We describe a new method to reconstruct the permittivity distribution, of an object to image, from the remotely measured electromagnetic field. We propose to use the remote fields measured before and after injecting locally in the medium plasmonic nanoparticles. Such a technique is known in the framework of imaging using contrast agents where, in optical imaging, the nanoparticles play the role of these contrast agents. The plasmonic nanoparticles are known to enjoy resonant effects, as enhancing the applied incident field, while excited at certain particular frequencies called plasmonic resonances. These resonant frequencies encode the values of the unknown permittivity at the location of the injected nanoparticles. The imaging methods we propose mainly use this resonant effect. We show that the imaging functional build up from contrasting the fields before and after injecting the nanoparticles, measured at one single back-scattered direction, and in an explicit band of incident frequencies, reaches its maximum values, in terms of the incident frequency, precisely at the mentioned plasmonic resonances. Such a behaviour allows us to recover these plasmonic resonances from which we recover the point-wise values of the permittivity distribution. In this work, we describe the method and provide the mathematical justification of this resonant effect and its use for the optical inversion using plasmonic nanoparticles as contrast agents.

We consider equations involving the truncated Laplacians ${\cal P}_k^\pm$ and having lower-order terms with singular potentials posed in punctured balls. We study both the principal eigenvalue problem and the problem of classification of solutions, in dependence on their asymptotic behaviour near the origin, for equations having also superlinear absorption lower order terms. In the case of ${\cal P}_k^+$, owing to the mild degeneracy of the operator, we obtain results which are analogous to the results for the Laplacian in dimension $k$. On the other hand, for operator ${\cal P}_k^-$, we show that the strong degeneracy in ellipticity of the operator produces radically different results.

Let $f$ be a normalized primitive holomorphic cusp form of even integral weight for the full modular group $\Gamma=SL(2,\mathbb{Z})$. Denote by $\lambda_{\text{sym}^{2}f}(n)$ the $n$th normalized coefficient of the Dirichlet series expansion of the symmetric square $L$-function $L(s,\text{sym}^{2}f)$. In this paper, we are interested in the shifted convolution sum

\begin{equation*}\sum_{h\leq H}\sum_{N \lt n\leq 2N}\lambda_{\textrm{sym}^{2}f}(n)\lambda_{\textrm{sym}^{2}f}(n+h).\end{equation*}

We establish a non-trivial bound for $H\gg N^{1/4}$.

We show that, for a finite spectrum $X$, Spanier–Whitehead duality induces an isomorphism between the cohomological and homological Atiyah–Hirzebruch spectral sequences. As an application, it follows that Poincaré duality for a Poincaré duality complex that is oriented over a ring spectrum $\mathcal{R}$ induces an isomorphism between the two spectral sequences.

In this short note, we prove that every closed, oriented, connected 3-manifold arises as Dehn surgery along a braid positive link.

We show that two simple, separable, nuclear, and ${\mathcal{Z}_0}$-stable $\mathrm{C}^{*}$-algebras are isomorphic if they are trace-preservingly homotopy equivalent. This result does not assume the Universal Coefficient Theorem and can be viewed as a tracial stably projectionless analogue of the homotopy rigidity theorem for Kirchberg algebras.

We study the class of transitive skew products associated with iterated function systems of circle diffeomorphisms. We approximate any of those skew products by maps in this class with a robustly zero Lyapunov exponent. In particular, we prove the existence of non-hyperbolic ergodic measures for an open and dense subset of transitive skew products. Moreover, these measures have full support and are the weak$^*$ limit of periodicmeasures.

In this paper, we use a degenerate reaction-diffusion system with free boundaries to model the spatial spread of rabies among foxes, whose population is divided into three sub-populations: susceptible ($S$), infected ($E$), and rabid ($I$). Based on established biological observations, susceptible and infected foxes are assumed to be territorial (random diffusion is ignored for $E$ and $S$ in the model), whereas rabid foxes disperse randomly (random diffusion is assumed for $I$), causing the spread of the disease. While $S$ evolves over the entire real line $\mathbb{R}$, $E$ and $I$ are found only in the infected region represented by an interval $[g(t), h(t)]$, which expands with moving fronts $x = g(t)$ and $x = h(t)$ as time $t$ increases. We show that this system admits a unique global solution and then analyse its dynamics and establish a spreading-vanishing dichotomy in certain natural parameter regimes. Moreover, we supply some simple sufficient conditions for the vanishing and spreading of the rabies, respectively. For example, we show that if the corresponding ODE system has basic reproduction number $\mathcal{R}_0 \gt 1$, then a spreading-vanishing dichotomy holds, and the outcome depends on the initial size of the infected region, while if a certain quantity $\mathcal R_0^* \in (\mathcal R_0,\infty)$ is no bigger than 1, then the rabies will always vanish. The degenerate nature of the model, combined with the evolving infected region, causes considerable difficulties in the mathematical treatment, both in proving the well-posedness and in understanding the long-time dynamics. This paper appears to be the first to treat a free boundary model where one reaction-diffusion equation is coupled with two ordinary differential equations.

In this paper, we establish general results for the asymptotic behaviour of solutions of dynamical systems in Banach spaces. We show that if the initial datum possesses a certain decay, then the corresponding solution emanating from the Cauchy problem studied inherits the same behaviour at any further time for which it exists. Our results are applied to a wide class of linear and non-linear models. In particular, we use our main results to show persistence properties for classical linear and non-linear ordinary differential equations (ODEs), the Benjamin–Bona–Mahony (BBM) equation, and the generalized Boussinesq equation.

Let $G(\mathbb {R})$ be a real reductive group. Suppose $\pi $ is an irreducible representation of $G(\mathbb {R})$ having a Whittaker model, and consider three invariants of $\pi $ related to nilpotent elements of the Lie algebra of G (or its dual): the associated variety, the wave-front set, and the set of Whittaker data for which $\pi $ has a Whittaker model. If $\pi $ is a discrete series representation, these invariants are known to determine each other. We provide a self-contained account of this and related matters. Many of the results were known: we give simplified proofs for several of them, for instance a simple proof (for generic discrete series) that the associated variety and the wave-front set are related by the Kostant–Sekiguchi correspondence.

We give a new definition of a Frobenius structure on an algebra object in a monoidal category, generalising Frobenius algebras in the category of vector spaces. Our definition allows Frobenius forms valued in objects other than the unit object and can be seen as a categorical version of Frobenius extensions of the second kind. When the monoidal category is pivotal, we define a Nakayama morphism for the Frobenius structure and explain what it means for this morphism to have finite order. Our main example is a well-studied algebra object in the (additive and idempotent completion of the) Temperley–Lieb category at a root of unity. We show that this algebra has a Frobenius structure and that its Nakayama morphism has order 2. As a consequence, we obtain information about Nakayama morphisms of preprojective algebras of Dynkin type, considered as algebras over the semisimple algebras on their vertices.

The present paper is concerned with the optimal weight of vibrating string equations with the first two eigenvalues $\lambda_1$ and $\lambda_2$ being given. Applying the method of critical equations in $L^p[0,1]$ for $p \gt 1$ and the inverse spectral theory of Sturm–Liouville problems with measure coefficients, we find that the optimal weight can be uniquely determined if and only if $\lambda_2 \ge 2\lambda_1$ provided that the weight is non-negative and symmetrical. As an application, we provide an estimation of the extremum for partial trace of the first two eigenvalues on a sphere in $L^1[0,1]$.

Biochemical reaction networks (RNs) are widely applied across scientific disciplines to model complex dynamic systems. We investigate the diffusion approximation of RNs with mass-action kinetics, focusing on the identifiability of the stochastic differential equations associated to the reaction network. We derive conditions under which the law of the diffusion approximation is identifiable and provide theorems for verifying identifiability in practice. Notably, our results show that some RNs have non-identifiable reaction rates, even when the law of the corresponding stochastic process is completely known. Moreover, we show that RNs with distinct graphical structures can generate the same diffusion law under specific choices of reaction rates. Finally, we compare our framework with identifiability results in the deterministic ordinary differential equation setting and the discrete continuous-time Markov chain models for RNs.

We study the dynamics of a delayed predator–prey system with Holling type II functional response, focusing on the interplay between time delay and carrying capacity. Using local and global Hopf bifurcation theory, we establish the existence of sequences of bifurcations as the delay parameter varies and prove that the connected components of global Hopf branches are nested under suitable conditions. A novel contribution is the demonstration that the classical limit cycle of the non-delayed system belongs to a connected component of the global Hopf bifurcation in Fuller’s space. Our analysis combines rigorous functional differential equation theory with continuation methods to characterize the structure and boundedness of bifurcation branches. We further demonstrate that delays can induce oscillatory coexistence at lower carrying capacities than in the corresponding ordinary differential equation model, yielding counterintuitive biological insights. The results contribute to the broader theory of global bifurcations in delay differential equations while providing new perspectives on nonlinear population dynamics.

For a $G$-equivariant fibration $p \colon E\to B$, we introduce and study the invariant analogue of Cohen, Farber, and Weinberger’s parametrized topological complexity, called the invariant parametrized topological complexity. This notion generalizes the invariant topological complexity introduced by Lubawski and Marzantowicz. When $G$ is a compact Lie group acting freely on $E$, we show that the invariant parametrized topological complexity of the $G$-fibration $p \colon E\to B$ coincides with the parametrized topological complexity of the induced fibration $\overline{p} \colon \overline{E} \to \overline{B}$ between the orbit spaces. Furthermore, we compute the invariant parametrized topological complexity of equivariant Fadell–Neuwirth fibrations, which measures the complexity of motion planning in the presence of obstacles with unknown positions, where the order of their placement is irrelevant. In addition, we study the equivariant sectional category and the equivariant parametrized topological complexity, which serve as essential tools for obtaining several results in this paper.

We continue our study of Ulam’s measure problem. In contrast to our previous works, we shift our focus from measures stratified by their additivity, to measures stratified by their indecomposability. The breakthrough here is obtained by replacing the classical ‘least’ function associated with ideals by a two-dimensional ‘last’ function associated with walks on ordinals. Consequently, we obtain conditions under which a measure admits not just infinite pairwise disjoint families of positive sets, but in fact families of maximum possible size. As an application we solve a problem left open in Shelah’s Cardinal Arithmetic book, proving that for every weakly inaccessible cardinal $\kappa $, if there exists a stationary subset of $\kappa $ that does not reflect at regulars, then the strong Ramsey relation $\kappa \nrightarrow [\kappa ]^2_\kappa $ holds.

We study the rationality of some geometrically rational three-dimensional conic and quadric surface bundles, defined over the reals and more general real closed fields, for which the real locus is connected and the intermediate Jacobian obstructions to rationality vanish. We obtain both negative and positive results, using unramified cohomology and birational rigidity techniques, as well as concrete rationality constructions.

We compute log canonical thresholds of reduced plane curves of degree $d$ at points of multiplicity $d-1$. As a consequence, we describe all possible values of log canonical threshold that are less than $2/(d-1)$ for reduced plane curves of degree $d$. In addition, we compute log canonical thresholds for all reduced plane curves of degree less than 6.

Beltran & Cladek use $L^r$ to $L^s$ bounds to prove sparse form bounds for pseudodifferential operators with Hörmander symbols in $S^m_{\rho,\delta}$ up to, but not including, the sharp end-point in decay $m$. We further develop their technique, obtaining pointwise sparse bounds for rough pseudodifferential operators that are merely measurable in their spatial variables and an alternative proof of their results, which avoids proving geometrically decaying sparse bounds. We also provide sufficient conditions for sparse form bounds to hold and use these to reprove known sparse bounds for pseudodifferential operators with symbols in $S^0_{1,\delta}$ for $\delta \lt 1$.

This note corrects an inaccuracy in the paper “Blow-up for the wave equation with nonlinear source and boundary damping terms” by the authors, appeared in Proc. Roy. Soc. Edinburgh Sect. A 145 (2015), no. 4, 759–778.