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We show that general triple planes with genus and irregularity zero belong to at most 12 families, that we call surfaces of type I to XII, and we prove that the corresponding Tschirnhausen bundle is a direct sum of two line bundles in cases I, II, III, whereas it is a rank 2 Steiner bundle in the remaining cases. We also provide existence results and explicit descriptions for surfaces of type I to VII, recovering all classical examples and discovering several new ones. In particular, triple planes of type VII provide counterexamples to a wrong claim made in 1942 by Bronowski. Finally, in the last part of the paper we discuss some moduli problems related to our constructions. [ABSTRACT FROM AUTHOR]

In this paper we introduce new densities on the set of primes of a number field. If K/K0 is a Galois extension of number fields, we associate to any element x ∈ ... a density δK/K0,x on the primes of K. In particular, the density associated to x = 1 is the usual Dirichlet density on K. We also give two applications of these densities (for x ≠ 1): the first is a realization result à la the Grunwald-Wang theorem such that essentially, ramification is only allowed in a set of arbitrarily small (positive) Dirichlet density. The second concerns the so-called saturated sets of primes, introduced by Wingberg. [ABSTRACT FROM AUTHOR]

In this paper we study various "abscissae" which one can associate to a given function f, or rather to the power moments of f. These are motivated by long-standing open problems in analytic number theory. We show how these abscissae connect to the distribution of values of f in a very elegant way using convex conjugates. This connection allows us to show which abscissae are realizable for both general and more specific arithmetical functions. Further it may give a new approach to, for example, Dirichlet's divisor problem. [ABSTRACT FROM AUTHOR]

At the 1987 Ramanujan Centenary meeting Dyson asked for a coherent group-theoretical structure for Ramanujan's mock theta functions analogous to Hecke's theory of modular forms. Many of Ramanujan's mock theta functions can be written in terms of R(ζ, q), where R(z, q) is the two-variable generating function of Dyson's rank function and ζ is a root of unity. Building on earlier work of Watson, Zwegers, Gordon, and McIntosh, and motivated by Dyson's question, Bringmann, Ono, and Rhoades studied transformation properties of R(ζ, q). In this paper we strengthen and extend the results of Bringmann, Rhoades, and Ono, and the later work of Ahlgren and Treneer. As an application we give a new proof of Dyson's rank conjecture and show that Ramanujan's Dyson rank identity modulo 5 from the Lost Notebook has an analogue for all primes greater than 3. The proof of this analogue was inspired by recent work of Jennings-Shaffer on overpartition rank differences mod 7. [ABSTRACT FROM AUTHOR]

Let μ be a Borel measure on a compactum X. The main objects in this paper are s-ideals Ipdimq, J0pμq, Jf pμq of Borel sets in X that can be covered by countably many compacta which are finite-dimensional, or of μ-measure null, or of finite μ-measure, respectively. Answering a question of J. Zapletal, we shall show that for the Hilbert cube, the s-ideal Ipdimq is not homogeneous in a strong way. We shall also show that in some natural instances of measures μ with nonhomogeneous s-ideals J0pμq or Jf pμq, the completions of the quotient Boolean algebras BorelpXq{J0pμq or BorelpXq{Jf pμq may be homogeneous. We discuss the topic in a more general setting, involving calibrated s-ideals. [ABSTRACT FROM AUTHOR]

In the 1960s Maurice Auslander proved the following important result. Let R be the commutative polynomial ring C[x1, . . ., xn], and let G be a finite small subgroup of GLn(C) acting on R naturally. Let A be the fixed subring RG:= {a ∈ R|g(a) = a for all g ∈ G}. Then the endomorphism ring of the right A-module RA is naturally isomorphic to the skew group algebra R* G. In this paper, a version of the Auslander theorem is proven for the following classes of noncommutative algebras: (a) noetherian PI local (or connected graded) algebras of finite injective dimension, (b) universal enveloping algebras of finite-dimensional Lie algebras, and (c) noetherian graded down-up algebras. [ABSTRACT FROM AUTHOR]

A quasi-infinitely divisible distribution on R is a probability distribution whose characteristic function allows a L'evy-Khintchine type representation with a "signed L'evy measure", rather than a L'evy measure. Quasiinfinitely divisible distributions appear naturally in the factorization of infinitely divisible distributions. Namely, a distribution μ is quasi-infinitely divisible if and only if there are two infinitely divisible distributions μ1 and μ2 such that μ1 * μ = μ2. The present paper studies certain properties of quasiinfinitely divisible distributions in terms of their characteristic triplet, such as properties of supports, finiteness of moments, continuity properties, and weak convergence, with various examples constructed. In particular, it is shown that the set of quasi-infinitely divisible distributions is dense in the set of all probability distributions with respect to weak convergence. Further, it is proved that a distribution concentrated on the integers is quasi-infinitely divisible if and only if its characteristic function does not have zeroes, with the use of the Wiener-L'evy theorem on absolutely convergent Fourier series. A number of fine properties of such distributions are proved based on this fact. A similar characterization is not true for nonlattice probability distributions on the line. [ABSTRACT FROM AUTHOR]

In this paper, we establish the following reverse Stein-Weiss inequality, namely the reversed weighted Hardy-Littlewood-Sobolev inequality, in Rn: ... for any nonnegative functions ... and ... such that .... We derive the existence of extremal functions for the above inequality. Moreover, some asymptotic behaviors are obtained for the corresponding Euler-Lagrange system. For an analogous weighted system, we prove necessary conditions of existence for any positive solutions by using the Pohozaev identity. Finally, we also obtain the corresponding Stein-Weiss and reverse Stein-Weiss inequalities on the ndimensional sphere Sn by using the stereographic projections. Our proof of the reverse Stein-Weiss inequalities relies on techniques in harmonic analysis and differs from those used in the proof of the reverse (non-weighted) Hardy-Littlewood-Sobolev inequalities. [ABSTRACT FROM AUTHOR]

In Theorem 3.1 of The rigidity theorems of self-shrinkers (2014), the author and Y. L. Xin proved a rigidity result for self-shrinkers under the integral condition on the norm of the second fundamental form. In this paper, we relax such a bound to any finite constant (see Theorem 4.4 for details). [ABSTRACT FROM AUTHOR]

In this paper we study some determinant inequalities and matrix inequalities which have a geometrical flavour. We first examine some inequalities which place work of Macbeath in a more general setting and also relate to recent work of Gressman. In particular, we establish optimisers for these determinant inequalities. We then use these inequalities to establish our Main Theorem, which gives a geometric inequality of matrix type which improves and extends some inequalities of Christ. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a sharp lower bound estimate for diameters with respect to an adapted metric in closed pseudohermitian (2n + 1)- manifolds when a sharp lower bound estimate for the first positive eigenvalue of the sublaplacian is achieved. As a consequence, we confirm the CR Obata Conjecture on a closed pseudohermitian (2n + 1)-manifold with an extra condition on covariant derivatives of torsion. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a restricted version of Bergman kernels for high powers of a big line bundle over a smooth projective variety. The geometric meaning of the leading term is specified. As a byproduct, we derive some integral representations for the restricted volume. [ABSTRACT FROM AUTHOR]

Let Sg (g ≥ 2) be a closed surface of genus g. Let K be any real number field, and let A be any quaternion algebra over K such that A ⊗K ℝ = M2(ℝ.). We show that there exists a hyperbolic structure on Sg such that K and A arise as its invariant trace field and invariant quaternion algebra. [ABSTRACT FROM AUTHOR]

We strengthen the volume inequalities for Lp zonoids of even isotropic measures and for their duals, which are originally due to Ball, Barthe, and Lutwak, Yang, and Zhang. The special case p = ∞ yields a stability version of the reverse isoperimetric inequality for centrally symmetric convex bodies. Adding to known inequalities and stability results for the reverse isoperimetric inequality of arbitrary convex bodies, we state a conjecture on volume inequalities for Lp zonoids of general centered (non-symmetric) isotropic measures. We achieve our main results by strengthening Barthe's measure transportation proofs of the rank one case of the geometric Brascamp--Lieb and reverse Brascamp--Lieb inequalities by estimating the derivatives of the transportation maps for a special class of probability density functions. Based on our argument, we phrase a conjecture about a possible stability version of the Brascamp--Lieb and reverse Brascamp--Lieb inequalities. We also establish some geometric properties of the distribution of general isotropic measures that are essential to our argument. In particular, we prove a measure theoretic analog of the Dvoretzky--Rogers lemma. [ABSTRACT FROM AUTHOR]

For the Dunkl transforms associated with the weight functions Hk²(x) = ∏j=1d ..., k1, ⋯, kd ≥ 0 on ℝd, it is proved that if p ≥ 2 + 1/λk and λk := d-1/2 + ∑j=1dkj, the maximal Bochner-Riesz operator B*δ(hk²; f) order δ > 0 is bounded on the space Lp(ℝd; hk² dx) if and only if δ > δk(p) := max{(2λk +1)(1/2 -- 1/p ) -- 1/2, 0}. This extends a well known result of M. Christ for the classical Fourier transforms (Proc. Amer. Math. Soc. 95 (1985), 16-20). The proof relies on a new local restriction theorem for the Dunkl transforms, which is stronger than the corresponding global restriction theorem, but significantly more difficult to prove. [ABSTRACT FROM AUTHOR]

Assume that D is a Krull-Schmidt, Hom-finite triangulated category with a Serre functor and a cluster-tilting object T. We introduce the notion of relative cluster tilting objects, and T[1]-cluster tilting objects in D, which are a generalization of cluster-tilting objects. When D is 2-Calabi-Yau, the relative cluster tilting objects are cluster-tilting. Let Λ = EndDop(T) be the opposite algebra of the endomorphism algebra of T. We show that there exists a bijection between T[1]-cluster tilting objects in D and support τ-tilting Λ- modules, which generalizes a result of Adachi--Iyama--Reiten [τ-tilting theory, Compos. Math. 150 (2014), no. 3, 415-452]. We develop a basic theory on T[1]-cluster tilting objects. In particular, we introduce a partial order on the set of T[1]-cluster tilting objects and mutation of T[1]-cluster tilting objects, which can be regarded as a generalization of "cluster-tilting mutation". As an application, we give a partial answer to a question posed in Adachi--Iyama--Reiten, loc. cit. [ABSTRACT FROM AUTHOR]

Let K be a number field with algebraic closure ..., and let S be a finite set of places of K containing all the infinite ones. Let Γ0 be a finitely generated subgroup of 픾m..., and let Γ ⊂ 픾m... be the division group attached to Γ0. Here is an illustration of what we will prove in this article. Fix a proper closed subinterval I of [0, ∞) and a nonzero effective divisor D on 픾m which is not the translate of any torsion divisor on the algebraic torus 픾m by any point of Γ with height belonging to I. Then we prove a statement which easily implies that the set of "integral division points on 픾m with height near I", i.e., the set of points of Γ with (standard absolute logarithmic Weil) height in J which are S-integral on 픾m relative to D, is finite for some fixed subinterval J of [0, ∞) properly containing I. We propose a conjecture on the nondensity of integral division points on semi-abelian varieties with prescribed height values, which generalizes some previously known conjectures as well as this finiteness result for 픾m. Finally, we also propose an analogous version for a dynamical system on P¹. [ABSTRACT FROM AUTHOR]

All SL(n) covariant vector valuations on convex polytopes in Rn are completely classified without any continuity assumptions. The moment vector turns out to be the only such valuation if n = 3, while two new functionals show up in dimension two. [ABSTRACT FROM AUTHOR]

We construct a computable sequence of computable reals (Xi) such that any real that can compute a subsequence that is maximal with respect to the finite intersection property can also compute a Cohen 1-generic. This is extended to establish the same result with 2IP in place of FIP. This is the first example of a classical theorem of mathematics that has been found to be equivalent, both proof theoretically and in terms of its effective content, to computing a 1-generic. [ABSTRACT FROM AUTHOR]

Using methods of descriptive theory it is shown that the classification problem for wild knots is strictly harder than that for countable structures. [ABSTRACT FROM AUTHOR]

What largeness and structural assumptions on A ? [R]ω can guarantee the existence of a non-empty perfect set P ? R such that [P]ω ? A? Such a set P is called A-homogeneous. We show that even if A is open, in general it is independent of ZFC whether for a cardinal κ, the existence of an A-homogeneous set H ε [R]κ implies the existence of a non-empty perfect A-homogeneous set. On the other hand, we prove an infinite dimensional analogue of Mycielski's Theorem: if A is large in the sense of a suitable Baire category-like notion, then there exists a non-empty perfect A-homogeneous set. We introduce fusion games to prove this and other infinite dimensional perfect set theorems. Finally we apply this theory to show that it is independent of ZFC whether Tukey reductions of the maximal analytic cofinal type can be witnessed by definable Tukey maps. [ABSTRACT FROM AUTHOR]

In this work, we are interested in developing new directions of the famous T(1)-theorem. More precisely, we develop a general framework where we look to replace the John-Nirenberg space BMO (in the classical result) by a new BMOL, associated to a semigroup of operators (ea-tL)t>0. These new spaces BMOL (including BMO) have recently appeared in numerous works in order to extend the theory of Hardy and BMO space to more general situations. Then we give applications by describing boundedness for a new kind of paraproduct, built on the considered semigroup. In addition we obtain a version of the classical T(1)-theorem for doubling Riemannian manifolds. [ABSTRACT FROM AUTHOR]

We prove that there exist only finitely many commuting squares of finite dimensional *-algebras of fixed dimension, satisfying a "large second relative commutant" condition. We show this by studying the local minima of w → dim(A ∩ wBw*), where A,B are fixed subalgebras of some *-algebra C and w ε C is a unitary. When applied to lattices arising from subfactors satisfying a certain extremality- like condition, our result yields Ocneanu's finiteness theorem for the standard invariants of such finite depth subfactors. [ABSTRACT FROM AUTHOR]

We give sharp sectional curvature estimates for complete immersed cylindrically bounded m-submanifolds φ: Mm → Nn-l Χ Rl, n+l ≤ 2m-1, provided that either φ is proper with the norm of the second fundamental form with certain controlled growth or M has scalar curvature with strong quadratic decay. The latter gives a non-trivial extension of the Jorge-Koutrofiotis Theorem. In the particular case of hypersurfaces, that is, m = n - 1, the growth rate of the norm of the second fundamental form is improved. Our results will be an application of a generalized Omori-Yau Maximum Principle for the Hessian of a Riemannian manifold, in its newest elaboration given by Pigola, Rigoli and Setti (2005). [ABSTRACT FROM AUTHOR]