Schrödinger’s cat and the conception of probability in item response theory

Although not presenting novel graphics or commenting on the quality of graphics, this article discusses how three similar-looking graphs can have quite different meanings. If careful interpretation is critical for such familiar graphs, then careful presentation and reading certainly is needed for more involved illustrations of data and models. This work was supported by the National Board of Medical Examiners. In addition, many of the key ideas contained here, as well as some of the explicit facts, were taken from Paul Holland’s 1989 presidential address to the Psychometric Society.     

Extensions of the disc algebra and of Mergelyanʼs theorem

We investigate the uniform limits of the set of polynomials on the closed unit disc $\overline{D}$ with respect to the chordal metric χ. More generally, we examine analogous questions replacing $\mathbb{C}\cup \{\infty \}$ by other metrizable compactifications of $\mathbb{C}$.     

A note on two Erdősʼs proofs of the infinitude of primes

P. Erdős found numerous theorems, problems, results and conjectures in elementary number theory. Some of them are two Erdősʼs proofs of of the famous Euclidʼs theorem on the infinitude of primes. As noticed below, one of these proofs immediately implies the fact that the number of primes smaller than x is $⩾\log x/(2\log 2)$.     

Derivation of Hartreeʼs theory for generic mean-field Bose systems

In this paper we provide a novel strategy to prove the validity of Hartree?s theory for the ground state energy of bosonic quantum systems in the mean-field regime. For the known case of trapped Bose gases, this can be shown using the strong quantum de Finetti theorem, which gives the structure of infinite hierarchies of k-particles density matrices. Here we deal with the case where some particles are allowed to escape to infinity, leading to a lack of compactness. Our approach is based on two ingredients: (1) a weak version of the quantum de Finetti theorem, and (2) geometric techniques for many-body systems. Our strategy does not rely on any special property of the interaction between the particles. In particular, our results cover those of Benguria–Lieb and Lieb–Yau for, respectively, bosonic atoms and boson stars.     

Revisiting the foundations of Barbilian?s metrization procedure

In the present work we prove that one of Barbilian?s theorems from 1960 regarding the metrization procedure in the plane admits a natural extension depending on a bilinear form and the relative position of two Apollonian hyperspheres. This result allows us to pursue two fundamental ideas. First, that all the distances with constant curvature can be described by Barbilian?s metrization principle. Secondly, that all the Riemannian metric corresponding to these distances can be obtained with the same unique procedure derived from the main theorem in the text (Theorem 2.5). We show how the hyperbolic metric of the disk, the hyperbolic metric on the exterior of the disk and the hyperbolic metric on the half-plane can be obtained in the same way using Theorem 2.5, which appears here for the first time and is an extension of a Barbilian classical result (Barbilian, 1960 [7]). Furthermore, we obtain metrics corresponding to quadratic forms with signature that includes minus. By considering the norms provided by either Lorentz or Minkowski (pseudo-)inner product as influence functions, two oscillant distances can be generated in some subsets of Lorentz or Minkowski plane. The extension of 1960 Barbilian?s theorem mentioned above allow us to obtain the metrics attached to these two Barbilian distances on corresponding subsets of Lorentz and Minkowski 2-dimensional spaces. The geometric study concludes that these metrics are generalized Lagrange metrics. A result concerning the distance induced by a Riemannian metric as a local Barbilian distance is also proved.     

Cheeger’s cut,maxcut and the spectral theory of 1-Laplacian on graphs

This is primarily an expository paper surveying up-to-date known results on the spectral theory of 1-Laplacian on graphs and its applications to the Cheeger cut, maxcut and multi-cut problems. The structure of eigenspace, nodal domains, multiplicities of eigenvalues, and algorithms for graph cuts are collected.     

Order of Selbergʼs and Ruelleʼs zeta functions for compact even-dimensional locally symmetric spaces

We prove that Selberg?s and Ruelle?s zeta functions considered by U. Bunke and M. Olbrich can be represented as quotients of two entire functions of order not larger than the dimension of the underlaying compact, even-dimensional, locally symmetric space.     

On A.I. Koshelev’s work on regularity theory of generalized solutions of quasilinear systems of elliptic and parabolic type

This is a survey of A.I. Koshelev’s studies in the theory of regular solutions of boundary value problems, based on iterative processes converging in both the energy norm and the strong norm as well as on a priori estimates in weighted function spaces. In numerous cases, Koshelev’s estimates contain explicitly computable and sometimes sharp (unimprovable) constants. The results obtained for a broad class of problems were adapted by Koshelev to the study of boundary value problems of nonlinear elasticity and problems of hydrodynamics of viscous fluids.     

Katoʼs residue homomorphisms and reciprocity laws on arithmetic surfaces

We explicitly study Kato?s residue homomorphisms in Milnor K-theory, which are closely related to Contou-Carrère symbols. As applications we establish several reciprocity laws for certain locally defined maps on K-groups that are associated to arithmetic surfaces.     

Maximum modulus points of meromorphic functions and Valironʼs defect

The initial question of Paul Erdös concerned the existence of a non-polynomial entire function with the number of maximum modulus points tending to infinity. Later on the issue was considered also for meromorphic functions, leading to some interesting results on separated maximum modulus points and, in particular, their connection with Petrenkoʼs deviation and Valironʼs defect.