R. Hindley. The equivalence of complete reductions. Transactions of the American Mathematical Society, 229:227--248, 1977.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of the lower bounds # # t , p # t to p # as t goes to infinity [11] In the special case when F is contained in or is equal to the set of points in a grid, Lasserre [12, 13] shows the finite convergence of the bounds # # t , p # t to p # . His proof is based on a result by Curto and Fialkow [5, 6] about flat extensions of moment matrices (which uses the Riesz representation theorem) see Corollary 13) As a consequence, every polynomial nonnegative on F has a decomposition (6) with a priori bounds on the degrees of the polynomials u j . When the ideal generated by the polynomials h 1 , ....

R.E. Curto and L.A. Fialkow. The truncated complex K-moment problem. Transactions of the American Mathematical Society, 352:2825--2855, 2000.

....then src(#) src(#) and tgt(#) tgt(#) and similarly for reductions (i.e. modulo the reduction identities) Remark 4.12 (i) Structural equivalence is studied in an abstract axiomatic 16 setting in [30] under the name of square permutation equivalence. ii) The results of the early days ([9,15,21]) mainly concern ordinary standardisation, i.e. sorting steps in textual left to right order, the reason being that both the # calculus, the prototypical higher order term rewrite system, and combinatory logic, the prototypical first order term rewrite system, are left normal orthogonal systems. ....

J.R. Hindley. Standard and normal reductions. Transactions of the American Mathematical Society, 241:253--271, July 1978.

....but to 5 provide a sequence of successively sharper relaxations covering the entire complexity spectrum, thus allowing the complexity sharpness tradeo# to be tuned. This is what we intend to describe in this section. We begin by recalling the construction of moment matrices as detailed in [CF00], Las01] and [Las02] Again, we let y x = 1, x 1 , x n , x n ) be the vector of all monomials in R[x 1 , x n ] up to degree m, listed in increasing graded lexicographic order. We note s(m) the size of the vector y x . Let y be the vector of moments (indexed according to y ....

R. Curto and A. Fialkow, The truncated complex K-moment problem, Transactions of the American mathematical society 352 (2000), no. 6, 2825--2855.

....receivers can significantly improve the performance. For the network capacity derivation, we use results from random graph theory [4] A random graph is characterized by the number N of nodes and the probability p of maintaining a link between two arbitrary nodes . Using the results proved in [3] we can obtain a constraint on the probability of maintaining a link such that the graph s diameter is D with high probability as the number of nodes goes to infinity. As these results are asymptotic in nature we also validate them through simulations for finite values of N . The graph diameter ....

....constraints are met. We characterize the ad hoc network asymptotic capacity for the case for which the number of nodes and the spreading gain go to infinity, while their ratio is fixed. 3 Asymptotic Capacity To characterize the ad hoc network capacity we rely on results from random graph theory [3], which determine the network diameter for a given link probability value p, when the number of nodes goes to infinity. We summarize the main results needed (from [3] in the following theorem: Theorem 1 [3] Let c be a positive constant, D = D(N) 2 a natural number, and define p = p(N, c, D) ....

[Article contains additional citation context not shown here]

B. Bollobas. The diameter of random graphs. Transactions of the American Mathematical Society, 267(1):41--52, September 1981.

....of anticommuting variables the simple superspace R must be replaced by a more general supermanifold. In this article it will be 10 sucient to consider supermanifolds constructed in a standard way from the data of a smooth vector bundle E over a smooth manifold M . A theorem of Batchelor [15] shows that all smooth supermanifolds may be obtained in this way. The idea of the construction is to patch together local pieces of the supermanifold using the change of coordinate functions of the manifold and the transition functions of the bundle. A careful de nition would need an excursion ....

M. Batchelor. The structure of supermanifolds. Transactions of the American Mathematical Society, 253:329-338, 1979.

....as G.m, p and I(l , n,p respectively. We establish some results on the likely structure of a random graph GN,p. For d 1 let = d) be defined by lande =de a , 11) and let g = g(d) i C(d) 1 g e dg = 0. 12) 4.3. 1 Component Structure The following lemma can be obtained from [8] [4] and [16] Lemma 7 Let p = din where 1 e = d= O(1) and lellogn With probability 1 O(N 1) the random graph G N, has the following component struc ture: For ko = Ae 2 log N where A is a sufficiently large constant, If d 1 then all components are of size at most ko. 10 If d i then ....

B.Bollobs, The evolution of random graphs, Transactions of the American Mathematical Society 286 (1984) 257-274.

....proof for this inclusion. The construction of Lasserre is motivated by results about representations of nonnegative polynomials as sums of squares and the dual theory of moments and his proof that the 0 1 polytope P is found after n steps relies on a nontrivial result of Curto and Fialkow [CF00] about truncated moment sequences. In fact, the Sherali Adams series of relaxations can also be formulated within this framework of moment matrices. The fact of formulating both Lasserre and Sherali Adams constructions in a common setting permits a better understanding of how they relate; both ....

....h i (x) 0 (i = 1, n) 47) setting h i (x) x i i for i = 1, n. Then, one can assume without loss of generality that each g # has degree at most 1 in every variable and the assumptions from Theorem 15 hold (with u(x) i=1 h i (x) Using a result of Curto and Fialkow [CF00] about rank extensions of moment matrices, Lasserre [Las01b] shows finite convergence of the successive relaxations t (F ) to conv(F ) namely, n v 1 (F ) conv(F ) 48) The set 0 (# = 1, m) 49) is a natural relaxation of F . As we see in Proposition 16 below, the ....

R.E. Curto and L.A. Fialkow. The truncated complex K-moment problem. Transactions of the American Mathematical Society, 352:2825--2855, 2000.

....constraints, while rule (10) is used to delete frames with inconsistent integer constraints. Hereby, the side condition j= of rule (10) is decidable. This can be shown by a simple reduction to the Diophantine problem for addition and divisibility that was proven decidable, for example, in [Lip78, Lip81]. 1 Altogether, S yields f( g for tautologies, whereas unsatis able formulae are eventually reduced to fg. Notice also that, in contrast to the rule system C in Figure 2, the frame transformation system S may introduce fresh variables a and b. Theorem 2 states a correctness result for ....

....can be reduced to solve non xed size equations on bit vectors built up from concatenation, extraction, and bitwise Boolean operators only [M ol98] Acknowledgements. We would like to thank Nikolaj Bj rner and David Cyrluk for their invaluable suggestions. In particular, Nikolaj pointed us to [Lip78]. Furthermore we thank the anonymous referees and Holger Pfeifer for their comments which helped to improve the presentation. ....

L. Lipshitz. The Diophantine Problem for Addition and Divisibility. Transactions of the American Mathematical Society, 235:271-283, January 1978.

....proof for this inclusion. The construction of Lasserre is motivated by results about representations of nonnegative polynomials as sums of squares and the dual theory of moments and his proof that the 0 Gamma 1 polytope P is found after n steps relies on a nontrivial result of Curto and Fialkow [CF00] about truncated moment sequences. In fact, the Sherali Adams series of relaxations can also be formulated within this framework of moment matrices. The fact of formulating both Lasserre and Sherali Adams constructions in a common setting permits a better understanding of how they relate; both ....

....0 (i = 1; n)g (47) setting h i (x) x i Gamma x 2 i for i = 1; n. Then, one can assume without loss of generality that each g has degree at most 1 in every variable and the assumptions from Theorem 15 hold (with u(x) P n i=1 h i (x) Using a result of Curto and Fialkow [CF00] about rank extensions of moment matrices, Lasserre [Las01b] shows finite convergence of the successive relaxations Q t (F ) to conv(F ) namely, Q n v Gamma1 (F ) conv(F ) 48) The set K : fx 2 [0; 1] n j g (x) 0 ( 1; m)g (49) is a natural relaxation of F . As we see in ....

R.E. Curto and L.A. Fialkow. The truncated complex K-moment problem. Transactions of the American Mathematical Society, 352:2825--2855, 2000.

.... topological and combinatorial features (cf. HS98] and [To97] In the late 1960 s, a partial ordering on the non principal ultrafilters fiN n N , the so called Rudin Keisler ordering, was established and small points with respect to this ordering were investigated rigorously (cf. Bo70] Bl73] Bl81 1 ] and [La89] The minimal points have a nice combinatorial characterization which is related to Ramsey s Theorem (cf. Ra29, Theorem A] and so, the ultrafilters which are minimal with respect to the Rudin Keisler ordering are also called Ramsey ultrafilters (for further ....

....g Gamma1 (X) By the definition of h we have h Gamma1 (Y ) g(Y ) and sinceU = g(V ) there is a Z 2U such that g(Y ) Z, which implies h(Z) Y , hence, h(U ) V . a The following proposition shows that . is directed upward (for a similar result concerning the Rudin Keisler ordering see [Bl73, p. 147] Fact 1.2.3 For any partition filters D ; E 2 PF Gamma ( Delta , there is a partition filter F 2 PF Gamma ( Delta , such that D .F and E .F . Proof: Let ae 1 and ae 2 be two functions from into defined by ae 1 (n) 2n and ae 2 (n) 2n 1. For a partition X ....

[Article contains additional citation context not shown here]

Andreas Blass: The Rudin-Keisler ordering of p-points. Transactions of the American Mathematical Society 179 (1973), 145--166.

....linkage. This notion was formalized by the third author with the idea that perhaps an arc could be straightened via an expansive motion. The tools that are applied here for the rst time come from the theory of mechanisms and rigid frameworks. Arcs and cycles can be regarded as frameworks. See [AR78, AR79, Con80, Con82, Con93, CW96, CW93, CW82, CW94, GSS93, RW81, Whi84a, Whi84b, Whi87, Whi88, Whi92a] for relevant information about this theory. Our approach is to prove that for any con guration there is an in nitesimal motion that increases all distances. Because of the nature of the arc and cycle set, this implies that there is a motion that works at least for a small expansive perturbation. ....

L. Asimow and B. Roth. The rigidity of graphs. Transactions of the American Mathematical Society, 245:279-289, 1978.

....linkage. This notion was formalized by the third author with the idea that perhaps an arc could be straightened via an expansive motion. The tools that are applied here for the rst time come from the theory of mechanisms and rigid frameworks. Arcs and cycles can be regarded as frameworks. See [AR78, AR79, Con80, Con82, Con93, CW96, CW93, CW82, CW94, GSS93, RW81, Whi84a, Whi84b, Whi87, Whi88, Whi92a] for relevant information about this theory. Our approach is to prove that for any con guration there is an in nitesimal motion that increases all distances. Because of the nature of the ....

L. Asimow and B. Roth. The rigidity of graphs. Transactions of the American Mathematical Society, 245:279-289, 1978.

....all clauses in C. 2 3 Language S 1 In this section we prove the decidability of simultaneous rigid E unification in the language S 1 . The proof is by reduction to the Diophantine problem for addition and divisibility that was proved decidable by Bel tyukov [4] Mart yanov [29] and Lipshitz [25]. The Diophantine problem for addition and divisibility is equivalent to the decidability of the class of formulas of the form 9x 1 : 9xn k i=1 A i (1) in natural numbers, such that A i have the form xm = x j x k , xm j x j or xm = p, where j is the divisibility predicate and p is a ....

L. Lipshitz. The Diophantine problem for addition and divisibility. Transactions of the American Mathematical Society, 235:271--283, Jan. 1978.

....constraints, while rule (10) is used to delete frames with inconsistent integer constraints. Hereby, the side condition Psi j= of rule (10) is decidable. This can be shown by a simple reduction to the Diophantine problem for addition and divisibility that was proven decidable, for example, in [Lip78,Lip81]. 1 Altogether, S yields f( g for tautologies, whereas unsatisfiable formulae are eventually reduced to fg. Notice also that, in contrast to the rule system C in Figure 2, the frame transformation system S may introduce fresh variables a and b. 1 The Diophantine problem for addition and ....

....can be reduced to solve non fixed size equations on bit vectors built up from concatenation, extraction, and bitwise Boolean operators only [Mol98] Acknowledgements. We would like to thank Nikolaj Bjrner and David Cyrluk for their invaluable suggestions. In particular, Nikolaj pointed us to [Lip78]. Furthermore we thank the anonymous referees and Holger Pfeifer for their comments which helped to improve the presentation. ....

L. Lipshitz. The Diophantine Problem for Addition and Divisibility. Transactions of the American Mathematical Society, 235:271--283, January 1978.

....In contrast, in Finite Family Developments in Section 4 distinct residuals can be contracted by distinct rules. Finiteness of developments (and strengthened versions of it) have been studied extensively in the literature for various classses of rewriting systems (see e.g. CR36, Sch65, Hin78, Klo80, Kha92, Raa, Oos94, Melon] In Subsection 3.1 a simple proof of FD is presented for the class of PRSs. In Subsection 3.2 upperbound information is added to the termination proof of Subsection 3.1, yielding (exact) upperbounds on the lengths of marked rewrite sequences. As a consequence ....

R. Hindley. Reductions of residuals are finite. Transactions of the American Mathematical Society, 240:345--361, June 1978.

.... type schemas in combinatory logic [Curry 58] Extending Curry s work, and collaborating with him, Hindley introduced the idea of a principal type schema, which is the most general polymorphic type of an expression, and showed that if a combinatorial term has a type, then it has a principal type [Hindley 69] In doing so, he used a result by Robinson about the existence of most general unifiers in the unification algorithm [Robinson 64] These results contained all the germs of polymorphic typechecking, including the basic algorithms. The existence of principal types means that a type inference ....

R.Hindley: The principal type scheme of an object in combinatory logic, Transactions of the American Mathematical Society, Vol. 146, Dec 1969, pp. 29-60.

....of the earlier one Likewise (dually) in case 3. As for the elementary diagrams, the consequence of this definition is the appearance of two new e.d. s as in Figure 14, corresponding to the two ways the patterns of a fi redex and j redex may overlap. Actually, these e.d. s also show 6 In Hindley [Hin77] the overlapping redexes R; S, in any of the situations 1 4 are suggestively called too close together . Descendants and Origins in Term Rewriting 17 (x:Ax)B fi j AB ; x: y:A(y) x fi j x:A(x) AB ; AB y:A(y) y:A(y) j ff x:A(x) Figure 14: New elementary diagrams that the ....

R. Hindley. The equivalence of complete reductions. Transactions of the American Mathematical Society, 229:227--248, 1977.

....the following is an incomplete list. The theorem was first proved by Church and Rosser [12, 13] for I ; they also sketch a proof for K . 6 Curry and Feys [15] and Schroer [61] give full proofs of the theorem for K . Other proofs were later given independently by Hyland [27] and Hindley [21]. Barendregt et al. 4] subsequently simplified Hyland s proof see also [2] Xi [82] gives a proof similar to the above using instead of the fundamental lemma of perpetuality for developments his characterization of SN fi see Remark 6.8. Van Oostrom [50, 51] shows that Lemma 6.9 can be ....

J.R. Hindley. Reductions of residuals are finite. Transactions of the American Mathematical Society, 240:345--361, 1978.

....axiomatized by (fi) j) and the equational theory of this algebra. An axiomatization of this theory is shown in Table 3. The reader may recognize this as the equational theory of a commutative ring with unit. Henkin showed that this is a complete axiomatization of validities in this algebra [9]. This is a decidable theory because it is a primitive recursive set of equations. Oriented left to right, any primitive recursive set of equations constitutes a Church Rosser and strong normalizing rewrite system. Such a set of equations gives rise to a decidable theory. Invoking Theorem 7.1, it ....

L. Henkin. The logic of equality. Transactions of the American Mathematical Society, 84:597--612, October 1977.

....(as far as we know) historical order we have: ffl TRS = Term Rewriting System. We don t know who introduced this name, but they were known at the end of the seventies. cf. also Rosen [Ros73] ffl CS = Contraction Scheme. Introduced by Aczel [Acz78] ffl (a) reductions were introduced by Hindley [Hin78]. ffl CRS = Combinatory Reduction System. Introduced by Klop [Klo80] ffl HOTRS = Higher Order Term Rewriting System. Introduced by Wolfram in his PhD thesis, see [Wol93] ffl ERS = Expression Reduction System. Introduced by Khasidashvili [Kha90] ffl (I)IN = Intuitionistic) Interaction Net. ....

R. Hindley. Reductions of residuals are finite. Transactions of the American Mathematical Society, 240:345--361, June 1978.

No context found.

R. Curto and A. Fialkow, The truncated complex K-moment problem, Transactions of the American mathematical society 352 (2000), no. 6, 2825--2855.

....equalization. This paper has been presented in part at the IEEE International Conference on Communications (ICC) Helsinki, Finland, June 2001. 1 Introduction The statistical properties of the zeros of random polynomials have been studied extensively in the mathematical literature (cf. e.g. [2, 1, 3, 4, 5, 6, 7, 8]) The results of this research have found rich application in biology [1] and di erent branches of physics, e.g. 4, 9, 10] however, they seem to be almost unknown in the communications and signal processing literature. Here, the only relevant reference appears to be a paper by the authors ....

L.A. Shepp and R.J. Vanderbei. The complex roots of random polynomials. Transactions of the American Mathematical Society, 347:4365-4384, 1995.

No context found.

R. Hindley. The equivalence of complete reductions. Transactions of the American Mathematical Society, 229:227--248, 1977.

No context found.

R.E. Curto and L.A. Fialkow. The truncated complex K-moment problem. Transactions of the American Mathematical Society, 352:2825--2855, 2000.

No context found.

, The theory of representations for Boolean algebras, Transactions of the American Mathematical Society 40 (1936), 37--111.

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