Papers

Let $R(G,p)$ be the normalized rigidity matrix of a framework $(G,p)$ in $\mathbb R^d$, and let \[ L(G,p)=R(G,p)R(G,p)^{T} \] be the associated stiffness matrix. We study the extremal eigenvalues of $L(K_n,p)$ for complete frameworks whose vertices lie on the unit sphere and have centroid at the origin. Our main result shows that, whenever $d\ge2$ and the image of $p$ contains at least three distinct points, the second largest eigenvalue of $L(K_n,p)$ is exactly $n/2$. This settles the eigenvalue part of a conjecture of Lew et al. [Israel J. Math. 256, 2023]. We further construct an infinite family of examples, given by regular polygons embedded in a two-dimensional subspace, for which the eigenvalue $n/2$ has multiplicity $2n-4$. Consequently, the multiplicity predicted in the conjecture is not correct in general. Our results reveal a dichotomy: the value of the second largest eigenvalue is universal, while its multiplicity is sensitive to the geometry of the underlying point configuration.

The reflection subgroups of a reflection group have a natural lattice structure given by the reflections that they contain. By considering the conjugation action orbits of the reflection subgroups for a given root line, we are able to give an essentially combinatorial way to calculate the lattice of all the normal reflection subgroups of a given (finite irreducible) reflection group, and natural generators for them. Moreover, we observe that every complex reflection group is a normal subgroup of the unique maximal reflection group which shares its collineation group. Hence, we are able to present the Shephard-Todd classification of the complex reflection groups as a collection of maximal reflection groups, together with appropriate (collineation preserving) normal reflection subgroups. We investigate the quotients by the normal reflection subgroups, which are known to be reflection groups. We also consider the action of the collineation group on some appropriate small systems of lines, and how these results extend to quaternionic reflection groups. Some novel techniques are introduced, including the notion of a "hidden reflection", a combinatorial-geometric description of the reflection subgroups and the size of their conjugacy class, and the role played by the abelianisation of the reflection group.

We prove that every mean-zero probability law on the real line is the law of X-Y for some identically distributed real-valued random variables X and Y.

Let $\Omega\subset\mathbb{R}^N$ be a bounded open convex set, and let $0=\mu_0(\Omega)<\mu_1(\Omega)\le \mu_2(\Omega)\le\cdots$ be the Neumann eigenvalues of the Laplacian, repeated according to multiplicity. We prove the sharp bounds $$ \mu_2(\Omega)\le 4\mu_1(\Omega),\qquad \mu_3(\Omega)\le 9\mu_1(\Omega). $$ The first estimate resolves a problem attributed to Henrot, while the second gives the next sharp case predicted by the one-dimensional model. The constants are optimal in every dimension.

A random $n\times k$ matrix $S$ is an \emph{$(r,\alpha)$-oblivious subspace injection} (OSI) if $\mathbb{E}\|S^\top x\|_2^2=\|x\|_2^2$ for every $x\in\mathbb{R}^n$, and for every fixed $r$-dimensional subspace $V\subset\mathbb{R}^n$, with probability close to one, one has $\alpha\|x\|_2^2\le\|S^\top x\|_2^2$ for all $x\in V$. In this work, we show that in the regime $r=\Omega(k)$ and $\alpha=\Omega(1)$, and under a mild additional structural assumption, no constant-row-sparsity matrix $S$ is OSI, thereby answering, in a strong form, a question raised by Camaño, Epperly, Meyer, and Tropp. We show that the failure of the OSI property for sparse random matrices stems from a general deterministic phenomenon, thereby reducing a probabilistic problem to a non-probabilistic one. This phenomenon is related to the restricted invertibility principle introduced in the seminal work of Bourgain–Tzafriri. Let $(n_k)_{k\in\mathbb{N}}$ be a sequence of integers satisfying $\frac{n_k}{k}\to\infty$. For each $k$, let $S^{(k)}$ be a $n_k\times k$ non-random matrix with $O(1)$ nonzero entries per row, whose nonzero entries have average magnitude $O(1)$, and such that the total number of pairs of rows with supports overlapping at two or more indices is $o({n_k}^2/k)$. We prove that for every constant $\varepsilon>0$, as $k\to\infty$, the overwhelming majority of $k\times \lfloor\varepsilon k\rfloor$ submatrices of $(S^{(k)})^\top$ have the smallest singular value $o(1)$. Thus, the well-invertible submatrices whose existence is guaranteed by the Bourgain–Tzafriri theorem are rare. The proof is itself based on probabilistic tools.

What does a discrete diffusion model actually learn? The literature trains denoisers, bridge plug-ins, and concrete scores, each deriving its own ELBO objective under different conventions. This fragmentation makes the object being learned unclear and ELBO values difficult to interpret and compare. All three methods learn the same object in different coordinates: the exact time-reversal of the noising process. We first prove the Oracle Distance identity: for any noising process, the negative ELBO is not just a bound on the likelihood; it equals the data entropy plus the path KL divergence from the reversed noising process to the learned one. Next, for token-factorizing noise, we show that denoiser, bridge plug-in, and concrete score are three coordinates of this reverse process, derive exact conversions between them, and identify the probability distribution learned by each. In particular, contrary to its standard interpretation, the bridge plug-in learns not a denoiser but the cavity distribution: the clean token given the noisy sequence excluding its own position. We confirm the theory by training all three coordinates with masked, uniform, and GIDD kernels on OpenWebText under a shared recipe, and by training uniform-diffusion solvers on four logic tasks. The learned coordinates agree after conversion, while confusing the cavity with the denoiser can severely degrade sampling quality. Together with exact boundary-term formulas, closed-form NELBO calibrations at initialization, and a clock interpretation of time schedules, these results provide a conceptual framework and a mechanical recipe for deriving, implementing, and comparing discrete diffusion processes.

We study the adversarial fake detection model introduced by Mendelson, Paouris and Vershynin. In this model, a genuine sample is $\pmb{X}\sim N(0,\pmb{I}_n)$, while a fake sample is produced as $\pmb{X}+r\pmb{t}({\pmb{X}})$, where the adversary first observes $\pmb{X}$ and then chooses an admissible perturbation $\pmb{t}({\pmb{X}})$ from a prescribed set $\mathscr{T}\subset\mathbb{R}^n$. The central quantity is the detectability radius $r(\mathscr{T})$, which formalizes the transition scale at which fake samples become reliably distinguishable from genuine ones. Mendelson, Paouris and Vershynin introduced the focused width $\widetilde{w}(\mathscr{T})$ as a geometric parameter for this radius and conjectured that, for every origin-symmetric set $\mathscr{T}$, it characterizes $r(\mathscr{T})$ up to universal constants. In this note, we disprove this conjecture for a broad class of discrete sets. More precisely, we consider any origin-symmetric set $\mathscr{T}_n$ lying between the hypercube and the odd integer grid: \begin{equation*} \{-1,1\}^n\subset\mathscr{T}_n\subset ( 2\mathbb{Z}+1)^n. \end{equation*} For every such $\mathscr{T}_n$, we prove that $\frac{\widetilde{w}(\mathscr{T}_n)}{r(\mathscr{T}_n) }\gtrsim \sqrt{\log n}$. Thus, in the Gaussian model, the focused width can overestimate the detectability radius by a $\sqrt{\log n}$ factor and therefore does not characterize it in general. We further show that this logarithmic scale is not intrinsic: in the corresponding non-Gaussian model with product Laplace data, the focused width benchmark can even exceed the detectability radius by at least a polynomial factor of order $n^{1/4}$.

We show that there exists a constant $\delta > 0$ such that for any positive integer $n$ there exists a set of $n$ points $P \subset \mathbb{R}^2$ with the following property: for every subset $A \subseteq P$ of size $|A| \geq 2$, \[ \max_{\lambda>0} \#\{(a,b)\in A \times A: a\ne b,\ \lvert a-b\rvert=\lambda\} \gtrsim \frac{|A|^2}{n^{1-\delta}}.\] Our result is a vertical amplification of a robust Ramanujan estimate recently established by Croot-Mao-Pohoata-Sheffer-Yip for arbitrary subsets of the ordinary square grid, and is inspired by recent constructions for the Erdős unit distance problem and the Elekes-Rónyai problem. Taking $A=P$, the inequality above gives a distance occurring $n^{1+\delta}$ times in $P$; thereby a scaled copy of $P$ is a counterexample for the unit-distance conjecture. In addition, the same inequality shows that (1) all subsets of $P$ of size $\gtrsim n^{1-\delta}$ must contain isosceles triangles, and (2) all subsets of $P$ of size $\gtrsim n^{1/2-\delta}$ must contain repeated distances. These features give polynomially improved estimates for old problems of Erdős. The existence of a set satisfying property (1) confirms a conjecture of Erdős from 1980, whereas the existence of a set with property (2) answers a question of Conlon-Fox-Gasarch-Harris-Ulrich-Zbarsky in the negative.

Energy storage operators often reserve usable capacity before uncertain market opportunities are realized, for example when a day-ahead operator commits capacity for an evening peak-spread or ancillary-service window. Price spikes, renewable forecast errors, activation calls and imbalance penalties can make the decision-relevant opportunity distribution asymmetric and heavy-tailed rather than Gaussian. This paper develops a tri-objective storage-newsvendor framework for this capacity-reservation decision. The model jointly maximizes expected profit, minimizes CVaR tail loss and minimizes maximum regret over candidate non-normal distributions. We derive a critical-fractile expected-profit solution and a finite-scenario convex scalarization that solves the three objectives together. In stylized log-NMVM experiments with matched mean and variance, a moment-matched normal policy over-reserves capacity by up to 1.337 units. In a lognormal policy comparison, the CVaR and maximum-regret policies reduce tail loss and candidate-set regret at the cost of lower expected profit.

We study the depth-weighted random recursive trees introduced by Leckey, Mitsche and Wormald in the case where the weights are bounded from above and from below. We establish the scaling limit of the depth profile of these trees when the weights satisfy a law of large numbers. In particular, we obtain the scaling limit of the depth of these trees, generalising some results of Lichev, Linker, Lodewijks and Mitsche. We also answer negatively a question left open by the same authors, showing that the scaling limit of the depth does not hold in general. Our main tools are appropriately defined martingales combined with the general Edgeworth expansion for the profiles introduced by Kabluchko, Marynych and Sulzbach.

We study the problem of reconfiguring a crossing-free embedding of a graph on a surface, with edges represented as curves, into another crossing-free embedding of the same graph on the same surface with the same fixed vertex positions. In this process, we reroute one edge at a time while maintaining crossing-free intermediate embeddings. This problem was introduced by Ito et al. [TALG 2025], who showed that even if the graph is a matching of two edges, reconfiguration is not always possible in the plane, but is always possible on the torus. For matchings of two or more edges, they gave a necessary and sufficient condition for reconfigurable embeddings in the plane, but not on the torus. Our main result is that for matchings, trees and forests, reconfiguration is always possible on the torus, and consequently, on any orientable surface of genus at least one. In addition, we provide sufficient conditions for reconfiguration on orientable surfaces of genus at least one and in the projective plane. For more general graphs, we show that reconfiguration is not always possible.

We prove, for every $\mathbb{E}_1$ algebra $A$, a formula describing the interaction of the action of the cap product on topological Hochschild homology of $A$ with the cyclotomic structure map, as well as a variant of this result relative to a ring $R$. Specializing to $R = \mathbb{F}_p$ gives a noncommutative analog of a formula of Cartier which describes the conjugation of interior product action on differential forms by the Cartier isomorphism, and which computes the $p$-curvature of the Getzler-Gauss-Manin connection in terms of an equivariant cap product. The motivation for this formula comes from symplectic geometry, where (in the case $R=\mathbb{F}_p$ or a Novikov analog) the symplectic analog of this formula explains the interaction between the cyclotomic structure on symplectic cohomology and the quantum Steenrod operations. We prove, under standard transversality and nondegeneracy assumptions on the Fukaya category, that for a Calabi-Yau symplectic manifold with rational symplectic form, the $p$-curvature of the quantum connection computes the Quantum Steenrod operations. In particular, the $p$-curvature of the quantum connections of projective Calabi-Yau hypersurfaces, and many other examples in mirror symmetry, can be interpreted in terms of $\mathbb{Z}/p\mathbb{Z}$-equivariant genus zero Gromov-Witten invariants.

Let $G$ be a graph and let $\PF_t(G)$ denote the simplicial complex whose faces are vertex subsets whose induced subgraphs contain no path on $t$ vertices. These complexes encode a forbidden-subgraph condition as a family of allowed vertex subsets. In this paper, we study $t$-path-free complexes of complete multipartite graphs. Let \[ G=K_{n_1,\dots,n_m}, \qquad n_1\le\cdots\le n_m. \] We first obtain an explicit structural decomposition of $\PF_t(G)$ as a union of join complexes, together with an additional lower-dimensional size-truncation term. Using this decomposition, we show that for $t\le 2n_{m-1}-2$ the complex $\PF_t(G)$ is not sequentially Cohen-Macaulay, while for $t\ge 2 n_{m-1}-1$ it is vertex decomposable. Consequently, we obtain a complete characterization for complete multipartite graphs: $\PF_t(G)$ is vertex decomposable if and only if $t\ge 2n_{m-1}-1$. Equivalently, this is also exactly the range in which $\PF_t(G)$ is shellable and sequentially Cohen-Macaulay. We further analyze the topology via a Mayer-Vietoris spectral sequence: for complete bipartite graphs, we determine the full homotopy type as an explicit wedge of spheres in all cases.

Let $x$ be sufficiently large and \[ N(x)=\big|\bigl\{n\le x:n\ \text{is odd and }n\ne p+2^a+2^b \textrm{ with } p \text{ a prime and } a,b\in \mathbb{N}\bigr\}\big|. \] Motivated by Crocker's result \[ N(x)\gg \log\log x, \] Erd\H os repeatedly asked whether there is an absolute constant $c_0$ such that $N(x)>c_0x$. Pan \cite{Pan} proved in 2011 that \[ N(x)\gg x\exp\!\left( -C_0\frac{\log\log\log\log x}{\log\log\log x}\log x \right), \] where $C_0>0$ is an absolute constant. We improve on Pan's result by showing that, given any $\eta>0$, for all sufficiently large $x$, \[ N(x)\gg_\eta x\exp\left(-(4+\eta)\frac{\log\log\log x}{\log\log x}\log x\right). \]

We introduce the microscopic weighting, a canonical signed measure of mass one that can be associated to almost any finite metric space. The microscopic weighting is obtained as the small-scale limit of the weightings used to define the magnitude function. We give general criteria for its existence, proving in particular that every finite space of strictly negative type admits a microscopic weighting; this includes every finite subset of Euclidean or hyperbolic space and every finite tree. Heuristically speaking, the microscopic weighting distributes its mass as widely as possible across a space, assigning greater weight to sparse or outlying regions and emphasizing points on the boundary. Indeed, we show that on a finite space of negative type the microscopic weighting can be characterized (when it exists) as an optimizing measure for an energy integral determined by the distance function. Alternatively, it can be characterized in terms of the geometry of the Schoenberg embedding. Each of these interpretations also clarifies the information carried by the derivative of the magnitude function at zero. Though our main focus in this paper is on finite metric spaces, we lay the groundwork to extend the theory to compact subsets of Euclidean space. In that setting, we observe that the microscopic weighting must be understood as a distribution rather than as a measure.

In the theory of mixing times, a famously wrong conjecture predicts that a sequence of Markov processes exhibits cutoff as soon as the product of their Poincaré constant and mixing time diverges. We prove that this statement becomes correct once the Poincaré constant $\gamma$ is replaced with its natural non-equilibrium refinement, which we denote by $\gamma_\star$. More precisely, we show that the width of the mixing window of any Markov process is $O(1/\gamma_\star)$. This estimate is sharp, and universal up to standard regularity assumptions: it holds on finite and infinite state spaces and from any initial condition, and it does not require reversibility, nor any kind of a chain rule. In addition, for deterministic initialization we show that $\gamma_\star\ge\kappa$, where $\kappa$ is the Bakry-Émery curvature, making our result broadly applicable. Finally, our proof is short and self-contained: we simply follow the classical idea of replacing the total variation distance by the more tractable $\chi^2$-divergence, but with the crucial novelty that the reference measure evolves in time, instead of being the equilibrium law.

We give a new proof of Li's theorem on the existence of geometric Bridgeland stability conditions on the bounded derived category of coherent sheaves on projective spaces. These stability conditions can then be restricted to induce Bridgeland stability conditions on arbitrary smooth projective varieties.

We construct a universal affine bundle $E$ over a compact complex manifold $X$ equipped with a probability measure $\mu$, which is affine over the vector space $H_{1,1}(X;\mathbb{R})$, and satisfies a number of natural properties. The bundle $E$ carries a natural action of the automorphism group of $(X,\mu)$, and provides potentials for all $dd^c$-closed $(1,1)$-forms on $X$. We relate our construction to lifts of the action of the automorphism group of a Calabi–Yau manifold to its universal torsor.

Let $V$ be a vector space of dimension $n+1$ over an algebraically closed field $\mathbb{k}$ of characteristic zero, and let $G_n = \operatorname{Gr}(2,\operatorname{Sym}^2V^\vee)$ be the Grassmannian parametrizing pencils of quadrics in $\mathbb{P}(V) \cong \mathbb{P}^n$. The determinant of the universal pencil defines a universal binary form of degree $n+1$. We prove that the divisor $\mathcal{D}_n\subseteq G_n$ of pencils whose determinant binary form has a multiple root has Chow class $[\mathcal{D}_n]=n(n+1)\sigma_1\in A^1(G_n),$ where $\sigma_1=c_1(S^\vee)$ and $S$ is the tautological rank-two subbundle on $G_n$. More generally, the higher-contact loci of determinant binary forms are computed by a universal jet formula. We also formulate the determinant-root collision strata as refined pullbacks of the universal collision strata for binary forms. For $n=3$, the main formula recovers the class $12\sigma_1$ for the boundary divisor in $\operatorname{Gr}(2,10)$ that the author established in a prior paper.

Using the theory of reproducing kernel Hilbert spaces introduced by L. de Branges and J. Rovnyak we prove a trace formula for pairs of operators in Hilbert space in terms of a Carathéodory function. We consider the special case, when the latter is rational. An application to the theory of first order discrete systems is given.

We derive necessary and sufficient conditions for a sequence of constants $(C_{n})_{n \in \mathbb{N}_0}$ to be a growth scale of a branching process in (possibly defective) varying environments $(Z_{n})_{n \in \mathbb{N}_0}$ in the sense that $(Z_{n}/C_{n})$ is bounded from above and below with positive probability. Along the way, we derive a Kesten-Stigum type result - necessary and sufficient conditions for branching processes with varying environments to converge in $L^1$ when normalised by their mean. The proofs exploit truncation and change of measure arguments, convergence of random series and martingale techniques.

After $n$ radial cuts of a circular cake, it is divided into $n$ slices. Call an adjacent pair of slices a portion. We exhibit an infinite sequence of cuts such that the ratio between the maximum and minimum sizes of a portion never exceeds 1.755. This improves on the trivial upper bound of 2, disproving a conjecture of Korsky.

We present the first constant-amortized-time algorithms for generating all length-$n$ necklaces and Lyndon words over a $k$-letter alphabet in colexicographic order, for arbitrary $k\geq 2$. Our approach introduces a novel class of words called quasinecklaces, which serve as an easily generated superset of necklaces through which all necklaces can be efficiently identified. We derive a formula for the number $Q_k(n)$ of length-$n$ quasinecklaces and show that $Q_k(n)$ is proportional to the number of length-$n$ necklaces, which is the key property needed to achieve constant amortized time. We also apply our results to efficiently generate a well-known de Bruijn sequence and efficiently generate necklaces and Lyndon words subject to a weight constraint.

In this paper, we define partial Gateaux and Frechet derivatives for multivariable and single-valued mappings between Banach spaces. We will prove some properties of partial Gateaux and Frechet derivatives. By these definitions, we find the explicit formulas for some polynomial type operators with two variables from lp by lp to lp with respect to each variable. Then, we will introduce the concepts of generalized partially critical points and partial ordered extrema in partially ordered Banach spaces. By these concepts, we will investigate the connection between generalized partially critical points and partial ordered-extrema of two-variable and single-valued mappings in partially ordered Banach spaces. These results extend the connection between critical points and extrema of real valued functions in calculus. We will give some applications of partial Gateaux and Frechet derivatives to ordered optimizations and variational inequality problems between partially ordered Banach spaces. Finally, we study the connection between partial Gateaux and Frechet derivatives and ordered monotone of two-variable and single-valued mappings in partially ordered Banach spaces.

The aim of this paper is to study the variation under quadratic twists of the analytic Iwasawa invariants of Sprung's sharp/flat 2-adic $L$-functions for elliptic curves over $\mathbb{Q}$ with good supersingular reduction at $2$. Under the hypothesis that the $\mu$-invariant vanishes, we obtain an explicit formula for the sharp/flat $\lambda$-invariants. This formula gives a supersingular analogue of Matsuno's formula in the good ordinary case. As an application, we show that the sharp/flat $\lambda$-invariants can be made arbitrarily large even among quadratic twists by single primes. Moreover, using the method of Hatley-Ray, we obtain an asymptotic lower bound for the number of quadratic twists with a prescribed sharp/flat 2-adic Iwasawa $\lambda$-invariant.

The goal of this note is to investigate quantitative stability properties of the critical Sobolev inequality in ${\sf CD}(N-1,N)$ metric measure spaces. Assuming that the optimal constant for the inequality is almost the same as the one of the round sphere, we show that the cumulative distribution of any almost extremal function is close, in Wasserstein distance, to the one of an Aubin-Talenti bubble on the round sphere. We obtain similar results for the log Sobolev inequality and the spectral gap under various curvature and dimension assumptions. In all cases we obtain a quantitative stability with sharp exponent.

Currently, among the open (unsolved) problems in number theory is the following: it is unknown what is the natural density of the sequence of Mersenne numbers in the set of natural numbers. In the paper, using methods of nonstandard mathematical analysis, we obtain the following equation: the natural density of Mersenne numbers (some infinitesimal value $e$) multiplied by the sum of the reciprocals of odd numbers (the infinitely large value {\omega} = 1 + 1/3 + 1/5 + ...) is equal to 1, or the equality $e$ = 1/{\omega} is true. In nonstandard analysis, the resulting infinitesimal numbers $e$ and 1/{\omega} are considered equivalent. We obtained this result by working with a two-dimensional matrix of non-negative integers, where odd numbers are separated from even ones by the Pepis-Kalmar pairing function.

At fixed Hamiltonian $H$, changing the initial state changes the cyclic pair and hence generally the Lanczos basis and spread. For every normalizable polynomial preparation $\lvert\psi_Q\rangle=N_Q^{-1/2}Q(H)\lvert K_0\rangle$, we reconstruct its state-Krylov problem exactly from reference cyclic data. The transfer assumes no integrability and applies on finite or infinite cyclic support. Reweighting the reference spectral measure by $\lvert Q\rvert^2$ gives a finite-band connector for every prepared amplitude and a finite-rank Christoffel-Darboux projection for cumulative probabilities and spread, without rerunning ambient-space Lanczos. Fixed-degree seeds preserve the limiting Jacobi coefficients of asymptotically constant chains. Every Charlier number-state jump obeys $K_r(t)\ge K_0(t)$, with strict inequality for $r\ge1$ away from revivals. At every fixed finite $r$, the Hermite endpoint has exact all-level amplitudes, whereas the continuous-$q$-Hermite chord chain of double-scaled SYK has an all-level root-free re-Lanczos reconstruction. Writing $D_r^{\rm H}$ and $D_r^{\mathrm{ch}}(t)$ for the mean absolute Fock and chord displacements, respectively, the bounds are $K_r\ge D_r^{\rm H}$ and $K_r(t)\ge D_r^{\mathrm{ch}}(t)\ge\lvert\overline n_r(t)-r\rvert$. Thus rebuilt spread bounds physical displacement and signed chord drift. At fixed $0\le q<1$ and time, $K_r(t)-D_r^{\mathrm{ch}}(t)\to0$ as $r\to\infty$, with both approaching the same folded-Bessel limit. In Liouville space, the exact unnormalized gap measure family on an open inverse-temperature interval determines the positive transition-resolved measure, assuming the thermal kernel is known and strictly positive and the requisite exponential moments exist. One cyclic solution can therefore be reused across polynomial seeds while separating physical propagation, basis response, and rebuilt spread.

For a class of polaron-type models, we establish a lower bound on the energy-momentum relation in terms of the vacuum overlap and the spectral gap of the total momentum zero Hamiltonian. We show convergence of the rescaled mean square displacement of the associated path measure to the inverse of the effective mass. We derive a probabilistic criterion for the absence of ground states at large total momentum.

An Edgeworth expansion of first order is established for general linear rank statistics under the null hypothesis with a remainder term that is usually of order $n^{-1}$. Furthermore, corresponding results for the second order are formulated, but not proved here. The proof for the first order is based on Stein's method and on an extension of the combinatorial method of Bolthausen. It is also shown that conditions of van Zwet imply up to a small factor our conditions for the validity of Edgeworth expansions. Moreover, our proof for the first order also provides us with a result about Edgeworth expansions for smooth functions.