Extending the notion of regularity introduced by Dickson in 1939, a positive definite ternary integral quadratic form is said to be spinor regular if it represents all the positive integers represented by its spinor genus (that is, all positive integers represented by any form in its spinor genus). Jagy conducted an extensive computer search for primitive ternary quadratic forms that are spinor regular, but not regular, resulting in a list of 29 such forms. In this paper, we will prove that there are no additional forms with this property.

In this paper we investigate the moments and the distribution of $L(1,\unicode[STIX]{x1D712}_{D})$, where $\unicode[STIX]{x1D712}_{D}$ varies over quadratic characters associated to square-free polynomials $D$ of degree $n$ over $\mathbb{F}_{q}$, as $n\rightarrow \infty$. Our first result gives asymptotic formulas for the complex moments of $L(1,\unicode[STIX]{x1D712}_{D})$ in a large uniform range. Previously, only the first moment has been computed due to the work of Andrade and Jung. Using our asymptotic formulas together with the saddle-point method, we show that the distribution function of $L(1,\unicode[STIX]{x1D712}_{D})$ is very close to that of a corresponding probabilistic model. In particular, we uncover an interesting feature in the distribution of large (and small) values of $L(1,\unicode[STIX]{x1D712}_{D})$, which is not present in the number field setting. We also obtain $\unicode[STIX]{x1D6FA}$-results for the extreme values of $L(1,\unicode[STIX]{x1D712}_{D})$, which we conjecture to be the best possible. Specializing $n=2g+1$ and making use of one case of Artin’s class number formula, we obtain similar results for the class number $h_{D}$ associated to $\mathbb{F}_{q}(T)[\sqrt{D}]$. Similarly, specializing to $n=2g+2$ we can appeal to the second case of Artin’s class number formula and deduce analogous results for $h_{D}R_{D}$, where $R_{D}$ is the regulator of $\mathbb{F}_{q}(T)[\sqrt{D}]$.

The main result of this paper establishes a bijection between the set of equivalence classes of simple transitive 2-representations with a fixed apex ${\mathcal{J}}$ of a fiat 2-category $\mathscr{C}$ and the set of equivalence classes of faithful simple transitive 2-representations of the fiat 2-subquotient of $\mathscr{C}$ associated with a diagonal ${\mathcal{H}}$-cell in ${\mathcal{J}}$. As an application, we classify simple transitive 2-representations of various categories of Soergel bimodules, in particular, completing the classification in types $B_{3}$ and $B_{4}$.

It is well known that if X is a CW-complex, then for every weak homotopy equivalence f : A → B, the map f* : [X, A] → [X, B] induced in homotopy classes is a bijection. In fact, up to homotopy equivalence, only CW-complexes have that property. Now, for which spaces X is f* : [B, X] → [A, X] a bijection for every weak equivalence f? This question was considered by J. Strom and T. Goodwillie. In this note we prove that a non-empty space inverts weak equivalences if and only if it is contractible.

Braces were introduced by Rump in 2007 as a useful tool in the study of the set-theoretic solutions of the Yang–Baxter equation. In fact, several aspects of the theory of finite left braces and their applications in the context of the Yang–Baxter equation have been extensively investigated recently. The main aim of this paper is to introduce and study two finite brace theoretical properties associated with nilpotency, and to analyse their impact on the finite solutions of the Yang–Baxter equation.

Let U be the unipotent radical of a Borel subgroup of a connected reductive algebraic group G, which is defined over an algebraically closed field k. In this paper, we extend work by Goodwin and Röhrle concerning the commuting variety of Lie(U) for Char(k) = 0 to fields whose characteristic is good for G.

The algebraic EHP sequences, algebraic analogues of the EHP sequences in homotopy theory, are important tools in algebraic topology. This note will outline two new proofs of the existence of the algebraic EHP sequences. The first proof is derived from the minimal injective resolution of the reduced singular cohomology of spheres, and the second one follows Bousfield's idea using the loop functor of unstable modules.

We investigate the structure of root data by considering their decomposition as a product of a semisimple root datum and a torus. Using this decomposition, we obtain a parametrization of the isomorphism classes of all root data. By working at the level of root data, we introduce the notion of a smooth regular embedding of a connected reductive algebraic group, which is a refinement of the commonly used regular embeddings introduced by Lusztig. In the absence of Steinberg endomorphisms, such embeddings were constructed by Benjamin Martin. In an unpublished manuscript, Asai proved three key reduction techniques that are used for reducing statements about arbitrary connected reductive algebraic groups, equipped with a Frobenius endomorphism, to those whose derived subgroup is simple and simply connected. Using our investigations into root data we give new proofs of Asai's results and generalize them so that they are compatible with Steinberg endomorphisms. As an illustration of these ideas, we answer a question posed to us by Olivier Dudas concerning unipotent supports.

In this paper we define almost gentle algebras, which are monomial special multiserial algebras generalizing gentle algebras. We show that the trivial extension of an almost gentle algebra by its minimal injective co-generator is a symmetric special multiserial algebra and hence a Brauer configuration algebra. Conversely, we show that any almost gentle algebra is an admissible cut of a unique Brauer configuration algebra and, as a consequence, we obtain that every Brauer configuration algebra with multiplicity function identically one is the trivial extension of an almost gentle algebra. We show that a hypergraph is associated with every almost gentle algebra A, and that this hypergraph induces the Brauer configuration of the trivial extension of A. Among other things, this gives a combinatorial criterion to decide when two almost gentle algebras have isomorphic trivial extensions.

We consider the problem of counting the number of rational points of bounded height in the zero-loci of Brauer group elements on semi-simple algebraic groups over number fields. We obtain asymptotic formulae for the counting problem for wonderful compactifications using the spectral theory of automorphic forms. Applications include asymptotic formulae for the number of matrices over $\mathbb{Q}$ whose determinant is a sum of two squares. These results provide a positive answer to some cases of a question of Serre concerning such counting problems.

We study a class of optimal transport planning problems where the reference cost involves a non-linear function G(x, p) representing the transport cost between the Dirac measure δx and a target probability p. This allows to consider interesting models which favour multi-valued transport maps in contrast with the classical linear case ($G(x,p)=\int c(x,y)dp$) where finding single-valued optimal transport is a key issue. We present an existence result and a general duality principle which apply to many examples. Moreover, under a suitable subadditivity condition, we derive a Kantorovich–Rubinstein version of the dual problem allowing to show existence in some regular cases. We also consider the well studied case of Martingale transport and present some new perspectives for the existence of dual solutions in connection with Γ-convergence theory.

In the study of the spectra of algebras of holomorphic functions on a Banach space E, the bidual E″ has a central role, and the spectrum is often shown to be locally homeomorphic to E″. In this paper we consider the problem of spectra of subalgebras invariant under the action of a group (functions f such that f ○ g = f). It is natural to attempt a characterization in terms of the space of orbits E″/~ obtained from E″ through the action of the group, so we pursue this approach here and introduce an analytic structure on the spectrum in some situations. In other situations we encounter some obstacles: in some cases, the lack of structure of E″/~ itself; in others, problems of weak continuity and non-approximability of functions in the algebra. We also define a convolution operation related to the spectrum.

In this article we are interested in the rigorous construction of WKB expansions for hyperbolic boundary value problems in the strip $\mathbb{R}^{d-1}\times [0,1]$. In this geometry, a new inversibility condition has to be imposed to construct the WKB expansion. This new condition is due to selfinteraction phenomenon which naturally appear when several boundary conditions are imposed. More precisely, by selfinteraction we mean that some rays can regenerated themselves after some rebounds against the sides of the strip. This phenomenon is not new and has already been studied in Benoit (Geometric optics expansions for hyperbolic corner problems, I: self-interaction phenomenon, Anal. PDE9(6) (2016), 1359–1418), Sarason and Smoller (Geometrical optics and the corner problem, Arch. Rat. Mech. Anal.56 (1974/75), 34–69) in the corner geometry. In this framework the existence of such selfinteracting rays is linked to specific geometries of the characteristic variety and may seem to be somewhat anecdotal. However for the strip geometry such rays become generic. The new inversibility condition, used to construct the WKB expansion, is a microlocalized version of the one characterizing the uniform in time strong well-posedness (Benoit, Lower exponential strong well-posedness of hyperbolic boundary value problems in a strip (preprint)). It is interesting to point here that such a situation already occurs in the half space geometry(Kreiss, Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math.23 (1970), 277–298).

We prove a gradient estimate for graphical spacelike mean curvature flow with a general Neumann boundary condition in dimension n = 2. This then implies that the mean curvature flow exists for all time and converges to a translating solution.

We establish several new metrical results on the distribution properties of the sequence ({xn})n≥1, where {·} denotes the fractional part. Many of them are presented in a more general framework, in which the sequence of functions (x ↦ xn)n≥1 is replaced by a sequence (fn)n≥1, under some growth and regularity conditions on the functions fn.

In this paper, we consider the existence of multiple solutions for the quasilinear Schrödinger equation

$$\left\{ {\matrix{ {-\Delta u-\Delta (\vert u \vert ^\alpha )\vert u \vert ^{\alpha -2}u = g(x,u) + \theta h(x,u),\;\;x\in \Omega } \hfill \cr {u = 0,\;\;x\in \partial \Omega ,} \hfill \cr } } \right.$$

Tissue engineering aims to grow artificial tissues in vitro to replace those in the body that have been damaged through age, trauma or disease. A recent approach to engineer artificial cartilage involves seeding cells within a scaffold consisting of an interconnected 3D-printed lattice of polymer fibres combined with a cast or printed hydrogel, and subjecting the construct (cell-seeded scaffold) to an applied load in a bioreactor. A key question is to understand how the applied load is distributed throughout the construct. To address this, we employ homogenisation theory to derive equations governing the effective macroscale material properties of a periodic, elastic–poroelastic composite. We treat the fibres as a linear elastic material and the hydrogel as a poroelastic material, and exploit the disparate length scales (small inter-fibre spacing compared with construct dimensions) to derive macroscale equations governing the response of the composite to an applied load. This homogenised description reflects the orthotropic nature of the composite. To validate the model, solutions from finite element simulations of the macroscale, homogenised equations are compared to experimental data describing the unconfined compression of the fibre-reinforced hydrogels. The model is used to derive the bulk mechanical properties of a cylindrical construct of the composite material for a range of fibre spacings and to determine the local mechanical environment experienced by cells embedded within the construct.

We examine situations, where representations of a finite-dimensional F-algebra A defined over a separable extension field K/F, have a unique minimal field of definition. Here the base field F is assumed to be a field of dimension ≼1. In particular, F could be a finite field or k(t) or k((t)), where k is algebraically closed. We show that a unique minimal field of definition exists if (a) K/F is an algebraic extension or (b) A is of finite representation type. Moreover, in these situations the minimal field of definition is a finite extension of F. This is not the case if A is of infinite representation type or F fails to be of dimension ≼1. As a consequence, we compute the essential dimension of the functor of representations of a finite group, generalizing a theorem of Karpenko, Pevtsova and the second author.