We introduce a new non-abelian quantum synchronisation model over the unitary group, represented as a gradient flow, where state matrices asymptotically converge to a common one up to phase translation. We provide a sufficient framework leading to quantum synchronisation based on Riccati-type differential inequalities. In addition, uniform time-delayed interaction is considered for modelling realistic communication, and we demonstrate that quantum synchronisation is persistent when a small time delay is allowed. Finally, numerical simulation is performed to visualise qualitative behaviours and support theoretical results.

In this paper, we present a sufficient framework to exhibit the sample path-wise asymptotic flocking dynamics of the Cucker–Smale model with unit-speed constraint and the randomly switching network topology. We employ a matrix formulation of the given equation, which allows us to evaluate the diameter of velocities with respect to the adjacency matrix of the network. Unlike the previous result on the randomly switching Cucker–Smale model, the unit-speed constraint disallows the system to be considered as a nonautonomous linear ordinary differential equation on velocity vector, which forces us to get a weaker form of the flocking estimate than the result for the original Cucker–Smale model.

We investigate different geometrical properties, related to Carleson measures and pseudo-hyperbolic separation, of inhomogeneous Poisson point processes on the unit disk. In particular, we give conditions so that these random sequences are almost surely interpolating for the Hardy, Bloch or weighted Dirichlet spaces.

This paper deals with the following quasilinear chemotaxis system with consumption of chemoattractant

\[ \left\{\begin{array}{@{}ll} u_t=\Delta u^{m}-\nabla\cdot(u\nabla v),\quad & x\in \Omega,\quad t>0,\\ v_t=\Delta v-uv,\quad & x\in \Omega,\quad t>0\\ \end{array}\right. \]

$\Omega \subset \mathbb {R}^N(N=3,\,4,\,5)$

$\partial \Omega$

$m>\max \{1,\,\frac {3N-2}{2N+2}\}$

$(\bar {u}_0,\,0)$

$t\rightarrow \infty$

$\bar {u}_0=\frac {1}{|\Omega |}\int _\Omega u_0$

$m>\frac {3N-2}{2N}$

$N\geq 3$

$m>\frac {3N}{2N+2}$

$N\geq 2$

In this paper, we consider the existence and limiting behaviour of solutions to a semilinear elliptic equation arising from confined plasma problem in dimension two

\[ \begin{cases} -\Delta u=\lambda k(x)f(u) & \text{in}\ D,\\ u= c & \displaystyle\text{on}\ \partial D,\\ \displaystyle - \int_{\partial D} \frac{\partial u}{\partial \nu}\,{\rm d}s=I, \end{cases} \]

$D\subseteq \mathbb {R}^2$

$\nu$

$\partial D$

$\lambda$

$I$

$c$

$f$

$k$

$\Gamma (x)=({1}/{2})h(x,\,x)- ({1}/{8\pi })\ln k(x)$

$\lambda \to +\infty$

$h(x,\,x)$

$-\Delta$

$D$

$f$

$k$

$k$

$measure$

A suitable notion of weak amenability for dual Banach algebras, which we call weak Connes amenability, is defined and studied. Among other things, it is proved that the measure algebra M(G) of a locally compact group G is always weakly Connes amenable. It can be a complement to Johnson’s theorem that $L^1(G)$ is always weakly amenable [10].

Given a graph $F$, we consider the problem of determining the densest possible pseudorandom graph that contains no copy of $F$. We provide an embedding procedure that improves a general result of Conlon, Fox, and Zhao which gives an upper bound on the density. In particular, our result implies that optimally pseudorandom graphs with density greater than $n^{-1/3}$ must contain a copy of the Peterson graph, while the previous best result gives the bound $n^{-1/4}$. Moreover, we conjecture that the exponent $1/3$ in our bound is tight. We also construct the densest known pseudorandom $K_{2,3}$-free graphs that are also triangle-free. Finally, we give a different proof for the densest known construction of clique-free pseudorandom graphs due to Bishnoi, Ihringer, and Pepe that they have no large clique.

We investigate here the behaviour of a large typical meandric system, proving a central limit theorem for the number of components of a given shape. Our main tool is a theorem of Gao and Wormald that allows us to deduce a central limit theorem from the asymptotics of large moments of our quantities of interest.

We generalize the influential $C^*$-algebraic results of Kawamura–Tomiyama and Archbold–Spielberg for crossed products of discrete groups actions to the realm of Banach algebras and twisted actions. We prove that topological freeness is equivalent to the intersection property for all reduced twisted Banach algebra crossed products coming from subgroups, and in the untwisted case to a generalized intersection property for a full $L^p$-operator algebra crossed product for any $p\in [1,\,\infty ]$. This gives efficient simplicity criteria for various Banach algebra crossed products. We also use it to identify the prime ideal space of some crossed products as the quasi-orbit space of the action. For amenable actions we prove that the full and reduced twisted $L^p$-operator algebras coincide.

The synchronization hierarchy of finite permutation groups consists of classes of groups lying between $2$-transitive groups and primitive groups. This includes the class of spreading groups, which are defined in terms of sets and multisets of permuted points, and which are known to be primitive of almost simple, affine or diagonal type. In this paper, we prove that in fact no spreading group of diagonal type exists. As part of our proof, we show that all non-abelian finite simple groups, other than six sporadic groups, have a transitive action in which a proper normal subgroup of a point stabilizer is supplemented by all corresponding two-point stabilizers.

In his work on modularity of elliptic curves and Fermat’s last theorem, A. Wiles introduced two measures of congruences between Galois representations and between modular forms. One measure is related to the order of a Selmer group associated to a newform $f \in S_2(\Gamma _0(N))$ (and closely linked to deformations of the Galois representation $\rho _f$ associated to f), whilst the other measure is related to the congruence module associated to f (and is closely linked to Hecke rings and congruences between f and other newforms in $S_2(\Gamma _0(N))$). The equality of these two measures led to isomorphisms $R={\mathbf T}$ between deformation rings and Hecke rings (via a numerical criterion for isomorphisms that Wiles proved) and showed these rings to be complete intersections.

We continue our study begun in [BKM21] of the Wiles defect of deformation rings and Hecke rings (at a newform f) acting on the cohomology of Shimura curves over ${\mathbf Q}$: It is defined to be the difference between these two measures of congruences. The Wiles defect thus arises from the failure of the Wiles numerical criterion at an augmentation $\lambda _f:{\mathbf T} \to {\mathcal O}$. In situations we study here, the Taylor–Wiles–Kisin patching method gives an isomorphism $ R={\mathbf T}$ without the rings being complete intersections. Using novel arguments in commutative algebra and patching, we generalize significantly and give different proofs of the results in [BKM21] that compute the Wiles defect at $\lambda _f: R={\mathbf T} \to {\mathcal O}$, and explain in an a priori manner why the answer in [BKM21] is a sum of local defects. As a curious application of our work we give a new and more robust approach to the result of Ribet–Takahashi that computes change of degrees of optimal parametrizations of elliptic curves over ${\mathbf Q}$ by Shimura curves as we vary the Shimura curve. The results we prove are not attainable using only the methods of Ribet–Takahashi.

In Communications in Contemporary Mathematics 24 3, (2022),the authors have developed a method for constructing G-invariant partial differential equations (PDEs) imposed on hypersurfaces of an $(n+1)$-dimensional homogeneous space $G/H$, under mild assumptions on the Lie group G. In the present paper, the method is applied to the case when $G=\mathsf{PGL}(n+1)$ (respectively, $G=\mathsf{Aff}(n+1)$) and the homogeneous space $G/H$ is the $(n+1)$-dimensional projective $\mathbb{P}^{n+1}$ (respectively, affine $\mathbb{A}^{n+1}$) space, respectively. The main result of the paper is that projectively or affinely invariant PDEs with n independent and one unknown variables are in one-to-one correspondence with invariant hypersurfaces of the space of trace-free cubic forms in n variables with respect to the group $\mathsf{CO}(d,n-d)$ of conformal transformations of $\mathbb{R}^{d,n-d}$.

Choosing ${\kappa }$ (horizontal ordinate of the saddle point associated to the homoclinic orbit) as bifurcation parameter, bifurcations of the travelling wave solutions is studied in a perturbed $(1 + 1)$-dimensional dispersive long wave equation. The solitary wave solution exists at a suitable wave speed $c$ for the bifurcation parameter ${\kappa }\in \left (0,1-\frac {\sqrt 3}{3}\right )\cup \left (1+\frac {\sqrt 3}{3},2\right )$, while the kink and anti-kink wave solutions exist at a unique wave speed $c^*=\sqrt {15}/3$ for $\kappa =0$ or $\kappa =2$. The methods are based on the geometric singular perturbation (GSP, for short) approach, Melnikov method and invariant manifolds theory. Interestingly, not only the explicit analytical expression of the complicated homoclinic Melnikov integral is directly obtained for the perturbed long wave equation, but also the explicit analytical expression of the limit wave speed is directly given. Numerical simulations are utilized to verify our mathematical results.

In this paper, we derive the effective model describing a thin-domain flow with permeable boundary through which the fluid is injected into the domain. We start with incompressible Stokes system and perform the rigorous asymptotic analysis. Choosing the appropriate scaling for the injection leads to a compressible effective model. In this paper, we derive the effective model describing a thin-domain flow with permeable boundary through which the fluid is injected into the domain. We start with incompressible Stokes system and perform the rigorous asymptotic analysis. Choosing the appropriate scaling for the injection leads to a compressible effective model.

Given any commutative Noetherian ring R and an element x in R, we consider the full subcategory $\mathsf{C}(x)$ of its singularity category consisting of objects for which the morphism that is given by the multiplication by x is zero. Our main observation is that we can establish a relation between $\mathsf{C}(x), \mathsf{C}(y)$ and $\mathsf{C}(xy)$ for any two ring elements x and y. Utilizing this observation, we obtain a decomposition of the singularity category and consequently an upper bound on the dimension of the singularity category.

We show a result on propagation of the anisotropic Gabor wave front set for linear operators with a tempered distribution Schwartz kernel. The anisotropic Gabor wave front set is parametrized by a positive parameter relating the space and frequency variables. The anisotropic Gabor wave front set of the Schwartz kernel is assumed to satisfy a graph type criterion. The result is applied to a class of evolution equations that generalizes the Schrödinger equation for the free particle. The Laplacian is replaced by any partial differential operator with constant coefficients, real symbol and order at least two.

The $d$-process generates a graph at random by starting with an empty graph with $n$ vertices, then adding edges one at a time uniformly at random among all pairs of vertices which have degrees at most $d-1$ and are not mutually joined. We show that, in the evolution of a random graph with $n$ vertices under the $d$-process with $d$ fixed, with high probability, for each $j \in \{0,1,\dots,d-2\}$, the minimum degree jumps from $j$ to $j+1$ when the number of steps left is on the order of $\ln (n)^{d-j-1}$. This answers a question of Ruciński and Wormald. More specifically, we show that, when the last vertex of degree $j$ disappears, the number of steps left divided by $\ln (n)^{d-j-1}$ converges in distribution to the exponential random variable of mean $\frac{j!}{2(d-1)!}$; furthermore, these $d-1$ distributions are independent.

We study the poor-biased model for money exchange introduced in Cao & Motsch ((2023) Kinet. Relat. Models 16(5), 764–794.): agents are being randomly picked at a rate proportional to their current wealth, and then the selected agent gives a dollar to another agent picked uniformly at random. Simulations of a stochastic system of finitely many agents as well as a rigorous analysis carried out in Cao & Motsch ((2023) Kinet. Relat. Models 16(5), 764–794.), Lanchier ((2017) J. Stat. Phys. 167(1), 160–172.) suggest that, when both the number of agents and time become large enough, the distribution of money among the agents converges to a Poisson distribution. In this manuscript, we establish a uniform-in-time propagation of chaos result as the number of agents goes to infinity, which justifies the validity of the mean-field deterministic infinite system of ordinary differential equations as an approximation of the underlying stochastic agent-based dynamics.

We show that for $n \neq 1,4$, the simplicial volume of an inward tame triangulable open $n$-manifold $M$ with amenable fundamental group at infinity at each end is finite; moreover, we show that if also $\pi _1(M)$ is amenable, then the simplicial volume of $M$ vanishes. We show that the same result holds for finitely-many-ended triangulable manifolds which are simply connected at infinity.

We prove the Singer conjecture for varieties with semismall Albanese map and residually finite fundamental group.