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In this article, we study scaling laws for singularly perturbed two-well energies with prescribed Dirichlet boundary data in settings where the wells and/or the boundary data are incompatible. Our main focus is the geometrically linear two-well problem, for which we characterize the energy scaling in two dimensions for nearly all combinations of linear boundary data and stress-free strains. In particular, we prove that if the boundary data enforces oscillations and the weight $ε$ of the surface energy is small, the minimal energy upon subtracting the zeroth-order contribution scales either as $ε^{{4}/{5}}$ or as $ε^{{2}/{3}}$, depending on whether the wells differ by a rank-one or a rank-two matrix, respectively. For the gradient and divergence-free two-well problem, we obtain analogous results, showing an $ε^{{2}/{3}}$-scaling behavior in two dimensions whenever oscillations are energetically favored. These results follow by deriving matching upper and lower scaling bounds. The lower scaling bounds are established in a general $\mathcal{A}$-free framework for incompatible two-well problems, which allows us to compute the excess energy and characterize boundary data which enforce o

We study classes of graphs with bounded clique-width that are well-quasi-ordered by the induced subgraph relation, in the presence of labels on the vertices. We prove that, given a finite presentation of a class of graphs, one can decide whether the class is labelled-well-quasi-ordered. This answers positively to two conjectures of Pouzet in the restricted case of bounded clique-width classes. Namely, we prove that being labelled-well-quasi-ordered by a set of size 2 or by a well-quasi-ordered infinite set are equivalent conditions, and that in such cases, one can freely assume that the graphs are equipped with a total ordering on their vertices. Finally, we provide a structural characterization of those classes as those that are of bounded clique-width and do not existentially transduce the class of all finite paths.

We establish Schauder-type estimates for linear parabolic systems driven by variable-coefficient nonlocal pseudo-differential operators of order $s>0$. These estimates are formulated in critical time-weighted Hölder/Besov-type spaces and are tailored to quasilinear equations at scaling-critical regularity. A key ingredient is a kernel-adapted freezing-coefficient method. After freezing the coefficients at a reference point, we derive explicit representation formulas through the corresponding fundamental kernels and then evaluate the resulting bounds at the physical point. This avoids treating the coefficient variation as a separate lower-order perturbation and yields robust control of the residual terms within the leading-order dynamics. As an application, we obtain a general well-posedness framework for a class of nonlocal quasilinear parabolic equations in critical spaces. In particular, we prove critical local and, in suitable regimes, global well-posedness for the Muskat equation with surface tension and for the two- and three-dimensional Peskin problems with nonlinear elastic tension. These results provide a unified critical framework for distinct nonlocal evolution equatio

We study the well-posedness and stability of an impedance passive infinite-dimensional linear system under nonlinear feedback of the form $u(t)=φ(v(t)-y(t))$, where $φ$ is a monotone function. Our first main result introduces conditions guaranteeing the existence of classical and generalised solutions in a situation where the original linear system is well-posed. In the absence of the external input $v$ we establish the existence of strong and generalised solutions under strictly weaker conditions. Finally, we introduce conditions guaranteeing that the origin is a globally asymptotically stable equilibrium point of the closed-loop system. Motivated by the analysis of partial differential equations with nonlinear boundary conditions, we use our results to investigate the well-posedness and stablility of abstract boundary control systems, port-Hamiltonian systems, a Timoshenko beam model, and a two-dimensional boundary controlled heat equation.

We consider coercive evolution equations with measure-valued delay on a Gelfand triple. The delayed feedback is induced by a bounded form on $V$ and may contain principal spatial derivatives, so it is naturally $V^*$-valued rather than bounded on the pivot space. For every finite signed Borel kernel with no atom at the origin, an exponential weight makes the causal history operator contractive relative to the coercive parabolic solution operator. This yields finite-time well-posedness without a smallness condition on the total variation and without any positivity assumption on the delayed form. An asymmetric residual identity gives a total-variation Lipschitz estimate for signed kernels and strong stability for non-negative retarded kernels under narrow, equivalently weak, convergence of finite measures. A delayed-diffusion realisation shows that the framework genuinely covers principal-part delays, and narrow stability yields qualitative distributed-to-discrete convergence at every fixed positive lag.

Frontier AI developers now train models against long written behavioral specifications, such as Anthropic's constitution (Anthropic, 2025a) and OpenAI's Model Spec (OpenAI, 2025a), integrated into post-training via methods like character training (Anthropic, 2024) and deliberative alignment (Guan et al., 2024). These documents serve a governance function, but it is unclear how well models actually follow them under adversarial, multi-turn pressure similar to what they would face in real-world deployment. We propose a multi-method audit pipeline that treats each lab's published specification as an auditable target: it decomposes the specification into atomic testable tenets (205 for Anthropic, 197 for OpenAI), generates multi-turn adversarial scenarios with the Petri auditing agent (Anthropic, 2025b), runs a modified SURF-style rubric search (Murray et al., 2026) to catch shallow single-turn failures Petri misses, validates flagged transcripts against the relevant specification, and compares the findings against the lab's own published system card. Applying the pipeline across seven models per specification, we find that models follow their own lab's specification substantially bett

This work investigates the well-posedness and optimal control of a sixth-order Cahn-Hilliard equation, a higher-order variant of the celebrated and well-established Cahn-Hilliard equation. The equation is endowed with a source term, where the control variable enters as a distributed mass regulator. The inclusion of additional spatial derivatives in the sixth-order formulation enables the model to capture curvature effects, leading to a more accurate depiction of isothermal phase separation dynamics in complex materials systems. We provide a well-posedness result for the aforementioned system when the corresponding nonlinearity of double-well shape is regular and then analyze a corresponding optimal control problem. For the latter, existence of optimal controls is established, and the first-order necessary optimality conditions are characterized via a suitable variational inequality. These results aim at contributing to improve the understanding of the mathematical properties and control aspects of the sixth-order Cahn-Hilliard equation, offering potential applications in the design and optimization of materials with tailored microstructures and properties.

We present a unified framework to construct well-posed formulations for large classes of linear operator equations including elliptic, parabolic and hyperbolic partial differential equations. This general approach incorporates known weak variational formulations as well as novel space-time variational forms of the hyperbolic wave equation. The main concept is completion and extension of operators starting from the strong form of the problem. This paper lays the theoretical foundation for a unified approach towards numerical approximation methods and also model reduction of parameterized linear operator equations which will be the subject of the following parts.

Sustainable Development Goals (SDGs) give the UN a road map for development with Agenda 2030 as a target. SDG3 "Good Health and Well-Being" ensures healthy lives and promotes well-being for all ages. Digital technologies can support SDG3. Burnout and even depression could be reduced by encouraging better preventive health. Due to the lack of patient knowledge and focus to take care of their health, it is necessary to help patients before it is too late. New trends such as positive psychology and mindfulness are highly encouraged in the USA. Digital Twins (DTs) can help with the continuous monitoring of emotion using physiological signals (e.g., collected via wearables). DTs facilitate monitoring and provide constant health insight to improve quality of life and well-being with better personalization. Healthcare DTs challenges are standardizing data formats, communication protocols, and data exchange mechanisms. As an example, ISO has the ISO/IEC JTC 1/SC 41 Internet of Things (IoT) and DTs Working Group, with standards such as "ISO/IEC 21823-3:2021 IoT - Interoperability for IoT Systems - Part 3 Semantic interoperability", "ISO/IEC CD 30178 - IoT - Data format, value and coding". T

A real potential Hamiltonian has real energy bound states below the scattering threshold and complex energy resonances above it. Scattering states are not square integrable, being instead delta function normalized. This lack of square integrability breaks the connection between Hermiticity and real eigenvalues, to thus allow for real bound state sector eigenvalues and complex scattering sector eigenvalues. When written as contour integrals delta functions take support in the complex plane, with the scattering amplitude being able to take support in the complex plane too. However, the scattering amplitude is CPT symmetric. For resonance scattering this antilinear symmetry requires the presence of a complex conjugate pair of energies, one to describe the excitation of the resonance and the other to describe its decay, with it being their interplay that enforces probability conservation. Each complex pair of energy eigenvalues corresponds to only one observable resonance not two, to thus modify the standard pure decaying complex energy pole discussion of resonances. We show that the non-relativistic real potential square-well Schrödinger equation possesses C and PT symmetry in both th

This paper studies the global well-posedness and optimal decay estimates to the Oldroyd-B model in $\mathbb R^d$ ($d\geq2$). By utilizing the special structure of this system, we give a simplified proof to the global existence of solutions for the case of initial data small in critical Besov spaces and non-small coupling parameters. Moreover, the optimal decay rates of the solutions under minimal small assumption on the initial data are established by fully making use of the effect of velocity dissipation and damping mechanism.

For most health or well-being interventions, the process of evaluation is distinct from the activity itself, both in terms of who is involved, and how the actual data is collected and analyzed. Tangible interaction affords the opportunity to combine direct and embodied collaboration with a holistic approach to data collection and evaluation. We demonstrate this potential by describing our experiences designing and using the Communal Loom, an artifact for art therapy that translates quantitative data to collectively woven artifacts.

An embedded variety is said to be well-poised when the associated initial ideal degenerations coming from points of the tropical variety are reduced and irreducible. Varieties with a well-poised embedding admit a large collection of explicitly constructible Newton-Okounkov bodies. This paper aims to study the well-poised property under torus quotients. Our first result states that GIT quotients of normal well-poised varieties by quasi-tori also have well-poised embeddings. As an application, we show that several Hassett spaces, $\overline{M}_{0,β}$, are well-poised under Alexeev's embedding. Conversely, given an affine $T$-variety $X$ with polyhedral divisor $\mathfrak{D}$ on a well-poised base $Y$, we construct an embedding of $X \subseteq \mathbb{A}^N$ and provide conditions on $Y$ and $\mathfrak{D}$ which if met, imply $X$ is well-poised under this embedding. Then we show that any affine arrangement variety meets the specified criteria, generalizing results of Ilten and the second author for rational complexity 1 varieties. Using this result, we explicitly compute many Newton-Okounkov cones of $X$ and provide a criterion for the associated toric degenerations to be normal. Our f

This paper is concerned with the Cauchy problem of the modified Kawahara equation (posed on $\mathbb T$), which is well-known as a model of capillary-gravity waves in an infinitely long canal over a flat bottom in a long wave regime \cite{Hasimoto1970}. We show in this paper some well-posedness results, mainly the \emph{global well-posedness} in $L^2(\mathbb T)$. The proof basically relies on the idea introduced in Takaoka-Tsutsumi's works \cite{TT2004, NTT2010}, which weakens the non-trivial resonance in the cubic interactions (a kind of smoothing effect) for the local result, and the global well-posedness result immediately follows from $L^2$ conservation law. An immediate application of Takaoka-Tsutsumi's idea is available only in $H^s(\mathbb T)$, $s > 0$, due to the lack of $L^4$-Strichartz estimate for arbitrary $L^2$ data, a slight modification, thus, is needed to attain the local well-posedness in $L^2(\mathbb T)$. This is the first low regularity (global) well-posedness result for the periodic modified Kwahara equation, as far as we know. A direct interpolation argument ensures the \emph{unconditional uniqueness} in $H^s(\mathbb T)$, $s > \frac12$, and as a byproduct

We consider the 2d $β$-plane stochastic Navier-Stokes equations in a periodic channel. We prove the well-posedness and existence of the stationary measure, as well as certain regularity estimates concerning the support of the stationary measure. The mentioned estimates are crucial for the rigorous study of the cascade phenomena in this equation [8]. To the best of our knowledge, this is the first mathematically rigorous treatment of these equations involving both the stochastic noise and the Coriolis force.

We discuss strong local and global well-posedness for the three-dimensional NLS equation with nonlinearity concentrated on $\mathbb{S}^2$. Precisely, local well-posedness is proved for any $C^2$ power-nonlinearity, while global well-posedness is obtained either for small data or in the defocusing case under some growth assumptions. With respect to point-concentrated NLS models, widely studied in the literature, here the dimension of the support of the nonlinearity does not allow a direct extension of the known techniques and calls for new ideas.

A graph $G$ is called well-covered if all maximal independent sets of vertices have the same cardinality. A well-covered graph $G$ is called uniformly well-covered if there is a partition of the set of vertices of $G$ such that each maximal independent set of vertices has exactly one vertex in common with each part in the partition. The problem of determining which graphs is well-covered, was proposed in 1970 by M.D. Plummer. Let $\cal G$ be the class of graphs with some disjoint maximal cliques covering all vertices. In this paper, some necessary and sufficient conditions are presented to recognize which graphs in the class $\cal G$ are well-covered or uniformly well-covered. This characterization has a nice algebraic interpretation according to zero-divisor elements of edge ring of graphs which is illustrated in this paper.

Artificial intelligence (AI) enabled products and services are becoming a staple of everyday life. While governments and businesses are eager to enjoy the benefits of AI innovations, the mixed impact of these autonomous and intelligent systems on human well-being has become a pressing issue. This article introduces one of the first international standards focused on the social and ethical implications of AI: The Institute of Electrical and Electronics Engineering (IEEE) Standard (Std) 7010-2020 Recommended Practice for Assessing the Impact of Autonomous and Intelligent Systems on Human Well-being. Incorporating well-being factors throughout the lifecycle of AI is both challenging and urgent and IEEE 7010 provides key guidance for those who design, deploy, and procure these technologies. We begin by articulating the benefits of an approach for AI centered around well-being and the measurement of well-being data. Next, we provide an overview of IEEE 7010, including its key principles and how the standard relates to approaches and perspectives in place in the AI community. Finally, we indicate where future efforts are needed.

It has become clear that early career astrophysics researchers (doctoral researchers, post-docs, etc) have a very diverse appreciation of their career, with some declaring it the best job that you can have and others suffering from overwork, harrassment and stress from the precarity of their job, and associated difficulties. In order to establish how astrophysics researchers, primarily in France, experience their career, we sent out a survey to understand the impact that their job has on their well-being. 276 people responded to the survey. Whilst around half of the respondents expressed pleasure derived from their career, it is clear that many (early career) researchers are suffering due to overwork, with more than a quarter saying that they work in excess of 50 hours per week and 2\% in excess of 90 h per week. Almost 30\% professed to having suffered harrassment or discrimination in the course of their work. Further, whilst only 20\% had suffered mental health issues before starting their career in astrophysics, $\sim$45\% said that they suffered with mental health problems since starting in astrophysics. Here we provide results from the survey as well as possible avenues to exp