Papers

In this paper, we study variational properties of the metric projection mapping onto isotone projection cones in finite-dimensional Euclidean spaces. We derive explicit formulas for both the Fréchet coderivative and the Mordukhovich coderivative of the projection operator. The analysis is based on a local description of the projection mapping via an associated generating system in a neighborhood of a reference point, which leads to computable coderivative characterizations. As an application, we compute the covering constant of the projection mapping, providing a quantitative description of its local regularity. Furthermore, we establish verifiable sufficient conditions for the Aubin property of the solution mapping associated with parametric nonlinear complementarity problems associated with isotone projection cones. The obtained results contribute to the variational analysis of metric projections in a general cone setting where orthogonality arguments are not available, and to the stability theory of complementarity systems in finite dimensions.

Adaptive beamforming is a cornerstone of array signal processing, yet its performance often collapses in the face of complex, rapidly changing interference. When interferers appear or move unpredictably, conventional estimators encounter a fundamental memory trade-off: short windows enable rapid tracking but suffer from high estimation variance, while long windows provide stable rejection but fail to adapt to shifts. This challenge is resolved by introducing the Universal Switching Beamformer (USB), which integrates competitive sequential prediction into the beamforming architecture. By employing a linear transition diagram, the USB implicitly maintains an exponentially large family of candidate covariance histories and dynamically re-weights them based on their cumulative output power. This mechanism allows the beamformer to automatically vary its effective memory length without explicit change detection or heuristic parameter tuning. A theoretical upper bound is proven on the regret relative to an omniscient oracle that selects the best piecewise-stationary covariance model in hindsight. Extensive simulations and experiments on the SwellEx-96 dataset demonstrate that the USB achieves the agility of short-window estimators and the precision of long-term integration, providing a principled solution for tracking highly non-stationary scenes.

We produce a model of ZF + DC in which there exists a discontinuous homomorphism from the real line to itself but no Hamel basis for the real line, and prove a generalization of this result in terms of internal direct sums.

In this paper, we study rational minimax approximation for continuous complex matrix-valued functions in the Frobenius norm, where all approximant entries share a common denominator. This generalizes classical scalar rational approximation, with applications in system modeling, microwave design, and nonlinear eigenvalue problems. We first prove the existence of such matrix-valued approximants on point sets dense in themselves, extending Walsh's foundational scalar result. Next, we establish characterizations of the local and global minimax approximants by deriving primal/dual matrix-valued Kolmogorov criteria and a Ruttan-type sufficient condition for global optimality. For analytic functions on a continuum, we link continuum minimax approximation to approximation on its boundary, and finite boundary samples via the maximum norm principle. We show that Ruttan's sufficient optimality condition provides a certificate under which a minimax approximant obtained from the boundary or from a discrete set of boundary nodes also solves the original continuum problem. Finally, for discrete approximation, we connect these conditions to a dual problem and the related dual-based numerical method m-d-Lawson: when the original minimax problem admits a solution, strong duality is equivalent to Ruttan's sufficient optimality condition, and, the optimality equations underlying the m-d-Lawson iteration coincide with Kolmogorov's dual criteria. These results provide a theoretical basis for certifying and computing matrix-valued rational minimax approximants.

We study stochastic fast-slow systems with noise acting exclusively on the slow variable near uniformly attracting, normally hyperbolic critical manifolds $x=X^\ast(y)$. The deviation equation contains transmitted diffusion and an Itô curvature forcing of order $O(\sigma^2\lVert D^2X^\ast\rVert)$. Using a parameter-dependent Lyapunov function and stopped martingale estimates, we obtain geometry-dependent concentration bounds under the sufficient noise restriction $\sigma\le C_0\min(\sqrt{\varepsilon},\varepsilon^{1/4}L_{\mathrm{geom}}/\sqrt{\lambda_0})$, where $\lambda_0$ is the uniform spectral gap and $L_{\mathrm{geom}}=\sqrt{\lambda_0/(\lVert D^2X^\ast\rVert\lVert h\rVert^2)}$. For fixed geometry and any fixed $p>0$, the confidence radius on $[0,T/\varepsilon]$ is $O(\varepsilon+\sigma^2+\sigma\sqrt{\log(1/\varepsilon)})$, with stopped fast-variable failure probability at most $\varepsilon^p$; slow-variable exit is controlled separately. We also establish an adiabatic error bound of order $O(\varepsilon+\sigma\sqrt{\varepsilon}+\sigma^2\lVert D^2X^\ast\rVert)$, with explicit failure probabilities. These estimates quantify the curvature dependence of fast-variable concentration and its $O(\varepsilon)$ contribution to the slow-error upper bound for fixed geometry.

This paper bridges combinatorial geometry and music theory to solve the fundamental challenge of embedding classical Western harmony into the microtonal 19-tone equal temperament (19-TET). Inspired by Roger Penrose's observations on the mathematical elegance of 19-TET, we provide the theoretical foundation for a physical 19-TET acoustic piano currently under construction. However, playing classical 12-TET music on such an instrument poses a topological problem: emvedding the classical Euler-Riemann Tonnetz into the 19-TET universe inevitably distorts structural chords, creating dissonant “wolves.” By formalizing these harmonic spaces as incidence configurations (the 12_3 and 19_3 graphs) and utilizing integer cuts in our optimization model, we exhaustively prove that exactly 32 out of 36 Neo-Riemannian harmonic connections can be preserved. We demonstrate a strict 5-fold degeneracy of this optimum: there exist exactly 5 mathematically equivalent local packings for the wolf chords. Among these, we identify a unique canonical realization in which the 14 excised vertices form a perfectly contiguous geometric void along the primary Hamiltonian cycle. We reveal that the 4 inevitably broken edges represent the exact topological scars of the historical enharmonic diesis, and we formulate the Vicentino Hypothesis regarding 16th-century microtonal composition. Finally, to make this theoretical geometry physically playable, we design a novel 19-TET split-key keyboard, formalized through a biomechanical cost function that optimizes the performer's hand span. This work provides the complete theoretical, historical, and ergonomic blueprint for the next generation of microtonal acoustic instruments.

We prove several conjectures recorded in the On-Line Encyclopedia of Integer Sequences. The conjectures considered here concern determinants and permanents of special matrices, such as Toeplitz matrices, cross matrices, Kronecker powers of matrices, and matrices whose entries are defined by powers of differences. As tools we use row and column operations, block determinant formulas, Cauchy determinants, Sylvester's determinant theorem, and $LU$-factorizations. We also obtain closed-form formulas for several related integer sequences for which no such formulas were conjectured.

We apply methods from derived algebraic geometry to supergeometry. In particular, we will define super fundamental classes, which allow us to reinterpret the $\Theta_{g,n}$-classes on the moduli of curves as the pushforward of the fundamental class of the moduli of SUSY-curves. Furthermore, we study the stacky approach to cohomology in the supergeometric setting. We introduce the classic transmutation stacks (Betti, de Rham and Dolbeault) due to Simpson, which are geometric avatars of locally constant sheaves, D-modules and Higgs bundles respectively. Moreover, we refine the notion of Betti stack to allow for $P$-constructible sheaves, and we give new proofs for results due to Penkov on D-Modules and the isomorphism between de Rham cohomology and super de Rham cohomology, which one observes to be the same theorem. To do this, we will develop the theory of derived categories on superstacks establishing, amongst others, base change and recollement theorems. This is achieved via a reduction argument from the $\mathbb{Z}$-graded commutative setting, also known as Dirac geometry which allows us to deduce these theorems, without having to give the analogous proofs in the $\mathbb{Z}_2$-graded setting compared to the $\mathbb{Z}$-graded setting.

We observe that the Kronecker product of tensors is the operation that converts the determinant polynomial into Cayley's first hyperdeterminant. We apply the Kronecker product to iterated matrix multiplication, which results in the hypercomputant, a VNP-complete and VW[1]-complete polynomial whose hardness we prove via the equivariance of the Kronecker product. The construction works over arbitrary commutative semirings and also for the tensor algebra and the exterior algebra. For the tensor algebra this gives a version of "noncommutative VNP", and for polynomials over the nonnegative real numbers this gives a version of "monotone VNP", each with the hypercomputant as the complete object. We take a parameterized complexity viewpoint and compare the noncommutative setting and the monotone setting. Using standard techniques we obtain optimal algebraic branching program width lower bounds in both settings, and these are notably not always the same. We also prove the polystability of the hypercomputant and that its isotypic components are characterized by their stabilizer.

Recently, it has been shown that the combinatorial complexity function $N_c(n)$ of a typical triangular billiard has weakly exponential growth, i.e., for almost any triangle and any $\varepsilon>0$ there is a constant $C$ such that $N_c(n)<Ce^{n^\varepsilon}$. We give the first example, an infinite family of explicitly described triangles with weakly exponential complexity growth. The simplest example in this family is a right triangle with integer side lengths. Moreover, we find one-parameter families of triangular tables with weakly exponential growth for generic parameters.

This paper proposes BPCOL+, an exact branch-and-price algorithm for the Graph Coloring Problem. The algorithm integrates enhanced pricing strategies based on Zero-Suppressed Binary Decision Diagrams (ZDDs) for solving the pricing problem associated with the maximal-stable-set-based set-covering formulation. After computing upper and lower bounds at the root node using heuristic procedures and column generation, BPCOL+ reduces the size of the ZDD by applying maximal stable set reduction techniques that exploit alternative dual vectors. Computational experiments on the 137 DIMACS benchmark instances and on 5,000 recently proposed Erdős–Rényi instances show that BPCOL+ outperforms existing exact branch-and-price algorithms and remains highly competitive with state-of-the-art SAT-based exact solvers. In particular, BPCOL+ solves 96 DIMACS instances within one hour and proves optimality for 4,641 of the 5,000 Erdős–Rényi instances.

We prove that a simple unital Jordan superalgebra of arbitrary dimension belongs to the list of known simple unital superalgebras or lies in a certain proper subvariety.

We study murmurations in the depth aspect for holomorphic cusp forms of conductor $\ell^{2a}$ and fixed weight, where $\ell$ is an odd prime. For both $\mathrm{GL}_2$ and the definite quaternion algebra ramified at $\{\infty,\ell\}$, we determine the murmuration density as $a\to\infty$ with $\ell$ fixed. The resulting density agrees with the one previously obtained for odd conductor exponents, and hence gives a uniform density for cusp forms of conductor $\ell^n$ as $n\to\infty$. We also consider the case of Maass forms of conductor $\ell^n$. Finally, we compute the murmuration density in conductor $\ell^n$ as $\ell\to\infty$ with $n\geq3$ fixed.

We study a finite-scale one-component regularity mechanism for suitable weak solutions of the three-dimensional incompressible Navier–Stokes equations. The results are organized in three layers. The first layer is unconditional. Under a fixed scale-invariant local bound Phi(1)=A(1)+E(1)+C(1)+D(1) <= M, smallness of the critical vertical-component quantity C_3(1)=int_{Q_1} |u_3|^3 dx dt yields a positive lower bound, depending only on M, for the local regularity radius at the origin. The proof converts one-component smallness into approximation by the two-and-a-half-dimensional limiting class and then into Caffarelli–Kohn–Nirenberg smallness at a smaller scale. The pressure approximation is measured in a quotient by spatially harmonic functions. This pressure topology reflects a genuine obstruction: time-dependent harmonic pressures may have bounded scale-invariant L^{3/2}-oscillation while their pointwise gradients lie beyond the control provided by the available scale-invariant quantities. The second layer is a conditional logarithmic refinement. A prepared two-shadow comparison package replaces the abstract compactness modulus by a logarithmic modulus and gives a logarithmic finite-scale decay. The third layer is a conditional relaxed-shadowing refinement. The comparison class is enlarged to smooth no-stretching horizontal flows V=(v_h,0), with the comparison pressure allowed to have partial_3 pi not equal to 0. The resulting vertical residual pairs with the small component u_3 in the relative-energy identity. Under the buffered strong-flow and localized relaxed stability inputs stated below, this gives a power-type relaxed harmonic approximation and a power-type finite-scale decay. The unconditional theorem is separated from the logarithmic and power-type assumptions; the latter two layers identify the quantitative stability mechanisms needed to upgrade the compactness modulus.

Stochastic nonlinear dynamics underlie many models in engineering and computational physics, yet accurate high-dimensional simulation remains challenging. We present a quantum algorithm for a broad class of $N$-dimensional stochastic differential equations with dissipation and quadratic drift. The algorithm applies to strongly nonlinear systems with all-to-all interactions, thereby extending the scope of previously known quantum algorithms that were limited to weak nonlinearity and sparse systems. For norm-preserving drifts, a condition satisfied by key fluid dynamics discretizations, our method approximates expectation values of low-order correlation functions with rigorous error bounds at a cost polynomial in $\log{(N)}$ and linear in the evolution time. Our main technical advance is a subroutine for simulating an auxiliary system of $N$ interacting quantum harmonic oscillators with cost polylogarithmic in $N$. Finally, we formulate turbulence models, including Navier-Stokes and damped Euler equations, within this framework, opening a route to quantum simulation of strongly nonlinear SDEs governing turbulence and nonlinear wave dynamics.

This work expands the Nash equilibrium algorithm to those systems that have a state delay within the deterministic case, a Linear-Quadratic (LQ) type performance index with generalized cross terms is used, following the constructive approach on optimal control: first, propose the strategies form, close the loop and find the Cauchy solution; secondly, express the cost function according to those expressions; and finally use the Bellman equation as restrictions to find the strategies parameters. The stability of the closed loop is demonstrated. The algorithm is applied to a thermal prototype with an ESP32 micro-controller and its performance is compared with optimal PI controls inside two commercial/industrial PID controllers REX-C100.

Reduced-precision arithmetic offers significant opportunities to improve performance, memory usage, and energy efficiency in numerical applications, provided that numerical accuracy is preserved. This work investigates automated precision tuning through customized floating-point formats with user-defined exponent and significand sizes, enabling the emulation of emerging low-precision formats and the exploration of non-standard precision configurations within a unified mixed-precision framework. The proposed methodology, implemented in the PROMISE precision autotuning tool, combines numerical validation with a systematic search to generate program variants that satisfy user-defined accuracy requirements. To address the computational cost of this exploration, a containerized benchmarking framework supports parallel execution across multiple algorithms and parameter configurations. The approach is evaluated on a suite of numerical programs, including linear solvers and applications from the Rodinia benchmark. Results show that a substantial proportion of variables can be safely reduced to lower precision while preserving accuracy, indicating that standard double precision is often over-provisioned. These findings highlight the potential of automated precision tuning to derive efficient mixed-precision configurations tailored to application-specific accuracy requirements.

The discrete-time treatment number of a graph $H$, denoted by $\tau(H)$, was introduced in arXiv:2408.0531(3) and arises from a deterministic process in which each vertex is assigned a color at each time-step. The pathwidth upper bound $\tau(H)\leq \lceil\frac{1+pw(H)}{2}\rceil$, is shown in arXiv:2408.0531(3), where $pw(H)$ denotes the pathwidth of graph $H$. Equality holds when $H$ is the complete binary tree of depth $d$ (denoted by $BT(d)$) and $1 \le d \le 6$. In this paper, we characterize the sizes of all subsets of vertices of $BT(d)$ whose boundary has $3$ or fewer vertices and use this result to prove that $\tau(BT(d))= 3$ for $8\leq d\leq 10$; in these cases, equality also holds in the pathwidth upper bound. By the hereditary property of the treatment number, all larger complete binary trees have treatment number at least $3$. In contrast, we provide an explicit construction to show that $\tau(BT(7))=2$, while the pathwidth upper bound only shows $\tau(BT(7))\le 3.$ We construct an infinite family of graphs, each with a cut-vertex, whose treatment number depends on the number of components when the cut-vertex is removed. We use a combination of pathwidth and vertex cuts to prove another upper bound on the treatment number and use this to construct an infinite family of graphs whose boundary size is limited, but whose treatment number is unlimited.

Let $F$ be a family of $n$ unit disks in $\mathbb{R}^2$ with $n$ being odd. We use $\mbox{OA}(F)$ to denote the area of the set of points that is covered by an odd number of disks. The purpose of this note is to disprove the conjecture $\mbox{OA}(F) \geq \pi$ which was suggested in the literature and to present some examples.

The central challenge in optimizing green ammonia systems is satisfying the minimum-load requirements of the Haber-Bosch (HB) process under renewable uncertainty. We develop a robust optimization framework consisting of a strategic capacity planning model and an operational flow model under solar and wind uncertainty. The strategic model is a mixed-integer optimization (MIO) problem with flexible HB operating modes, namely hot-idling and shutdowns. To address the resulting computational challenges, we propose a robust scenario-reduction framework that combines k-means clustering with robust optimization to generate adversarial renewable trajectories. For the operational model, we develop adaptive robust rolling-horizon formulations under forecast uncertainty. Computational results show that the proposed framework produces feasible capacity plans under out-of-sample simulation, whereas existing approaches based on constraint aggregation fail to satisfy HB minimum-load requirements. Adaptive policies achieve higher ammonia production than static robust policies for a given robustness level, but provide weaker protection against realizations outside the uncertainty set.

We discuss the relationship between the chain rule for the divergence operator and the renormalization property for weak solutions of the continuity equation. We construct an example of bounded divergence-free vector field on the plane, which demonstrates that in general the first property is not sufficient for the second one.

Let $X=\sum_{n\geq1}\xi_n2^{-n} $ be a random number where we model the digits $\xi_n$ as independent Bernoulli random variables with possibly non-identical parameters $p_n=\mathbb{P}(\xi_n=1)$. For any polynomial $P\in\mathbb{R}[X]$ with degree $d\geq2$, we prove almost sure absolute normality of $P(X)$ under the condition $p_n(1-p_n)\geq (\log n)^{\Gamma(d)} n^{-(d-1)/d}$ for a suitable constant $\Gamma(d)$ depending only on the degree $d$. Our analysis reveals the sharp power law $n^{-(d-1)/d}$, which is suggested by an elementary heuristics regarding carrier interactions. Our results establish a transition as we further show that the pure critical power law is insufficient, but the precise critical window remains an interesting open problem. As far as we know, this is the first sharp result on digit mixing. We complement our main results by structurally convenient summability criteria, which turns out to be sharp at least for $X^2$, and we formulate a more general conjecture for higher degrees. Our proofs rely on Fourier decay estimates which we obtain by probabilistic argument involving conditioning and non-resonancy estimates combined with a subtle triangularization argument.

In this manuscript we develop the direct and inverse scattering problem for the cubic focusing nonlinear Schrödinger equation and for initial data that are asymptotic to an elliptic travelling wave with distinct phase at $\pm \infty$. We consider the case in which the spectral bands intersect the real axis. We then show that this class of initial data has non zero intersection with the full soliton gas initial data.

This paper benchmarks sequential feedback optimization (SFO) for wind farm power maximization using a medium-fidelity dynamic flow model. We compare SFO with two well-established approaches, adjoint-based economic model predictive control (AMPC) and extremum seeking control (ESC), under a common nine-turbine layout and identical operating constraints. The comparison focuses on steady-state power production and computational efficiency, both relevant for real-time implementation. The simulation results illustrate that SFO achieves higher steady-state power while preserving real-time feasibility, AMPC provides a better transient performance at a higher online computational cost and without guarantees of convergence to the steady-state optimum, and ESC offers a computationally inexpensive model-free baseline that may converge to locally optimal solutions. These results provide a practical reference for selecting wind farm control strategies and for designing scalable, real-time optimization methods.

We investigate the relationship between the analytical properties of functions of bounded variation and the statistical behavior of hyperuniform point processes. We establish several characterization formulas for the jump part of the gradient of a bounded variation function, extending and unifying previous results by Beretti–Gennaioli and Dávila. In particular, we provide new expressions for the $L^2$-jump of the gradient using both difference quotients and Fourier transform methods. Furthermore, we connect these analytic structures to the theory of hyperuniform point processes. By analyzing the variance of linear statistics associated with bounded variation functions, we provide asymptotic estimates that depend on the specific classification of the hyperuniformity of the point process. The results show how the regularity and jump discontinuities of a function dictate the growth rate of fluctuations in point processes. Finally, we introduce an averaged quadratic BMO-type oscillation functional over translated and rotated cube partitions, similar to the one recently studied by Ambrosio et al., and prove, using results from point process, that it converges to an explicit dimensional constant times the $L^2-$jump, giving in particular a further new characterization of the perimeter of a set.

We study a canonical $G_2$-equivariant operator $h:\mathbb{O}\otimes_{\mathbb{R}}V\to \mathbb{O}\otimes_{\mathbb{R}}V$ defined using only octonion multiplication, where $V$ is the standard $7$-dimensional $G_2$-module. We first compute its ordinary real spectrum using the $G_2$-decomposition of $\mathbb{O}\otimes_{\mathbb{R}}V$. We then analyze the octonionic right-eigenvalue problem $$ h(\widehat w)=\widehat w\lambda, \qquad \lambda\in\mathbb{O}. $$ After fixing a complex slice $\mathbb{R}\oplus\mathbb{R}\widehat u\subset\mathbb{O}$, the problem becomes a real spectral problem for $H_{u,Q}=h-Q R_{\widehat u}$, whose residual symmetry is $\mathrm{SU}(3)$. The resulting $\mathrm{SU}(3)$-block decomposition yields two explicit spectral loci in each slice: a quartic curve and a circle. The equations defining these loci are independent of the slice, and the full right spectrum is obtained by allowing $\widehat u$ to vary over the unit sphere in $\operatorname{Im}\mathbb{O}$.

In this paper, we propose a second-order-in-time, structure-preserving, and mesh-robust parametric finite element method for surface diffusion and volume-preserving mean curvature flow. We first reformulate the original evolution equations into new systems in which the tangential motion is governed by a harmonic map heat flow. This heat flow maps a fixed reference surface onto the unknown evolving surface and drives points on the evolving surface to move in their tangent spaces so as to reduce the associated harmonic energy. As a result, in the discrete setting, the mesh quality can be maintained at a level comparable to that of the reference surface, unless singularities occur. The volume-preserving property is theoretically guaranteed by the careful design of the scheme, while energy dissipation is enforced through a Lagrange multiplier. We present several numerical experiments to demonstrate second-order convergence in time and the advantage of the proposed method in preserving mesh quality. The structure-preserving properties are further confirmed by the numerical results. Finally, the proposed framework can be readily extended to other geometric flows.

We show that every probability-measure-preserving equivalence relation generated by a locally-finite Borel graph with planar connected components is sofic in the sense of Elek–Lippner. In particular, every unimodular random planar graph is sofic. This removes the additional assumptions in the works of Angel–Hutchcroft–Nachmias–Ray and Timár on the soficity of unimodular random planar maps and graphs. To prove this, we investigate Borel simplicial complexes with planar components and approximate them by treeable covering spaces. To construct these coverings, we use a canonical family of tracks on planar simplicial complexes introduced by Dunwoody.

We prove the Ashbaugh–Benguria conjecture for bounded domains with Lipschitz boundary in $\mathbb R^m$. More precisely, among all bounded Lipschitz domains of fixed volume, the ball minimizes the sum of the reciprocals of the first $m$ nonzero Neumann eigenvalues. Equality is attained precisely by balls.

We consider bounded positive semi-definite Hankel operators $H$, realised as integral operators on the positive semi-axis. For each value of $E$, not necessarily in the spectrum of $H$, we analyse solutions $f$ of the eigenvalue equation $Hf=Ef$, understood as an integral equation on the semi-axis. Our analysis reveals strong analogies with properties of solutions of the one-dimensional Schrödinger equation.