Non-Classical Stems from Classical: N. A. Vasiliev’s Approach to Logic and his Reassessment of the Square of Opposition

In the XIXth century there was a persistent opposition to Aristotelian logic. Nicolai A. Vasiliev (1880–1940) noted this opposition and stressed that the way for the novel – non-Aristotelian – logic was already paved. He made an attempt to construct non-Aristotelian logic (1910) within, so to speak, the form (but not in the spirit) of the Aristotelian paradigm (mode of reasoning). What reasons forced him to reassess the status of particular propositions and to replace the square of opposition by the triangle of opposition? What arguments did Vasiliev use for the introduction of new classes of propositions and statement of existence of various levels in logic? What was the meaning and role of the “method of Lobachevsky” which was implemented in construction of imaginary logic? Why did psychologism in the case of Vasiliev happen to be an important factor in the composition of the new ‘imaginary’ logic, as he called it?      

God, king, and geometry: revisiting the introduction to Cauchy’s Cours d’analyse

This article offers a systematic reading of the introduction to Augustin-Louis Cauchy’s landmark 1821 mathematical textbook, the Cours d’analyse. Despite its emblematic status in the history of mathematical analysis and, indeed, of modern mathematics as a whole, Cauchy’s introduction has been more a source for suggestive quotations than an object of study in its own right. Cauchy’s short mathematical metatext offers a rich snapshot of a scholarly paradigm in transition. A close reading of Cauchy’s writing reveals the complex modalities of the author’s epistemic positioning, particularly with respect to the geometric study of quantities in space, as he struggles to refound the discipline on which he has staked his young career.     

Maximum Number of Constant Weight Vertices of the Unit <Emphasis Type="Italic">n</Emphasis>-Cube Contained in a <Emphasis Type="Italic">k</Emphasis>-Dimensional Subspace

We introduce and solve a natural geometrical extremal problem. For the set E (n,w) = {x ⁿ {0,1} ⁿ : x ⁿ has w ones } of vertices of weight w in the unit cube of ⁿ we determine M (n,k,w) max{|U _k ⁿ E(n,w)|:U _k ⁿ is a k-dimensional subspace of ⁿ . We also present an extension to multi-sets and explain a connection to a higher dimensional Erds–Moser type problem.     

Products of Nevanlinna-Pick kernels and operator colligations

An example is given which clarifies the present situation of the operator norm convergence of Trotter-Kato product formula. It shows that the rate of convergence of the formula with respect to the operator norm obtained in [NZ2] is best possible. It also yields a counter-example of the operator norm convergence of the formula in another case.     

Solving equations in groups: A survey of Frobenius' theorem

A fundamental result of Frobenius states that in a finite group the number of elements which satisfy the equationx ⁿ=1, wheren divides the order of the group, is divisible byn. Here 1 denotes the identity of the group. This theorem and several generalizations were obtained by Frobenius at the turn of the century. These results have stimulated a great amount of interest in counting solutions of equations in groups. This article discusses these results and traces the various developments which these fundamental papers have generated.LetG be a finite group of order |G|. Leto(g) denote the order ofg( G). LetH(s, k)={xG:k|o(x)| sk} wherea/b meansa dividesb and leth(s,k)=|H(s,k)|. Using this notation the simplest of Frobenius' results states ifn/|G|, then/h(n, 1). The minimum value ofh(n, 1) is discussed in the first section. Various conditions are known to insure thath(n, 1)=n. A long standing conjecture of Frobenius states ifn=h(n, 1) thenH(n, 1) is a subgroup (where of coursen/|G|). This conjecture is valid for solvable groups, as well as for various arithmetic conditions.In the second section other divisibility conditions arising from Frobenius' Theorem are discussed. One direction covers more general arithmetic divisibility condition. Another direction has a much wider scope, involving a finite number of equations of an unspecified form and is mainly due to P. Hall. Recently some divisibility conditions involving all groups of a given order have been obtained. Divisibility conditions also hold in infinite groups, and for automorphism analogues of element order. In the next section generalizations to group characters relating back to Frobenius are given. Some of these expressions are used in analyzing properties of group representations and have applications in quantum theory. In the last section clear evidence is established for the combinatorial rather than group-theoretic nature of these results. In particular, some recent work of Snapper links the counting of solutions of equations with the cycle indices in combinatorial theory. Counting solutions of equations in the symmetric groups is also discussed.     

Parametrized Partitions of Products of Finite Sets*

For every infinite sequence of positive integers and every Borel partition c : ×[]{0, 1} there is H[] and a sequence of subsets of , with |H_i|=m_i for every i, such that c is constant on .* Research partially supported by CNRS-FONACIT Project PI 2000001471. This author thanks the University of Paris VII for hospitality.     

Rothberger bounded groups and Ramsey theory

We show that:

(1)

Rothberger bounded subgroups of σ-compact groups are characterized by Ramseyan partition relations (Corollary 4).

(2)

For each uncountable cardinal κ there is a T₀ topological group of cardinality κ such that ONE has a winning strategy in the point-open game on the group and the group is not a closed subspace of any σ-compact space (Theorem 8).

(3)

For each uncountable cardinal κ there is a T₀ topological group of cardinality κ such that ONE has a winning strategy in the point-open game on the group and the group is σ-compact (Corollary 17).

     

Extending partial isomorphisms on finite structures

We prove the following theorem:Let A be a finite structure in a fixed finite relational language,p ₁,...,p _m partial isomorphisms of A. Then there exists a finite structure B, and automorphismsf _i of B extending thep _i 's. This theorem can be used to prove the small index property for the random structure in this language. A special case of this theorem is, if A and B are hypergraphs. In addition we prove the theorem for the case of triangle free graphs.     

Countable partition ordinals

The structure of ordinals of the form ω^ωβ for countable β is studied. The main result is:

Theorem 1. Ifβ<ω₁is the sum of one or two indecomposable ordinals, then

ω^ωβ→(ω^ωβ,3)².     

Resolvability vs. almost resolvability

A space X is κ-resolvable (resp. almost κ-resolvable) if it contains κ dense sets that are pairwise disjoint (resp. almost disjoint over the ideal of nowhere dense subsets of X).Answering a problem raised by Juhász, Soukup, and Szentmiklóssy, and improving a consistency result of Comfort and Hu, we prove, in ZFC, that for every infinite cardinal κ there is an almost κ²-resolvable but not ω₁-resolvable space of dispersion character κ.     

Some consequences of MA + ¬wKH

Let wKH denote the following statement: There exists a ω₁-tree of power ω₁ with >ω₁ ω₁-branches. First, using methods of [6] and [13], we shall prove theconsistency of MA + ¬wKH. Then we shall prove that MA + ¬wKH implies the following: (a) There is no θ-dense (splitting) poset of power ω₁; (b) Every LOTS of density ω₁ has a θ-disjoint π-base; (c) There is no Baire LOTS of power ω₁ without isolated points; (d) Every perfectly normal non-Archimedian space of weight ω₁ is metrizable. These results are connected to problems from [4], [10], [7] and [15], respectively. A part of these results was announced in [17].     

Resolvability properties via independent families

The authors give a consistent affirmative response to a question of Juhász, Soukup and Szentmiklóssy: If GCH fails, there are (many) extraresolvable, not maximally resolvable Tychonoff spaces. They show also in ZFC that for ω<λ?κ, no maximal λ-independent family of λ-partitions of κ is ω-resolvable. In topological language, that theorem translates to this: A dense, ω-resolvable subset of a space of the form (DI(λ)) is λ-resolvable.     

Tychonoff expansions with prescribed resolvability properties

The recent literature offers examples, specific and hand-crafted, of Tychonoff spaces (in ZFC) which respond negatively to these questions, due respectively to Ceder and Pearson (1967) [3] and to Comfort and García-Ferreira (2001) [5]: (1) Is every ω-resolvable space maximally resolvable? (2) Is every maximally resolvable space extraresolvable? Now using the method of KID expansion, the authors show that every suitably restricted Tychonoff topological space (X,T) admits a larger Tychonoff topology (that is, an “expansion”) witnessing such failure. Specifically the authors show in ZFC that if (X,T) is a maximally resolvable Tychonoff space with S(X,T)?Δ(X,T)=κ, then (X,T) has Tychonoff expansions U=Ui (1?i?5), with Δ(X,Ui)=Δ(X,T) and S(X,Ui)?Δ(X,Ui), such that (X,Ui) is: (i=1) ω-resolvable but not maximally resolvable; (i=2) [if κ^′ is regular, with S(X,T)?κ^′?κ] τ-resolvable for all τ<κ^′, but not κ^′-resolvable; (i=3) maximally resolvable, but not extraresolvable; (i=4) extraresolvable, but not maximally resolvable; (i=5) maximally resolvable and extraresolvable, but not strongly extraresolvable.     

Stillman's question for exterior algebras and Herzog's conjecture on Betti numbers of syzygy modules

Let K be a field of characteristic 0 and consider exterior algebras of finite dimensional K-vector spaces. In this short paper we exhibit principal quadric ideals in a family whose Castelnuovo–Mumford regularity is unbounded. This negatively answers the analogue of Stillman's Question for exterior algebras posed by I. Peeva. We show that, via the Bernstein–Gel'fand–Gel'fand correspondence, these examples also yields counterexamples to a conjecture of J. Herzog on the Betti numbers in the linear strand of syzygy modules over polynomial rings.     

Ploucquet’s “Refutation” of the Traditional Square of Opposition

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.     

Density results for uniform families

A familyF of subsets is calledk-dense if there exists ak-element setA such that all 2 ^k of its subsets can be obtained in the formAF for someFF. Best possible bounds are obtained for the maximum number of sets in notk-densek-partite families. This is a consequence of a new rank formula for inclusion matrices.     

Some modifications of Newton’s method with higher-order convergence for solving nonlinear equations

In [YoonMee Ham etal., Some higher-order modifications of Newton’s method for solving nonlinear equations, J. Comput. Appl. Math., 222 (2008) 477–486], some higher-order modifications of Newton’s method for solving nonlinear equations are constructed. But if p=2$p=2$, then their main theorem did not hold. In this paper, we first give an example to show that YoonMee Ham etal.’s methods are not always correct in the case p=2$p=2$. Then, we present the condition that H(x,y)$H(x,y)$ should satisfy such that the order of convergence increases three or four or five units. Per iteration they only need two additional function evaluations to increase the order. Based on this and multi-step Newton’s scheme, we give further modifications of the method to obtain higher-order convergent iterative methods. Finally, several examples are given to demonstrate the efficiency and performance of our modified methods and compare them with some other methods.     

Spectral shift function, amazing and multifaceted

Several formula representations for the I. M. Lifshits — M. G. Kreîn spectral shift function (SSF) are discussed and intercompared. It is pointed out that the equivalence of these representations is not apparent, and different properties of the SSF are revealed by different formulas. The presentation is informal and contains no proofs.To the memory of the great mathematician Mark Grigor'evich Kreîn