Given any generically étale morphism of varieties $f \colon X \to Y$, we define the relative Mather discrepancy function on the arc space $X_\infty$ of the domain and show that this function computes the dimension of the kernel of the differential map of the induced morphism on arc spaces $f_\infty \colon X_\infty \to Y_\infty$. We relate this result to the change-of-variable formula in motivic integration. We introduce the notion of $\widehat K$-equivalence, which agrees with $K$-equivalence for smooth varieties, and prove that $\widehat K$-equivalent varieties of arbitrary characteristic define the same class in the motivic ring.

QCD-like theories are of interest in various areas of beyond-Standard-Model phenomenology, including composite Higgs models or pionic dark matter. The effective field theories provide a framework for describing the dynamics of such strongly coupled gauge theories at low energies. In this work, we present next-to-leading order (NLO) expressions for masses, condensates, decay constants, and scattering amplitudes in the chiral expansion of QCD-like theories with $N_F=2$ fermions with non-degenerate masses in both real and pseudoreal representations of the gauge group. We further apply the NLO formulas to fit existing lattice spectroscopic and scattering data for $Sp(N_c=4)$ gauge theory with $N_F=2$ fermions in fundamental representation, extracting the NLO low-energy constants of the theory. Using these fits, we refine the NLO formulas describing the $2\to 2$ pion self-interactions and confirm that the NLO contributions play a crucial role in determining the viable parameter space of pion dark matter scenarios like the strongly interacting massive particles (SIMP).

The coupling between real-space inhomogeneities coordinates and spin (r-s) provides an alternative route to achieve efficient spin manipulation in spintronics beyond the conventional momentum-spin (k-s) coupling paradigm. Here we demonstrate an unexpected manifestation of one-dimensional (1D) r-s coupling in two-dimensional (2D) altermagnetic second-order topological insulators, where the spin-split floating edge states -- energetically isolated within the bulk band gap -- emerge and exhibit both Neel-vector-dependent and electrically tunable behaviors. The 1D edge-spin r-s coupling ensures carrier transport to be exclusively carried by the edge states with quantized spin conductance, giving rise to an unconventional edge tunnel magnetoresistance (edge-TMR) effect that can be switched On or Off. As a proof of concept, we computationally design an edge-TMR device based on Cr_2Se_2O monolayer to demonstrate its edge transportation and controllability via the Néel order or electric field. Our findings propose a general prototype altermagnetic device for next-generation low-dimensional spintronics.

We study actions of (infinitely presented) graphical small cancellation groups on the Gromov boundaries of their coned-off Cayley graphs. We show that a class of graphical small cancellation groups, including (infinitely presented) classical small cancellation groups, admit hyperfinite boundary actions, more precisely, the orbit equivalence relation that they induce on the boundaries of the coned-off Cayley graphs is hyperfinite.

We prove that for any countable group acting virtually specially on a CAT(0) cube complex, the orbit equivalence relation induced by its action on the Roller boundary is hyperfinite. This can be considered as a generalization of hyperfiniteness of the boundary action of cubulated hyperbolic groups by Huang-Sabok-Shinko.

Expressions of dissent against authority are an important feature of most societies, and efforts to suppress such expressions are common. Modern digital communications, social media, and Internet surveillance and censorship technologies are changing the landscape of public speech and dissent. Especially in authoritarian settings, individuals must assess the risk of voicing their true opinions or choose self-censorship, voluntarily moderating their behavior to comply with authority. We present a model in which individuals strategically manage the tradeoff between expressing dissent and avoiding punishment through self-censorship while an authority adapts its policies to minimize both total expressed dissent and punishment costs. We study the model analytically and in simulation to derive conditions separating defiant individuals who express their desired dissent in spite of punishment from self-censoring individuals who fully or partially limit their expression. We find that for any population, there exists an authority policy that leads to total self-censorship. However, the probability and time for an initially moderate, locally-adaptive authority to suppress dissent depend critically on the population's willingness to withstand punishment early on, which can deter the authority from adopting more extreme policies.

A graph $G(V, E)$ is word-representable if there exists a word $w$ over the alphabet $V$ such that for distinct letters $x,y\in V$, $x$ and $y$ alternate in $w$ if and only if they are adjacent in $G$. In general, determining whether a graph is word-representable is an NP-complete problem. A graph is co-bipartite if its complement is bipartite. Therefore, the vertex set of a co-bipartite graph can be partitioned into two disjoint subsets $X$ and $Y$ such that the subgraphs induced by $X$ and $Y$ are cliques. In this paper, we obtain necessary and sufficient conditions for a co-bipartite graph to be word-representable in terms of a vertex ordering. Based on this ordering, we study the representation number of word-representable co-bipartite graphs and analyse the speed and entropy of this graph class. We show that the representation number of any word-representable co-bipartite graph is at most $3$, and that permutation graphs are the only co-bipartite graphs with representation number $2$. We prove that the speed is $2^{O(n \log n)}$ and the entropy is $0$. This provides an asymptotic bound on the number of labelled graphs in this class, which is significantly smaller than the known bound for the class of all co-bipartite graphs. These results provide a better understanding of the structure and enumeration of word-representable co-bipartite graphs and show that vertex ordering is an effective tool for studying this class.

Wave erosion of ice cliffs is one of the main mechanisms for waterline ice melt for icebergs, glacier fronts, and ice-shelf fronts. Despite its importance, this process is fundamentally not well understood or extensively tested in controlled experiments, and therefore coarsely parameterized in geophysical and climate models. In this study, we examine the melting of a vertical ice wall caused by surface waves using both theory and laboratory experiments, with an emphasis on the flow-induced heat transport in the theory and measurements of the melt rate profile under different wave conditions in the experiments. In both the theory and the experiments, we find that there is enhanced melting near the surface that decays with depth. Via an analysis the oscillatory boundary layer flow, we find that an approximate, leading-order, wave-averaged balance of heat transport is given by the vertical advection due to an Eulerian boundary layer streaming current and horizontal diffusion. By solving for this balance and obtaining the wave-averaged temperature field, we find an explicit expression for the wave-induced melt rate. Experimental data show a good match to this expression, especially for larger wave amplitudes and colder water temperatures, though we find that the ambient melt can be a significant contributor to the waterline melt rate.

Among the extensive studies of fractional quantum anomalous Hall (FQAH) states, there recently appears a growing interest in the topological states with coexisting charge density wave (CDW) orders. Such states are referred to as Hall crystals. However, compared to those with integer Hall conductivities, the FQAH crystal (FQAHC) is still elusive even at the level of microscopic model. In this work, we numerically study a topological flat-band model on triangular lattice with spinless fermions. At fractional filling of the Chern band, the nearest-neighbor interaction leads to a commensurate and topologically trivial CDW state. Interestingly, the folded mini-band above the CDW gap is non-trivial, and we focus on the doping of it without any projection. A series of (F)QAHC states at (fractional) integer fillings of this mini-band are discovered and some FQAHC state might even exist in less "ideal" conditions. The ground-state degeneracies of such (F)QAHC states are enlarged by the CDW degeneracy and the Hall conductivities -- determined by the fillings of the mini-band -- are different from the fillings of the original Chern band. We also study the thermodynamics of an FQAHC state and find a compressible CDW phase at intermediate temperatures, which might serve as a precursor of lower temperature FQAHC phase. Moreover, we numerically demonstrate that such a generic scheme of doping CDW-folded topological mini-band could be applied to bosonic systems, broadening the platforms of Hall-crystal physics and motivating its exploration in quantum moire and cold-atom systems.

Crystal Structure Prediction (CSP) remains a fundamental challenge with significant implications for materials discovery and the advancement of various scientific disciplines. Recent advances have demonstrated that generative models, particularly diffusion models, are especially promising for CSP. However, traditional graph-based representations, where atomic bonds are modeled as pairwise graph edges, fail to capture the intricate high-order interactions essential for accurately describing crystal structures. To address this limitation, we propose leveraging hypergraphs to represent crystal structures, enabling more expressive modeling of multi-way atomic interactions. Hypergraphs naturally encode complex high-order relationships and respect key symmetries -- such as permutation and periodic translation invariance -- that are crucial for characterizing crystalline materials. Building on this representation, we propose the \textbf{E}quivariant \textbf{H}ypergraph \textbf{Diff}usion Model (\textbf{EH-Diff}), a generative framework designed to exploit the symmetry-preserving properties of hypergraphs. EH-Diff provides an efficient and accurate method for predicting crystal structures, with rigorous theoretical guarantees on invariance preservation. Empirically, we conduct extensive experiments on four benchmark datasets, and the results demonstrate that EH-Diff outperforms state-of-the-art CSP methods even with a single diffusion sample.

In this work, we constrain the parameter space of the Ricci-Cubic Holographic Dark Energy (RCHDE) model using several observational datasets, including Hubble parameter measurements, cosmic chronometer (CC) data, Baryon Acoustic Oscillation (BAO) data, and recent DESI observations. The RCHDE model is constructed from a cubic curvature invariant formed through cubic contractions of the Ricci and Riemann tensors. To estimate the model parameters, we employ the Markov Chain Monte Carlo (MCMC) sampling technique within a Bayesian inference framework. The resulting likelihood contours provide both marginalized and joint posterior distributions of the model parameters. The best-fit cosmological evolution predicted by the RCHDE model is reconstructed and compared with observational $H(z)$ measurements as well as with the standard $Λ$CDM cosmological model. The best-fit value obtained in our model exhibits a moderate Hubble tension of approximately $2.3σ$ with respect to the reference value for $Λ$CDM. While this indicates a noticeable discrepancy, it remains significantly lower than the $\sim 5σ$ tension typically reported between early- and late-Universe measurements, suggesting a partial alleviation of the tension. In addition to the statistical parameter estimation, we perform an enhanced machine learning analysis using observational Hubble parameter data. We have done a comparative stability analysis between different holographic dark energy models using the squared speed of sound, where it is seen that the RCHDE model does not have any upper hand over its counterparts. Finally, the cosmic coincidence problem is tested to compare the efficiency of the RCHDE model in comparison to other models. It is found that the RCHDE model produced a significant alleviation to the cosmic coincidence problem, outshining its counterparts.

The detection of compact binary mergers with sub-solar masses at gravitational-wave observatories could mark the groundbreaking discovery of primordial black holes (PBHs). Concurrently, evidence for a nHz stochastic gravitational wave background observed by pulsar timing arrays (PTAs) could suggest a non-astrophysical origin, potentially arising from scalar-induced gravitational waves (SIGW). In this work, we analyze the connection between the two phenomena in the case where they share a common origin: the collapse of large primordial curvature perturbations in the early universe. We focus on sub-solar PBH populations within reach of upcoming experiments, including the current and future runs of LIGO-Virgo-KAGRA as well as the third generation observatories such as the Einstein Telescope and Cosmic Explorer. Using a Bayesian framework with physically motivated priors, we perform a consistent model comparison that incorporates existing astrophysical bounds together with the discovery potential of future detectors. Our analysis lends stronger support for the SIGW interpretation over the astrophysical one, as the narrowed priors place greater weight on the region of highest likelihood. Ultimately, we illustrate that combining PTA data with interferometer searches can deliver correlated evidence for new physics across multiple gravitational-wave bands.

Topological materials are characterized by integer invariants that underpin robust quantized electronic properties, as exemplified by the Chern number in the integer quantum Hall effect. Yet, for most candidate systems, the observable linked to the topological invariant remains unknown, precluding direct verification of their topological nature. We present a general method to identify topological materials by connecting the local electronic density~$δρ(\bm{r})$ to Atiyah-Singer index theorems. This method offers a concrete protocol for determining the winding number, the topological invariant associated with chiral-symmetric Hamiltonians. It also identifies a contour-independent wavefront dislocation pattern in $δρ(\bm{r})$ arising from interference induced by topological defects and demonstrates its application to numerical simulations and to existing STM data. The method clearly distinguishes topological states from non-topological ones through a unified, standardized filtering step, offering a definitive approach for identifying and characterizing quantum topological states and opening the door to their use as robust, entangleable building blocks in quantum technologies.

We report the results of a simultaneous X-ray and optical spectroscopy campaign on the Galactic black hole X-ray binary V4641 Sgr, carried out with XRISM and the Seimei telescope during a low-luminosity phase towards the end of its 2024 outburst. Despite a very low X-ray luminosity of $10^{34}$ erg s$^{-1}$, the continuum spectrum is well reproduced by a disk blackbody model with a high inner disk temperature ($1.8$ keV). XRISM/Resolve provides the highest-resolution X-ray spectrum ever obtained from the source, and several strong, narrow emission lines were detected, resolved and characterized at a high significance level. The continuum shape and narrow emission lines both indicate that the inner disk region is obscured by the surrounding high-density gas, and the intrinsic luminosity is several orders of magnitude higher. In the simultaneous optical observation from the Seimei telescope, the line features are largely dominated by the optical companion. Although we detect a clear emission component in H$α$ that could originate from a cold outflow or the disk atmosphere, there are no signs of the strong outflow signatures historically detected in this source. In X-rays, the combination of significantly redshifted ($\sim 700$ km s$^{-1}$) and weakly blueshifted ($\sim-250$ km s$^{-1}$) components, all varying strongly on ks timescales, along with a marginally significant (99.2%) highly blueshifted ($\sim-1200$ km s$^{-1}$) component, indicates a complex, inhomogeneous outflow geometry. This is corroborated by the erratic long-term evolution of the source seen in the complementary X-ray monitoring, and radio detections spanning 3 orders of magnitude.

In this paper, we investigate an example of summation of non-logarithmic singularities of a specific type in a two-dimensional non-linear sigma model. As a result of the study, we obtained an explicit formula, which, upon formal expansion in terms of the coupling constant, reproduces a particular part of the quantum action. Additionally, the paper introduces a new auxiliary function and discusses some of its properties.

A transition matrix can be constructed through the partial contraction of two given quantum states. We analyze and compare four different definitions of entropy for transition matrices, including (modified) pseudo entropy, SVD entropy, and ABB entropy. We examine the probabilistic interpretation of each entropy measure and show that only the distillation interpretation of ABB entropy corresponds to the joint success probability of distilling entanglement between the two quantum states used to construct the transition matrix. Combining the transition matrix with preceding measurements and subsequent non-unitary operations, the ABB entropy either decreases or remains unchanged, whereas the pseudo-entropy and SVD entropy may increase or decrease. We further apply these entropy measures to transition matrices constructed from several ensembles: (i) pairs of independent Haar-random states; (ii) bi-orthogonal eigenstates of non-Hermitian random systems; and (iii) bi-orthogonal states in $PT$-symmetric systems near their exceptional points. Across all cases considered, the SVD and ABB entropies of the transition matrix closely mirror the behavior of the subsystem entanglement entropy of a single random state, in contrast to the (modified) pseudo entropy, which can exceed the bound of subsystem size, fail to scale with system size, or even take complex values.

We investigate the convergence to (quasi--)equilibrium of a density dependent Markov chain in~${\mathbb Z}^d$, whose drift satisfies a system of ordinary differential equations having an attractive fixed point. For a sequence of such processes~${\mathbb X}^N$, indexed by a size parameter~$N$, the time taken until the distribution of~${\mathbb X}^N$, started in some given state, approaches its (quasi--)equilibrium distribution~$π^N$ typically increases with~$N$. To first order, it corresponds to the time~$t_N$ at which the solution to the drift equations reaches a distance of~$\sqrt N$ from their fixed point. However, the length of the time interval over which the total variation distance between ${\mathcal L} ({\mathbb X}^N(t))$ and its (quasi--)equilibrium distribution~$π^N$ changes from being close to~$1$ to being close to zero is asymptotically of smaller order than~$t_N$. In this sense, the chains exhibit `cut--off', and we are able to prove that the cut-off window is of (optimal) constant size.

We construct a new example of an immortal mean curvature flow of smooth embedded connected surfaces in $\mathbb R^3$, which converges to a plane with multiplicity $2$ as time approaches infinity.

The simplest way to make a dynamical system out of a finite connected graph $G$ is to give it a polarization, that is to say a cyclic ordering of the edges incident to a vertex, for each vertex. The phase space $\mathcal{P}(G)$ then consists of all pairs $(v,e)$ where $v$ is a vertex and $e$ is an edge incident to $v$. Such an initial condition gives a position and a momentum. The data $(v,e)$ is of course equivalent to an edge endowed with an orientation $e_{\mathcal O}$. With the polarization, each initial data leads to a leftward walk defined by turning left at each vertex, or making a rebound if there is no other edge. A leftward walk is called complete if it goes through all edges of $G$, not necessarily in both directions. As usual, we define the valence of a vertex as the number of edges incident to it, and we define the valence of a graph as the average of the valences of its vertices. In this article, we prove that if a graph which is embedded in a closed oriented surface of genus $g$ admits a complete leftward walk, then its valence is at most $1 + \sqrt{6g+1}$. We prove furthermore that this result is sharp for infinitely many genera $g$, and that it is asymptotically optimal as $g \to + \infty$. This leads to obstructions for the embeddability of graphs on a surface in a way which admits a complete leftward walk. Since checking that a polarized graph admits a complete leftward walk or not is done in time $4N$, where $N$ is the cardinality of the edges, this obstruction is particularly efficient in terms of computability. This problem has its origins in interesting consequences for what we will call here the topological ergodicity of conservative systems, especially Hamiltonian systems $H$ in two dimensions where the existence of a complete leftward walk corresponds to a topologically ergodic orbit of the system, i.e. an orbit of $H$ visiting all the topology of the surface.

We show that on every non-$G_2$ complex symmetric space of rank two, there are complete Calabi-Yau metrics of Euclidean volume growth with prescribed horospherical singular tangent cone at infinity, providing the first examples of affine Calabi-Yau smoothings of singular and irregular tangent cone. As a corollary, we obtain infinitely many examples of Calabi-Yau manifolds degenerating to the tangent cone in a single step, supporting a recent conjecture by Sun-Zhang, which was only proved when the tangent cone at infinity has only an isolated singularity.

We study the vortex formation in extreme type-II superconductors immersed in strong magnetic fields in the framework of the the Ginzburg-Landau theory. We focus on the regime where superconductivity survives in the bulk of the material but the magnetic field penetrates the sample, i.e., for applied field much larger than the first critical one, but below the transition to surface superconductivity. Through a two-scale vortex construction, we obtain precise estimates for the vortex distribution and prove the existence of isolated defects with non-trivial winding numbers. In this respect, our work provides the first rigorous mathematical proof of the existence of isolated vortices for fields comparable to the second critical one.

Density functional theory (DFT) is a widespread and effective tool in electronic structure calculations for ground-state electron systems. Its success has prompted exploration into the use of DFT for non-collective excited states. The delta self-consistent field ($Δ$SCF) method allows for the extension of DFT to excited-state energies by restricting the Kohn-Sham orbital occupations, producing an excited-state electron density, and then computing its energy. In this paper, we examine the performance of the LSDA, PBE generalized-gradient approximation (GGA), and SCAN/r2SCAN meta-GGA for the excitation energies of several important systems. We consider the energies of atoms with atomic number 1-18. For the hydrogen atom, where we use the exact electron density and have no multiplet splitting, we find significant improvement up the ladder from LSDA to PBE to SCAN. For the uniform gas, we find an effective mass different from the bare mass only with r2SCAN. We split the case of multi-electron atoms into non-aufbau excitations, where the highest-energy electron is excited to the lowest state in the next nl subshell (where accuracy is least limited by available basis sets), and spin-flip excitations, where the spin of an electorn is flipped, leading to a higher-energy state of the same nl configuration. We find reasonably accurate approximate excitation energies, except for the spin-flip cases where the auxiliary non-interacting wavefunction of a non-Hund's-rule spin state is not well-described by a single determinant.

The real-time data collection and automation capabilities offered by the Internet of Things (IoT) are revolutionizing and transforming Business Processes (BPs) into IoT-enhanced BPs, showing high potential for improving sustainability. Although already studied in Business Process Management (BPM), sustainability research has primarily focused on environmental concerns. However, achieving a holistic and lasting impact requires a systematic approach to address sustainability beyond the environmental dimension. This work proposes a conceptual model and a structured methodology with the goal of analyzing the potential of IoT to measure and improve the sustainability of BPs. The conceptual model formally represents key sustainability concepts, linking BPM and IoT by highlighting how IoT devices support and contribute to sustainability. The methodology guides the systematic analysis of existing BPs, identifies opportunities, and implements sustainability-aware, IoT-enhanced BPs. The approach is illustrated through a running example from the tourism domain and a controlled case study in healthcare.

Supersymmetry algebras can be used to obtain algebraic expressions for constant Yang-Baxter solutions, also known as braid group generators. This was done for non-invertible braid operators in \cite{maity2025non}. In this work we extend this construction for the invertible ones. The resulting expressions are then shown to obey relations analogous to those satisfied by quotients of braid groups. Examples of the latter include the Iwahori-Hecke algebra and the Birman-Murakami-Wenzl (BMW) algebra. As a result, we can Baxterize the constant Yang-Baxter solutions to yield spectral parameter dependent $R$-matrices. The regularity of these $R$-matrices depends on the representation of SUSY generators. In some cases they are regular in the usual sense and in the remaining they are `almost' regular. In the latter case they are also non-invertible. Nevertheless, we show that they can still help us construct integrable models in all dimensions of the local Hilbert space. These models can be described by Hamiltonian densities that are either local or non-local, depending on the representation chosen for the SUSY generators. We demonstrate this for all constant $4\times 4$ invertible Yang-Baxter solutions. Apart from finding new nearest-neighbor interaction spin $\frac{1}{2}$ systems, we also find their higher spin analogs due to the algebraic [representation independent] approach.

We prove the existence of a localization spectral sequence for the hat variant of Guth and Manolescu's recent construction of real Heegaard Floer homology, and apply it to branched double covers and strongly invertible knots. Our construction applies to real Lagrangian Floer homology in exact symplectic manifolds equipped with anti-symplectic involutions more generally, and may be of independent interest to symplectic geometers.

Graph neural networks (GNNs) excel in processing non-Euclidean data, but traditional spectral GNNs rely on static bases and fundamentally lack active spectral regulation. Although the graph fractional Fourier transform (GFRFT) introduces cross-domain modulation, it applies a uniform fractional parameter across all frequencies. This ignores frequency heterogeneity and restricts the models' adaptive capacity in graph node classification tasks. In the paper, we propose two novel types of multiple-parameter GFRFTs (MPGFRFTs) and establish their corresponding theoretical frameworks, including essential properties, computational complexity, and parameters differentiability. By assigning independent, learnable fractional parameters to distinct frequency bands, MPGFRFTs enable fine-grained spectral regulation. Then, we operationalize this mathematical framework by designing the adaptive multiple-parameter fractional spectral regulation (MPFSR) module, a plug-and-play component for mainstream spectral models. We also establish rigorous theoretical bounds on the spectral stability of this module, guaranteeing a stable and reliable convergence during the end-to-end parameters optimization. Experiments demonstrate that integrating the proposed MPFSR module alleviates the constraints of static bases and yields performance gains in node classification on complex graphs, advancing a novel paradigm for active spectral modulation in graph representation learning.

This work establishes a step forward in advancing data-driven trajectory-based methods for stochastic systems with unknown mathematical dynamics. In contrast to scenario-based approaches that rely on independent and identically distributed (i.i.d.) trajectories, this work develops a data-driven framework where each trajectory is gathered over a finite horizon and exhibits temporal dependence, referred to as a non-i.i.d. trajectory. To ensure safety of dynamical systems using such trajectories, the current body of literature primarily considers dynamics subject to unknown-but-bounded disturbances, which facilitates robust analysis. While promising, such bounds may be violated in practice and the resulting worst-case robust analysis tends to be overly conservative. To overcome these key challenges, this paper considers stochastic systems with unknown mathematical dynamics, influenced by process noise with arbitrary distributions. In the proposed framework, data is collected from stochastic systems under multiple realizations within a finite-horizon experiment, where each realization generates a non-i.i.d. trajectory. Leveraging the concept of stochastic control barrier certificates constructed from data, this work quantifies probabilistic safety guarantees with a certified confidence level. To achieve this, the proposed conditions are formulated as a sum-of-squares (SOS) optimization problem, relying solely on empirical average of the collected trajectories and statistical features of the process noise. The efficacy of the approach has been validated on three stochastic benchmarks with unknown models and arbitrary noise distributions. In one case study, it is shown that while no safety controller exists for the robust analysis of the system under bounded disturbances, the proposed stochastic framework yields a safety controller together with quantified probabilistic safety guarantees.

The equivalence between the inequalities of Babuška-Aziz and Friedrichs for sufficiently smooth bounded domains in the plane has been shown by Horgan and Payne 30 years ago. We prove that this equivalence, and the equality between the associated constants, is true without any regularity condition on the domain. For the Horgan-Payne inequality, which is an upper bound of the Friedrichs constant for plane star-shaped domains in terms of a geometric quantity known as the Horgan-Payne angle, we show that it is true for some classes of domains, but not for all bounded star-shaped domains. We prove a weaker inequality that is true in all cases.

The simplest modeling of planar quantum waveguides is the Dirichlet eigenproblem for the Laplace operator in unbounded open sets which are uniformly thin in one direction. Here we consider V-shaped guides. Their spectral properties depend essentially on a sole parameter, the opening of the V. The free energy band is a semi-infinite interval bounded from below. As soon as the V is not flat, there are bound states below the free energy band. There are a finite number of them, depending on the opening. This number tends to infinity as the opening tends to 0 (sharply bent V). In this situation, the eigenfunctions concentrate and become self-similar. In contrast, when the opening gets large (almost flat V), the eigenfunctions spread and enjoy a different self-similar structure. We explain all these facts and illustrate them by numerical simulations.

We prove weighted anisotropic analytic estimates for solutions of second order elliptic boundary value problems in polyhedra. The weighted analytic classes which we use are the same as those introduced by Guo in 1993 in view of establishing exponential convergence for hp finite element methods in polyhedra. We first give a simple proof of the known weighted analytic regularity in a polygon, relying on a new formulation of elliptic a priori estimates in smooth domains with analytic control of derivatives. The technique is based on dyadic partitions near the corners. This technique can successfully be extended to polyhedra, providing isotropic analytic regularity. This is not optimal, because it does not take advantage of the full regularity along the edges. We combine it with a nested open set technique to obtain the desired three-dimensional anisotropic analytic regularity result. Our proofs are global and do not require the analysis of singular functions.