Analyticity and the Deviant Logician: Williamson’s Argument from Disagreement

One way to discredit the suggestion that a statement is true just in virtue of its meaning is to observe that its truth is the subject of genuine disagreement. By appealing to the case of the unorthodox philosopher, Timothy Williamson has recast this response as an argument foreclosing any appeal to analyticity. Reconciling Quine’s epistemological holism with his treatment of the ‘deviant logician’, I show that we may discharge the demands of charitable interpretation even while attributing trivial semantic error to Williamson’s philosophers. Williamson’s effort to generalize the argument from disagreement therefore fails.     

Preface of the Special Issue on Symplectic Geometry and Mathematical Physics

<正>Symplectic geometry is a branch of differential geometry and differential topology and has its origins in the Hamiltonian formulation of classical mechanics. In the last few decades, symplectic geometry has experienced enormous progress and has had interactions with many other branches of mathematics, including enumerative geometry, low-dimensional topology, mathematical physics, Hamiltonian dynamics,     

Pál-type interpolation on nonuniformly distributed nodes on the unit circle

We study here the regularity of certain Pál-type interpolation problems involving the Möbius transforms of the zeros of zⁿ+1 and zⁿ−1 along with one additional point ζ or two additional points ζ and 1.     

The Halász Inequality for Additive Function of Rational Argument

We obtain an upper bound for the quantity . Here I is an interval, is the set of rational numbers q=m/n I such that nx, and f(q) is an arbitrary real-valued additive function of rational argument. The interval I and function f may depend on x3.     

A short note on the derivation of the elastic von Kármán shell theory

We derive the Γ-limit of scaled elastic energies h^?4E ^h (u ^h ) associated with deformations u ^h of a family of thin shells \({S^h} = \left\{ {z = x + t\vec n\left( x \right);x \in S, - g_1^h\left( x \right) < t < g_2^h\left( x \right)} \right\}\). The obtained von Kármán theory is valid for a general sequence of boundaries g ₁ ^h , g ₂ ^h converging to 0 in an appropriate manner as h vanishes. Our analysis relies on the techniques and extends the results in [10] and [11].     

Ball polytopes and the Vázsonyi problem

Let V be a finite set of points in the Euclidean d-space (d ≧ 2). The intersection of all unit balls B(υ, 1) centered at υ, where υ ranges over V, henceforth denoted by $ \mathcal{B} $ (V) is the ball polytope associated with V. After some preparatory discussion on spherical convexity and spindle convexity, the paper focuses on two central themes. (a) Define the boundary complex of $ \mathcal{B} $ (V), i.e., define its vertices, edges and facets in dimension 3, and investigate its basic properties. (b) Apply results of this investigation to characterize finite sets of diameter 1 in the (Euclidean) 3-space for which the diameter is attained a maximal number of times as a segment (of length 1) with both endpoints in V. A basic result for such a characterization goes back to Grünbaum, Heppes and Straszewicz, who proved independently of each other, in the late 1950’s by means of ball polytopes, that the diameter of V is attained at most 2|V| ? 2 times. Call V extremal if its diameter is attained this maximal number (2|V| ? 2) of times. We extend the aforementioned result by showing that V is extremal iff V coincides with the set of vertices of its ball polytope $ \mathcal{B} $ (V) and show that in this case the boundary complex of $ \mathcal{B} $ (V) is self-dual in some strong sense. The problem of constructing new types of extremal configurations is not addressed in this paper, but we do present here some such new types.     

A remark on the conjecture of Erdös, Faber and Lovász

The celebrated Erd?s, Faber and Lovász Conjecture may be stated as follows: Any linear hypergraph on ν points has chromatic index at most ν. We show that the conjecture is equivalent to the following assumption: For any graph , where ν(G) denotes the linear intersection number and χ(G) denotes the chromatic number of G. As we will see for any graph G = (V, E), where denotes the complement of G. Hence, at least G or fulfills the conjecture.      

The Kalām Cosmological Argument,the Big Bang,and Atheism

While there has been much work on cosmological arguments, novel objections will be presented against the modern day rendition of the Kalām cosmological argument as standardly articulated by William Lane Craig. The conclusion is reached that this cosmological argument and several of its variants do not lead us to believe that there is inevitably a supernatural cause to the universe. Moreover, a conditional argument for atheism will be presented in light of the Big Bang Theory.     

A lower bound on the Chvátal-rank of Antiwebs

In [Holm, E., L. M. Torres and A. K. Wagler, On the Chvátal-rank of linear relaxations of the stable set polytope, International Transactions in Operational Research 17 (2010), pp. 827–849; Holm, E., L. M. Torres and A. K. Wagler, On the Chvátal-rank of Antiwebs, Electronic Notes in Discrete Mathematics 36 (2010), pp. 183–190] we study the Chvátal-rank of the edge constraint and the clique constraint stable set polytopes related to antiwebs. We present schemes for obtaining both upper and lower bounds. Moreover, we provide an algorithm to compute the exact values of the Chvátal-rank for all antiwebs with up to 5,000 nodes. Here we prove a lower bound as a closed formula and discuss some cases when this bound is tight.     

Combinatorial results on the fitting problems of the multivariate gamma distribution introduced by Prékopa and Szántai

Generalizations of the results of an earlier paper of the second author, related to the problem of fitting a multivarite gamma distribution to empirical data, are discussed in the paper. The multivariate gamma distribution under consideration is the one that was introduced in the paper of Prékopa and Szántai (in Water Resources Research, 14:19?C24, 1978), some earlier results on the fitting problem were given in the paper of Szántai (in Alkalmazott Matematikai Lapok 10:35?C60, 1984). In the present paper it is proved that the necessary conditions given earlier are not sufficient and some further new, mostly computational results are provided, too. Using the more efficient computation tools we are able now to give the sufficient conditions for dimensions 5 and 6 as well. For higher dimensions we have only necessary conditions and the invention of a suitable necessary and sufficient condition remains an open problem when n is greater than 6. The miscellaneousness of the necessary and sufficient conditions obtained in our new project for n=6 indicates that finding necessary and sufficient conditions in general should be a very hard problem.     

Glimpses of the Octonions and Quaternions History and Today’s Applications in Quantum Physics

Before we dive in this essay into the accessibility stream of nowadays indicatory applications of octonions and quaternions to computer and other sciences and to quantum physics (see for example [50-53], [41], [33]) and to Clifford algebras (see for example [17,16], 18) let us focus for a while on the crucially relevant events for today’s revival on interest to nonassociativities while the role of associative quaternions in eight periodicity constructive classification of associative Clifford algebras is now a text-book knowledge.     

The Turán-type inequality in the space L0 on the unit interval

Analysis Mathematica - Let P(x) be an arbitrary algebraic polynomial of degree n with all zeros in the unit interval ?1 ≤ x ≤ 1. We establish the Turán-type inequality...     

On the Lovász theta function and some variants

The Lovász theta function of a graph is a well-known upper bound on the stability number. It can be computed efficiently by solving a semidefinite program (SDP). Actually, one can solve either of two SDPs, one due to Lovász and the other to Grötschel et al. The former SDP is often thought to be preferable computationally, since it has fewer variables and constraints. We derive some new results on these two equivalent SDPs. The surprising result is that, if we weaken the SDPs by aggregating constraints, or strengthen them by adding cutting planes, the equivalence breaks down. In particular, the Grötschel et al. scheme typically yields a stronger bound than the Lovász one.     

On the Ramsey-Turán numbers of graphs and hypergraphs

Let t be an integer, f(n) a function, and H a graph. Define the t-Ramsey-Turán number of H, RT _t (n,H, f(n)), to be the maximum number of edges in an n-vertex, H-free graph G with α _t (G) ≤ f(n), where α _t (G) is the maximum number of vertices in a K _t -free induced subgraph of G. Erd?s, Hajnal, Simonovits, Sós and Szemerédi [6] posed several open questions about RT _t (n,K _s , o(n)), among them finding the minimum ? such that RT _t (n,K _t+? , o(n)) = Ω(n ²), where it is easy to see that RT _t (n,K _t+1, o(n)) = o(n ²). In this paper, we answer this question by proving that RT _t (n,K _t+2, o(n)) = Ω(n ²); our constructions also imply several results on the Ramsey-Turán numbers of hypergraphs.     

On the projective Finsler metrizability and the integrability of Rapcsák equation

A. Rapcsák obtained necessary and sufficient conditions for the projective Finsler metrizability in terms of a second order partial differential system. In this paper we investigate the integrability of the Rapcsák system and the extended Rapcsák system, by using the Spencer version of the Cartan-Kähler theorem. We also consider the extended Rapcsák system completed with the curvature condition. We prove that in the non-isotropic case there is a nontrivial Spencer cohomology group in the sequences determining the 2-acyclicity of the symbol of the corresponding differential operator. Therefore the system is not integrable and higher order obstruction exists.     

A Remark on the Weak Topology and the Degree

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A note on the Structure of Turán Densities of Hypergraphs

Let r ≥ 2 be an integer. A real number α ∈ [0, 1) is a jump for r if there exists c > 0 such that no number in (α, α + c) can be the Turán density of a family of r-uniform graphs. A result of Erd?s and Stone implies that every α ∈ [0, 1) is a jump for r = 2. Erd?s asked whether the same is true for r ≥ 3. Frankl and Rödl gave a negative answer by showing an infinite sequence of non-jumps for every r ≥ 3. However, there are still a lot of open questions on determining whether or not a number is a jump for r ≥ 3. In this paper, we first find an infinite sequence of non-jumps for r = 4, then extend one of them to every r ≥ 4. Our approach is based on the techniques developed by Frankl and Rödl.