Let $\mu _{M,D}$ be the self-similar measure generated by $M=RN^q$ and the product-form digit set $D=\{0,1,\ldots ,N-1\}\oplus N^{p_1}\{0,1,\ldots ,N-1\}\oplus \cdots \oplus N^{p_s}\{0,1,\ldots ,N-1\}$, where $R\geq 2$, $N\geq 2$, q, $p_i(1\leq i\leq s)$ are integers with $\gcd (R,N)=1$ and $1\leq p_1<p_2<\cdots <p_s<q$. In this paper, we first show that $\mu _{M,D}$ is a spectral measure with a model spectrum $\Lambda $. Then, we completely settle two types of spectral eigenvalue problems for $\mu _{M,D}$. In the first case, for a real t, we give a necessary and sufficient condition under which $t\Lambda $ is also a spectrum of $\mu _{M,D}$. In the second case, we characterize all possible real numbers t such that $\Lambda '\subset \mathbb {R}$ and $t\Lambda '$ are both spectra of $\mu _{M,D}$.

Let S and T be smooth projective varieties over an algebraically closed field k. Suppose that S is a surface admitting a decomposition of the diagonal. We show that, away from the characteristic of k, if an algebraic correspondence $T \to S$ acts trivially on the unramified cohomology, then it acts trivially on any normalized, birational and motivic functor. This generalizes Kahn’s result on the torsion order of S. We also exhibit an example of S over $\mathbb {C}$ for which $S \times S$ violates the integral Hodge conjecture.

In this article, we study the Johnson homomorphisms of basis-conjugating automorphism groups of free groups. We construct obstructions for the surjectivity of the Johnson homomorphisms. By using it, we determine their cokernels of degree up to four and give further observations for degree greater than four. As applications, we give the affirmative answer to the Andreadakis problem for the basis-conjugating automorphism groups of free groups at degree four. Moreover, we calculate twisted first cohomology groups of the braid-permutation automorphism groups of free groups.

Given a Hamiltonian torus action on a symplectic manifold, Teleman and Fukaya have proposed that the Fukaya category of each symplectic quotient should be equivalent to an equivariant Fukaya category of the original manifold. We lay out new conjectures that extend this story – in certain situations – to singular values of the moment map. These include a proposal for how, in some cases, we can recover the non-equivariant Fukaya category of the original manifold starting from data on the quotient.

To justify our conjectures, we pass through the mirror and work out numerous examples, using well-established heuristics in toric mirror symmetry. We also discuss the algebraic and categorical structures that underlie our story.

We construct an fpqc gerbe $\mathcal {E}_{\dot {V}}$ over a global function field F such that for a connected reductive group G over F with finite central subgroup Z, the set of $G_{\mathcal {E}_{\dot {V}}}$-torsors contains a subset $H^{1}(\mathcal {E}_{\dot {V}}, Z \to G)$ which allows one to define a global notion of (Z-)rigid inner forms. There is a localization map $H^{1}(\mathcal {E}_{\dot {V}}, Z \to G) \to H^{1}(\mathcal {E}_{v}, Z \to G)$, where the latter parametrizes local rigid inner forms (cf. [8, 6]) which allows us to organize local rigid inner forms across all places v into coherent families. Doing so enables a construction of (conjectural) global L-packets and a conjectural formula for the multiplicity of an automorphic representation $\pi $ in the discrete spectrum of G in terms of these L-packets. We also show that, for a connected reductive group G over a global function field F, the adelic transfer factor $\Delta _{\mathbb {A}}$ for the ring of adeles $\mathbb {A}$ of F serving an endoscopic datum for G decomposes as the product of the normalized local transfer factors from [6].

In this paper, we deal with the following nonlinear time-harmonic Maxwell’s equations

\begin{equation*}\nonumber\begin{cases} \nabla \times(\nabla \times u_1)=\mu_1|u_1|^4u_1+\beta|u_1||u_2|^3u_1 & \text {in } \mathbb{R}^3, \\ \nabla \times(\nabla \times u_2)=\mu_2|u_2|^4u_2+\beta|u_1|^3|u_2|u_2 & \text {in } \mathbb{R}^3, \end{cases}\end{equation*}

$\nabla\times$

$\mathbb{R}^3$

$\mu_1,\mu_2 \gt 0$

$\beta\in\mathbb{R}\backslash\{0\}$

$\beta\in\mathbb{R}\backslash\{0\}$

In this article, we investigate the possibility of generating all the configurations of a subshift in a local way. We propose two definitions of local generation, explore their properties and develop techniques to determine whether a subshift satisfies these definitions. We illustrate the results with several examples.

We establish two complementary results about the regularity of the solution of the periodic initial value problem for the linear Benjamin–Ono equation. We first give a new simple proof of the statement that, for a dense countable set of the time variable, the solution is a finite linear combination of copies of the initial condition and of its Hilbert transform. In particular, this implies that discontinuities in the initial condition are propagated in the solution as logarithmic cusps. We then show that, if the initial condition is of bounded variation (and even if it is not continuous), for almost every time the graph of the solution in space is continuous but fractal, with upper Minkowski dimension equal to $\frac32$. In order to illustrate this striking dichotomy, in the final section, we include accurate numerical evaluations of the solution profile, as well as estimates of its box-counting dimension for two canonical choices of irrational time.

We give a notion of boundary pair $(\mathcal{B}_-,\mathcal{B}_+)$ for measured groupoids which generalizes the one introduced by Bader and Furman [BF14] for locally compact groups. In the case of a semidirect groupoid $\mathcal{G}=\Gamma \ltimes X$ obtained by a probability measure preserving action $\Gamma \curvearrowright X$ of a locally compact group, we show that a boundary pair is exactly $(B_- \times X, B_+ \times X)$, where $(B_-,B_+)$ is a boundary pair for $\Gamma$. For any measured groupoid $(\mathcal{G},\nu )$, we prove that the Poisson boundaries associated to the Markov operators generated by a probability measure equivalent to $\nu$ provide other examples of our definition. Following Bader and Furman [BF], we define algebraic representability for an ergodic groupoid $(\mathcal{G},\nu )$. In this way, given any measurable representation $\rho \,:\,\mathcal{G} \rightarrow H$ into the $\kappa$-points of an algebraic $\kappa$-group $\mathbf{H}$, we obtain $\rho$-equivariant maps $\mathcal{B}_\pm \rightarrow H/L_\pm$, where $L_\pm =\mathbf{L}_\pm (\kappa )$ for some $\kappa$-subgroups $\mathbf{L}_\pm \lt \mathbf{H}$. In the particular case when $\kappa =\mathbb{R}$ and $\rho$ is Zariski dense, we show that $L_\pm$ must be minimal parabolic subgroups.

Several classical knot invariants, such as the Alexander polynomial, the Levine-Tristram signature, and the Blanchfield pairing, admit natural extensions from knots to links, and more generally, from oriented links to so-called colored links. In this note, we explore such extensions of the Arf invariant. Inspired by the three examples stated above, we use generalized Seifert forms to construct quadratic forms and determine when the Arf invariant of such a form yields a well-defined invariant of colored links. However, apart from the known case of oriented links, these new Arf invariants turn out to be determined by the linking numbers.

In [5], a particular family of real hyperplane arrangements stemming from hyperpolygonal spaces associated with certain quiver varieties was introduced which we thus call hyperpolygonal arrangements ${\mathscr H}_n$. In this note, we study these arrangements and investigate their properties systematically. Remarkably, the arrangements ${\mathscr H}_n$ discriminate between essentially all local properties of arrangements. In addition, we show that hyperpolygonal arrangements are projectively unique and combinatorially formal.

We note that the arrangement ${\mathscr H}_5$ is the famous counterexample of Edelman and Reiner [17] of Orlik’s conjecture that the restriction of a free arrangement is again free.

We consider the two-dimensional nonlinear Schrödinger equation with point interaction and we establish a local well-posedness theory, including blow-up alternative and continuous dependence on the initial data in the energy space. We provide proof by employing Kato’s method along with Hardy inequalities with logarithmic correction. Moreover, we establish finite time blow-up for solutions with positive energy and infinite variance.

In this paper, we study the following high-order elliptic equation involving the GJMS operator:

\begin{align*}\alpha P_{\mathbb{S}^n}v_{\alpha}+2Q_{g_{\mathbb{S}^n}}=2Q_{g_{\mathbb{S}^n}}e^{nv_{\alpha}}.\end{align*}

We establish that if α > 1 and $n\geq3$ or if $\alpha\in (1-\epsilon_0, 1)$ with $n=2m\geq4$, then $v_{\alpha}\equiv0$. As an application, we present a new proof of the classical Beckner inequality.

In this article, we prove the local-in-time existence of regular solutions to dissipative Aw–Rascle system with the offset equal to gradient of some increasing and regular function of density. It is a mixed degenerate parabolic-hyperbolic hydrodynamic model, and we extend the techniques previously developed for compressible Navier–Stokes equations to show the well-posedness of the system in the $L_2-L_2$ setting. We also discuss relevant existence results for offset involving singular or non-local functions of density.

This paper considers two commuting smooth transformations on a Banach space and proves the sub-additivity of the measure theoretic entropies under mild conditions. Furthermore, some additional conditions are given for the equality of the entropies. This extends Hu’s work [Some ergodic properties of commuting diffeomorphisms. Ergod. Th. & Dynam. Sys. 13(1) (1993), 73–100] about commuting diffeomorphisms in a finite dimensional space to the case of systems on an infinite dimensional Banach space.

We consider one-parameter families of smooth circle cocycles over an ergodic transformation in the base, and show that their rotation numbers must be log-Hölder regular with respect to the parameter. As an immediate application, we get a dynamical proof of the one-dimensional version of the Craig–Simon theorem that establishes that the integrated density of states of an ergodic Schrödinger operator must be log-Hölder.

Special Ricci–Hessian equations on Kähler manifolds $(M,g)$, as defined by Maschler [‘Special Kähler–Ricci potentials and Ricci solitons’, Ann. Global Anal. Geom. 34 (2008), 367–380], involve functions $\tau $ on M and state that, for some function $\alpha $ of the real variable $\tau\kern-0.8pt $, the sum of $\alpha \nabla d\tau\kern-0.8pt $ and the Ricci tensor equals a functional multiple of the metric g, while $\alpha \nabla d\tau\kern-0.8pt $ itself is assumed to be nonzero almost everywhere. Three well-known obvious ‘standard’ cases are provided by (non-Einstein) gradient Kähler–Ricci solitons, conformally-Einstein Kähler metrics, and special Kähler–Ricci potentials. We show that, outside of these three cases, such an equation can only occur in complex dimension two and, at generic points, it must then represent one of three types, for which, up to normalizations, $\alpha =2\cot \tau\kern-0.8pt $, $\alpha =2\coth \tau\kern-0.8pt $, or $\alpha =2\tanh \tau\kern-0.8pt $. We also use the Cartan–Kähler theorem to prove that these three types are actually realized in a ‘nonstandard’ way.

The Bray–Liebhafsky reaction is one of many intricate chemical systems that is known to exhibit periodic behaviour. Although the underlying chemistry is somewhat complicated and involves at least ten chemical species, in a recent work we suggested a reduced two-component model of the reaction involving the concentrations of iodine and iodous acid. Although it is drastically simplified, this reduced system retains enough structure so as to exhibit many of the oscillatory characteristics seen in experimental analyses. Here, we consider the possibility of spatial patterning in a nonuniformly mixed solution. Since many practical demonstrations of chemical oscillations are undertaken using circular containers such as beakers or Petri dishes, we develop both linearized and nonlinear pattern solutions in terms of cylindrical coordinates. These results are complemented by an analysis of the patterning that might be possible within a rectangular domain. The simulations give compelling evidence that spatial patterning may well be feasible in the Bray–Liebhafsky process.