Papers

We study the one-dimensional Dirichlet spectrum and obtain new results concerning the origin $\mu$ of the ray contained in the spectrum and its isolated points. First, we prove the improved upper bound $\mu \leq (448875+1611\sqrt{67773})/46982 = 18.4809\ldots$. We also exhibit an explicit gap $(L,R)$ in the Dirichlet spectrum, which yields the improved lower bound $\mu \geq R = 5.9596\ldots$. The proof of the lower bound includes an exact computer-assisted finite search over admissible continued-fraction words. We further prove that all points $Z_k$, $k\geq 2$, introduced by Ivanov are isolated points of the Dirichlet spectrum. Finally, we determine the asymptotic behavior of the neighboring gaps around $Z_k$ and describe the equivalence classes of all irrational numbers whose Dirichlet value is $Z_k$.

We study tilings of closed surfaces by congruent most curvilinear polygons. By relating such tilings to branched coverings determined by planar diagrams and vertex types, we establish necessary and sufficient conditions for a given closed topological surface to admit a metric of constant curvature supporting a tiling with prescribed combinatorial data and number of tiles. We also characterize minimal tilings, which do not descend through nontrivial unbranched coverings to tilings with fewer tiles, in terms of common block decompositions of the associated permutations. Explicit permutation constructions yield infinite families of minimal tilings with prescribed vertex types, even when the planar diagram and vertex types are fixed. In the hyperbolic setting, these results also give finite-index subgroups of the universal tiling symmetry groups that are maximal among torsion-free subgroups, with the index equal to the number of tiles.

For a positive integer $k$, let $\beta_k(G)$ be the maximum cardinality of a vertex set inducing maximum degree less than $k$, and let $SF_k(G)$ be the maximum cardinality of a minimal $k$-star-forming set. The known inequality $\beta_k(G)\le SF_k(G)$ suggests asking when equality holds. We place this question in the framework of upper domination: at $k=1$, $\beta_1(G)=\alpha(G)$ and $SF_1(G)=\Gamma(G)$, so the classical equality $\Gamma=\alpha$ on bipartite graphs is exactly the first member of the proposed hierarchy. We prove the equality for complete bipartite graphs for every $k$, obtaining \[ \beta_k(K_{a,b})=SF_k(K_{a,b})=\max\{a,b,2k-2\}\qquad(a,b\ge k), \] and record the elementary low-degree case $\Delta(G)<k$. For $k=2$ we derive certificate restrictions for minimal $2$-star-forming sets in bipartite graphs. For chain graphs we go further: we prove $\beta_2(G)=SF_2(G)$ and obtain an exact formula for their common value. The proof uses the nested-neighborhood structure together with a classification of witnesses to the indispensability of a high internal-degree vertex. We retain the equality problem for chain graphs as a conjecture only for $k\ge3$, and formulate the broader bipartite conjecture. We also determine the upper domination number of every rectangular grid and combine it with the known exact dissociation number to compare $\Gamma$, $\beta_2$, and $SF_2$. In particular, $\Gamma=\beta_2$ on every even-by-even rectangular grid, while $\beta_2\le SF_2$ always; this motivates a grid equality conjecture whose even-by-even case would yield a three-parameter identity.

Three neural networks are compared as solvers of nonlinear ordinary differential equations: a multilayer perceptron (MLP), a Kolmogorov-Arnold network (KAN) whose edges are B-splines, and a KAN whose edges are Chebyshev series. Each network is trained only on the residual of the equation and on its initial or boundary conditions. The exact solution, or a numerical solution accurate to $10^{-13}$, is used only afterward to measure the error. Five test problems of differential orders one to four are investigated: the logistic equation, a singular Lane-Emden type initial value problem, a singular boundary value problem, the Falkner-Skan equation with suction, a fourth-order problem. The same training budget and the same choice of five activation functions are allocated to all networks. No numerical failures are encountered in any of the \nRuns{} training runs. For every problem, the best KAN is found to be at least as accurate as the best MLP while \paramsKANmin{} to \paramsKANmax{} parameters are used instead of \paramsMLP{}; the margin ranges from a tie to a factor of three. In the Falkner-Skan setting of an earlier study, the B-spline KAN is shown to be 34 to 220 times more accurate at every published test point. Much smaller MLPs are also found to be competitive, whereas at orders three and four, failures are experienced more frequently by KANs, and a coarse basis is required. The derivatives in the equation are distorted by the usual grid extension of a B-spline KAN; consequently, it is replaced with knot insertion, by which the logistic error is lowered more than a hundredfold, though no improvement is provided at order four.

This paper studies flasque resolutions for linear reductive group schemes over arbitrary bases, extending Colliot-Thelene's theory. A flasque resolution exists for any linear reductive group scheme over a connected base scheme (resp. a quasi-compact, quasi-separated base scheme). The construction relies on Borovoi's fundamental group and descent theory involving automorphisms of Chevalley groups. Applications include R-equivalence for reductive group schemes over rings and over a field of cohomological dimension at most 2.

We study spontaneous scalarization of static, spherically symmetric black holes in scalar–Gauss–Bonnet gravity with a Ricci coupling $F=1 \beta\phi^{2}/4+\gamma\beta^{2}\phi^{4}+O(\phi^{6})$ that nonlinearly completes the quadratic one. For arbitrary couplings and potential, we derive the near-horizon regularity condition and horizon curvature invariants in closed form; scalarized branches terminate where this condition degenerates, at finite horizon curvature. A weakly nonlinear expansion fixes the bifurcation direction exactly: for a massless scalar, its governing coefficient is quadratic in $\beta$, changes sign at $\beta=1.1156$, and depends linearly on a quartic self-interaction. At equal horizon values of the scalar, completions differ in mass, to leading order, in proportion to $\gamma$. A radial-perturbation master equation for arbitrary couplings confirms the exchange of stability at the bifurcation and turning points. Consequently, the domain of radially stable black holes shrinks as $\gamma$ increases over the range studied; it is smaller for the exponential completion than for the quadratic truncation.

We study online convex optimization with one unbiased stochastic subgradient per round and an unknown finite conditional $p$th noise moment, $1<p\le2$. For every fixed interval $I$ of length $n$ and comparator path with $\Lambda_I=1+P_I/D$, one learner achieves \[ E[Regret_I(u)]\le\min(GDn, C[GD\sqrt{n(\Lambda_I+\log^2(2T))} +\sigma Dn^{1/p}(\Lambda_I+\log^2(2T))^{(p-1)/p}]). \] The learner uses none of $G,\sigma,p,I,P_I$, and the constant is universal. Interval adaptation adds to comparator complexity, preserving the distinct mean-gradient and noise exponents. The analysis controls calibration in expectation and limits the cost of observation-scale changes. Its general theorem compares to distributions over predictably available experts with relative-entropy dependence on a nonuniform prior. A common prior favors long windows and long restart lengths. With the statistics supplied, the interval cost becomes $1+\log(T/n)$, including the optimal full-horizon static rate. A change-of-measure lower bound identifies the noise power of this logarithm for learners retaining a full-horizon optimal guarantee, under explicit conditions. Static comparisons and deterministic partitions follow from the same decisions.

A dagger on a category assigns to every morphism $f \colon X \to Y$ a morphism $f^\dagger \colon Y \to X$, contravariantly and involutively, in the way the transpose reverses a real matrix. We show that on the categories of linear algebra and differential geometry, daggers encode metric and Riemannian structures. As is known, on a single vector space, an anti-involution of the endomorphism ring is the adjoint of a form that is determined up to a scalar. But more generally, on the category of all finite-dimensional vector spaces, a dagger is the adjoint for an involution of the scalars together with a unique unimodular Hermitian form on every space. The same holds for finitely generated projective modules over a commutative ring, and hence, through the Serre-Swan correspondence, for vector bundles, where a dagger is a choice of bundle metrics. The dagger structure expresses notions of orthogonality categorically: orthogonal complements and normal bundles acquire universal properties, and Hermitian modules form a dagger compact category. Moreover, daggers on Lie groups, Lie groupoids and their infinitesimal counterparts capture unitary representations and metric connections: a connection is metric precisely when its parallel transport is a dagger functor, or, infinitesimally, when its splitting of the Atiyah sequence commutes with the daggers. Finally, we classify the daggers on the locally graded category of differential operators of a Lie-Rinehart algebra. They correspond to metric-divergence structures, consisting of a real structure, a divergence, and a unimodular Hermitian form on every finitely generated projective module, and they act on operators as formal adjoints; Green's formula expresses the difference between an operator and its adjoint as a divergence.

The universal approximation property of dropout neural networks does not by itself describe the network size required for an accurate random realization. In this work, we study approximation of the unit ball of $W^{n,\infty}([0,1]^d)$ by ReLU networks whose edges are retained independently with probability $p$. The approximation error is measured uniformly over the input domain, and the guarantee holds with probability at least $1-\delta$ for a single sampled network. We construct networks of constant depth and size $\widetilde O_{n,d}(p^{-9}\varepsilon^{-\max\{d/n,2\}} \log(1/\delta))$. The construction combines bounded local subnetworks, localization on a successful approximation event, and a multiscale Taylor decomposition. Conversely, Sobolev capacity imposes a lower bound on the number of surviving edges, while approximation of a fixed affine function requires an output-layer cost of order $((1-p)/p)\varepsilon^{-2}\log(1/\delta)$ at sufficiently high confidence. For fixed $p\in(0,1)$ and $\delta<\min\{1/2,1-p\}$, the upper and lower bounds match in the accuracy exponent under a fixed or logarithmic depth budget. When $d\leq2n$, they also match in confidence up to logarithms of accuracy. We extend the lower bounds to $W^{n,r}$ targets with $L^s$ error, and distinguish this extension from the upper bound for $W^{n,\infty}$. The optimal retention dependence and logarithmic factors remain open.

Adapting the circle method developed by Frantzikinakis–Klurman–Moreira, we establish a sufficient criterion for a set of natural numbers to be multiplicatively recurrent. We apply this criterion to Piatetski–Shapiro sets with small exponents and to regular irrational nil-Bohr sets. We also study the value distribution of unimodular completely multiplicative functions along these sets.

This paper develops a new moment-matching order reduction method for multi-input quadratic-bilinear descriptor systems. The proposed approach accommodates multiple inputs acting on both the differential and algebraic equations and constructs separate projection spaces associated with different input channels and input combinations. The method uses a matrix-free algorithm, allowing the reduced-order matrices and tensors to be computed without explicitly assembling or storing the full-order system matrices and tensors. This feature makes the proposed method particularly suitable for large-scale computational fluid dynamics problems. Numerical experiments carried out for two-dimensional flow problems demonstrate that the resulting reduced-order models accurately reproduce the transient responses of the full-order models under multiple time-varying inputs.

Let $K_t$ be obtained by attaching a pendant path of length $t$ to a fixed rooted graph $K$. We prove that $\mathrm{Mo}(L(K_t))-\mathrm{Mo}(K_t)$, the difference between its line-graph and original Mostar indices, is exactly affine on each parity class beyond a sharp uniform cutoff. The slope depends on distance-level neighbor counts and is positive for every non-bipartite core. Triangle chains give infinitely many graphs with $\mathrm{Mo}(L(G))>\mathrm{Mo}(G)$ at every positive cyclomatic number $c$, with maximum degree three, resolving Alex-Indulal Problem 3.3. At zero slope, a cactus branch-mass formula decides equality; identical vertex profiles need not give identical intercepts. Fixed cores giving equality for all sufficiently long attachments exist exactly when $c$ is odd. Maximum degree three suffices, and at every even $c$ a binary-tree construction gives equality on one eventual parity class.

We show that the spectral radius introduced by Shalit and Shamovich uniquely determines the underlying operator space structure: If E_1 and E_2 are operator space structures on C^d such that the corresponding spectral radii \rho_{E_1} and \rho_{E_2} agree on every d-tuple of operators, then the identity map is completely isometric. This answers a question of Scherer, Shalit and Shamovich. Moreover, we obtain an analogous result for the minimal spectral radii, and showcase an application to the completely bounded isomorphism problem for algebras of bounded non-commutative functions on operator space unit balls.

This work introduces a hyperbolic relaxation system designed for the simulation of kinetic equations in the low-Mach number limit. The methodology is built upon a micro-macro decomposition of the scaled BGK model, which reformulates the kinetic distribution function into a coupled system consisting of a macroscopic equilibrium part and a microscopic non-equilibrium remainder. By projecting the microscopic deviations onto a set of orthogonal polynomials, we derive a closed moment relaxation system. The resulting system is a version of Grad's 13 moment system with a linear hyperbolic part and a relaxation adapted to the incompressible limit. We prove the model's structural stability in the incompressible Navier-Stokes limit. Moreover, we develop a high order Asymptotic-Preserving (AP) numerical framework using Implicit-Explicit (IMEX) Runge Kutta schemes for temporal accuracy and finite difference WENO reconstructions as well as central difference approximations for high order spatial resolution. The proposed scheme ensures uniform stability and consistency across different physical regimes, automatically degenerating into a consistent high order discretization of the incompressible thermal Navier-Stokes limit as the scaling parameter vanishes. Numerical experiments in one and two dimensions corroborate the theoretical findings, demonstrating significant reductions in computational cost and robust performance across a wide range of Knudsen and Mach numbers.

Let $A$ and $B$ be polynomials with integer coefficients. We study nonzero integer solutions of $$y\mid A(x),\qquad x\mid B(y),$$ using quotient transformations that may change the polynomial pair. We call this process congruence jumping. We establish a general reciprocal-polynomial rule governing such jumps, without coprimality, monicity, or unit assumptions, and give exact criteria for when the resulting polynomial states can be normalized integrally. Companion-surface identities provide a mechanism for constructing infinite integral quotient chains, including an explicit mixed-degree example. The quadratic case is considerably more rigid. We obtain a quantitative denominator bound for finite chains, classify the exceptional nonconstant one-sided infinite chains with nonintegral conic parameter, and show that changing quadratic states reduce to ordinary Vieta dynamics on a fixed conic. We further construct a genuinely nonunit recurrent four-cycle with infinitely many positive integral points and show that its dynamics admits a uniform Pell-type linearization. A motivating nonunit divisibility system is analyzed through Pell orbits and changing-state ladders. Finally, an independent relation-lattice criterion reduces certain solution sets to a finite divisor search.

We prove that if $F$ is a noncyclic free group, $R\triangleleft F$ and $S\le F$, then the inclusion $R'\le S'$ implies $R\le S$. Thus the normality hypothesis on the second subgroup can be omitted from the Auslander–Lyndon theorem. No restrictions are imposed on ranks or indices.

Antipodal self-duality (ASD) is a remarkable property that relates an amplitude to itself, after performing a suitable kinematic map and reversing the letters of the symbol (more precisely, acting with the antipode of the Hopf algebra of multiple polylogarithms). ASD has been seen in only one process in planar $\mathcal{N}=4$ super-Yang-Mills theory, and its physical origin is unknown, motivating a search for ASD in other contexts. Recently, ASD was found to hold for square fishnet integrals in a strongly-deformed version of planar $\mathcal{N}=4$. In this paper, we investigate the properties of polynomials in ladder integrals under the ASD transformation that leaves the square fishnet invariant. Our basic building blocks are determinants of certain matrices of ladder integrals, that are labeled by partitions or Young diagrams, and automatically obey the Steinmann relations. We employ integrability-based representations of such determinants, which are essentially matrix-model integrals, and the theory of symmetric polynomials to find many more ASD quantities – linear combinations of determinants whose coefficients are expressed using the combinatorics of partitions. As a byproduct, we provide an alternate proof that square fishnet integrals obey ASD.

We prove that the graph of a standard one-dimensional Brownian motion, $\Gamma_B=\{(t,B_t):t \in [0,1]\}$, is almost surely not conformally removable. This resolves the question left open by Doherty and Miller in their study of the Sobolev removability of the Brownian graph~\cite{DM26}. We also construct non-removable $\alpha$-Hölder graphs for every $\alpha \in (0,2/3)$, including the previously open range $\alpha \in [1/2, 2/3)$. Together with the known removability result for $\alpha>2/3$, this identifies $2/3$ as the threshold for conformal removability of Hölder graphs.

We introduce source-anchored algebraic amenability for Hausdorff ample groupoids, a Følner condition formulated in the Steinberg algebra and localized at compact open subsets of the unit space. The unanchored bisectional condition is exactly algebraic amenability of the Steinberg algebra, whereas the anchored version recovers ordinary amenability for one-unit groupoids and detects behavior invisible to the unanchored theory. We establish permanence under isomorphisms, open invariant reductions, disjoint unions, finite products, directed unions and AF approximations, while arbitrary subgroupoids need not inherit the property. For countable transitive discrete groupoids, infinite unit spaces are amenable through range escape, whereas finite unit spaces are amenable precisely when their isotropy groups are. This yields a characterization of hereditary algebraic amenability in terms of isotropy.

In this paper we define the ascending number and bridge number for virtual and welded spatial graphs. We prove that the former admits a lower bound in terms of the latter. This relationship allows us to compare different notions of unknotting virtual spatial graphs by crossing changes.

We obtain a characterization of finite non-$p$-groups with five automorphism orbits. In the solvable case of Fitting height $3$, the groups are precisely $S_4$ and $C_2^4\rtimes D_{10}$. Moreover, the corresponding fixed-point subspaces form a spread in a finite projective space. For Fitting height $2$, we distinguish the possible structures according to the number of prime divisors of $|G|$.

We study two Riordan arrays associated with the Motzkin-prefix sequence (\href{https://oeis.org/A005773}{OEIS A005773}). By connecting their production matrices through a simple boundary factorization, we obtain a period-three transformation on Motzkin endpoint distributions and exact formulas for paths by final height modulo~$3$. In addition, a unified factorization for the shifted Hankel matrices leads directly to explicit closed forms for the third and fourth Hankel determinants, $\Delta_n^{(3)}$ and~$\Delta_n^{(4)}$.

A $k$-arrangement of pseudolines is a set of bi-infinite curves in the plane such that any two of them intersect each other in exactly $k$ points, at which they cross, and it is simple if no three curves meet at a common point. Cyclic arrangements are the only simple $1$-arrangements of pseudolines that are unavoidable, in the Ramsey spirit: for each fixed $m\ge 1$, every sufficiently large simple $1$-arrangement of pseudolines has a cyclic subarrangement of size $m$. We show that, for every $m\ge 3$, the number of unavoidable simple $k$-arrangements of pseudolines of size $m$ grows exponentially with $k$, independently of $m$. For even $k$, we prove an analogous result for $k$-arrangements of pseudocircles.

We establish duality relations for conditional entropies relative to subalgebras of arbitrary von Neumann algebras, including the Umegaki relative entropy as well as the sandwiched- and Petz-Rényi divergences. Even in finite dimensions, these results extend the familiar dualities for conditional entropies from tensor-product subsystems to general subalgebras, while also lifting the framework to the fully infinite-dimensional setting. As an application, we establish an infinite-dimensional extension of quantum Stein's lemma when the alternative hypothesis is the state space of a finite-index subalgebra, which gives an operational interpretation of the relative entropy to a subalgebra as the optimal asymptotic error exponent in composite quantum hypothesis testing-the first beyond finite dimensions.

We summarize recent developments in the theory of the Benjamin-Ono equation, a well-known long-wave asymptotic model for internal waves in deep water, focusing on results obtained by means of an explicit formula for the solution of the Cauchy problem in terms of given initial data. We give a full description of this formula, and provide simple examples, with the aim of bringing it into the realm of applied mathematics. We also show how the formula simplifies further upon restriction to rational initial data. We then show how the formula and its rational restriction explain in great detail, and in a strikingly simple fashion, the asymptotic behavior of solutions of the Benjamin-Ono equation in the small-dispersion and long-time limits. In the setting of long-time asymptotics, we prove some new results related to soliton resolution yielding pointwise convergence with a concrete decay rate under the assumption of rational initial data. The first half of the paper is a stand-alone general survey and user's guide, while the appendices constituting the second half are intended for readers who want to explore the topics in greater depth, and it includes the proofs of our new results.

We consider the augmentation of randomized experiments or trials with data from observational studies for the purpose of improving statistical precision. In particular, we focus on the Hybrid Augmented Inverse Probability Weighting estimator, designed to integrate predictions from several foundation models while preserving valid statistical inference. In this article, we extend the Augmented Inverse Probability Weighting framework to the estimation of the risk ratio, the quotient of the absolute risk of an exposed to an unexposed group. Our approach allows one to use information from black-box foundation models trained on external and possibly unstructured data, yielding an estimator of the risk ratio whose asymptotic variance is never larger than the one of the default estimator based on experimental data alone.

Let $k$ be a positive integer, and consider the sequences of positive rationals with $x_0\in\N$ and $x_{n+1}=k/(x_0+x_1+\dots+x_n)$. Write $x_n=a_n/b_n$ in lowest terms. We show that there is a rational constant $c>0$ such that $c\,a_n+b_n$ is a perfect power for every such sequence and every $n\ge2$ if and only if $k=h^2$ and every prime factor of $h$ is congruent to $3$ modulo $4$; in that case $c=2/h$ and the powers are squares. The proof rests on the observation that $t_n=(x_0+\dots+x_n)/h$ satisfies $t_{n+1}=t_n+1/t_n$, a recursion under which reduced fractions never cancel. This yields an exact formula for $c\,a_n+b_n$, shows that along any single sequence each prime factor of $h$ spoils at most one term, and leads to two generalizations. For arbitrary $k$ the invariant $b_n^2-\frac4k a_n^2$ is always a rational square, and it is always an integer square exactly when the primes dividing the square part of $k$ satisfy an inertness condition in $\Q(\sqrt{-k})$. For the recursions $x_{n+1}=h^2/(x_0+\dots+x_n+n\mu h)$ the role of the Gaussian integers is played by the quadratic fields $\Q(\sqrt{\mu^2-4})$, which include $\Q(\sqrt{-3})$ and every real quadratic field.

Molecular neural interfaces that deliver and sense the same neurotransmitter must estimate endogenous release in the presence of their own chemical input. We study two exchanging regions with shared saturable uptake and concentration change measurements. Known delivery and exchange permit identification of apparent uptake at the driven region, but an unknown resting concentration difference leaves the source region's incremental uptake uncertain. We characterize all admissible source histories consistent with both ideal records and show how delivery and baseline information affect release classification. A dopamine-inspired example produces identical ideal concentration-change records for mean release-rate changes of -8.99 and +1.31 nanomolar per second. Stronger delivery changes whether all compatible sources imply suppression, with the true source held fixed. Under a Gaussian observation model, we bound discrimination based on chemical fluctuations. A baseline reference with an assumed standard deviation of 3.3 nanomolar reduces root-mean-square error from 10.41 to 1.21 nanomolar per second compared with an incorrect fixed baseline under exact gain calibration. Classification errors remain substantial with weak excitation, gain errors or responses near the category boundaries.

This paper studies a finite-horizon mixed $H_2/H_\infty$ feedback Nash game for stochastic evolution equations on a separable Hilbert space. The drift generator is unbounded, the remaining coefficients are bounded, and the one-dimensional Brownian diffusion depends on the state, control, and disturbance. The $H_2$ channel is an LQ state–control energy. For the disturbance channel, the stochastic LQ uniform-convexity characterization yields equivalence between strict induced $L^2$ attenuation and unique strongly regular mild Riccati solvability. Simultaneous bounded-generator approximation of the Lyapunov equation and the state justifies quadratic identities for strongly continuous mild operator solutions. These identities verify both Nash inequalities and full-output strict attenuation from a strongly regular coupled Riccati pair. An invertible feedback block and a contraction argument establish locally unique coupled solutions on a sufficiently short terminal interval for every positive attenuation level. For a specified stochastic heat-equation model at $\gamma=0.09$, a uniform invariant rectangle further proves full-horizon existence, bounded infinite-dimensional feedbacks, and operator-norm spectral convergence. Numerical computations reproduce the projected gains and compare selected best responses. Global coupled solvability for general coefficients remains an explicit hypothesis.

Let $(\xi_t)_{t \geq 0}$ be a continuous-time Markov process taking values in a countable state space $E$. We consider $k$ copies of $\xi$ started from distinct positions, and write $X = (X_t)_{t \geq 0}$ for the resulting process of configurations in $E^k$. We compare two forms of interaction between the particles: the first is conditioning the particles to never collide (noncolliding dynamics), and the second is simply switching off any jumps that would cause two particles to occupy the same position (exclusion dynamics). We show under mild conditions on the tail probabilities of the collision time that the dynamics are related by an explicit exponential martingale change of measure of one another. This generalises a recent result in the integrable probability literature relating TASEP on the ring to asymmetric walkers conditioned to never collide.