Papers

Let $X$ carry a transversely holomorphic foliation, equivalently an elliptic involutive structure $V\subset T_{\mathbb C}X$, and let ${\mathcal O}_V$ be its sheaf of leafwise-constant, transversely holomorphic functions. We construct a finite superconnection model for the derived category of coherent ${\mathcal O}_V$-modules. The key input is a mixed local reduction for finite Maurer–Cartan objects over the mixed de Rham–Dolbeault dga $(\wedge^\bullet V^\vee,d_V)$: a multiplicative homotopy contracts the real directions, after which Block's Dolbeault gauge theorem removes the positive transverse form degrees. For compact $X$, this gives an exact equivalence between the homotopy category of bounded finite-rank flat $V$-superconnections and $D^b_{\mathrm{coh}}(X,{\mathcal O}_V)$, interpolating between the de Rham and Dolbeault realizations. We prove locally finite Čech descent under necessary uniform amplitude and rank bounds, identify the coherent heart with equivariant coherent analytic sheaves on a transverse monodromy groupoid, and characterize descent to ordinary holonomy. For a holomorphic suspension, we identify the full superconnection category, up to Morita equivalence, with the homotopy fixed points of the Dolbeault category of the transversal and derive an equivariant Ext spectral sequence. Examples on $S^1$ and $S^2$ delimit when ordinary monodromy $1$-groupoids can recover the derived category.

In this note, we give an explicit description of countable products in $KK$- and $E$-theory and provide several applications.

In this work, we study policy optimization under domain randomization for linear quadratic control, focusing on learning a single state-feedback controller that minimizes the average cost across systems with uncertain dynamics. We propose a policy iteration algorithm with a step-size rule that preserves stability across all sampled systems at each iteration. We show that the method yields monotonic improvement of the sample-average objective and that a stabilizing step size always exists. Under standard smoothness assumptions, the iterates converge subsequentially to stationary points, and under a gradient-dominance condition, we obtain a global linear convergence rate.

We establish Gaussian free field fluctuations for uniformly random lozenge tilings of simply connected polygonal domains with $3d$ sides whose directions cycle through the three lattice directions. More precisely, assuming that the liquid region is connected and that the boundary data do not force the height at any interior point, we prove that the fluctuations of centered height function converge to the Gaussian free field in the liquid region, confirming a prediction of Kenyon and Okounkov from 2007. We introduce a tiling action function that encodes the geometry of the limit shape through its critical points. The action function has a complex conjugate pair of critical points in the liquid region, repeated real critical points on the arctic boundary, and distinct real critical points in the frozen region. Using this tiling action function, we construct an approximation to the inverse Kasteleyn matrix in terms of explicit single-contour and double-contour integrals and prove that the approximation is uniform throughout the polygonal domain. The convergence to the Gaussian free field then follows from standard kernel computations.

We derive a family of degenerate vortices with non trivial swirls and fat tails which can be used to bifurcate both stationary and self similar solutions to the three dimensional axi-symmetric incompressible Euler equations.

We study the Liouville property for nonnegative weak solutions of the $(p,q)$-Laplacian elliptic differential inequality $$\Delta^{m}_{p}u(x)+\Delta^{m}_{q}u(x)+V(x)u^{s}(x)\leq 0$$ and the associated parabolic differential inequality $$\partial_t u(x,t) \geq \Delta^{m}_{p}u(x,t)+\Delta^{m}_{q}u(x,t)+V(x,t)u^{s}(x,t)$$ on a forward geodesically complete noncompact Finsler measure space $(M,F,m)$ with finite reversibility. Under several sets of integral growth conditions on the positive potential function over certain annular domains, we prove that any nonnegative weak solution vanishes almost everywhere. The proofs of our results are essentially based on the nonlinear capacity method.

Albertson's conjecture asserts that every finite simple graph $G$ with $\chi(G) \ge r$ satisfies $\operatorname{cr}(G) \ge \operatorname{cr}(K_r)$. Building on Cranston's verification for $r \le 24$ and his reduction of $r \in \{25,26\}$ to three residual orders, we eliminate those residual cases and then prove the cases $r=27,28,29$. The first structural ingredient is a Kempe-chain construction: if a $k$-critical graph has a vertex of degree $k-1$, then it contains a branch-clean essential immersion of $K_k$. Essential immersions are crossing-monotone, so a critical counterexample must have minimum degree at least $k$. For $r=27$, this one-unit degree gain, Gallai's join structure, critical-graph edge bounds, and induced-subgraph averaging close every possible order. For $r=28$ and $r=29$, the remaining near-$2r$ orders are converted to dense complements. Stehlík's coloring theorem makes the odd-order complements factor-critical; a clique-partition obstruction yields an anti-tight matching property; and Tutte barriers, Hall-type expansion, and deficit bookkeeping eliminate the final cases. At order 58 for $r=29$, Rabern's coloring inequality handles the regular case, while the last degree-deficit-two case is reduced to two disjoint triangles and a finite barrier analysis.

The resistance Laplacian is a graph matrix associated with the effective resistance metric and provides a global counterpart of the classical graph Laplacian. Although it inherits several fundamental properties of the ordinary Laplacian, including a connected graph partitioning theorem analogous to that of Fiedler, its intrinsic spectral structure has remained largely unexplored. In this paper, we develop a structural theory of the resistance Laplacian. We derive a canonical decomposition that separates its intrinsic, average, and deviation components, thereby revealing how the global geometry induced by effective resistance differs from the local geometry encoded by the ordinary Laplacian. Building upon this decomposition, we establish several structural and spectral properties of the associated deviation operator, obtain variational characterizations of the largest eigenvalue and its corresponding eigenspace, and express the resistance Laplacian in Laplacian coordinates, thereby elucidating the relationship between the eigenspaces of the two operators. Finally, we formulate resistance-based graph partitioning objectives whose spectral relaxations recover the dominant eigenvector of the resistance Laplacian, providing a variational interpretation of the connected partition theorem. Experimental results on synthetic and real-life datasets demonstrate the effectiveness of the proposed framework for graph partitioning, data clustering, and exploratory anomaly detection.

We investigate finite group structures that yield complex irreducible characters induced simultaneously from two or more distinct subgroups. When such a group is realized as the Galois group of a Galois extension of number fields, this phenomenon implies that multiple subfields share the same Artin $L$-function.

Due to resource constraints or external and internal uncertainties, clients in real-world federated learning systems are often intermittently available edge devices. In highly dynamic environments, the parameter server lacks prior real-time knowledge of clients' availability, making it challenging to adapt traditional federated learning algorithms to be resilient to uncertainties in client availability. If not carefully addressed, complex client availability can introduce significant bias, potentially harming the performance of the trained model. Most prior work either fails to account for non-stationary client availability dynamics or demands significant memory and computational overhead. This paper aims to develop efficient federated learning algorithms that are provably resilient to heterogeneous and non-stationary stochastic client availability. We propose FedSWE, which admits novel algorithmic structures to (i) compensate for missed computations, (ii) stabilize and diffuse the global updates over rounds, and (iii) evenly mix the local updates through implicit gossiping, despite being agnostic to non-stationary dynamics. Compared with the standard FedAvg, FedSWE introduces light additional memory and computation overhead. We show that FedSWE converges to a stationary point of non-convex objectives while achieving the desired linear speedup property in certain special cases. We corroborate our analysis with numerical experiments over diversified client unavailability dynamics on real-world data sets.

We study the time derivative of the matrix exponential $B(t)=\mathrm{exp}(A(t))$, where $A(t)$ is a time-parametrized curve in $\mathrm{Mat}(n)$. Starting from the Taylor series expansion, we derive a nested summation formula, which is then reformulated into a double summation. This expression is converted into an integral representation using Euler's beta function, and further expressed in terms of Lie brackets and the adjoint representation. Later, we extend this to a family of problems and verify a well-known result. Finally, we establish connections to the Gateaux and Fréchet derivatives, proving the latter's existence. These results offer a unified and explicit framework for understanding the derivative of the exponential map.

Let $f: X\rightarrow S$ be a faithful affine map. In this article we construct a group homomorphism $\theta_{{fppf}}: \text{Br}(f) \longrightarrow H^1_{{fppf}}\!\left(S, (f_*\mathcal{O}_X^\times / \mathcal{O}_S^\times)_{{fppf}}\right)$, extending the formerly defined morphism over an étale site $\theta_{et}$. In doing so, we eliminate the characteristic constraint previously found in the relative version of Kummer's exact sequence. Furthermore, given a finite subintegral extension of noetherian rings $A \hookrightarrow B$, we show that all the $fppf$ cohomology groups, $H^i_{fppf}(Spec(A),\mu^f_n); \;\forall i\geq 0$ vanish whenever $n$ does not divide the characteristic of $A$.

A Dehornoy structure is a tool used to obtain a left order in the Garside group~$G$. Arcis and Paris introduced two conditions, known as Condition~$A$ and Condition~$B$. When they are both satisfied, they ensure that~$G$ possesses a Dehornoy structure. These two conditions describes how the product in the corresponding monoid behaves with an alternating decomposition. Here, we consider the case of a spherical type Artin group~$G$. We prove that Condition A almost always holds, but Condition~$B$ is rarely satisfied.

In this article, we characterize when a Toeplitz operator and a Hankel operator on the vector-valued Hardy space are skew commutators of each other, and determine necessary and sufficient conditions under which their product is self-adjoint. These characterizations extend the results in \cite{LZD}. In addition, we completely classify the skew commutators of the unilateral shift $S$, its adjoint $S^*$ and $S\oplus S^*.$ We also characterize the class of bounded linear operators that have $S$, $S^*$ and $S\oplus S^*$ as skew-commutators.

Let $K$ be a field and let $\T=L_K(E_T)$ be the algebraic Toeplitz algebra. We classify the Lie ideals of $\T^-$. If $w$ is the sink of the Toeplitz graph and $F=I(w)$, then $F\cong M_\infty(K)$, $\T/F\cong K[t,t^{-1}]$, and $[\T,\T]=F$. We prove that every noncentral Lie ideal contains $\mathfrak s=[F,F]$, the finitary trace-zero Lie algebra. Thus the classification reduces to the Heisenberg-type quotient $\T/\mathfrak s$. The Lie ideals are precisely $0$, $K1$, the spaces $\mathfrak s+U$, where $U\leq P_K\oplus Kw$ and $w\notin U$, and the inverse images $\rho^{-1}(W)$, where $W\leq K[t,t^{-1}]$. Here $P_K=K1$ in characteristic $0$, and $P_K=\operatorname{span}_K\{1,e^{m\ell},(e^*)^{m\ell}:m\geq1\}$ if $\operatorname{char}K=\ell>0$. We also describe the Lie derivations of $\T^-$ as sums of associative derivations of $\T$ and central Lie derivations factoring through $\T/F$.

Using the language of q-opers, we propose an algebro-geometric description of the duality between Bethe Ansatz equations for quantum XXZ spin chains and many-body trigonometric Ruijsenaars-Schneider/Macdonald systems for classical groups. Upon this duality, the energy level sets of the classical B/C/D-type Hamiltonians are found to be in bijection with the set of solutions of the XXZ Bethe Ansatz equations of type A, albeit with open boundary conditions. This dictionary generalizes previous results in which a GL(N) XXZ spin chain with twisted periodic boundary conditions was dual to an N-body Ruijsenaars-Schneider system. Our construction implements a $\mathbb{Z}/2\mathbb{Z}$ folding both at the level of the (GL(N),q)-oper data as well as at the level of the N-body trigonometric Ruijsenaars-Schneider model. The nonreduced BC-type systems appear in our analysis as well.

In this paper, we study games involving a continuum of heterogeneous players in the discrete-time setting with finite state spaces and continuous action spaces. We introduce a new model that incorporates joint state-action interactions within the graphon-weighted aggregate, described by a coupled forward-backward system. We establish the existence of graphon mean-field equilibria and characterize them through this forward-backward system. Additionally, we provide uniqueness results under monotonicity and contraction conditions. To illustrate the practical relevance of our framework, we solve an example of portfolio liquidation with price impact. We provide numerical results for three different graphons and two different initial distributions, illustrating the impact of the network's structure and the distribution heterogeneity on the distribution and the policy.

Let $G=\operatorname{Gr}(k,V)$ be a complex Grassmannian in its Pl$\"u$cker embedding, and let $Y_E$ be a general codimension-$r$ linear section. We study the open Hilbert scheme $R_2(Y_E)$ of smooth conics using the kernel and span of a conic. These define a flag variety $B=\operatorname{Fl}(k-2,k+2;V)$ and a relative Grassmannian of three-planes in a rank-six bundle. For $0\le r\le3$, we prove that $R_2(Y_E)$ is nonempty, smooth, irreducible and rational of dimension $e=2\dim V+k(\dim V-k)-3-3r$, with birational model $B\times\operatorname{Gr}(3,6-r)$. For $r\ge4$, a general-position argument gives a componentwise birational correspondence with a rank-three degeneracy locus and two smooth projective incidence resolutions. Every component meets the locus of smooth minimal-envelope conics; no additional density assumption is needed for general $E$. If $0\le e\le r$, the degeneracy locus itself is smooth. In dimension zero we obtain a top Chern class formula, recovering the known counts $1225$ and $1176$ by exact localization.

The classical unsolved Gauss circle problem concerns estimating the error between the number of lattice points inside a circle and the area of the circle as its radius tends to infinity. About a century ago, Hardy proposed a conjecture concerning this problem. In this paper, we attempt to provide numerical evidence in support of the Hardy conjecture through large-scale numerical computations.

This paper provides the instability counterpart to our recent construction of nonspherically symmetric approximating $\kappa$-self-similar naked-singularity solutions for the Einstein–scalar field system. These singular solutions contain pervasive nonspherical borderline terms, and to prove instability the delicate renormalization procedure developed in [2] does not extend to the more singular setting considered here. To overcome these difficulties, we introduce a new iteration scheme adapted to singular backgrounds whose leading-order geometry depends on the angular variables. At each step, the nonlinear coefficients are frozen using the preceding double-null geometry, and the resulting equations are solved in a triangular order. In this way, the nonspherical borderline terms are incorporated into the approximate geometry rather than treated as perturbative errors, yielding successively sharper estimates. After sufficiently many iterations, the scheme controls the singular angular structure and produces an approximate spacetime whose difference from the exact solution satisfies the required bounds. In particular, these bounds provide an existence region large enough to carry out the instability argument. We further prove that anisotropic perturbations of the outgoing data, arbitrarily small in a scale-critical norm, lead to the formation of a trapped surface. We also formulate and verify a matter-focusing condition under which a parabolic flow argument guarantees the existence of a corresponding marginally outer trapped surface (MOTS). Together, these results establish the nonlinear instability of $\kappa$-self-similar naked singularities beyond spherical symmetry in the Einstein–scalar field system and introduce a framework for applications across Einstein systems.

We study the annealed survival probability of a random walk in an inhomogeneous Poisson environment of mobile traps on $\mathbb{Z}^d$. In dimensions $d=1,2$, we determine the asymptotics for the annealed survival probability of the walker under suitable assumptions on the average trap intensity and study how inhomogeneity in the initial trap configuration affects these asymptotics. Our results extend the asymptotics proved for the homogeneous setting in \cite{DGRS2012}. We also establish a law of large numbers, central limit theorem and large deviation principle for the inhomogeneous Poisson trap environment in $d\ge 1$, generalising the corresponding results proved for the homogeneous case considered in \cite{CG1984}. We present a class of examples of inhomogeneous trap environments where the decay rate of survival probability can be identified and also instances where it does not decay with time.

In this study, we demonstrate the characteristics of the stochastic Loewner evolution (SLE) as a one-dimensional (1D) turbulence model. First, we show that the diffusion process $x(t)$ obtained by the time coordinate change in the SLE becomes the time dependent diffusion process that satisfies Richardson's law of turbulence.Subsequently, we derive the condition on the diffusivity parameter $\kappa$ such that the diffusion process $x(t)$ satisfies Kolmogorov's 4/5th law, which determines the relation between the mean energy dissipation and velocity in the particles of turbulence. Further mathematical analysis showed that this condition is seen as the condition on Loewner entropy $S_{\mathrm{Loew}}$, which includes the white Gaussian noise term $W_{s}$ in the Loewner driving function of the present model. In addition, the numerical simulations were performed to verify the scaling relations of this 1D turbulence model. These results suggest that the standard SLE becomes a beneficial model of 1D turbulence by introducing an appropriate time coordinate change and imposing the conditions on $\kappa$ and $S_{\mathrm{Loew}}$.

The size-Ramsey number $\widehat{R}_k(\mathcal G)$ of a $k$-uniform hypergraph $\mathcal G$ is the minimum number of edges in a $k$-uniform hypergraph $\mathcal H$ such that every $2$-edge-coloring of $\mathcal H$ contains a monochromatic copy of $\mathcal G$. Fox asked whether, for fixed $2\le \ell<k$, the size-Ramsey number of every $k$-uniform relaxed $\ell$-tree on $n$ vertices is bounded by a polynomial in $n$, as recorded by Dudek, La Fleur, Mubayi and Rödl. We give a negative answer when $\ell+1\le k\le 2\ell-2$. More precisely, for every sufficiently large $n$, we construct a $k$-uniform relaxed $\ell$-tree $\bar{\mathcal T}_{n,\ell}^{(k)}$ on exactly $n$ vertices such that \[ \widehat{R}_k\bigl(\bar{\mathcal T}_{n,\ell}^{(k)}\bigr) \ge 2^{c_{k,\ell}n^{\,1-1/(2\ell-k)}}, \] where $c_{k,\ell}>0$ depends only on $k$ and $\ell$. Moreover, any two distinct edges of the construction intersect in at most $2\ell-k-1$ vertices. In contrast, for $k\ge 2\ell$, we prove that every $k$-uniform relaxed $\ell$-tree $\bar{\mathcal T}_{n,\ell}^{(k)}$ on $n$ vertices satisfies \[ \widehat{R}_k\bigl(\bar{\mathcal T}_{n,\ell}^{(k)}\bigr) =O_{k,\ell}(n^{2k}). \]

Transaction costs can make or break a trading strategy, particularly in relative-value trading of commodity and macro markets, where edges are a few basis points. Price impact is a central component of transaction cost. Price impact models usually include self-impact (a trade in a contract moves that contract's price) but omit two well-documented effects: cross-impact (a trade in one contract also moves the prices of related contracts) and transient impact (price impact decays over time, so an unwind recovers part of the entry cost). A model without these effects overprices the impact of relative-value trades, whose correlated legs are built and unwound over days, and so forgoes potentially profitable trades. This paper models both effects with a convex quadratic cost. In each period, a positive semidefinite matrix built from volatility, volume, and correlation forecasts couples trades across contracts. A power-law decay kernel then couples trades across periods. The resulting cost admits no price manipulation even when liquidity varies over the planning horizon. The model is demonstrated empirically on calendar spread trading of crude oil futures around the commodity index roll.

We give a new Hecke-theoretic method to control the growth of monodromy in $p$-adic families of automorphic Galois representations for unitary groups and quasisplit classical groups.

We prove the codegree isomorphism conjecture for finite simple groups. Thus every nonabelian finite simple group is determined, among finite groups, by its set of irreducible character codegrees.

We investigate the spectral properties of a class of inhomogeneous self-similar measures, which does not admit a non-trivial infinite convolution structure. A central example is the golden-mean self-similar measure $\mu$, for which the existence of an exponential orthonormal basis in the associated $L^2$-space has remained a long-standing open problem. The usual approach for homogeneous self-similar measures does not apply here, new methods are required. We establish several basic properties of the measure and then carry out a detailed numerical study of the zero set of its Fourier transform. Using a scanning and refinement algorithm that combines uniform grid sampling, quadratic interpolation, and golden-section search, we examine a wide range and find no real zeros of $\widehat{\mu}$, which provides concrete evidence that $\mu$ is very likely non-spectral, suggesting that inhomogeneity may serve as a natural obstruction to the existence of exponential orthonormal bases. To the best of our knowledge, our paper is the first attempt to study the spectrality of such measures through a combined analytic and numerical framework.

For a graph $G$, let $i_k(G)$ denote the number of independent sets of cardinality $k$, and let \[ I(G,z)=\sum_{k\ge0} i_k(G)z^k \] be its independence polynomial. Following Brown and Cameron \cite{BrownCameron2018}, a graph is called stable if all zeros of its independence polynomial lie in the closed left half-plane. They proved that every star is stable, but also constructed nonstable trees. They then asked for a characterization of stable trees. In this paper, we extend and strengthen their result by proving that every spider, obtained from a star by arbitrary and possibly nonuniform subdivisions of its edges, has all its independence roots in the open left half-plane. Hence, every spider is stable.

There is a body of research devoted to understanding how complex singularities of solutions of nonlinear partial differential equations (pdes) spontaneously emerge at $t=0^+$ and propagate for $t>0$, and how their behaviour affects the solution on the real axis. Despite the importance of the small-time limit in these studies, there is still a lack of understanding of how complex singularities are born at $t=0^+$, including for initial conditions that are not analytic functions of the spatial variable. In this paper, we use Burgers' equation as a prototype nonlinear pde and study the complex-plane singularities for initial conditions that are piecewise smooth. Using matched asymptotic expansions, we show how infinitely many singularities emerge from points of discontinuity in a pattern that can be described using branches of the Lambert-$W$ function. For various initial conditions, we observe how these singularities rearrange themselves to align with the appropriate exactly-described long-time behaviour, including sigmoid-shaped travelling waves, constant-area (triangular wave) similarity solutions and $N$-wave solutions. In terms of Burgers' equation, our small-time asymptotic analysis of the singularity propagation for piecewise-continuous initial conditions illustrates the types of generic behaviours that arise for inner regions when diffusion dominates advection. More generally, this work is a step towards understanding complex-plane behaviour of solutions of nonlinear partial differential equations with non-analytic initial conditions.

In this paper, we provide explicit formulas for the average decoding error probabilities of the ensemble of all $(n,M)_q$ codes over the erasure channel under unambiguous, list and maximum-likelihood decoding principles. Moreover, we derive the asymptotic error exponents for the ensemble of all $(n,q^{Rn})_q$ codes.