Papers

Predicting the structure and response of inhomogeneous polymer electrolytes requires a description of ion correlations that retains molecular-scale accuracy while remaining transferable across spatial scales and geometries. We develop a neural density functional for electrolytes that preserves spatial symmetries, thermodynamic integrability and the Noether identities, with perfect screening recovered in stable, noncritical bulk states. Its nonlinear density dependence captures the concentration-dependent correlations missed by a pair closure, including a crossover from enhanced to suppressed long-wavelength number fluctuations at strong coupling. The functional describes density profiles at an untrained salt concentration and predicts bulk structure factors and the long-wavelength number response. Trained solely on planar density and internal-force profiles from molecular dynamics, the functional predicts ionic structure in larger domains and in two-dimensional external fields. On the same ion data, it is more accurate than three other neural density-functional architectures and keeps its accuracy with a quarter of the training runs, where the errors of the best alternative grow by about two thirds. The spatial transferability provides a necessary foundation for connecting molecular correlations to continuum predictions at larger scales.

Let $A(X)$ denote the free Abelian topological group over a Tychonoff space $X$. We prove that if $A(X)$ and $A(Y)$ are topologically isomorphic and $X$ is countably compact, then $Y$ is also countably compact. Thus countable compactness is an $A$-invariant and, consequently, an $M$-invariant; this answers Open Problem~7.10.1 of Arhangel'skii and Tkachenko.

The Simplex method is among the most widely used approaches for solving linear programs. Starting at a vertex solution of the feasible region, the algorithm proceeds through a sequence of basis exchanges (known as pivots), each corresponding to a move along an improving edge of the polyhedron toward a better vertex. A central challenge affecting the efficiency of the Simplex method is degeneracy, which can cause long sequences of pivot operations that fail to change the current vertex solution. In this paper, we prove the existence of a pivot rule which is able to escape degeneracy and follow any given shadow edge-direction when initialized at some compatible basis, with a linear number of degenerate Simplex pivots. As a byproduct of our result, we obtain an improved bound on the number of degenerate Simplex pivots needed to solve linear programs defined on 0/1 polytopes.

We prove the lower bounds $\mathrm{R}(6,8)\ge 135$ and $\mathrm{R}(8,10)\ge 345$, improving the bounds 134 and 343 listed in the April 2026 revision of Radziszowski's dynamic survey. We give explicit red/blue colorings of $K_{134}$ with no red $K_6$ or blue $K_8$, and of $K_{344}$ with no red $K_8$ or blue $K_{10}$, and verify them with two independently written exhaustive clique checkers. Starting from published colorings, we find these witnesses by adding vertices and repairing the resulting monochromatic cliques through local search. The search uses exact conflict counts, preparation moves aimed at making remaining cliques cheaper to break, and mutations followed by repair. A loss based on the largest monochromatic clique containing each edge yielded a useful intermediate state for the $\mathrm{R}(6,8)$ construction. Guided by the author, an AI coding agent wrote and ran the search code; we describe the methods and document the ancestry of the resulting colorings.

Let $(A,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d$, residue field $k$ and let $I$ be an $\mathfrak{m}$-primary ideal. Assume $k$ is infinite. Let $N$ be a perfect $A$-module of dimension $t \geq 1$. Let $M$ be a MCM $A$-module. The function $n \rightarrow \ell(\text{Tor}^A_j(M, N/I^{n+1}N))$ is of polynomial type and let $r_{I, N}^j(M)$ be its degree. In general we have $r_{I, N}^j(M) \leq r_{I,N}^2(\text{Syz}^d_A(k))$ for $j \geq d + 2$. If $A$ is also dominant in the sense of Takahashi and $M$ is non-free then we show $\limsup_{j \rightarrow \infty} r_{I, N}^j(M) = r_{I,N}^2(\text{Syz}^d_A(k))$. We prove an analogous result when $V$ is a Cohen-Macaulay module of finite injective dimension and the polynomial type function $n \rightarrow \ell(\text{Ext}_A^j(M, V/I^{n+1}V))$

For a Hausdorff locally quasi-convex abelian group $G$, let $\C(G)$ be the poset of all Hausdorff locally quasi-convex group topologies on its underlying group having the same continuous characters as $G$. We prove that, whenever $G$ is non-precompact, $\C(G)$ contains an order-isomorphic copy of $(\Pow(\cont),\subseteq)$, where $\cont=2^{\aleph_0}$. The embedding takes values between the Bohr topology and the original topology. Consequently, every infinite discrete abelian group $D$ satisfies $|\C(D)|=\width\C(D)=2^{2^{|D|}}$. The discrete reduction for locally compact abelian groups then yields exact cardinality and width formulas for all such groups; in particular, both invariants equal $2^{\cont}$ for every non-compact $\sigma$-compact locally compact abelian group. These results answer Questions 6.1–6.3 and 6.5–6.7, and the locally compact case of Question 6.4, posed by L.~Außenhofer and D.~Dikranjan in \cite{AD20}. The embedding also applies to non-compact complete metrizable locally quasi-convex groups. A non-compact precompact nuclear group with a unique compatible topology shows that non-compactness alone does not suffice in the nuclear setting. Finally, the compatible poset of $\R^{\N}$ is not order-isomorphic to that of any discrete abelian group.

This work is concerned with a stabilizing adaptive dynamic programming (ADP) approach to approximate solution of a given infinite-horizon optimal control problem. Since the latter problem cannot, in general, be solved exactly, a parametrized function approximator for the infinite-horizon cost function is introduced in ADP (so called “critic”). This critic is used to adapt the parameters of the function approximator. The so called “actor” in turn derives the optimal input of the system. It is a notoriously hard problem to guarantee closed-loop stability of ADP due to the use of approximation structures in the control scheme. Since at least stabilizability is always assumed in the analyses of ADP, it is justified to invoke a respective Lyapunov function. The proposed ADP scheme explicitly uses the said Lyapunov function to simultaneously optimize the critic and guarantee closed-loop stability. A Hessian-free optimization routine is utilized for the actor and critic optimization problems. Convergence to prescribed vicinities of the optima is shown. A computational study showed significant performance improvement for the critic-based approach compared a nominal stabilizing controller for a range of initial conditions.

For a graph $G$, let $e(G)$ denote the number of edges of $G$, and let $c(G)$ be the number of distinct cycles in $G$. Morrison, Roberts and Scott asked whether, for every fixed graph $H$ and all large $n$, some $n$-vertex $H$-free graph maximizes both $e(G)$ and $c(G)$. In this paper, we show that the answer is no for every possible chromatic number. Let $T_{n,r}$ be the complete $r$-partite Turán graph on $n$ vertices. We find a constant $\gamma_r>0$ such that if $n$ is sufficiently large and $\mathcal{H}$ is a finite family with $\chi(\mathcal{H})=r+1$ satisfying \[ ex(n,\mathcal{H})<e(T_{n,r})+\gamma_r n, \] then every cycle-maximal $\mathcal{H}$-free graph is edge-extremal, and $\gamma_r$ is best possible.

Let $M$ be a smooth manifold and $TM$ its tangent bundle. A $C^0$-Finsler structure of $M$ is a continuous function $F: TM \to [0, \infty)$ such that restricted to each tangent space is an asymmetric norm. A mollifier smoothing of $F$ is a family of Finsler structures $F_\varepsilon: TM \to [0, \infty)$, parameterized by $\varepsilon > 0$ and constructed using the standard mollifier, such that $F_\varepsilon \to F$ uniformly on compact subsets as $\varepsilon \to 0$. In this work, we construct two mollifier smoothings of $F$. In the first, we assume that $F: TM \to [0, \infty)$ is strongly convex, that is, that $F$ restricted to each tangent space is strongly convex. In the second, we assume that $M=G$ is a Lie group and that $F:TG \to [0, \infty)$ is strongly convex and left-invariant, and in this case, $F_\varepsilon$ is also left-invariant. In both cases, we prove that if $F$ is a Finsler structure, objects such as the fundamental tensor, the Chern, Cartan, Hashiguchi and Berwald connections, and the flag curvature of $(M, F_\varepsilon)$ converge uniformly on compact subsets to the corresponding objects of $(M, F)$ as $\varepsilon \to 0$.

We study the parameter dependence and transversality properties of trajectories of the controlled ODE $$ U_x = P, \qquad P_x = - f(U) - \beta P + \tilde \alpha(x)U, $$ arising from traveling-wave solutions of a bistable reaction-diffusion equation with control. Optimal profiles minimizing the $L^1$-norm of the control $\tilde \alpha$ are determined by the unstable and stable manifolds $\Gamma_u$, $\Gamma_s$, the critical curve $$ P^*(U) = \sqrt{Uf(U)}, $$ and the associated function $$ F(U) = \frac{dP^*}{dU} + \frac{f(U)}{P^*(U)}. $$ We establish generic transversality properties for these objects with respect to perturbations of the speed $\beta$ and the nonlinear source $f$, together with quantitative estimates on the first- and second-order dependence of the invariant manifolds on these parameters. We also analyze the loss of regularity at the critical speed $\beta^{**}$, where the unstable manifold approaches the strongly stable manifold of the equilibrium $(U^*,0)$. These results provide the ODE ingredients used in the analysis of the structure and regularity of optimal profiles in paper [S. Bianchini and C. Trifone. Traveling profiles and control cost for a PDE describing the evolution of invasive species. 2026. arXiv: 2606.23979 [math.OC]].

We establish a conditional theory of genuinely non-perturbative global dynamics for the cutoff Boltzmann equation with hard potentials on the three-dimensional torus. We prove that any global mild solution satisfying \[ \sup_{t\ge0}\int_{\mathbb R^3}W(v)\sup_{x\in\mathbb T^3}F(t,x,v)\,dv<\infty, \] for a suitable stretched-exponential weight \(W\), enjoys uniform weighted \(L^\infty\) control, is unique in the dissipative mild class, and converges exponentially to the global Maxwellian. In particular, these conclusions hold for arbitrary-amplitude solutions admitting a uniform stretched-exponential velocity upper bound. The key mechanism is a *tail-to-coercivity principle*: uniform velocity-tail control generates positive lower bounds for the local density and nonlinear collision frequency, without imposing macroscopic coercivity a priori. A double-Duhamel positivity argument also generates a Gaussian lower bound at arbitrarily short positive times, allowing initial data with local vacuum. We establish a polynomially weighted counterpart under an additional spatially uniform lower velocity profile, obtaining analogous weighted \(L^\infty\) bounds, uniqueness, and exponential relaxation. Our results provide a low-regularity alternative to conditional theories requiring smoothness and pointwise macroscopic bounds. They reveal how a single velocity-tail condition controls large-data Boltzmann dynamics and suggest a route toward problems with physical boundary conditions.

Using skein-theoretic methods, we study the SL2(C) character variety of the five-punctured sphere with arbitrary conjugacy class at each of the five punctures. We show that a Fricke-Klein-Vogt-type relation is realized as a symmetric determinantal hypersurface. We also give a realization of the character variety in terms of cluster variables. We explicitly derive the Poisson structure of the simple closed curves on the surface, and prove that cluster mutations induce automorphisms of the character variety.

We study the kernels of Lefschetz operators arising from complete intersections of nef classes. When the classes are semiample, we provide a complete characterization of mixed Lefschetz properties below a fixed degree – all obstructions are detected by irreducible subvarieties of relevant codimensions. We further show that the kernels in the next degree range vanish off the diagonal, while on the diagonal they admit bases consisting of fundamental classes of irreducible subvarieties annihilated by the Lefschetz operator. We also establish a kernel decoupling theorem for nef classes on complex Lefschetz modules. As an application, we obtain a kernel characterization for supercritical collections of nef classes, thereby proving a conjecture of Shenfeld-van Handel and Hu-Xiao.

Finite element simulations carried out on meshes issued from real-world geometries - patient-specific anatomies, scanned industrial parts, fractured media - almost never satisfy the shape-regularity assumptions under which classical a posteriori error analysis is written. Practitioners are therefore left without a quantitative answer to a very legitimate question: how wrong is my simulation when my mesh is coarse and contains badly shaped elements? In this work, we study to which extent equilibrated flux estimators can provide such an answer. Indeed, they deliver a guaranteed and fully computable upper bound of the energy-norm discretization error, with a constant equal to one; and, crucially for the present study, its derivation uses no shape-regularity, quasi-uniformity or angle condition. The only geometric ingredient is the Poincaré constant on a single cell, which Payne and Weinberger showed to be at most hK /pi on any convex cell K, however distorted (hK being the cell size). Mesh quality affects the local efficiency - the sharpness of the estimator - but never its reliability, so that the failure mode under distortion is pessimism rather than false confidence.

Let $S$ denote a $k$-subset of $\mathbb{Z}$. Let $G$ be a (multiplicative) group of size $k$. Assume that there exists a function $\varphi\colon S \to G$ that is injective and that satisfies the following: If $x, y, z \in S$ and $z = xy$, then $\varphi(z) = \varphi(x) \varphi(y)$. In 1990, Forcade and Pollington conjectured that if $S = \{ 1, 2, \ldots, k \}$, then $G$ is abelian. This has been a long-standing conjecture highlighted by Richard K. Guy as an unsolved problem, and this problem appears to have remained open, with its open status having been noted by Caicedo et al. in 2021 [Electron. J. Combin.]. We succeed in proving the Forcade-Pollington conjecture in the affirmative and in full generality.

Let $p>3$ be a prime number. If $A$ and $B$ are two principally polarized superspecial abelian surfaces over $\mathbb{F}_{p^{2}}$ with $p^{2}$-Frobenius $[-p]$, we prove that there exists a separable polarized isogeny between them with multiplier at most $p/\sqrt{2}$. The multiplier bound is uniform in the pair and asymptotically optimal up to a small multiplicative constant. Given quaternionic coordinates, we provide a deterministic algorithm that satisfies this upper bound in polynomial time. For unrestricted multipliers, $\mathrm{KLPT}^{2}$ provides a heuristic algorithm with a multiplier upper bound of order $p^{6+o(1)}$ for arbitrary pairs and $p^{3+o(1)}$ when one polarization matrix is the identity matrix. Our algorithm's improvement is an explicit upper bound below $p$ with an unconditional deterministic guarantee.

In this work, we propose a mathematical description of the critical point of boundary phase transitions in 2+1D topological orders. Given two boundary phases, $\mathcal{C}_A$ and $\mathcal{C}_B$, with corresponding Lagrangian algebras $A$ and $B$ in a 2+1D topological order $\mathcal{C}$, respectively, we propose that the critical point of the phase transition between these two phases, if it exists, corresponds to $D = A \times_{M} B$, which is the fiber product of $A$ and $B$ over $A$-$B$-algebra $M$. We prove that $D$ is also a condensable algebra. Additionally, we present an alternative classification of condensable algebras in $Z(Vec_G^\omega)$ and reveal splitting phenomenon that arises during the condensation of certain algebras. We also perform explicit computations for the condensable algebras within several examples of $Z(Vec_G^\omega)$, covering both Abelian and non-Abelian groups $G$.

The locus $S_d(X)$ of $d$-secant lines to a projective variety $X$ plays an important role in the study of projective varieties via projections (Lazarsfeld 1987, Kwak 1998, Beheshti-Eisenbud 2010). For a projective scheme $X\subseteq\mathbb{P}^r$ satisfying $\textbf{N}_{d,2}$, we show that $S_d(X)\cup X$ is cut out set-theoretically by the $r$-minors of a matrix $M$ constructed from $d$-forms scheme-theoretically defining $X$ and their linear syzygies. In particular, $S_d(X)=\mathbb{P}^r$ if and only if every $r$-minor of $M$ vanishes. Using this, we find a singular threefold $X\subseteq\mathbb{P}^5$ with $S_4(X)\ne \mathbb{P}^5$ and $(I_X)_3=0$, and a normal fourfold $X\subseteq\mathbb{P}^7$ with $S_3(X)\ne \mathbb{P}^7$ and $(I_X)_2=0$. The nonexistence of such varieties in the smooth case is an open question raised by the second author and by Gruson-Peskine. Next, we consider the locus $S_{k,d}(X)$ of $k$-planes $L$ such that $L\cap X$ contains a hypersurface of degree $\ge d$ in $L$ so that $S_{1,d}(X)=S_d(X)$. The locus $S_{k,d}(X)\cup X$ is set-theoretically defined by the $(r+1-k)$-minors of $M$. As a result, we obtain determinantal equations vanishing on the $(q+1)$-secant variety $\sigma_{q+1}X$ from the equations of $\sigma_qX$ and their linear syzygies when $\sigma_qX$ satisfies $\textbf{N}_{q+1,2}$. Finally, we extend the $d$-secant lemma for lines to the locus $S_{k,d}(X)$ of $k$-planes. This yields a lower bound on the number of linear syzygies.

For every quartic Salem number $\beta$, we prove that the equal-weight Bernoulli convolution $\nu_{\beta^{-1}}$ has Hausdorff dimension strictly less than one. The main step converts large Fourier coefficients at geometrically separated frequencies into a deficit in the Shannon entropy of a finite convolution. A positive trigonometric product gives the entropy estimate, while the algebraic norm controls the number of distinct atoms in each spatial cell. For the two quartic Salem numbers in $(1,2)$, reciprocal differences at suitable return times produce the required Fourier coefficients. A finite block cover then gives the dimension bound. This proof is independent of numerical computation. Separate exact calculations give $\dimH\nu_{\beta_1^{-1}}<1-1.4\cdot10^{-23}$ and $\dimH\nu_{\beta_2^{-1}}<1-7.8\cdot10^{-22}$, where $\beta_1<\beta_2$ are the two parameters in $(1,2)$. The accompanying verification programs use rational interval arithmetic and the Python standard library.

It was known that the canonical bundle of a smooth toroidal compactification of a noncompact finite-volume ball quotient is nef in dimension $n\geq3$ and ample in dimension $n\geq6$. We prove that the canonical bundle is ample in dimension $n\geq4$ and show that this dimension bound is sharp. We construct infinitely many pairwise noncommensurable noncompact Picard modular threefolds whose smooth projective toroidal compactifications have non-ample canonical bundle. Each compactification contains a smooth rational curve meeting the interior and having canonical degree zero. More generally, in every dimension $n\geq2$, we construct infinitely many pairwise noncommensurable noncompact Picard modular varieties whose smooth toroidal compactifications contain smooth rational curves meeting the interior. In particular, such curves persist in arbitrarily high dimension, even when the canonical bundle is ample.

Multiple candidate flight paths between the same vertiports provide flexibility for managing noise exposure and cruise separation in on-demand urban air mobility (UAM). This paper introduces the fleet scheduling problem with alternative flight paths (FSP-AFP). We develop a time-expanded fleet-flow formulation for a finite aircraft fleet, in which each service or empty repositioning flight is assigned a planar path from a predefined set of alternatives connecting its origin and destination vertiports. These path choices are jointly optimized with cruise altitudes, departure times, and repositioning movements, while maintaining aircraft flow balance and enforcing geometric separation between flights cruising concurrently at the same altitude. The objective combines flight costs, service departure delays, and population-weighted noise exposure. A tailored branch-and-cut algorithm incorporates three complementary clique constructions based on a request-augmented conflict graph, exploiting physical flight conflicts, request exclusivity, and request-route relationships. Computational experiments show that, compared with a baseline implemented using Gurobi's native lazy-constraint functionality, the proposed algorithm generally obtains stronger lower bounds and finds feasible solutions for all test instances, while generating fewer cuts on average and achieving a 43.16-fold reduction in the mean number of processed search nodes. An operational case study shows that increasing the number of candidate planar routes per vertiport pair from one to four reduces the weighted objective by 6.71% and modeled noise exposure by 15.14% under the cruise-separation constraints. Further increases in the number of candidate routes yield only marginal improvement in the objective value. These findings show the value of coordinating alternative flight paths, scheduling, and fleet circulation in UAM.

Faster-than-Nyquist (FTN) signaling can improve spectral efficiency by transmitting symbols closer together than the Nyquist limit, at the cost of additional inter-symbol interference (ISI). In this letter, we study the finite-alphabet symbol-wise achievable information rate (AIR) of FTN signaling for BPSK, QPSK, and 16-QAM under a transmit-power constraint and a common minimum mean square error (MMSE) channel-shortening (CS) detection framework, cross-validated for BPSK against an independent reduced-state Ungerboeck Bahl-Cocke-Jelinek-Raviv (BCJR) benchmark. Our results show a clear trend. Lower-order constellations can benefit from more aggressive time acceleration, while higher-order constellations become less tolerant of acceleration and achieve their best spectral efficiency closer to the Nyquist limit. At a reference signal-to-noise ratio (SNR) of 6 dB the optimum acceleration factor shifts from $\tau^\star\approx0.65$ for BPSK to 0.75 for QPSK and 0.90 for 16-QAM. We further show that this trend is preserved in coded systems and remains robust to the detector memory. The impact of FTN operation on the instantaneous-to-average power ratio (IAPR) is also examined. Overall, the results highlight how constellation order should be considered when selecting the operating point for FTN signaling.

The following inequality for vectors $\mathbf{v}, \mathbf{w} \in \mathbb{R}^n$ was considered in [Linear and Multilinear Algebra 74:1779–1798 (2026)]: \[\|\mathbf{v}^p\|\|\mathbf{w}^p\| - \langle\mathbf{v}^p,\mathbf{w}^p\rangle \leq \|\mathbf{v}\|^p\|\mathbf{w}\|^p - \langle\mathbf{v},\mathbf{w}\rangle^p,\] where $\mathbf{v}^p$ is the vector that is obtained by raising each entry of $\mathbf{v}$ to the power $p$. It was shown that this inequality holds for all positive integers $p$, and it was conjectured that it holds for all real $p \geq 2$ when the entries of $\mathbf{v}$ and $\mathbf{w}$ are positive. A formula for the maximum amount by which the inequality's right-hand side can exceed its left-hand side for nonnegative unit vectors was also conjectured. We prove both of these conjectures, and we determine exactly when the inequality holds with equality. We also present a new generalization of this inequality to three or more vectors.

Given a finite-type orientable surface without boundary, outside of a few low-complexity surfaces, we show that any measurable embedding of the associated mapping class group into itself is a measure equivalence. This is a measure-theoretic analog to the result of Ivanov and McCarthy that mapping class groups are co-Hopfian.

We determine the percolation threshold of the Brownian loop soup on a transient locally finite weighted tree with proper cable metric and no killing. It equals $1-D_G/2$, where $D_G\in[0,2]$ is the Hausdorff dimension of the boundary in the metric induced by the normalized Green function. This gives an analogue of Lyons' branching-number formula for Bernoulli percolation, with Green distance replacing graph distance. Spherically symmetric trees with unit conductances and branching number two realize every threshold in $[1/2,1]$, while geometric conductances give explicit positive thresholds below one-half and a subdivided binary tree has threshold zero. Under the capacity assumption of Drewitz, Prévost and Rodriguez, we characterize the threshold one-half by a finite-energy condition, without a minimum-degree assumption. Non-percolation at one-half then follows from their theorem and Lupu's coupling. For supercritical Galton-Watson trees with unit conductances, conditioned on survival, the boundary dimension is almost surely one and the threshold is one-half, with non-percolation at criticality. No offspring moment assumption is required, and leaves and vertices with one child are allowed. The classification also applies intrinsically to proper real trees spanned by their infinite rays.

Random-walk loop-catchers (RWLC) are a one-parameter family of random subsets of the trace of a random walk, recently introduced in [Cai 2026, arXiv:2607.18070], which interpolates between loop-erased random walk (LERW) and the full walk trace, satisfying a recovery property. We prove that the continuous version of this recovery property uniquely characterizes the scaling limit of LERW and RWLC in three dimensions, using a Green-function test via entangled multipath LERW, as outlined in [Cai 2026, arXiv:2607.18070]. For LERW, this resolves the 3D case of the conjecture of Sapozhnikov–Shiraishi [Sapozhnikov–Shiraishi 2018, Probab. Theory Related Fields 172, 615–662] and gives a new and axiomatic proof of the existence of the scaling limit, which was first proved in [Kozma 2007, Acta Math. 199, 29–152]. For RWLC, this gives the construction of 3D Brownian loop-catchers. The characterization also yields inversion invariance of the 3D LERW scaling limit.

We prove the hyperplane winning property of weighted badly approximable vectors inside nondegenerate analytic manifolds and affine subspaces satisfying almost optimal Diophantine conditions. This, in particular, answers a conjecture of Badziahin–Velani for higher-dimensional manifolds, including affine cases that were understood only in $\mathbb{R}^2$.

We prove that for each $d \geq 2$, there exist infinitely many symmetric abstract $d$-rigidity matroids that are not the $d$-rigidity matroid, the $d$-hyperconnectivity matroid, or the $d$-cofactor matroid. When $d=2$, our constructed matroids are pathological, in the sense that they are not contained within any abstract 2-rigidity matroid family. Our construction additionally provides a counter-example to a conjecture of Tyomkyn.

We study the capacity of the primitive diamond relay channel and derive an improved upper bound compared with existing upper bounds by employing the Csiszár–Körner–Marton sum identity and Gallager-type identification. For Gaussian primitive diamond relay channels, we show that our proposed bound coincides with the existing strengthened Gaussian upper bound of Wu, Özgür, Peleg, and Shamai. For discrete memoryless channels, our bound can be strictly tighter than both the cut-set bound and the existing upper bound, and we assess the improvement using different channel instances, together with a decode–forward/compress–forward time-sharing inner bound.

We show that there exists $c > 0$ such that any subset of $\{1,\ldots,N\}$ having size $\gg N / \exp( (\log\log\log N)^c )$ contains a nontrivial pattern of the form $x,x+y,x+2y,x+y^3$. It is the first configuration of complexity strictly greater than $0$, other than refinements of arithmetic progressions, for which quantitative bounds over integers were obtained.