Papers

We consider the problem of finding lower bounds for integer linear combinations of $\sqrt{a_1},\ldots,\sqrt{a_K}$, where $a_1,\ldots,a_K$ are positive integers such that their square roots are linearly independent over the rationals. We use a probabilistic approach and prove that for $K\geq 8$ and nonzero integers $m_1,\ldots,m_K$, $$ \bigg|\sum_{n\leq K} m_n\sqrt{a_n}\bigg| > e^{\frac{2^{K-1}-1}{K} - \frac{1}{2}} \bigg(\max_{n\leq K}|m_n|\sqrt{a_n}\cdot \sqrt{K}\bigg)^{-(2^{K-1}-1)}. $$ This inequality improves the dependence on $K$ in the classical product bound.

We study the one-dimensional Zakharov-Rubenchik system with periodic initial conditions. This system models the interaction of low-frequency waves with high-frequency waves in different physical phenomena. Using semigroup theory, we prove the global well-posedness of the associated linear equation in $H^q(\mathbb{T})\times H^s(\mathbb{T})\times H^{s+1}(\mathbb{T})$, $q,s \in \mathbb{R}$. Subsequently, we consider the modified Zakharov-Rubenchik system and, by means of a change of variables and the Banach Fixed Point Theorem, we obtain local well-posedness in the space $H^s(\mathbb{T})\times H^s(\mathbb{T})\times H^{s+1}(\mathbb{T})$, $s>3/2$.

We study a generalization of the classical optimal insulation problem by replacing the standard Laplacian with the \ti{p-Laplacian}, leading to a \ti{p-Poisson equation} with Robin boundary conditions. We also investigate the corresponding eigenvalue problem. This work extends and complements some results from \cite{bu1,pietra1,pietra2} to the nonlinear setting.

Positive integer solutions to $x^2+y^2+z^2- x y z =D$ correspond to the important Markov (Markoff) numbers when $D=0$. From a given solution triple, three more are found with Vieta involutions, making an infinite tree of solutions. Starting instead with three real numbers greater than $2$ gives lengths of closed geodesics in a punctured torus. In this paper we study these tree structures for any real $D$. A continuous function, originally related to a norm on homology, encodes all the numbers on each of these trees. The properties of this function are developed here in general, with a self-contained exposition, showing that the usual $D=0$ case is part of a bigger picture. Encoding function graphs are shown to be convex for $D<4$, straight lines for $D=4$ and concave for $D>4$. Among other consequences are generalizations to all $D$ of: estimates for counting numbers on these trees, descriptions of the geometry of the corresponding lattice curves, uniqueness conditions, and identities of McShane and Hines.

In this paper we provide a positive version of Miljutin's theorem. Namely, we show that for any uncountable compact metric spaces $K$ and $L$ there exists a positive isomorphism $T:C(K)\longrightarrow C(L)$. Moreover, the inverse of $T$ is a regular operator, and $T$ may be chosen so that $\|T\|_\mathrm{r} \cdot \|T^{-1}\|_\mathrm{r} \leq 9+6\sqrt3$. The method also gives positive versions of classical nonmetrizable results for Miljutin and Dugundji compacta. In particular, if $G$ and $H$ are infinite compact groups of the same weight, then $C(G)$ and $C(H)$ are positively and regularly isomorphic with the same uniform quantitative estimate.

We study a class of distributionally robust optimization (DRO) problems for the statistical risk problem, formulated as minimax problems over the product of a Euclidean space and a Riemannian manifold. Because the resulting minimax landscape is nonconvex nonconcave in general, no globally convergent first order method is known to be available. We instead introduce the notion of a basin saddle point, a Nash equilibrium defined locally on the Cartesian product of a $\delta$ basin around a connected component of the local minima critical set and a geodesic ball on the measure manifold. We develop an abstract convergence framework for a Riemannian gradient ascent multistep descent iteration to a basin saddle point under a local Łojasiewicz type growth condition, with exponent $\beta \in (1,2]$, in the $\delta$ basin around connected components of the local minima critical sets. Under Lipschitz regularity of critical sets we establish linear convergence for $\beta = 2$ and polynomial convergence for $\beta \in (1,2)$ to a basin saddle point, with explicit dependence on the sectional curvature of the manifold. We then instantiate this framework for the statistical risk DRO problem over Gaussian measures, where the ambiguity set is naturally modeled as the product of Euclidean space and the Bures Wasserstein manifold of covariance matrices, which we relax to a penalized DRO formulation. We derive nonasymptotic Hessian estimates for the resulting Lagrangian, establish existence and local uniqueness of its maximizer, and prove that an alternating Riemannian gradient scheme converges to a basin saddle point of the penalized DRO problem, recovering the linear and polynomial rates of the abstract theory with all constants explicit in terms of data dimension, loss moments, and the reference covariance.

We study the number of limit cycles of the piecewise smooth differential system \[ \dot{x}=y, \quad \dot{y}=-x-\varepsilon\bigl(f(x)y+\operatorname{sgn}(x)g(x,y)\bigr),\] where $f(x)$ is a polynomial of degree $n\geq 1$, $g(x,y)$ is a bivariate polynomial of degree $m\geq 2$, and $\varepsilon$ is a sufficiently small parameter. Using the first-order averaging theorem for discontinuous systems, we obtain that the number $H_{m,n}=\lfloor n/2\rfloor+\lfloor m/2\rfloor$ is a lower bound for the number of limit cycles bifurcating from the linear center $\dot{x}=y, \ \dot{y}=-x$. The bound is sharp, and we provide an explicit example that attains $H_{m,n}$ limit cycles.

Let $f:\tilde{C}\to C$ be an étale cyclic cover of odd prime degree $d$ of a hyperelliptic curve of genus $g\geq 2$. We revisit the theta-duality reconstruction argument for the generic injectivity of the associated Prym map by Naranjo-Ortega-Pirola-Spelta and identify an additional residual locus arising from the fixed divisor of the line bundles occurring in that argument. For $d\ge5$, this locus does not affect reconstruction, giving injectivity under the numerical assumption $(d-1)(g-1)\geq 7$. For $d=3$, the geometry is governed instead by trigonal pencils on the quotient of $\tilde{C}$ by a lift of the hyperelliptic involution; this yields generic degree $2$ in genus $5$ and injectivity for $g\ge6$.

Hidden-free-fermion (HFF) Hamiltonians exhibit free-fermion spectra despite lacking an explicit quadratic representation, yet their algebraic provenance and periodic-boundary behavior remain open questions. Here we establish that a broad class of HFF models arises from distinct Lax operator realizations of a twisted multistate Perk–Schultz \(R\) matrix. Its reflected degeneration generates a closed finite hierarchy of mutually commuting transfer matrices, whose functional relations determine the complete many-body spectrum. As a decisive example, we solve the inhomogeneous free-fermions-in-disguise (FFD) chain, which exhibits HFF, nested HFF, or interacting spectra, depending on its coupling. The homogeneous FFD chain with both one-wrap and periodic boundary conditions exhibits an emergent cubic finite-size gap while retaining the exact \(z=3/2\) bulk behavior. The construction extends to higher-state and face-type realizations, providing a common algebraic framework for coupling-dependent spectral organization and multiscale behavior in interacting quantum systems.

We give a desingularization-free proof of Voevodsky's identification of bistable mod-$\ell$ motivic cohomology operations with the motivic Steenrod algebra over fields of characteristic zero. This supplies the direct argument anticipated by Voevodsky in his study of motivic Eilenberg–MacLane spaces. As an application, we study Thom's topological obstructions to desingularization, which arose in his work on the Steenrod realization problem. Without invoking desingularization, we prove that the primary and all higher Thom obstructions vanish on complex algebraic cycles. The proof uses the motivic bidegrees of the successive $k$-invariants in the Brown–Peterson tower to show that every lifting obstruction vanishes.

Structured feedback controllers provide rigorous stability guarantees, but often require manual parameter tuning to achieve good closed-loop performance. Policy-gradient methods offer a systematic approach to parameter optimization; however, conventional gradient evaluation requires sequential forward state rollout and backward costate propagation. This letter develops a time-parallel policy-gradient framework for discrete-time nonlinear control-affine systems. We derive the policy-gradient expression where the state and costate rollouts required for policy-gradient evaluation are formulated as residual-minimization problems and solved using Gauss-Newton (GN) iterations with parallel associative scans. For closed-loop systems that are globally asymptotically stable and locally exponentially stable, we show that the residual-minimization problems satisfy a local Polyak-Lojasiewicz (PL) inequality and that the GN iterates converge locally at a quadratic rate. Moreover, the PL constant, the size of the convergence neighborhood, and the quadratic convergence bound are independent of the rollout horizon T. We also prove that, for any finite horizon T, the state solver recovers the exact trajectory from any initialization in at most T iterations. Finally, an inertia-wheel pendulum example with interconnection and damping assignment passivity-based control (IDA-PBC) demonstrates improved closed-loop performance and the computational benefits of the proposed parallel policy-gradient framework.

Let $G$ be a standard reductive group over an algebraically closed field of positive characteristic and let $\mathfrak{g} = \operatorname{Lie}(G)$. The category of Harish-Chandra bimodules consists of $G$-equivariant $U(\mathfrak{g})$-bimodules such that the adjoint $\mathfrak{g}$-action integrates to a $G$-action. In this paper we classify the simple Harish-Chandra bimodules and determine their central characters.

We study the behaviour of a given volume of liquid confined between two rough solid plates. When the separation between the plates is small relative to the liquid volume, capillary bridges are expected to form, which minimise Gauss's capillary energy locally. We derive a $\Gamma$-expansion for the energy as the plate separation approaches zero, yielding a dimensionally reduced problem in terms of the wetted regions on the plates. At leading order, the energy is determined by the area of the wetted regions, while the next order term is given by their perimeter, weighted by appropriate functions of the relative adhesion coefficients. This provides a framework for a subsequent phase-field approximation, which is employed in numerical simulations to study the evolution of the droplets under the normal movement of the plates. The theory also gives justification to models for capillarity that assume that liquids fill up the rough topography to the Kelvin radius.

For a finite set $S$ with $\lvert S\rvert = N$, the number of families $\mathcal{B} \subseteq \mathcal{P}(S)$ that are topological bases is $\#(N) = \sum_{\mathcal{T} \in \operatorname{Top}(S)} 2^{\lvert\mathcal{T}\rvert - \lvert\mathcal{M}_{\mathcal{T}}\rvert}$, where $\mathcal{M}_{\mathcal{T}}$ is the canonical minimal basis of minimal open neighborhoods. The identity is proved in Lean 4 / Mathlib (`CARDB.lean`): bases generating $\mathcal{T}$ are exactly the sets with $\mathcal{M}_{\mathcal{T}} \subseteq \mathcal{B} \subseteq \mathcal{T}$. The small-$N$ table and the discrete-dominance sandwich are proved in `CARDB/SmallN.lean` and `CARDB/Asymptotics.lean`.

In this paper, we study the computational complexity of finding fixed points for set-valued contractions. We first formulate a computational problem for Nadler's fixed-point theorem: Projected-Nadler, and prove that it is $\mathsf{CLS}$-complete by showing its equivalence to Continuous-LocalOpt. We then establish a stronger converse for Nadler's fixed point theorem that can be applied as a tool to analyze the convergence rate of set-valued basic iteration procedure. Finally, we reduce large-margin triplet stationarity problem to Projected-Nadler. Together with its $\mathsf{CLS}$-hardness introduced in [arXiv:2509.16898], this yields $\mathsf{CLS}$-completeness of large-margin triplet stationarity.

The deformed \(q\)-exponential function \[ e_q(z,u) = \sum_{n=0}^{\infty} u^{\binom{n}{2}} \frac{z^n}{(q;q)_n} \] provides a common framework containing several classical \(q\)-exponential functions as particular cases, including the Jackson \(q\)-exponentials \(e_q(z)\) and \(E_q(z)\). In this paper we investigate the multiplicative inversion problem \[ e_q(z,u)e_q(-z,v)=1, \] and determine all pairs of deformation parameters \((u,v)\) for which this identity holds. To this end, we introduce a family of coefficient polynomials whose common zeros characterize the inversion property. A geometric analysis of the first nontrivial coefficients reduces the problem to two parameter branches. The symmetric branch is excluded through a parity phenomenon, while the nonsymmetric branch is completely determined by the first two coefficient constraints. As a consequence, we prove a rigidity theorem showing that \[ e_q(z,u)e_q(-z,v)=1 \] if and only if \[ (u,v)=(1,q) \qquad\text{or}\qquad (u,v)=(q,1). \] Thus the classical Jackson inversion identity is rigid within the deformed family and no new multiplicative inversion identities arise from the deformation parameter.

In this paper we introduce a potentially infinite process of reflections applied to a sequence of points in the dual of the Tits cone associated with an arbitrary finitely generated infinite Coxeter group. This process is infinite whenever the starting point is not in the imaginary cone of the given Coxeter group. We show that this process exhibits well-defined asymptotic behaviour. We study the resulting limits, and give a characterisation of these limits. We then apply the techniques developed from this asymptotic process to the study of infinite sequences of roots totally ordered by the dominance partial ordering.

Isotonic regression and mean estimation over a simplex are among the most classical denoising problems under geometric constraints. Chatterjee and Lafferty generalized both to the recovery of a flow on a rooted tree from Gaussian noise, with isotonic regression on a path and the simplex on a star. Their results reach only special trees; on a general tree the minimax risk remained unknown for a decade. We settle the question in full. On every tree with $n$ vertices, at every budget $V$ and noise level $\sigma$, the minimax risk is of order $\min\{V^2H_T,\ \sigma^2k_0\}$, with $H_T$ the diameter and $k_0$ the crossing index of a truncated ancestor profile. A deterministic estimator, an exponentially weighted aggregate over integer states, attains it in $O(n^2\log(n+1))$ arithmetic operations. Least squares at a known budget, the natural convex program for the model, is suboptimal in the worst case by a factor of order $(\log n)^{2/5}$. One functional, the truncated ancestor profile, carries the rate, the estimator, and the least squares comparison. It serves as covering radius, as packing entropy in the positive cone, and as the range of the one-dimensional state behind the exact computation. Two classical theories become one on trees, sharp and computable throughout.

The Russell cubic threefold is the hypersurface $\mathcal X\subset\mathbb A^4_{\mathbb C}$ defined by $x^2y+z^2+x+t^3=0$. We prove that its cylinder $\mathcal X\times\mathbb A^1$ is flexible. More precisely, after replacing its standard embedding in $\mathbb A^5$ by an equivalent one, we construct four explicit algebraic $\mathbb G_a$-actions on the ambient affine space whose restrictions generate a group acting transitively on the cylinder.

Let $R$ be an affine domain of dimension $d$ over an infinite field, with normalisation $\tilde{R}$ smooth, and let $\mathfrak{c}$ be the conductor. We answer a question of Nori in the affirmative when $R$ is singular only in high codimension.

Quantum computing offers the potential to solve computational mechanics problems with a more favourable complexity scaling than classically possible. To realise this potential, a quantum-compatible formulation of the finite element method is essential. We present such a formulation for Poisson problems with spatially varying coefficients on general domains discretised by an immersed uniform Cartesian grid. The resulting banded sparse system matrices are encoded as the sum of diagonal matrices with polynomial entries and their products with shift matrices. The sum of the matrices is formed using the linear combination of unitaries (LCU) technique. The resulting block-encoded system matrix can be implemented using a polylogarithmic number of elementary gates. For immersed boundaries, the block-encoded system matrix is further processed using a diagonal indicator matrix with zeros and ones. The indicator matrix is defined in terms of an integer-valued level-set function composed of simple geometric primitives via Boolean set operations. The required integer operations are computed on the fly using quantum arithmetic. The resulting block-encoded linear system of equations is solved using quantum singular value transformation (QSVT). The overall gate count, including the QSVT, is $O(N_{\text{tot}}^{2/d} \operatorname{polylog}(N_{\text{tot}}))$, where $d$ is the problem dimension and $N_{\text{tot}}$ the number of grid points. In numerical examples that confirm the theoretical complexity estimates, the solutions converge approximately linearly for immersed problems and quadratically for simply connected domains.

This article investigates the duality of abelian topological groups by delving into the interplay between the algebraic and topological properties of a group $G$ and those of its dual group. In particular, extending the notion of a $g$-barrelled group, we introduce and study the classes of $\kappa$-barrelled and $\omega$-barrelled groups. Our first main result establishes that every $\omega$-barrelled group is $\aleph_0$-barrelled (i.e., if every weakly convergent sequence in the dual group is equicontinuous, then every metrizable weakly compact subset of the dual group is also equicontinuous). From this, we deduce that the free topological abelian group $A(K)$ over a metrizable compact space $K$ is determined by the subgroup generated by any of its dense subsets. Furthermore, we prove that if $G$ is a $\sigma$-compact, $\omega$-barrelled group, then its dual group $\widehat{G}_\kappa$ equipped with the compact-open topology is sequentially complete (complete if $G$ is hemicompact). We also solve an open question posed by Trigos-Arrieta by constructing an explicit example of an $\omega$-barrelled metrizable group that fails to be $g$-barrelled. Finally, we investigate the dualization of groups lacking infinite compact subsets, proving that for a totally bounded abelian group $G$, every compact subset of $G$ is finite if and only if its dual group $\widehat{G}$, endowed with the finite-open topology, is unordered Baire-like.

Cutting stock problems are of large relevance to a variety of industry branches. Here, we present a linear programming formulation for a restricted version of the multistage 2D cutting stock problem with Guillotine cuts and test it on published benchmark instances. After this, we derive a non-exact reformulation in quadratic unconstrained binary optimization (QUBO) form via unbalanced penalization, which we show to have possible benefits in modeling the problem with usable leftovers. This work mainly considers the restricted formulation, in which the dimensions of each cut are determined by those of a single required piece. We further discuss how the approach can be extended to the unrestricted problem, which allows more general cutting patterns involving cuts with dimensions of multiple required pieces and naturally gives rise to quadratic terms in the inequalities of the problem formulation. We showcase solutions via a Simulated Annealing solver for the restricted problem class, where we mainly study the performance of the iterative augmented Lagrangian method. We also discuss possible applications of Quantum Annealing in this setting, as a strong motivation to research QUBO formulations.

This paper studies the non-stationary marked Hawkes processes in which the base intensity function is time-dependent and the kernel function is governed by an external random factor called mark. The marks are assumed to be determined by random events occurrence over time resulting in a non-identical distribution. These facts introduce two kinds of difficulties: one from the absence of the i.i.d. (independent and identical distributed) and the other is from the non-stationarity. In this framework, an asymptotic regime, often referred to as the high intensity regime, is considered, under which the intensity increases with time. This can be obtained by multiplying the time parameter by n alpha, for some alpha > 0. Under the high intensity regime, the functional law of large numbers and the functional central limit theorem are established. The rescaled (centered and normalized) marked Hawkes process is proved to converge in distribution to a Gaussian process, which is the accumulation of Gaussian noise and a diffusion process. In addition, a shot-noise process is studied, and the functional Limit Theorems under appropriate hypotheses are established.

Let $R$ be non-commutative ring with unity. In this paper, we prove that a given matrix in $M_n(R)$ is a sum of $p$-th powers in $M_n(R)$, where $n < p$ if and only if its trace can be written as a sum of $p$-th powers in $M_n(R)$, for $n = 2,3,4$ and $p = 5$ and $n = 4,5,6$ for $p = 7$. This paper extends the results of S. A. Katre and Kshipra Wadikar, $(p=3)$ and Garge $(n \geq p), (p = 5, 7)$, and S. A. Katre and Deepa Krishnamurthi $(n \geq p)$.

In 1974, E. M. Brown introduced the $\wp$-functor to study the proper homotopy groups of an end, suggesting a parallel proper homology theory that has since remained undeveloped. In this paper, we construct this missing proper homology and relate it to proper homotopy by developing a proper Hurewicz theorem. Bypassing abstract pro-categorical machinery, we show that $\wp$ has major advantages over the classical limits $\varprojlim$ and $\varprojlim^1$: it is exact, detects pro-triviality, satisfies a cardinality dichotomy, and connects to these classical limits through a 4-term exact sequence. As an application, we prove that the Brown–Grossman proper fundamental group of any open contractible $3$-manifold other than $\mathbb{R}^3$ is uncountable and perfect.

For a crystallographic root system ${\mathfrak D}$ we consider the simplicial complex $\Delta_{\mathfrak D}$ of all antichains in the root poset of ${\mathfrak D}$. We show that $\Delta_{\mathfrak D}$ is shellable if and only if ${\mathfrak D}$ is $A_n$, $B_n$, $D_3$ or $G_2$. Since antichains in types $A_n$ and $B_n$ can be identified with Dyck paths and symmetric Dyck paths, respectively, this yields a simplicial complex on Dyck paths. Indeed, in type $A_n$, shellability can be extended to rational Dyck paths. The $f$- and $h$-triangles then yield statistics on (symmetric/rational) Dyck paths. We determine these statistics for $A_n$ and $B_n$ and leave the case of rational Dyck paths as an open problem.

We consider the problem of objective inference in the context of receding-horizon linear-quadratic regulator (LQR). In this setting, we are given sequential state-action observations, where each observed action is the first control of a newly solved finite-horizon LQR problem. We characterize when the objective of that problem is uniquely identifiable from these observations and when additional observations provide no new information about the objective. We then analyze action prediction at unseen states and show that all objectives reproducing the observed actions yield identical actions throughout the affine hull of the observed states; outside this hull, we derive an upper bound on the prediction error. % and deriving a prediction-error bound outside this hull. Additionally, we show that, when only the linear objective terms are unknown, exact prediction holds at every state. Finally, numerical results show that, even under stochastic observation noise, re-optimizing an inferred objective enables accurate action prediction at unseen states across different planning horizons.

This paper introduces a new polynomial model for the exact variant of the Syndrome Decoding Problem (SDP) in the binary case. The model is based on elementary symmetric polynomials. We estimate the computational complexity of solving the corresponding polynomial system by establishing bounds on the degree of regularity and on the solving degree of the ideal associated to the model. The complexity estimate is lower than for previous polynomial models. We also provide a variant of the model whose complexity depends directly on the specific instance of the SDP and is lower than for the first model. Finally, we discuss how to apply our approach to solve other variants of the SDP.

We introduce defect measures for the heat flow of a class of harmonic maps with free boundary and initiate their analysis. Our flow is motivated by singular perturbations on the boundary of the domain, emphasizing structural differences between harmonic maps with free boundary and their classical counterpart. The harmonic maps we consider are a reformulation of maps introduced by Da Lio and Rivière and arise in several problems in the spectral geometry of Steklov eigenvalues. Contrary to the classical case, the flow we consider exhibits features of global nature and the blow-up analysis turns out to be much more challenging.