Papers

We investigate the behaviour of a primal–dual penalty scheme applied to all possible instances, i.e., boundary conditions, of an optimal control problem involving the double integrator. The proposed duality-based framework enables us to derive closed-form expressions for the associated dual functions as well as for the optimal control variables. Furthermore, we provide a complete characterization of the least exact penalty parameter for this problem. The iteration dynamics of the algorithm are examined by constructing and analyzing the trajectories of the constraint functions for every admissible instance of the problem. Finally, we compare the two step-size rules employed in our primal–dual algorithm with two alternative Polyak-type step-size strategies.

In this work we propose, by linearizing the equations of fully nonlinear poroelasticity, a consistent model in which only the solid phase is incompressible. This reformulation circumvents some inconsistency issues encountered in standard primal formulations of nonlinear poroelasticity while still retaining its key physical coupling mechanisms. We show a well-posed and consistent discretization strategy and also formulate scalable solvers based on a Schur complement formalism. A distinctive feature of the model is that it allows for a lowest order, inf-sup stable family of Finite Elements (FE) spaces. Numerical tests in two and three dimensions are provided to validate the proposed method and solver framework.

The transition function of the unlabeled infinitely-many-neutral-alleles diffusion model, as expressed by Zhou (2015), is derived from the transition function of the labeled infinitely-many-neutral-alleles diffusion model, slightly simplifying the derivation by Feng (2010).

Quantum locally recoverable codes (QLRCs) have recently gained attention as a framework for achieving efficient quantum storage with local recovery capabilities. Analogous to their classical counterparts, QLRCs allow a lost qudit to be reconstructed using only a small subset of other qudits, thereby reducing the resource and operational overhead in recovery. In this work, we extend the study of QLRCs by considering $(r,\delta)$ QLRCs characterized by locality parameter $r$ and local distance $\delta \geq 2$. We present constructions of both random and explicit $(r,\delta)$ QLRCs, including explicit families based on the quantum Tamo–Barg construction. We also present an efficient decoding algorithm for these quantum Tamo–Barg codes. Furthermore, we introduce quantum hierarchical locally recoverable codes (QHLRCs), which extend local recovery to multiple hierarchical levels. For any integer $h\geq 2$, we construct both random and explicit $h$-level QHLRCs, the latter being $h$-level quantum Tamo–Barg codes, and establish a Singleton-like bound for these codes using a CSS framework built from dual-containing classical codes. These results advance the theoretical foundations of quantum erasure recovery and contribute to the design of efficient quantum storage architectures.

We settle the projectedness problem for the compactified unpunctured supermoduli stack in every genus at least two. In genus two, the odd component is split, whereas the even component is non-projected. In every genus $g\geq 3$, both compactified parity components are non-projected.

We show that the small-data scattering map uniquely determines the nonlinearity for a class of nonlinear Schrödinger equations with radial, Hartree-type nonlinearities. Our assumptions on the convolution kernel require only a mild decay condition at infinity and permit a locally integrable singularity at the origin.

We develop a data-driven framework for estimating optimal linear response of nonlinear dynamical systems. Our approach is based on kernel-smoothed approximations of the transfer/Koopman operators of the system, built from possibly high-dimensional observations along trajectories. Combining these operator approximations with the theory developed in [Antown et al. (2018), J. Stat. Phys., 170(6), 1051-1087], we formulate a computationally tractable optimization problem for the optimal infinitesimal perturbation realising any desired manipulation of the spectrum. We also introduce a notion of optimal-response vector fields for visualising, and physically interpreting, the system's response to the optimal perturbation under arbitrary observations. Our focus is on finding perturbations that optimally increase the frequency or optimally suppress the decay of correlations of almost-cycles or almost-invariant sets associated with the eigenvalues of the kernel-smoothed transfer operator. We illustrate our approach with applications to low-dimensional periodic and chaotic systems, as well as a high-dimensional example involving the El Nino Southern Oscillation in a comprehensive Earth system model. In these examples our approach discovers nontrivial optimal perturbations of the system, which are post hoc natural and consistent with the desired dynamical objectives.

We propose and study a simple and scalable Fourier-based spectral method for a continuum model of network formation under periodic boundary conditions. The model provides the evolution of the pressure $p$ and the conductivity $m$ over time. The evolution of $p$ is given by an anisotropic Poisson equation, while the equation for $m$ contains three terms corresponding to a diffusion and an activation term of the network – that depends on the gradient of the pressure – as well as a relaxation term that acts as a decaying term. This system arises as a formal $L^2$-gradient flow of a non-convex energy functional. Our algorithm combines two ingredients: (i) a splitting method for the equation for $m$, where the activation and relaxation parts are solved analytically, and the diffusion part is solved via Fast Fourier Transform (FFT), and (ii) an FFT combined with the Conjugate Gradient (CG) method applied to the equation for the pressure. This makes the scheme easy to implement compared to implicit schemes and naturally extensible to three dimensions on uniform periodic grids. To showcase the method, we recover the previously documented influence of the activation strength $c$, the diffusion coefficient $D$, and the metabolic exponent $\gamma$ on the morphology of emergent networks, and report grid convergence results.

We study a semidiscrete single-field Galerkin approximation of the Cahn-Hilliard equation using high-order conforming finite element spaces. More specifically, globally $C^1$ finite elements with $H^2$-conforming trial spaces, including Argyris, Bell, and Bogner-Fox-Schmit elements, allow a direct discretization of the fourth-order formulation and preserve mass exactly. The main structural result is an exact energy balance for the physical Cahn-Hilliard energy. Besides the expected discrete dissipation, the balance contains an explicitly computable energy defect. This defect vanishes for Laplacian-invariant periodic spaces, such as Fourier spaces, but is generally nonzero for classical $C^1$ finite elements. It therefore quantifies the precise loss of a discrete gradient-flow structure. We prove semidiscrete a priori error estimates by a relative-energy argument. The estimate is closed using an augmented relative energy and a discrete elliptic reconstruction bound for the inverse discrete Laplacian. The resulting convergence rates match the expected approximation orders. Numerical experiments with Bell and Argyris elements confirm the rates and demonstrate the defect mechanism by comparison with a Fourier reference discretization.

We give a product formula for the Tamagawa numbers of Jacobians over a discrete valuation field with perfect residue field k. It comes as a product of four terms - unipotent, toric, arithmetic and (somewhat intricate) cohomological. It is proved by (1) extending the classical flow-cut construction from semistable to arbitrary curves, which, by Raynaud's results, gives the formula when k is algebraically closed; (2) observing that this formula respects the natural metric on the edges of the dual graph, which allows to quotient out the Galois action; (3) extending Bosch-Liu's description of the cohomological term when k is finite. In particular, this answers a question of Bosch-Liu, and gives an alternative description of Poonen-Stoll's cohomological obstruction in terms of characters.

Gao and Wang proved a strong clopen marker theorem for finite generating sets of $\mathbb Z^n$ under the assumption that each generator has support of size either $1$ or $n$. We show that this support assumption can be removed. The proof is a short conjugacy argument: after a unimodular change of coordinates, any finite set of nonzero lattice vectors can be put in full-support position, allowing one to apply the theorem of Gao and Wang and conjugate the resulting marker set back.

This paper develops the Dynamic Gaussian Process (DGP), a framework for estimating functions governed by integro-difference equations (IDEs). IDEs model continuous functions that evolve with discrete-time dynamics and arise naturally from time-discretization of linear partial differential equations (PDEs). The DGP extends Gaussian process regression to time-varying functions and extends Kalman filtering to infinite-dimensional states. The DGP posterior remains a Gaussian process with closed-form mean and covariance updates, and separable kernel structure reduces the problem to a finite-dimensional Kalman filter on basis function coefficients. This paper extends the DGP to vector-valued states, enabling the treatment of higher-order PDEs, and provides a stability and approximation error analysis for the basis function approximation. The functional L2 estimation error decomposes exactly into in-subspace and out-of-subspace contributions, and all approximation errors vanish as the number of basis functions grows. The framework is demonstrated on the heat equation and on the wave equation, the latter with a vector-valued state. Code is available at https://github.com/JvHulst/Dynamic_Gaussian_Processes.

Explicit arithmetic progressions modulo primes $p \equiv 1 \pmod{4}$ are derived in which the coefficients in the expansions of products of quintuple products vanish. In particular, if $p = m^{2} + n^{2}$, and $b$ is a positive integer, and $$\sum_{n=0}^{\infty} a_{n}q^{n} = \frac{(q^{2bm},q^{p-2bm};q^{2bn},q^{p-2bn};q^p)_{\infty}}{(q^p,-q^{b m},-q^{p-bm},-q^{bn},-q^{p-bn};q^p)_{\infty}^2},$$ we determine $\alpha = \alpha(m,n,p)$ such that $a_{pt+ \alpha}=0$. Our results are proven using involutive transformations on integer lattices.

A linear optical medium can delay, mix, and superpose light, but never make two pulses multiply: multiplication is nonlinear, and a linear system has no such operation. This roots a sharp limit on continuous-variable quantum reservoir computers (QRCs) built from Gaussian optics. Within the reservoir they cannot form genuine products of the input at different past times, the cross-time nonlinear correlations many temporal computations require; they can only fake them by storing each past input separately and multiplying in the readout, forcing an exponentially harder high-order measurement. We show that a single Kerr (intensity-dependent phase) element in a time-delayed feedback loop removes this limit. The Kerr effect makes phase depend on intensity, a true multiplication inside the medium; feedback makes the light revisit that element repeatedly, so one mode mixes its own history against itself once per round-trip. Feedback turns time into space: D passes through one nonlinear mode replace D parallel linear modes. We prove an unbounded resource separation (Theorem 3, Corollary 2): an N-mode Gaussian reservoir reaches cross-time nonlinear rank at most 2N, a hardware ceiling, while a single Kerr mode reaches rank equal to its feedback depth D, costing no extra modes. For every N, one Kerr mode performs a computation no N-mode linear reservoir can. Loss is the counterintuitive ingredient: each round-trip dims the light, so the nonlinear phase differs pass to pass, giving every echo its own fingerprint; without loss the passes would be redundant. We confirm activation on an exact open-system simulation and ground the separation in nonlinear channel equalization. Achievable D is 30 to 230 on integrated platforms, so one nonlinear mode replaces up to about 100 linear ones, at the price of measurement time.

We generalize the Ahlswede–Daykin inequality (1978) to a Schur positive ADS inequality, which also contains the Lam–Postnikov–Pylyavskyy inequality (2007) as a special case. We then present a number of further generalizations and applications. Notably, we resolve Mihalcea's conjecture on log-supermodularity of stable Grothendieck polynomials.

Let $D$ be a quaternion division algebra over a non-Archimedean local field $F$ of characteristic zero, and let $G_n=GL_n(D)$. Let $H_{1,n-1}$ denote the subgroup of $G_n$ consisting of block-diagonal matrices of the form $diag(g_1,g_2)$, where $g_1\in G_1$ and $g_2\in G_{n-1}$. In this article, we formulate a conjectural classification of irreducible smooth $H_{1,n-1}$-distinguished representations of $G_n$ for $n>2$. We prove this conjecture in the cases $n=3$ and $n=4$. When $n=2$, the results are well known due to the contributions by various authors.

We discuss self improving properties of some local and nonlocal, elliptic and parabolic, equations on bounded domains. We employ a functional analytic approach wherein the solution space sits in a suitable interpolation scale. Utilizing a classical analytic perturbation result, we extrapolate the invertibility of the main operator from the base space to nearby spaces within the interpolation family.

We introduce and study a unified framework for density-functional theory and its variants for quantum systems on finite-dimensional Hilbert spaces. These theories seek to reduce the complexity inherent in the many-body quantum problem by describing ground states through reduced variables. The central ingredients of our unified framework are a generalized choice of basic observables, whose expectation values define precisely those reduced variables, and a fixed part of the Hamiltonian characterizing the class of quantum systems under consideration. It is this minimal structure, which we call the scope of a functional theory, that is necessary and sufficient for the formulation of a functional theory. In particular, it allows one to define the universal functionals, establish their convexity and differentiability properties, address representability questions, and prove a Hohenberg-Kohn-type uniqueness result. A purification construction also relates ensemble and weighted-ensemble functionals to the pure-state variant. Particular emphasis is placed on functional theories with Lie-algebra observable structures, connecting the variational framework to symplectic geometry. The result of this work is a systematic mathematical formulation in which structural results can be proved once and applied across a broad class of finite-dimensional functional theories.

For a locally compact group G we introduce and study the reduced Beurling-Fourier-Stieltjes algebra, a weighted analogue of the reduced Fourier-Stieltjes algebra, together with the algebra of completely bounded multipliers of the associated weighted Fourier algebra. We show, in particular, that these two algebras coincide when G is amenable. For a general locally compact group G, we identify them as subspaces of the reduced Fourier-Stieltjes algebra and of the space of functions that locally belong to the Fourier algebra, respectively. Furthermore, we establish sufficient conditions on the group and the weight under which the algebra of completely bounded multipliers of the weighted Fourier algebra embeds into its unweighted counterpart.

Let $G$ be a compact abelian group whose dual group $\Gamma=\widehat{G}$ has bounded torsion. In 1967, Malliavin-Brameret and Malliavin proved that every Sidon set in $\Gamma$ is a finite union of quasi-independent sets when $\Gamma$ has prime exponent. This was later extended to squarefree exponents in work of Varopoulos and Bourgain. We prove the remaining bounded-torsion case. Consequently, if $\widehat{G}$ has bounded torsion, then a subset $\Lambda\subset \widehat{G}\setminus{0}$ is Sidon if and only if it is a finite union of quasi-independent sets.

We study the convergence of SU(1,1) and SU(2) nonlinear Fourier transform with sparse square-summable data. The asymptotics of the associated polynomials orthogonal on the unit circle is obtained as a corollary.

In this article, we study the sine-Gordon model with logarithmic correlations in arbitrary dimension $d \ge 1$. The model is defined as a Euclidean field theory whose interaction term in the classical action is \[ S_{\mathrm{int}}(\varphi) := 2z \int_{\Lambda} \cos(\sqrt{\beta}\,\varphi)\,\dx, \] where $\varphi$ is a logarithmically correlated Gaussian field on a compact domain $\Lambda \subset \mathbb{R}^d$ that coincides with the Gaussian free field when $d=2$, and where $z \in \mathbb{R}$ and $\beta>0$. We treat both the massive and massless cases, with the main emphasis on the massless regime. We construct the field without imposing boundary conditions, together with its charge and gradient correlation functions, and we show that the partition function is renormalizable. The results are non-perturbative, holding for all $z \in \mathbb{R}$. If $d=1$, our results are valid for the full subcritical range $\beta\in (0,4\pi)$ and if $d\geq 2$, the results are valid for $\beta\in (0,(d+1)2\pi)$. Our analysis is carried out directly in the continuum: we perform renormalization via a scale decomposition and control the partition function by controlling the renormalized potential that arises in this procedure, using the iterated Mayer expansion of Brydges and Kennedy~\cite{BrKe87a}.

This work considers combinatorial and statistical aspects of {\em{shifts of finite type}}, which are families of words over a finite alphabet which avoid a fixed class of {forbidden} sub-words. The overarching question we are interested in is: how do local statistics of a uniformly random element of the shift space depend on combinatorial features of the forbidden set? We focus on the binary alphabet $\{0,1\}$, the class of shift spaces where a single pattern is forbidden, and the average frequency of $1$s (equivalently, the probability of observing $1$ at a given position). In this case, the relevant combinatorial information is encoded by a two-variable auto-correlation polynomial associated to the forbidden word, which we call the {\em{border polynomial}}. We present several results and examples characterizing the ordering of all words by their letter frequencies: for example, we describe the set of patterns which, when forbidden, cause the frequency of $1$s to increase, decrease, or stay exactly $1/2$. Our methodologies include novel explicit local injections and bijections, generating function analysis, and a connection with a probabilistic notion of letter frequency.

We address the Euler immersion problem, a fluid-structure interaction problem in which an elastic body is immersed in an incompressible inviscid fluid governed by the Euler equations. We show that the system exhibits a loss of one derivative and formulate the problem in analytic function spaces. We then prove local well-posedness in analytic spaces under velocity-matching boundary conditions. Finally, by means of an example, we show that existence fails in analytic spaces when both velocity and stress-matching boundary conditions are prescribed.

This paper develops a binary-gamble framework for characterizing risk sensitivity and loss aversion in Cumulative Prospect Theory (CPT). The proposed probabilistic risk-sensitivity metric is defined as a probability-threshold ratio that determines acceptance and preference thresholds in choice problems involving either a certain outcome and a binary gamble or two binary gambles. We show how standard notions of symmetric and non-symmetric bet aversion can be recovered within this framework, and we compare the resulting threshold-based conditions with utility premia, probability premia, and Arrow–Pratt curvature measures. The analysis clarifies when these criteria coincide and when they diverge, particularly for increasing aversion conditions, binary gambles with unequal probability distributions, and settings involving probability weighting functions. We also identify technical restrictions that arise when CPT-utility functions are used to represent loss aversion at the reference point. The resulting framework provides a decision-theoretic interpretation of risk sensitivity that is directly tied to probability thresholds and complements existing premium-based approaches.

Let $\mathbf{k}$ be a field and let $\Lambda$ and $\Gamma$ finite dimensional $\mathbf{k}$-algebras. Assume that ${_\Gamma}X_\Lambda$ and ${_\Lambda}Y_\Gamma$ are bimodules that define a singular equivalence of Morita type with level (in the sense of Z. Wang) between $\Lambda$ and $\Gamma$ and which also induce an equivalence between the stable categories of finitely generated Gorenstein-projective modules $\Lambda$-$\underline{\text{Gproj}}$ and $\Gamma$-$\underline{\text{Gproj}}$. We prove that if $V$ is an indecomposable object in $\Lambda$-$\underline{\text{Gproj}}$ with $\underline{\mathrm{End}}_\Lambda(V)\cong \mathbf{k}$, then $X\otimes_\Lambda V$ is an object in $\Gamma$-$\underline{\text{Gproj}}$ such that $\underline{\mathrm{End}}_\Gamma(X\otimes_\Lambda V)\cong \mathbf{k}$ and the universal deformation rings (in the sense of F.M. Bleher and the second author) $R(\Lambda,V)$ and $R(\Gamma, X\otimes_\Lambda V)$ are isomorphic. This result generalizes the one obtained by the second author assuming that $\Lambda$ and $\Gamma$ are Gorenstein $\mathbf{k}$-algebras.

We introduce two new relations involving the pentagon and the quadrilateral for the evaluation of the Penrose polynomial at $n=4$ that is proven using a new type of ribbon graph polynomial. Additionally, we extend several relations for the evaluation of the Penrose polynomial at $n=3$ to all $n$.

In this note, we present two perturbations of sequences of orthogonal polynomials related to two iterations of the Geronimus transformation, with the feature that the resulting sequences of polynomials are missing a single degree. We relate these sequences to quasi-orthogonal polynomials of order 2 and investigate properties including recurrence relations, completeness, and associated differential equations.

In this paper, we derive a new two-dimensional rod-Kadomtsev-Petviashvili (rod-KP) equation from the incompressible and irrotational three-dimensional Euler equation under the shallow water scaling. We establish the local well-posedness of the Cauchy problem in a suitable Sobolev space and derive the blow-up criterion for strong solutions via the energy method. Then, by exploring two appropriate conservation laws of the rod-KP equation, we are able to construct its global strong solution when the physical dimensionless parameter $\sigma=0$, and on the other hand produce the finite-time blow up solutions under some certain conditions when $\sigma \neq 0$. Furthermore, we present a uniqueness continuation property for the solutions. Finally, we investigate the existence of traveling-wave solutions in order to highlight the influence of weak transverse effects on wave stability, and we also exhibit the symmetry of solitary waves in the propagation direction.

In the round $N$-sphere, we calculate the Morse index and nullity of all immersed minimal 2-spheres having constant Gauss curvature. We also obtain bounds on the stability index of the associative cone in $R^7$ whose link is the Boruvka sphere in $S^6$.