Papers

Spacecraft collision avoidance for low-thrust satellites often requires determining not only how to maneuver, but also how late a maneuver can begin while still ensuring safety. This paper presents a greedy time-optimal (GTO) backward-sweep method to find the latest maneuver initiation time. The method starts from the nominal time of closest approach and iteratively propagates the maneuver backward in time, selecting at each step the thrust direction that locally minimizes the chosen danger metric. Differential algebra is used to efficiently propagate state sensitivities and update the time of closest approach online. The method is tested on a large dataset of conjunctions, using both miss distance and probability of collision as safety metrics. The approach achieves accurate results and only a small loss of optimality relative to an optimal-control benchmark, while retaining runtimes suitable for on-board implementation.

Ramesh's 2023 dissertation introduces the categorical notions of introspective theories and geminal categories, which formalize "self-internalizing" structures sharing the form of Löb's theorem ($\Box A \vdash A$ implies $\vdash A$). We reorganize the theory of geminal categories in a self-contained manner by introducing "code structures on fibrations," which serve as a categorical abstraction of Gödel coding. This framework leads to a significant simplification of the proof of Löb's theorem for geminal categories, as well as to a new categorical counterpart of the Gödel-Löb axiom ($\Box(\Box A \to A) \to \Box A$). This formulation offers an accessible framework for Ramesh's approach and suggests connections to modal type theories, where similar meta- and object-level interactions arise.

We formulate a factorization hypothesis for linear orders definable by oracle polynomial-time machines. Assuming this hypothesis, we provide an approach to constructing a nonstandard model satisfying the minimization scheme for polynomial-time predicates while violating the tournament principle for graphs with polynomial-time edge relation.

We consider a semi-blind deconvolution problem arising in the decoding of blurred linear barcodes. Building on the Piecewise Constant Level Set (PCLS) framework introduced in [De\,Cezaro et al., Inv.\,Probl., 29 (2013), 015003], we propose and analyze a solution method based on augmented Lagrangians to obtain stable approximate solutions to the corresponding inverse problem with respect to noisy measurements. We establish the existence of generalized multipliers for the augmented Lagrangian functional under consideration, as well as the absence of duality gaps. These results provide the theoretical foundation required to prove regularization properties of the approximate solutions produced by the proposed strategy. Furthermore, we present an associated ADMM-type iterative scheme for the explicit computation of approximate barcodes. Numerical experiments are carried out for various variance values (responsible for the blurred effect) and several levels of noise, validating the effectiveness of the proposed method.

Let $X$ be a Banach space and $f\in\mathcal{C}^1(X)$ be bounded from below. We show that if for some $m\geq 1$, the function $x\mapsto \|\nabla f(x)\|^m$ is convex, then $f$ is convex. We also establish a more general version of this result: if $f$ is continuous and bounded from below, then it is convex, provided $x\mapsto s_f(x)^m$ is convex for some $m\geq 1$, where $s_f$ denotes the (De Giorgi) metric slope of $f$.

Motivated by the works of Gromov and LeBrun in Riemannian geometry, we study the analogous phenomena in complex geometry. We first show that both $\int_M |S_C^-(g)|^ndV_g$ and ${\rm vol}_g(M)$ (normalized by $S_C(g)\ge -1$) are bounded below by $\frac{(n\pi)^n}{n!}\mathrm{CanVol}(M)$ for any Hermitian metric $g$ on a compact complex $n-$manifold $M$. Here $S_C$ denotes the Chern scalar curvature, $S_C^-=\max\{-S_C,0\}$ and ${\rm CanVol}(M)$ is the canonical volume of $M$, i.e., the volume of the canonical line bundle $K_M$. Moreover, if ${\rm vol}_g(M)=\frac{(n\pi)^n}{n!}\mathrm{CanVol}(M)$ holds for some Kähler metric with $S_C\ge -1$, then it has to be the Kähler-Einstein metric of negative scalar curvature. The completely new phenomenon is that if $M$ is a compact Kähler manifold such that $K_M$ is nef, then ${\rm MinVol}_C(M)=\mathcal{I}_C(M)=\mathcal I_C^-(M)=\frac{(n\pi)^n}{n!}\mathrm{CanVol}(M)$, where ${\rm MinVol}_C(M)$ is the infimum of ${\rm vol}_g(M)$ with $S_C(g)\ge -1$ and $\mathcal I_C^-(M)=\inf_g \int_M |S_C^-(g)|^ndV_g$, $\mathcal I_C(M)=\inf_g \int_M |S_C(g)|^ndV_g$. It remains unknown whether the nef condition is superfluous. The answer is positive when $M$ is obtained by blowing up a finite number of points from a projective manifold with big and nef canonical line bundle. The arguments are based on the asymptotic behaviour of the Bergman kernel of $mK_M$ as $m\rightarrow \infty$, the theory of Kähler-Ricci flow and singular Kähler-Einstein metric, as well as a very delicate gluing technique, using the Burns-Simanca metric.

Gasper gave a two-step extension of the Askey-Roy formula for a beta-integral type defined on the unit circle. This involved increasing the number of parameters and adding a balancing condition. We derive a multidimensional generalization of the final formula in Gasper's extension. By considering a two-step degeneration involving parameters our generalization becomes Tarasov-Varchenko's formula – itself a multidimensional generalization of the original Askey-Roy formula. The proof is done by Aomoto's method. This generalized integration by parts strategy makes use of a class of functions known as the fundamental invariants of type $BC$.

We study non-radial Dunkl multipliers via chamber lifting. For an arbitrary finite reflection group $G$, the chamber lifting records all reflected values of a function and conjugates a multiplier into a finite matrix-valued operator on the chamber. If the dyadic matrix entries admit off-diagonal kernels satisfying the chamber $L^2$ Hörmander condition $\operatorname{CH}^2_{s,\eta}$ with $s>N_\kappa/2$, then the original multiplier is bounded on $L^p(\mathbb R^N,d\omega)$ for every $1<p<\infty$. For the product reflection group $\Sigma_N=A_1^N\simeq\mathbb Z_2^N$ this chamber condition follows from scalar Sobolev conditions on the Walsh pieces of the multiplier. The tensor product of the one-dimensional even/odd Dunkl decompositions, together with the finite Walsh transform, identifies each lifted matrix entry with a Hankel multiplier acting between parity components. Wall separation and a scale-invariant $L^2$ Sobolev condition of order $\sigma>N_\kappa/2$ therefore imply $L^p$ boundedness, for all $1<p<\infty$, for a genuinely non-radial class of symbols. The order $N_\kappa/2$ is forced already by the rank-one Bessel transform. The same chamber theorem also applies to non-product examples once the matrix kernel condition is known, including the dihedral groups $I_2(q)$ and hence $A_2\simeq I_2(3)$ and $B_2\simeq I_2(4)$. The scalar Walsh–Sobolev verification is specific to $A_1^N$. In non-product groups such as $A_2$, $A_{N-1}$, and $B_N$, the product parity calculus is absent, so a scalar theorem of the same form would require additional transform estimates.

The Laplace-Beltrami formalism, in which the Ricci tensor in the Einstein field equations (EFEs) is formulated at leading-order in terms of the partial-differential Laplace-Beltrami operator, was previously applied to coalescing compact binaries (CCBs) generating gravitational waves (GWs). Supposing that the CCB is an effective singular body – a hollow mass-shell – that follows a Kerr metric Ansatz, the EFEs were approached variationally such that the Ansatz geometric signature dictates the energetic output via $G_{\mu\nu}=8\pi GT_{\mu\nu}$. For the CCB mass-shell representation, the generated GW energy is treated as radiated surface energy via $E:=T_{00}V$. This surface energy yielded a close approximation to the cataloged GW coalescence energy, as previously shown in past comparisons. Given this success, it is logical to ask whether the Laplace-Beltrami formalism can be applied to other general relativistic systems, whether “simple" or “perturbative", beyond CCBs. This heuristic work focuses broadly on the EFEs themselves under the Laplace-Beltrami formalism, considering all differential orders up to second-order. This namely includes a deeper analysis on the variational methodology employed on the EFEs in the second-order sector, utilized in previous works, and the benchmark analysis of the lower first- and zeroth-order terms. This all-order report utilizes representative examples and select metric Ansätze to explore the formalism's practicality and its limitations; this is shown that the first-order decomposition showcases heuristically the mechanics of vector and scalar fields upon a curved spacetime.

The aim of this paper is to revisit, in Souriau's book "Structure of Dynamical Systems", the chapter devoted to the geometric quantization where the justifications of important results and formulae are not given and are difficult to prove. After recalling the coadjoint orbit method and its application to the relativistic particle with spin, we state and prove two keystone theorems that allow to equip its prequantum manifold with the symplectic and contact structures. We apply them to the relativistic particle with spin 1/2, leading to the Dirac equation and the conservation of the probability current. We propose also an identity of conservation of the spin current and, invoking the Kaluza-Klein theory, a systematic construction of the symmetries of charge conjugation, parity transformation and time reversal which seems to us more convenient and readable than that of the classical presentations.

In this paper we further develop the theory of MCS spaces. Our main result shows that MCS spaces, as defined by Perelman, are CS sets with respect to their MCS stratification, and that in fact, the intrinsic stratification agrees with the MCS stratification. As a consequence, we improve on Perelman's result and answer affirmatively a question by Fujioka.

The existence of primes in a short interval, which asks if there are prime numbers in the interval $[x, x + x^\theta]$, is a core problem in number theory. Guth and Maynard proved the best known result for this problem with an asymptotic formula while Baker, Harman and Pintz proved the best lower bound result. In this article, we focus on Piatetski-Shapiro primes in a short interval. The study of Piatetski-Shapiro primes of the form $\lfloor n^c \rfloor$ is an approximation of the well-known conjecture that there exist infinitely many primes of the form $n^2+1$. We prove the existence of such primes under restrictions on $\theta$ and $c$ with an asymptotic formula and a lower bound, respectively.

We prove an integral type Gauss-Green formula on non-collapsed RCD spaces using the strong locality of the Laplacian and an eigenfunction approximation method. As applications, we generalize Colding's monotonicity formulas and prove an asymptotic formula linking the mean curvature of a hypersurface at a given point to the volume of small balls centered at that point.

We study fitting problems, sometimes called “training problems”, where we have a finite sample consisting of inputs and outputs, and we want to know whether there is a function in a certain class that could produce these outputs, exactly or approximately, on the given inputs. We focus on the computational and descriptive complexity of fitting for logically-defined classes in common decidable structures, like the real ordered field and Presburger arithmetic, and also for broader classes defined via combinatorial or model-theoretic properties. We isolate the complexity of these fitting problems, with particular attention to cases where we can use queries in a natural query language over the sample to determine whether a sample is fittable.

Let $\Lambda=\{\lambda_n\}_{n=1}^{\infty}\subset\mathbb{N}$ with $\lambda_n$ strictly increasing and such that $\sum_{n=1}^{\infty}\lambda_n^{-1}<\infty$. We show that a Hardy subspace $H^2 (\mathbb{D}, \Lambda)$ consisting of functions with sparse Fourier spectrum $\Lambda$ coincides with a Müntz space $\overline{M^2_{\Lambda}}(\mathbb{D})$ characterized by square-summability of coefficients relative to a biorthogonal family. As consequences, we obtain a new characterization of the Hardy norm in $H^2 (\mathbb{D}, \Lambda)$ and an integral representation formula for the Fourier coefficients. The proof uses the biorthogonal representation developed in a previous work of the author.

In this paper, we extend Mazzola's model of two-voice counterpoint to three-voice first-species counterpoint. The construction combines a fiber product over a shared lower voice with a harmonic mask and a two-stage maximization defining admitted successors. For the Fuxian dichotomy, we compute the successor relation and investigate connections with the Riemann dichotomy and neo-Riemannian transformations. Among pairs of same-mode triads, the model admits the most transporter realizations exactly at the pairs that generate Mazzola's Riemann monoid, but it does not single out the dominant-tonic pair, and it admits only 12 of the 192 parsimonious neo-Riemannian realizations, largely because it excludes transitions that keep a pair of voices stationary.

Satellite coverage analysis commonly relies on intensive numerical computations to evaluate access and revisit times, resulting in high computational costs, particularly in long-term performance assessment and large-scale constellation analysis involving numerous satellites. To address this issue, this paper presents a semi-analytic method for rapid coverage analysis for a satellite using piecewise ellipse models. We introduce the epoch longitude of the ascending node (ELAN), defined as the longitude of the ascending node at the beginning of each nodal period. Since the shape of the ground track over one nodal period remains invariant, target visibility within the period is uniquely determined by ELAN. By identifying the feasible range of ELAN, the proposed method excludes invisible nodal periods, thereby substantially reducing search space. Within the feasible range, we transform the visibility-boundary relationship into a modified ELAN (MELAN) and approximate it using analytically invertible piecewise ellipse models, enabling direct evaluation of the entry and exit times without repeated time-domain visibility searches. By exploiting the periodic evolution of ELAN, the proposed method efficiently computes access and revisit times over a given time horizon. Numerical case studies over a wide range of orbital configurations and target latitudes show that the proposed method achieves sub-0.1% error in most cases while significantly reducing computational effort.

For near-field communications, it is a hardware-efficient means to form an extremely large-scale array (XL-array) by concatenating multiple modular arrays (also referred to as subarrays). In this letter, we aim to investigate the effect of time synchronization errors among transmissions of different subarrays on the beam-focusing performance. To this end, we first characterize the beam pattern function when the transmit beamforming is designed based on maximum ratio transmission (MRT) under the premise of perfect time synchronization. As this function is highly difficult for analysis, we then consider a typical case with two subarrays. Interestingly, we show that for this case, the beam-focusing effect still persists even in the presence of time synchronization errors, while the focused location is deviated from the user location with an angle offset upper-bounded by 1/M, where M denotes the number of antennas in each subarray. Subsequently, for the general case with multiple subarrays, despite analytical intractability, we numerically show that time synchronization errors give rise to an imbricated (instead of focused) beam pattern. This may significantly degrade multi-user communication performance in practice due to the reduced desired signal power and increased inter-user interference.

Bands and idylls are algebraic structures introduced recently by M. Baker, N. Bowler, T. Jin, and O. Lorscheid in the context of matroid theory. Bands generalize hyperrings and provide a new approach to geometry over the field with one element $\mathbb{F}_1$. In the first part of this paper, we develop the theory of modules over a band, establishing several of its fundamental properties. In particular, we prove that the category of modules over a band is a closed symmetric monoidal category that is both complete and cocomplete. We then apply this theory in two directions. First, we prove that the category of band schemes is equivalent to the category of schemes relative to the category of modules over a band, in the sense of B. Toën and M. Vaquié. Second, we investigate the relationship between band schemes and the affine hyperring schemes as developed by R. Procesi-Ciampi, R. Rota, and J. Jun. Although bands generalize hyperrings, we see that the corresponding scheme theories are not compatible. Finally, we prove that the category of modules over a band admits a proto-exact structure, generalizing a result of J. Jun for hypermodules over hyperrings.

Let $p$ be an odd prime and let $S$ be a set of tame primes. We denote by $G_S$ the Galois group of the maximal pro-$p$ extension of $\mathbb{Q}$ unramified outside $S$. We prove that for every finite set of tame primes $S_0$ with $|S_0|\geq 2$, there exists a set $S_1$ consisting of two tame primes such that $G_{S_0\cup S_1}$ has cohomological dimension $2$. This refines a result of Labute. More generally, we establish an analogous result for number fields not containing a primitive $p$-th root of unity, under a suitable splitting condition. Our approach answers a question of Labute, from his seminal paper on mild groups, and combines weighted Zassenhaus filtrations, graph-theoretic methods, and Koch-type presentations. As an application, we solve several cohomological Galois inverse problems with prescribed ramification and splitting. We also provide numerical examples and statistics.

We study \(\Gamma_{2^n}(K)\), the least positive number \(\gamma>0\) such that an \(n\)-dimensional convex body \(K\) can be covered by \(2^n\) translates of \(\gamma K\). For \(n\)-simplices \(\Delta_n\), we prove that \(\Gamma_{2^n}(\Delta_n)\), as a sequence in \(n\), tends to \(1/2\). For the cross-polytope \(B_1^n\), we show that \(\Gamma_{2^n}(B_1^n)\leq5/6\) holds for all \(n\geq2\), and that \(\limsup_{n\to\infty}\Gamma_{2^n}(B_1^n)\leq0.641\cdots\). Finally, we prove the existence of a constant \(\kappa_*<1\) such that \(\Gamma_{2^n}(B_p^n)\leq\kappa_*\) for all \(n\geq2\) and all \(p\in[1,\infty]\).

We study the Riemann problem for the compressible Euler equations with a stationary coupling interface across which a discontinuity in the heat flux is prescribed. This coupling gives rise to non-conservative effects and models heat addition mechanisms such as condensation-induced waves. Without imposing restrictions on sonic states, we analyze the problem in all Mach number regimes. Lax weak entropy solutions are constructed via half-Riemann problems, and we show that non-uniqueness occurs for a large class of initial data. To address this, we introduce an admissibility criterion derived from the evolutionarity criterion, and we characterize the full structure of admissible Riemann solutions. Our analysis establishes local existence of admissible Riemann solutions provided the heat flux jump is sufficiently small, while also identifying families of initial data for which admissible Riemann solutions cannot exist for any fixed, nonzero heat flux jump. Numerical experiments are included to illustrate the theoretical findings.

Let $X$ be a real Banach space and $\lambda \in[0,1]$. Motivated by orthogonal versions of the James constant, we introduce the weighted Birkhoff orthogonal James-type constant $$J_\lambda^{\perp}(X)=\sup \left\{\min \{\|\lambda x+(1-\lambda) y\|,\|\lambda x-(1-\lambda) y\|\}: x, y \in S_X, x \perp_B y\right\},$$ where \(\lambda\in[0,1]\) and $x \perp_B y$ stands for Birkhoff orthogonality. We establish its basic bounds, stability properties, and reduction principles, and clarify its relations with the orthogonal James constant $J_{\perp}(X)$. The 2 -Lipschitz continuity of $J_\lambda^{\perp}(X)$ with respect to $\lambda$ is proved. New characterizations of uniformly nonsquare spaces are obtained; in particular, $J_\lambda^{\perp}(X)=1$ for some $\lambda \in(0,1)$ if and only if $X$ is not uniformly nonsquare. We also discuss connections with strict convexity, uniform convexity, modulus of smoothness, and the von Neumann-Jordan constant.

In this paper, we study some geometric constants in complete $p$-normed spaces with $0 < p \leq 1$. We introduce a new symmetric geometric constant associated with isosceles orthogonality, establish its sharp bounds, and provide an orthogonal characterization of the generalized von Neumann-Jordan constant in such spaces. We also investigate two Milman-type moduli in complete $p$-normed spaces, including their fundamental properties and sharp product inequalities. Finally, we extend the relation between the James constant and the generalized von Neumann-Jordan constant .

This paper investigates the infinite-horizon classical stochastic optimal control problem with input delay under an entropy-regularized relaxed control framework. In particular, by constructing a relaxed system and introducing an entropy regularization term, we reformulate the classical optimal control problem into an entropy regularized formulation, and derive the optimal controller that follows a Gaussian distribution. Furthermore, we show that the optimal Gaussian control distribution converges to the optimal Dirac measure as the exploration weight tends to zero. Numerical simulation is provided to validate the effectiveness of the proposed method.

Bilevel optimization is a fundamental framework for hierarchical decision-making, but its solution is challenging due to the implicit and typically set-valued nature of the lower-level optimality condition. In this paper, we study bilevel optimization problems through an exact-penalty reformulation based on the $\ell_1$-norm of the lower-level gradient. Under suitable regularity assumptions, we show that this penalty defines a distance-bound function and yields an exact penalty property for sufficiently large penalty parameters. To solve the resulting nonsmooth penalized problem, we propose an exact-penalty prox-linear (EPPL) method and establish a stationarity-oriented convergence guarantee. We further specialize the method to the simple bilevel setting, where the subproblem admits an explicit dual reformulation as a box-constrained quadratic program. This structure leads to a dual spectral projected gradient method with closed-form primal recovery, for which convergence of the inner dual iterates is proved. Numerical experiments on a minimum-norm least-squares bilevel model show that the proposed method is effective in reducing both the lower-level and upper-level gaps to high accuracy. Compared with several existing methods, the proposed approach attains the best final solution accuracy on the tested instance.

This paper uses the Baire category theorem, the Riemannian manifold structure of Grassmannians, and the Khintchine Groshev theorem in Diophantine analysis, to relate properties of lattice subsets to those of associated subset of a Grassmannian.

Let $n\geq 0$ be an integer and $q$ a prime power. The polynomial $g_{n,q}$ was introduced in [10] with the purpose of finding new classes of permutation polynomials over finite fields. We investigate the permutation behaviour of the polynomial $g_{n,q}(X)$ over finite fields of even characteristic. We introduce the multivariate case of the polynomial $g_{n,q}$, and study the permutation polynomials in several variables and local permutation polynomials resulting from the polynomials $g_{n,q}(X_1,X_2,\ldots , X_k)$. We also present several new identities of $g_{n,q}(X)$, and present some open questions on the permutation property of $g_{n,q}(X)$.

We utilize multi-virtual knot theory where there are a multiplicity of virtual crossings to study strict virtual linkoids. In strict virtual linkoid theory, local moves define all virtual moves and Reidemeister moves. In the strict equivalence, no moves, classical or virtual, can transfer an arc across a linkoid endpoint. By taking closures of strict virtual linkoids that are multi-virtual knots and links, we obtain new invariants for strict virtual linkoids. Generalized bracket polynomial invariants and generalized loop bracket polynomial invariants (for planar strict virtual linkoids) are studied in this context. The paper defines virtual polar links where there are degree two nodes in virtual link diagrams across which isotopies are forbidden. The paper shows how multi-virtual theory and its concepts can be applied to obtain invariants for polar virtual links.

A spin-orbit Hamiltonian with an effective gauge structure carries two distinct loop objects that are routinely conflated: an energy-independent Wilson holonomy, which organizes interference and internal spin transport, and an energy-dependent monodromy, which quantizes the spectrum. We show that cleanly separating these objects supplies a precise, computable bridge between the loop/holonomy representation of gauge theories and condensed-matter spin-orbit transport. The construction maps a spin-orbit Hamiltonian to an effective $U(1)$ plus internal non-Abelian connection, reduces it to a first-order transport problem, and reads physical predictions from holonomy, monodromy, curvature, and eigenphase data. Two rings make the separation explicit. For a Dirac (graphene) ring with Rashba coupling and Aharonov-Bohm flux, the total holonomy factorizes exactly into a commuting $U(1)$ flux phase times an internal spin/pseudospin holonomy, and the spectrum follows from a holonomy-eigenvalue condition. For a Rashba-Dresselhaus ring, the internal $SU(2)$ transport is genuinely non-Abelian away from the $\alpha=\pm\beta$ pure-gauge locus, where curvature controls path ordering; spectral quantization then requires an explicit first-order reduction obtained by phase-space doubling of the second-order Schrödinger problem. A non-Abelian Stokes formulation and Magnus expansion serve as ordering diagnostics rather than spectral tools. Spin-network ideas enter only as historical geometric motivation, not as a dynamical import into spintronics.