Papers

We investigate infinite-exponent partition relations on arbitrary relational structures, with a focus on linear orders and graphs. Any such relation contradicts the Axiom of Choice. We show that there are some such relations which are consistent with ZF which imply the failure not just of Choice but also of the Kinna-Wagner Selection Principle KWP$_1$ and the Ordering Principle O.

$a_n=[x^n](1-x)^{-n}(1-x^2)^{-n}$ is the sequence A348410 in the Encyclopedia of Integer Sequences. Using a method from Hautus and Klarner from 1971 and the software Gfun we find an algebraic equation for the generating function $g(z)=\sum_na_nz^n$. The second identity uses the generalized binomial series, popularized in the textbook Concrete mathematics.

We introduce a random matrix framework for studying statistical-mechanical lattice systems through spectral observables. Equilibrium configurations sampled from a Boltzmann measure are mapped to matrix ensembles whose covariance structure is inherited from the spatial correlations of the underlying model. This construction maps real-space correlation functions to a momentum-space variance profile, providing a direct bridge between statistical-mechanical correlations and correlated random matrix ensembles. We derive this variance profile in finite-correlation-length and critical regimes, and compute spectral moments within a Wick-contraction expansion. A complementary self-consistent description of the bulk density is developed using the resolvent formalism. These analytical methods are benchmarked against Monte Carlo data for the two-dimensional Ising model and three-dimensional Edwards–Anderson spin glasses. In both cases, the spectra evolve from the semicircle law at high temperature to model-dependent critical forms reflecting the structure of correlations. The framework, therefore, provides a quantitative spectral route to probing collective behavior in ordered and disordered statistical systems, while also defining a class of physically motivated correlated random matrix ensembles.

In this paper we introduce the concept of a nonnegative rank of a positive operator $T\colon X\to Y$ between ordered vector spaces. In the case of nonnegative matrices, our definition agrees with the standard definition of a nonnegative rank. Under some natural and mild assumptions on the cone $Y_+$, we prove that the nonnegative rank and the rank agree whenever the rank is at most two. This can be considered as the infinite-dimensional version of \cite[Theorem 4.1]{CR93}. We also provide an example of a positive rank-three operator on the Banach lattice $C[0,1]$ with an infinite nonnegative rank.

We solve a fifty-year-old conjecture of Erdős and Graham concerning whether the binomial coefficient $\binom{n}{k}$ with $1 \leq k \leq \frac{n}{2}$ must always have a divisor $\leq n$ that is “close” to $n$: that is, bigger than a constant times $n$. We show this is the case when $k$ is sufficiently large as a function of $n$. However, we show it is possible to find binomial coefficients $\binom{n}{k}$, where $k$ is small compared to $n$, such that $\binom{n}{k}$ does not have divisors $\leq n$ close to $n$. This latter, more substantial argument involves a restricted covering problem with residue classes, sieve methods, and various exponential sum estimates.

We investigate a family of relational systems arising from interval partitions of $\omega$, inspired by Vojtáš's characterization of the bounding and dominating numbers. By varying the underlying asymptotic quantifiers and interval constraints, we obtain several natural interval-type generalizations. We show that the universal variants are remarkably robust: in all the discrete, colored, restricted, bounded, and measure-theoretic settings considered here, the associated bounding and dominating numbers coincide with the classical invariants $\mathfrak b$ and $\mathfrak d$. In contrast, the existential variants systematically reverse these invariants, yielding that the bounding number coincides with $\mathfrak d$ and the dominating number coincides with $\mathfrak b$.

In this work, we present quantum reinforcement learning (RL) as a solution strategy for process synthesis problems. Building on our prior work, we develop a generalized framework that formally poses process synthesis as a Markov decision process and introduces quantum-enhanced RL algorithms to solve it with improved scalability. Earlier implementations of quantum-based RL for process synthesis were limited by qubit requirements, which scaled poorly with problem complexity. This work overcomes this challenge by introducing state encoding algorithms to decouple qubit requirements from problem size. A classical RL-based solution strategy is used as a baseline to benchmark the quantum algorithms under identical training conditions. All algorithms are evaluated across a flowsheet synthesis problem of increasing unit counts to analyze their performance and scalability. Results show that all approaches are capable of identifying the optimal flowsheet designs in small design spaces. For moderate-scale unit counts, quantum approaches demonstrate competitive performance on a per-episode basis and improved efficiency on a per-parameter basis versus the classical RL benchmark. This work provides a foundation for future quantum computing applications within process systems engineering, establishes a controlled benchmark for comparing classical and quantum algorithms, and shows that the proposed quantum variants remain competitive for the process synthesis problem examined in this work.

In this work we present an efficient and practically implementable approach for the application of reinforcement learning (RL)-based control in chemical process systems. This is an area that has yet to widely adopt RL-based control largely due to inherent challenges in trusting RL algorithms and the time-consuming process of training reliable agents. To address these challenges, we leverage a class of RL algorithms termed Y-wise Affine Neural Network (YANN)- RL, which we have developed in our prior work (Braniff and Tian, 2025a). By strategically initializing actor and critic networks YANN-RL algorithms provide confident and interpretable starting points within control schemes. We apply this RL-based control approach to three different process engineering case studies publicly available on the PC-Gym library (Bloor et al., 2026): (i) a continuous stirred tank reactor (CSTR), (ii) a four-tank system, and (iii) a multistage extraction column. Our approach is compared to several popular RL algorithms (PPO, SAC, DDPG, and TD3) and is benchmarked against nonlinear model predictive control (NMPC). These case studies demonstrate that YANN-RL can greatly reduce the training time and data needed, can be deployed with confidence for chemical process systems, and can approach the performance of NMPC without the knowledge of a full nonlinear model.

We establish results for the first sensitivity analysis of the stochastic fluid models (SFMs). We derive expressions for the sensitivity analysis of the key stationary and transient (time-dependent) quantities of this class of models. We also construct numerical examples to demonstrate the application potential of our methodology in queueing systems, such as deteriorating systems and insurance risk processes. This work forms foundation for the sensitivity analysis of other Markovian modulated models, which are generalisations of the SFMs, and have widespread applications.

We study the connection between the degree sequence of a $k$-uniform hypergraph and the size of its largest matching. Let $\mathcal{F}$ be a $k$-uniform hypergraph on $n$ vertices and let $d_1(\mathcal{F})\ge d_2(\mathcal{F})\ge\cdots\ge d_n(\mathcal{F})$ be the vertex degrees arranged in non-increasing order. For integers $k\ge 2$, $s\ge 2$ and $n > 2sk$, we prove that if the $2sk$-th largest degree satisfies $d_{2sk} > \binom{n-1}{k-1} - \binom{n-s}{k-1},$ then $\mathcal{F}$ contains a matching of size at least $s$. Moreover, by relaxing the range of $n$, we obtain the same bound for the $(k+2s-2)$-th largest degree vertex. Note that the number $k+2s-2$ is optimal. For a $k$-set of vertices $S \subseteq [n]$, the degree of $S$ is defined as $\deg(S) = \sum_{v \in S} \deg(v)$, and the minimum of $\deg(S)$ over all non-edge $k$-subsets $S \notin E(\mathcal{F})$ of $V(\mathcal{F})$ is the Ore-degree of $\mathcal{F}$, denoted by $\sigma_k(\mathcal{F})$. Balogh, Palmer and Raeisi proved: for $s \ge 2$ and $n \ge 3k^2(s-1)$, if $\sigma_k(\mathcal{F}) > k\left(\binom{n-1}{k-1} - \binom{n-s}{k-1}\right),$ then $\mathcal{F}$ contains a matching of size $s$. They also conjectured that the result holds when $n > sk$. As a corollary, we prove that the bound on $n$ can be taken to be linear in $sk$ ($ n \geq 3sk $).

We investigate the blow-up analysis and quantitative behavior for a sequence of maps $\{u_n\}_{n=1}^\infty$ from degenerating tori $(T^2,g_n)$ or from degenerating cylinders $(S^1\times [0,\pi],g_n)$ with free boundary conditions $u_n(S^1\times \{0,\pi\})\subset K$ to a compact Riemannian manifold $(N,h)$ satisfying $$E(u_n)+\|\tau(u_n,g_n)\|_{L^2}\leq \Lambda<\infty,$$ where $\tau(u_n,g_n)$ is the tension field of $u_n$, $K\subset N$ is a smooth submanifold. We establish generalized energy identities and prove that away from bubbles, the asymptotic limit of the necks are either some geodesics on $N$ or some geodesic-like curves on $K$ where some length formulas are given. This partially confirms a conjecture by Ding-Li-Liu \cite{Ding-Li-Liu} in the sense of approximate sequence case. Moreover, we study an evolution system to seek minimal cylinders in a compact Riemannian manifold with free boundary and with arbitrary codimensions. By studying the convergence of the flow at infinity, we obtain some existence results of minimal cylinders with free boundary. Compared with the closed case in, an interesting new phenomenon here is that the neck may converges to a geodesic-like curve on $K$.

We consider the case of scattering by several obstacles in $\mathbb{R}^d$ for $d \geq 2$. We establish a relative trace formula for Neumann and transmission boundary conditions analogous to the one obtained in arXiv:2002.07291 for Dirichlet boundary conditions. In the case of $f(x) = x^{1/2}$ the trace has the interpretation of the Casimir energy of the obstacle configuration. In the one-dimensional case, we recover a rigorous version of the Lifshitz formula for the Casimir energy of parallel plates with frequency-independent electric permittivity and magnetic permeability. We thereby strengthen the mathematical foundations of the Casimir effect and demonstrate the flexibility of the rigorous approach established in arXiv:2104.09763 and arXiv:2002.07291.

We prove ergodicity for random dynamics satisfying some expansion and irreducibility conditions. As a particular application, we show that if $R_1,R_2\in \mathrm{SO}(d+1)$, $d\ge 2$, generate a dense subgroup, then the random dynamics of $R_1$ and $R_2$ on $S^d$ is stably ergodic. Previously this was only known to hold in even dimensions. As a consequence, we deduce spectral gap and statistical limit theorems for such systems. In particular, our results apply in the presence of zero Lyapunov exponents.

Given a countable Turing ideal $\mathcal{I} \subseteq \omega^{\omega}$, we say that $x$ is a list (resp. weak list) of $\mathcal{I}$ if $\mathcal{I}=\{x^{[n]} : n \in \omega\}$ (resp. if $\mathcal{I} \subseteq \{x^{[n]} :n \in \omega\}$). We show that, for several natural ideals $\mathcal{I}$, $x$ computes a list of $\mathcal{I}$ if and only if it computes a function dominating all the functions in $\mathcal{I}$. On the other hand, we provide reals which are $\mathsf{HYP}$-strongly null engulfing (and hence $\mathsf{HYP}$-dominating, by results of Greenberg, Kuyper and Turetsky) but which cannot compute a weak list for $\mathsf{HYP}$, solving a problem left open in a recent paper by Greenberg and the second author. This result can be generalized to any countable ideal which is downward closed under $\leq_{\mathsf{HYP}}$. We also give a characterization of reals which compute a list of $\mathsf{HYP}$: $x$ computes a list of $\mathsf{HYP}$ if and only if $x$ is $\mathsf{HYP}$-dominating and $\mathcal{O}$ is $\Sigma^0_2(x)$.

We prove the sharp Gaussian isoperimetric inequality for conjugate heat-kernel measures along a Ricci flow via a monotonicity formula. As consequences, we obtain the exact Gaussian enlargement theorem and a Gaussian-quantile two-set concentration estimate. In particular, this recovers the exponential concentration estimate of Hein–Naber from a sharper isoperimetric profile. We also derive Gaussian rearrangement inequalities, recover the sharp Hein–Naber log-Sobolev inequality, and identify the universal Gaussian-model constants in Bamler's \(L^p\)-Poincaré inequalities. Further applications include Gaussian-profile localization near Bamler's \(H_n\)-centers, convex-order and moment estimates for logarithmic derivatives of the conjugate heat kernel, reverse hypercontractivity, entropy-regular profile stability, and a path-space Bobkov inequality.

We construct a generic extension of $L$ satisfying Martin's Axiom, $2^{\aleph_0}=\aleph_3$, a lightface $\Delta^1_3$ wellorder of the reals, and $\Sigma^1_n$-uniformization for every $n\geq 2$ simultaneously.

We study privacy guarantees in the framework of pointwise maximal leakage (PML) that satisfy two requirements: they are robust under post-processing and upper bound the failure probability, i.e., the probability that the information leakage exceeds a given threshold. We first examine two candidate definitions inspired by (approximate) differential privacy and show that neither one satisfies both requirements simultaneously. We then introduce the notion of the PML envelope, which quantifies the largest amount of information leakage about a secret after arbitrary post-processing of a mechanism's output. By construction, the PML envelope satisfies both requirements. We discuss basic structural properties of the envelope, such as monotonicity, and derive general upper and lower bounds. We further analyze the envelope for two widely used privacy mechanisms: the PML-extremal mechanisms in the high-privacy regime and randomized response. Overall, this work establishes the PML envelope as a natural and operationally meaningful definition for providing privacy guarantees that are preserved under arbitrary downstream transformations.

It is well known that the graph of a total $\mathbf{\Sigma}^1_n$-function is $\mathbf{\Pi}^1_n$. We prove the consistency of the dual assertion at the third projective level: there is a model of $\ZFC$ in which the graph of every total $\mathbf{\Pi}^1_3$-function is $\mathbf{\Sigma}^1_3$. This principle is incompatible with $\mathbf{\Pi}^1_3$-uniformization and hence with the usual projective-determinacy picture. The construction also repairs the final step of the failure-of-uniformization argument from~\cite{HOFFELNER2023103292}.

Semi-blind joint channel estimation and data detection (JCD) is a promising approach to mitigate pilot contamination in cell-free massive multiple-input multiple-output (CF-MaMIMO) networks. The effectiveness of such methods fundamentally depends on identifiability, i.e., the ability to unambiguously recover the unknown channel coefficients and transmitted data signals from the received uplink observations. In this work, we investigate the identifiability of semi-blind JCD from a large-scale system design perspective. We consider a CF-MaMIMO network in which access points (APs) and user equipments (UEs) are spatially distributed according to Poisson point processes (PPPs). The resulting network topology is modeled as bipartite random geometric graph (BRGG) that captures local connectivity induced by wireless propagation. To enable a tractable analysis, the spatially dependent graph model is approximated by a surrogate independent-edge random graph with matched degree distributions. Building on this model, we develop a recursive probabilistic analysis that characterizes the conditions under which semi-blind recovery succeeds with high probability. The proposed analysis reveals an identifiability region as a function of key system parameters, including AP and UE densities and the connectivity radius beyond which channel coefficients are assumed negligible. Monte Carlo simulations validate the predicted identifiability region and assess the accuracy of the proposed graph approximation. The proposed framework provides system level insights into how network density and connectivity affect identifiability in large-scale CF-MaMIMO systems and offers guidelines for selecting deployment parameters and pilot sequence lengths that enable reliable semi-blind recovery.

We prove that it is consistent that every two disjoint boldface $\mathbf{\Sigma}^1_1$ subsets of $\omega_1^{\omega_1}$ can be separated by a boldface $\mathbf{\Delta}^1_1$ set. The forcing starts from $L$ and preserves CH and therefore also $\omega_1^{<\omega_1}=\omega_1$.

The purpose of this paper is to develop a new effective approach to higher-order mixing in the semisimple setting. We prove effective exponential mixing of all orders for partially hyperbolic algebraic actions, under a strong spectral-gap assumption. The decay rates are explicit in the Lyapunov and spectral-gap data, and the required Sobolev orders are explicit. Already at order two, our estimates require only partial Sobolev/Hölder regularity along weak stable and unstable subgroup directions, with no transverse derivatives. For representations admitting better-than-tempered decay, the resulting order-two estimate attains the optimal matrix-coefficient exponent. The proof introduces a new fractional-cohomological method in the semisimple setting. The central analytic input is a solvability theory for multiple fractional cohomological equations of Type~$II$ (sum-of-product type). These equations are solvable in a cohomology-free range governed by the spectral behavior near the edge \(0\), and the solutions satisfy estimates in partial Sobolev norms. This mechanism converts fractional solvability into order-two decay of correlations under partial regularity, and then into effective higher-order mixing, yielding a quantitative form of Rokhlin's multiple-mixing problem.

We introduce three new model comparison games that characterize separability by first-order formulas with generalized quantifiers. The first is built on the Ehrenfeucht-Fraïssé game, the second is a formula-size game, and the third unifies them both and incorporates minor quantifiers.

Distributed optimization is widely used in large-scale and privacy-preserving machine learning, where each agent stores a local objective and communicates only with its neighbors in a connected network. We study decentralized second-order optimization and focus on consensus procedures that approximately average local iterates, gradients, and Hessians through neighbor-to-neighbor communications. We propose a general Decentralized Cubic Newton method for convex optimization under $L_1$-smoothness of gradients and $L_2$-Lipschitz continuity of Hessians, and develop a theory that accurately tracks the inaccuracies caused by consensus and by disagreement between local iterates. Under these assumptions, the method matches the iteration complexity of the exact Cubic Newton method and requires only additional polylogarithmic communication-round overhead to reach the necessary consensus accuracy. We further propose an Accelerated Decentralized Cubic Newton method for strongly convex objectives and show that it matches the iteration complexity of the exact Accelerated Cubic Newton method, again with only additional polylogarithmic communication-round overhead. Finally, although the general method requires exchanging full $d \times d$ Hessian matrices, we show how it can be implemented for generalized linear models by transmitting only vectors, making the approach substantially more practical in high dimensions.

We review non-autonomous Hamiltonian systems, polynomial in two dependent variables, with the property that all of their solutions are meromorphic functions in the complex plane. These are related to known Hamiltonian systems with the Painlevé property, for which the solutions are single-valued outside a set of fixed singularities. Our systems are equivalent to them in the absence of fixed singularities, and give modified Painlevé equations otherwise. Using the geometric approach by computing the Okamoto's spaces of initial conditions for certain Hamiltonian systems with general coefficient functions, we obtain differential constraints on these functions for the systems to have only meromorphic solutions. Guided by the Newton polygon of the Hamiltonian function, we obtain all such systems with polynomial Hamiltonian of degree three, four, five, and seven, up to affine equivalence in the dependent variables, while there are none for degree six or degree higher than seven. We thus obtain a list of 12 standard polynomial Hamiltonians that can serve as reference for the Painlevé equivalence problem. This list contains also some new Hamiltonians not previously written down, such as quartic Hamiltonians for Painlevé I and II, quartic Hamiltonians for the modified Painlevé III and V equations, a quintic Hamiltonian for Painlevé IV and quintic and septic Hamiltonians for a modified Painlevé VI equation.

An edge of a graph of order $n$ is pancyclic if it lies in a cycle of every length $3,\ldots,n$. A graph of order $n$ is vertex-pancyclic if every vertex lies in a cycle of every length $3,\ldots,n$. Recently, Li and Zhan proved that every $2$-connected $[4,2]$-graph of order at least seven contains a pancyclic edge. Zhan asked whether there exists a positive integer $k$ such that every $k$-connected vertex-pancyclic graph contains a pancyclic edge. We answer this question by showing that for every positive integer $k$, there is a $k$-connected vertex-pancyclic graph containing no pancyclic edge.

In this note, we prove a recent conjecture of Amdeberhan, Andrews and Ballantine concerning a double Lambert series (J. Combin. Theory Series A 221 (2026), Paper No. 106154). More precisely, they conjectured that \[ [q^{N2^a}] \sum_{m,k\geq 1} \frac{q^{mk2^a}}{(1+q^{k2^{a-1}})(1-q^{2m-1})} =\sigma_1(N), \] where $\sigma_1(N)$ is the sum of all the positive divisors of $N$. We provide two proofs of this conjecture. One of the approach leads us to derive a new representation of quasi-modular forms $E_2(q)$.

This paper introduces and rigorously analyzes a least-squares weak Galerkin (LS-WG) finite element method for the severely ill-posed Cauchy problem associated with the Helmholtz equation. By utilizing a weak Laplacian operator defined on a space of discontinuous functions, the proposed framework facilitates the seamless treatment of complex boundary conditions and internal interfaces. We emphasize the geometric flexibility of the LS-WG scheme on general polygonal and polyhedral partitions. Furthermore, we prove the uniqueness of the numerical solution and derive optimal-order error estimates with respect to a specifically designed discrete energy norm. Extensive numerical experiments validate the theoretical convergence rates and demonstrate the algorithm's robustness and efficiency over traditional Galerkin approaches.

Let $(M,g)$ be a Riemannian manifold. Choose a pair $(\alpha,H)$, where $\alpha$ is a calibration and $H$ is a calibrated distribution. Using these data, we define a 1-parameter family of forms $\alpha_\varepsilon$ and study its adiabatic limit as $\varepsilon\rightarrow 0$. We show that (i) the limit is a calibration in a generalized sense, (ii) under the usual closedness assumptions, the adiabatic calibrated submanifolds are anisotropic minimal in the classical sense defined in the Calculus of Variations/PDE theory. We apply this construction to $G_2$-manifolds endowed with an associative distribution. Here, one can also define the notion of Fueter maps. We prove that, in the case of isometric immersions, adiabatic calibrated submanifolds coincide with Fueter maps: this is a first-order analogue of the classical relationship between minimal submanifolds and harmonic maps. We provide explicit examples and prove local analytic existence theorems for adiabatic calibrated submanifolds. Applying mirror symmetry as described by the real Fourier-Mukai transform in the standard “toy model” situation, the picture is as follows: adiabatic limits correspond to large radius limits, calibrated (associative) submanifolds correspond to deformed Donaldson-Thomas connections, adiabatic calibrated submanifolds correspond to $G_2$-instantons.

For a complex reductive group $G$, we prove a homological mirror symmetry between the wrapped Fukaya category of the affine Toda system for $G$ and coherent sheaves on the regular centralizer group scheme for the Langlands dual group $G^\vee$. This can be interpreted as a geometric Langlands equivalence for $\mathbb{P}^1$ with mildest wild ramification at $0$ and $\infty$.

We are concerned with the reconstruction of inclusions in elastic bodies based on measurements from a laboratory experiment. In doing so, we solve the inverse problem of the time-harmonic elastic wave equation, in contrast to the stationary wave equation and the corresponding lab experiment proposed earlier in Eberle and Moll (2021). The investigation of the harmonic problem leads to a better reconstruction compared to the stationary one. Since we deal with real measurement data, we have to take into account, that those measurements always include measurement errors, so that we have to handle noisy data. Thus, we consider the linearized monotonicity method for noisy data and introduce a modified version of this method. Based on this, we reconstruct the inclusions numerically.