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We establish a theory of $Q$-valued functions minimizing a suitable generalization of the Dirichlet integral. In a second paper the theory will be used to approximate efficiently area minimizing currents $\mathrm{mod}(p)$ when $p=2Q$, and to establish a first general partial regularity theorem for every $p$ in any dimension and codimension., 37 pages. First part of a two-papers work aimed at establishing a first general partial regularity theory for area minimizing currents modulo p, for any p and in any dimension and codimension. v3 is the final version, to appear on Comm. Pure Appl. Math

In this paper, we show how to use the framework of mod-Gaussian convergence in order to study the fluctuations of certain models of random graphs, of random permutations and of random integer partitions. We prove that, in these three frameworks, a generic homogeneous observable of a generic random model is mod-Gaussian under an appropriate renormalisation. This implies a central limit theorem with an extended zone of normality, a moderate deviation principle, an estimate of the speed of convergence, a local limit theorem and a concentration inequality. The universal asymptotic behavior of the observables of these models gives rise to a notion of mod-Gaussian moduli space., Comment: New version: the paper has been slightly shortened, and a few references were added. 52 pages, 13 figures

Let R be a commutative noetherian ring. Lindo and Pande have recently posed the question asking when every ideal of R is isomorphic to some trace ideal of R. This paper studies this question and gives several answers. In particular, a complete answer is given in the case where R is local: it is proved in this paper that every ideal of R is isomorphic to a trace ideal if and only if R is an artinian Gorenstein ring, or a 1-dimensional hypersurface with multiplicity at most 2, or a unique factorization domain.

The purpose of this document is to connect two maps related to certain graphs embedded in the disc. The first is Postnikov's boundary measurement map, which combines partition functions of matchings in the graph into a map from an algebraic torus to an open positroid variety in a Grassmannian. The second is a rational map from the open positroid variety to an algebraic torus, given by certain Plucker coordinates which are expected to be a cluster in a cluster structure. This paper clarifies the relationship between these two maps, which has been ambiguous since they were introduced by Postnikov in 2001. The missing ingredient supplied by this paper is a twist automorphism of the open positroid variety, which takes the target of the boundary measurement map to the domain of the (conjectural) cluster. Among other applications, this provides an inverse to the boundary measurement map, as well as Laurent formulas for twists of Plucker coordinates.

In this paper we prove a sharp distortion property of the Cassinian metric under Mobius transformations of a punctured ball onto another punctured ball. The paper also deals with discussion on local convexity properties of the Cassinian metric balls in proper subdomains of Rn. Inclusion properties of the Cassinian metric balls with other hyperbolic-type metric balls are also investigated. In particular, several conjectures are also stated in response to sharpness of the inclusion relations.

This paper is devoted to one-dimensional interpolation Gagliardo-Nirenberg-Sobolev inequalities. We study how various notions of duality, transport and monotonicity of functionals along flows defined by some nonlinear diffusion equations apply. We start by reducing the inequality to a much simpler dual variational problem using mass transportation theory. Our second main result is devoted to the construction of a Lyapunov functional associated with a nonlinear diffusion equation, that provides an alternative proof of the inequality. The key observation is that the inequality on the line is equivalent to Sobolev's inequality on the sphere, at least when the dimension is an integer, or to the critical interpolation inequality for the ultraspherical operator in the general case. The time derivative of the functional along the flow is itself very interesting. It explains the machinery of some rigidity estimates for nonlinear elliptic equations and shows how eigenvalues of a linearized problem enter in the computations. Notions of gradient flows are then discussed for various notions of distances. Throughout this paper we shall deal with two classes of inequalities corresponding either to p>2 or to p

We give a possible generalization of Lutz twist to all dimensions. This reproves the fact that every contact manifold can be given a non-fillable contact structure and also shows great flexibility in the manifolds that can be realized as cores of overtwisted families. We moreover show that $R^{2n+1}$ has at least three distinct contact structures. This version of the paper contains both the texts of the published version of the paper together with an Erratum to the published version appended to the end., Comment: 19 pages, 2 figures, and erratum to the published version of the paper

In a landmark paper, D.Y. Kleinbock and G.A. Margulis established the fundamental Baker-Sprindzuk conjecture on homogeneous Diophantine approximation on manifolds. Subsequently, there has been dramatic progress in this area of research. However, the techniques developed to date do not seem to be applicable to inhomogeneous approximation. Consequently, the theory of inhomogeneous Diophantine approximation on manifolds remains essentially non-existent. In this paper we develop an approach that enables us to transfer homogeneous statements to inhomogeneous ones. This is rather surprising as the inhomogeneous theory contains the homogeneous theory and so is more general. As a consequence, we establish the inhomogeneous analogue of the Baker-Sprindzuk conjecture. Furthermore, we prove a complete inhomogeneous version of the profound theorem of Kleinbock, Lindenstrauss & Weiss on the extremality of friendly measures. The results obtained in this paper constitute the first step towards developing a coherent inhomogeneous theory for manifolds in line with the homogeneous theory., 37 pages: a final section on further developments has been added

For a smooth projective curve, the cycles of e-secant k-planes are among the most studied objects in classical enumerative geometry and there are well-known formulas due to Castelnuovo, Cayley and MacDonald concerning them. Despite various attempts, surprisingly little is known about the enumerative validity of such formulas. The aim of this paper is to completely clarify this problem in the case of the generic curve C of given genus. Using degeneration techniques and a few facts about the birational geometry of moduli spaces of stable pointed curves we determine precisely under which conditions the cycle of e-secant k-planes in non-empty and we compute its dimension. We also precisely determine the dimension of the variety of linear series on C carrying e-secant k-planes. In a different direction, in the last part of the paper we study the distribution of ramification points of the powers of a line bundle on C having prescribed ramification at a given point., Comment: 25 pages. Numerous changes suggested by the referee, several proofs explained in more detail. To appear in the Journal of the London Mathematical Society

Twenty-five years ago, Conway and Norton published their remarkable paper `Monstrous Moonshine', proposing a completely unexpected relationship between finite simple groups and modular functions. This paper reviews the progress made in broadening and understanding that relationship., 33 pages. 10 references added; minor changes to text

In the present paper, we construct a cusped hyperbolic $4$-manifold with all cusp sections homeomorphic to the Hantzsche-Wendt manifold, which is a rational homology sphere. By a result of Gol\'enia and Moroianu, the Laplacian on $2$-forms on such a manifold has purely discrete spectrum. This shows that one of the main results of Mazzeo and Phillips from 1990 cannot hold without additional assumptions on the homology of the cusps. This also answers a question by Gol\'enia and Moroianu from 2012. We also correct and refine the incomplete classification of compact orientable flat $3$-manifolds arising from cube colourings provided earlier by the last two authors., Comment: 15 pages, 1 figure, 1 table; SageMath worksheets available at https://github.com/sashakolpakov/24-cell-colouring

The mod two cohomology of the three connective covering of S3 has the form F2[X2n] ⊗ E(Sq1X2n) where x2n is in degree 2n and n = 2. If F denotes the homotopy theoretic fibre of the map S3 → B2S1 of degree 2, then the mod 2 cohomology of F is also of the same form for n = 1. Notice (cf. Section 7 of the present paper) that the existence of spaces whose cohomology has this form for high values of n would immediately provide Arf invariant elements in the stable stem. Hence, it is worthwhile to determine for what values of n the above algebra can be realized as the mod2 cohomology of some space. The purpose of this paper is to construct a further example of a space with such a cohomology algebra for n = 4 and to show that no other values of n are admissible. More precisely, we prove the following.

In this paper, we further Meirong Zhang, et al.'s work by computing the number of weighted eigenvalues for Sturm-Liouville equations, equipped with general integrable potentials and Dirac weights, under Dirichlet boundary condition. We show that, for a Sturm-Liouville equation with a general integrable potential, if its weight is a positive linear combination of $n$ Dirac Delta functions, then it has at most $n$ (may be less than $n$, or even be $0$) distinct real Dirichlet eigenvalues, or every complex number is a Dirichlet eigenvalue; in particular, under some sharp condition, the number of Dirichlet eigenvalues is exactly $n$. Our main method is to introduce the concepts of characteristics matrix and characteristics polynomial for Sturm-Liouville problem with Dirac weights, and put forward a general and direct algorithm used for computing eigenvalues. As an application, a class of inverse Dirichelt problems for Sturm-Liouville equations involving single Dirac distribution weights is studied., 23 pages

The main object of the present paper is to consider the binomial transforms for Horadam quaternion sequences. We gave new formulas for recurrence relation, generating function, Binet formula and some basic identities for the binomial sequence of Horadam quaternions. Working with Horadam quaternions, we have found the most general formula that includes all binomial transforms with recurrence relation from the second order. In the last part, we determined the recurrence relation for this new type of quaternion by working with the iterated binomial transform, which is a different type of binomial transform. WOS:000638679800001 2-s2.0-85104113068

The Markov evolution is studied of an infinite age-structured population of migrants arriving in and departing from a continuous habitat $X \subseteq\mathds{R}^d$ -- at random and independently of each other. Each population member is characterized by its age $a\geq 0$ (time of presence in the population) and location $x\in X$. The population states are probability measures on the space of the corresponding marked configurations. The result of the paper is constructing the evolution $\mu_0 \to \mu_t$ of such states by solving a standard Fokker-Planck equation for this models. We also found a stationary state $\mu$ existing if the emigration rate is separated away from zero. It is then shown that $\mu_t$ weakly converges to $\mu$ as $t\to +\infty$.

In this paper, we define and study Gevrey spaces associated with a Hormander family of (globally defined) vector fields and its corresponding sub‐Laplacian. We show some natural relations between the various Gevrey spaces in this setting on general manifolds, and more particular properties on Lie groups with polynomial growth of the volume. In the case of the Heisenberg group and of SU(2), we show that all our descriptions coincide.

The paper is on the vanishing topology of singular Milnor fibres of holomorphic families of arbitrary square, symmetric and skew-symmetric matrices with sufficiently many parameters. We define vanishing cycles on such fibres, prove an extended form of the Damon-Pike $\mu=\tau$ conjecture about the families of a special type, and make first steps towards understading of the monodromy of matrix singularities. We also prove a Lyashko-Looijenga type theorem for simple matrix families, and point out a surprising relationship between certain Shephard-Todd groups, simple odd functions and a sporadic part of the Bruce-Tari simple matrix classification.

The main goal of this paper is to study some properties of an extension of valuations from classical invariants. More specifically, we consider a valued field $(K,\nu)$ and an extension $\omega$ of $\nu$ to a finite extension $L$ of $K$. Then we study when the valuation ring of $\omega$ is essentially finitely generated over the valuation ring of $\nu$. We present a necessary condition in terms of classic invariants of the extension by Hagen Knaf and show that in some particular cases, this condition is also sufficient. We also study when the corresponding extension of graded algebras is finitely generated. For this problem we present an equivalent condition (which is weaker than the one for the finite generation of the valuation rings).

In this paper we consider a class of chemotaxis models with two arbitrary constitutive functions g(u) and f(v). After having performed a complete symmetry group classification with respect to them the reduced systems are derived. By considering g(u) of the logistic form wide classes of exact solutions are found.

In this paper, we consider discrete groups in ${\rm PGL}_d(\mathbb{R})$ acting convex co-compactly on a properly convex domain in real projective space. For such groups, we establish an analogue of the well known flat torus theorem for ${\rm CAT}(0)$ spaces., Minor revisions, to appear in the Journal of the London Mathematical Society. 20 pages. Comments welcome

In this paper we present a unified simple approach to anisotropic Hardy inequalities in various settings. We consider Hardy inequalities which involve a Finsler distance from a point or from the boundary of a domain. The sharpness and the non-attainability of the constants in the inequalities are also proved., 31 pages. We add "Note added to Proof" in Introduction and several references

In this paper, we consider some generalized holomorphic maps between pseudo-Hermitian manifolds and Hermitian manifolds. By Bochner formulas and comparison theorems, we establish related Schwarz type results. As corollaries, Liouville theorem and little Picard theorem for basic CR functions are deduced. Finally, we study CR Carath\'eodory pseudodistance on CR manifolds., Comment: 16 pages

Let $\mathscr J$ be the set of inner functions whose derivative lies in the Nevanlinna class. In this paper, we discuss a natural topology on $\mathscr J$ where $F_n \to F$ if the critical structures of $F_n$ converge to the critical structure of $F$. We show that this occurs precisely when the critical structures of the $F_n$ are uniformly concentrated on Korenblum stars. The proof uses Liouville's correspondence between holomorphic self-maps of the unit disk and solutions of the Gauss curvature equation. Building on the works of Korenblum and Roberts, we show that this topology also governs the behaviour of invariant subspaces of a weighted Bergman space which are generated by a single inner function., 45 pages

In this paper, the weak Harris theorem developed in \cite{HMS11} is illustrated by using a straightforward Wasserstein coupling, which implies the exponential ergodicity of the functional solutions to a range of neutral type SDEs with infinite length of memory. A concrete example is presented to illustrate the main result.

We present a new proof of the sphere covering inequality in the spirit of comparison geometry, and as a byproduct we find another sphere covering inequality which can be viewed as the dual of the original one. We also prove sphere covering inequalities on surfaces satisfying general isoperimetric inequalities, and discuss their applications to elliptic equations with exponential nonlinearities in dimension two. The approach in this paper extends, improves, and unifies several inequalities about solutions of elliptic equations with exponential nonlinearities.

ith respect to the analytic‐algebraic dichotomy, the theory of Siegel modular forms of half‐integral weight is lopsided; the analytic theory is strong, whereas the algebraic lags behind. In this paper, we capitalise on this to establish the fundamental object needed for the analytic side of the Iwasawa main conjecture — the p ‐adic L ‐function obtained by interpolating the complex L ‐function at special values. This is achieved through the Rankin–Selberg method and the explicit Fourier expansion of non‐holomorphic Siegel Eisenstein series. The construction of the p ‐stabilisation in this setting is also of independent interest.

In this paper we prove transference inequalities for regular and uniform Diophantine exponents in the weighted setting. Our results generalize the corresponding inequalities that exist in the `non-weighted' case.

We study how small is the set of critical values of the distance function from a compact (resp. closed) set in the plane or in a connected complete two-dimensional Riemannian manifold. We show that for a compact set, the set of critical values is compact and Lebesgue null (which is a known result) and that it has "locally" (away from 0) bounded sum of square roots of lengths of gaps (components of the complement). In the planar case, these conditions of local smallness are shown to be optimal. These results improve and generalize those of Fu (1985) and of our earlier paper from 2012. We also find an optimal condition for the smallness of the whole set of critical values of a planar compact set.

In this paper we prove sufficient conditions for the Fredholm property of a non-smooth pseudodifferential operator $P$ which symbol is in a Holder space with respect to the spatial variable. As a main ingredient for the proof we use a suitable symbol-smoothing.

The paper extends some well-known results for analytic functions onto solutions of the Vekua equation $\partial _{\overline{z}}W=aW+b\overline{W}$ regarding the existence and construction of the Bergman kernel and of the corresponding Bergman projection operator.

In this paper, we study the existence and multiplicity results of nontrivial positive solutions to a quasilinear elliptic equation in $\RN$, when $N\geq2$, as \begin{equation} \Lp u+u^{p-1}=\lambda\hspace{0.2mm}k(x)u^{r-1}-h(x)u^{q-1}.\nonumber \end{equation} Here, $h(x),k(x)>0$ are Lebesgue measurable functions, $10$ is a parameter.

International audience; In the paper, the authors introduce a notion "multivariate exponential polynomials" which generalize exponential numbers and polynomials, establish explicit formulas, inversion formulas, and recurrence relations for multivariate exponential polynomials in terms of the Stirling numbers of the first and second kinds with the help of the Fàa di Bruno formula, two identities for the Bell polynomials of the second kind, and the inversion theorem for the Stirling numbers of the first and second kinds, construct some determinantal inequalities and product inequalities for multivariate exponential polynomials with the aid of some properties of completely monotonic functions and other known results, derive the logarithmic convexity and logarithmic concavity for multivariate exponential polynomials, and finally find an application of multi-variate exponential polynomials to white noise distribution theory by confirming that multivariate exponential polynomials satisfy conditions for sequences required in white noise distribution theory.

In this paper, we study the generalized Lebesgue-Nagell equation \[ x^2+7^{2k+1}=y^n. \] This is the last case of equations of the form $x^2+q^{2k+1}=y^n$ with $k\geq0$ and $q>0$ where $\mathbb{Q}(\sqrt{-q})$ has class number one. Our proof is based on the modular method and the symplectic argument.

In this paper we study the relationship between two fundamental regularity properties of an $s$-dimensional Calder\'{o}n-Zygmund operator (CZO) acting on a Borel measure $\mu$ in $\mathbb{R}^d$, with $s\in (0,d)$. In the classical case when $s=d$ and $\mu$ is equal to the Lebesgue measure, Calder\'on and Zygmund showed that if a CZO is bounded in $L^2$ then the principal value integral exists almost everywhere. However, there are by now several examples showing that this implication may fail for lower-dimensional kernels and measures, even when the CZO has a homogeneous kernel consisting of spherical harmonics. We introduce sharp geometric conditions on $\mu$, in terms of certain scaled transportation distances, which ensure that an extension of the Calder\'{o}n-Zygmund theorem holds. These conditions are necessary and sufficient in the cases of the Riesz transform and the Huovinen transform. Our techniques build upon prior work by Mattila and Verdera, and incorporate the machinery of symmetric measures, introduced to the area by Mattila., Comment: 32 pages

International audience; Orbital varieties are the irreducible components of the intersection between a nilpotent orbit and a Borel subalgebra of the Lie algebra of a reductive group. There is a geometric correspondence between orbital varieties and irreducible components of Springer fibers. In type A, a construction due to Richardson implies that every nilpotent orbit contains at least one smooth orbital variety and every Springer fiber contains at least one smooth component. In this paper, we show that this property is also true for the other classical cases. Our proof uses the interpretation of Springer fibers as varieties of isotropic flags and van Leeuwen's parametrization of their components in terms of domino tableaux. In the exceptional cases, smooth orbital varieties do not arise in every nilpotent orbit, as already noted by Spaltenstein. We however give a (nonexhaustive) list of nilpotent orbits which have this property. Our treatment of exceptional cases relies on an induction procedure for orbital varieties, similar to the induction procedure for nilpotent orbits.

Adjoint functor theorems give necessary and sufficient conditions for a functor to admit an adjoint. In this paper, we prove general adjoint functor theorems for functors between ∞‐categories. One of our main results is an ∞‐categorical generalization of Freyd's classical General Adjoint Functor Theorem. As an application of this result, we recover Lurie's adjoint functor theorems for presentable ∞‐categories. We also discuss the comparison between adjunctions of ∞‐categories and homotopy adjunctions, and give a treatment of Brown representability for ∞‐categories based on Heller's purely categorical formulation of the classical Brown representability theorem.

In this paper, we propose a new approach to Cwikel estimates both for the Euclidean space and for the noncommutative Euclidean space.

In this paper we show rigidity results for super-solutions to fully nonlinear elliptic conformally invariant equations on subdomains of the standard $n$-sphere $\mathbb S^n$ under suitable conditions along the boundary. We emphasize that our results do not assume concavity assumption on the fully nonlinear equations we will work with. This proves rigidity for compact connected locally conformally flat manifolds $(M,g)$ with boundary such that the eigenvalues of the Schouten tensor satisfy a fully nonlinear elliptic inequality and whose boundary is isometric to a geodesic sphere $\partial D(r)$, where $D(r)$ denotes a geodesic ball of radius $r\in (0,\pi/2]$ in $\mathbb S^n$, and totally umbilical with mean curvature bounded below by the mean curvature of this geodesic sphere. Under the above conditions, $(M,g)$ must be isometric to the closed geodesic ball $\overline{D(r)}$. As a side product, in dimension $2$ our methods provide a new proof to Toponogov's Theorem about the rigidity of compact surfaces carrying a shortest simple geodesic. Roughly speaking, Toponogov's Theorem is equivalent to a rigidity theorem for spherical caps in the Hyperbolic three-space $\mathbb H^3$. In fact, we extend it to obtain rigidity for super-solutions to certain Monge-Amp\`ere equations.

We consider Sturm-Liouville operators on geometrical graphs without cycles (trees) with singular potentials from the class $W_2^{-1}$. We suppose that the potentials are known on a part of the graph, and study the so-called partial inverse problem, which consists in recovering the potentials on the remaining part of the graph from some parts of several spectra. The main results of the paper are the uniqueness theorem and a constructive procedure for the solution of the partial inverse problem. Our method is based on the completeness and the Riesz-basis property of special systems of vector functions, and the reduction of the partial inverse problem to the complete one on a part of the graph.

The purpose of the paper is to obtain estimates for differences of functions of two pairs of commuting contractions on Hilbert space. In particular, Lipschitz type estimates, H\"older type estimates, Schatten--von Neumann estimates are obtained. The results generalize earlier known results for functions of self-adjoint operators, normal operators, contractions and dissipative operators.

A biperiodic alternating link has an alternating quotient link in the thickened torus. In this paper, we focus on semi-regular links, a class of biperiodic alternating links whose hyperbolic structure can be immediately determined from a corresponding Euclidean tiling. Consequently, we determine the exact volumes of semi-regular links. We relate their commensurability and arithmeticity to the corresponding tiling, and assuming a conjecture of Milnor, we show there exist infinitely many pairwise incommensurable semi-regular links with the same invariant trace field. We show that only two semi-regular links have totally geodesic checkerboard surfaces; these two links satisfy the Volume Density Conjecture. Finally, we give conditions implying that many additional biperiodic alternating links are hyperbolic and admit a positively oriented, unimodular geometric triangulation. We also provide sharp upper and lower volume bounds for these links., 25 pages, 15 figures. V2: Minor changes. Added reference, fixed typos, and clarified proof of Theorem 5.1

We consider the problem of reconstructing a signal $f$ from its spectrogram, i.e., the magnitudes $|V_\varphi f|$ of its Gabor transform $$V_\varphi f (x,y):=\int_{\mathbb{R}}f(t)e^{-\pi (t-x)^2}e^{-2\pi iy t}dt, \quad x,y\in \mathbb{R}.$$ Such problems occur in a wide range of applications, from optical imaging of nanoscale structures to audio processing and classification. While it is well-known that the solution of the above Gabor phase retrieval problem is unique up to natural identifications, the stability of the reconstruction has remained wide open. The present paper discovers a deep and surprising connection between phase retrieval, spectral clustering and spectral geometry. We show that the stability of the Gabor phase reconstruction is bounded by the reciprocal of the Cheeger constant of the flat metric on $\mathbb{R}^2$, conformally multiplied with $|V_\varphi f|$. The Cheeger constant, in turn, plays a prominent role in the field of spectral clustering, and it precisely quantifies the `disconnectedness' of the measurements $V_\varphi f$. It has long been known that a disconnected support of the measurements results in an instability -- our result for the first time provides a converse in the sense that there are no other sources of instabilities. Due to the fundamental importance of Gabor phase retrieval in coherent diffraction imaging, we also provide a new understanding of the stability properties of these imaging techniques: Contrary to most classical problems in imaging science whose regularization requires the promotion of smoothness or sparsity, the correct regularization of the phase retrieval problem promotes the `connectedness' of the measurements in terms of bounding the Cheeger constant from below. Our work thus, for the first time, opens the door to the development of efficient regularization strategies.

Geodesic balls in a simply connected space forms Sn, Rn or Hn are distinguished manifolds for comparison in bounded Riemannian geometry. In this paper we show that they have the maximum possible boundary volume among Miao–Tam critical metrics with connected boundary provided that the boundary of the manifold has a lower bound for the Ricci curvature. In the same spirit we also extend a rigidity theorem due to Boucher et al. and Shen to n‐dimensional static metrics with positive constant scalar curvature, which gives us a partial answer to the Cosmic no‐hair conjecture.

In complex dynamics, the bungee set is defined as the set points whose orbit is neither bounded nor tends to infinity. In this paper we study, for the first time, the bungee set of a quasiregular map of transcendental type. We show that this set is infinite, and shares many properties with the bungee set of a transcendental entire function. By way of contrast, we give examples of novel properties of this set in the quasiregular setting. In particular, we give an example of a quasiconformal map of the plane with a non-empty bungee set; this behaviour is impossible for an analytic homeomorphism.

We present an existence result for L\'evy-type processes which requires only weak regularity assumptions on the symbol $q(x,\xi)$ with respect to the space variable $x$. Applications range from existence and uniqueness results for L\'evy-driven SDEs with H\"older continuous coefficients to existence results for stable-like processes and L\'evy-type processes with symbols of variable order. Moreover, we obtain heat kernel estimates for a class of L\'evy and L\'evy-type processes. The paper includes an extensive list of L\'evy(-type) processes satisfying the assumptions of our results.

A bosonic Laplacian is a conformally invariant second order differential operator acting on smooth functions defined on domains in Euclidean space and taking values in higher order irreducible representations of the special orthogonal group. In this paper, we firstly introduce the motivation for study of the generalized Maxwell operators and bosonic Laplacians (also known as the higher spin Laplace operators). Then, with the help of connections between Rarita-Schwinger type operators and bosonic Laplacians, we solve Poisson's equation for bosonic Laplacians. A representation formula for bounded solutions to Poisson's equation in Euclidean space is also provided. In the end, we provide Green's formulas for bosonic Laplacians in scalar-valued and Clifford-valued cases, respectively. These formulas reveal that bosonic Laplacians are self-adjoint with respect to a given $L^2$ inner product on certain compact supported function spaces.

A powerful method pioneered by Swinnerton-Dyer allows one to study rational points on pencils of curves of genus 1 by combining the fibration method with a sophisticated form of descent. A variant of this method, first used by Skorobogatov and Swinnerton-Dyer in 2005, can be applied to study rational points on Kummer varieties. In this paper we extend the method to include an additional step of second descent. Assuming finiteness of the relevant Tate-Shafarevich groups, we use the extended method to show that the Brauer-Manin obstruction is the only obstruction to the Hasse principle on Kummer varieties associated to abelian varieties with all rational 2-torsion, under mild additional hypotheses.

Kato introduced the exotic nilpotent cone to be a substitute for the ordinary nilpotent cone of type C with cleaner properties. Here we describe the irreducible components of exotic Springer fibres (the fibres of the resolution of the exotic nilpotent cone), and prove that they are naturally in bijection with standard bitableaux. As a result, we deduce the existence of an exotic Robinson–Schensted bijection, which is a variant of the type C Robinson–Schensted bijection between pairs of same-shape standard bitableaux and elements of the Weyl group; this bijection is described explicitly in the sequel to this paper. Note that this is in contrast with ordinary type C Springer fibres, where the parametrisation of irreducible components, and the resulting geometric Robinson–Schensted bijection, are more complicated. As an application, we explicitly describe the structure in the special cases where the irreducible components of theexotic Springer fibre have dimension 2, and show that in those cases one obtains Hirzebruch surfaces.

Positivstellensatz is a fundamental result in real algebraic geometry providing algebraic certificates for positivity of polynomials on semialgebraic sets. In this article Positivstellensatze for trace polynomials positive on semialgebraic sets of $n\times n$ matrices are provided. A Krivine-Stengle-type Positivstellensatz is proved characterizing trace polynomials nonnegative on a general semialgebraic set $K$ using weighted sums of hermitian squares with denominators. The weights in these certificates are obtained from generators of $K$ and traces of hermitian squares. For compact semialgebraic sets $K$ Schmudgen- and Putinar-type Positivstellensatze are obtained: every trace polynomial positive on $K$ has a sum of hermitian squares decomposition with weights and without denominators. The methods employed are inspired by invariant theory, classical real algebraic geometry and functional analysis. Procesi and Schacher in 1976 developed a theory of orderings and positivity on central simple algebras with involution and posed a Hilbert's 17th problem for a universal central simple algebra of degree $n$: is every totally positive element a sum of hermitian squares? They gave an affirmative answer for $n=2$. In this paper a negative answer for $n=3$ is presented. Consequently, including traces of hermitian squares as weights in the Positivstellensatze is indispensable.

The first part of the paper studies the expression for, and the properties of, the logarithm map on an orthant space, which is a simple stratified space, with the aim of analysing Frechet means of probability measures on such a space. In the second part, we use these results to characterise Frechet means and to derive various of their properties, including the limiting distribution of sample Frechet means.