The difference between common knowledge of formulas and sets

Common knowledge can be defined in at least two ways: syntactically as the common knowledge of a set of formulas or semantically, as the meet of the knowledge partitions of the agents. In the multi-agent S5 logic with either finitely or countably many agents and primitive propositions, the semantic definition is the finer one. For every subset of formulas that can be held in common knowledge, there is either only one member or uncountably many members of the meet partition with this subset of formulas held in common knowledge. If there are at least two agents, there are uncountably many members of the meet partition where only the tautologies of the multi-agent S5 logic are held in common knowledge. Whether or not a member of the meet partition is the only one corresponding to a set of formulas held in common knowledge has radical implications for its topological and combinatorial structure.     

Multilinear enumerative techniques

When S is a finite set and G a finite group acting on S, we consider the problem of rejecting isomorphs in a G-stable subset of S. In previous work we developed a linear algebraic context for this problem by constructing the finite dimensional vector spaceF ^s whereF is a field of characteristic zero. When S is a finite function space, or a finite direct product of finite function spacesF ^s acquires a multilinear structure By various specializations ofG and S and by applications of results which have appeared elsewhere, identities of Sheehan, deBruijn and P61ya are obtained. Furthermore, these same techniques are applied to examples which do not have a clear resolution using the more common formulas     

Finite sets which contain their Radon points

A point r is a Radon point of a finite set S if r=conv Aconv B, where A and B are disjoint subsets of S. Two characterizations are given for those finite sets in R ^d which contain their Radon points, and the convex hull of such a set is described.     

A-Solvability and Mixed Abelian Groups

An abelian group A is an S-group (S ⁺-group) if every subgroup B ≤ A of finite index is A-generated (A-solvable). This article discusses some of the differences between torsion-free S-groups and mixed S-groups, and studies (mixed) S- and S ⁺-groups, which are self-small and have finite torsion-free rank.     

Nonadditive Set Functions on a Finite Set and Linear Inequalities

A set function is a function whose domain is the power set of a set, which is assumed to be finite in this paper. We treat a possibly nonadditive set function, i.e., a set function which does not satisfy necessarily additivity, ?(A) + ?(B) = ?(A ∪ B) forA ∩ B = ∅, as an element of the linear space on the power set. Then some of the famous classes of set functions are polyhedral in that linear space, i.e., expressed by a finite number of linear inequalities. We specify the sets of the coefficients of the linear inequalities for some classes of set functions. Then we consider the following three problems: (a) the domain extension problem for nonadditive set functions, (b) the sandwich problem for nonadditive set functions, and (c) the representation problem of a binary relation by a nonadditive set function, i.e., the problem of nonadditive comparative probabilities.     

A solution to a problem of Cameron on sum-free complete sets

For any subsets A and B of an additive group G, define A + B = { a + b: a ε A and b ε B} and −A = {−a: a ε A}. A subset S of G is said to be sum-free, complete, and symmetric respectively if S + S S^c, S + S S^c, and S = −S. Cameron asked if for all sufficiently large moduli m there exists a sum-free complete set in Z/mZ that is not symmetric. We answer Cameron's question by showing there exists such a set for all moduli greater than or equal to 890626. We also show that every sum-free complete set in Z/mZ that is not symmetric can be used to construct a counter-example to a conjecture of Conway disproved by Marica. Conway conjectured that for any finite set S of integers, |S + S| |S --- S|.     

Generating random elements of abelian groups

Algorithms based on rapidly mixing Markov chains are discussed to produce nearly uniformly distributed random elements in abelian groups of finite order. Let A be an abelian group generated by set S. Then one can generate ?‐nearly uniform random elements of A using 4|S|log(|A|/?) log(|A|) additions and the same number of random bits. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2005     

The endomorphism spectrum of a monounary algebra

The endomorphism spectrum specA of an algebra A is defined as the set of all positive integers, which are equal to the number of elements in an endomorphic image of A, for all endomorphisms of A. In this paper we study finite monounary algebras and their endomorphism spectrum. If a finite set S of positive integers is given, one can look for a monounary algebra A with S = specA. We show that for countably many finite sets S, no such A exists. For some sets S, an appropriate A with spec A = S are described. For n ∈ ? it is easy to find a monounary algebra A with {1, 2, ..., n} = specA. It will be proved that if i ∈ ?, then there exists a monounary algebra A such that specA skips i consecutive (consecutive eleven, consecutive odd, respectively) numbers. Finally, for some types of finite monounary algebras (binary and at least binary trees) A, their spectrum is shown to be complete.     

Finite phase transitions in countable abelian groups

Let A be an infinite set that generates a group G. The sphere S _A (r) is the set of elements of G for which the word length with respect to A is exactly r. We say G admits all finite transitions if for every r ≥ 2 and every finite symmetric subset W ì G\{e}{W \subset G{\setminus}\{e\}}, there exists an A with S _A (r) = W. In this paper we determine which countable abelian groups admit all finite transitions. We also show that \mathbbRⁿ{\mathbb{R}^n} and the finitary symmetric group on \mathbbN{\mathbb{N}} admit all finite transitions.     

On the bounded quasi‐degrees of c.e. sets

We study the degree structure of bQ‐reducibility and we prove that for any noncomputable c.e. incomplete bQ‐degree a, there exists a nonspeedable bQ‐degree incomparable with it. The structure $\mathcal {D}_{\mbox{bs}}$ of the $\mbox{bs}$‐degrees is not elementary equivalent neither to the structure of the $\mbox{be}$‐degrees nor to the structure of the $\mbox{e}$‐degrees. If c.e. degrees a and b form a minimal pair in the c.e. bQ‐degrees, then a and b form a minimal pair in the bQ‐degrees. Also, for every simple set S there is a noncomputable nonspeedable set A which is bQ‐incomparable with S and bQ‐degrees of S and A does not form a minimal pair.     

Some Results on Systems of Finite Sets That Satisfy a Certain Intersection Condition

Let S be a finite set, and fix K>2. Let F be a family of subsets of S with the property that whenever A₁,...,A_k are sets in F, not necessarily distinct, and A₁ ? ? ? A_k = ?, then A₁ ? ? ? A_k = S. We prove here that the maximum size of such a family is 2^|S|?1 + 1. If we require that the sets A₁,...,A_k be distinct, then the maximum size of F is again 2^|S|?1 + 1, provided that |S| ≥ log₂(K?2)+3.     

Nevanlinna Functions,Perturbation Formulas,and Triplets of Hilbert Spaces

Let S be a closed symmetric operator with defect numbers (1,1) in a Hilbert space ?? and let A be a selfadjoint operator extension of S in ??. Then S is necessarily a graph restriction of A and the selfadjoint extensions of S can be considered as graph perturbations of A, cf. [8]. Only when S is not densely defined and, in particular, when S is bounded, 5 is given by a domain restriction of A and the graph perturbations reduce to rank one perturbations in the sense of [23]. This happens precisely when the Q - function of S and A belongs to the subclass No of Nevanlinna functions. In this paper we show that by going beyond the Hilbert space ?? the graph perturbations can be interpreted as compressions of rank one perturbations. We present two points of view: either the Hilbert space ?? is given a one-dimensional extension, or the use of Hilbert space triplets associated with A is invoked. If the Q - function of S and A belongs to the subclass N₁ of Nevanlinna functions, then it is convenient to describe the selfadjoint extensions of S including its generalized Friedrichs extension (see [6]) by interpolating the original triplet, cf. [5]. For the case when A is semibounded, see also [4]. We prove some invariance properties, which imply that such an interpolation is independent of the (nonexceptional) extension.     

A game-theoretic equivalence to the Hahn-Banach theorem

Let Γ_X() = X, A (X), υ be a cooperative von Neumann game with side payments, where X is a nonempty set of arbitrary cardinality, A(X) the Boolean ring generated from P(X) with the operations Δ and ∩ for addition and multiplication, respectively, such that S² =S for all S ε A (X), and with ;() = 0. The Shapley-Bondareva-Schmeidler Theorem, which states that a game of the form Γ_X() = X, A (X), is weak if and only if the core of Γ_X(),ζ(Γ_X()), is normal, may be regarded as the fundamental theorem for weak cooperative games with side-payments. In this paper we use an ultrapower construction on the reals, , to summarize a common mathematical theme employed in various constructions used to establish the Shapley-Bondareva-Schmeidler Theorem in the literature (Dalbaen, 1974; Kannai, 1969; Schmeidler, 1967, 1972). This common mathematical theme is that the space L, comprised of finite, real linear combinations of the collection of functions, {χ_a : a ε A (X)}, possesses a certain extension property that is intimately related to the Hahn-Banach Theorem of functional analysis. A close inspection of the extension property reveals that the Shapley-Bondareva-Schmeidler Theorem is in fact equivalent to the Hahn-Banach Theorem.     

Invariant class sizes and solvability of finite groups under coprime action

Suppose that A and G are finite groups of relatively prime orders such that A acts on G via automorphisms, and then, A acts on the set of conjugacy classes of G. We study several arithmetical properties on the sizes of certain classes which are left fixed by A and how they reflect on the A‐invariant structure and may imply the solvability of G.     

Locally finite groups with all subgroups either subnormal or nilpotent-by-Chernikov

Let G be a locally finite group satisfying the condition given in the title and suppose that G is not nilpotent-by-Chernikov. It is shown that G has a section S that is not nilpotent-by-Chernikov, where S is either a p-group or a semi-direct product of the additive group A of a locally finite field F by a subgroup K of the multiplicative group of F, where K acts by multiplication on A and generates F as a ring. Non-(nilpotent-by-Chernikov) extensions of this latter kind exist and are described in detail.     

A Note on Blockers in Posets

The blocker A* of an antichain A in a finite poset P is the set of elements minimal with the property of having with each member of A a common predecessor. The following is done: (1) The posets P for which A** = A for all antichains are characterized.(2) The blocker A* of a symmetric antichain in the partition lattice is characterized.(3) Connections with the question of finding minimal size blocking sets for certain set families are discussed.AMS Subject Classification: 05C35, 05D05, 06A07.     

Spectral theory of minimax estimation

The set ofS ₁-estimates of solutions of systems of linear equations with random parameters is found. It is proved that the maximal eigenvalue in the goodness criterion is not simple. For the purpose of finding estimates from theS ₁ set, the perturbation formulas for eigenvalues and formulas for distribution density of random matrices are used.     

Simultaneous similarity of matrices

In this paper we solve completely and explicitly the long-standing problem of classifying pairs of n × n complex matrices (A, B) under the simultaneous similarity (TAT⁻¹, TBT⁻¹). Roughly speaking, the classification decomposes to a finite number of steps. In each step we consider an open algebraic set ⁰_n,2,r,π M_n × M_n (M_n = the set of n × n complex-valued matrices). Here r and π are two positive integers. Then we construct a finite number of rational functions ø₁,…,ø_s in the entries of A and B whose values are constant on all pairs similar in _n,2,r,π to (A, B). The values of the functions ø_i(A, B), I = 1,…, s, determine a finite number (at most κ(n, 2, r)) of similarity classes in _n,2,r,π. Let S_n be the subspace of complex symmetric matrices in M_n. For (A, B) ε S_n × S_n we consider the similarity class (TAT^t, TBT^t), where T ranges over all complex orthogonal matrices. Then the characteristic polynomial |λI − (A + xB)| determines a finite number of similarity classes for almost all pairs (A, B) ε S_n × S_n.     

Multiplicative Invariants and Semigroup Algebras

Let G be a finite group acting by automorphism on a lattice A, and hence on the group algebra S=k[A]. The algebra of G-invariants in S is called an algebra of multiplicative invariants. We present an explicit version of a result of Farkas stating that multiplicative invariants of finite reflection groups are semigroup algebras.     

Un analogue elliptique du théorème de Roth

We present here quantitative versions, in dimension one, of Faltings' theorem according to which the set of K-rational points (where K is a given number field) of an Abelian variety A defined over K, which are close (with respect to a v-adic distance on K) to some K-subvariety X of A, but do not belong to X, is finite. More precisely, we treat the case where A is an elliptic curve and X is reduced to a point of A and we give (in this case) explicit bounds for the cardinal of the exceptional finite set. We consider also, more generally, not only one place v of K, but also a finite set S of places of K and the distance from the point of A to X, which takes into account all the places of S. To cite this article: B. Farhi, C. R. Acad. Sci. Paris, Ser. I 341 (2005).