Papers

We propose the Coding Conception of ordinals and sets, which takes Cantor's three generating principles as its sole foundation. Bounded sets of ordinals are generated synchronously with the ordinals themselves through a bijective encoding function that, at each stage, selects only the finitely many bounded sets actually required by the successor, limit, and restriction principles. This selective coding yields the first-order theory $SC^{reg}$, which we establish is the metamathematically correct theory of the ordinals: it is bi-interpretable with $ZFGC^+$, yet makes no claim about the general concept of set. Extending the conception to full set theory via a monadic second-order ordinal theory with arithmetic and class comprehension produces two mutually inconsistent first-order set theories according to distinct maximality intuitions: a Type-A universe $MC_A$, in which the power set of every ordinal is a set and the universe satisfies $ZFC$; and a Type-B universe $MC_B^+$, in which sets are strictly more than ordinals and a “largeness cardinal” exists, beyond which power sets remain unencodable. We prove that this Power Set Dichotomy is unavoidable, even under potentialism, and conclude that $ZFC^-+$“every cardinal has a successor” is the only philosophically uncontroversial common fragment of any true set theory; the status of the full power-set axiom remains the sole open philosophical choice point.

We study a densely connected system of excitatory integrate-and-fire neurons which are subject to common noise. The system incorporates a gradual transmission of action potentials and captures the re- and hyperpolarization phases through a random refractory period followed by a reset to a randomized level below the rest potential. As the number of neurons tends to infinity, we show that there is weak convergence to a unique membrane potential density governed by a stochastic Fokker–Planck equation with a well-defined spike transmission rate. The latter is driven by the mean cumulative spike count, which is non-differentiable but shown to satisfy a generalized flux condition. We obtain the uniqueness of the Fokker–Planck equation from energy estimates in the dual of the first Sobolev space. Finally, we give a conditional McKean–Vlasov representation of the membrane potential density as the law of a representative neuron given the common noise.

We refer to the variety $\operatorname{Susp}(X, f, k_1, \dots, k_m) = \mathbb{V}(y_1^{k_1} \dots y_m^{k_m} - f(x)) \subset X \times \mathbb{A}^m$ as an $m$-suspension over affine variety $X$, constructed via a local slice $f(x) \in \mathbb{K}[X]$, if there exists a locally nilpotent derivation $\delta$ on $X$ such that $\delta (f) \neq 0, \delta^2 (f) = 0$. In this paper, we determine the sufficient conditions under which such a variety is generically flexible and those under which it is flexible. Furthermore, for a flexible $X$ we propose a construction of a local slice $f$ that guarantees the flexibility of the suspension $\operatorname{Susp}(X, f, 1, k_2, \dots, k_m)$.

Lecture notes for a course on convex optimization taught by Andreas Habring at Graz University of Technology in 2026. The notes cover mathematical preliminaries, fundamental results about existence and of solutions for minimization problems, projected subgradient descent, proximal-gradient methods, heavy ball gradient descent, Nesterov accelerated gradient descent and FISTA, primal-dual methods, and a short intro to optimal transport.

We propose an algorithm and its implementation for trajectory planning and certification for 3-DOF robot manipulators. The method uses Real Quantifier Elimination (QE) based on Comprehensive Gröbner Systems (CGS), also known as the CGS-QE method. The main advantage of the proposed method is its efficiency in trajectory planning and solution certification. This efficiency comes from the effective use of the CGS. First, for trajectory planning, we solve the inverse kinematics problem at each point along the trajectory via Gröbner basis computation. This usually requires recalculating the Gröbner basis at every point, which is time-consuming. We avoid this by computing the CGS for a parametric system. Here, the end-effector coordinates are parameters. This approach streamlines the algorithm. Second, for solution certification, the CGS-QE method certifies that an inverse kinematics solution exists at any point along the end-effector's trajectory. Our method also certifies solutions for trajectories composed of line segments and cubic natural splines. The algorithm is implemented within the computer algebra system Risa/Asir.

Write $(3/2)^n = m_n + \varepsilon_n$ with $m_n$ the nearest integer and $\varepsilon_n\in[-\tfrac12,\tfrac12)$, and let $T=(t_n)$, $t_n=2m_{n+1}-3m_n$, be the resulting steering word: the step-by-step record of the map $x\mapsto\tfrac32 x$ on the orbit of 1, coded by nearest-integer rounding. We prove that the subword complexity $p_T(k)$ of $T$ is superlinear, $p_T(k)/k\to\infty$. A pigeonhole argument reduces this to exponential pair repulsion of the orbit modulo one, which we establish by two applications of the $p$-adic Subspace Theorem: once in three variables, which handles all gaps between the two orbit points at once, and once in two variables (Ridout's theorem) for the single degenerate family of constant gap. The argument is completely formalized in Lean~4.

We introduce the chain replacement of a poset flow: it is obtained by considering the simplicial nerves of the posets of strictly increasing chains in the given poset, ordered by refinement. It maps posets to q-cofibrant flows and inclusions of posets to q-cofibrations. Using the combinatorial properties of the chain replacement, we prove that pushouts along the chain replacement of an order-reflecting inclusion of posets induce trivial h-cofibrations on spaces of execution paths between states of the original flow. By introducing the Hurewicz model structure on flows (or H-model structure), we deduce the same property for any q-cofibrant replacement of an order-reflecting inclusion of posets.

In this paper, we introduce and study the class $\mathcal{W}_{\mathcal{H}_n^0}(\alpha)$ of normalized pluriharmonic mappings, characterized by a suitable bound on their second-order partial derivatives. We establish a one-to-one correspondence between this pluriharmonic class and an associated class of holomorphic functions, thereby extending a result of Ghosh and Vasudevarao \cite{Ghosh-Allu-2019} to the setting of several complex variables. Furthermore, we obtain sharp coefficient bounds, growth estimates and a convex combination theorem for functions in $\mathcal{W}_{\mathcal{H}_n^0}(\alpha)$. Finally, we introduce sections (partial sums) of pluriharmonic mappings and investigate their properties for functions belonging to $\mathcal{W}_{\mathcal{H}_n^0}(\alpha)$.

In 1992, Billera and Sturmfels introduced coherent monotone paths on polytopes as part of their description of the fiber polytope construction, and later in 1994 showed with Kapranov that these coherent monotone paths capture the topology of the space of all monotone paths, paths from a minimum to a maximum, in the directed graph of a polytope with orientation induced by a linear function. Those results motivate the following analog of the polynomial Hirsch conjecture: Does there always exist a coherent monotone path of polynomial length on a polytope for any choice of orientation induced by a linear function? We show this is not the case by exhibiting a family of polytopes and corresponding linear functions for which every coherent monotone path is exponentially long. As applications, we strengthen longstanding results pertaining to lower bounds for the shadow simplex method, geometric transversals in discrete geometry, and parametric linear optimization.

We study the policy convergence of unregularized policy mirror descent (PMD) with arbitrary constant step sizes for finite discounted Markov decision processes. We focus on decomposable mirror maps of the form $h(p)=\sum_a \psi(p(a))$, where $\psi$ satisfies standard Legendre-type assumptions. Under these conditions, we prove that the policy sequence generated by PMD converges in the policy domain to a limiting optimal policy, even when the optimal policy set is not a singleton. This result covers a broad class of commonly used mirror maps, including the squared Euclidean mirror map underlying projected Q-ascent, the negative Shannon entropy underlying softmax natural policy gradient, Tsallis entropy, the Hellinger mapping, and the Fermi-Dirac entropy. Although policy convergence has been established previously for specific PMD instances or for regularized variants such as homotopic PMD, to the best of our knowledge, this is the first systematic and unified policy convergence theory for unregularized PMD under general decomposable mirror maps and arbitrary constant step sizes. Our analysis further reveals that the convergence behavior is governed by the differentiability of $\psi$ at $0$ and $1$, leading to different behaviors, including finite-time termination, asymptotic convergence, and an MDP-dependent dichotomy. When $\psi$ is twice continuously differentiable with strictly positive finite curvature, we further establish local policy convergence rates for the asymptotic convergence cases, covering the standard mirror maps mentioned above.

We study expansions of quasiminimal pregeometry structures with a dense codense unary predicate and their relation with the complexity properties of the pregeometry of the underlying structure. We consider beautiful pairs as well as $H$-structures. We show each of these expansions can be axiomatized with a single $L_{\omega_1 \omega}(Q)$-sentence and that both expansions are $\omega$-stable. For $H$-structures we provide a natural notion of independence in the expansion and when the underlying structure is modular, we also provide a natural notion of independence for beautiful pairs. Then we relate the complexity of the pregeometry to properties of the expansions.

We construct homogeneous solutions with non-Lipschitz Hessian for finite, constant-coefficient Bellman equations. First, for every $\sigma\in(0,1)$, we find two uniformly elliptic matrices $A_1,A_2\in\mathcal{S}^4$ and a nonzero $(2+\sigma)$-homogeneous solution $u$ of \[\max\bigl\{{\rm tr}\,(A_1D^2u),{\rm tr}\,(A_2D^2u)\bigr\}=0 \qquad\text{in }\mathbb{R}^4.\] Second, in $\mathbb{R}^2$ we construct three matrices satisfying ${\rm Id}_2\leq A_j\leq3{\rm Id}_2$ for which the corresponding Bellman equation admits a homogeneous solution with a non-Lipschitz Hessian. In particular, solutions to convex fully nonlinear uniformly elliptic equations are not in $C^{2,1}$, and not even in $C^{2, 1-\varepsilon}$ for $\varepsilon > 0$ small.

We develop a non-nested Bethe ansatz description of rational $\mathfrak{gl}_\ell$ spin chains in the vector representation. Starting from the quantum spectral curve and the separation-of-variables framework, we derive closed systems of Bethe equations involving only the momentum-carrying Bethe roots. The construction is worked out explicitly for the $\mathfrak{gl}_3$ and $\mathfrak{gl}_4$ spin chains and then generalized to arbitrary rank. A central result of this work is the identification of a recursive hierarchy associated with the fundamental transfer matrices. The hierarchy is generated by regularity conditions of the lower transfer matrices and closes through a universal rank-$\ell$ equation $\mathcal{R}_{\ell}=0$. This equation replaces the final level of the conventional nested Bethe ansatz and eliminates all auxiliary Bethe roots. Consequently, the complete spectral data of an eigenstate are encoded solely in the first Baxter polynomial $Q_{1}(u)$. We further obtain explicit expressions for the eigenvalues of all fundamental transfer matrices in terms of the momentum-carrying roots alone. The resulting formulation provides a compact characterization of the spectrum of rational $\mathfrak{gl}_\ell$ spin chains and reveals a direct connection between the quantum spectral curve, transfer-matrix fusion relations, and a truncated $Q$-system underlying the non-nested description. Finally, we investigate the quasi-classical (Gaudin) limit of the non-nested Bethe equations. For the $\mathfrak{gl}_3$ spin chain, we show that the leading non-trivial contribution gives rise to Gaudin equations whose pole-free form naturally defines a scalar third-order $\mathfrak{gl}_3$ oper.

We construct a $C^1$-diffeomorphism of the circle having an invariant Cantor minimal set such that, when the lengths of the connected components of its complement are arranged in decreasing order, the ratios of consecutive lengths do not converge to 1. This gives a negative answer to a longstanding question posed by Dusa McDuff.

We derive a formula for the sectional curvature on an arbitrary two-dimensional or three-dimensional smooth manifold equipped with a Lorentzian metric.

In this paper, we study the Gibbs measures on Euclidean spaces associated to the focusing nonlinear Schrödinger equation with harmonic potential and critical non linearity whose coupling constant tends to 0, a question initially posed by Brydges-Slade (1996) for the $\Phi^4_2$-model on $\mathbb{T}^2$. In dimension one and in the higher dimensional cases (with radial assumption), we establish a critical threshold below which the frequency-truncated measures converge to the base Gaussian measure (possibly with a renormalized $L^2$ cut-off) while, in the supercritical regime, we prove non-convergence of the frequency-truncated measures, even up to a subsequence.

We give upper and lower bounds for the number of solutions of the equation $e_n(x,y) = g$ in the group $W_k=(C_p\wr C_{p^k})^2$, where $e_n(x,y)$ is the $n$-th Engel word and $g\in W_k$. We obtain several corollaries from this. First, we prove a stronger version of the Amit-Ashurst conjecture for Engel words in $W_k$. We also prove that Engel words are not probabilistic identities in profinite groups with arbitrarily large wreath product quotients $W_k$. To conclude, we construct closed subsets of $(C_p\wr\Z_p)^2$ with positive Haar measure, empty-interior, and which are the preimage of an Engel word map.

Let $p\in[1,\infty)$, and let $X=(X,d,\mu)$ be a metric measure space such that $\mu$ is uniformly locally doubling and $X$ supports a local weak $(1,p)$-Poincaré inequality. Given $\theta\in(0,p)$ and an Ahlfors–David codimension-$\theta$ regular set $E\subset X$, the trace-space of the Besov space $B^{\theta/p}_{p,1}(X)$ to $E$ can be identified with $L_p(E,\mathcal{H}_{\theta}\lfloor_E)$. If there exists a measurable set $A\subset E$ with $0<\mathcal H_\theta(A)<\infty$ such that $\mathcal H_\theta\lfloor_A$ is nonatomic, we prove that there is no bounded linear extension operator $\operatorname{Ext}:L_p(E,\mathcal{H}_{\theta}\lfloor_E) \to B^{\theta/p}_{p,1}(X)$.

This paper introduces numerical optimizations for maximizing throughput on GPU when solving large batches (10,000 to over 100,000) of sequential quadratic programming (SQP) iterations, where all problems have the same structure. The optimizations are implemented in a toolbox WarpMPC for model-predictive control (MPC) in JAX and Warp. Based on the insight that all MPC problem instances in a batch share the same sparsity in time, cost, and constraints, we propose unrolling sparse linear factorizations and solves, which dominate alternating direction method of multipliers (ADMM) solver runtime. We avoid memory access bottlenecks and wasting computations via optimized memory layout, padding-reducing segmentation of the unrolled factorization, and dependency level scheduled backsolves, additionally accelerating sensitivity computation. We achieve throughputs of 8,000 to 250,000 SQP iterations per second on nonlinear cartpole, quadrotor, and humanoid robot benchmarks, outperforming baselines by 3$\times$ to 25$\times$. We illustrate practical usefulness by synthesizing a dataset and training a neural network approximation of an MPC in under 4 minutes that stabilizes a nano quadrotor in hardware experiments.

Block encoding is the standard technique for accessing matrix data in quantum linear-algebra algorithms. Its implementation directly affects its subnormalization, which in turn controls the algorithm's success probability, simulation time, and downstream costs. Explicit construction of block encodings with provably optimal subnormalization exists for only a handful of operators, with bespoke calculations used in its design. In this work, we develop a framework to block encode translation-invariant finite-difference operators on a periodic grid. These operators are the finite-difference discretizations of the constant-coefficient partial differential equations that sit at the core of scientific computing. We show that the moment order of these stencils can be used to simultaneously determine the continuum operator approximated, the vanishing order of the Fourier symbol, and the cost of the block encoding. From there, we derive a closed-form optimality criterion as a function of the stencil coefficients, which certifies whether the construction attains the optimal subnormalization for an entire operator family, uniformly in grid size, and quantifies the gap when it does not. The framework subsumes optimal constructions for the Laplacian operator in the literature and can be used to certify new instances at higher even orders, including the biharmonic operator. Furthermore, we derive success-probability floors parameterized by spectral properties of the operator's symbol and find explicit constants for the block encoding of the advection-diffusion family for which no prior explicit spatial block encoding exists.

Character-polynomial codes are constructed by evaluating finite field polynomials and mapping the results to complex roots of unity through additive characters. This paper shows that, over extension fields, the original polynomial family may contain redundancies: distinct polynomials can generate the same codeword. We identify the source of this non-injectivity through the trace map and cyclotomic cosets, determine the exact code cardinality, and construct a refined polynomial family that parametrizes the code without redundancy. These results give corrected parameters for CP codes and clarify their algebraic structure.

Two-stage stochastic mixed-integer programs are important tools for decision-making under uncertainty. Representing the uncertainty with many scenarios, however, can make them challenging to solve. Scenario reduction addresses this by finding a distribution supported on fewer scenarios that still yields similar optimal first-stage decisions. In this paper, we revisit the classical scenario reduction theory based on distances between probability distributions and the optimal mass transportation problem. The transportation problem's cost function captures scenario similarity and is central to the effectiveness of scenario reduction. We then review and compare various transportation cost functions from the literature and propose a new one. Using the Forward Selection Algorithm, we prove that our proposed cost function selects the best possible scenario from a given sample on the first draw with respect to the relative approximation error. To reduce the computational cost of evaluating this cost function, we further propose a hybrid algorithm with a scenario pre-selection phase. We assess solution quality and computational complexity on the two-stage stochastic unit commitment problem for small 24-bus and large 300-bus case studies. With only around five scenarios, the proposed cost function approximates the full-distribution optimum to within roughly 2.1% and 0.4% error for the small and large cases, respectively. In contrast, prevalent cost functions often need 25 scenarios or more to achieve that solution quality. The hybrid algorithm achieves similar solution quality while reducing wall-clock time by a factor of 18 and work (per Gurobi solver) by a factor of 66 on the large case study.

We consider a left-invariant Lorentzian problem on SU(2). The following questions are addressed: complete controllability, existence of length maximizers, extremals, curvature.

Due to the nonconvex binary constraints of unit commitment (UC), no uniform linear pricing scheme supports the optimal dispatch. Convex hull pricing (CHP) and copositive duality pricing (CDP) both address this problem. CHP derives the price from the subgradient of the value function of the convex hull relaxation of UC. CDP refers to several different pricing mechanisms that can be constructed from the dual multipliers of the completely positive programming reformulation. In this work, we define a centralized convex hull price over the joint feasible set of UC and prove that, under non-degeneracy, it coincides with the marginal copositive duality price. Numerical experiments on the Scarf example validate this equivalence and quantify the pricing gap introduced by the semidefinite restriction.

Accurate, spatially resolved flow field measurements are essential for the reliable assessment of hemodynamic quantities in cardiovascular research and clinical practice. Experimental techniques, such as 4D flow MRI, PIV, or Doppler ultrasound, often yield data that are sparse, noisy, or under-resolved, particularly near vessel walls and in regions of complex flow. This limits the fidelity of distributed or derived hemodynamic indicators such as the wall shear stress and the clinical utility of such measurements. To address these challenges, we propose a physics-informed neural network (PINN) framework that integrates the incompressible Navier-Stokes equations with velocity measurements coming from experimental flow field data. By embedding physical laws into data, PINN enhances the reconstruction of velocity fields, enables the estimation of unmeasured quantities such as pressure and wall shear stress, and improves the spatial resolution of hemodynamic indicators. We show the effectiveness of our approach using both in silico and experimental data. First, we apply our method to the FDA nozzle benchmark, leveraging both control particle image velocimetry (PIV) measurements and computational fluid dynamics (CFD) simulations. Next, we apply our method to the more complex case of blood flow in an aneurysm model, exploiting in vitro 4D flow MRI data. In both cases, the synergy between data-driven learning and physics-based regularization yields results that align more closely with ground truth observations than standard CFD or pure data-driven approaches. Our findings highlight the potential of PINNs to improve the fidelity of under-resolved flow field measurements and yield spatially resolved hemodynamic indicators.

We propose a new strategy for gradient-based unconstrained optimization, involving two parallel sequences of iterates that cooperate to determine their stepsizes via a Twin-Step principle. Rather than minimizing the objective function individually, the algorithm selects steplengths that minimize the Euclidean distance between the two gradient based processes occurring simultaneously at each iteration. The theoretical analysis shows that the convergence of the mutual distance is governed by the angle between the search directions. In particular the effectiveness of the overall process degrades as the directions approach parallelism. To ensure robustness against collinearity, we introduce a hybrid framework, Twin-ABB$_{\min}$, which switches to the Adaptive Barzilai–Borwein method when the geometric cooperation becomes ineffective. Extensive and very promising numerical results evidence that the Twin phase creates favorable initial conditions for subsequent BB-type iterations.

In this paper, we study obstructions to smoothing nodal orbicurves with orbighosts mapped into cyclic quotient singularities. As an application, we show, under some assumptions, that the orbifold Hecke algebra of a complex global quotient orbifold $[X/G]$ is isomorphic to the degree $0$ cohomology of the $A_\infty$-algebra of endomorphisms of a regular cotangent fiber $T_{[x]}^*[X/G]$ regarded as an object of the bulk-deformed wrapped Fukaya category of the orbifold $T^*[X/G]$ for a compact complex manifold $X$ and finite group $G \subset \operatorname{Aut}(X)$.

In this paper, we propose the Algebraic Invariant Quadratization (AIQ) framework for rational-like energy functions by introducing auxiliary variables, which are interpreted as Casimir functions of the extended system. Combining AIQ with symplectic Runge–Kutta (SRK) methods in time and Fourier pseudo-spectral discretization in space, we obtain fully discrete schemes. The resulting schemes are applied to Cahn–Hilliard equations in both the isotropic and anisotropic cases. We analyze the discrete dispersion relation, spinodal instability, coarsening behavior, and missing-orientation phenomena. Numerical comparisons demonstrate the improved performance superiority of the proposed method over the stabilized invariant energy quadratization (S-IEQ) and scalar auxiliary variable (SAV) methods in preserving the original energy evolution and capturing the underlying physical phenomena.

For normalized sequences $\left(g(n)\right)_{n\in\mathbb{N}}$ we consider recursively defined polynomials $P_n^g(x)$. In this paper we study their submultiplicative property, viewed as a Bessenrodt–Ono type inequality for the partition function, and provide an effective criterion for establishing it.

We provide a combinatorial characterization for pairs of order-theoretic trees with homeomorphic ray spaces, answering an open problem proposed by Kurkofka ad Pitz. This solution is inspired by the introduction of a transfinite topological game, which allows us to characterize not only ray spaces through the existence of winning strategies for one of the players, but also their homeomorphic classes. As applications of these results, we obtain a new topological characterization for graph-theoretic end spaces (thus obtaining yet another solution to a recently solved problem of Diestel), as well as for edge-end spaces and completely ultrametrizable spaces. We also introduce a generalization of the class of ray spaces (which is strict, as witnessed by the Sorgenfrey line). Furthermore, we establish that, for subspaces with cardinality less than continuum of end spaces, the scattered property is equivalent to the property of being, itself, an end space. At last, we determine that ray spaces in a couple of classes fail to have their product with any non-discrete space as a ray space.