Papers

Using divisibility relations between the generators of a square-free monomial ideal $I$, we describe divisibility relations between the generators of the second power $I^2$. We then employ discrete Morse theory to produce a cellular free resolution of $I^2$ which is minimal for specific ideals that are extremal with respect to a given divisibility relation. In particular, we provide sharp bounds on the projective dimension of $I^2$ when the generators of $I$ satisfy at least one divisibility relation.

Let $M^n\subset\mathbb R^d$ be a closed, connected, orientable $C^4$-smooth Riemannian submanifold of dimension $n\ge3$. We construct, for each degree $0\le k\le n$, a family of deformed Hodge Laplacians $\Delta_t^k$, $t>0$, defined in terms of the extrinsic geometry of $M^n$, and prove that $\Delta_t^k$ converges uniformly to the classical Hodge Laplacian $\Delta^k$ as $t\to0^+$. Given an i.i.d. uniformly distributed point cloud $S_m\subset M^n$, we define empirical Hodge operators $\widehat\Delta_{t,S_m}^k$. Under the scaling $t=m^{-1/(2n)}$, we prove uniform consistency in probability and compact Mosco convergence in probability of the associated quadratic forms. Consequently, the empirical spectral cluster near zero contains exactly the $k$-th Betti number $b_k$ of eigenvalues, counted with multiplicity, and converges in the transported discrete $L^2$-sense to the space of harmonic $k$-forms. We also construct consistent empirical estimators of the tangent projection, the second fundamental form, the Riemannian curvature tensor, and the Weitzenböck curvature endomorphisms. As applications, we obtain consistent recovery of the Betti numbers and harmonic representatives of de Rham cohomology, as well as of the Pontryagin forms and their periods. Evaluating these periods on a homology basis recovers the corresponding real Pontryagin classes from sampled data.

This work develops a conceptual framework for the foundations of quantum physics, linking two main approaches: the algebraic formulation and quantum probability. Rather than proposing new axioms or theories, the text reorganizes and synthesizes existing models, highlighting their assumptions, conceptual structures, and operational significance. The analysis begins with von Neumann's measurement theory and its subsequent developments by Mackey, emphasizing the role of experimentally feasible procedures and the need for a statistical model grounded in laboratory practice. The work adopts an operational perspective, according to which physical quantities are defined solely through experimental measurement methods, and the corresponding probabilistic measures are derived from measurement outcomes. The introduction critically examines the limitations of purely mathematical formulations - such as the algebraic method - when separated from experimental interpretation. The text argues for a clear distinction between axioms, postulates, and presuppositions, and for a reconstruction of quantum theory that respects both empirical constraints and conceptual clarity. Overall, the goal is to provide a coherent path from operational principles to algebraic structures, offering a basis for an axiomatic reformulation of quantum mechanics that remains faithful to physical practice.

This paper investigates the fundamental limits of information-theoretic decentralized secure aggregation (DSA) with user dropouts. We consider a fully decentralized network where $K$ users communicate over broadcast channels without a trusted aggregation server. Each user holds a private input and aims to recover the sum of the surviving users' inputs (users may drop) while ensuring that no additional information about individual inputs is revealed to that user, even if it can collude with other users. A two-round communication protocol is considered, where we assume at least $U$ users survive and each user can collude with at most $T$ other users. For this setting, the optimal communication rate region is fully characterized: we show that DSA is infeasible if $U\le T+1$; otherwise, the optimal rate region is given by $R_1\geq 1$ and $R_2\geq \frac{1}{U-T-1}$, where $R_1$ and $R_2$ denote the first- and second-round communication rates, respectively. The proposed aggregation scheme is based on correlated secret keys constructed from $(T+1)$-private maximum distance separable (MDS) matrices, which simultaneously provide robustness against user dropouts and security against collusion. We also derive tight converse bounds that establish the optimality of the proposed scheme. Our result shows that the optimal second-round communication rate depends only on the effective redundancy level $U-T-1$ regardless the total number of users.

We study online optimization problems in which the cost function depends on latent, time-varying parameters that are unmeasurable and governed by unknown dynamics. Specifically, we consider a strongly convex cost function whose linear term evolves according to unknown linear stochastic dynamics about an unknown nominal operating point, while the algorithm has access only to finite noisy gradient measurements. We propose a solution that uses control theoretic tools to reconstruct the latent parameters from gradient observations using a Gauss-Markov estimator, then identifies the parameter dynamics using a bias-reduced instrumental variable estimator, and finally forecasts the parameters to compute the minimizer. We provide a high probability, finite-horizon bound on the expected tracking error. We illustrate our algorithm on a series of numerical examples.

We prove that being of klt type is not an open condition in flat families of varieties. This answers a question of Han and Jiang. The construction in this paper substantially uses generative AI: the general idea for the counterexample was suggested by ChatGPT Pro 5.5, and the explicit example was found and proved by the Rethlas system.

In this paper, we study the dynamics of commuting transcendental entire functions $f$ and $g$, where $g$ is of the form $af^p + b$ with $a,b \in \C$, $p \in \N$, and $a \neq 0,1$. We establish that the escaping sets, filled Julia sets, and bungee sets of $f$ and $g$ all coincide. As an immediate consequence, we obtain in particular that the Julia sets of $f$ and $g$ are identical. Our theorem extends the 1998 result of Poon and Yang. Furthermore, following Wang and Yang, we consider a non-constant polynomial $Q$ and permutable entire functions $f$ and $g$ satisfying the relation $Q(g)=aQ(f)+b$, where $a(\neq 0,1), b \in \C$. In this more general setting, we also prove that the escaping sets, the filled Julia sets, and the bungee sets of $f$ and $g$ are equal.

We provide a complete combinatorial and asymptotic analysis of positive linear systems of equations in one catalytic variable that appear in several combinatorial problems such as in lattice path counting or stack-sortable permutation counting. We show that the corresponding generating functions satisfy a positive polynomial system of equations (which is associated to a context-free grammar). Furthermore we prove a universal asymptotic behaviour.

We introduce the concept of integrable observables and propose them as alternatives to the standard Witten's psi classes (a.k.a. descendants in $2D$ quantum gravity) to be coupled with cohomological field theories and their generalisations. The main property of integrable observables is that they retain the integrability properties. We present three examples of integrable observables. The first two recover the Dubrovin-Zhang and double ramification hierarchies, while revealing new structural features in this framework. The third, a new example, builds on recently established properties of the so-called $\mathbb{\Pi}$-class, extending them and placing this class naturally within the theory of integrable systems. Notably, our integrable observables framework yields a proof that the new $\mathbb{\Pi}$-hierarchies are Miura equivalent both to the Dubrovin-Zhang hierarchies and to the double ramification hierarchies. A new very short proof of Witten's conjecture is also provided.

Complex dynamical systems governed by holomorphic maps such as $z^2 + c$ exhibit fractal boundaries with extreme sensitivity to initial conditions. Accurately modelling these structures from data requires methods that respect the underlying complex-analytic geometry, yet Multi-Layer Perceptrons (MLPs) within Neural Ordinary Differential Equations (Neural ODEs) lack complex-analytic priors, violate the Cauchy–Riemann conditions, and function as opaque approximators incapable of yielding governing equations. We introduce Holomorphic KAN-ODE, a framework that replaces the MLP with a Kolmogorov-Arnold Network (KAN) whose learnable B-spline activations reside on network edges, and incorporates Cauchy–Riemann equations as a differentiable regularization to preserve holomorphic structure. We evaluate on six families of complex dynamical systems spanning polynomial and transcendental classes. With only 280 parameters ($16\times$ fewer than the MLP baseline), the network achieves velocity-field $R^2 > 0.95$ on all six systems, correctly identifies all six governing symbolic families through automatic spline-to-formula fitting, and reconstructs Julia set fractal boundaries with up to 98.0% agreement. Crucially, the model exhibits only 4% MSE degradation under 10% observation noise versus $15.2\times$ for MLPs, and achieves 90.4% improvement in transfer learning from quadratic to cubic dynamics. While the MLP attains lower pointwise reconstruction error due to its larger capacity, the KAN uniquely provides interpretable symbolic equations, enforced holomorphic structure, and superior noise resilience, capabilities that are entirely absent in black-box architectures. These results establish KANs as a parameter-efficient, interpretable alternative to MLPs for physics-informed discovery of holomorphic dynamics.

A problem of Erdős asks for extremal conditions forcing edge-disjoint cycles with a prescribed nested structure. In the geometric version, the nesting is required to be noncrossing with respect to the cyclic orders. Fernández, Kim, Kim and Liu proved that constant average degree forces two such cycles. We prove a polylogarithmic bound for the natural multi-layer version: for every fixed $k\ge 3$, every sufficiently large $n$-vertex graph with at least \[ C_k n(\log n)^{k-1}(\log\log n)^{k-3} \] edges contains $k$ pairwise edge-disjoint nested cycles without geometric crossings. The proof combines the robust sublinear expander framework of Alon, Bucić, Sauermann, Zakharov and Zamir with a controlled wrapping lemma that permits the layers to be built successively with controlled length.

This paper investigates the geometry and singularities of parallel surfaces of cuspidal cross caps, the fundamental non-front frontal singularities. We establish a criterion for the degeneracy of the distance squared function in terms of known geometric invariants and describe the resulting configuration of singularities. Our main result demonstrates that while the parallel surface is generically $\mathcal{A}$-equivalent to a cuspidal cross cap, it degenerates into a degenerated cuspidal $S_1$ singularity at specific distances characterized by the equation $C_2(\varepsilon)=0$. These distances act as a novel analogue of the principal radii of curvature. Indeed, although the Gaussian and mean curvatures diverge at the singularity, their asymptotic expansions reveal that their constant terms correspond to the product and average of the reciprocals of these distances, respectively.

We determine the $L^p$ spectrum, $1<p<\infty$, of the Stokes operator $A=-\PP\LD$ on divergence-free vector fields of hyperbolic 3-space of curvature $-a^2$, where $\LD=\Delta_B+\Ric$ is the Ebin-Marsden deformation Laplacian and $\PP$ is the Leray projector. The spectrum is the parabolic region $a^2\{4+z^2:\ z\in\C,\ |\im z|\le|1-2/p|\}$. For $p>2$ every interior point is an eigenvalue, realized by an explicit family of divergence-free swirl fields which leave $L^p$ exactly at the boundary of the region. It follows that the Stokes semigroup on $\Lp$ has exponential type exactly $-\mu_p$ with $\mu_p=a^2\bigl(4-(1-2/p)^2\bigr)$. At $p=3$ this is $35a^2/9$, which lies strictly above the rates $26a^2/9$ and $10a^2/3$ that follow from pointwise domination by the scalar heat semigroup and from interpolation, and strictly below the $L^2$ gap $4a^2$. For the Navier-Stokes equations with small data in $L^3$, whose global solvability on $\HH^3$ is due to Pierfelice, the solutions decay in $L^3$ and in $L^6$ at every rate below $\mu\cdot35a^2/9$, where $\mu$ is the viscosity, and no faster rate holds for the linearized flow. The exponent $3$ is the usual scaling threshold. Curvature fixes the time scale of the decay and introduces no new threshold.

We study the homogenization of first-order Hamilton-Jacobi equations on an infinite-dimensional Hilbert space, motivated by systems of infinitely many indistinguishable particles on the torus. A central difficulty is that the analysis takes place in an infinite-dimensional setting, where the compactness arguments available in finite dimensions break down. The problem is further complicated by the possible nonconvexity of the Hamiltonian, which prevents the direct use of variational methods. Under suitable assumptions on the Hamiltonian and the initial data, we characterize the effective Hamiltonian through an associated cell problem and prove that the solutions converge to those of the limiting equation at rate $O(\varepsilon^{1/3})$. This yields a qualitative and quantitative homogenization result for a class of possibly nonconvex Hamilton-Jacobi equations in infinite dimensions.

Let $\mathrm{PG}(n-1,q)$ denote the $(n-1)$-dimensional projective space over $\mathbb{F}_q$. We investigate the intersection of two Desarguesian $(h-1)$-spreads of $\mathrm{PG}(kh-1,q)$ and show that it is determined by a subgeometry over a suitable extension field. Our approach combines a characterization of subsets of points of $\mathrm{PG}(k-1,q^h)$ closed under $q$-order subgeometries with a matrix model for Desarguesian spreads based on Moore matrices. This leads naturally to the notion of generalized Segre varieties $\mathcal S^r_{kr-1,h-1}(q)$ and a geometric description of their maximal subspaces. As a main application, we prove that if two distinct Desarguesian $(h-1)$-spreads of $\mathrm{PG}(kh-1,q)$ contain a common pseudo-arc of size $k+1$, then their intersection is precisely the system $\mathcal R^r_{h,q}$ of $(h-1)$-dimensional subspaces of $\mathcal S^r_{kr-1,h-1}(q)$, for some proper divisor $r$ of $h$.

We give a rigorous derivation of the Fourier transform of the Heaviside function within a framework for tempered distributions that is suitable for undergraduate engineering and mathematics students. The proofs rely on fundamental concepts typically taught in a freshman-level calculus course, including limits, generalized integrals, integration by parts and the Taylor Remainder Theorem. In passing, we examine the Principle Value of $\frac{1}{x}$ and the relationship between its derivative of order $n$ and the Principle Value of $\frac{1}{x^{n+1}}$.

Bandit convex optimization (BCO) is a fundamental online learning framework with partial feedback, where the learner observes only the loss incurred at the chosen decision point in each round. In this work, we investigate whether optimistic gradient predictions can improve worst-case regret guarantees in a prediction-adaptive manner. Specifically, given gradient predictions $m_t$, we seek regret bounds that scale with the cumulative prediction error $S_T=\sum_{t=1}^T \|\nabla f_t(x_t)-m_t\|^2.$ We first establish a negative result: under the single-point feedback protocol, an unavoidable $\Omega(\sqrt{T})$ regret lower bound persists even when $S_T=o(T)$, showing that the variance of gradient estimation fundamentally obscures the benefit of accurate predictions. To overcome this barrier, we propose Two-Point Variance-Reduced Optimistic Gradient Descent (TP-VR-OPT) for the two-point feedback setting. The key idea is a novel variance-reduced gradient estimator whose variance scales with the prediction error rather than the gradient norm. This yields a regret bound of $O\big(\sqrt{d\,\mathbb{E}[S_T]}\big),$ where $d$ is the decision dimension. Complementing this result, we establish an information-theoretic lower bound that scales as $\Omega(\sqrt{\mathbb{E}[S_T]})$, providing a fundamental characterization of the best achievable prediction-adaptive regret and showing that TP-VR-OPT is optimal up to a factor of $\sqrt d$. We further develop adaptive variants that eliminate the need for prior knowledge of $\mathbb{E}[S_T]$ or the horizon $T$, and extend our framework to non-stationary environments, establishing dynamic regret guarantees that adapt simultaneously to the cumulative prediction error and the comparator path length.

GPUs have significantly accelerated first-order methods for large-scale optimization, especially in continuous optimization. However, this success has not transferred cleanly to problems with discrete variables, combinatorial structure, and nonlinear objectives, such as certifying optimal solutions for cardinality-constrained generalized linear models. Major challenges include the sequential processing of heterogeneous nodes in branch and bound (BnB) and frequent data movement between the CPU and GPU. We propose a simple, generic, and modular CPU–GPU framework that processes multiple BnB nodes in batches on GPUs. The framework is built around a small set of GPU-efficient routines and uses padding together with lightweight custom kernels to handle irregular node data structures. Experiments show one to two orders of magnitude speedups and zero optimality gap on challenging instances. The framework can also be extended to collect the entire Rashomon set, enabling downstream statistical analysis such as variable-importance analysis and model selection under secondary user-specific measures (e.g., AUC in classification).

We study the Cox rings of smooth anticanonical Calabi-Yau hypersurfaces in smooth toric Fano varieties. Using the combinatorics of primitive pairs of the ambient Fano polytope and the description of Cox rings of embedded varieties via localizations, we identify several configurations for which the hypersurface is a Mori dream space and obtain explicit presentations of its Cox ring. We also exhibit combinatorial configurations forcing the birational automorphism group to be infinite, yielding in dimensions three and four a dichotomy between finite generation of the Cox ring and infinite birational automorphism group. Finally, for a class of non-Mori dream examples, we prove the Morrison-Kawamata cone conjecture for the movable cone.

In this paper, we establish gradient bounds for $p(\cdot)$-harmonic differential forms subject to a Coulomb-type gauge condition. For variable exponents satisfying the log-Hölder continuity assumption, we derive higher integrability estimates of Meyers type, ensuring improved regularity beyond the natural energy space. Furthermore, under the stronger assumption of Hölder continuity of the exponent function, we prove that the gradient of solutions exhibits Hölder continuity. These results extend classical regularity theory for constant-exponent $p$-harmonic systems to the variable-exponent setting, which is essential for modeling nonhomogeneous and anisotropic media.

High dimensional expanders simultaneously satisfying spectral and combinatorial (coboundary) expansion have recently played a major role in breakthroughs in PCP and coding theory, but the only known construction of such complexes is extremely involved, requiring deep algebraic number theory. In this work, we give an extremely simple combinatorial construction of a sub-polynomial degree complex based on projections of the flags complex (subspace chains) that is (i) a local spectral expander, (ii) a coboundary expander, and (iii) a swap coboundary expander. As a corollary, we also give the first near-linear size combinatorial hypergraphs with good agreement tests in the '1%' regime, and a simple PCP construction with near-linear size.

Based on Landauer's principle, we provide a geometrical definition for the entropy of a given static, spherically symmetric spacetime. Considering a congruence of geodesics across a surface, one defines the entropy of a congruence as the surface integral of the entropy of the constituent geodesics. Under certain mild assumptions, we establish a second law for the entropy function thus defined (Landauer entropy), and relate it to Bekenstein-Hawking entropy.

In this paper, we compute minimum second neighborhood degree spectrum and energy of commuting graphs of certain finite non-commutative rings. In particular, we consider non-commutative rings of order $p^2, p^3, p^4, p^5, p^2q$ and $p^3q$, where $p$ and $q$ are primes. We shall also show that the commuting graphs of these rings are MSN-integral but not MSN-hyperintegral. Finally, employing the techniques used in this paper, we prove Conjecture 3 of [Nath, R. K., Fasfous, W. N. T., Das, K. C. and Shang, Y. Common neighbourhood energy of commuting graphs of finite groups, Symmetry 13(9), Article No. 1651, 2021.] and Conjecture 3.12 of [W. N. T. Fasfous and Nath, R. K. Common neighborhood spectrum and energy of commuting graphs of finite rings, Palestine J. Math. 13(1), 66–76, 2024.]. We conclude this paper with two open problems.

A novel boundary element method (BEM) removes the classical dependence on explicit fundamental solutions and extends quasi-optimal BEM discretisations to strongly elliptic operators with variable coefficients. The approach constructs a computable approximation of the boundary operator from a Galerkin discretisation of the underlying elliptic differential operator in a one-time preprocessing step, for instance by conforming finite elements. The resulting algebraic formulation retains the dimension reduction intrinsic to boundary integral methods and is compatible with standard data-sparse matrix compression techniques.

We study the largest gaps between successive zeros of a smooth stationary Gaussian process. Our main result is that, if correlations decay at least polynomially, then after suitable rescaling of the locations and sizes of the largest gaps in a growing interval, the resulting joint process converges to a Poisson point process. The main novel step in the proof is to establish an approximate splitting property, with multiplicative error, for gap events in well-separated intervals; notably we achieve this for processes with arbitrarily slow polynomial decay of correlations.

A weave is a type of textile that consists of vertical and horizontal threads, and typically it has a periodic structure. In this paper, we regard a weave as a link in the thickened torus with a diagram consisting of closed geodesics. As main results, we characterize isotopies and hyperbolicity of weaves to determine them from diagrams. Moreover, we show that there does not exist an essential Conway sphere for a weave. We use normal positions of essential surfaces of weave complements to describe them.

Let $G$ be a finite group and $T(G)$ be the sum of the degrees of its irreducible complex representations. We investigate the relationship between $T(G)$ and the number of twisted involutions $m_\sigma = |\{g \in G \mid \sigma(g) = g^{-1}\}|$ for an automorphism $\sigma$. While it is known that $T(G) = m_e$ for the identity automorphism $e$ in certain cases (e.g., real characters), we analyze this relation for non-identity automorphisms of groups of order $p, 2p, p^2$. We prove that for the family of Dihedral groups $D_n$, the inequality $T(D_n) \geq m_\sigma$ holds for all $\sigma \in \mathrm{Aut}(D_n)$. We provide a complete classification of $m_\sigma$ using number-theoretic properties of the automorphism parameters.

This is a verbatim copy of a technical report I wrote in 2017-2018 to obtain the law of the iterated logarithm using the guarantee on the wealth of an online betting strategy.

In this paper, we study tensor spaces beyond the boundary format and analyze whether the general critical space coincides with the general span of singular vector tuples. For all tensor spaces exceeding the boundary format by one in an arbitrary number of factors, we relate the codimension of this span within the critical space to the dimension of the kernel of a map in cohomology. Furthermore, we exhibit an infinite family of order-three tensors with a defective behavior: the general span of singular vector tuples achieves the maximum possible codimension rather than the expected minimum. Finally, we conjecture a classification of the behavior of critical spaces in this regime and draw a connection to Koszul cohomology.

The Ramsey number $\mathrm{R}(G_1,G_2)$ is the smallest integer $N$ such that any red-blue coloring of the edges of the complete graph $K_N$ contains either a red copy of $G_1$ or a blue copy of $G_2$. In 2022, the third author and others gave lower and upper bounds of the Ramsey number $\mathrm{R}(W_n,W_n)$, where $W_n$ is the wheel graph with $n$ vertices. In this paper, we improve their bounds by showing that $3n-2\leq \mathrm{R}(W_n,W_n)\leq 6n-6$ for even $n\geq 8$ and $2n\leq \mathrm{R}(W_n,W_n)\leq \frac{9n-7}{2}$ for odd $n\geq 7$. Furthermore, we give recursive bounds for the $k$-colored Ramsey number for $W_n$.