( Selecta Mathematica 19 :4 (2013), 865–877), we produce a new structure without the canonical base property, which is interpretable in Baudisch’s group. Said structure is, in particular, CM-trivial, and thus at the lowest possible level of the ample hierarchy.

Model Theory: Index

59.4 T h e C u t Elimination Theorem for Second Order Logic 171 (respectively, v . ) . In particular, (a) guarantees t h a t *"-PIND or c ~ ' ~ I N D , and C ~ ' ~ C R inferences are still valid after the substitution of V for a. However, P ( V ) may fail t o be a proof in t h a t there may be initial sequents of P ( V ) of the form where V={Z')A(Z'). However, sequents of this form are easy t o prove without free cuts. S o we merely tack onto P ( V ) free cut free proofs of these initial sequents and thus obtain a proof Q of I'(V)+A(V). Q is not necessarily free cu t free, as Q may contain free cu ts with principal formula A(3). B u t since V is a first order abstract, every free cut inference in Q has a first order principal formula. Hence, by (b), there is a free cut free proof of I'(V)--+A(V). Theorem 10: (Cut Elimination Theorem). Let P be a firproof o r a VTproof. Then there is a proof P * in the same theory such tha t P * has the same endsequent as P and there are no free cu ts in P * . Furthermore, each principal formula of an induction inference in P * is a substitution instance of a principal formula of an induction inference in P and each principal abstract of a comprehension inference in P * is a substitution instance of a principal abstract of a comprehension inference in P. Hence, for all i 2 0 , if P is a fii(or vi-) proof then so is P*. Proofi We shall modify Takeuti 's exposition on pages 22-29, 112, 143-144 of [28]. T h e reader should have [28] available as he reads the proof. Following [28], we define the grade of a formula A t o be the number of logical symbols in A . T h e level of A is the number of second order quantifiers in A . A mix inference with principal formula A is of the form where l l* and A * are obtained from rI and A by removing all occurrences of A . A mix inference is free iff all of the occurrences of A in A and II are free. Since a mix inference and a cut inference are so similar, i t suffices t o prove Theorem 10 for proofs which use mix inferences instead of cu t inferences. Suppose P is a proof whose last inference is a mix with principal formula A as shown above. Define the distance of a sequent in P t o be the number of inferences separating i t from the endsequent of P. T h e right rank of P is defined t o be the maximum distance of a sequent containing a direct ancestor of an occurrence of A in the cedent ll. T h e left rank of P is the maximum distance of a sequent containing a direct ancestor of an occurrence of A in the cedent 172 Foundations of Second Order Bounded Arithmetic A . T h e rank of P is the sum of the right rank and left rank. I t suffices t o consider P with a single mix inference as the last inference. T h e proof is by ordinal induction on where level(P) and grade(P) are the level and grade of the principal formula of t he final mix of P, and w is the first infinite ordinal. T h u s it suffices t o show tha t if P is a proof with no free mixes except for the final inference of P and if Theorem 10 is satisfied for all proofs P' with ord(P' )<ord(P) , then P satisfies Theorem 10. W e modify the proof of Lemma 5.4 of Takeut i [28]: Case (1): rank(P)=2. Cases (1 .I)-(1.5. ii): Similar t o pages 2 4 2 7 of [28]. Case (1.5.iii): Suppose A=(V+)B(q5) and the last inferences of P are where V is a ~ d ~ ~ a b s t r a c t , and since rank(P)=f the indicated occurrences of (V+)B(+) are the only ones. By Lemma 9(c), we can obtain a free mix free proof of rA,B( V) from the free mix free proof of I'-+A,B(cr). T h u s we have a proof Q such t h a t the only free mix in Q is i ts last inference: where II# and A# are II and A minus all occurrences of B(V). By the induction hypothesis, there is a free mix free proof Q* of ~,II#-A#,A since ord(Q)<ord(P). By adding weak inferences t o the end of Q* we obtain the desired proof P* . Case (1.5. i~) : Suppose A=(34)B(4). This case is very similar t o Case (1.5.iii). Case (2): rank(P) >2. Case (2.1): T h e right rank of P is >l. Cases (2.1.1)-(2.1.9. ii): Similar t o 1281. T h e C u t Elimination Theorem for Second Order Logic 173 Case (2.1.J.iii): Suppose A=(V4)B(4), V is a ~:p~-abstract and the last inferences of P are: where now A and II, b u t not r , may contain occurrences of (Vc$)B(4). T h e cedents TI* and A* are II and A minus all occurrences of (V+)B(c$). Modify the end of P t o obtain a proof P1 which ends as T h e right rank of P1 is one less than the right rank of P so by the induction hypothesis there is a free mix free proof Pg of the endsequent of P I . Now consider the proof which ends T h e right rank of this is one, so by the induction hypothesis and some exchanges and contractions we obtain a free mix free proof of r , n * + A*,A. T h e rest of the cases are similar. Q.E.D. 9.5. C:yb-Defined Functions and A:yb-Defined Predicates. T h e second order theories of Bounded Arithmetic are in many respects analogous t o the first order theories s~~ and T;. One of the most fundamental properties of second order Bounded Arithmetic is t ha t new function and predicate symbols may be introduced into the language of Bounded Arithmetic; under certain conditions, these new function and predicate symbols may be used freely in the principal formulae of induction axioms and comprehension axioms. 174 Foundations of Second Order Bounded Arithmetic Definition: Let R be a second order theory of Bounded Arithmetic. Suppose A(2,y) is a formula with all free variables indicated and tha t Then we say t h a t R can define the function j such t h a t N t= ('d$)A(Z',j($)). T h e ~ t ) ~ d e f i n e d function symbols and the Atpb-defined predicate symbols play the same role in the second order theories of Bounded Arithmetic as the C:-defined function symbols and the A/-defined predicate symbols d o in the first order theories Si and T;'. In particular, the analogues of Theorems 2.2, 2.3 and 2.4 hold for second order Bounded Arithmetic. Definition: Let 7 and ji be new function and predicate symbols. T h e sets C,'tb(7,ji) and ll,'*b(7,j?) are sets of bounded formulae in the language of second order Bounded Arithmetic plus the symbols f' and ji. These sets are defined by counting alternations of second order quantifiers ignoring the first order, bounded quantifiers. Theorem 11: Let R be a second order theory of Bounded Arithmetic. Suppose R can C;lb-define each of the functions f' and can define each of the predicates ji. Let R * be the theory obtained from R by adjoining the new symbols 7 and j3 and their defining axioms. Then, if i > O and B is a C;'lb(7,j?)(or a Il,'lb@,j?)-) formula, then there is a formula B*Ec;'~~ (or II,'pb, respectively) such t h a t R * t B** B. T h e proof of Theorem 11 is similar t o the proofs of Theorems 2.2 and 2.4. Definition: Let R be a theory of Bounded Arithmetic and let 7 be a vector of defined function symbols of R and j3 be a vector of defined predicate symbols. Then ~(7, j i ) is the conservative extension of R obtained by enlarging the language t o include 7 and 3 and including the defining axioms for these symbols. Corollary 12: ( i 2 1): (a) Let f' be a vector of ~ t * ~ d e f i n e d function symbols of U; (respectively, v;) and let ji be a vector of At*b-defined predicate symbol-. of u;' (respectively, v;' ). Then U;@,j3) (respectively, vfl,j?) ) has as theorems the E : ~ ~ ~ , ~ ~ ) P I N D axioms (respectively, the c,"~(~!,~~)-IND axioms). (b) Let 7 be a vector of C:rb-defined function symbols of fi; (respectively, F;' ) and let ji be a vector of A:.b-defined predicate symbols of 6; (respectively, ) Then fi;'(],j?) (respectively, ?i@,j?)) has as theorems the E:I*(~!,~I)-PIND axioms (respectively, the E,'P~@,~~)-IND axioms). (c) Let f' be a vector of C:lb-defined function symbols of 8; (respectively, 8;:) and let j? be a vector of A:rb-defined predicate symbols of 8; (respectively, ) Then 8i(],jI) s9.5 zipb-Defined Functions and airb-Defined Predicates 175 (respectively, ) has as theorems the E:V~-PIND v , a ) axioms (respectively, the c~'~~-IND (7,jJ) axioms) and the i :9b(7,$)-~~ axioms. Corollary 12 tells us t h a t C:~~-defined function symbols and a:1b-defined predicate symbols may be used freely in the principal formulae of induction inferences. Furthermore, if we are working in the theory 82 or 82 we may use such function and predicate symbols freely in principal abstracts of comprehension inferences. T h e next two theorems give an application of ~ : ~ ~ c o m ~ r e h e n s i o n t show t h a t 82 and 8: can define the iteration of defined predicates. I t is an open question whether these theorems hold for the theories U; and v;. Theorem 18: Let ~(a,i?,?) and ~ ( a , b,t,a:,q) be formulae of 8:, where a: is a unary predicate variable. Let t (b , t ) be a term which contains only the free variables b and t. Then the predicate ~ ( a , b,i?,;) which satisfies if b=O and a < t ( b , t ) ~ ( a , b , t , < ) if a > t ( b , t ) B ( ~ , b , t , {z}K(z, [ ;b ~, t , ; j ) ,3 otherwise Proofi T h e idea, of course, is t o define K ( a , b , t , 3 by induction on the length of b . Let B*(a, b , t , c u ~ , ~ ) be the formula and let D(u ,4) be the formula I t is easy t o see t h a t

Bounded Arithmetic と計算量の根本問題

We give an unconditional proof of the Andre-Oort conjecture for arbitrary products of modular curves. We establish two generalizations. The first includes the Manin-Mumford conjecture for arbitrary products of elliptic curves defined over Q as well as Lang’s conjecture for torsion points in powers of the multiplicative group. The second includes the Manin-Mumford conjecture for abelian varieties defined over Q. Our approach uses the theory of o-minimal structures, a part of Model Theory, and follows a strategy proposed by Zannier and implemented in three recent papers: a new proof of the Manin-Mumford conjecture by Pila-Zannier; a proof of a special (but new) case of Pink’s relative Manin-Mumford conjecture by Masser-Zannier; and new proofs of certain known results of Andre-Oort–Manin-Mumford type by Pila. 2010 Mathematics Subject Classification: 11G18, 03C64

/pdf/o-minimality-and-the-andre-oort-conjecture-for-mathbb-c-n-12audtvy2y.pdf

O-minimality and the André-Oort conjecture for $\mathbb{C}^{n}$

We develop a method for computing a nearly singular integral, such as a double layer potential due to sources on a curve in the plane, evaluated at a point near the curve. The approach is to regularize the singularity and obtain a preliminary value from a standard quadrature rule. Then we add corrections for the errors due to smoothing and discretization, which are found by asymptotic analysis. We prove an error estimate for the corrected value, uniform with respect to the point of evaluation. One application is a simple method for solving the Dirichlet problem for Laplace's equation on a grid covering an irregular region in the plane, similar to an earlier method of A. Mayo [SIAM J. Sci. Statist. Comput., 6 (1985), pp. 144--157]. This approach could also be used to compute the pressure gradient due to a force on a moving boundary in an incompressible fluid. Computational examples are given for the double layer potential and for the Dirichlet problem.

A Method for Computing Nearly Singular Integrals

Expansions of the real field with power functions

We prove that no restriction of the sine function to any (open and nonempty) interval is definable in (R, +, ., <, exp, constants), and that no restriction of the exponential function to an (open andnonempty) intervalisdefinablein (R, <, <sin0, constants), wheresino(x) = sin(x) forx E [-7t, 7r], and sino (x) = O for all x f [ , 7T]. ?

Nondefinability Results for Expansions of the Field of Real Numbers by the Exponential Function and by the Restricted Sine Function

We establish conditions for the existence of solutions of the linear integral equation of Volterra \begin{equation} x\left( t\right) +^{\ast }\int\nolimits_{[ a,t] }\alpha ( s) x ( s )\, ds=f ( t ) ,\quad t\in [ a,b ] ,\tag{$V_{\ast}$} \end{equation} where the functions are Banach space-valued and $^{\ast }\int $ denotes either the Bochner-Lebesgue or the Henstock integral In some cases it is possible to calculate the solution of $( V)_{\ast }$ explicitly We give several examples

/pdf/linear-integral-equations-of-volterra-concerning-henstock-59efmcrbz1.pdf

Linear Integral Equations of Volterra Concerning Henstock Integrals

We apply the Kurzweil-Henstock integral setting to prove a Fredholm Alternative-type result for the integral equation Z (a;b) (t;s)x (s)ds = f (t); t2 (a;b); where x and f are Kurzweil integrable functions (possibly highly oscil- lating) dened on a compact interval ( a;b) of the real line with values on Banach spaces. An application is given.

/pdf/linear-fredholm-integral-equations-and-the-integral-of-55hf75wmb8.pdf

Linear fredholm integral equations and the integral of kurzweil

Undefinability results in o-minimal expansions of the real numbers

We consider the multidimensional abstract linear integral equation of Volterra type

 1
 $$
x{\left( t \right)} + {\left( * \right)}{\int_{R_{t} } {\alpha {\left( s \right)}x{\left( s \right)}ds = f{\left( t \right)},{\kern 1pt} t \in R} }
$$
 , as the limit of discrete Stieltjes-type systems and we prove results on the existence of continuous solutions. The functions x, α and f are Banach space-valued defined on a compact interval R of $$\mathbb{R}^{\text n}$$
 , R
 t
 is a subinterval of R depending on t ∈ R and (⋆) ∫ denotes either the Bochner-Lebesgue integral or the Henstock integral. The results presented here generalize those in [1] and are in the spirit of [3]. As a consequence of our approach, it is possible to study the properties of (1) by transferring the properties of the discrete systems. The Henstock integral setting enables us to consider highly oscillating functions.

/pdf/linear-volterra-integral-equations-as-the-limit-of-discrete-1y57lpnfqn.pdf

Linear Volterra Integral Equations as the Limit of Discrete Systems

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