Papers

Signed total Roman domination is a variant of the classic Roman domination-problem in graphs. A signed total Roman dominating function (STRD function) on a graph $G=(V,E)$ is a function $f: V \to \{-1,1,2\}$ such that (i) $\sum_{u \in N(v)} f(u) \geq 1$ for all $v \in V$, where $N(v)$ denotes the neighborhood of $v$, and (ii) every vertex $v$ with $f(v) = -1$ is adjacent to a vertex $u$ with $f(u) = 2$. The weight of $f$ is $\sum_{v \in V} f(v)$. The signed total Roman domination number of $G$ is the minimum weight among all its STRD functions. A signed total Roman dominating family (STRD family) on $G$ is a family $\{f_1, \ldots, f_d\}$ of pairwise distinct STRD functions such that $\sum_{i=1}^{d} f_i(v) \leq 1$ for all $v \in V$. The signed total Roman domatic number of $G$ is the maximum size among all its STRD families. In this paper, we relate the signed total Roman domination number of a cubic graph to its open packing number, 2-tuple total domination number, and signed total domination number, allowing us to derive sharp bounds on the first invariant and to establish new $\mathcal{NP}$-completeness results for all four invariants. We demonstrate that having a degree-3 vertex determines a graph's signed total Roman domatic number. Combined with known results this implies that the associated decision problem is easy for graphs with maximum degree at most three and $\mathcal{NP}$-complete otherwise. To contrast these general hardness results, we determine signed total Roman domination and domatic numbers in complete multipartite graphs. Despite their simple structure, this is a non-trivial task.

We adapt ideas of Kim and Roush [15], originally developed in the study of automorphisms of sofic subshifts, to obtain sufficient conditions under which a subshift has a huge automorphism group. We apply this approach to non-sofic subshifts defined by sets of multiples. In particular, we establish a dichotomy for the $\mathscr{B}$-admissible subshift: its automorphism group is either trivial or contains an embedded copy of the automorphism group of the full shift $\{0,1\}^{\mathbb Z}$. In the latter case, we say that the automorphism group is huge. We further show that the automorphism group of the hereditary closure of the $\mathscr{B}$-free subshift is huge whenever $\mathscr{B}\subset \mathbb N$ is infinite and contains no infinite pairwise coprime subset.

In 1796, Gauss succeeded in solving the problem of constructing the regular 17-gon with compass and straightedge. Later he proved that, using a compass and straightedge, it is possible to construct the regular polygons with $n=2^m n_1\cdots n_l$ sides if $n_1,\cdots, n_l$ are different prime numbers of the form $n_k=2^{2^{\nu_k}}+1$. Gauss also knew that only these regular polygons can be constructed but did not prove it.\linebreak P. Wantzel completed the result of Gauss and proved it in 1837. The present paper provides a new proof for Wantzel's theorem.

We construct a contact form on a three dimensional CR manifold such that the CR Yamabe flow fails to converge. More precisely, on small Rossi deformations of the standard CR three-sphere, we exhibit an example whose corresponding CR Yamabe flow develops a one-bubble concentration regime. The construction is based on the negativity of the pseudohermitian mass on the Rossi spheres. This shows that mass positivity is not merely a technical assumption in the known convergence results for the CR Yamabe flow, but is genuinely connected to the large-time dynamics of the flow.

For a graph $G$, a proper $k$-coloring of $G$ is equitable if the sizes of any two color classes differ by at most one. The \textsc{Equitable $k$-Coloring} problem asks, for a given graph $G$ and integer $k$, whether $G$ admits an equitable $k$-coloring. Bodlaender and Fomin showed that it is polynomial-time solvable on graphs of bounded treewidth, while it remains $\NP$-hard on cographs, and thus on graphs of constant clique-width. Fellows et al. showed that the problem becomes $\mathsf{W[1]}$-hard when parameterized by tree-width (and hence clique-width) plus the number of colors~$k$. We first show that, for every fixed $k$, counting equitable $k$-colorings is polynomial-time solvable on graph classes of bounded clique-width, given a clique-width expression. We then show that, under $\mathsf{SETH}$, the dependence on clique-width in this algorithm is essentially optimal. As a consequence, our results provide a fairly tight picture of the complexity of \textsc{Equitable $k$-Coloring} with respect to the combined parameter $k$+clique-width. Second, we refine our clique-width algorithm for the linear setting. We show that there exists an algorithm, given an integer $k\ge 1$ and an $n$-vertex graph $G$ together with a linear $w$-expression constructing $G$, computes the number of equitable $k$-colorings of $G$ in time $\max\{1,2^k-2\}^w\cdot n^{k+O(1)}$. Third, we consider a different structural restriction, namely the class of $P_t$-free graphs. A graph is called $P_t$-free if it does not contain the path on $t$ vertices as an induced subgraph. This is a different setting from bounded clique-width; in particular, already $P_5$-free graphs have unbounded clique-width. Nevertheless, we show that for every $P_t$-free graph $G$, the number of equitable list $3$-colorings of $G$ can be computed in subexponential time.

We present a new proof of the Positive Mass Theorem and the Riemannian Penrose Inequality for three-dimensional asymptotically flat Riemannian manifolds whose metrics fail to be $C^1$ across a hypersurface $\Sigma$, first proven by Miao and McCormick-Miao, respectively. Unlike these approaches, ours recovers these results directly, without relying on their original formulations for smooth metrics. The proofs are based on a unified argument which applies to both theorems. We achieve this by establishing an approximate monotonicity for the quantity introduced by Agostiniani-Mantegazza-Mazzieri-Oronzio, employing the approximation scheme of Miao, for metrics with $C^{2,\alpha}$ regularity up to $\Sigma$.

We prove a limited asymptotic expansion up to order $t^4\log t$ of the logarithmic derivative of the Selberg zeta function for the spin Dirac operator on compact surfaces for families of hyperbolic metrics degenerating towards complete hyperbolic metrics with cusps in a pinching process where the lengths $l_j(t)$ of certain disjoint simple closed geodesics converge smoothly to $0$ as $t\to 0$.

In earlier work \cite{Mah19}, the first author constructed, for each finite $m\geq 1$, a computable tree $\A_{m+1}$ of rank $m+1$ whose strong degree of categoricity is $\mathbf{0}^{(2m)}$, and showed this degree is optimal at each rank. Those results are lightface: they concern Turing degrees of isomorphisms between computable copies. In this paper we determine the boldface content of the construction. We isolate a transfer principle: a degree-of-categoricity lower bound that holds uniformly relative to every oracle defeats $L_{\omega_1\omega}$-definability of automorphism orbits outright. We verify that the construction of \cite{Mah19} has this uniformity, and deduce that $\A_{m+1}$ has Scott rank exactly $2m+1$, with Scott sentence complexity one of $\Sin{2m+1}$, $\dSin{2m+1}$, or $\Pin{2m+2}$. For rank $2$ we carry out a complete, computability-free Scott analysis: the orbit of the level-one nodes of infinite degree is $\Pin{2}$- but not $\Sin{2}$-definable, and $\SSC(\A_2)=\Pin{4}$ exactly, witnessed by a computable $\Pc{4}$ Scott sentence. We then prove $\SSC(\A_{m+1})=\Pin{2m+2}$ for all finite $m$: among the three candidates, only $\Pin{2m+2}$ is consistent with the parameterized Scott rank, and a parameter-reservation argument – naming any finite tuple reserves only finitely many top-level subtrees, and the relativized coding survives on the infinitely many spare subtrees – pins $\pSR(\A_{m+1})=2m+1$, selecting that candidate. These results begin a classification of the Scott sentence complexities of trees of finite rank, in analogy with the Gonzalez–Rossegger analysis of linear orders, and connect the $2\alpha$-jump phenomenon in degrees of categoricity to Scott spectral-gap questions for trees.

We study endogenous reinsurance pricing in a competitive insurance market with one strategic reinsurer and many heterogeneous insurers. The reinsurer acts as a Stackelberg leader by choosing a common premium rate and an investment strategy, while insurers decide how much risk to retain and how to invest, taking into account their own performance, their performance relative to the insurer population, and common insurance-claim and financial-market noise. This creates a feedback loop absent from standard reinsurance models with exogenous premiums: a premium change affects insurers directly through the cost of reinsurance, and indirectly through the population's aggregate exposure to common insurance-claim risk. For a fixed premium, we characterize the insurers' equilibrium retention through a scalar fixed point and establish its monotone premium response. This characterization reveals a spillover mechanism generated by relative performance concerns and leads to a threshold structure in which insurers move from full cession to partial retention and then to full retention as the premium increases. Using this structure, we reduce the reinsurer's premium problem to a one-dimensional optimization over a compact premium interval and characterize Stackelberg equilibria in both finite-player and mean field models. In the finite-player case, we develop an efficient threshold continuation procedure that determines equilibrium premiums without enumerating all retention configurations. We also prove convergence from finite-player equilibria to mean field equilibria without requiring the mean field equilibrium premium to be unique. Numerical illustrations show how relative performance concerns amplify spillover effects and can induce retention even when reinsurance remains actuarially favorable. They also demonstrate that Stackelberg equilibria need not be unique in either setting.

In their 2010 article entitled “How to invert $n$ cups $m$ at a time" in Mathematics Today, Man-Keung Siu and Ian Stewart extend a classic trick in which $3$ cups are flipped $2$ at a time to any number of cups, flipped any number at a time. Here, we generalize another feature of the problem. A typical cup in our universe, as far as we know, has two “sides", right side up and upside down. But what if the cup had $k$ “sides" with $k\ge 2$? We call these $k-$hypercups. The classic trick depends on parity. In this article, we show the analogous hypercups trick depends on greatest common divisors. We generalize all of Siu and Stewart's results to $k-$hypercups. Our results imply the known results for $2-$hypercups, also known as cups.

We classify the sets of $L$-space surgeries on all two-bridge links, providing the first examples of hyperbolic links for which such sets cannot be described as unions of finitely many rectangles in $\mathbb{Q}^2$. The proof relies on several different techniques, each of which is applicable in greater generality: we introduce a sufficient diagrammatic condition for links in $S^3$ to be persistently foliar, a property that implies that every non-trivial surgery on such links supports a coorientable taut foliation. We define a simplified model for the Heegaard Floer homology of rational surgeries on two-component $L$-space links, following the work of Manolescu-Ozsváth, Liu, and Zemke, and use it to obtain obstructions to $L$-space surgeries. Finally, we use explicit computations of Turaev torsions to determine $L$-space surgeries in the case of generalised $L$-space links. Among the consequences of our results, we obtain an optimal uniform bound on the volume of any hyperbolic $L$-space that is surgery on a two-bridge link, together with a classification of all $L$-space satellite knots whose associated two-component pattern link is a two-bridge link.

We establish the Banach-range boundedness of bilinear rough singular integral operators, together with their maximal and maximally truncated forms, under the fractional geometric condition on the mean-zero angular kernel \[ \sup_{\xi \in \mathbb{S}^{1}}\int_{\mathbb{S}^{1}} \frac{|\Omega(\theta)|}{|\theta \cdot \xi|^{a}} \, d\sigma(\theta) < \infty, \qquad \frac12 < a < 1. \] This condition imposes integrability strictly weaker than the $L^q(\mathbb{S}^1) (q>1)$ constraints considered by Grafakos, He, Honzík (Adv. Math., 2018), Dosidis and Slavíková (Math. Ann., 2024), while defining a class of functions that is neither contained in nor contains the classical Orlicz space $L(\log L)^\alpha(\mathbb{S}^1) $ ($\alpha>1$). Our proof avoids traditional wavelet decompositions of the multiplier, instead using local Fourier series expansions of the input functions.

This paper is devoted to $\mathbb{N}$-graded supermanifolds $\mathcal{M}$ whose grading is induced by a homogeneity action, i.e., a smooth action $\mathbb{R}\ni t\mapsto h_t$ of the multiplicative monoid of real numbers on $\mathcal{M}$. We show that the map $h_0$ is a smooth retraction onto a submanifold $M=h_0(\mathcal{M})$, and that $h_0:\mathcal{M}\to M$ is a fiber bundle with typical fiber $\mathbb{R}^{m|n}$. Using this homogeneity approach, we obtain a simple proof of a homogeneous version of the Frobenius theorem. If, in addition, $h_{-1}$ acts as the parity operator on $\mathcal{M}$, we provide a geometric characterization equivalent to the recent definition of $\mathbb{N}$-manifolds due to Bursztyn, Cueca, and Mehta in terms of sheaves of graded algebras with prescribed local models. As a consequence, the homogeneous Frobenius theorem for $\mathbb{N}$-manifolds proved by these authors appears as a special case of our more general result.

Dual-space multilevel kernel-splitting (DMK) is a fast summation framework that combines ideas from the fast multipole method, Ewald summation, and multilevel summation. Originally formulated for free-space problems, and later extended to fully periodic problems on a cube, it decomposes the kernel interaction into a smooth global contribution and a hierarchy of localized interactions evaluated on an octree. We extend DMK to problems on rectangular cuboids with periodic boundary conditions in one, two, or three coordinate directions. The periodization leverages the fact that interactions on all tree levels below the root are localized, allowing for their evaluation with minimal modification on a cubical tiling of the domain. The remaining smooth root-level far-field contribution is evaluated in Fourier space, with Fourier series in the periodic directions and Fourier integrals in the free directions. For reduced periodicity, truncated kernels are used to regularize singular and near-singular Fourier kernels, yielding rapidly convergent trapezoidal discretizations and a unified treatment of all periodicities. For large-aspect-ratio cuboids, the root-level sum can be accelerated using the fast Fourier transform. We validate the method for the electrostatic potential and Stokeslet, stresslet and rotlet potentials, for all periodicities and a wide range of aspect ratios. Numerical experiments show that the periodization adds only a small overhead to the original free-space DMK algorithm, also for high-aspect-ratio cuboids. The resulting method provides a framework for applying DMK to problems with mixed periodicity on rectangular cuboids, and extends naturally to other non-oscillatory kernels for which a kernel split is available.

A graph with maximum degree $\Delta$ is \emph{$\Delta$-critical} if it is connected, has chromatic index $\Delta+1$, and the deletion of any edge reduces its chromatic index to $\Delta$; it is nontrivial if it has no subgraph $H$ with $|E(H)|>\Delta\lfloor |V(H)|/2\rfloor$. We prove a characterization of all nontrivial $3$-critical graphs: every such graph is obtained either from smaller $3$-critical graphs by replacing a vertex with a triangle, by a Hajós-join, or by replacing a vertex with a suitable class~$1$ graph, or from a snark by deleting at most one vertex and some independent edges. The third operation, which we call a Meredith-type extension, generalizes the Meredith extension by allowing other class~$1$ graphs in place of $K_{2,3}$, and we give a sufficient condition under which it produces new nontrivial $3$-critical graphs from smaller ones. To the best of our knowledge, this is the first characterization of this kind for edge-chromatic $3$-critical graphs, and the Meredith-type extension has not previously been considered in this generality. The characterization was found by analyzing the output of an exhaustive computer search for nontrivial $3$-critical graphs of orders $13$ to $24$, and we make the resulting data publicly available. Based on these data, we also propose two conjectures that are closely related to the case $\Delta=3$ of Seymour's exact conjecture and to Grötzsch's conjecture on the $3$-edge-colorability of planar graphs.

We construct a rate-$2/3$ quantum low-density parity-check (LDPC) code from a $(3,18)$-regular two-branch finite-field base and a circulant-permutation-matrix (CPM) lift of degree $P=101$. The resulting code is a Calderbank–Shor–Steane (CSS) code with parameters $[[34542,23032,18]]$. Its distance is established by a computer-assisted proof: an explicit nontrivial logical operator gives the upper bound, and a complete symmetry-reduced enumeration excludes every nonzero vector of weight below 18 in both parity-check kernels. We also prove that every affine-in-$t$ CPM lift of the same base satisfying CSS orthogonality has distance at most 18 for every prime lift degree $P>19$. The construction has row weight 18 and column weight 3, and the Tanner graphs of $H_X$ and $H_Z$ separately have girth 8. Decoder experiments with log-likelihood-ratio (LLR) joint belief propagation (BP) and deterministic post-processing show no failures in $10^8$ trials at $p=0.01$, and a finite-length frame error rate (FER) sweep estimates the transition near $p=0.029$.

The Mordell conjecture asserts that there are only finitely many rational points on a smooth projective curve of genus at least two over a number field. The uniform Mordell problem asks for suitable upper bounds on the number of rational points in the Mordell conjecture, and has been solved by combining works of Vojta, Dimitrov–Habegger–Gao and Kuhne. In this survey, we will introduce a quantitative version of the uniformity problem proved by the recent work of Yu–Yuan–Zhou.

We study the Threshold Group Testing (TGT) problem without a gap in the noiseless, non-adaptive setting, where the goal is to exactly recover a sparse binary vector from pooled test outcomes using as few tests as possible. In TGT, a test applied to a subset of items returns a positive outcome if the number of defective items in the subset reaches a prescribed threshold, and a negative outcome otherwise. Under the assumption of an analytic condition, TGT has been shown to undergo a sharp information-theoretic phase transition for exact recovery on the class of constant-column test designs. In this paper, we develop an efficient inference algorithm that achieves exact recovery with high probability using the minimum number of non-adaptive tests that are needed for the constant-column design, thereby matching the information-theoretic threshold of a natural benchmark test design. Our approach is based on a spatially coupled test design and admits a significantly simpler analysis than existing algorithms for related group testing problems. In particular, unlike previous methods for binary group testing, our algorithm does not rely on the analysis of intricate weighted sums. This leads to a more straightforward proof technique, while still allowing near-optimal performance guarantees.

We prove that the differential of the log of the partition function for the $2$-matrix model with quartic interactions converges in a certain double-scaling regime to the differential of the $\boldsymbol{\tau}$-function for the $(3,4)$ string equation. This confirms the convergence of the critical Ising model on random surfaces to the $(3,4)$ topological minimal model, which was stated in the works of Douglas and Shenker, Brézin and Kazakov, and Gross and Migdal. Our analysis is based on a steepest-descent analysis of a Riemann-Hilbert problem associated to a family of biorthogonal polynomials. New features in the matching problem in the construction of local parametrices appear.

The Maker-Breaker sabotage game is played on a graph $G$ by Runner and Blocker. They play in turns, Runner first moves along a not yet traversed edge from her current position, afterwards Blocker removes one edge. The goal of Runner is to visit as many vertices of $G$ as possible, Blocker's goal is opposite. Assuming that both players use optimal strategies, the number of vertices visited by Runner determines an invariant called the sabotage number ${\rm sab}(G)$ of $G$. A formula for the sabotage number of an arbitrary tree is proved which can be evaluated in polynomial time. For a unicyclic graph $G$ it is proved that ${\rm sab}(G)\in \{{\rm sab}^-(G), {\rm sab}^-(G)+1\}$, where ${\rm sab}^-(G)$ is the lower sabotage number of $G$. The sabotage number of a bridgeless subcubic graph is sharply bounded from the above by the maximum girth. The sabotage number is also bounded for complete bipartite graphs and generalized Sierpiński graphs, and determined exactly in some special cases.

For a smooth algebraic curve defined over a number field, one can associate a bipartite graph called a dessin d'enfant. We study the regularity and automorphism groups of dessins with uniform passports. In a previous paper, we proved that every passport of the form $[n,b^{q},n]$ of genus at least 2 admits a dessin with trivial automorphism group. Here we prove the analogous result for passports of the form $[b^{q},b^{q},n]$. We also construct examples of uniform passports of genus at least 2 for which every dessin with that passport has nontrivial automorphism group, and others for which every dessin with that passport has trivial automorphism group. Finally, we give an alternative proof of the $[n,b^{q},n]$ case using counting arguments based on centralizers of permutations.

We study the occupation-time fluctuations of a critical age-dependent branching particle system with immigration in $\mathbb{R}^d$. Immigrants arrive according to a homogeneous Poisson random measure in space and time. Each particle moves independently according to a symmetric $\alpha$-stable process and, at the end of its lifetime, either dies or splits into two offspring with equal probability. The lifetime distribution is allowed to have either finite mean or a heavy tail of index $\gamma\in(0,1]$. We investigate the asymptotic behavior of the centered occupation-time process under a suitable space-time scaling. Assuming $ \alpha<d<(1+\gamma)\alpha, $ we prove that the rescaled occupation-time fluctuations weakly converge as processes with values in the space of tempered distributions to a centered Gaussian process with an explicitly identified covariance structure. The normalization and the covariance depend on both the stability index $\alpha$ and the tail exponent $\gamma$. The limiting process is self-similar, possesses long-range dependence, and is neither Markovian nor a semimartingale. In contrast with the corresponding age-dependent branching system without immigration, the contribution of the initial population vanishes in the limit, and the asymptotic fluctuations are entirely determined by the immigration mechanism. When $\gamma=1$, our results recover the covariance structure previously obtained for branching systems with immigration and finite-mean lifetimes. The proofs rely on the space-time random field approach, Fourier analytic techniques, and asymptotic properties of renewal functions associated with the lifetime distribution.

This article is an expository account of the basics of adelic line bundles on quasi-projective varieties, arithmetic positivity of adelic line bundles, and applications of positivity to an equidistribution theorem and the uniform Bogomolov conjecture.

In this paper we study the dynamics of relativistic detonation waves theoretically and numerically. The reaction is physically accounted for by an extra term in the definition of the total energy density and by an additional equation for the evolution of the mass fraction of the reactant, while leaving formally unmodified the equations of mass and energy-momentum conservation. In this way, the Rankine-Hugoniot relations maintain the same formal structure of the inert version. For the numerical solution we use a finite volume ALE scheme with a second order TVD space-time reconstruction for the homogeneous part of the system, where the mesh velocity is chosen equal to the shock speed, and we combine it through operator splitting with a locally implicit algorithm for the potentially stiff reaction source terms that arise in the equation of the reactant. We furthermore propose a particularly efficient algorithm for the conversion from the conserved to the primitive variables, which for the relativistic Euler equations is known to be nontrivial. Following this approach, we can successfully solve the Zel'dovich-von Neumann-Doering profile of a relativistic detonation wave, up to Lorentz factors of the shock front $\gamma_s\sim 7$. Our analysis allowed us to highlight a new special relativistic effect, which has remained unnoticed so far. While in Newtonian detonations the Zel'dovich pressure jump decreases monotonically with the mass flux through the shock front, in the relativistic case it shows a minimum and then rises monotonically as a function of the mass flux. This may have interesting physical implications on the amount of energy that can be extracted from a relativistic detonation wave.

In the pinwheel scheduling problem, each task $i$ is associated with a positive integer $a_i$ called its period, and we want to (perpetually) schedule one task per day so that each task $i$ is performed at least once every $a_i$ days. An obvious necessary condition for schedulability is that the density, defined as the sum of execution rates $1/a_i$, does not exceed $1$. We prove that all instances with density not exceeding $5/6$ are schedulable, as was conjectured by Chan and Chin in 1993. Like some of the known partial progress towards the conjecture, our proof involves computer search for schedules for a large but finite set of instances. A key idea in our reduction to these finite cases is to generalize the problem to fractional (non-integer) periods in an appropriate way. As byproducts of our ideas, we obtain a simple proof that every instance with two distinct periods and density at most $1$ is schedulable, as well as a fast algorithm for the bamboo garden trimming problem with approximation ratio $4/3$.

Extending a little-known construction of Ruzsa, we characterize the quadruples $(s,t,u,v)$ of nonnegative integers such that the inequality $|sA-tA| \leq |uA-vA|$ holds for all finite sets $A \subseteq \mathbb{Z}$

The superiorization method (SM) is situated between feasibility-seeking and constrained optimization. Instead of aiming at the minimum of a given objective function over a constraint set, it seeks a feasible point at which the objective function value is reduced - though not necessarily minimal - rather than hard targets, or in which a mathematically optimal solution is not strictly required. While the method has been investigated for several applications in physics, its broader use has been limited, in part due to the lack of openly available software for researchers wishing to explore it. In this work we apply superiorization to three problems from applied physics: seismic image reconstruction, low-dose CT reconstruction and intensity-modulated radiotherapy treatment planning. These experiments are conducted with SupPy, an open-source modularized Python toolbox developed for this work, which supports execution of feasibility-seeking algorithms and their superiorized version on both the CPU and the GPU. In all three cases the superiorized algorithms achieve favorable results compared to feasibility-seeking alone, with reduced noise in the imaging examples and lowered body dose in the radiotherapy plans. For the radiotherapy case we further observe that superiorization produces clinically viable plans on infeasible constraint sets.

We prove that commensurators of a geometrically rigid residually finite hyperbolic group have bounded average distortion.

We study the forward discrete space-time wave equation on the homogeneous $(q+1)$-regular tree $T_{q+1}$ associated with a two-parameter generalized Laplacian. Under the natural nonnegativity assumption on this operator, we derive explicit formulas for the two fundamental wave kernels. The formulas are given in terms of discrete $I$-Bessel functions and yield convolution representations for solutions with general initial conditions. In the boundary case corresponding to the bottom of the spectrum, we obtain another explicit representation of the wave kernel in terms of discrete $J$-Bessel functions. This representation leads to a discrete analogue of the classical $I\!\leftrightarrow\!J$ relation. We also perform both analytic and numerical studies of the asymptotic behavior of the wave kernels, including large radial distance, large time, and large degree of the tree. An important feature of our analysis is that the wave kernels are expressed as finite sums; hence, the propagation formulas remain finite for every discrete time.

We study the problem of finding stationary points of non-convex functions when access to the objective is provided only through a comparison oracle that, given two points, outputs which has the larger function value. For a twice differentiable $f\colon\mathbb R^n\to\mathbb R$ with Lipschitz gradient and Hessian, we develop an algorithm that visits an $\epsilon$-stationary point using $\widetilde O(n^2/\epsilon^{1.5})$ queries. Our approach uses a subroutine that estimates the normalized Hessian to accuracy $\delta$ using $\widetilde O(n^2\log(1/\delta))$ queries. We further study this problem with a quantum comparison oracle model where queries can be made in superpositions, and develop the first quantum algorithm that finds an $\epsilon$-stationary point, which takes $\widetilde O(n/\epsilon^{1.5})$ queries.