On weak* convergent sequences in duals of symmetric spaces of <Emphasis Type="Italic">τ</Emphasis>-measurable operators

It is shown that the pre-dual of a σ-finite von Neumann algebra has property (k) in the sense of Figiel, Johnson and Pelczyński [12]. This resolves in the affirmative an open question raised in [12]. It is shown further that a weakly sequentially complete symmetric space E of τ-measurable operators affiliated with a semifinite σ-finite von Neumann algebra has property (k).     

Topological nature of the inertiality of a nonequilibrium system far from equilibrium

We consider inertial and topological properties of a nonequilibrium system far from the equilibrium. We show that from the topological standpoint, the layered structure of the energy levels of a nonequilibrium system leads to the concept of connectivity. We state an assumption on the key role of the dimension of the hypersurface of the full entropy production in the study of the inertiality of an irreversible process in a nonequilibrium system including in the region of unstable states.     

Automorphisms of the mapping class group of a nonorientable surface

Let \(\mathcal {X}_S\) denote the class of spaces homeomorphic to two closed orientable surfaces of genus greater than one identified to each other along an essential simple closed curve in each surface. Let \(\mathcal {C}_S\) denote the set of fundamental groups of spaces in \(\mathcal {X}_S\). In this paper, we characterize the abstract commensurability classes within \(\mathcal {C}_S\) in terms of the ratio of the Euler characteristic of the surfaces identified and the topological type of the curves identified. We prove that all groups in \(\mathcal {C}_S\) are quasi-isometric by exhibiting a bilipschitz map between the universal covers of two spaces in \(\mathcal {X}_S\). In particular, we prove that the universal covers of any two such spaces may be realized as isomorphic cell complexes with finitely many isometry types of hyperbolic polygons as cells. We analyze the abstract commensurability classes within \(\mathcal {C}_S\): we characterize which classes contain a maximal element within \(\mathcal {C}_S\); we prove each abstract commensurability class contains a right-angled Coxeter group; and, we construct a common CAT(0) cubical model geometry for each abstract commensurability class.     

A maximum of the first eigenvalue of semibounded differential operator with a parameter

We consider a self-adjoint differential operator in Hilbert space. Then the domain of the operator is changed by the perturbation of the boundary conditions so that a given neighborhood “is cleared” from the points of the spectrum of the perturbed operator. For the Sturm–Liouville operator on the segment and the Laplace operator on the square such a possibility is attained via integral perturbations of boundary conditions.     

Bounded gaps between primes and the length spectra of arithmetic hyperbolic 3-orbifolds

In 1992, Reid asked whether hyperbolic 3-manifolds with the same geodesic length spectra are necessarily commensurable. While this is known to be true for arithmetic hyperbolic 3-manifolds, the non-arithmetic case is still open. Building towards a negative answer to this question, Futer and Millichap recently constructed infinitely many pairs of non-commensurable, non-arithmetic hyperbolic 3-manifolds which have the same volume and whose length spectra begin with the same first m geodesic lengths. In the present paper, we show that this phenomenon is surprisingly common in the arithmetic setting. In particular, given any arithmetic hyperbolic 3-orbifold derived from a quaternion algebra, any finite subset S of its geodesic length spectrum, and any $k\ge 2$, we produce infinitely many k-tuples of arithmetic hyperbolic 3-orbifolds which are pairwise non-commensurable, have geodesic length spectra containing S, and have volumes lying in an interval of (universally) bounded length. The main technical ingredient in our proof is a bounded gaps result for prime ideals in number fields lying in Chebotarev sets which extends recent work of Thorner.     

Computer-Assisted Analysis of the Anderson–Hájek Ontological Controversy

A universal reasoning approach based on shallow semantical embeddings of higher-order modal logics into classical higher-order logic is exemplarily employed to analyze several modern variants of the ontological argument on the computer. Several novel findings are reported which contribute to the clarification of a long-standing dispute between Anderson and Hájek. The technology employed in this work, which to some degree realizes Leibniz’s dream of a characteristica universalis and a calculus ratiocinator for solving philosophical controversies, is ready to be fruitfully adopted in larger scale by philosophers.