Abstract In this article, we investigate the existence and multiplicity of solutions to the Robin problem where () is a smooth bounded domain, and . Our main assumption is that is a locally Lipschitz function, possibly sign‐changing, such that for every , where are two zeros of . Without any further conditions, we establish the existence of two nonnegative solutions whose maximum lies in for sufficiently large . Moreover, we analyse the limiting behaviour of the solution set of this Robin problem, showing that it degenerates into that of the associated Neumann problem as and into that of the associated Dirichlet problem as .

Abstract We investigate the Cauchy problem for the Hunter–Saxton equation with a general continuous forcing term . For weak solutions satisfying , we determine as the sharp critical exponent governing energy conservation and uniqueness. When , every weak solution conserves the ‐energy and is unique for arbitrary continuous . Conversely, for , we construct continuous forcings that admit non‐conservative weak solutions and exhibit failure of uniqueness. Our counterexample further shows that dissipative solutions with identical initial data may be non‐unique, thereby distinguishing the conservative and dissipative frameworks. The analysis relies on Littlewood–Paley theory, Besov embeddings, and delicate commutator estimates.

Abstract For nonempty subsets and of a group , we say that is a tiling of if every element of can be uniquely expressed as for some and . In 1966, Rothaus and Thompson studied whether the symmetric group with admits a tiling , where consists of the identity and all the transpositions in . They showed that no such tiling exists if is divisible by a prime number at least . In this paper, we establish a new necessary condition for the existence of such a tiling: the subset must be partition‐transitive with respect to certain partitions of . This generalizes the result of Rothaus and Thompson, as well as a result of Nomura in 1985. We also study whether can be tiled by the set of all the transpositions, which finally leads us to conjecture that neither nor tiles for any .

Abstract Under the generalized Riemann hypothesis, we use Beurling–Selberg extremal functions to bound the mean and mean square of the argument of Dirichlet ‐functions for a large prime modulus . As applications, we give alternative proofs of several results on low‐lying zeros of and obtain a new lower bound on the proportion of modulo with zeros close to the central point . In particular, we show conditionally that for any , there exist a positive proportion of Dirichlet ‐functions whose first zero has height less than times the average spacing between consecutive zeros.

Abstract We show that the path algebra of a quiver satisfies the same polynomial identities (PI) of an algebra of matrices, if any. In particular, the algebra of matrices is PI‐equivalent to the path algebra of the oriented cycle with vertices.

Abstract We prove that a Kakeya set in a vector space over a finite field of size always supports a probability measure, whose Fourier transform is bounded by for all non‐zero frequencies. We show that this bound is sharp in all dimensions at least 2. In particular, this provides a Fourier analytic proof that a Kakeya set in dimension 2 has size at least (which is asymptotically sharp). We also establish analogous results for sets containing ‐planes in a given set of orientations.

Abstract Fix a prime number . The post‐critically finite polynomials of the form play a fundamental role in polynomial dynamics. While many results are known in the complex dynamical setting, much less is understood about the arithmetic properties of these polynomials. In this paper, we describe the factorization of the iterates of post‐critically finite polynomials over their fields of definition. As a consequence, we prove new cases of a conjecture of Andrews and Petsche on abelian arboreal Galois representations.

Abstract We show that a ‐algebra with topological dimension zero has the Global Glimm Property (every hereditary subalgebra contains an almost full nilpotent element) if and only if it is nowhere scattered (no hereditary subalgebra admits a finite‐dimensional representation). This solves the Global Glimm Problem in this setting. It follows that nowhere scattered ‐algebras with finite nuclear dimension and topological dimension zero are pure.