Papers

Let $\bar{p}_{s,t}(N)$ be the number of overpartitions of $N$ into parts that are not divisible by $s$ or $t$. In this paper, we obtain asymptotic formulas for $\bar{p}_{s,t}(N)$ for relatively prime integers $s>t\ge 10$ via the saddle-point method. Our results cover different ranges of the parameters $s$ and $t$ relative to $N$. As an application, we show that $\bar{p}_{s,t}(N)$ satisfies higher-order Turán inequalities for all sufficiently large $N$ by using a result of Griffin, Ono, Rolen, and Zagier.

Godsil and McKay (1978) showed that the rooted product is a powerful tool for constructing non-isomorphic cospectral pairs of graphs. Despite lacking a convenient tensor product structure, we show that the rooted product is useful for constructing graphs with good quantum state transfer properties. In particular, we prove a simple transference principle: if a graph $X$ has quantum state transfer and $Y$ is a controllable graph, their rooted product $X^Y$ has quantum state transfer (inherited from $X$). This complements a folklore property of Cartesian product which preserves perfect state transfer. However, the rooted product is a significantly sparser graph and, more importantly, can be easily used to construct efficient high-fidelity state transfer even if $X$ has no quantum state transfer. Our proof exploits the fact that a rooted product creates a large number of strongly cospectral pairs of vertices and that its condition number can be controlled by its pendant subgraph.

We prove linear ill-posedness in tangential Sobolev spaces for the three-dimensional resistive MHD boundary layer equations. We assume that a suitable linear combination of the initial tangential velocity components has a non-degenerate critical point and that both initial tangential magnetic components have a double zero at the same point. Compared with the corresponding two-dimensional MHD result, our construction requires only this double-zero condition, rather than a higher-order degeneracy of the magnetic shear. The proof combines the three-dimensional Prandtl critical layer construction with a vector correction to the two-dimensional magnetic quotient. This correction cancels the leading terms in both induction equations while preserving the magnetic divergence constraint. The resulting approximate solutions grow like $\exp(\sigma t/\sqrt{\varepsilon})$, and their residuals have a prefactor of order $O(\varepsilon^{-3/8})$, up to logarithmic factors. A Duhamel argument then yields linear ill-posedness for every tangential derivative loss $\mu<\frac18$. Thus the three-dimensional Prandtl instability persists when the stabilizing tangential magnetic field degenerates.

Recently, René Peschmann proved perfect-cuboid nonexistence on 1,072 explicit master-tuple fibers with $\max(m,n)\le100$, leaving 968 fibers in the same bounded parameter range. In this paper, I eliminate all 968 remaining fibers. Most fibers fall to fairly short rank-zero or covering arguments, while the final cases need the stronger local-global sieves. Together with Peschmann's theorem this excludes perfect cuboids on all 2,040 admissible fibers with $m\le100$; the result is still a bounded-fiber theorem rather than a proof of the full perfect-cuboid conjecture.

We study how far a fixed Ramsey upper bound can be improved by descending through blue neighborhoods in one vertex set while keeping a second set fixed. A weighted inequality in the two set sizes determines when the descent can stop. For the source bound specified here, the infimum diagonal exponent over all finite derivations lies in $[1.305,\,1.307]$. A finite derivation gives $R(k,k)\le3.69507^k$ for all sufficiently large $k$; a concave polygon proves the lower bound for every finite depth. We also characterize the infimum as a greatest fixed point and show that every larger exponent has a finite derivation valid uniformly for nearby clique-size ratios.

The $\Delta$-unknotting number for a knot is defined as the minimum number of $\Delta$-moves needed to deform the knot into the trivial knot. In this paper, we determine the $\Delta$-unknotting numbers for certain families of Montesinos knots. Using results on pretzel knots and two-bridge knots, we prove that for these families the $\Delta$-unknotting number equals the absolute value of the second coefficient of the Conway polynomial. In particular, every positive pretzel knot belongs to these families, and certain two-bridge knots also belong to them.

We give a lower bound for the unknotting number under the L-space satellite operations. In the case of braided L-space satellite operations, including cabling operations, we obtain a stronger bound. The proof uses a lower bound for the unknotting number in knot Floer homology due to Alishahi-Eftekhary. The computational tool is a formula for computing the knot Floer complex under the L-space satellite operations, which was defined in the authors' previous work.

Motivated by quantum foundations and complexity theory, Aaronson formalized a hidden-variable theory inspired by a proposal by Schrödinger. To any quantum state and unitary, this Schrödinger theory assigns a joint probability distribution via the Sinkhorn algorithm: rescale the columns and rows of the entrywise modulus of the unitary so that the marginals match the Born rule for the initial and final quantum states, respectively. He conjectured that this map is robust, i.e., inverse polynomially small perturbations in the inputs lead to inverse polynomially small perturbations of the joint distribution, which is important for complexity theoretic applications. We present a counterexample to this conjecture: We construct a pure state and two unitaries that are exponentially close, yet the joint probability distributions assigned by the Schrödinger theory differ by at least an inverse-linear term in at least one entry, and by a constant in total variation distance. We also propose a modified version of Schrödinger's theory that satisfies robustness, while retaining all of its other desirable properties. This yields a complete picture of which axioms of Aaronson can be simultaneously satisfied by hidden-variable theories.

A tensor randomized average block coordinate descent method with heavy-ball momentum is proposed for the tensor least squares problem with respect to the t-product. Theoretical analysis for tensor block coordinate descent method is established and average techniques are applied to avoid the computation of tensor Moore–Penrose inverse. To further accelerate convergence, a heavy-ball momentum scheme is incorporated into the block coordinate descent method, where the step size and momentum parameters are adaptively determined by a two-dimensional minimal residual projection. Theoretical analyses give the convergence of the new method and provide an improved bound on the linear convergence rate. Numerical experiments further verify the efficiency of the proposed method in terms of the number of iterations and the better video recovery performance.

Giuliano Basso asked whether every $n$-dimensional integral current space admits an integral current structure with the same characteristic set and multiplicity one almost everywhere. We give a negative answer already in dimension one. The obstruction is a parity phenomenon at odd-valence branch points. More precisely, if a one-dimensional integral current has unit multiplicity on the arms incident to an isolated vertex $v$, then the coefficient of its boundary at $v$ has the same parity as $\deg(v)$. Hence infinitely many odd-valence vertices force infinite boundary mass for every full-support unit-multiplicity current. We construct a compact geodesic, doubling, $1$-Ahlfors regular graph-like continuum $G$ of finite length and a boundaryless current $T\in\I_1(G)$ with $\set(T)=G$, while no $S\in\I_1(G)$ with $\set(S)=G$ can have multiplicity one $\Hh^1$-almost everywhere. We classify all full-support integral cycles on $G$. In particular, the least possible essential maximal multiplicity is exactly two, and the minimum mass of such a cycle is $3/2$. For finite metric graphs we identify a complementary sharp defect: the minimum boundary mass among unit-multiplicity currents equals the number of odd-degree vertices. Finally, by taking products with flat tori and using the precise Ambrosio–Kirchheim slice representation, we obtain boundaryless compact geodesic Ahlfors-regular counterexamples in every dimension.

We show that the fractional perimeter of any order $0<s<1$ is maximal, among all planar sets of fixed perimeter, for the disk.

We establish necessary and sufficient conditions under which a binary operation on a bounded trellis is a proper nullnorm. The representation combines a t-conorm on the lower interval, a t-norm on the upper interval, two order-preserving maps, and a commutative, increasing function on $I_a^3\times I_a^3$, where $I_a^3$ consists of the elements incomparable with the absorbing element $a$ that neither reach $a$ nor are reachable from $a$. Unlike earlier range-restricted constructions, this function may take values anywhere in the trellis. To preserve associativity for such unrestricted values, we introduce a mixed interaction function that evaluates every pair with at least one component in $I_a^3$. We also derive the specializations in which this region is empty or consists of a single element, including an exact description of the admissible value in the singleton case. A five-element lattice example shows that the mixed associativity condition is independent of the remaining hypotheses, while a fourteen-element nontransitive trellis example demonstrates the necessity of allowing the unrestricted range. Finally, the principal range-restricted subclasses and the bounded-lattice case are recovered as specializations of the general representation.

A covariance matrix rarely distorts in a single direction: a shift can expand all variances simultaneously, shift variance along one or two principal directions, displace spectral mass without changing marginal means, or rotate the dependence structure. Default mean-shift detectors are blind to these effects, and no existing online procedure calibrates a family of spectral deviation tests with provable false alarm control across both time and tracking window selection; this paper fills that gap. We consider four spectral deviations: $D_{KL}(P_{1}\|P_{0})$, $D_{KL}(P_{0}\|P_{1})$, Jeffreys, and Bhattacharyya, each evaluated on the eigenvalues of the empirical relative covariance operator $\widehat{\Sigma}_{0}^{-1/2}\widehat{\Sigma}_{t}\widehat{\Sigma}_{0}^{-1/2}$. Critical values come from a conditional parametric bootstrap that accounts for estimation uncertainty in both the past and tracking windows, an aspect asymptotic approaches typically overlook. The procedure controls the false alarm rate family-by-family over a predefined monitoring period and a range of candidate window sizes; when a single operational window is needed, a power-based criterion selects it. We prove consistency under constant alternatives via local spectral expansions with second-order sensitivity near the null. Simulations are conservative under the null and show detection power depends largely on spectral shape rather than magnitude: $D_{KL}(P_{1}\|P_{0})$ excels under global inflation, while Jeffreys and Bhattacharyya cover a broader range of alternatives. We illustrate the approach on three financial applications: European stock indices, Fama-French sector portfolios, and large-cap technology stocks.

We prove the Cartan-Hadamard conjecture (the sharp Euclidean isoperimetric inequality) in every dimension, assuming that the sectional curvature is pinched relative to a reference radius of curvature. The reference radius of curvature may vary, and the condition includes examples where the radius of curvature blows up, and the manifold has both unbounded (negative) and asymptotically vanishing sectional curvature. The key step in the proof is to control the entire convex hull of a quotient minimiser at its mean-curvature scale and compare the metric distortion with the surplus in the normalised hyperbolic isoperimetric profile at that same scale.

We start from a simple problem in geometric optics. A particle moves between two homothetic ellipses, with refraction at the inner interface and reflection at the outer one. In a suitable nearly circular regime, the small geometric anisotropy of the ellipses is amplified by the strong refraction, and the corresponding return map develops a transverse heteroclinic connection between the two axial motions. We turn this geometric mechanism into a construction for a two-mode Kirchhoff system while keeping the standard quadratic elastic variable throughout. The singular optical model is first reduced to an explicit kick–drift map, for which the heteroclinic connection is obtained by a contraction argument. The same argument also gives exponential convergence along the two tails and transversality. We then show that the connection persists for the genuine elliptic billiard and, subsequently, through a smooth regularization of the reflecting and refracting interfaces. Finally, a small positive background is added to the Kirchhoff coefficient, making it uniformly positive without destroying the hyperbolicity of the axial modes or the transverse connection. This produces a smooth and uniformly positive Kirchhoff coefficient admitting a transverse heteroclinic orbit between two simple modes. Combined with the road-map theorem developed in our previous work, the construction yields a finite-time blow-up example for a forced abstract Kirchhoff equation with a regular forcing term.

Finite-horizon prediction is challenging when observations are limited and a quantity of interest depends on several coupled nonlinear measurement channels. This paper develops a three-phase data-driven twin framework that integrates finite-horizon Hankel–Koopman energy decomposition, inverse-calibrated reconstruction using nonlinear autoregressive models with exogenous inputs, and forecast composition. Channel dynamics are identified and simulated in Hankel coefficient space, with candidate models assessed after reconstruction in physical measurement space. The resulting reduced-order model of the channel dynamics drives recursive prediction of the quantity of interest. Three theorems support the framework, establishing the energy decomposition and modal orthogonality; existence, uniqueness, and inverse stability of the auxiliary continuous inverse-calibration solution; and finite-horizon stability of the composed forecast for a fixed selected model under stated assumptions. A retrospective solar-power case study yields high correlations between predicted and reference trajectories across the considered forecast horizons, demonstrating reproduction of a constructed channels-dependent quantity of interest.

We develop a graph-compatible batched wavefront formulation for upwind discontinuous Galerkin discretisations of linear transport. For sweepable discretisations, the cell equations form a block lower-triangular system under a topological ordering of the directed upwind dependency graph. This causal structure can be shared across families of fixed-source problems whose local operators, sources and inflow data vary. We exploit this observation by grouping channels with a common dependency graph into graph-compatible sweep classes and carrying samples, right-hand sides, energy groups and sign-compatible angular ordinates as tensor dimensions within a common wavefront schedule. We prove that the resulting batched wavefront algorithm is algebraically equivalent to independent classical DG sweeps for every channel in a class, while exposing parallelism simultaneously across wavefront cells and channel dimensions. We realise the formulation as a GPU tensor program based on batched cell-local solves and quantify its work, depth and storage requirements, including the throughput–memory tradeoff introduced by sample microbatching. We further characterise the discrete parameter-to-observable map on fixed face-sign branches and show that reverse-mode differentiation through the wavefront algorithm reproduces the corresponding discrete adjoint action. Numerical experiments verify the expected DG convergence and adjoint consistency, demonstrate substantial GPU execution gains from sample batching, and illustrate the method in a material-shadowing uncertainty-quantification problem and a coupled multigroup C5G7/KAIST-inspired power-iteration workload.

In 1965, Chowla conjectured that the signs of the Liouville function become asymptotically uncorrelated at any fixed collection of distinct shifts. In 2015, Matomäki, Radziwiłłand Tao proved an averaged form of Chowla's conjecture. In 2022, Lichtman proved a variant of this conjecture over primes on average. In the same year, Lichtman and Teräväinen proved the Hardy-Littlewood-Chowla conjecture on average. In this article, motivated by the recent work of Bergelson and Richter on the dynamical generalizations of the prime number theorem, we will first establish the analogous dynamical generalizations of these three results related to Chowla's conjecture. In the proofs, we follow an approach of Qi and Zheng who established a variant of Bergelson and Richter's theorem over irreducible binary cubic forms. In 2016, Tao proved that the two-point logarithmic Chowla's conjecture holds. Recently, Charamaras and Richter generalized Tao's theorem to bounded arithmetic functions and proposed a conjecture that generalizes Chowla's conjecture to bounded multi-variable arithmetic functions. Building on their work, we will also prove a dynamical generalization of Tao's theorem and some variants for the composition of the sum-of-digits function with the prime Omega function.

The functions $\varphi_\ell(z)=(e^z-s_\ell(z))/z^\ell$, with $s_\ell$ the Taylor polynomial of $e^z$ of degree $\ell-1$, are what exponential integrators evaluate, and they evaluate them through rational approximations on the negative real axis $\mathbb{R}_-$. Schmelzer and Trefethen (2007/08) conjectured that the best such approximations converge at the rate of the classical `1/9'-problem: $E_n(\varphi_\ell)^{1/n}\to H$, Halphen's constant, for every $\ell$. We prove it. The theorem behind it gives the same rate for $f=u_0+u_1\exp$ with $u_0$ rational and $u_1\not\equiv0$ meromorphic with finitely many poles and $\log|u_1(z)|=o(|z|)$, the lower bound required only away from $\mathbb{R}_-$, whenever $f$ has no pole on $\mathbb{R}_-$ (Theorem~4.4). For rational $u_1$ this is Theorem 1 of Stahl and Schmelzer (2009), announced with its proof deferred to a manuscript that never appeared; the hypotheses admit, for instance, $u_1(z)=\sinh(\pi\sqrt z)/(\pi\sqrt z)$, whose zeros are infinite in number. The proof is a reading of Gonchar and Rakhmanov (1989) rather than an extension of them. Their theorem concerns a sequence given by a Cauchy-type integral with a varying weight $\Phi_n$, and a contour identity of Schmelzer and Trefethen presents $u_1\exp$, once its principal parts are removed, in exactly that form, with $\Phi_n(t)=u_1(-nt)e^{-nt}$. The factor $u_1(-nt)$ grows subexponentially in $n$, and their hypothesis on $\Phi_n$ divides by $2n$, so it drops out of the external field: field, extremal arc, $S$-property and constant are all theirs, unchanged. For $\varphi_\ell$ the factor is $t^{-\ell}$, whose pole sits at the origin, a point of the approximation set and hence at positive distance from every admissible contour.

Retrieval-Guided Fine-Tuning (RAG-FT) incorporates retrieved data directly into the training objective, but the statistical consequences of noisy retrieval during training remain theoretically undercharacterized. We study this question by modeling RAG-FT as an estimation problem in a multi-task linear regression framework, using an OLS proxy for single-layer linear self-attention to obtain finite-sample risk bounds. Under homoscedastic retrieval noise, we show that retrieval failure decays exponentially with task separation relative to noise, and derive explicit finite-sample conditions under which RAG-FT achieves lower risk than both target-only and full-corpus training. We then introduce a Distance-Proportional Noise (DPN) model, in which retrieval quality degrades with rank, and compare two estimators under the same retrieval process: the OLS proxy and the literal, uniform-weight forward pass of linear self-attention. We prove that the attention estimator's bias diverges as $\Theta(n^{2q})$ even under exact retrieval, while OLS risk remains $\Theta(d/n)$ for every noise exponent $q>0$. These results locate the instability not in noisy retrieval itself, but in the fixed, unweighted aggregation of the literal LSA forward pass, which reweighting by reliability empirically removes. We validate the predicted rate separation through direct simulation of the DPN model.

We classify all smallest sets of edge deletions that destroy every nonnested perfect matching in a complete ordered graph. The vertices have a fixed linear order. A perfect matching selects edges that use each vertex exactly once; it is nonnested when no selected edge has both endpoints strictly between those of another. On $2k$ vertices, exactly $k$ deletions are necessary, and we describe all $2^k+k-2$ minimum deletion sets for every $k\ge2$. Allowing a matching to leave vertices unused changes the extremal problem. Barát, Freschi and Tóth proposed that an $n$-vertex ordered graph avoiding a nonnested $k$-edge matching can have at most $(k-1)n$ edges. We construct counterexamples for every $k\ge5$ and $n\ge2k+1$. For $n\ge3k$, our constructions exceed the proposed value by a number of edges proportional to $k^2$, matching the order of the known upper bound on this excess. The classification follows from cuts between consecutive intervals; the larger constructions coordinate deletions at the two ends of the vertex order. The exact extremal value remains open in general.

In this paper, we study some geometric properties of the disk bundles over the complete Kähler manifolds. When the base space is the complete Kähler-Einstein manifold, we compute the holomorphic sectional curvature, Riemannian sectional curvature of the complete Kähler-Einstein metric on the disk bundle. Moreover, we present a parametric criteria for whether the manifold admits negative pinched property. When the base space is a bounded pseudoconvex domain equipped with its complete Kähler metric, the corresponding ball bundle in the $k$-direct sum of line bundles is a bounded pseudoconvex Hartogs domain with fiber dimension $k$ . If the base domain is simply-connected and $k\geq2$, we prove that the Bergman metric on such a Hartogs domain is Kähler-Einstein if and only if the domain is biholomorphically equivalent to a unit ball.

For every negative fundamental discriminant $-f$ and every $k\geq1$, we construct a rational function $R_{f,2k}\in\mathbb{Q}(x_1,\ldots,x_{2k})$ and a constant $r_{f,2k}\in\mathbb{Q}^\times$ such that \[ m(R_{f,2k})=r_{f,2k}L'(\chi_{-f},1-2k), \] where $m$ denotes the logarithmic Mahler measure and $\chi_{-f}$ is the quadratic Dirichlet character associated with $-f$. This proves the weak Chinburg conjecture. An independent construction using Bloch cycles yields a stronger result in two variables: there exists a nonzero polynomial $R_f\in\mathbb{Q}[x,y]$ satisfying \[ m(R_f)=4wL'(\chi_{-f},-1), \] where $w$ is the number of roots of unity in $\mathbb{Q}(\sqrt{-f})$.

The minimum distance and the covering radius are two fundamental properties of the code. Both have been extended: the former to the generalized Hamming weights hierarchy, and the latter to the generalized covering radii hierarchy. In both cases, the lowest level of the hierarchies corresponds to the classical minimum distance and covering radius, respectively. From a geometric point of view, the minimum distance of the code determines the packing radius, which is upper bounded by the covering radius. It was conjectured this relation extends to all other orders of the hierarchy, namely, that the generalized packing radii are upper bounded by the generalized covering radii of the same order. In this paper we prove this conjecture is true for the second order radii. We also prove the conjecture holds for all orders when the code rate is at most $3/5$. Finally, we show that for any code rate in $(0,1)$, for all sufficiently long codes the conjecture holds for all orders.

Assume the Generalised Riemann Hypothesis. If $A\subseteq \{1,...,N\}$ contains no two elements differing by $p+1$, $p$ prime, then $|A|\ll N^{\frac{89}{90} + o(1)}$.

We prove the Erdős–Chvátal simplex conjecture across its entire parameter range. If $2\le d<k$ and $(d+1)k\le dv$, then every $d$-simplex-free family of $k$-subsets of $[v]$ has at most $\binom{v-1}{k-1}$ members, with equality only for a full star.

Although flexible resource aggregators (FRAs), represented by electric vehicle aggregators (EVAs) and load aggregators (LAs), possess inherent flexibility to mitigate three-phase unbalance, the key challenge lies in how to effectively incentivize them to actively participate in unbalance mitigation. To address this, this paper proposes a generalized Nash bargaining (GNB)-based chance-constrained coordinated operation model for the distribution system operator (DSO) and FRAs. In this framework, the DSO and FRAs cooperate to mitigate three-phase unbalance, where FRAs provide flexibility to reduce the DSO's unbalance mitigation costs, and the DSO offers economic incentives to FRAs in return. Bargaining power is tailored according to each participant's contribution to unbalance mitigation to ensure fair profit allocation. Moreover, a scenario-based chance-constrained formulation is integrated to handle photovoltaic (PV) output uncertainties. To solve the model, a distributed proximal decomposition algorithm (PDA) is introduced for the independent model, while the coordinated model is decomposed into a social welfare maximization subproblem and a payment bargaining subproblem, with an improved bilinear Benders decomposition algorithm developed to solve the former. Numerical results validate the effectiveness of the proposed method in mitigating unbalance, ensuring fair profit distribution, and enhancing robustness against uncertainties.

Periodic Bloom-filter transmission can impose substantial overhead in communication-constrained distributed systems. Lossless compression preserves membership behavior but provides a single transmission size, whereas established OR folding produces smaller representations with higher false-positive rates (FPRs) while preserving the no-false-negative property. This paper investigates channel-aware selection among OR-folded representations. The sender retains an unchanged canonical filter, constructs a catalog satisfying a maximum FPR, and selects the FPR-qualified representation with the largest retained length supported by the communication resources available at each reporting opportunity. Unlike folding driven principally by cardinality and false-positive constraints, selection is driven by time-varying communication conditions. Using two phishing URL datasets, the framework is evaluated under Five-State Markov Capacity, Gilbert–Elliott burst-error, and Rayleigh block-fading models. Channel-aware folding improves communication efficiency and receiver freshness relative to complete-filter and lossless-compression baselines when communication opportunities vary substantially. Under the more favorable Gilbert–Elliott model, it remains competitive in efficiency while maintaining the freshest receiver state. These results show that FPR-qualified folded views provide useful transmission operating points when a recent lower-fidelity update is preferable to delaying a larger representation.

We prove a Filippov-type stability theorem for integro-differential sweeping processes of Volterra type with an outer multivalued perturbation, in a separable Hilbert space, under the assumptions that the moving sets are uniformly prox-regular and Lipschitz continuous with respect to the Hausdorff distance and that the perturbation is one-sided Lipschitz. Given an absolutely continuous solution of the system perturbed both in the state argument of the multivalued term and by an outer integrable term, we show the existence of a solution of the original Volterra sweeping process and estimate the distance between the two trajectories explicitly in terms of the data of the problem. The estimate becomes sharper when only the outer perturbation is present, and it recovers the Lipschitz dependence on the initial condition of Filippov's classical theorem. The proof combines a reduction of the constrained dynamics to an unconstrained differential inclusion, measurable selection arguments, and an enhanced version of Grönwall's inequality established in this work. As an application, we obtain the Lipschitz dependence of the attainable set on the initial set with respect to the Hausdorff distance, together with a one-sided estimate quantifying the effect of the perturbations. We also work out a spatially distributed fishery model with ecological memory whose harvesting rule, triggered by the aggregate biomass, is one-sided Lipschitz but is not Lipschitz continuous with respect to the Hausdorff distance, so that the classical framework does not apply. For this model, our estimates produce a stability constant governed only by the biological data.

We prove that every Igusa–Todorov Artin algebra satisfies the Auslander–Reiten conjecture. More precisely, let $V$ be an $n$-Igusa–Todorov witness, and let $t$ be the number of isomorphism classes of nonprojective indecomposable summands of $V$. If a finitely generated module $M$ satisfies $\Ext_A^i(M,A)=0$ for every $i>0$ and $\Ext_A^q(M,M)=0$ for $1\leq q\leq 2t+1$, then $M$ is projective. As applications, algebras of representation dimension at most three and algebras satisfying $J^{2m+1}=0$ for which $A/J^m$ has finite representation type satisfy the Auslander–Reiten conjecture.