Papers

The increasing deployment of separate biowaste collection across Europe raises questions regarding environmentally and economically optimal treatment strategies. From this perspective, the balance between decentralised and centralised infrastructure is a potential lever to consider. The choice of technologies is another. Composting and anaerobic digestion (AD) offer viable circular bioeconomy pathways, but their performance depends on transport distance, facility size, seasonal waste fluctuations, and technological efficiency. Threshold distances and scale-dependent interactions affecting the environmental and economic performance of treatment plants remain poorly characterised. This study develops a unified techno-economic and life cycle assessment framework to quantify how transport constraints and treatment capacities shape biowaste valorisation systems. Three facility scales for composting and AD are evaluated across a range of transport distances and compared with local incineration in terms of environmental and economic performance. Transport burdens are modelled using vehicle life cycle assessment (LCA) and realistic heavy-duty truck data for electric-, gas-, and diesel-powered vehicles. Results reveal non-linear trade-offs between scale and transport, with zero-distance composting yielding the lowest combined impacts. Across all scenarios, 45 configurations outperform local incineration economically and environmentally, while 111 remain environmentally preferable despite higher costs, thus providing quantitative guidance for infrastructure planning.

In multi-objective and multi-criteria decision-making under risk, especially in settings involving individual behavior, risk-aware analysis based on subjective evaluation has become increasingly important. Moving beyond risk-neutral modeling and the constraints of Expected Utility Theory (EUT), Cumulative Prospect Theory (CPT) provides a behaviorally grounded framework for capturing how individuals perceive and evaluate risky prospects. This paper conducts a rigorous theoretical analysis of Neilson's definitions of aversion. We provide a gamble-based interpretation, sharpen key conceptual distinctions, and make explicit the conditions under which the weak and strong notions coincide as well as when they diverge. Furthermore, we examine the Kobberling-Wakker utility function and related standard CPT specifications, highlighting structural limitations and inconsistencies that arise when these forms are required to satisfy Neilson-type aversion conditions. To address these issues, we propose a generalized CPT-utility function that retains the canonical reference-dependent shape while offering additional flexibility. This generalization extends the descriptive scope of CPT and provides an explicit functional form that is useful for sensitivity analysis and utility function-based optimization.

We define the lift of a partial cohomological field theory with respect to a Frobenius algebra and the corresponding lift of local polyvector fields, extending the lift procedure proposed by Della Vedova, Lorenzoni, and Savoldi. This allows us to systematically produce examples of non-semisimple homogeneous partial cohomological field theories whose integrable systems possess a second Hamiltonian structure, thus confirming the conjecture of Buryak et al. on an explicit formula for the second Poisson bracket in new non-semisimple cases. Moreover we present some relations of these lift constructions with the Morimoto theory of the lift of geometric structures to the Weil bundle of infinitely near points associated to a local algebra.

We study an inverse obstacle problem for the fractional Schrödinger operator $(-\Delta)^s+q$, $0<s<1$. For each exterior datum, the state is constrained by a prescribed obstacle in a bounded domain and satisfies the fractional Schrödinger equation only in the associated noncontact set. This set is unknown and depends on the coefficient, so the exterior Dirichlet-to-Neumann map is nonlinear. We show that the nonlocal character of the equation gives a direct way around this moving-free-boundary difficulty. Equality of one obstacle measurement on an exterior open set forces equality of the two corresponding obstacle states in the whole space. In their common noncontact set one then obtains $(q_1-q_2)u=0$. This identity yields recovery of the potential in the exposed region by measurable unique continuation when $s\in[1/4,1)$, and by the usual unique continuation principle together with continuity when the potentials are continuous. We also prove a geometric coverage theorem for nonnegative potentials: rational positive scalings of one nontrivial nonnegative exterior datum expose the whole domain up to a null set. Consequently, under the corresponding assumptions, a countable family of nonlinear exterior obstacle measurements determines the potential globally.

The category of constituent pieces of Dubrovin connection of varieties is introduced and shown to be representable in affine space.

This paper investigates a variational model that describes a system of N coupled sphere-valued fields through a penalization term imposing the constraint that their sum coincides with a prescribed sphere value map. We analyze the asymptotic behavior and regularity of minimizers as the penalization parameter tends to infinity. A fundamental dichotomy is established according to the number of interacting fields. For N = 2, topological obstructions may prevent the existence of exact sphere-valued decompositions, leading to the blow-up of the minimal energy in the large-penalization regime. In contrast, for N $\ge$ 3, exact decompositions always exist, yielding uniform energy bounds and allowing the identification of the limiting constrained problem via $\Gamma$-convergence. We further derive an explicit characterization of the limiting energy by decomposing admissible configurations into an average field and fluctuation components. Finally, we establish a gap phenomenon showing that, under suitable topological assumptions, every minimizer is necessarily singular for sufficiently large values of the penalization parameter. These results highlight the fundamental role of topology in the asymptotic behavior and regularity of coupled sphere-valued variational systems and extend previous results on single-field models to the multi-field setting.

Let f be either a generalized interval exchange transformation or a locally monotone homeomorphism of a Cantor subset of the real line. In this article, we prove that the following are equivalent. 1. The number of discontinuities of f^n is bounded. 2. There exists n $\ge$ 1 such that the element f is conjugate to the restriction to a closed invariant subset of a disjoint union of n circles of a homeomorphism of this disjoint union of circles. 3. The element f is distorted in the group of generalized interval exchange transformations or in the group of locally monotone homeomorphisms of the Cantor subset.

We introduce a two-relaxation-times (TRT) kinetic approximation for scalar non-linear conservation laws in one space dimension, addressing the convergence of relaxation approximations to entropy solutions. The proposed TRT system, derived from a lattice Boltzmann scheme, generalizes the classical BGK (single-relaxationtime) framework. Requesting quasi-monotonicity of the relaxation operator, we establish global existence of solutions for the TRT system and prove their convergence, using a lattice Boltzmann scheme, to the entropy solution of the original conservation law as the relaxation times vanish. Moreover, we propose a qualitative analysis, clarifying the role of relaxation parameters and equilibrium coefficients in shaping solutions, via Chapman-Enskog expansion and looking at travelling wave solutions.

HyperLogLog is a now classic probabilistic algorithm that provides an approximation of the number of distinct elements in a massive dataset, using only one pass over the data. In the original article, Flajolet, Fusy, Gandouet, Meunier (2007) provided a sharp analysis of the expectation and variance of the output, using explicit formulas analyzed using poissonization and Mellin transform. In this short article, we revisit the analysis of HyperLogLog with a more probabilistic viewpoint. This allows us to establish exponential deviation inequalities for the HyperLogLog estimator. The methods are elementary, but the estimates are non-asymptotic and totally explicit.

This article is concerned with the study of the focusing energy-critical wave equation on the three-dimensional Riemannian manifold $\mathcal{M}=\mathbb{R}^3$ equipped with a smooth time-independent Riemannian metric $g$, which coincides with the Euclidean metric outside a compact set. Our focus is on Type II blow-up solutions, namely finite-time solutions that remain bounded in the energy space, with the goal of revealing their fundamental properties. We establish in particular lower bounds for the energy norm as the solution approaches its maximal time of existence in terms of the bounds of the ground state of the Euclidean problem.

Intracranial saccular aneurysms (ISAs) pose severe health risks, yet conventional population-based risk stratification scores (PHASES, UIATS, and ELAPSS) offer limited capacity for patient-specific rupture risk assessment. Image-based computational approaches have gained prominence, but traditional surface reconstruction relies on mathematical smoothing (e.g., L-curve criteria) that indiscriminately suppresses both imaging artifacts and genuine pathological features such as rupture-prone blebs. Although Laplace's membrane equilibrium ($\kappa_1 T_1 + \kappa_2 T_2 = P$) has long governed aneurysm wall mechanics (Humphrey and Kyriacou [Neurol. Res., 18 (1996)]), its integration into geometric reconstruction pipelines remains unexplored. This work introduces a physics-informed neural network (PINN) framework with B-spline representations, whose key contributions are: (i) embedding the variational equilibrium condition ($\delta \Pi = 0$) as a physics-informed loss that replaces the mathematical L-curve criterion with a biomechanically grounded artifact discrimination, (ii) developing a Manifold-Consistent CNN ansatz that preserves the closed-surface topology of vascular geometries, and (iii) establishing a Laplace equilibrium-driven reconstruction that filters imaging noise while preserving diagnostically critical high-curvature features. Application to patient-specific clinical datasets demonstrates that the framework eliminates non-physical concave artifacts without compromising genuine geometric anomalies. Clinical evaluation by a practicing neurosurgeon confirms that the resulting risk map—with rupture risk concentrated at the dome apex—is consistent with intraoperative observations, establishing a computational biomarker foundation for patient-specific rupture risk assessment of intracranial saccular aneurysms.

In this paper we establish the pointwise-in-time dispersive decay for solutions to the mass-critical nonlinear Schrödinger equation in spatial dimensions $d\geq3$. Our argument relies on a delicate decomposition of the nonlinearity and an improved linear estimate, which together enable us to control the nonlinear contribution. This work unifies a framework for extending the foundational results established in an earlier paper by Fan, Killip, Visan, and Zhao [Math. Z. 311(1), Paper No. 21, 16 pp (2025)], where the same problem was addressed for spatial dimensions $d=1,2,3$, to the general higher-dimensional setting $d\geq3$.

Segmentation of adjacent structures with similar intensity distributions remains a challenging problem in image analysis, particularly when object boundaries are weak or ambiguous. Under such conditions, classical variational models may suffer from degenerated image-driven forces, leading to boundary leakage or undesired merging of neighboring regions. To address these limitations, we propose a smooth phase-separation variational model based on the Cahn–Hilliard equation for weak-boundary segmentation of homogeneous-appearance structures. The proposed framework integrates softmax-based region fitting with Cahn–Hilliard phase-field regularization to maintain interface discrimination under weak image-driven forces. We further introduce a mixed $L^2-H^{-1}$ gradient flow, which preserves higher-order interfacial regularization while allowing adaptive changes of phase masses, establish the continuous energy dissipation law, and prove the existence and uniqueness of weak solutions in the natural solution class. For numerical computation, we develop a stabilized scalar auxiliary variable (SAV) scheme that is linear, FFT-based, and satisfies a modified discrete energy dissipation law. Numerical experiments on synthetic and medical images demonstrate that the proposed method effectively separates adjacent homogeneous structures across weak boundaries and achieves competitive segmentation accuracy and improved boundary localization compared with representative variational, phase-field, and deep learning methods.

For an oriented graph $D$, let $\vec{\alpha}(D)$ be the maximum order of an induced acyclic subdigraph, $\vec{\chi}(D)$ its dichromatic number, and $\mathrm{fas}(D)$ the minimum number of arcs whose deletion makes $D$ acyclic. We prove that for every fixed $\zeta \in (0, 1/2)$, there are triangle-free graphs $G_n$ on $n$ vertices such that a uniformly random orientation $D_n$ satisfies, $$ \left( \frac{1}{2} - \zeta \right) e(G_n[U]) < \mathrm{fas}(D_n[U]) \leq \frac{1}{2} e(G_n[U]) $$ with probability at least $1-\exp\!\left[-\Omega_\zeta\!\left(\sqrt n\,(\log n)^{3/2}\right)\right]$ simultaneously for every vertex set $U$ of size at least $C_\zeta\sqrt{n\log n}$. The upper bound is universal, so the feedback-arc ratio can be made arbitrarily close to the largest possible value, uniformly over all sufficiently large induced subdigraphs. In particular, $\vec{\alpha}(D_n) = O(\sqrt{n \log n})$, and every linear-size induced subdigraph has dichromatic number $\Omega(\sqrt{n/\log n})$. This yields $\vec{\alpha}(n) = \Theta(\sqrt{n \log n})$ and $\vec{t}(n) = \Theta\left(\sqrt{\frac{n}{\log n}}\right)$, where $\vec{\alpha}(n)$ and $\vec{t}(n)$ denote, respectively, the minimum of $\vec{\alpha}(D)$ and the maximum of $\vec{\chi}(D)$ over all oriented triangle-free graphs $D$ of order $n$. This confirms two conjectures of Aboulker, Havet, Pirot, and Schabanel.

Let $g_n$ be the largest number of Euclidean balls of diameter $1$ which may be needed to cover a set of diameter $1$ in $\mathbb{R}^n$. We study this problem for finite sets invariant under all coordinate permutations. We prove that the exponential growth rate in this symmetric problem can be characterized exactly as a finite-alphabet squared-error rate-distortion supremum $\alpha_0$. Specialized to the two-point case, i.e., for subsets of Boolean cubes, this gives the explicit lower bound \[g_n\ge (1.160235457\ldots-o(1))^n,\] improving the previous best bound $(2/\sqrt3-o(1))^n$. Using Fix's Gaussian characterization of the rate-distortion problem, we give a numerical three-point construction with exponent base greater than $1.160497831$. Finally, we show that $\alpha_0$ is not attained by any finitely supported distribution.

Upstream suppliers often set wholesale prices and stocking levels without direct access to detailed end-market demand information, which is instead held by downstream retailers. We study how a supplier can learn from a small number of historical transactions consisting of wholesale prices, retailer-chosen retail prices, and the resulting demand responses. We develop a robust, nonparametric framework that combines inverse optimization with distributionally robust optimization. Historical retailer decisions are used to restrict the set of demand distributions consistent with the observed transactions, after which the supplier optimizes against the least favorable demand response within this ambiguity set. We further accommodate imperfect retailer decisions through two error models (condition error and price error) and derive tractable worst-case response computations based on linear programs and one-dimensional price searches. Numerical experiments across six demand environments show that incorporating historical decision information can generate substantial profit improvements with only a few transactions and that these gains persist under moderate decision errors.

This paper focus on a class of singular or degenerate fully nonlinear elliptic equations with Hamiltonian terms under oblique boundary conditions on $C^{1}$ domains. Under quite general conditions on the singularity/degeneracy of the model, we establish the sharp $C^{1,\alpha}$ regularity up to the boundary within a unified framework.

In this paper, we classify all six-dimensional real nilpotent Lie bialgebras of symplectic type. The Poisson structures on all of the related six-dimensional Poisson-Lie groups are obtained. Some new integrable Hamiltonian systems for which the Poisson-Lie group plays the role of a phase space and its dual Lie group plays the role of a symmetry group of the system are obtained.

High-resolution analysis and design of large-scale heterogeneous structures require accurate reduced-order models and efficient online computation. Existing multiscale methods must repeatedly construct local basis functions for different material distributions, whereas substructure-based problem-independent machine learning (PIML) methods can be limited by prescribed boundary displacement interpolation. We propose PIML-OFEM, an overlapping finite element method accelerated by problem-independent machine learning. Each substructure retains only its corner-node degrees of freedom. Oversampled numerical basis functions are constructed by solving local elasticity problems on extended domains and restricting the solutions to the target substructure, eliminating prescribed displacement interpolation on its boundary. Independently constructed local bases are blended through a partition-of-unity overlapping formulation to obtain a globally continuous displacement field. A U-Net learns the mapping from local Young's modulus distributions to numerical basis functions, replacing repeated online local solves and allowing the model to be reused across load cases and global boundary conditions. Numerical examples show close agreement with fine-scale finite element results in displacement and elemental strain energy. PIML-OFEM reduces online computational cost relative to direct finite element analysis and improves accuracy over PIML substructure models based on linear boundary interpolation. In topology optimization, the method supports stable high-resolution iterations with small filter radii and preserves fine-scale features, including local patterns resembling rank-2 microstructures. The framework provides an efficient physics-data approach for large-scale heterogeneous structural analysis and high-resolution topology optimization.

Very recently, the notion of rough family has been introduced in [Leonetti, P., J. Convex Anal. 32(4):1083-1090, 2025] to explore rough ideal convergence in topological spaces where the limit of a sequence may not be unique. This raises the question of whether $T_2$ topological spaces can be characterized using rough families. In this article, we prove that a topological space is $T_2$ if and only if it can never be a rough topological space. In this context, we first introduce the notions of rough interior and rough closure of a set from the perspective of a rough family, which leads to the definition of rough open sets (rough closed sets). As a consequence, we generate a new topology, termed rough topology, as well as rough homeomorphism. Our main contribution presents the novelty of this new class; in particular, we explicitly construct several examples which ensure that two non-homeomorphic spaces can be roughly homeomorphic under certain roughness. Additionally, we extend the concepts of compactness as well as connectedness, where our findings diverge from existing literature in these areas, in a nutshell, providing new insights and perspectives.

We prove that for every fixed $\lambda>0$ and all sufficiently large $n$, any $z_1,\dots,z_n\in\C$ with $|z_j|\geq1$ satisfy $\max_{2\leq k\leq n+1}|\sum_j z_j^k|>e^{-\lambda n}$. Consequently, the $n$th root of the optimal maximum tends to $1$, so no constant $C>1$ in Erdős 973 can exist. The proof combines a truncated exponential factorization with overconvergence on an open set outside the unit disk and a normal-family obstruction for Cauchy transforms.

We study block polytopes whose variables are divided into equal-size blocks and whose local inequalities bound the total contribution of adjacent blocks. For blocks arranged along a path, we develop a transfer-matrix enumeration in the length direction. We also carry out an Ehrhart-theoretic analysis in the dilation direction. The original transfer matrix admits a compression to a weighted height matrix, and the numerator and denominator of the length generating function are described by explicit recurrences and determinant formulas. We also study the cyclic analogue, where the length generating function is governed by the logarithmic derivative of the same determinant. On the Ehrhart side, the path polytopes with at least two blocks and the even cyclic polytopes are stable-set polytopes of perfect graphs; consequently they are Gorenstein of codegree $2a + 1$ (independent of the number of blocks $m \ge 2$), satisfy an explicit Ehrhart–Macdonald reciprocity, and have palindromic unimodal $h^*$-polynomials of degree $a(m-2)$. Odd cyclic polytopes have denominator exactly two, and their lattice-point enumerators are Ehrhart quasipolynomials of period dividing two. Further combinatorial interpretations and open problems are discussed.

We study the sharpness of recent sufficient conditions for polynomial Fourier decay of self conformal measures on the line. First, we construct a $C^\infty$ iterated function system which is not $C^1$-conjugate to self-similar, but which nevertheless admits a stationary measure that is not Rajchman. Second, for every strongly separated Bernoulli convolution $\mu$ and every $1\leq r<\infty$, we construct a $C^r$-diffeomorphism $h$ such that $h'$ is constant on $\operatorname{supp}\mu$, yet the image measure $h\mu$ has polynomial Fourier decay. All constructions are within the framework of pseudo-affine iterated function systems, previously introduced by the authors.

In this paper, we aim to define a new variable Hardy spaces in the Dunkl setting by the discrete Littlewood–Paley square functions. We also consider the atomic decomposition characterization for the new variable Hardy spaces with the help of the Littlewood–Paley theory and Hardy spaces on spaces of homogeneous type in the sense of Coifman and Weiss. As applications, we obtain the variable Hardy spaces boundedness for Dunkl–Calderón–Zygmund singular integrals involving by the Euclidean metric and the Dunkl “metric” which is associated with the finite reflection groups on the Euclidean space.

In this paper, we establish a doubling argument to obtain Hessian estimates for convex solutions to the Hessian quotient equation $\frac{\sigma_n}{\sigma_k}(D^2u) = f(x,u,Du)$ for $k=n-1$ and $k=n-2$ under the condition that $\log f$ is convex in the $Du$ variable. In particular, our approach is pointwise and does not make use of the Legendre transform or integral-based local maximum principles. We provide a counterexample demonstrating that interior estimates can fail if no structural assumption is imposed on $f$ in the $Du$ variable. Finally, we extend our doubling argument to general Hessian quotient equations $\frac{\sigma_l}{\sigma_k}(D^2u) = f(x,u,Du)$ for $k \in \{l-1, l-2\}$, under a similar structural condition imposed on $f$ in the $Du$ variable, alongside an additional structural concavity assumption on the operator introduced by Lu-Tsai 2026, which has very recently been established in independent works.

The classical gamma function is singled out among solutions of Euler's recurrence by the Bohr–Mollerup theorem. We develop an analogous normalization theory for the Bhargava factorial attached to the rational primes. Using factorial calculi, filtered prime-layer completion, centered cyclotomic quadrature, and orbitwise Stirling normalization, we construct a canonical zero-free entire function $\displaystyle \Gamma_{\mathbb P}^{\mathrm{cyc}}$ interpolating the prime Bhargava factorial. It satisfies the Euler-type reflection law \[ \Gamma_{\mathbb P}^{\mathrm{cyc}}(z)\Gamma_{\mathbb P}^{\mathrm{cyc}}(1-z)=1, \qquad \Gamma_{\mathbb P}^{\mathrm{cyc}}(1/2)=1. \] At positive integers we obtain the unsmoothed Stirling formula \[ \log (n+1)!_{\mathbb P} =n\log n+(C_{\mathbb P}-1)n+\tfrac12\log(2\pi n) +\mathcal R_{\mathbb P}(n)+O(n^{-1}), \qquad C_{\mathbb P}=\sum_p\frac{\log p}{(p-1)^2}, \] where $\displaystyle \mathcal R_{\mathbb P}(n)=o(n)$ and admits an exact prime-local fractional-part expansion. This gives a prime-case response to Questions 31 and 33 of Bhargava's paper The Factorial Function and Generalizations. Hankel positivity rules out positive Euler–Mellin representations for the function and its reciprocal; the cyclotomic construction instead motivates a prime Hankel conjecture for a canonical contour or distributional representation of the reciprocal. Finally, all but one centered subcritical Vaughan character moment are controlled unconditionally. Assuming the corresponding square-root estimate, we prove \[ \sum_{n\le X}\mathcal R_{\mathbb P}(n)^2 \sim-\frac43\zeta(-1/2)X^{3/2}\log X. \]

The well-known characterization of Hardy spaces $\mathrm{H}_{p}(\mathbb D)$, $0 < p < \infty$, in terms of the Littlewood-Paley g-function $$ S f (\zeta) = \left(\int_0^1 |f' (r \zeta)|^2 (1 - r) \mathrm dr\right)^{1/2} \in \mathrm{L}_{p} $$ is generalized to Hardy-type spaces $X_A$ corresponding to quasi-Banach lattices $X$ on the unit circle $\mathbb T$ under the assumption that the Hardy-Littlewood maximal operator $M$ is bounded in $(X^\delta)'$ with some $\delta > 0$. As an application to composition operators $C_\varphi$, we derive an exact criterion for the boundedness and compactness of $C_\varphi : \mathcal{B}^\omega \to X_A$, where $\mathcal{B}^\omega = \{f \mid \sup |f'|/\omega < \infty\}$ is the weighted Bloch space with a log-convex radial weight $\omega$, generalizing recent results in the one-dimensional setting.

In analogy to the point-to-set principle for Hausdorff and packing dimension of Lutz, Lutz, and Mayordomo, we develop a Kolmogorov complexity point-to-set principle framework for upper and lower topological pressure and BS dimension in dynamical systems (and by extension for upper and lower entropy). As a demonstration of this framework we expand upon the Bowen pressure equation results of Climenhaga, as well as prove a pointwise version of the Bowen pressure equation.

Low-rank adaptation (LoRA) optimizes $J(B,A)=\mathcal L(W_\mathrm{base}+sBA)$ over two adapters $B \in \mathbb{R}^{m \times r}$ and $A \in \mathbb{R}^{r \times n}$ that form a low-rank update to a frozen pretrained weight matrix $W_\mathrm{base} \in \mathbb{R}^{m \times n}$. The prior analysis shows LoRA-GD takes $\exp\{\mathcal{O}(\epsilon^{-2})\}$ oracle calls to find an $\epsilon$-stationary point such that $\|\nabla J(B,A)\|\leq \epsilon$ in the deterministic setting. We sharpen the analysis and show that $\mathcal{O}(\epsilon^{-4})$ full-gradient evaluations suffice for the same first-order criterion. We further study stochastic LoRA under unbiased gradient estimates and finite variance. We propose LoRA-NSGDM, which finds an $\epsilon$-stationary point with $\mathcal{O}(\epsilon^{-8})$ stochastic oracle complexity. Under the additional mean-square smoothness condition, we use variance reduction strategy and propose LoRA-STORM, which improves the stochastic oracle complexity to $\mathcal{O}(\epsilon^{-6})$.

We initiate the study of the nonlocal to local limit for linear transport equa- tions. A nonlocal version of the linear transport equation is introduced alongside an appropriate well-posedness theory for distributional solutions with measure initial data. We proceed by deriving the associated linear transport equation as a nonlocal-to-local limit. Some of the results apply in arbitrary dimension and for measure initial data.