Cantor set and interpolation

In 1998, Y. Benyamini published interesting results concerning interpolation of sequences using continuous functions ℝ → ℝ. In particular, he proved that there exists a continuous function ℝ → ℝ which in some sense “interpolates” all sequences (x _n ) _n∈ℤ ∈ [0, 1]^ℤ “simultaneously.” In 2005, M.R. Naulin and C. Uzcátegui unified and generalized Benyamini’s results. In this paper, the case of topological spaces X and Y with an Abelian group acting on X is considered. A similar problem of “simultaneous interpolation” of all “generalized sequences” using continuous mappings X → Y is posed. Further generalizations of Naulin-Uncátegui theorems, in particular, multidimensional analogues of Benyamini’s results are obtained.     

Towards variational analysis in metric spaces: metric regularity and fixed points

The main results of the paper include (a) a theorem containing estimates for the surjection modulus of a “partial composition” of set-valued mappings between metric spaces which contains as a particlar case well-known Milyutin’s theorem about additive perturbation of a mapping into a Banach space by a Lipschitz mapping; (b) a “double fixed point” theorem for a couple of mappings, one from X into Y and another from Y to X which implies a fairly general version of the set-valued contraction mapping principle and also a certain (different) version of the first theorem.     

Uniform version of Weyl–von Neumann theorem

We prove a “quantified” version of the Weyl–von Neumann theorem, more precisely, we estimate the ranks of approximants to compact operators appearing in Voiculescu’s theorem applied to commutative algebras. This allows considerable simplifications in uniform K-homology theory, namely it shows that one can represent all the uniform K-homology classes on a fixed Hilbert space with a fixed *-representation of C ₀(X), for a large class of spaces X.     

On log canonical divisors that are log quasi-numerically positive

Let (X Δ) be a four-dimensional log variety that is projective over the field of complex numbers. Assume that (X, Δ) is not Kawamata log terminal (klt) but divisorial log terminal (dlt). First we introduce the notion of “log quasi-numerically positive”, by relaxing that of “numerically positive”. Next we prove that, if the log canonical divisorK _X+Δ is log quasi-numerically positive on (X, Δ) then it is semi-ample.     

Expressing and Describing Experiences. A Case of Showing Versus Saying

Experiences are interpreted as conscious mental occurrences that are of phenomenal character. There is already a kind of (weak) intentionality involved with this phenomenal interpretation. A stricter conception of experiences distinguishes between purely phenomenal experiences and intentional experiences in a narrow sense. Wittgenstein’s account of psychological (experiential) verbs is taken over: Usually, expressing mental states verbally is not describing them. According to this, “I believe” can be seen as an expression of one’s own belief, but not as an expression of a belief about one’s belief. Hence, the utterance “I believe it is raining” shows that I believe that it is raining, although it is not said by these words that I believe that it is raining. Thinking thoughts such as “I believe it is raining, but it is not raining” (a variant of Moore’s paradox) is an absurdity between what is already said by silently uttering “It is not raining” and what is shown by silently uttering “I believe it is raining.” The paper agrees with a main result of Wittgenstein’s considerations of Moore’s paradox, namely the view that logical structure, deducibility, and consistency cannot be reduced solely to propositions—besides a logic of propositions, there is, for example, a logic of assertions and of imperatives, respectively.     

Tableau complexes

Let X, Y be finite sets and T a set of functions X → Y which we will call “ tableaux”. We define a simplicial complex whose facets, all of the same dimension, correspond to these tableaux. Such tableau complexes have many nice properties, and are frequently homeomorphic to balls, which we prove using vertex decompositions [BP79]. In our motivating example, the facets are labeled by semistandard Young tableaux, and the more general interior faces are labeled by Buch’s set-valued semistandard tableaux. One vertex decomposition of this “Young tableau complex” parallels Lascoux’s transition formula for vexillary double Grothendieck polynomials [La01, La03]. Consequently, we obtain formulae (both old and new) for these polynomials. In particular, we present a common generalization of the formulae of Wachs [Wa85] and Buch [Bu02], each of which implies the classical tableau formula for Schur polynomials.     

<Emphasis Type="Italic">g</Emphasis>-metrizable spaces and the images of semi-metric spaces

In this paper, we prove that a space X is a g-metrizable space if and only if X is a weak-open, π and σ-image of a semi-metric space, if and only if X is a strong sequence-covering, quotient, π and mssc-image of a semi-metric space, where “semi-metric” can not be replaced by “metric”. This project was supported by NNSF of China (No. 10571151 and 10671173).     

Exotic baker and wandering domains for Ahlfors islands maps

Let X be a compact Riemann surface of genus at most 1, i.e., the Riemann sphere or a torus, and let W ⊊ X be an arbitrary domain. We construct a variety of examples of holomorphic functions g: W → X that satisfy Epstein’s Ahlfors islands property and that have “pathological” dynamical behaviour. In particular, we show that the accumulation set of any curve tending to the boundary of W can be realized as the ω-limit set of a Baker domain of such a function. We furthermore construct Ahlfors islands maps

Entire functions having a concordant value sequence

A classic theorem of Pólya shows that 2 ^z is, in a strong sense, the “smallest” transcendental entire function that is integer valued on ℕ. An analogous result of Gel’fond concerns entire functions that are integer valued on the setX _a={a ⁿ:n ∈ ℕ}, wherea ∈ ℕ,|a|≥ 2. LetX=ℕ orX=X _a andκ ∈ ℕ orκ=∞. This paper pursues analogous results for entire functionsf having the following property: on any finite subsetD ofX with#D≤κ+1, the valuesf(z),z ∈D admit interpolation by an element of ℤ[z]. The results obtained assert that if the growth off is suitably restricted then the restriction off toX must be a polynomial. WhenX=X _a andκ<∞ a “smallest” transcendental entire function having the requisite property is constructed.     

The Geometry of Standard Deontic Logic

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).      

Disagreement point axioms and the egalitarian bargaining solution

We provide new characterizations of the egalitarian bargaining solution on the class of strictly comprehensive n-person bargaining problems. The main axioms used in all of our results are Nash’s IIA and disagreement point monotonicity—an axiom which requires a player’s payoff to strictly increase in his disagreement payoff. For n = 2 these axioms, together with other standard requirements, uniquely characterize the egalitarian solution. For n > 2 we provide two extensions of our 2-person result, each of which is obtained by imposing an additional axiom on the solution. Dropping the axiom of anonymity, strengthening disagreement point monotonicity by requiring player i’s payoff to be a strictly decreasing function of the disagreement payoff of every other player j ≠ i, and adding a “weak convexity” axiom regarding changes of the disagreement point, we obtain a characterization of the class of weighted egalitarian solutions. This “weak convexity” axiom requires that a movement of the disagreement point in the direction of the solution point should not change the solution point. We also discuss the so-called “transfer paradox” and relate it to this axiom.     

Graph factors

This exposition is concerned with the main theorems of graph-factor theory, Hall’s and Ore’s Theorems in the bipartite case, and in the general case Petersen’s Theorem, the 1-Factor Theorem and thef-Factor Theorem. Some published extensions of these theorems are discussed and are shown to be consequences rather than generalizations of thef-Factor Theorem. The bipartite case is dealt with in Section 2. For the proper presentation of the general case a preliminary theory of “G-triples” and “f-barriers” is needed, and this is set out in the next three Sections. Thef-Factor Theorem is then proved by an argument of T. Gallai in a generalized form. Gallai’s original proof derives the 1-Factor Theorem from Hall’s Theorem. The generalization proceeds analogously from Ore’s Theorem to thef-Factor Theorem.     

Instability of the wet x soap film

For idealized, infinitely thin (“dry”) soap films, an Xis unstable, while for very thick (“wet”) soap films it is minimizing. We show that for soap films of relatively small but positive wetness, the Xis unstable. Full stability diagrams for the constant liquid fraction case and the constant pressure case are generated. Analogous questions about other singularities remain controversial.     

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.     

“Naturadeus”, a metaphor of the Perfect Diamond

In this essay, the author outlines his re-construction of Spinoza’s ontological monism by re-presenting the system of Ethica, ordine geometrico demonstrata, in an “intuitive” model of the Perfect Diamond, called NATURADEUS. So, for example, ordo et connexio idearum et rerum, is presented to the inner eye in the forms of two parallel structures, of rays and of facets within the NATURADEUS, respectively. The conceptual background of the proposed model is mostly analytic, the author essays to develop some ideas of Jonathan Bennett’s Spinoza’s metaphysics (especially “transattributive mode identity”), with strong emphasis on ethical issues of ontological monism or pantheism. This essay is written as a dialogue between master Bruno and his student John (physically absent at the moment). More philosophical dialogues of this kind can be found on the author’s web page and in his book Four Seasons (in Slovenian, 2002).     

The sweet mystery of compatibilism

Any satisfactory account of freedom must capture, or at least permit, the mysteriousness of freedom—a “sweet” mystery involving a certain kind of ignorance rather than a “sour” mystery of unintelligibility, incoherence, or unjustifiedness. I argue that compatibilism can capture the sweet mystery of freedom. I argue first that an action is free if and only if a certain “rationality constraint” is satisfied, and that nothing in standard libertarian accounts of freedom entails its satisfaction. Satisfaction of this constraint is consistent with the universal causal predetermination of action (UCP). If UCP is true and the rationality constraint satisfied, there’s a sense in which our actions are explanatorily (though not necessarily causally) overdetermined. While it seems plausible (given UCP) that our actions are so overdetermined, it seems utterly mysterious why they should be so overdetermined. Compatibilism’s capacity to accommodate this mystery is a mark in its favor.     

Independent sequences in Banach spaces

In every ∞-dimensional separable Banach spaceX there is a fundamental sequence such that no subsequence of it, which is fundamental inX, is independent (“{x _n} is fundamental inX” meansX=span {x _n}).     

Fibrewise completion and unstable Adams spectral sequences

We describe a tower of spaces whose inverse limit is a “fiberwise completion” of a fibrationE →B, and study the resulting spectral sequence converging to the homotopy groups of the space of lifts of a mapX →B. This is used to give a proof of the “generalized Sullivan conjecture”. All three authors were supported in part by the National Science Foundation.     

Local Search Proximal Algorithms as Decision Dynamics with Costs to Move

Acceptable moves for the “worthwhile-to-move” incremental principle are such that “advantages-to-move” are higher than some fraction of “costs-to-move”. When combined with optimization, this principle gives raise to adaptive local search proximal algorithms. Convergence results are given in two distinctive cases, namely low local costs-to-move and high local costs-to-move. In this last case, one obtains a dynamic cognitive approach to Ekeland’s ϵ-variational principle. Introduction of costs-to-move in the algorithms yields robustness and stability properties.     

Transfer of complex structures and topological invariance of property of being holomorphic

The following question by V. I. Arnold is answered in affirmative. Let X, Y, and Z be three complex manifolds of equal dimension, let p: X → Y be a universal covering, and let g: Y → Z be a nondegenerate holornorphic mapping. Assume that the term Y in the chain X\xrightarrowpY\xrightarrowgZ X\xrightarrow{p}Y\xrightarrow{g}Z is “forgotten,” while the complex structures on X and Z are changed so that the mapping g ∘ p remains holomorphic. Can one recover the “forgotten” term Y? Bibliography: 2 titles.