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1.
The main goal of the article is to show that Paley-Wiener functions ƒ ∈ L
2(M) of a fixed band width to on a Riemannian manifold of bounded geometry M completely determined and can be reconstructed
from a set of numbers Φi (ƒ), i ∈ ℕwhere Φi
is a countable sequence of weighted integrals over a collection of “small” and “densely” distributed compact subsets. In particular, Φi, i ∈ ℕ,can be a sequence of weighted Dirac measures δxi, xi ∈M.
It is shown that Paley-Wiener functions on M can be reconstructed as uniform limits of certain variational average spline
functions.
To obtain these results we establish certain inequalities which are generalizations of the Poincaré-Wirtingen and Plancherel-Polya
inequalities.
Our approach to the problem and most of our results are new even in the one-dimensional case. 相似文献
2.
Alex M. McAllister 《Mathematical Logic Quarterly》2002,48(2):245-259
We examine the relationship between two different notions of a structure being Scott set saturated and identify sufficient conditions which guarantee that a structure is uniquely Scott set saturated. We also consider theories representing Scott sets; in particular, we identify a sufficient condition on a theory T so that for any given countable Scott set there exists a completion of T that is saturated with respect to the given Scott set. These results extend Scott's characterization of countable Scott sets via models and completions of Peano arithmetic. 相似文献
3.
M. Tetruashvili 《Georgian Mathematical Journal》1994,1(5):561-565
The unquantified set theory MLSR containing the symbols ∪, ∖, ≠, ∈,R (R(x) is interpreted as a rank ofx) is considered. It is proved that there exists an algorithm which for any formulaQ of the MLSR theory decides whetherQ is true or not using the spacec|Q|3 (|Q| is the length ofQ). 相似文献
4.
We study the filter ℒ*(A) of computably enumerable supersets (modulo finite sets) of an r-maximal set A and show that, for some such set A, the property of being cofinite in ℒ*(A) is still Σ0
3-complete. This implies that for this A, there is no uniformly computably enumerable “tower” of sets exhausting exactly the coinfinite sets in ℒ*(A).
Received: 6 November 1999 / Revised version: 10 March 2000 /?Published online: 18 May 2001 相似文献
5.
6.
Larry Michael Manevitz 《Israel Journal of Mathematics》1976,25(3-4):211-232
Robinson (or infinite model theoretic) forcing is studied in the context of set theory. The major result is that infinite
forcing, genericity, and related notions are not absolute relative to ZFC. This answers a question of G. Sacks and provides
a non-trivial example of a non-absolute notion of model theory. This non-absoluteness phenomenon is shown to be intrinsic
to the concept of infinite forcing in the sense that any ZFC-definable set theory, relative to which forcing is absolute,
has the flavor of asserting self-inconsistency. More precisely: IfT is a ZFC-definable set theory such that the existence of a standard model ofT is consistent withT, then forcing is not absolute relative toT. For example, if it is consistent that ZFC+ “there is a measureable cardinal” has a standard model then forcing is not absolute
relative to ZFC+ “there is a measureable cardinal.” Some consequences: 1) The resultants for infinite forcing may not be chosen
“effectively” in general. This answers a question of A. Robinson. 2) If ZFC is consistent then it is consistent that the class
of constructible division rings is disjoint from the class of generic division rings. 3) If ZFC is consistent then the generics
may not be axiomatized by a single sentence ofL
w/w.
In Memoriam: Abraham Robinson 相似文献
7.
Alessio Moretti 《Logica Universalis》2009,3(1):19-57
Whereas geometrical oppositions (logical squares and hexagons) have been so far investigated in many fields of modal logic
(both abstract and applied), the oppositional geometrical side of “deontic logic” (the logic of “obligatory”, “forbidden”,
“permitted”, . . .) has rather been neglected. Besides the classical “deontic square” (the deontic counterpart of Aristotle’s
“logical square”), some interesting attempts have nevertheless been made to deepen the geometrical investigation of the deontic
oppositions: Kalinowski (La logique des normes, PUF, Paris, 1972) has proposed a “deontic hexagon” as being the geometrical
representation of standard deontic logic, whereas Joerden (jointly with Hruschka, in Archiv für Rechtsund Sozialphilosophie
73:1, 1987), McNamara (Mind 105:419, 1996) and Wessels (Die gute Samariterin. Zur Struktur der Supererogation, Walter de Gruyter,
Berlin, 2002) have proposed some new “deontic polygons” for dealing with conservative extensions of standard deontic logic
internalising the concept of “supererogation”. Since 2004 a new formal science of the geometrical oppositions inside logic
has appeared, that is “n-opposition theory”, or “NOT”, which relies on the notion of “logical bi-simplex of dimension m” (m = n − 1). This theory has received a complete mathematical foundation in 2008, and since then several extensions. In this paper,
by using it, we show that in standard deontic logic there are in fact many more oppositional deontic figures than Kalinowski’s
unique “hexagon of norms” (more ones, and more complex ones, geometrically speaking: “deontic squares”, “deontic hexagons”,
“deontic cubes”, . . ., “deontic tetraicosahedra”, . . .): the real geometry of the oppositions between deontic modalities
is composed by the aforementioned structures (squares, hexagons, cubes, . . ., tetraicosahedra and hyper-tetraicosahedra),
whose complete mathematical closure happens in fact to be a “deontic 5-dimensional hyper-tetraicosahedron” (an oppositional
very regular solid).
相似文献
8.
A maximal antichain A of poset P splits if and only if there is a set B ⊂ A such that for each p ∈ P either b ≤ p for some b ∈ B or p ≤ c for some c ∈ A\B. The poset P is cut-free if and only if there are no x < y < z in P such that [x,z]P = [x,y]P ∪ [y,z]P . By [1] every maximal antichain in a finite cut-free poset splits. Although this statement for infinite posets fails (see
[2])) we prove here that if a maximal antichain in a cut-free poset “resembles” to a finite set then it splits. We also show
that a version of this theorem is just equivalent to Axiom of Choice.
We also investigate possible strengthening of the statements that “A does not split” and we could find a maximal strengthening.
* This work was supported, in part, by Hungarian NSF, under contract Nos. T37846, T34702, T37758, AT 048 826, NK 62321. The
second author was also supported by Bolyai Grant. 相似文献
9.
“Setting” n-Opposition 总被引:1,自引:1,他引:0
Régis Pellissier 《Logica Universalis》2008,2(2):235-263
Our aim is to show that translating the modal graphs of Moretti’s “n-opposition theory” (2004) into set theory by a suited device, through identifying logical modal formulas with appropriate
subsets of a characteristic set, one can, in a constructive and exhaustive way, by means of a simple recurring combinatory,
exhibit all so-called “logical bi-simplexes of dimension n” (or n-oppositional figures, that is the logical squares, logical hexagons, logical cubes, etc.) contained in the logic produced
by any given modal graph (an exhaustiveness which was not possible before). In this paper we shall handle explicitly the classical
case of the so-called 3(3)-modal graph (which is, among others, the one of S5), getting to a very elegant tetraicosahedronal
geometrisation of this logic.
相似文献
10.
Benedikt Löwe 《Archive for Mathematical Logic》2001,40(8):651-664
We investigate Turing cones as sets of reals, and look at the relationship between Turing cones, measures, Baire category
and special sets of reals, using these methods to show that Martin's proof of Turing Determinacy (every determined Turing
closed set contains a Turing cone or is disjoint from one) does not work when you replace “determined” with “Blackwell determined”.
This answers a question of Tony Martin.
Received: 6 December 1999 / Revised version: 28 June 2000 Published online: 3 October 2001 相似文献
11.
Ilya A. Krishtal Benjamin D. Robinson Guido L. Weiss Edward N. Wilson 《Journal of Geometric Analysis》2007,17(1):87-96
An orthonormal wavelet system in ℝd, d ∈ ℕ, is a countable collection of functions {ψ
j,k
ℓ
}, j ∈ ℤ, k ∈ ℤd, ℓ = 1,..., L, of the form
that is an orthonormal basis for L2 (ℝd), where a ∈ GLd (ℝ) is an expanding matrix. The first such system to be discovered (almost 100 years ago) is the Haar system for which L
= d = 1, ψ1(x) = ψ(x) = κ[0,1/2)(x) − κ[l/2,1)
(x), a = 2. It is a natural problem to extend these systems to higher dimensions. A simple solution is found by taking appropriate
products Φ(x1, x2, ..., xd) = φ1 (x1)φ2(x2) ... φd(xd) of functions of one variable. The obtained wavelet system is not always convenient for applications. It is desirable to
find “nonseparable” examples. One encounters certain difficulties, however, when one tries to construct such MRA wavelet systems.
For example, if a = (
1-1
1 1
) is the quincunx dilation matrix, it is well-known (see, e.g., [5]) that one can construct nonseparable Haar-type scaling
functions which are characteristic functions of rather complicated fractal-like compact sets. In this work we shall construct
considerably simpler Haar-type wavelets if we use the ideas arising from “composite dilation” wavelets. These were developed
in [7] and involve dilations by matrices that are products of the form ajb, j ∈ ℤ, where a ∈ GLd(ℝ) has some “expanding” property and b belongs to a group of matrices in GLd(ℝ) having |det b| = 1. 相似文献
12.
Dorothy Maharam 《Israel Journal of Mathematics》1997,98(1):15-28
Let (X, A) be a set with a countably σ-generated “Borel” field of subsets; letW be a “Borel” subset of the product of (X, A) with the real line ℝ and its Borel fieldB; and for eachx∈X let γ
x
be a measure on the “slice”W
x={(w, t)∈W:w=x}. It is shown that, under reasonable conditions, the σ-field A⊗B|W can be generated by a real-valued functiong in such a way that, given any measurablef:W→ℝ,g can be chosen to be arbitrarily close tof and so that its “slice-integrals”
coincide with those off. This theorem is the first step in a study of monotonic sequences of countably generated σ-fields. 相似文献
13.
14.
All induced connected subgraphs of a graphG contain a dominating set of pair-wise adjacent vertices if and only if there is no induced subgraph isomorphic to a path
or a cycle of five vertices inG. Moreover, the problem of finding any given type of connected dominating sets in all members of a classG of graphs can be reduced to the graphsG∈G that have a cut-vertex or do not contain any cutsetS dominated by somes∈S.
This research was supported in part by the “AKA” Research Fund of the Hungarian Academy of Sciences. 相似文献
15.
This paper gives simple proofs for “G
k
∈? implies G
k
+1∈?” when ? is the family of all interval graphs, all proper interval graphs, all cocomparability graphs, or all m-trapezoid graphs.
Received: November 21, 1997 Final version received: October 5, 1998 相似文献
16.
In a graphG, which has a loop at every vertex, a connected subgraphH=(V(H),E(H)) is a retract if, for anya, b ∈V(H) and for any pathsP, Q inG, both joininga tob, and satisfying |Q|≧ ≧|P|, thenP ⫅V(H) wheneverQ ⫅V(H). As such subgraphs can be described by a closure operator we are led to the investigation of the corresponding complete
lattice of “closed” subgraphs. For example, in this complete lattice every element is the infimum of an irredundant family
of infimum irreducible elements.
The work presented here was supported in part by N.S.E.R.C. Operating Grant No. A4077. 相似文献
17.
An arrangement of oriented pseudohyperplanes in affined-space defines on its setX of pseudohyperplanes a set system (or range space) (X, ℛ), ℛ ⊑ 2
x
of VC-dimensiond in a natural way: to every cellc in the arrangement assign the subset of pseudohyperplanes havingc on their positive side, and let ℛ be the collection of all these subsets. We investigate and characterize the range spaces
corresponding tosimple arrangements of pseudohyperplanes in this way; such range spaces are calledpseudogeometric, and they have the property that the cardinality of ℛ is maximum for the given VC-dimension. In general, such range spaces
are calledmaximum, and we show that the number of rangesR∈ℛ for whichX - R∈ℛ also, determines whether a maximum range space is pseudogeometric. Two other characterizations go via a simple duality concept
and “small” subspaces. The correspondence to arrangements is obtained indirectly via a new characterization of uniforom oriented
matroids: a range space (X, ℛ) naturally corresponds to a uniform oriented matroid of rank |X|—d if and only if its VC-dimension isd,R∈ℛ impliesX - R∈ℛ, and |ℛ| is maximum under these conditions.
Part of this work was done while the first author was a member of the Graduiertenkolleg “Algorithmische Diskrete Mathematik,”
supported by the Deutsche Forschungsgemeinschaft, Grant We 1265/2-1. Part of this work has been supported by the German-Israeli
Foundation for Scientific Research and Development (G.I.F.). 相似文献
18.
Anthony W. Hager Chawne M. Kimber Warren Wm. McGovern 《Rendiconti del Circolo Matematico di Palermo》2003,52(3):453-480
An archimedean lattice-ordered groupA with distinguished weak unit has the canonical Yosida representation as an ℓ-group of extended real-valued functions on a
certain compact Hausdorff spaceY A. Such an ℓ-groupA is calledleast integer closed, orLIC (resp.,weakly least integer closed, orwLIC) if, in the representation,a ∈A implies [a] ∈A (resp., there isa′ ∈A witha′=[a] on a dense set inY A), where [r] ≡ the least integer greater than or equal tor. Earlier, we have studiedLIC groups, with an emphasis on their a-extensions. Here, we turn towLIC groups: we give an intrinsic (though awk-ward) characterization in terms of existence of certain countable suprema. This
results also in an intrinsic characterization ofLIC, previously lacking. Also,wLIC is a hull class (whichLIC is not), and the hullwlA is “somewhere near” the projectable hullpA. The best comparison comes from a (somewhat novel) factoringpA=loc(wpA), wherewpA is the “weakly projectable” hull (defined here), andlocB is the “local monoreflection”; then,wpA≤wlA≤loc(wpA), andpA≤loc(wlA), while with a strong unit, all these coincide. Numerous examples and special cases are examined. 相似文献
19.
If π is a set of primes, a finite group G is block π-separated if for every two distinct irreducible complex characters α, β ∈ Irr(G) there exists a prime p ∈ π such that α and β lie in different Brauer p-blocks. A group G is block separated if it is separated by the set of prime divisors of |G|. Given a set π with n different primes, we construct an example of a solvable π-group G which is block separated but it is not separated by every proper subset of π.
Received: 22 December 2004 相似文献
20.
Wolfgang Lenzen 《Logica Universalis》2008,2(1):43-58
In the 18th century, Gottfried Ploucquet developed a new syllogistic logic where the categorical forms are interpreted as
set-theoretical identities, or diversities, between the full extension, or a non-empty part of the extension, of the subject
and the predicate. With the help of two operators ‘O’ (for “Omne”) and ‘Q’ (for “Quoddam”), the UA and PA are represented
as ‘O(S) – Q(P)’ and ‘Q(S) – Q(P)’, respectively, while UN and PN take the form ‘O(S) > O(P)’ and ‘Q(S) > O(P)’, where ‘>’ denotes set-theoretical disjointness. The use of the symmetric operators ‘–’ and ‘>’ gave rise to a new conception
of conversion which in turn lead Ploucquet to consider also the unorthodox propositions O(S) – O(P), Q(S) – O(P), O(S) > Q(P), and Q(S) > Q(P). Although Ploucquet’s critique of the traditional theory of opposition turns out to be mistaken, his theory of the “Quantification
of the Predicate” is basically sound and involves an interesting “Double Square of Opposition”.
My thanks are due to Hanno von Wulfen for helpful discussions and for transforming the word-document into a Latex-file. 相似文献