Papers

In multi-task learning (MTL) negative transfer is often considered as an optimization artifact, but it can also be viewed as a consequence of limited shared capacity and weak task redundancy. We investigate this effect through a Capacity–Redundancy (CR) identity that decomposes the sum of per-task predictive informations into joint predictive information that includes label redundancy defined via total correlation (TC), and a residual coupling term that quantifies interference left unresolved by the shared representation. Additionally, we show two key results: (i) a clustering-gap decomposition that gives a necessary and sufficient condition for clustered sharing to outperform global sharing, and (ii) a gradient–TC bridge in a Gaussian multi-task model that formally justifies gradient cosine similarity as a proxy for redundancy ordering. Empirically, we estimate the residual coupling $\Delta$ from validation residual correlations, showing that clustered LoRA substantially reduces $\widehat{\Delta}$, outperforms size-matched random partitions, and results in statistically significant gains with multi-seed confidence intervals.

We establish a Catino-type integral inequality for closed Bach-flat $\mathcal{A}_2$-manifolds, namely Riemannian manifolds with nonnegative scalar curvature and constant nonnegative second Schouten curvature. In the equality case, we derive rigidity results showing that the manifold is either Einstein or isometrically covered by $\mathbb{S}^{1}\times\mathbb{S}^{n-1}(\kappa)$ endowed with the product metric.

Random basis expansion (RBE) search the span of a randomly sampled basis to find the best approximation of a target function. They are equivalent to single layer neural networks where the hidden layer weights are chosen randomly. Universal approximation properties have been established for RBEs using continuous weight sampling distributions and real-valued activation functions. Our results extend the universal approximation property to RBEs that use non-continuous weight distributions with dense support in the weight space and that use complex-valued activation functions. The result shows such random bases have the universal approximation property with arbitrarily high probability.

Maximum-entropy reference distributions are usually constructed on the normalized probability simplex. This formulation is less natural for unnormalized statistical models, in which positive multiples represent the same shape, and it does not directly explain how a prescribed admissible region should determine the deformation parameter of a bounded-support reference distribution. We formulate maximum entropy on the projective space of nonnegative measures and establish three results of statistical relevance. First, a universality theorem shows that every admissible monotone transform of the same normalized power functional has exactly the same optimizer under linear moment constraints. The result unifies the maximum-entropy implications of Tsallis and Rényi entropies, Hölder composite scores, pseudo-spherical scores, Bregman–Hölder constructions, and related homogeneous divergences without asserting a new distribution family. Second, the common optimizer is characterized as a $q$-exponential density; under mean and covariance constraints it is a compactly supported $q$-Gaussian for positive deformation and a Student-type density for negative deformation. Third, a prescribed Mahalanobis acceptance region with squared radius $R^2>d+2$ uniquely determines the deformation parameter $\gamma_R=2/(R^2-d-2)$. The resulting affine-equivariant reference density is the unique projective maximum-entropy solution, and its support coincides with the specified ellipsoid without an additional support constraint. This provides a principled method for constructing bounded-support statistical reference distributions from robust location and scatter estimates or from externally specified admissible regions.

Optimal Polynomial Intersection (OPI) is a structured optimization problem for which Decoded Quantum Interferometry (DQI) attains a satisfaction guarantee governed by the semicircle law. Sun and Wootters recently showed that, for balanced OPI over prime fields, a Fourier-defined distribution $P_u$ gives a strict worst-case improvement from limiting rate $0.6225$ onward and asymptotically perfect solutions from rate $0.7496$ onward, and asked whether $P_u$ can be sampled efficiently. We answer this question for Reed–Solomon OPI parameters satisfying their exponent condition strictly below the dual Johnson radius. Under coherent membership-oracle access, we give a bounded-error polynomial-time quantum sampler for $P_u$. The ideal circuit samples $P_u$ exactly conditioned on success, while a finite-precision implementation achieves any prescribed inverse-polynomial total-variation error. Consequently, every fixed limiting rate $0.6225\le r<1$ admits a strict worst-case improvement over the DQI semicircle value, and every limiting rate $r\ge 3/4$ admits solutions of satisfaction $1-o(1)$ with high probability. The algorithm coherently sums the amplitudes of all low-weight errors in each syndrome class using deterministic complete list decoding. Complete Reed–Solomon list decoding and the Sun–Wootters denominator estimate make the list size and postselection overhead polynomial. In concurrent and independent work, Horinaga and Yamakawa obtain worst-case OPI algorithms over prime-power fields and exact satisfaction at every fixed rate strictly above $3/4$.

In finite horizon dynamic programming with history-dependent preferences, the relevant state may be the entire realized history, even when the physical state is Markov. This paper develops a behavioral state-reduction theory for such Markov decision processes. Under behavioral axioms and a certainty-equivalent richness condition, the full-history problem admits a recursive representation composed of time and risk aggregators. We then derive a canonical preference-augmented (PA) state by quotienting histories that have the same current physical Markov state, are indifferent under every common continuation plan, and remain equivalent after every common one-step extension. This canonical PA state is minimal among reachable recursive factorizations of the underlying preferences. Under Markov feasibility and standard dynamic-programming regularity, a PA Bellman selector induces an optimal full-history policy. With additional rectangularity and exhaustiveness conditions, we reparameterize the preference memory into distinct belief and taste coordinates, and obtain a separated representation and Bellman recursion. We give a taxonomy of examples to illustrate the scope of our framework.

We introduce a positive-allocation companion construction for Koopman-inspired finite-dimensional prediction of nonlinear dynamical systems. The method determines recurrence coefficients by representing a target observable snapshot as a nonnegative, normalized combination of earlier training snapshots. These coefficients define a companion matrix whose spectral structure is induced by the allocation constraints at the construction stage. We prove that normalized positive allocation places the companion spectrum in the closed unit disk and, because the coefficients sum to one, includes $1$ as an eigenvalue. Additionally, we develop modal and non-modal diagnostics for the resulting model trajectory. When the companion matrix is diagonalizable, the modal representation shows that first and second finite differences act as spectral filters through factors of $\lambda_\ell-1$ and $(\lambda_\ell-1)^2$. We also derive $C$-based finite-difference bounds that avoid diagonalization and can be evaluated directly from the training data and companion matrix. Numerical experiments on the FitzHugh–Nagumo and susceptible–infectious–recovered (SIR) models illustrate the behavior of the construction in oscillatory and transient dissipative settings. The examples demonstrate both the interpretability of the companion recurrence and its limitations, particularly when pointwise trajectory agreement degrades while finite-difference and modal diagnostics remain informative.

Batteries are becoming a central part of modern energy systems, especially as electricity, transport and heat are decarbonised. In Great Britain, batteries already play an important role by providing flexibility and acting as a buffer for the system, and their importance will continue to grow as the system moves towards net zero by 2050. This perspective reviews how the role of batteries in Great Britain is evolving, their current value, and the growing contribution of batteries in the transport sector.It argues that the full value of electric vehicle batteries will only be realised through coordinated dispatch and better integration into system operation. Evidence from future scenarios developed by the National Energy System Operator (NESO) shows that while electric vehicles could provide most of the storage capacity, unmanaged charging could create challenges for the system rather than benefits. Unlocking this value will require improvements in market design, optimisation, and control strategies.

The field of moduli of a variety $X$ over an algebraically closed field $K$ is defined as the fixed field of those automorphisms $\sigma$ of $K$ for which $X\simeq X^{\sigma}$. A fundamental question is under what conditions a variety admits a model over its field of moduli. We give a complete answer for smooth del Pezzo surfaces in characteristic $0$: every smooth del Pezzo surface of degree at least $3$ has a model over its field of moduli, whereas in degrees $1$ and $2$ there exist smooth complex del Pezzo surfaces with field of moduli $\mathbb{R}$ which do not admit a real model.

We study a fully discrete scheme for nonlinear wave equations on general bounded polygonal/polyhedral domains with initial data $(u^0,v^0)\in H^\gamma(\Omega)\times H^{\gamma-1}(\Omega)$, $0<\gamma\le 1$, subject to the natural compatibility condition associated with the homogeneous Dirichlet boundary condition. The scheme combines an exponential Euler time integrator with a finite element spatial discretization. In contrast to existing low-regularity error analyses, which are mostly based on Fourier spectral discretizations, our approach applies to general bounded domains and finite element spatial discretizations. We prove rigorous error estimates for low-regularity solutions. The analysis is based on a frequency decomposition of the underlying elliptic operator, used solely as an analytical regularization device and not in the actual implementation, which allows low-regularity techniques to be extended beyond the Fourier spectral framework. The results also indicate that higher-order finite element methods remain advantageous in spatial approximation even for solutions of limited Sobolev regularity. Numerical experiments on different domains and with different polynomial degrees confirm the predicted convergence behavior.

Integrated sensing and communication (ISAC) and AI-and-communication (AIAC) are identified as separate usage scenarios in the ITU IMT-2030 vision for sixth-generation (6G) networks. In practice, however, these two directions are already beginning to merge. ISAC gives the network a way to observe the physical world, while AI gives the network a way to learn from those observations and act on them. This article introduces AI-integrated sensing and communication (AISAC) as a closed-loop framework for this merger. In AISAC, AI is not only a tool used to optimize an ISAC system. ISAC is also the physical substrate through which AI receives data, context, and connectivity. The key technical message is that AISAC requires a new physical-layer design principle, in which the ISAC waveform, beam, power, bandwidth, and sensing mode should be configured for learning alignment, not for sensing distortion or communication rate alone. In particular, the sensing configuration that is most accurate from a classical estimation viewpoint need not be the one that is most useful for training or inference. We present the AISAC landscape, explain why imperfect sensing changes the learning problem, develop the closed-loop architecture and its three-way sensing-communication-learning tension, and outline a vehicular edge-intelligence use case together with open problems for theory, implementation, and standardization.

We study precise asymptotics for high-level exceedance probabilities of path functionals of continuous vector-valued Gaussian processes. The probabilities have the form $$ \mathbb{P}\{\Gamma_{[0,T]}(\check{\boldsymbol{u}}(\boldsymbol{X}-u\boldsymbol{b}))>L_u\}, \qquad u\to\infty, $$ where $\boldsymbol{X}$ is a centered $\mathbb R^d$-valued Gaussian process and $\Gamma$ belongs to a broad class satisfying natural monotonicity, scaling, no-atom, and continuity assumptions. The class covers classical sojourn times, Choquet-type sojourn integrals, area-under-the-curve functionals, and, through a local-footprint extension, shrinking-window Parisian persistence functionals. We obtain exact asymptotics in the stationary case and in the non-stationary case when the inverse generalized variance has a unique minimizer at the boundary point. The non-stationary theorem covers the regimes $\alpha<\beta$, $\alpha=\beta$, and $\alpha>\beta$, which lead respectively to Pickands-type, Piterbarg-type, and deterministic limiting constants. We also derive conditional limit laws for the first exceedance time. The main claims are stated in the body of the paper, while the proofs and auxiliary estimates are collected in the appendices.

Time-of-flight magnetic resonance angiography (TOF-MRA) is widely used for nonin-vasive visualization of arterial anatomy, but extracting hemodynamics like blood velocity typically requires supplementary phase-contrast scans, tagging or multi-TE images. This study proposes a novel, physics-informed computational framework to extract variable fluid velocity directly from standard TOF-MRA signal profiles. We analytically expand the approach-to-steady-state Bloch equations to include convective flow, establishing a mathematical relationship between the spatial decay of longitudinal magnetization and fluid velocity. The velocity derivation was further extended to pointwise estimation over a 1-D centerline, overcoming the limitations of constant-velocity assumptions. To validate and solve this problem, a MATLAB (R2025b) simulation framework was developed to model fluid flow in two variable-geometry flowing tube cases, i.e., continuous narrowing and focal stenosis, under synthetic scanner noise. A global inverse optimization approach utilizing Dual-Tikhonov regularization was applied to stably invert the ill-posed transit time integral, actively penalizing high-frequency numerical ringing while preserving structural curves. The computational simulations successfully recovered ground-truth point-wise velocities, tracking gradual hemodynamic accelerations and sharp stenotic jets. This theoretical framework and the example centerline TOF-MRA signal intensity pro-vide a robust mathematical proof-of-concept that quantitative, localized functional he-modynamic metrics can be extracted from standard structural MRA imaging, establishing a foundation for advanced flow quantification without requiring additional scan time.

Point vortices represent an important reduced model describing two-dimensional ideal fluid dynamics. It is well known that there exist three-vortex configurations on the Euclidean plane $\mathbb{R}^2$ that exhibit finite-time singularities, i.e., collapse to a single point. Moreover, in $\mathbb{R}^2$, such collapses occur only self-similarly. Here, we investigate the extent to which this phenomenon persists on curved surfaces. We show that self-similar collapse is a universal feature of surfaces of nonnegative constant curvature, namely the plane and the sphere. In contrast, on the hyperbolic plane, it is shown that self-similar collapsing solutions do not exist with respect to any distance variable defined by an analytic function of the geodesic distance. Finally, we establish the existence of nearly self-similar collapse of three vortices on arbitrary smooth surfaces embedded in $\mathbb{R}^3$.

We study the behaviour of heat semigroups on quantum graphs under Benjamini-Schramm convergence. For quantum graphs with uniformly bounded geometry, equipped with continuity and Kirchhoff vertex conditions, we show that the heat semigroup, transported to a common Hilbert space by a canonical breadth-first identification of the edges, depends continuously on the underlying rooted quantum graph with respect to a local Benjamini-Schramm-type metric. As a consequence, root-averaged pairings of the semigroup converge along Benjamini-Schramm convergent sequences of finite quantum graphs. Combining this with a Trotter-Kato-type approximation of the semigroup by semigroups on metric balls, we obtain a double-limit theorem interchanging the truncation radius and the graph limit.

We introduce analogues of partial derivatives with respect to the generators of the enveloping algebras U(gl(N)) and their q-deformations, the so-called modified reflection equation algebras. Using these quantum partial derivatives, we introduce the corresponding differential algebras, define the quantum de Rham operator, and exhibit the Leibniz rule for its action. Our Leibniz rule differs from its classical analogue and stems from the construction of a quantum double. In the case of the algebra U(gl(N)), we present another form of the Leibniz rule that is more convenient for prolonging the differential calculus to certain extensions of this algebra. We then use analogues of the Cayley-Hamilton identity for the generating matrices of U(gl(N)) and of reflection equation algebras to construct projective modules over these algebras. For these projective modules, we introduce Grassmannian connections and define the corresponding Chern classes.

We consider a supercritical two-type continuous-time linear birth-death process with mutation and selection, in which wild-type individuals give rise to mutant offspring with a larger net growth rate $\lambda_1$. In this setting, we investigate a component of the site frequency spectrum (SFS), describing the number of driver mutations present at any given frequency in the population. We establish asymptotic convergence results at large times and detection sizes. We examine three distinct regions of the SFS: (1) the small frequency region occupied by clones of size $\mathcal{O}(1)$, (2) the intermediate frequency region occupied by clones of size $1\ll j(t)\ll \exp(\lambda_1 t)$, and (3) the large frequency region occupied by clones of size at least $\exp(\lambda_1 t).$ We obtain strong laws of large numbers for the driver SFS in the small- and large-frequency regions by constructing suitable $L^2$-approximations. In the intermediate frequency region, we identify a cutoff frequency $j(t)$ growing with time at which there are an order-one number of mutant clones. At this frequency, we obtain a Poisson convergence theorem as $t\to\infty$. Overall, our results provide quantitative insights into how selection shapes the site frequency spectrum both at small and large frequencies, which can in principle be leveraged to construct estimators of relevant evolutionary parameters. As an example of this, we propose and show asymptotic consistency of a new estimator of the fitness increase.

We develop a new approach to the problem of the motion of a large number of rigid bodies immersed in a viscous fluid. The leading idea is the concept of cluster - a collection of individual rigid objects that may be grouped or even connected in such a way that their collective impact on the bulk motion of the system is similar to that of a single body. The applications of the new approach include: 1. Improving the critical value of the number of balls of small radius such that their cloud has no impact on the limit system represented by the incompressible Navier–Stokes equations. 2. The balls follow the fluid flow in the asymptotic limit of vanishing radius and increasing number even if a gravitational force is imposed.

Fulek defined the $0$-$1$ matrix \[ L_3=\begin{pmatrix} 1&0&0&1&0\\ 0&0&0&0&1\\ 0&1&1&0&0 \end{pmatrix} \] and asked whether $\text{ex}(n,L_3) = O(n)$. We prove that every $r\times s$ $0$-$1$ matrix avoiding $L_3$ has at most $27r+2s$ $1$ entries. Fulek's general lower bound construction has $6n-8$ $1$ entries, so \[ 6n-8\leq \text{ex}(n,L_3)\leq29n \] for $n\geq5$. The same argument applies to an infinite family. If $Q_{a,b,k,\ell}$ is the light three-row matrix with column word $1^a3^k1^b2^\ell$, where $a,b,\ell\geq1$ and $k\geq2$, then \[ \text{ex}(r,s,Q_{a,b,k,\ell}) \leq\bigl(5(k-1)(4b+1)+a+b+\ell-1\bigr)r+2s. \] This verifies a conjecture of Pettie and Tardos on linear light patterns for an infinite family that includes the previously unresolved weight-five pattern $L_3$. The proof assigns matrix entries to edges of a bar $1$-visibility hypergraph, cuts gaps to control the multiplicity of these edges, and charges the cuts to a noncrossing graph on the rows.

This paper surveys different topics where the theory of quantitative rectifiability plays a central role. First, it reviews the characterization of rectifiability in terms of square functions involving $\beta$ type coefficients and the $\varepsilon^2$ conjecture of Carleson. It also discusses the deep connections between rectifiability and the $L^2$ boundedness of Riesz transforms and their application to the Painlevé problem for Lipschitz harmonic functions. Finally, the paper explores recent major advances in connection with harmonic measure and the $L^p$ solvability of the Dirichlet, regularity, and Neumann problems for the Laplace equation in rough domains, emphasizing the key role of quantitative rectifiability in these developments.

While non-intrusive reduced-order emulators successfully substitute for costly finite-element simulations, certification-grade applications demand a rigorous error assessment relative to the solver's own discretization uncertainty. This work introduces a verification framework designed to evaluate prediction accuracy directly against this numerical uncertainty floor, rather than physical observation. We demonstrate this approach on a carbon-carbon aircraft brake disc during a rejected take-off scenario, predicting peak temperature and peak von Mises stress fields. Operating over a fixed mesh of 60914 nodes, the emulator relies on a pipeline coupling proper orthogonal decomposition (POD) and Gaussian process regression (GPR) parameterized by 5 inputs. We formulate an algebraically exact, a posteriori error budget that isolates the emulator's signed error into distinct truncation and regression components at the extremum. This error is then evaluated against a discretization floor established via mesh-convergence studies on the peak quantities of interest. On a completely independent test set, our emulator matches the high-fidelity solver's peak values within its own discretization uncertainty for both fields, a benchmark we define as surrogate-to-code fidelity. Notably, the residual error stems primarily from linear reduction limits rather than regression inaccuracies. Beyond the main framework, we present two secondary contributions: the concurrent, non-intrusive prediction of the thermal and mechanical peaks, and a probabilistic approach to spatial zone localization for these extrema. This methodology applies to non-intrusive reduced-order models of parameterized finite-element simulations wherever a discretization-error estimate for the quantity of interest can be obtained.

We classify profinite groups that have a Poincaré-like duality between their homology and cohomology. Our proofs work over every profinite coefficient ring, and for profinite as well as discrete coefficients. We do not only unify and generalise existing results, but also construct two novel examples of duality groups. We prove our results via the framework of condensed mathematics. This recently developed setting is a natural home for profinite objects, and our work is among the first to leverage this for the study of the (co)homology of profinite groups. Establishing our duality results involves an investigation of homological finiteness properties in condensed mathematics.

Self-assemblies of tunable units are being intensively studied as physical systems with signal-processing capabilities. Specifically, silver nanowire networks (AgNWNs) have demonstrated accumulation, non-linearity, and memory retention with multiple timescales, features that enable a wide variety of neuromorphic implementations. In this study, we aim to extract the interconnection scheme to analyze the experimentally obtained network architecture and, eventually, use it as input to a previously developed simulation platform. By post-processing photomicrographs of AgNWN, we present a pipeline optimized to extract the interconnection diagram, recognizing the intersections formed among the nanowires, to determine the associated graph for each physical sample. A graph is a collection of nodes and edges whose properties can be associated with different electrical responses. It is thus possible to study graphs' metrics, such as the degree distribution, community size, clustering coefficient, and path-length, to compare the experimental assemblies' attributes to those of topological models of reference. Small-world, modular, and scale-free are well-known structures in the field of mathematical graphs. By analyzing the degree distribution, the adjacency matrices, among other useful representation means, the experimental assemblies reveal similarities to both small-world and modular topologies. All the mentioned analyses were conducted considering the interconnection scheme obtained from zenithal-view optical images, which overestimates the number of NWs' interconnections (due to the impossibility of distinguishing real junctions from spurious cross-points between vertically displaced NWs). For that reason, this communication also studies the impact of artificially removing junctions from the resulting graphs on the previously calculated clustering coefficient and path length.

We study regularity properties of minimizers for the one-phase Alt–Caffarelli problem associated with the fractional $p$-Laplacian in the range $p\geq2$. We consider minimizers of the fractional $p$-energy penalized by the measure of the positivity set, with prescribed nonnegative exterior datum. We prove existence of minimizers and derive their basic variational properties: minimizers are nonnegative and are weak subsolutions of the homogeneous fractional $p$-Laplace equation. The main regularity argument combines fractional $p$-harmonic replacements, energy-gap estimates, nonlocal tail bounds, and a Campanato-type iteration. This yields local Hölder continuity of minimizers and implies that they solve the homogeneous equation in their positivity set. Finally, we prove the optimal free boundary growth estimate, showing that the sharp order known in the linear fractional Bernoulli problem persists in the fractional $p$-Laplacian setting.

Stochastic Mckean-Vlasov models have a substantial importance in different fields such as finance, biology and control. This paper puts the light on stochastic proportional delay Mckean-Vlasov model with Lévy jump where the non-jump coefficients are granted the permission to grow beyond linearity. The truncated Euler-Maruyama algorithm is then applied to our addressed model where the convergence rate and almost sure exponential stability of the aforementioned numerical algorithm are being investigated. Finally, numerical examples are presented to foster the theoretical analysis done throughout the paper

We investigate two-color, two dimensional spatially localized light modes in a resonant Kerr third-harmonic generation model. Using computational tools, we identify two distinct families of localized states. Unlike the single-component 2D nonlinear Schrödinger equation and previously studied non-resonant two-color systems, the dynamics are not dictated by a universal critical power, but depend on the distribution of power between the harmonics. The "fundamental-dominated" family acts as a "dynamical separatrix" between simultaneous collapse and joint diffraction, whereas the "third-harmonic-dominated" family does not. We further identify resonant collapse events accompanied by strong oscillations of the third harmonic, revealing a collapse mechanism absent from standard Kerr self-focusing.

The integration of distributed energy resources (DERs) into the power grid has introduced new challenges to AC optimal power flow (AC-OPF) problems. Traditional OPF optimize consider transmission systems, treating distribution networks as static loads. However, the growing presence of DERs makes accurate distribution system modeling crucial for grid operations. Consequently, efficiently solving the resultant large-scale, nonconvex transmission and distribution (T&D) AC-OPF problem remains a significant challenge. This paper proposes a Smoothed Two-Stage Decomposition Optimizer (StsDOpt) to address these complexities by decomposing the T&D AC-OPF problem into a master-subproblem(s) structure, enabling parallel solving. Unlike traditional methods, StsDOpt does not rely on approximations or relaxations. It uses a smoothing technique to render the subproblems responses differentiable with respect to the master problem, leveraging the barrier problem properties inherent in primal-dual interior point methods. This approach is crucial for accurately modeling and solving distribution systems, which are multiphase, unbalanced, and nonlinear, distinguishing StsDOpt apart from other methods. Integrated into the PowerModelsITD framework, StsDOpt has been validated through numerical experiments, demonstrating reduced wall-clock solve time and increased scalability. Results highlight its efficacy as a robust, scalable solution for large-scale T&D AC-OPF problems, facilitating the reliable integration of DERs into complex T&D systems.

We investigate the geometric content of the first-order part of the Laplace–Beltrami operator on an oriented Riemannian manifold. Relative to an arbitrary local orthonormal frame, the first-order part of the Laplace–Beltrami operator defines a distinguished vector field, whose coefficients are expressed through the Levi–Civita connection forms. The corresponding one-form determines a covariant derivative whose connection factorization removes the first-order part of the Laplace–Beltrami operator and replaces it by a natural scalar potential. The construction is frame dependent. We show that its transformation under local rotations of the orthonormal frame compensates the corresponding variation of the sum-of-squares part, leaving the full Laplace–Beltrami operator invariant. These results reveal the geometric role of the first-order part of the Laplace–Beltrami operator.

Let $G=(V,E)$ be a finite simple graph of order $n\geq 1$, and let $\ell:V\to\{0,1\}$ be a prescribed parity labeling. A set $S\subseteq V$ is called $\ell$-admissible if $d_S(v)\equiv \ell(v)\pmod 2$ for every $v\in S$, where $d_S(v)=|N_G(v)\cap S|$. Let $h_\ell(G)$ be the maximum order of an $\ell$-admissible set and let $f_{\rm oe}(G)=\min_\ell h_\ell(G)$. For $x\in\mathbb R$, define the weighted counting polynomial $$ M_{\ell,x}(G)=\sum_{S\in {\cal A}_\ell(G)}x^{|S|}, $$ where ${\cal A}_\ell(G)$ is the collection of all $\ell$-admissible sets in $G$. For $R\subseteq V$, let $z_\ell(R)$ be the number of vertices $v\in V\setminus R$ for which $d_R(v)\equiv\ell(v)\pmod 2$. We prove the exact identity $$ M_{\ell,x}(G) =2^{-n}\sum_{R\subseteq V} x^{|R|}(2+x)^{z_\ell(R)}(2-x)^{n-z_\ell(R)-|R|}. $$ If $G$ has no isolated vertices, then, for every $\ell$ and every $x\in(0,2)$, $ M_{\ell,x}(G)>x^{n/2}(4-x^2)^{n/4}. $ Combining this estimate with a binary-entropy upper bound and optimizing $x$ gives $$ f_{\rm oe}(G)>c_*n>\frac{2n}{21}, $$ where $c_*\approx0.095862615$. Ferber and Krivelevich (Adv. Math. 2022) proved that $h_{\mathbf{1}}(G)\ge 10^{-4}n$, where $\mathbf{1}$ is the all-one labeling. Since $h_{\mathbf{1}}(G)\ge f_{\rm oe}(G)$, our result improves coefficient in their bound by almost three orders of magnitude, and does so simultaneously for every labeling.

We consider a four-field formulation of Biot's quasi-static model of consolidation with the (effective) elastic stress, the solid displacement, the fluid flux and the fluid pressure as unknown physical quantities of interest. The weak form of this system composed of the momentum and mass balance equations complemented by the stress-strain relation and Darcy's law yields a two-fold perturbed saddle-point problem. Its well-posedness is proven for a natural choice of inifinite-dimensional Hilbert spaces, where the displacement-flux pair is sought in a novel Hilbert space. Next, based on recent works on $H(\rm div \, \rm div)$-conforming finite elements, we propose a family of mixed-mixed finite element methods that preserve the angular momentum as well as the fluid mass balance pointwise, i.e., in a strong sense. We prove the well-posedness of the arising discrete problem and establish optimal a priori error estimates for the related family of fully conservative mixed-mixed finite element methods.