Given any unital, finite, classifiable $\mathrm{C}^*$-algebra A with real rank zero and any compact simplex bundle with the fibre at zero being homeomorphic to the space of tracial states on A, we show that there exists a flow on A realizing this simplex. Moreover, we show that given any unital $\mathrm{UCT}$ Kirchberg algebra A and any proper simplex bundle with empty fibre at zero, there exists a flow on A realizing this simplex.

Let µ be a finite positive Borelmeasure on $[0,1)$ and $\alpha \gt -1$. The generalized integral operator of Hilbert type $\mathcal {I}_{\mu_{\alpha+1}}$ is defined on the spaces $H(\mathbb{D})$ of analytic functions in the unit disc $\mathbb{D}$ as follows:

\begin{equation*}\mathcal {I}_{\mu_{\alpha+1}}(f)(z)=\int_{0}^{1} \frac{f(t)}{(1-tz)^{\alpha+1}}d\mu(t),\ \ f\in H(\mathbb{D}),\ \ z\in \mathbb{D} .\end{equation*}

In this paper, we give a unified characterization of the measures µ for which the operator $\mathcal {I}_{\mu_{\alpha+1}}$ is bounded from the Bloch space to a Bergman space for all $\alpha \gt -1$. Additionally, we also investigate the action of $\mathcal {I}_{\mu_{\alpha+1}}$ from the Bloch space to the Hardy spaces and the Besov spaces.

We show that a compact complex manifold X has no non-trivial nef $(1,1)$-classes if there is a non-biholomorphic bimeromorphic map $f\colon X\dashrightarrow Y$, which is an isomorphism in codimension 1 to a compact Kähler manifold Y with $h^{1,1}=1$. In particular, there exist infinitely many isomorphic classes of smooth compact Moishezon threefolds with no nef and big $(1,1)$-classes. This contradicts a recent paper (Strongly Jordan property and free actions of non-abelian free groups, Proc. Edinb. Math. Soc., 65(3) (2022), 736–746).

In this paper, we will show the Carleson measure characterizations for the boundedness and compactness of the Cesàro-type operator

\begin{equation*}\mathcal{C}_{\mu}(f)(z)=\sum^{\infty}_{n=0}\left( \int_{[0,1)}t^nd\mu(t)\right) \left(\sum^{n}_{k=0}a_k \right)z^n, \quad z\in \mathbb{D},\end{equation*}

acting on a number of important analytic function spaces on $\mathbb{D}$, where µ is a positive finite Borel measure. The function spaces are some newly appeared analytic function spaces (e.g., Bergman–Morrey spaces $A^{p,\lambda}$ and Dirichlet–Morrey spaces $\mathcal{D}_p^{\lambda}$) . This work continues the lines of the previous characterizations by Blasco and Galanopoulos et al. for classical Hardy spaces and weighted Bergman spaces and so forth.

In this paper,the linear space $\mathcal F$ of a special type of fractal interpolation functions (FIFs) on an interval I is considered. Each FIF in $\mathcal F$ is established from a continuous function on I. We show that, for a finite set of linearly independent continuous functions on I, we get linearly independent FIFs. Then we study a finite-dimensional reproducing kernel Hilbert space (RKHS) $\mathcal F_{\mathcal B}\subset\mathcal F$, and the reproducing kernel $\mathbf k$ for $\mathcal F_{\mathcal B}$ is defined by a basis of $\mathcal F_{\mathcal B}$. For a given data set $\mathcal D=\{(t_k, y_k) : k=0,1,\ldots,N\}$, we apply our results to curve fitting problems of minimizing the regularized empirical error based on functions of the form $f_{\mathcal V}+f_{\mathcal B}$, where $f_{\mathcal V}\in C_{\mathcal V}$ and $f_{\mathcal B}\in \mathcal F_{\mathcal B}$. Here $C_{\mathcal V}$ is another finite-dimensional RKHS of some classes of regular continuous functions with the reproducing kernel $\mathbf k^*$. We show that the solution function can be written in the form $f_{\mathcal V}+f_{\mathcal B}=\sum_{m=0}^N\gamma_m\mathbf k^*_{t_m} +\sum_{j=0}^N \alpha_j\mathbf k_{t_j}$, where ${\mathbf k}_{t_m}^\ast(\cdot)={\mathbf k}^\ast(\cdot,t_m)$ and $\mathbf k_{t_j}(\cdot)=\mathbf k(\cdot,t_j)$, and the coefficients γm and αj can be solved by a system of linear equations.

In this paper, we investigate Kolmogorov-type theorems for small perturbations of degenerate Hamiltonian systems. These systems are index by a parameter ξ as $ H(y,x,\xi) = \langle\omega(\xi),y\rangle {+ \bar h(y,\xi)}+\varepsilon P(y,x,\xi,\varepsilon) $, where ɛ > 0. We assume that the frequency mapping $\omega(\cdot)$, $\bar h(y,\cdot)=O(|y|^2)$ and the perturbation $\varepsilon P(y,x,\cdot, \varepsilon)$ maintain Hölder continuity about ξ. We prove that the persistent invariant tori retain the same frequency as those of the unperturbed tori, under a certain topological degree condition and a weak convexity condition for the frequency mapping. Notably, this paper presents, to our understanding, pioneering results on the KAM theorem under such conditions with only assumption of Hölder continuous dependence of frequency mapping ω on the parameter.

Well-posedness in time-weighted spaces of certain quasilinear (and semilinear) parabolic evolution equations $u'=A(u)u+f(u)$ is established. The focus lies on the case of strict inclusions $\mathrm{dom}(f)\subsetneq \mathrm{dom}(A)$ of the domains of the nonlinearities $u\mapsto f(u)$ and $u\mapsto A(u)$. Based on regularizing effects of parabolic equations it is shown that a semiflow is generated in intermediate spaces. In applications this allows one to derive global existence from weaker a priori estimates. The result is illustrated by examples of chemotaxis systems.

For commuting contractions $T_1,\dots,T_n$ acting on a Hilbert space $\mathscr{H}$ with $T=\prod_{i=1}^n T_i$, we find a necessary and sufficient condition such that $(T_1,\dots,T_n)$ dilates to a commuting tuple of isometries $(V_1,\dots,V_n)$ on the minimal isometric dilation space of T with $V=\prod_{i=1}^nV_i$ being the minimal isometric dilation of T. This isometric dilation provides a commutant lifting of $(T_1, \dots, T_n)$ on the minimal isometric dilation space of T. We construct both Schäffer and Sz. Nagy–Foias-type isometric dilations for $(T_1,\dots,T_n)$ on the minimal dilation spaces of T. Also, a different dilation is constructed when the product T is a $C._0$ contraction, that is, ${T^*}^n \rightarrow 0$ as $n \rightarrow \infty$. As a consequence of these dilation theorems, we obtain different functional models for $(T_1,\dots,T_n)$ in terms of multiplication operators on vectorial Hardy spaces. One notable fact about our models is that the multipliers are all analytic functions in one variable. The dilation when T is a $C._0$ contraction leads to a conditional factorization of T. Several examples have been constructed.

We study the geometry of tropical extensions of hyperfields, including the ordinary, signed, and complex tropical hyperfields. We introduce the framework of ‘enriched valuations’ as hyperfield homomorphisms to tropical extensions and show that a notable family of them are relatively algebraically closed. Our main results are hyperfield analogues of Kapranov’s theorem and the Fundamental theorem of tropical geometry. Utilizing these theorems, we introduce fine tropical varieties and prove a structure theorem for them in terms of their initial ideals.

We study the mixing time of the single-site update Markov chain, known as the Glauber dynamics, for generating a random independent set of a tree. Our focus is obtaining optimal convergence results for arbitrary trees. We consider the more general problem of sampling from the Gibbs distribution in the hard-core model where independent sets are weighted by a parameter $\lambda \gt 0$; the special case $\lambda =1$ corresponds to the uniform distribution over all independent sets. Previous work of Martinelli, Sinclair and Weitz (2004) obtained optimal mixing time bounds for the complete $\Delta$-regular tree for all $\lambda$. However, Restrepo, Stefankovic, Vera, Vigoda, and Yang (2014) showed that for sufficiently large $\lambda$ there are bounded-degree trees where optimal mixing does not hold. Recent work of Eppstein and Frishberg (2022) proved a polynomial mixing time bound for the Glauber dynamics for arbitrary trees, and more generally for graphs of bounded tree-width.

We establish an optimal bound on the relaxation time (i.e., inverse spectral gap) of $O(n)$ for the Glauber dynamics for unweighted independent sets on arbitrary trees. We stress that our results hold for arbitrary trees and there is no dependence on the maximum degree $\Delta$. Interestingly, our results extend (far) beyond the uniqueness threshold which is on the order $\lambda =O(1/\Delta )$. Our proof approach is inspired by recent work on spectral independence. In fact, we prove that spectral independence holds with a constant independent of the maximum degree for any tree, but this does not imply mixing for general trees as the optimal mixing results of Chen, Liu, and Vigoda (2021) only apply for bounded-degree graphs. We instead utilize the combinatorial nature of independent sets to directly prove approximate tensorization of variance via a non-trivial inductive proof.

The fixed points of the generalized Ricci flow are the Bismut Ricci flat (BRF) metrics, i.e., a generalized metric (g, H) on a manifold M, where g is a Riemannian metric and H a closed 3-form, such that H is g-harmonic and $\operatorname{Rc}(g)=\tfrac{1}{4} H_g^2$. Given two standard Einstein homogeneous spaces $G_i/K$, where each Gi is a compact simple Lie group and K is a closed subgroup of them holding some extra assumption, we consider $M=G_1\times G_2/\Delta K$. Recently, Lauret and Will proved the existence of a BRF metric on any of these spaces. We proved that this metric is always asymptotically stable for the generalized Ricci flow on M among a subset of G-invariant metrics and, if $G_1=G_2$, then it is globally stable.

In this paper, we define and study an equivariant analogue of Cohen, Farber and Weinberger’s parametrized topological complexity. We show that several results in the non-equivariant case can be extended to the equivariant case. For example, we establish the fibrewise equivariant homotopy invariance of the sequential equivariant parametrized topological complexity. We obtain several bounds on sequential equivariant topological complexity involving the equivariant category. We also obtain the cohomological lower bound and the dimension-connectivity upper bound on the sequential equivariant parametrized topological complexity. In the end, we use these results to compute the sequential equivariant parametrized topological complexity of equivariant Fadell–Neuwirth fibrations and some equivariant fibrations involving generalized projective product spaces.

We establish new results on complex and $p$-adic linear independence on a class of semiabelian varieties. As applications, we obtain transcendence results concerning complex and $p$-adic Weierstrass sigma functions associated with elliptic curves.

We prove interior boundedness and Hölder continuity for the weak solutions of nonlocal double phase equations in the Heisenberg group $\mathbb{H}^n$. This solves a problem raised by Palatucci and Piccinini et al. in 2022 and 2023 for the nonlinear integro-differential problems in Heisenberg setting. Our proof of the a priori estimates bases on De Giorgi–Nash–Moser theory, where the important ingredients are Caccioppoli-type inequality and Logarithmic estimate. To achieve this goal, we establish a new and crucial Sobolev–Poincaré type inequality in local domain, which may be of independent interest and potential applications.

In [CDD22], we investigated the structure of $\ast $-isomorphisms between von Neumann algebras $L(\Gamma )$ associated with graph product groups $\Gamma $ of flower-shaped graphs and property (T) wreath-like product vertex groups, as in [CIOS21]. In this follow-up, we continue the structural study of these algebras by establishing that these graph product groups $\Gamma $ are entirely recognizable from the category of all von Neumann algebras arising from an arbitrary nontrivial graph product group with infinite vertex groups. A sharper $C^*$-algebraic version of this statement is also obtained. In the process of proving these results, we also extend the main $W^*$-superrigidity result from [CIOS21] to direct products of property (T) wreath-like product groups.

In network science, one of the significant and challenging subjects is the detection of communities. Modularity [1] is a measure of community structure that compares connectivity in the network with the expected connectivity in a graph sampled from a random null model. Its optimisation is a common approach to tackle the community detection problem. We present a new method for modularity maximisation, which is based on the observation that modularity can be expressed in terms of total variation on the graph and signless total variation on the null model. The resulting algorithm is of Merriman–Bence–Osher (MBO) type. Different from earlier methods of this type, the new method can easily accommodate different choices of the null model. Besides theoretical investigations of the method, we include in this paper numerical comparisons with other community detection methods, among which the MBO-type methods of Hu et al. [2] and Boyd et al. [3], and the Leiden algorithm [4].

We study Morita equivalence for idempotent rings with involution. Following the ideas of Rieffel, we define Rieffel contexts, and we also introduce Morita $*$-contexts and enlargements for rings with involution. We prove that two idempotent rings with involution have a joint enlargement if and only if they are connected by a unitary and full Rieffel context. These conditions are also equivalent to having a unitary and surjective Morita $*$-context between those rings. We also examine how the mentioned conditions are connected to the existence of certain equivalence functors between the categories of firm modules over the given rings with involution.

In this work, the Riemann–Hilbert (RH) problem is employed to study the multiple high-order pole solutions of the cubic Camassa–Holm (cCH) equation with the term characterizing the effect of linear dispersion under zero boundary conditions and nonzero boundary conditions. Under the reflectionless situation, we generalize the residue theorem and obtain the multiple high-order pole solutions of cCH equation by solving an algebraic system. During the process of establishing the solution of RH problem, to simplify the calculations involving the implicitly expressed of variables (x, t) in the solution, we introduce a new scale (y, t) to ensure the solution of RH problem is explicitly expressed with respect to it. Finally, the exact solutions are obtained for cases involving one high-order pole and N high-order poles.

We present a one-parameter family Fλ of transcendental entire functions with zeros, whose Newton’s method yields wandering domains, coexisting with the basins of the roots of Fλ. Wandering domains for Newton maps of zero-free functions have been built before by, e.g. Buff and Rückert [23] based on the lifting method. This procedure is suited to our Newton maps as members of the class of projectable functions (or maps of the cylinder), i.e. transcendental meromorphic functions f(z) in the complex plane that are semiconjugate, via the exponential, to some map g(w), which may have at most a countable number of essential singularities. In this paper, we make a systematic study of the general relation (dynamical and otherwise) between f and g, and inspect the extension of the logarithmic lifting method of periodic Fatou components to our context, especially for those g of finite-type. We apply these results to characterize the entire functions with zeros whose Newton’s method projects to some map g which is defined at both 0 and $\infty$. The family Fλ is the simplest in this class, and its parameter space shows open sets of λ-values in which the Newton map exhibits wandering or Baker domains, in both cases regions of initial conditions where Newton’s root-findingmethod fails.

The article studies an initial boundary valueproblem (ibvp) for the radial solutions of the nonlinear Schrödinger (NLS) equation in a radially symmetric region $\Omega\in \mathbb R^n$ with boundaries. All such regions can be classified into three types: a ball Ω0 centred at origin, a region Ω1 outside a ball, and an n-dimensional annulus Ω2. To study the well-posedness of those ibvps, the function spaces for the boundary data must be specified in terms of the solutions in appropriate Sobolev spaces. It is shown that when $\Omega = \Omega_1$, the ibvp for the NLS equation is locally well-posed in $ C( [0, T^*]; H^s(\Omega_1))$ if the initial data is in $H^s(\Omega_1)$ and boundary data is in $ H^{\frac{2s+1}{4}}(0, T)$ with $s \geq 0$. This is the optimal regularity for the boundary data and cannot be improved. When $\Omega = \Omega_2$, the ibvp is locally well-posed in $ C( [0, T^*]; H^s(\Omega_2))$ if the initial data is in $ H^s(\Omega_2)$ and boundary data is in $ H^{\frac{s+1}{2}}(0, T)$ with $s \geq 0$. In this case, the boundary data requires $1/4$ more derivative compared to the case when $\Omega = \Omega_1$. When $\Omega = \Omega_0$ with n = 2 (the case with n > 2 can be discussed similarly), the ibvp is locally well-posed in $ C( [0, T^*]; H^s(\Omega_0))$ if the initial data is in $ H^s(\Omega_0)$ and boundary data is in $ H^{\frac{s+1}{2}}(0, T)$ with s > 1 (or $s \gt n/2$). Due to the lack of Strichartz estimates for the corresponding boundary integral operator with $ 0 \leq s \leq 1$, the local well-posedness can only be achieved for s > 1. It is noted that the well-posedness results on Ω0 and Ω2 are the first ones for the ibvp of NLS equations in bounded regions of higher dimension.