Papers

We study rollout policies constructed from an approximate value function for stochastic shortest path problems with an absorbing terminal state. Our main bound weights how much the selected action exceeds the optimal one-step value by expected state visits before termination; the uniform hitting-time estimate follows as a corollary. The bound separates value-approximation and numerical-optimization errors from the error caused by replacing an expectation with a nominal disturbance, and uses approximation errors only at states that can occur after one transition from states visited by the implemented policy. A positive lower bound on nonterminal stage costs also gives conditions under which the generated policy reaches the terminal state almost surely with finite expected total cost. In the minimum expected hitting-time problem, a uniform value-approximation error smaller than one half guarantees almost-sure termination, finite expected cost, and a multiplicative hitting-time bound, and the value one half cannot be increased. A deterministic construction shows that the factor two in the uniform rollout bound is asymptotically sharp. Certainty-equivalent rollout is covered by the same analysis.

In the paper, in light of the generating function of the complete Bell polynomials and other techniques, the author presents concise and elegant proofs of three formulas for the complete Bell polynomials.

A function $F:\mathbb{F}_{2}^{n}\to \mathbb{F}_{2}^{m}$ is called $k$th-order sum-free if the sum of its values over any $k$-dimensional affine subspace of $\mathbb{F}_2^n$ is non-zero. Carlet recently introduced this notion and constructed such functions for every $2\le k\le n$. We prove that, for $2\le k\le n-2$ and $m \leq n$, the existence of a (non-degenerate) $\mathbb{F}_{2}^{m}$-valued $k$th-order sum-free function on $\mathbb{F}_{2}^{n}$ is equivalent to the existence of a codimension $m$ linear subcode of the Reed-Muller code $\mathrm{RM}(n-k,n)$ with minimum distance $3\cdot 2^{k-1}$. In particular, this yields a new family of Reed-Muller subcodes that avoid all minimum weight codewords of $\mathrm{RM}(n-k,n)$, and thus have minimum distance $3/2$ times that of $\mathrm{RM}(n-k,n)$. We also derive new necessary conditions for the existence of $k$th-order sum-free functions and present the first nontrivial lower bound on $m$. Finally, we observe that $k$th-order sum-free functions lead to a partition of the Grassmannian of all $k$-dimensional (linear) subspaces of $\mathbb{F}_2^n$ into constant-dimension subspace codes. Under the assumption that functions exist that are $k$th-order sum-free for multiple values of $k$, we obtain an improved partitioning result and a stronger upper bound on the chromatic number of the Grassmann graphs.

Given two subsets $X,Y$ of a finite group $G$, we write $\Pr(X,Y)$ for the probability that random elements $x \in X$ and $y \in Y$ commute. If $X,Y$ are subgroups, we denote by $\Pr^*(X,Y)$ the maximum real number $\epsilon$ with the property that for every pair of distinct primes $p\in\pi(X)$ and $q\in\pi(Y)$ there is a Sylow $p$-subgroup $P$ of $X$ and a Sylow $q$-subgroup $Q$ of $Y$ such that $\Pr(P,Q) \geq \epsilon$. In this paper we handle, among other things, finite groups $G$ with high probabilities $\Pr^*(T,G)$, where $T$ is either a term of the lower central series of $G$ or the generalized Fitting subgroup $F_i^*(G)$. Our main results show that the structure of such groups is similar, in some precise sense, to that of nilpotent groups.

Score matching is an alternative to maximum likelihood estimation when the normalizing constant is unknown or too costly to evaluate. However, vanilla score matching has shown to be inefficient relative to maximum likelihood estimation for multimodal distributions with well-separated modes, which are commonly encountered in practical applications. We compare a novel diffusion-based denoising score matching estimator (DDSME) to the vanilla score matching estimator (SME) in this scenario. In particular, we prove statistical guarantees for both estimators, showing that the error bound for the vanilla SME worsens when the separation between the modes increases, which can be avoided in case of the DDSME with suitable hyperparameter tuning. This provides a novel theoretical explanation for the superior behavior of diffusion-based score matching over the vanilla version. We support our theoretical findings by numerical experiments.

We present a graph-theoretic reformulation of pitch-class set theory in which each set in $\mathbb{Z}_n$ is represented as a complete weighted graph whose edge weights are interval classes. We show that this construction is invariant under the dihedral group $D_n$, and that the full interval structure is encoded by a cyclic step composition, from which all interval data are recovered via an additivity principle. This framework yields a direct correspondence between T/I equivalence and graph isomorphism, and reinterprets Z-relation as non-isomorphic graphs with identical edge-weight multisets. We extend the model to weighted clique complexes, linking higher-order homometry to simplex-weight structure, and introduce a cent-weighted formulation enabling comparisons across different equal temperaments. Finally, we define a polynomial invariant derived from antipodal step pairings for algebraic analysis of pitch class space.

We identify topological symmetric homology as the free $\mathbb{E}_\infty$-algebra on an $\mathbb{E}_1$-algebra and topological braid homology as the free $\mathbb{E}_2$-algebra on an $\mathbb{E}_1$-algebra. In this way, topological symmetric homology and topological braid homology can be regarded as variants of $1$-dimensional representation homology. In order to identify topological braid homology as the free $\mathbb{E}_2$-algebra on an $\mathbb{E}_1$-algebra, we prove that the $\mathbb{E}_2$-monoidal envelope of the associative operad can be identified with the braided crossed simplicial group. Using this, we also compute the topological braid homology of grouplike $\mathbb{E}_1$-spaces. Further, we develop computational tools for topological symmetric and braid homologies. These tools allow us to perform low-degree computations of topological symmetric homology and prove that it is not Morita invariant. We also compute the topological $\Delta \mathbf{G}$-homology of Thom spectra in general and produce explicit formulas in the case of topological symmetric and braid homologies.

We introduce a new family of invariants of real algebraic sets defined in terms of the topology of their complexifications and compute some of these invariants for spheres. This allows us to completely classify topological isomorphism classes of algebraic vector bundles over products of two spheres. We also obtain new results concerning both the existence and nonexistence of regular maps from products of spheres into spheres. Additionally, we show that the newly defined invariants provide obstructions to weak algebraic approximation of smooth submanifolds of real algebraic sets, disproving a conjecture of Kucharz and Kurdyka.

In this article, we determine the Rogosinski radii for certain subclasses of close-to-convex functions defined on open unit disc $\mathbb{D}= \{z \in \mathbb{C}: |z| < 1\}$. Furthermore, we establish improved versions of the classical Bohr inequality and the Bohr-Rogosinski inequality pertaining to these subclasses. We demonstrate that all results derived in the study are sharp.

We introduce Prox-ITEM, an optimal proximal gradient method for minimizing $f+g$, where $f$ is smooth and strongly convex, and $g$ is convex, proper, and lower semicontinuous. In the smooth case $g=0$, Prox-ITEM reduces to the information-theoretic exact method (ITEM). We prove an exact distance-to-solution bound for Prox-ITEM with the same distance-convergence rate as ITEM, and show that this rate is minimax optimal among span-based first-order methods using the same number of gradient-oracle calls for $f$ and an arbitrary number of proximal-oracle calls for $g$. We also identify the stationary limit of Prox-ITEM, denoted Prox-TMM, which gives a proximal extension of the triple momentum method (TMM) to the composite setting and achieves the corresponding TMM distance-convergence rate.

We prove that every smooth compact submanifold of $\R^n$ can be approximated up to a small isotopy by the real locus of a nonsingular complex algebraic subset of $\C^n$ defined over $\R$. This settles a version of a conjecture posed in 1952 by Nash. Moreover, we show that if the codimension of the manifold being approximated is at least two, then the approximating real algebraic sets can be chosen to all have the same predetermined biregular isomorphism type.

Flows of vector fields are an essential tool in differential geometry, with countless applications in both theory and practice. While they have been extensively studied for ordinary manifolds and supermanifolds, a treatment of flows in $\mathbb{Z}$-graded geometry is currently missing. $\mathbb{Z}$-graded manifolds constitute a natural generalization of supermanifolds, allowing functions in coordinates of arbitrary integer degree. In this paper, flows of vector fields on $\mathbb{Z}$-graded manifolds are defined, and it is proved that every vector field admits a unique maximal flow (in the odd case, under the assumption that the vector field is homological). Vector fields invariant under flows are examined. Conditions under which the flows of two vector fields commute are investigated, and the interaction between flows corresponding to related vector fields is studied. The main body of the paper is intentionally kept short, and we hope it is accessible to readers with only minimal prior knowledge of $\mathbb{Z}$-graded geometry.

In this expository article we show explicitly how to compute Gamma(p/q) in terms of Beta function values which in turn are Kontsevich-Zagier Periods.

AI-native wireless receivers based on deep learning exhibit remarkable performance under stationary channel conditions, yet their resilience to distributional shifts remains poorly characterized by conventional metrics such as bit error rate (BER). To overcome these limitations, this paper proposes a novel real-time metric, the Topological Resilience Index (TRI), grounded in persistent homology and persistence exponents. TRI quantifies the structural stability of a neural network receiver's parameter space during online adaptation to non-stationary channels. Specifically, TRI captures resilience through three complementary dimensions: (i) validation-loss resilience measuring model-channel mismatch, grounded in the topological persistence of loss-landscape sublevel sets; (ii) channel impulse response (CIR) distribution shift, tracking geometric drift of CIR vectors from the calibration reference distribution; and (iii) channel manifold topology, quantified by the spectral gap of the Gaussian kernel matrix normalized by the Olivier-Ricci curvature norm. We establish theoretical guarantees showing that TRI is bounded, monotonic under performance degradation, and Lipschitz-stable with respect to perturbations in channel distributions measured in Wasserstein distance. Simulation results for an OFDM deep-learning receiver adapting across ten ITU-R inter-environment transitions at three shift rates demonstrate that TRI provides a consistent mean warning lead of more than one OFDM symbol over gradient-norm and validation-loss baselines, whereas the gradient-norm baseline achieves zero lead in every scenario. Furthermore, the proposed TRI-guided burst re-adaptation reduces post-shift BER by 80% relative to no adaptation within 200 OFDM symbols.

For a graph $G$ on $[n]$, the $k$-cut complex $\Delta_k(G)$ has facets $[n]\setminus T$, where $T$ ranges over the $k$-subsets that induce disconnected subgraphs of $G$. Bayer, Denker, Jelić Milutinović, Sundaram, and Xue proved that $\Delta_k(P_n^2)$ is shellable for $n\ge k+3$ and conjectured a finite-difference recurrence for its top reduced Betti number along each diagonal $n-k=r$. We prove the exact formula \[ \beta(k,n)=\binom{n-1}{k-1} -\sum_{j=0}^{\min\{k-1,n-k\}}\binom{k-1}{j}(n-k-j+1)+(n-k) \] for $k\ge2$ and $n\ge k+3$. The proof rests on a complete classification of complements of cardinality at least $k$ that contain no disconnected $k$-subset: every such complement has size $k$ or $k+1$ and is, respectively, a connected $k$-subset of $P_n^2$ or an interval of $k+1$ consecutive vertices. For fixed $r\ge3$, the formula is the restriction of a polynomial in $k$ of degree exactly $r-1$. Hence its $r$th backward difference vanishes identically and its $(r-1)$st backward difference is the constant $r-2$. On the topologically defined sequence, these identities hold for $k\ge r+2$ and $k\ge r+1$, respectively. We also obtain the conjectured closed forms for $k=4,5$, the complete face enumerator, and the associated $h$-polynomial and Stanley–Reisner Hilbert series.

We introduce $p$-uniformity to characterize the scaling of density fluctuations in spatial random systems in $\mathbb R^d$, ranging from hyperfluctuation to stealthy hyperuniformity. Our central theorem establishes sufficient conditions to preserve $p$-uniformity under transport. The first condition, a finite $(d+p)$-th moment of the transport distance, allows for a Taylor expansion of the transport. The second condition controls the corresponding terms. We thus solve a previously stated open problem; indeed we extend it, since our result applies to a general $p$-uniform source in any dimension, and the source and transport may be dependent. As an application, we construct new classes of point processes that are isotropic and $p$-uniform with arbitrarily high $p$, and that can be simulated in linear time. We thus achieve three-dimensional isotropic hyperuniform samples with an unprecedented system size of $10^9$ points. We conclude with an outlook on a converse statement.

We develop the differential theory of complex spinorial forms associated with irreducible complex spinors across all dimensions and signatures. This framework enables the study of constrained parallelicity conditions for irreducible complex spinors by reformulating them as equivalent differential systems for exterior forms within a prescribed semi-algebraic body of the Kähler-Atiyah bundle. To illustrate this approach, we first apply it to the spin-c Killing spinor equation in low dimensions, refining existing results by relaxing standard assumptions of simply connectedness and completeness. Then, we proceed to apply our framework to supersymmetry conditions in supergravity, and we prove that every quasi-supersymmetric solution of Freedman's gauged supergravity belongs to an explicit four-parameter family of geodesically complete, globally hyperbolic gyratonic Brinkmann waves with spherical wave fronts. Finally, we study the quasi-supersymmetric solutions of six-dimensional minimal supergravity, defined by a system that couples a self-dual curving on a bundle gerbe to a Lorentzian metric with an irreducible chiral spinor parallel under a metric connection with totally skew-symmetric torsion given by the curvature of the aforementioned curving. Along the way, we prove that a Lorentzian six-manifold admits a skew-torsion parallel spinor with an integrable screen bundle only if it admits a foliation whose leaves are locally conformally Kähler complex surfaces.

We analyze finite-particle drifting models for one-step generative modeling. For a conservative velocity given by the difference of the kernel-smoothed data and model scores, a joint-entropy identity yields continuous-time bounds for the smoothed Fisher discrepancy and the squared particle velocity. The finite-particle correction involves reciprocal kernel density estimates. We give local-occupancy conditions and exact expectation bounds, and show that these expectations diverge for full-support initial densities in a shrinking-bandwidth regime. Keeping the bandwidth dependence of the quadrature constants explicit yields a conditional root residual-velocity rate of $N^{-1/(d+4)}$ under uniform regularity, with a corresponding rate under weaker quadrature growth conditions. We also analyze the original displacement field with the exact Laplace kernel. A companion kernel gives a weighted coercivity estimate that accommodates the unbounded empirical scale and the kernel's nondifferentiability. Localization controls the particle velocity, while relative scale alignment absorbs the mismatch into dissipation. Both analyses quantify the size of an additional drift correction under explicit regularity and stability assumptions.

We study the equivalence classes of slice-regular functions $f:\Omega\to\mathbb{H}$ on a symmetric slice domain $\Omega$, and of their subclass made of polynomial slice-regular functions, with respect to the natural action of $\mathrm{PGL}(2,\mathbb{H})$ and its subgroups, by employing the twistor construction. In particular, we characterize slice–regular functions whose twistor lift is planar and belongs to a given orbit, and we find normal classes of slice-regular polynomials with respect to the action of a parabolic subgroup of $\mathrm{GL}(2,\mathbb{H})$.

We prove a complete characterization of the extremal invariant measures for half-space geometric last-passage percolation with an arbitrary boundary parameter. This is the first result of its kind for a model in the KPZ universality class that has boundary effects and an unbounded domain. A description of a class of invariant measures was previously given in a work of Barraquand and Corwin, where it was conjectured that these should comprise all extremal invariant measures. To complete the classification, we prove a one force–one solution principle: when started in the distant past from an arbitrary initial condition with a given asymptotic slope at $\infty$, the recentered solution at time $0$ converges to a process which is distributed as the associated invariant measure with the specified slope. This limiting process is called the Busemann process, the first of its kind constructed for a half-space model. The Busemann process across all slopes is distributed as the joint invariant measure for geometric half-space LPP, recently constructed by Dauvergne and Zhang. There, it was conjectured that the constructed family of jointly invariant measures comprises all extremal jointly invariant measures; our analysis also confirms this conjecture. When the model has a strong (attractive) boundary, the collection of slopes for the invariant measures has a discontinuity, which does not arise in the full-space case. To handle this difficulty, we combine the control of the directions of semi-infinite geodesics with techniques from the theory of half-space Gibbsian line ensembles. Along the way, we classify the set of directions of semi-infinite geodesics for half-space geometric LPP, confirming a recent conjecture of Dauvergne and Zhang.

We present a unified algebraic framework utilizing the formal Bell transform to bridge the Dirichlet convolution of arithmetic functions with the combinatorial structure of infinite Euler-type products. By analyzing the logarithmic derivative of exponential generating functions, we establish explicit mappings between Bell exponents and Möbius inversions. We apply this framework to derive exact vanishing properties and congruence inheritances for classical sequences, including Ramanujan's tau function and prime-colored partitions. Furthermore, we demonstrate that the inverse Bell transform seamlessly recovers classical partition recurrences and provides a discrete combinatorial engine for generating special polynomial families, including classical Appell and Sheffer sequences.

We show that the total Cartier index of varieties with rational singularities in a bounded family is bounded. This solves a problem of Han and Jiang. The overall structure of the proof, which treats the surface case and the higher-dimensional case separately, was originated by generative AI, particularly the Rethlas system, and was substantially corrected and elaborated by hand.

The Flexible Job Shop Scheduling Problem (FJSP) is the optimal allocation of a set of jobs to machines. Two primary challenges persist in FJSP: the unpredictable arrival of future jobs and the combinatorial complexity of the problem, rendering it intractable for conventional mixed-integer linear programming solvers. This paper proposes an event-based \gls{DRL} approach to solve FJSP with random job arrivals. Specifically, we employ the Proximal Policy Optimization algorithm and use lightweight Multi-Layer Perceptrons to train the \gls{DRL} agent for minimizing the total completion time of all jobs. We design the state representation to be directly accessible from the environment, and limit the learning agent to selecting from among a set of well-established dispatching rules. Simulations show that our \gls{DRL} approach outperforms any of the individual dispatching rules on datasets with varying heterogeneity and job arrival rates. We benchmark our \gls{DRL} against an arrival-triggered mixed-integer linear programming solution and show that our method achieves good performance especially when the datasets are heterogeneous.

Every transitive cellular automaton (CA) is sensitive to initial conditions. We study this implication in the more general context of non-uniform cellular automata (NUCA) with finitely many different local update rules assigned to cells. We construct a two-dimensional NUCA that is minimal – and hence transitive – but that is not sensitive to initial conditions. The construction is based on an odometer NUCA on $\{0,1,2\}^\mathbb{N}$ which is nearly uniform in the sense that only the first cell uses a different local rule. Then we show that if the assignment of local rules in the cells is recurrent then transitivity implies sensitivity.

We study the high excursion probability of a centered Gaussian field on a square. Writing \(\sigma\) and \(r\) for its standard deviation and correlation function, we assume that \(\sigma\) has a unique maximum at the corner \(\boldsymbol{0}=(0,0)\) and \[ 1-\sigma(\boldsymbol{t}) \sim t_1^\beta+t_2^\beta+t_1^a t_2^a , \qquad \boldsymbol{t}=(t_1,t_2)\to\boldsymbol{0} \] in \(\mathbb R_+^2\). The local correlation is assumed to satisfy \[ 1-r(\boldsymbol{t},\boldsymbol{s})\sim |t_1-s_1|^\alpha+|t_2-s_2|^\alpha, \qquad 0<\alpha<\beta . \] This product form of the standard-deviation loss is not covered by the usual locally additive assumptions. In the range \(a<\beta/2\), the classical essential rectangle at the variance-loss scale no longer captures the leading contribution; the relevant localization becomes side-attached and, in one regime, effectively one-dimensional. We determine the corresponding high-level asymptotics, including the logarithmic and side-dominated regimes which do not arise in the locally additive case.

The prime number graph is the set of points $(n,p_n)$ where $p_n$ denotes the $n^{\rm th}$ prime. Let $L(n)$ be the minimum number of straight line segments needed to cover the first $n$ points in this set. Let $B(n)$ be the largest number of points $(k,p_k)$ with $k\le n$ covered by a single line. Recently Sloane conjectured that $L(n) = O(n/\log n)$. We show that $L(n)=O(n \log \log n / \log n)$ and $B(n)\ge c\log n$ for a constant $c>0$ and all large $n$. Under RH we show that for large $n$ we have $B(n)=O(n^{3/4}(\log n)^{1/2})$ and $ L(n)\ge c' n^{1/4} (\log n) ^{-1/2}$ for some constant $c'>0.$

We define the grove polynomials, a set-valued extension of forest polynomials. We show that they are $K$-theoretically dual to the quasisymmetric Schubert cells which pave the quasisymmetric flag variety, in the same way that Grothendieck polynomials are dual to Schubert cells in the complete flag variety. As a consequence, the finite truncations of the multi-fundamental quasisymmetric functions of Lam-Pylyavskyy acquire a geometric interpretation as $K$-theoretic representatives of quasisymmetric Schubert cells.

We develop a first and second fundamental theorem for $n$–tuples of bosonic and fermionic matrices, by developing graded analogues of the classical case.

This paper investigates the non-resistive compressible magnetohydrodynamic (MHD) equations in $\mathbb{R}^2$. We establish the global existence and stability of classical solutions for initial data sufficiently close to a constant equilibrium state. A distinguishing feature of our result is that global stability is derived solely from pure $H^s$ energy estimates and an intrinsic $L^2$ time-decay mechanism, thereby bypassing the traditional requirement for the initial data of $L^1$ integrability or negative-order Sobolev norm regularity. To achieve this goal, we first introduce a specific quantity motivated by the effective viscous flux, which intrinsically couples the density and magnetic field perturbations. Secondly, to overcome the critical time-decay obstacle arising from the absence of negative-order regularity, we develop a novel pseudo-negative-derivative technique. Moreover, we regard the wildest nonlinear term as a whole and bypass the need to obtain time-decay estimate for individual components. These approaches enable us to close the higher-order energy estimate entirely within standard Sobolev spaces.

Optimization over the intersection of two manifolds arises in a broad range of applications, but is hindered by the coupled geometry of the feasible region. In this paper, we prove that the regularities – clean intersection and intrinsic transversality – are equivalent, which yields a tractable projection onto the tangent space of the intersection. Therefore, we propose a geometric method that employs a retraction on only one manifold and updates the iterate along two orthogonal directions. Specifically, the iterates stay on one manifold, and the two directions are responsible for asymptotically approaching the other manifold and decreasing the objective function, respectively. Under intrinsic transversality, we derive the convergence rate for both the feasibility and optimality measures, and show that every accumulation point is first-order stationary. Numerical experiments on problems stemming from sparse and low-rank optimization, including fitting spherical data, approximating hyperbolic embeddings on real data, and computing compressed modes, demonstrate the effectiveness of the proposed method.