Papers

Deriving analytic solutions for optimal mixed strategies in zero-sum linear-quadratic differential games (ZSLQDGs) remains an open problem. In this paper, we analytically synthesize near-optimal mixed strategies for ZSLQDGs and establish rigorous performance certifications. Specifically, we construct a surrogate pure-strategy stochastic differential game (SDG) by matching the first two moments of the mixed strategies. This method achieves an $\mathcal{O}(\bar{\pi}^2)$ weak approximation of state distributions and expected costs with respect to the maximum commitment delay $\bar{\pi}$. By analytically resolving the surrogate SDG, we derive closed-form optimal control laws for the matched moments. Crucially, we reveal that the surrogate game is governed by a Generalized Riccati Differential Equation (GRDE), which explicitly dictates a dynamic energy allocation law for variance injection. Building on these solutions, we propose a robust dual-routing architecture to execute the near-optimal mixed strategies. Furthermore, we certify that both the global value approximation error and the strategy suboptimality gaps are bounded by $\mathcal{O}(\bar{\pi}^{\frac{1}{2}})$. Finally, numerical experiments on a double-integrator pursuit-evasion game illustrate the induced physical behaviors and validate the theoretical bounds.

This paper introduces a novel framework for polar codes, designed for flexible Incremental Redundancy Hybrid Automatic Repeat Request (IR-HARQ). By generalizing the decoding order beyond the standard 1$\to$N sequence, we enable a capacity-aware scheduling strategy that prioritizes the decoding of reliable subblocks. The framework integrates nested parity-check polar construction and reverse bit-mapping to support continuous and arbitrary transmission lengths $E \in [N_{\min}, N_{\max}]$. Simulation results show that the proposed rateless codes match the coding gain of independently optimized fixed-rate codes across the entire range of rates and lengths. With a validated hardware implementation, this work provides a practical solution for next-generation wireless data channels.

We prove that any Fano threefold of genus 9 and 10 contains a cylinder, i.e. an open subset isomorphic to the product of a quasiprojective variety and the affine line. Moreover, we show that any Fano threefold of genus 10 has a point such that the Hilbert scheme of lines through the point has length three.

Using the operator method and properties of $O(2,1)$ algebra, the integral representation for the two-dimensional Green's function of an electron in the field of a magnetic monopole is found. This representation is valid in all complex plane of the electron energy.

We show that Young translation has dense orbits in the space of rough paths: for any two geometric rough paths, one can translate the first by a sequence of smooth paths so that it converges to the second in rough path topology. As applications, we obtain full-support criteria for rough paths arising from random series and Gaussian processes, including non-centered fractional Brownian rough paths.

For $1<p\le q<\infty$ and $n\geq 4$, we prove that the Poisson-like semigroup $(P_t)_{t\in \mathbb{R}_+}$ on $\mathbb{Z}_n$, associated with the word length $\psi_n(k)=\min(k,n-k)$, is hypercontractive from $L_p$ to $L_q$ if and only if $t\ge \tfrac{1}{2}\log\big(\tfrac{q-1}{p-1}\big)$. To this end, we establish the corresponding sharp Log–Sobolev inequalities with the optimal constant $2$.

A central aim of deep learning theory is to characterize how neural networks make predictions in the regime of simultaneously large model and training set size. Since the limits of diverging number of model parameters and dataset size do not commute it is not clear a priori what limits exist. In this work, we shed new light on these questions by studying Bayesian inference in deep non-linear MLPs in the regime where the number of training samples ($P$), the input dimension ($N_0$), the hidden layer width ($N$), and the number of hidden layers ($L$) can all be large. We build on the Neural Covariance SDE (Li et al., 2022) to analyze predictive posteriors in the regime where $LP/N\in\Theta(1)$, playing the role of an effective network depth. Our framework covers both smooth and ReLU activation functions and applies to arbitrary temperature. We find to first order in $LP/N$ a simple criterion for which data generating processes benefit from depth in the sense that larger $LP/N$ increases the Bayesian model evidence. We also give a novel derivation of a prior result from the physics literature that at least to first order in $LP/N$, the Bayesian predictive posterior is remarkably simple and is simply equivalent to that of a data-dependent kernel method.

Bloch equations are a powerful tool in describing the dynamics of nuclear magnetization in magnetic resonance phenomena. The fractional generalization of the Bloch equation effectively captures the anomalous relaxation and diffusion in porous, heterogeneous, and complex media. These equations describe how nuclear magnetization evolves under the influence of magnetic fields and relaxation processes. This work effectively employs a hybrid approach, the Laplace residual power series method, to investigate and analyze the fractional Bloch equation. A series solution is derived as the approximate solution for magnetization components. The influence of fractional order on each magnetization component in magnetization dynamics is analyzed and illustrated graphically. We conduct an error analysis to demonstrate the reliability and effectiveness of the proposed approach. The superiority of the suggested approach is shown using a comparative study with existing methods. The findings indicate the potential of the suggested approach as a reliable tool in understanding fractional magnetic resonance systems arising in applications such as NMR spectroscopy, MRI, MRF, and other complex heterogeneous materials.

We introduce a diffusion-based uncertainty model for robust optimization on directed graphs, in which perturbations of edge weights propagate along adjacent edges and satisfy conservation constraints at nodes. This topology-aware structure is natural in networked systems where uncertainty is induced by flows and local interactions, including transportation, logistics, communication, and energy networks. We analyze how such diffusive uncertainty reshapes the computational landscape of robust graph optimization. We focus on two canonical combinatorial graph problems, shortest path and the traveling salesman problem (TSP), which provide complementary benchmarks: shortest path is polynomial-time solvable in the nominal setting, whereas TSP is already NP-hard. We show that, for shortest path, propagation depth induces a sharp transition between tractable and intractable robust counterparts. For the traveling salesman problem, robustness often adds no computational complexity beyond ordinary TSP, because the structure of Hamiltonian cycles makes the fixed-tour adversarial problem collapse to explicit formulas. Together, these results show that topology-aware uncertainty can fundamentally change robust combinatorial optimization, with tractability governed by the interaction between propagation, budget geometry, and the structure of feasible solutions.

Model-based derivative-free trust-region methods build local interpolation models and restrict trial steps to regions where those models are reliable. This paper studies the shape of that region. When an objective is poorly scaled or locally anisotropic, a Euclidean ball can be governed by the steepest local direction and can restrict progress along directions of slow variation. We propose MATRO (Metric-Aware Trust-Region Optimization), a fully quadratic interpolation framework in which the trust region is the ellipsoid s^T M_k s <= Delta_k^2. For any positive definite metric M_k, the induced variable y = M_k^{1/2}s converts the ellipsoidal subproblem into a standard Euclidean trust-region subproblem, so model decrease, ratio tests, radius updates, poisedness, and fully quadratic error bounds can be stated in induced coordinates under a uniform metric contract. The metric is selected from the interpolation Hessian: positive definite quadratics yield a unique volume-normalized curvature metric that isotropizes the induced Hessian and gives a truncated Newton step, while indefinite fitted Hessians motivate an absolute-curvature metric that balances curvature magnitudes without changing curvature signs. Under the standard fully quadratic assumptions and the metric contract, MATRO retains the first-order evaluation-complexity order O(n^2 epsilon^{-2}). Experiments on More-Wild benchmarks, controlled anisotropy tests, and two-dimensional trajectories show that curvature-shaped regions are most effective when the interpolation Hessian captures stable local anisotropy, while dense linear algebra is most visible at loose accuracies or on inexpensive analytic tests.

Underdetermined quadratic interpolation is a central model-construction tool in model-based derivative-free trust-region methods: it limits sampling costs but leaves an affine family of interpolating quadratics. Classical solvers select one element of this family by prescribing a fixed norm or model-change measure, such as the least-Frobenius-change Hessian update in Powell-type methods. We introduce BUP-TR (Bayesian Underdetermined Projection Trust-Region), which instead completes the model by projecting a prior quadratic onto the affine interpolation set in the precision norm supplied by the prior. The same precision matrix defines a spectral geometry certificate, MAP-poisedness, and a repair mechanism for interpolation sets. Under standard smoothness assumptions, uniform precision bounds, MAP-poisedness, and a trust-region-scale prior-accuracy condition, the hard-MAP models are fully linear. Consequently, BUP-TR attains global first-order convergence and O(epsilon^{-2}) evaluation complexity, with geometry-repair evaluations included. A NEWUOA-style implementation, BUP-NEWUOA, improves fixed-budget performance on the reported benchmark suite at moderate and stringent accuracy targets while retaining the computational structure of a Powell-type trust-region method.

A group $G$ is called a Howson group if the intersection of any two finitely generated subgroups of $G$ is again finitely generated. It is called strongly Howson if, in addition, the rank of such an intersection is bounded only in terms of the ranks of the two subgroups. Strongly Howson groups are indeed Howson. Recently, Zhang and Zhao showed that the converse fails, by constructing the first examples of infinitely generated Howson groups which are not strongly Howson. In this note, we provide finitely generated examples with and without torsion elements, thereby answering a question posed by Zhang and Zhao.

In this paper, we propose and analyze a novel two-field symmetric formulation with solid displacement and fluid pressure as main unknowns for the Biot's consolidation model in poroelasticity. Firstly, we prove the well-posedness of the new formulation and then show the existence and uniqueness of optimal control where the fluid sources in the model act as a control variable. We prove a priori error estimates for the fully discrete scheme with backward Euler time discretization and a variational approximation of the control variable. A numerical example is presented to validate the performance of the proposed novel scheme.

Recent SVD based compression methods for large language models like SVD LLM and Basis Sharing can be unified under one optimization problem. While mathematical proofs and tests on Pythia models show this unified approach improves weight reconstruction error by up to 46% percent it fails in practical tasks. Downstream metrics like perplexity and accuracy severely degrade compared to standard per layer SVD LLM. The authors explain this failure mechanistically. Although the bundle method mathematically couples adjacent layers the transformer residual stream actually decouples them during forward passes. Thus per layer optimality matters more than joint cross layer optimization. The paper concludes that weight space reconstruction is a flawed objective for cross layer compression and future methods must focus on per layer activation reconstruction instead.

The group $\E(X)$ of homotopy self-equivalences of a topological space $X$ is a well-known group in homotopy theory and has been studied by many people since it was first introduced in the late 1950s. $\E$ is not a functor in the usual sense. In this paper we show that $\E$ is a Lax functor from the category $\mathscr Top$ of topological spaces to a strict $2$-category $\op{Corr}_{\mathscr Gr}$ of correspondences of groups.

Under what conditions does the sign of the Ollivier Ricci curvature on a graph of a certain order change? In this paper, we discuss the curvature of graphs obtained by removing edges from complete graphs, as complete graphs have a stable positive curvature. We defined graphs obtained by removing matching edges, the set of edges incident with the vertex, and cycle edges from complete graphs, and then analyzed the Ollivier Ricci curvature of those graphs. The results show that the curvature of the graphs in the above three patterns is equal to the value obtained by dividing the number of triangles, including two vertices, by the maximum degree of the two vertices. This result also indicates that the curvature of the above graphs is zero or positive. This study concludes that the Ollivier Ricci curvature is predicted to be positive even if some edges are removed from a complete graph, and we suggest that these discussions are suitable for investigating the conditions under which the sign of the Ollivier Ricci curvature on a graph.

This work introduces a novel structure-preserving methods for conservative systems based on a predictor-corrector strategy. The framework applies a discrete gradient correction to predictions generated by explicit one-step or multi-step schemes, which preserves nonlinear invariants while maintaining the accuracy order of the original predictor. This approach extends naturally to problems requiring simultaneous conservation of multiple invariants. Under mild conditions, conservation properties, solvability, numerical accuracy, and stability are established. Long-term numerical simulations on Lotka-Volterra systems, sine-Gordon equations, rigid body dynamics, and Kepler problems demonstrate improved robustness and conservation properties compared to existing projection and relaxation methods.

Unlearning in diffusion models aims to remove undesirable data or concepts while preserving the utility of pretrained models – two fundamentally conflicting objectives. We propose a principled constrained optimization framework that formulates unlearning as minimizing the deviation from a pretrained model, subject to explicit separation constraints from the unlearning distributions. Specifically, we formulate three constrained optimization problems based on reverse and forward KL divergences, and likelihood constraints. The first two generalize existing approaches for concept and data unlearning, while the third offers a novel and natural formulation for unlearning. Despite the nonconvexity of the KL constraints, we establish strong duality for all three problems, enabling us to explicitly characterize their optimal solutions as unlearning targets and develop primal-dual algorithms for each formulation. Experimental results demonstrate that our KL-constrained approach achieves superior retention-unlearning tradeoffs compared to weight-based baselines for concept and data unlearning, and that our likelihood-based approach matches unlearning effectiveness while better preserving retained concepts compared to baselines.

We prove interior $C^{2}$ regularity result for convex viscosity solutions of the quadratic Hessian equation $\sigma_2(D^2u) = f(x)$, under the assumption that $f\in C^{0,1}$ with $\inf f>0$. The result is almost sharp: if $f$ are merely continuous, there exist convex viscosity solutions that fail to be $C^{1,1}$. When $f\in C^{\alpha}$ for some $\alpha\in (0,1)$, the corresponding interior regularity remains open.

We introduce a spectral version of the classical inducibility problem. Given an $\ell$-vertex graph $F$ and an $n$-vertex graph $G$, let $H_F(G)$ be the $\ell$-uniform hypergraph whose edges are the $\ell$-sets inducing a copy of $F$ in $G$. We study the maximum possible $\alpha$-spectral radius of $H_F(G)$ over all $n$-vertex graphs $G$. For fixed $G$, this spectral parameter tends to $\ell!$ times the number of induced copies of $F$ in $G$ as $\alpha\to\infty$, and therefore refines the usual induced-copy count. Our main result is a spectral analogue of the Brown–Sidorenko reduction: for every complete multipartite graph $F$, every $n$, and every $\alpha\ge1$, a spectral extremal graph can be chosen to be complete multipartite. We also show that the leading asymptotic constant is the ordinary inducibility $i(F)$, and obtain exact multipartite reductions for stars $K_{1,t}$ and balanced complete $r$-partite graphs $K_{a,\ldots,a}$ with $r\le 2^a-1$.

We study Dehn fillings on two-bridge knots via non-abelian representations into $\mathrm{PSL}(2,\mathbb{R})$ whose meridian image is hyperbolic. For each fixed nontrivial two-bridge knot, we prove that the set of surgery slopes admitting such representations is bounded. Equivalently, Dehn fillings along slopes with sufficiently large absolute value admit no non-abelian $\mathrm{PSL}(2,\mathbb{R})$ representations with hyperbolic meridian image. The proof combines the Riley polynomial with Khoi's surgery-slope formula. On each admissible real algebraic branch, we express the meridian and longitude translation parameters as functions of the branch parameter and derive uniform endpoint estimates for their quotient. The resulting bound is effective in principle but not optimized. We also provide examples illustrating the parameter sets and endpoint behavior.

Let $B(\Lambda_a) \, (a=0,1)$ be the crystal of the level 1 integrable irreducible highest weight representation of the affine quantum group $U_q(\widehat{\mathfrak{sl}}_2)$. We consider the crystal graphs of degree $n$ associated with the irreducible $(2r+1)$-dimensional (resp. $(2r+2)$-dimensional) $U_q(\mathfrak{sl}_2)$ module in $B(\Lambda_0)$ (resp. $B(\Lambda_1)$). In this paper, we construct an explicit combinatorial procedure providing a bijection between the set of highest weight paths in these graphs with respect to the action of the Kashiwara operator $\tilde{f}_{1}$, and the set of integer partitions of $n$ with sqrank (resp. rerank) $r$, which is a recently introduced partition statistic. As a byproduct, we also obtain a precise interpretation of the motif description of spinons suggested by Bernard-Pasquier-Serban in the spinon picture for Wess-Zumino-Witten conformal field theory models.

Particulate Stokes flow in confined, periodic geometries underlies a broad class of problems in biophysics, microfluidics, and the rheology of complex fluids. Boundary integral equation (BIE) methods are a natural tool for such problems, but existing periodization schemes rely either on periodic Green's functions, which are restrictive for complex confining geometries, or on free-space schemes that solve auxiliary proxy strengths alongside the surface densities in an extended linear system whose cost scales unfavorably in three dimensions. We present a BIE framework for three-dimensional particulate Stokes flow in periodic pipes with circular cross-sections, wall-bounded doubly-periodic, and triply-periodic geometries that uses only the free-space Green's function and avoids both Ewald summation and the extended linear system. Proxy sources placed on equivalent surfaces of the kernel-independent FMM (KIFMM) form the auxiliary basis, and contributions from far image boxes are captured by a hierarchical proxy sum made absolutely convergent by a net-force-zero compatibility condition. The resulting periodization precomputation depends only on the periodic-box geometry, independent of the kernel and of the surfaces inside the box, and is reused verbatim across the Stokeslet, stresslet, and rotlet. Combined with high-order adaptive surface discretizations, the method achieves high-order accuracy at $\mathcal{O}(N)$ cost with a single layer of image boxes in the near field. Numerical examples on dense polydisperse suspensions with thousands of particles and on flow through complex periodic channels, together with strong and weak scaling studies, demonstrate efficient performance on systems with millions of degrees of freedom on distributed-memory architectures.

We develop a systematic method to express generating functions for moments of combinatorial statistics in terms of partition traces. We employ an algebraic approach based on the complete Bell polynomials and their inversion formula, alongside an analytic approach via Faà di Bruno's formula. Our approach can be applied to a wide class of combinatorial statistics, such as the largest part of an integer partition, the partition crank and rank, and the unimodal sequence rank.

The Moebius-Kantor graph MK=G(8,3) is a Cayley graph of three non-abelian groups, the Pauli group P(1), the semi-dihedral group SD(16), as well as the dihedral group D(16) of order 16. In topological graph theory, it illustrates the Heawood number 7 of the torus and leads to the Tucker group Aut(MK), the unique group of genus 2. We compute the Lefschetz numbers to illustrate the Brouwer-Lefschetz fixed point theorem. MK is also the dual of the 2-skeleton complex of the 3-sphere G. The graph represents one of flat Clifford tori of a Hopf fibration in the 3-sphere G=K(2,2,2,2) reflecting that Coxeter saw that MK is a subgraph of the tesseract G*. It carries a metric d so that (MK,d) has only one algebraic group structure (P(1),*) that preserves the metric. It makes the Pauli group natural, similarly as the Moebius ladder M(16) makes the dihedral group D(16) natural, forcing the algebraic structure from the metric structure.

We prove, at arbitrary loop order and including arbitrary tadpole decorations, the shifted-tree decomposition of two-site cosmological wavefunction coefficients proposed in our previous work, and uncover the combinatorial and geometric structure underlying it. For conformally coupled scalars with polynomial interactions, this decomposition expresses each loop coefficient as a finite signed sum of tree coefficients evaluated at shifted energies. For bubble-like graphs, the alternating sum over a Boolean lattice can be written in terms of finite differences or maximal chains. We prove the maximal-chain identity algebraically by induction on loop order and geometrically by decomposing a cubical Laplace integral into simplices. The maximal chains correspond to permutohedron vertices, whose incident facets encode the chain-dependent divisors. For tadpole-decorated graphs, the finite-difference Leibniz rule gives an explicit expansion indexed by ordered partitions and local maximal chains, which glue into global maximal chains and organize the coefficient as a combinatorial "path integral". A natural ordering of the divisors yields a term-by-term match with graph tubings and completes the all-loop proof, while other orderings give alternative tubing representations of the same coefficient. These identities hold before spatial loop-momentum integration and extend to power-law FRW backgrounds by linearity of the twisted integral. Two-site loop coefficients can therefore be obtained directly from known tree coefficients without solving new loop-level differential systems.

This paper presents a rigorous convergence analysis for the lightning plus polynomial approximation scheme, which employs rational approximations constructed with preassigned tapered, exponentially clustered poles. This pole placement strategy was originally introduced by Trefethen and his collaborators for the resolution of corner singularities. Ample numerical results indicate that this scheme achieves root-exponential convergence, and in particular attains the same optimal convergence rate as the best rational approximation to $x^\alpha$ on $[0,1]$ established by Stahl.% which is conjectured in [SIAM J. Numer. Anal., 61:2580-2600, 2023]. In this work, we establish optimal root-exponential convergence for the class of prototype functions of the form $g(z)z^\alpha$ or $g(z)z^\alpha\log z$, where $g$ is analytic on a neighborhood of the sector domain. These results confirm the validity of Conjectures 3.1 and 5.3 stated in [SIAM J. Numer. Anal., 61:2580-2600, 2023], and demonstrate that the choice $\sigma_{\mathrm{opt}} =\frac{\sqrt{2(2 - \beta)}\pi}{\sqrt{\alpha}}$ achieves the theoretically optimal convergence rate $\mathcal{O}\left(e^{-\sqrt{2(2 - \beta)N\alpha}\pi}\right)$. Notably, for the specific case of $\beta = 0$, the scheme recovers Stahl's optimal convergence rate for $x^\alpha$. Furthermore, working within the decomposition framework for corner domains proposed by Gopal and Trefethen, this paper provides a rigorous proof of optimal root-exponential convergence for lightning plus polynomial approximation problems on corner domains, and explicitly derives the optimal pole clustering parameter.

We study the lowest eigenvalue of the Neumann magnetic Laplacian in a planar domain divided into two regions, with piecewise constant magnetic fields that may scale differently in the inner and outer parts. Our aim is to describe the high-flux limit and determine when the ground-state energy is eventually monotone and when it continues to oscillate. We identify several asymptotic regimes according to the relative strength of the outer field. When the outer field is fixed, the lowest eigenvalue exhibits persistent oscillations and the low-energy states localize in the outer region. When the outer field grows more slowly, the behavior depends strongly on the geometry: it is eventually monotone for non-circular domains, while oscillations may persist for disks. In the critical regime, where the two fields are comparable, geometry and flux distribution both play a decisive role. When the outer field dominates, the problem reduces asymptotically to an effective operator on the inner region. These results show how uneven magnetic scaling, topology, and geometry shape the high-flux spectral behavior.

Let $\bigtriangleup$ be a simplicial complex and let $\delta_{\mathcal{NF}}$ denote the NF-operator. The NF-complex $\delta_{\mathcal{NF}}(\bigtriangleup)$ is defined as the Stanley–Reisner complex of the facet ideal of $\bigtriangleup$. Iterating $\delta_{\mathcal{NF}}$ gives a periodic orbit (up to isomorphism), and the smallest positive integer $t$ for which $\delta_{\mathcal{NF}}^{\,t}(\bigtriangleup)\cong \bigtriangleup$ is called the NF-number of $\bigtriangleup$ (Habi and Mahmood, Algebra Colloquium, 2022). In this work, we provide various results and determine explicit formulas for the NF-number for several families of graphs. In particular, we compute the NF-number for dumbbell graphs. We also prove that the NF-number of the complete split graph $S_{n,m}$ equals $m+n+2$, and that the NF-number of the double star $D_{p+q}$ equals $p+q+4$. We conclude with remarks, open problems, and conjectures to guide future research.

We investigate the eigenvector distribution at the soft edge for Gaussian random matrices with finite-rank deformations, in the critical regime of BBP transition. For finite-rank deformations of the GOE and GUE with critical spikes, we find that the squared overlap between a leading eigenvector and a spike, rescaled by \(N^{1/3}\), converges weakly to the negative reciprocal of the derivative of an Airy-Green function evaluated at the corresponding soft-edge root. For the rank-one critically spiked Gaussian \(\beta\)-ensemble, \(\beta>0\), we obtain an analogous result involving an Airy-Green function. In both cases, the Airy-Green functions are generalizations of the one introduced by Bykhovskaya–Gorin–Sodin \cite{Bykhovskaya-Gorin-Sodin25}. The proofs are both based on an eigenvector–eigenvalue identity and a resolvent-differentiation mechanism.