By examining asymptotic behavior of certain infinite basic ( -) hypergeometric sums at roots of unity (that is, at a ‘ -microscopic’ level) we prove polynomial congruences for their truncations. The latter reduce to non-trivial (super)congruences for truncated ordinary hypergeometric sums, which have been observed numerically and proven rarely. A typical example includes derivation, from a -analogue of Ramanujan's formula of the two supercongruences valid for all primes , where denotes the truncation of the infinite sum at the -th place and stands for the quadratic character modulo 3.

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 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.

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.

We establish a Gagliardo-Nirenberg-type inequality in for functions which decay fast as . We use this inequality to derive upper bounds for the decay rates of solutions of a degenerate parabolic equation. Moreover, we show that these upper bounds, hence also the Gagliardo-Nirenberg-type inequality, are sharp in an appropriate sense.

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.

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.

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 .

The Kronecker coefficients and the Littlewood- Richardson coefficients are nonnegative integers depending on three partitions , , and . By definition, (resp. ) are the multiplicities of the tensor product decomposition of two irreducible representations of symmetric groups (resp. linear groups). By a classical Littlewood-Murnaghan's result the Kronecker coefficients extend the Littlewood-Richardson ones. The nonvanishing of the Littlewood-Richardson coefficient implies that satisfies some linear inequalities called Horn inequalities. In this paper, we extend the essential Horn inequalities to the triples of partitions corresponding to a nonzero Kronecker coefficient. Along the way, we describe the set of tripless of partitions such that and , and , for some given positive integers and . This set is the natural analogue of the classical Horn semigroup when one thinks about as the branching multiplicities for the subgroup of .

We establish the existence and uniqueness of smooth solutions with large vorticity and weak solutions with vortex sheets/entropy waves for the steady Euler equations for both compressible and incompressible fluids in arbitrary infinitely long nozzles. We first develop a new approach to establish the existence of smooth solutions without assumptions on the sign of the second derivatives of the horizontal velocity, or the Bernoulli and entropy functions, at the inlet for the smooth case. Then the existence for the smooth case can be applied to construct approximate solutions to establish the existence of weak solutions with vortex sheets/entropy waves by nonlinear arguments. This is the first result on the global existence of solutions of the multidimensional steady compressible full Euler equations with free boundaries, which are not necessarily small perturbations of piecewise constant background solutions. The subsonic–sonic limit of the solutions is also shown. Finally, through the incompressible limit, we establish the existence and uniqueness of incompressible Euler flows in arbitrary infinitely long nozzles for both the smooth solutions with large vorticity and the weak solutions with vortex sheets. The methods and techniques developed here will be useful for solving other problems involving similar difficulties.