We give sharp remainder terms of and weighted Hardy and Rellich inequalities on one of most general subclasses of nilpotent Lie groups, namely the class of homogeneous groups. As consequences, we obtain analogues of the generalised classical Hardy and Rellich inequalities and the uncertainty principle on homogeneous groups. We also prove higher order inequalities of Hardy–Rellich type, all with sharp constants. A number of identities are derived including weighted and higher order types.

In this paper, we study the birational geometry of the Hilbert scheme P-2[n] of n-points on P-2. We discuss the stable base locus decomposition of the effective cone and the corresponding birational models. We give modular interpretations to the models in terms of moduli spaces of Bridgeland semi-stable objects. We construct these moduli spaces as moduli spaces of quiver representations using G.I.T. and thus show that they are projective. There is a precise correspondence between wall-crossings in the Bridgeland stability manifold and wall-crossings between Mori cones. For n <= 9, we explicitly determine the walls in both interpretations and describe the corresponding flips and divisorial contractions. (C). 2012 Elsevier Inc. All rights reserved.

A skew Calabi–Yau algebra is a generalization of a Calabi–Yau algebra which allows for a non-trivial Nakayama automorphism. We prove three homological identities about the Nakayama automorphism and give several applications. The identities we prove show (i) how the Nakayama automorphism of a smash product algebra is related to the Nakayama automorphisms of a graded skew Calabi–Yau algebra and a finite-dimensional Hopf algebra that acts on it; (ii) how the Nakayama automorphism of a graded twist of is related to the Nakayama automorphism of ; and (iii) that the Nakayama automorphism of a skew Calabi–Yau algebra has trivial homological determinant in case is noetherian, connected graded, and Koszul.

Since the early 2000s physicists have developed an ingenious but non-rigorous formalism called the to put forward precise conjectures on phase transitions in random problems (Mézard et al., 2002 ). The cavity method predicts that the satisfiability threshold in the random -SAT problem is , with (Mertens et al., 2006 ). This paper contains a proof of the conjecture.

The affine Yangian of gl(1) has recently appeared simultaneously in the work of Maulik-Okounkov [11] and Schiffmann-Vasserot [20] in connection with the Alday-Gaiotto-Tachikawa conjecture. While the presentation from [11] is purely geometric, the algebraic presentation in [20] is quite involved. In this article, we provide a simple loop realization of this algebra which can be viewed as an "additivization" of the quantum toroidal algebra of gl(1) in the same way as the Yangian Y-h(g) is an "additivization" of the quantum loop algebra U-q(Lg) for a simple Lie algebra g. We also explain the similarity between the representation theories of the affine Yangian and the quantum toroidal algebras of gl(1) by generalizing the main result of [10] to the current settings. (C) 2016 Elsevier Inc. All rights reserved.

We investigate rates of decay for -semigroups on Hilbert spaces under assumptions on the resolvent growth of the semigroup generator. Our main results show that one obtains the best possible estimate on the rate of decay, that is to say an upper bound which is also known to be a lower bound, under a comparatively mild assumption on the growth behaviour. This extends several statements obtained by Batty et al. (2016) . In fact, for a large class of semigroups our condition is not only sufficient but also necessary for this optimal estimate to hold. Even without this assumption we obtain a new quantified asymptotic result which in many cases of interest gives a sharper estimate for the rate of decay than was previously available, and for semigroups of normal operators we are able to describe the asymptotic behaviour exactly. We illustrate the strength of our theoretical results by using them to obtain sharp estimates on the rate of energy decay for a wave equation subject to viscoelastic damping at the boundary.

Our main purpose in this paper is to establish the existence and nonexistence of extremal functions (also known as maximizers) and symmetry of extremals for several Trudinger-Moser type inequalities on the entire space , including both the critical and subcritical Trudinger-Moser inequalities (see Theorems 1.1, 1.2, 1.3, 1.4, 1.5). Most of earlier works on existence of maximizers in the literature rely on the complicated blow-up analysis of PDEs for the associated Euler-Lagrange equations of the corresponding Moser functionals. The new approaches developed in this paper are using the identities and relationship between the supremums of the subcritical Trudinger-Moser inequalities and the critical ones established by the same authors in , combining with the continuity of the supremum function that is observed for the first time in the literature. These allow us to establish the existence and nonexistence of the maximizers for the Trudinger-Moser inequalities in different ranges of the parameters (including those inequalities with the exact growth). This method is considerably simpler and also allows us to study the symmetry problem of the extremal functions and prove that the extremal functions for the subcritical singular Truddinger-Moser inequalities are symmetric. Moreover, we will be able to calculate the exact values of the supremums of the Trudinger-Moser type in certain cases. These appear to be the first results in this direction.

In this paper we prove the existence of solutions to the viscous, non-resistive magnetohydrodynamics (MHD) equations on the whole of , , for divergence-free initial data in certain Besov spaces, namely and . The a priori estimates include the term on the right-hand side, which thus requires an auxiliary bound in . In 2D, this is simply achieved using the standard energy inequality; but in 3D an auxiliary estimate in is required, which we prove using the splitting method of Calderón (1990) . By contrast, our proof that such solutions are unique only applies to the 3D case.

The (p), dual curvature measure was introduced by Lutwak, Yang & Zhang in an attempt to unify the L-p Brunn Minkowski theory and the dual Brunn-Minkowski theory. The characterization problem for L-p dual curvature measure, called the L-p dual Minkowski problem, is a fundamental problem in this unifying theory. The L-p dual Minkowski problem contains the L-p Minkowski problem and the dual Minkowski problem, two major problems in modern convex geometry that remain open in general. In this paper, existence results on the L-p dual Minkowski problem in the weak sense will be provided. Moreover, existence and uniqueness of the solution in the smooth category will also be demonstrated. (C) 2018 Elsevier Inc. All rights reserved.

In this paper we develop a theory of slice regular functions on a real alternative algebra . Our approach is based on a well-known Fueter's construction. Two recent function theories can be included in our general theory: the one of slice regular functions of a quaternionic or octonionic variable and the theory of slice monogenic functions of a Clifford variable. Our approach permits to extend the range of these function theories and to obtain new results. In particular, we get a strong form of the fundamental theorem of algebra for an ample class of polynomials with coefficients in and we prove a Cauchy integral formula for slice functions of class .

We consider the Yangs–Mills equations in dimensions. This is the energy critical case and we show that it admits a family of solutions which blow up in finite time. They are obtained by the spherically symmetric ansatz in the gauge group and result by rescaling of the instanton solution. The rescaling is done via a prescribed rate which in this case is a modification of the self-similar rate by a power of . The powers themselves take any value exceeding 3/2 and thus form a continuum of distinct rates leading to blow-up. The methods are related to the authors' previous work on wave maps and the energy critical semi-linear equation. However, in contrast to these equations, the linearized Yang–Mills operator (around an instanton) exhibits a zero energy eigenvalue rather than a resonance. This turns out to have far-reaching consequences, amongst which are a completely different family of rates leading to blow-up (logarithmic rather than polynomial corrections to the self-similar rate).

A new family of geometric Borel measures on the unit sphere is introduced. Special cases include the L-p surface area measures (which extend the classical surface area measure of Aleksandrov and Fenchel & Jessen) and L-p-integral curvature (which extends Alkesandrov's integral curvature) in the L-p, Brunn-Minkowski theory. It also includes the dual curvature measures (which are the duals of Federer's curvature measures) in the dual Brtum- Minkowski theory. This partially unifies the classical theory of mixed volumes and the newer theory of dual mixed volumes. (C) 2018 Elsevier Inc. All rights reserved.

In a recent paper Komornik et al. (2017) proved a long-conjectured formula for the Hausdorff dimension of the set of numbers having a unique expansion in the (non-integer) base , and showed that this Hausdorff dimension is continuous in . Unfortunately, their proof contained a gap which appears difficult to fix. This article gives a completely different proof of these results, using a more direct combinatorial approach.

Gillespie's Theorem gives a systematic way to construct model category structures on , the category of chain complexes over an abelian category . We can view as the category of representations of the quiver with the relations that two consecutive arrows compose to 0. This is a self-injective quiver with relations, and we generalise Gillespie's Theorem to other such quivers with relations. There is a large family of these, and following Iyama and Minamoto, their representations can be viewed as generalised chain complexes. Our result gives a systematic way to construct model category structures on many categories. This includes the category of -periodic chain complexes, the category of -complexes where , and the category of representations of the repetitive quiver with mesh relations.

In this paper, we derive Li-Yau inequality for unbounded Laplacian on complete weighted graphs with the assumption of the curvature dimension inequality , which can be regarded as a notion of curvature on graphs. Furthermore, we obtain some applications of Li-Yau inequality, including Harnack inequality, heat kernel bounds and Cheng's eigenvalue estimate. These are the first kind of results in this direction for unbounded Laplacian on graphs.

Many of the conjectures of current interest in the representation theory of finite groups in characteristic are local-to-global statements, in that they predict consequences for the representations of a finite group given data about the representations of the -local subgroups of . The local structure of a block of a group algebra is encoded in the fusion system of the block together with a compatible family of Külshammer-Puig cohomology classes. Motivated by conjectures in block theory, we state and initiate investigation of a number of seemingly local conjectures for arbitrary triples consisting of a saturated fusion system on a finite -group and a compatible family .

We investigate the problem of characterising the family of strongly quasipositive links which have definite symmetrised Seifert forms and apply our results to the problem of determining when such a link can have an L-space cyclic branched cover. In particular, we show that if is the dual Garside element and is a strongly quasipositive braid whose braid closure is definite, then implies that is one of the torus links or pretzel links . Applying we deduce that if one of the standard cyclic branched covers of is an L-space, then is one of these links. We show by example that there are strongly quasipositive braids whose closures are definite but not one of these torus or pretzel links. We also determine the family of definite strongly quasipositive 3-braids and show that their closures coincide with the family of strongly quasipositive 3-braids with an L-space branched cover.

We study a variational model for transition layers with an underlying functional that combines an Allen-Cahn type structure with an additional nonlocal interaction term. A transition layer is represented by a map from to . Thus it has a topological invariant in the form of a winding number, and we study minimisers subject to a prescribed winding number. As shown in our previous paper , the nonlocal term gives rise to solutions that would not be present for a functional including only the (local) Allen-Cahn terms. We complete the picture here by proving existence of minimisers in all cases where it has been conjectured. We also prove non-existence in some other cases. Finally, in addition to existence, we prove a result for the structure of minimizers.