A Combinatorial Proof of Kneser’s Conjecture*

Knesers conjecture, first proved by Lovász in 1978, states that the graph with all k-element subsets of {1, 2, . . . , n} as vertices and with edges connecting disjoint sets has chromatic number n–2k+2. We derive this result from Tuckers combinatorial lemma on labeling the vertices of special triangulations of the octahedral ball. By specializing a proof of Tuckers lemma, we obtain self-contained purely combinatorial proof of Knesers conjecture.* Research supported by Charles University grants No. 158/99 and 159/99 and by ETH Zürich.     

A Simpler Proof of Mirzakhani’s Simple Curve Asymptotics

Maryam Mirzakhani (in her doctoral dissertation) has proved the author’s conjecture that the number of simple closed curves of length bounded by L on a hyperbolic surface S is asymptotic to a constant times L^d, where d is the dimension of the Teichmüller space of S. In this note we clarify and simplify Mirzakhani’s argument.     

On a Modification of Olver’s Method: A Special Case

We consider the asymptotic method designed by Olver (Asymptotics and special functions. Academic Press, New York, 1974) for linear differential equations of the second order containing a large (asymptotic) parameter \(\Lambda \): \(x^my''-\Lambda ^2y=g(x)y\), with \(m\in \mathbb {Z}\) and g continuous. Olver studies in detail the cases \(m\ne 2\), especially the cases \(m=0, \pm 1\), giving the Poincaré-type asymptotic expansions of two independent solutions of the equation. The case \(m=2\) is different, as the behavior of the solutions for large \(\Lambda \) is not of exponential type, but of power type. In this case, Olver’s theory does not give many details. We consider here the special case \(m=2\). We propose two different techniques to handle the problem: (1) a modification of Olver’s method that replaces the role of the exponential approximations by power approximations, and (2) the transformation of the differential problem into a fixed point problem from which we construct an asymptotic sequence of functions that converges to the unique solution of the problem. Moreover, we show that this second technique may also be applied to nonlinear differential equations with a large parameter.     

A New Proof of Calabi-Yau＇s Theorem

We give a new proof of Calabi-Yau＇s theorem on the volume growth of Riemannian manifolds with non-negative Ricci curvature.     

A Simple Proof of Perelman’s Collapsing Theorem for 3-manifolds

We will simplify earlier proofs of Perelman’s collapsing theorem for 3-manifolds given by Shioya–Yamaguchi (J. Differ. Geom. 56:1–66, 2000; Math. Ann. 333: 131–155, 2005) and Morgan–Tian ( [math.DG], 2008). A version of Perelman’s collapsing theorem states: “Let {M³_i}\{M^{3}_{i}\} be a sequence of compact Riemannian 3-manifolds with curvature bounded from below by (−1) and $\mathrm{diam}(M^{3}_{i})\ge c_{0}>0$\mathrm{diam}(M^{3}_{i})\ge c_{0}>0 . Suppose that all unit metric balls in M³_iM^{3}_{i} have very small volume, at most v _i →0 as i→∞, and suppose that either M³_iM^{3}_{i} is closed or has possibly convex incompressible toral boundary. Then M³_iM^{3}_{i} must be a graph manifold for sufficiently large i”. This result can be viewed as an extension of the implicit function theorem. Among other things, we apply Perelman’s critical point theory (i.e., multiple conic singularity theory and his fibration theory) to Alexandrov spaces to construct the desired local Seifert fibration structure on collapsed 3-manifolds.     

Monotonic and Non-monotonic Embeddings of Anselm’s Proof

A consequence relation \(\vdash \) is monotonic iff for premise sets \(\Gamma , \Delta \) and conclusion \(\varphi \), if \(\Gamma \vdash \varphi \), \(\Gamma \subseteq \Delta \), then \(\Delta \vdash \varphi \); and non-monotonic if this fails in some instance. More plainly, a consequence relation is monotonic when whatever is entailed by a premise set remains entailed by any of its supersets. From the High Middle Ages through the Early Modern period, consequence in theology is assumed to be monotonic. Concomitantly, to the degree the argument formulated by Anselm at Proslogion 2–4 is taken up by later commentators, it is accepted or rejected in accordance with a monotonic notion of consequence. Examining Anselm’s use of parallelism in the Proslogion, I show Anselm embeds his famous argument in Proslogion 2–4 in a non-monotonic context. The results here presented challenge some deeply ingrained ideas governing the historiography of the long twelfth century, particularly concerning how the theology of the later eleventh through the twelfth century relates to the scholasticism of the thirteenth.     

On a Constructive Proof of Kolmogorov’s Superposition Theorem

Kolmogorov (Dokl. Akad. Nauk USSR, 14(5):953–956, 1957) showed that any multivariate continuous function can be represented as a superposition of one-dimensional functions, i.e., $$f(x_{1},\ldots,x_{n})=\sum_{q=0}^{2n}\varPhi _{q}\Biggl(\sum_{p=1}^{n}\psi_{q,p}(x_{p})\Biggr).$$ The proof of this fact, however, was not constructive, and it was not clear how to choose the outer and inner functions Φ _q and ψ _q,p , respectively. Sprecher (Neural Netw. 9(5):765–772, 1996; Neural Netw. 10(3):447–457, 1997) gave a constructive proof of Kolmogorov’s superposition theorem in the form of a convergent algorithm which defines the inner functions explicitly via one inner function ψ by ψ _p,q :=λ _p ψ(x _p +qa) with appropriate values λ _p ,a∈?. Basic features of this function such as monotonicity and continuity were supposed to be true but were not explicitly proved and turned out to be not valid. Köppen (ICANN 2002, Lecture Notes in Computer Science, vol. 2415, pp. 474–479, 2002) suggested a corrected definition of the inner function ψ and claimed, without proof, its continuity and monotonicity. In this paper we now show that these properties indeed hold for Köppen’s ψ, and we present a correct constructive proof of Kolmogorov’s superposition theorem for continuous inner functions ψ similar to Sprecher’s approach.     

A Reader’s Comment on: “Hysteresis Model of Unconscious-Conscious Interconnection: Exploring Dynamics on m-Adic Trees”

p-Adic Numbers, Ultrametric Analysis and Applications - This comment is aimed to point out that the recent work due to H. Kim, J-Y. Moon, G. A. Mashour and U. Lee ([22]), in which the clinical and...     

Proof of the Existence of Universals—and Roman Ingarden’s Ontology

The paper ends with an argument that says: necessarily, if there are finitely spatially extended particulars, then there are monadic universals. Before that, in order to characterize the distinction between particulars and universals, Roman Ingarden’s notions of “existential moments” and “modes (ways) of being” are presented, and a new pair of such existential moments is introduced: multiplicity–monadicity. Also, it is argued that there are not only real universals, but instances of universals (tropes) and fictional universals too.     

A Direct Proof of the Gale–Nikaido–Debreu Lemma Using Sperner’s Lemma

Journal of Optimization Theory and Applications - The Gale–Nikaido–Debreu lemma plays an important role in establishing the existence of competitive equilibrium. In this paper, we use...     

On a Uniform Treatment of Darboux’s Method

Let F(z) be an analytic function in |z| < 1. If F(z) has only a finite number of algebraic singularities on the unit circle |z| = 1, then Darbouxs method can be used to give an asymptotic expansion for the coefficient of zⁿ in the Maclaurin expansion of F(z). However, the validity of this expansion ceases to hold, when the singularities are allowed to approach each other. A special case of this confluence was studied by Fields in 1968. His results have been considered by others to be too complicated, and desires have been expressed to investigate whether any simplification is feasible. In this paper, we shall show that simplification is indeed possible. In the case of two coalescing algebraic singularities, our expansion involves only two Bessel functions of the first kind.     

Toward a Culture of Creativity: A Personal Perspective on Logo’s Early Years and Ongoing Potential

This paper describes the creation and early uses of Logo. It includes a brief summary of more recent work on Logo-derived languages and learning applications. It closes with a lament on the unrealized potential of student programming languages as empowering tools for knowledge construction and learning.     

Modification of Rissanen’s Method in Linear Memory

The problem of solving a linear system with a Hankel or block-Hankel matrix, as well as Rissanen’s algorithm and its generalization to the block case, are considered. Modifications of these algorithms that use less memory (O(n) against O(n²)).     

Toward Homological Characterization of Semirings: Serre’s Conjecture and Bass’s Perfectness in a Semiring Context

Among other results on homological characterization of semirings, we prove that the classes of projective and free right (left) semimodules over the polynomial semiring R[x₁, x₂,..., x_n] over an additively regular division semiring R coincide iff R is a field. Also it is shown that an additively regular commutative semiring R is perfect (in H. Basss sense) iff R is a perfect ring.In Celebration of the Sixtieth Birthday of Ralph N. McKenzieReceived July 27, 2003; accepted in final form April 2, 2004.     

A Geometric Proof of Bourgain’s $$L^2$$ Estimate of Maximal Operators Along Analytic Vector Fields

Bourgain (A remark on the maximal function associated to an analytic vector field. Analysis at Urbana, Cambridge University Press, Cambridge, 1989) proved that the maximal operator associated with an analytic vector field is bounded on \(L^2\). In the present paper, we give a geometric proof of Bourgain’s result by using the tools developed by Lacey and Li in (Trans Am Math Soc 358(9):4099–4117, 2006) and (Mem Am Math Soc 205 (965):viii+72, 2010).