Papers

Dickman's function $\rho(u)$ denotes the natural density of integers $n$ whose largest prime factor does not exceed $n^{1/u}$. We establish numerically explicit upper and lower bounds for $\rho(u)$, resulting in estimates with a relative error of less than $0.005/u^2$ for all $u\ge 5$. This allows for an approximate evaluation of $\rho(u)$ without the need to solve the delay differential equation numerically.

We propose a new framework for optimisation over non-Archimedean spaces inspired by Berkovich geometry. Specifically, we introduce polydisc spaces, which consists of products of closed balls over a non-Archimedean field. These spaces retain the rigid hierarchical structure of the non-Archimedean field whilst acquiring many desirable geometric features absent from it. We show that metric trees embed naturally into these spaces, demonstrating their capacity to represent hierarchical data. We study their metric geometry, establishing properties such as geodesic uniqueness, confirming their comaptibility with classical optimisation techniques. We further propose a class of real-valued functions given by linear combinations of absolute values of polynomials. These functions admit a piecewise polynomial description along geodesics and satisfy a universal approximation property. We formulate a theory of optimisation on polydisc spaces: we prove existence of minimisers and explore algorithms for finding them. We provide an accompanying open-source Julia library implementing the core objects and optimisation procedures introduced.

We define a quasilocal mass of Bartnik type, and establish its positivity and temporal monotonicity properties for two classes of domains associated with black holes. More precisely, we first show that the quasilocal mass is strictly positive for spacelike hypersurfaces that are: compact with apparent horizon boundary or noncompact with asymptotically flat ends and containing an apparent horizon in any admissible extension. Secondly, we show that the quasilocal mass is monotonically nondecreasing in time within evolutionary scenarios related to the two aforementioned settings.

We develop a hypercohomology theory of a group with coefficients in a crossed module, and apply it to define abelianization maps for Galois cohomology of reductive algebraic groups.

We show that vectorial absolute minimisers of general $k$-th order supremal functionals in $W^{k,\infty}(\Omega,\mathbb R^N)$ satisfy a maximum principle of the form $$ \max_{\overline U} \mathrm{H} \big(\cdot, u, \mathrm D u, ..., \mathrm D^{k}u\big)=\max_{\partial U}\mathrm{H} \big(\cdot, u, \mathrm D u, ..., \mathrm D^{k}u\big), \qquad\forall\ U\subseteq\Omega \mbox{ open}, $$ suitably interpreted. This is only necessary for absolute minimisers, whilst it characterises a relevant weaker notion of absolute minimality involving compactly supported variations. Further, we obtain an existence result to the Dirichlet problem for such weaker absolute minimisers, as an application of the Baire Category method. Finally, via different methods, we supplement our results by establish a gradient maximum principle for $p$-harmonic maps for $p<\infty$.

Data-driven control of nonlinear systems from input-output measurements remains a fundamental challenge, as existing approaches with rigorous closed-loop guarantees predominantly require access to full state measurements. In this paper, we address this gap by proposing a data-driven output-feedback controller design method for nonlinear systems that provides provable closed-loop guarantees while operating solely on measured input-output data. Our approach combines Koopman operator theory with an extended state representation of the nonlinear system constructed from input-output trajectories. This allows us to obtain a bilinear surrogate model directly from data, on which robust state-feedback design methods can be applied. By exploiting the observability of the underlying nonlinear system, we establish exponential stability of the extended state, which in turn implies exponential convergence of the original system state to the origin. Finally, we validate our theoretical findings in numerical simulations.

We study the Hamiltonian $H_n(h,\sigma)=\sum_{i=1}^n h_i(\sigma_i-m), $ where $(h_i)$ are i.i.d.\ real random variables and $(\sigma_i)$ are i.i.d.\ Ising spins. We consider the energy levels obtained after an independent thinning that retains an exponential number of configurations ($e^{O(n)}$). We prove that, after an $(h_i)$-dependent centering, the resulting point process converges in distribution to a Poisson point process with exponential intensity. Thus, the energy levels asymptotically has the one of the Random Energy Model (REM). Our results extend previous ones, where REM universality for this model was established only either for energy fluctuations of order $e^{-O(n)}$ or for $e^{o(\sqrt n)}$ randomly selected configurations. We also identify the limiting Gibbs weights, which converge to a Poisson–Dirichlet law, and the quenched free energy, which exhibits a freezing transition at $\beta=\tilde\lambda$. The proofs are presented here in compressed form; full details are given in the companion preprint.

In this note, we introduce the partial order decomposition number of a digraph $D$, denoted $pod(D)$, defined as the minimum integer $k$ such that $A(D)=A(P_1)\cup\cdots\cup A(P_k)$, where $P_1,\ldots,P_k$ are partial orders on $V(D)$. We prove that $\dic(D)\le \diomega(D)^{pod(D)}$ for every digraph $D$. In particular, every class of digraphs with bounded $pod$ is polynomially $\dic$-bounded. We apply this to tournaments, showing that if $\mathcal C$ is a class of tournaments with bounded dichromatic number, then the closure of $\mathcal C$ under substitution is polynomially $\dic$-bounded, thereby making progress on a question of Aubian, Charbit, Lopes, and the first author. As further applications of $pod$, we prove that poset tournaments of bounded dimension are $\dic$-bounded, derive polynomial lower bounds on the directed clique number of an explicit family of tournaments, thereby answering a conjecture of Gutowski and Rams, and show that tournaments with bounded $pod$ have bounded domination number.

Wilson lines are fundamental probes of gauge theories. We show that, in sufficiently rich representations, they support a large new class of operator insertions. For half-BPS lines in $\mathcal{N}=4$ SYM, some have the quantum numbers of the displacement supermultiplet and define natural defect deformations. By relating their beta-functions to chiral ring OPE coefficients, we show that these deformations are marginally irrelevant, meaning the half-BPS line is IR stable for all representations. We connect these deformations to multidimensional RG flows and find evidence for new conformal line defects.

We provide a complete classification of twisting supercharges for type IIA supergravity in a flat background by determining the orbits of square-zero elements in the $\mathcal{N}=(1,1)$ super Poincaré algebra in ten dimensions under the action of the Spin group. Together with recent results for type IIB, this completes the description of the moduli space of twists for type II supergravities. We show that the type IIA nilpotence variety is the union of the eleven-dimensional nilpotence variety and a second component, confirming a conjecture of Eager, Saberi, and Walcher, and determine which twists of type IIA arise via dimensional reduction. Using the pure spinor formulation of the superstring, we construct maps from the moduli spaces of worldsheet twists to those of target space twists for both type IIA and type IIB, and find that these maps are not surjective. Precisely those twists of supergravity whose stabilizers are special holonomy groups—$G_2$ in type IIA, $\mathrm{Spin}(7)$ and $\mathrm{Sp}(4)$ in type IIB—have no corresponding background on the pure spinor worldsheet. Finally, we work out the action of T-duality relating twists of type IIA and type IIB, and find an obstruction to dualizing the $G_2$-twist of type IIA along certain directions.

Motivated in part by anomaly cancellation for integral charge spectra in chiral gauge theory, we study the symmetric locus in the ideal degree-three Prouhet-Tarry-Escott problem. A symmetric integer solution is one whose entries are paired about a common center $c\in \frac12\mathbb Z$. This symmetry reduces the problem to a sum-of-two-squares equation, $x^2+y^2=u^2+v^2$, in integer variables, subject to the appropriate parity conditions. Thus the problem is governed by representations as sums of two squares. For the full symmetric locus, let $N_{\mathrm{sym}}(H)$ denote the number of nontrivial symmetric integer solutions of height at most $H$, counted with unordered multiset conventions and summed over the admissible centers. Then \begin{align*} N_{\mathrm{sym}}(H) = \frac{4\log 2}{3\pi^2}H^3\log H+O(H^3). \end{align*} The logarithmic enhancement comes from the second moment of the sum-of-two-squares representation function. In particular, the symmetric locus is larger than one would expect from the naive $H^3$ degree-weighted box-counting scale alone. This asymptotic identifies a large arithmetically structured subfamily of the ideal degree-three solution space, and suggests that paired anomaly-free integral charge spectra reflect a fundamental number-theoretic structure.

The ensemble Kalman filter (EnKF), originally developed in the geophysical sciences, is now widely used for state and parameter estimation problems in various domains of application. It may be viewed as a robust, cheap-to-implement, alternative to the optimal, Bayesian, filter. However, despite its empirical successes, theoretical understanding of its properties, in relation to the optimal filter, is in its infancy. In this paper we contribute novel theoretical understanding, studying the behavior of the mean-field limit of the ensemble Kalman filter (MFEnKF) in the linear setting. In this setting the MFEnKF coincides with the Kalman filter itself, for Gaussian initial data. We study the MFEnKF with general, non-Gaussian, initial data. Under the assumptions of nondegeneracy of the signal noise and detectability of the signal-observation pair we make three specific contributions. First, we derive the MFEnKF via a variational approximation of the Bayes update using a covariance-weighted optimal transport metric. Secondly, we use this metric to show that the MFEnKF is a strict contraction towards the subspace of Gaussian measures, giving explicit geometric rates in terms of the covariances, in both discrete and continuous time. Finally, in the continuous-time setting, we deduce a stable form of almost sure accuracy of the MFEnKF: given any pair of (nondegenerate) initial data for the MFEnKF and the optimal filter, any uniformly continuous moments of the two filtering distributions coincide in large time, for almost every observation path.

Quantum combinatorial optimization offers theoretical advantages for complex financial modeling, but physical implementation on Noisy Intermediate Scale Quantum (NISQ) devices is severely constrained by hardware topology. This study presents a hardware benchmarking analysis between a Hardware Efficient Variational Quantum Neural Network (HE-VQNN) and the Warm Start Quantum Approximate Optimization Algorithm (WS-QAOA) for a hybrid Mean Variance and Conditional Value at Risk (CVaR) portfolio objective. By implementing a novel classical quantum hybrid proxy matrix to bypass the CVaR auxiliary qubit bottleneck, we map up to 16 assets from the NIFTY 50 index onto an IBM heavy hex processor. We systematically quantify algorithmic resilience to the "SWAP tax" incurred during routing. Empirical results reveal a critical operational trade-off: WS-QAOA provides exact theoretical mapping but suffers catastrophic hardware decoherence due to exponential nonlocal gate overhead. Conversely, HE-VQNN preserves hardware coherence but lacks the mathematical expressibility to capture dense tail risk asset correlations. This study exposes the limitations of dense financial optimization on current architectures forces an nonviable choice between algorithmic inexpressibility and hardware decoherence. This is indicative of a deeper limitation as to what can and cannot be done with NISQ computers lacking in all-to-all connectivity.

We point out three inaccuracies in paper [M.V. Moreira, T.C. Jesus, and J.C. Basilio. Polynomial time verification of decentralized diagnosability of discrete event systems. IEEE Transactions on Automatic Control, 56(7):1679-1684, July 2011]. First, the authors wrongly claimed that their algorithm for verifying (co-)diagnosability of labeled finite-state automata (LFSAs) did not depend on assumptions. We give an LFSA that is not deadlock-free or divergence-free such that their algorithm cannot correctly verify its diagnosability. Because diagnosability is a special case of co-diagnosability, their algorithm cannot correctly verify co-diagnosability either when LFSAs are not deadlock-free or divergence-free. Second, they wrongly claimed that adding at each dead state an unobservable self-loop can help verifying diagnosability for an LFSA that is not deadlock-free or divergence-free, but this is wrong, because such a modification sometimes changes the diagnosability of an LFSA. Third, they wrongly claimed that their algorithm for verifying co-diagnosability ran in polynomial time. A polynomial-time algorithm unlikely exists, because the problem of verifying co-diagnosability of LFSAs is PSPACE-hard.

Transfer learning addresses the challenge of transfering knowledge from one domain to another. Traditional transfer learning focuses on adapting models trained on a source domain (with a lot of observations) to improve performance on a target domain (with few observations). In this work we consider the case of a model shift and we focus on the transfer learning applied to a causal forest namely HTERF. This causal forest aims to estimate the Conditional Average Treatment Effect (CATE). The approach considered is the offset method presented by Wang (2016) adapted to a causal context. This method relies on the use of intermediate models in order to estimate the offset between source and target distributions. Our main result is a bound on the CATE error of HTERF on target depending on the error of the intermediate models. Simulation studies show the good performances of this approach in different settings on simulations and on a real-world dataset.

This manuscript contains a collection of counterintuitive problems in discrete probability, together with detailed solutions. The dataset was constructed as part of a broader research project investigating the capabilities of the latest-generation Large Language Models (LLMs) in solving discrete probability problems, in order to assess whether LLMs tend to make systematic reasoning errors associated with known cognitive biases. The problems collected here are specifically designed to challenge heuristic reasoning strategies that often lead to intuitively appealing but mathematically incorrect conclusions. The dataset combines several types of problems. Some are adapted from classical probabilistic paradoxes and cognitive-bias literature, while others originate from recreational mathematics sources or were developed by ourselves following similar principles. The primary purpose of this document is to provide a transparent and publicly accessible reference for the problems used in our experimental evaluation of language models, as well as providing detailed human-made solutions. At the same time, we believe that this collection may also prove useful for future research on probabilistic reasoning, cognitive biases, and the evaluation of reasoning capabilities in artificial intelligence systems.

We investigate the probabilistic reasoning capabilities of large language models through a controlled benchmarking study on discrete probability problems. We constructed two datasets, respectively a set of standard exercises and a set of counterintuitive exercises, designed to trigger heuristic reasoning, and evaluated 8 state-of-the-art models, each tested with and without Chain-of-Thought prompting. Models achieve an average accuracy of 0.96 on standard problems but only 0.59 on counterintuitive ones. We further provide empirical evidence of token bias: performance drops by over 20% when canonical formulations are replaced by disguised variants. Embedding misleading suggestions in the prompt reduces performance by up to 34%, with no model proving immune. Taken together, the reported findings suggest that current LLMs are not yet genuine probabilistic reasoners, despite their success in advanced mathematical problems.

Springer fibers are subvarieties of the flag variety parameterized by nilpotent matrices. They are central objects of study in geometry representation theory. This paper focuses on two-row Springer fibers, those corresponding to nilpotent matrices with two Jordan blocks. Irreducible components of two-row Springer fibers are in bijection with two-row standard Young tableaux and also with noncrossing matchings. Inspired by the combinatorial commutative algebra of matrix Schubert varieties, we define a polynomial ideal for each noncrossing matching and prove that these ideals define the corresponding components of the Springer fiber. Our proofs leverage geometric descriptions of Springer fibers established by Fung, Stroppel–Webster, Fresse, and Goldwasser–Nadeem–Sun–Tymoczko. Using these ideals to compute examples, we give two conjectural formulas for the cohomology class of each component of a two-row Springer fiber. We apply commutative algebra techniques to prove these conjectures for a specific family of two-row tableaux.

In this paper we construct numerically a new class of unstable self-similar solutions of the incompressible Navier-Stokes equations in $\mathbb{R}^3$. Our solutions are axially symmetric and homogeneous of degree $-1$ at $\infty$, and are unstable in the sense that the linearization around these solutions contains unstable modes. Solutions of this type have been discovered numerically by Guillod and Šverák and Hou, Wang, and Yang, and have applications to proving non-uniqueness results. The main novelty in this paper is that we discover the existence of such solutions in the space of axially symmetric swirl-free (ASSF) vector fields. These approximate solutions are defined on all of $\mathbb R^3$ and achieve global pointwise residuals of order $10^{-10}$. We discuss the numerical construction of these solutions in detail, as well as their relevance to the problem of non-uniqueness of solutions of the incompressible Navier-Stokes equations in 3D, in the space of ASSF solutions.

We derive asymptotic multivariate normal limits and explicit non-asymptotic normal approximation bounds for group sequential quasi-maximum likelihood estimators under possible model misspecification and within-group dependence. The bounds, obtained using Stein's method, have known constants and apply to a class of dependent-data estimating problems in which the likelihood used for estimation may differ from the true data-generating mechanism. We compute the limiting covariance structure and finite-sample bound explicitly for a Poisson generalized linear mixed model with random group effects and illustrate the results using data from an epilepsy clinical trial.

Decentralized stochastic optimization is a fundamental paradigm for large-scale learning over networks, where agents communicate only with their neighbors and no central coordinator is required. For strongly convex problems, communication efficiency is mainly determined by the condition number \(\kappa=L/\mu\) and the network spectral gap \(1-\beta\). Although deterministic decentralized methods can simultaneously achieve accelerated \(\sqrt{\kappa}\) and \(1/\sqrt{1-\beta}\) dependences, no existing stochastic method attains both improvements at once. In this paper, we propose Multi-Gossip Accelerated DSGD (MG-ADSGD), a decentralized stochastic algorithm that combines Nesterov-type primal–dual extrapolation with multi-round fast gossip averaging. The key idea is to couple the gossip depth with the mini-batch size so that additional communication rounds simultaneously improve consensus accuracy and reduce gradient variance. We show that MG-ADSGD achieves the communication complexity \[ \widetilde{\mathcal O}\!\left( \frac{\sigma^2}{\mu n\epsilon}\log\frac{1}{\epsilon} + \sqrt{\frac{\kappa}{1-\beta}}\log\frac{1}{\epsilon} \right), \] where \(\epsilon\) denotes the target accuracy, \(n\) is the number of nodes, and \(\sigma^2\) is the gradient variance. To the best of our knowledge, this bound yields the best currently available communication complexity for decentralized stochastic strongly convex optimization, up to logarithmic factors that are independent of $\epsilon$.

We compare two cluster algebras related to a positive element $\mathtt{b}$ in the braid group of finite $ADE$ type. One is the localized bosonic extension ${\widetilde{\mathbb{A}}}_\mathbb{C}(\mathtt{b})$ equipped with an initial seed arising from an admissible chain $\mathfrak{C}$ of $i$-boxes, which is deeply connected to monoidal categorification. The other is the coordinate ring $\mathbb{C}[X({\underline{\Delta}} {\boldsymbol{i}})]$ of the braid variety $X({\underline{\Delta}} {\boldsymbol{i}})$ equipped with an initial seed arising from a Demazure weave $\mathfrak{W}$, where ${\boldsymbol{i}}$ and ${\underline{\Delta}}$ are expression sequences of $\mathtt{b}$ and the half twist $\Delta$, respectively. We explicitly construct a Demazure weave $\mathfrak{W}_{{\underline{\Delta}}}(\mathfrak{C})$ for each admissible chain $\mathfrak{C}$ associated with ${\boldsymbol{i}}$, and prove that there exists an algebra isomorphism $\varphi_{{\boldsymbol{i}}}\colon {\widetilde{\mathbb{A}}}_\mathbb{C}(\mathtt{b})\to\mathfrak{C}[X({\underline{\Delta}} {\boldsymbol{i}})]$ which is compatible with the two seeds arising from $\mathfrak{C}$ and $\mathfrak{W}_{{\underline{\Delta}}}(\mathfrak{C})$. Moreover, the isomorphism $\varphi_{{\boldsymbol{i}}}$ sends the PBW vectors ${\overline{\mathsf{p}}}_{{\boldsymbol{i}},k} \in {\widetilde{\mathbb{A}}}_\mathbb{C}(\mathtt{b})$ to the coordinates $z_k \in \mathfrak{C}[X({\underline{\Delta}} {\boldsymbol{i}})]$ indexed by the letters of ${\boldsymbol{i}}$. As applications, we investigate a connection between Demazure weaves and signed words via the $i$-boxes and interpret the isomorphism $\varphi_{{\boldsymbol{i}}}$ from the viewpoint of monoidal categorification using Hernandez–Leclerc categories.

The singular cohomology ring of a matroid is an algebraic invariant which generalizes the Chow ring of a matroid. We study combinatorial and Lefschetz properties of the singular cohomology ring of a uniform matroid. Combinatorially, we construct an explicit basis for the singular cohomology ring in terms of Koszul homology. From this basis we derive multiple formulas for the Hodge numbers of the cohomology ring that recover and extend known formulas for the Chow polynomial of a uniform matroid. We also use this basis to show that the singular cohomology ring of a uniform matroid satisfies the "quasi-projective strong Lefschetz property" – a slight weakening of the Hard Lefschetz property found in the Chow ring of a matroid.

We study two probabilistic questions for stochastic Volterra equations arising in rough volatility. These equations underlie some of the most popular non-Markovian stochastic volatility models in mathematical finance. First, we establish subcritical positive moment bounds for stochastic exponentials driven by Gaussian Volterra processes. In the Gaussian Volterra-Bergomi setting, we prove that if $\rho\in[-1,0)$, then $\mathbb{E}[S_T^p]<\infty$ for every $0<p<p_\rho$, where $p_{-1}=\infty$ and $p_\rho=(1-\rho^2)^{-1}$ for $-1<\rho<0$. For the fractional rough Bergomi kernel, we additionally prove explosion at the critical exponent $p=p_\rho$. Combined with the known explosion above the threshold, this yields the exact criterion $\mathbb{E}[S_T^p]<\infty$ if and only if $0<p<p_\rho$ in the fractional rough Bergomi model. Second, for the fractional Volterra square-root process, equivalently the rough Heston variance process, we prove that its law has a positive atom at zero at every positive time. In particular, no Feller-type condition can make the zero boundary inaccessible in the fractional rough Heston regime.

The classical Erdős-Kac theorem gives a central limit theorem for the number of prime divisors of a random integer. We prove an analog for the number of ramified primes in a random $G$-extension of a number field when $G$ is abelian. This builds on previous work of Lemke Oliver and Thorne in the cases $G = S_d$ ($2 \le d \le 5$), and provides the first examples where local ramification events at distinct primes are not independent. We develop probability results that can be used "out of the box" to prove Erdős-Kac theorems for sequences of ideals in a number field, subject to Tauberian hypotheses involving finite sums of Euler products.

For a finite group $G$, let $\beta(G)$ denote the minimum cardinality $|P|$ among finite posets $P$ whose automorphism group $\Aut(P)$ is isomorphic to $G$. While every finite group is realizable as the automorphism group of some finite poset, exact values of $\beta(G)$ are known only in special cases, most notably for cyclic groups. In this paper we prove that $\beta(\mathbb{Z}_2 \times \mathbb{Z}_4) = 14$; in particular, the product bound $\beta(G \times H) \le \beta(G) + \beta(H)$ is sharp in this case. The upper bound is realized by an explicit $14$-element poset $P_{14}$, whose automorphism group is computed by a height-function argument together with a rigidity analysis of its covering relations. The lower bound, which constitutes the substantive part of the proof, is established by a case analysis of the orbit decompositions of a hypothetical poset on at most $13$ points under a faithful $G$-action, organized according to the largest orbit size; in each case we construct an order-automorphism outside the given copy of $G$, contradicting $\Aut(P) \cong G$. Among non-cyclic groups, to our knowledge this is the first exact determination of $\beta(G)$ whose lower bound requires a structural analysis of this kind: for the other non-cyclic abelian groups of order at most $8$, namely $\mathbb{Z}_2 \times \mathbb{Z}_2$ and $\mathbb{Z}_2^3$, the value of $\beta$ is elementary. The arguments are closely adapted to the subgroup lattice of $\mathbb{Z}_2 \times \mathbb{Z}_4$.

We introduce and analyze a mixed Virtual Element Method for the $p$-Laplace equation in a non-Hilbertian setting, covering the full range $p \in (1, \infty)$. The discrete framework combines standard mixed Virtual Element spaces with a novel non-linear stabilization term designed to mimic the power-law structure of the continuous operator. We establish discrete inf-sup stability under non-Hilbertian norms and rigorously prove the continuity and coercivity of the discrete form. This guarantees the well-posedness of the problem and allows us to derive a priori error estimates for the primal variable and the flux. A set of numerical tests supports the theoretical derivations.

We show that, for every $\varepsilon>0$ and all sufficiently large $k$, any spanning subgraph of the $k$th power of a cycle with minimum degree at least $(1+\varepsilon)k$ contains a Hamilton cycle. This asymptotically settles a conjecture of Espuny Díaz, Lichev, and Wesolek.

This paper sets out the most general computational and econometric implications of the rigorous stochastic equilibrium theory from SELCKE (Staines (2024a)) arXiv:2312.16214. The analytical backbone is the discovery that the system converges geometrically to long-run equilibrium, at a rate given by the greater of the eigenvalue or inverse eigenvalue (from outside) closest to the unit circle and the maximum shock persistence. High-order shocks converge faster. I develop a simulation procedure to test, with asymptotic power, whether stochastic equilibrium exists for a particular model. The fundamental approximation result asserts that, whatever the order of expansion or loss function, the stochastic steady state delivers the most accurate perturbation solution. I also show that super-consistent parameter estimators $O(1/T)$ arise whenever second-order terms vanish. Besides Calvo, I study stochastic equilibrium in two alternative pricing models. Dynamics simplify considerably. I bound the time the impulse response peaks, by the maximum lag in the errors. This lends empirical support to Taylor contracts, although there are issues surrounding unit roots and the strong cost-channel. For menu costs, I demonstrate that the initial price distribution decays away super-exponentially, producing a system equivalent to Calvo with an endogenous reset probability. The impact of idiosyncratic disturbances appears as an additional wedge between actual and efficient output. Blow-up of the objective function at the boundary is proven, with the help of new distributional arguments, so the model meets existing eigenvalue existence conditions for the recursive equilibrium. Along the way, new light is shone on existing theoretical models and statistical procedures.

We construct locally minimizing $(1,2)$-clusters whose exterior interfaces are asymptotic to various prescribed singular area-minimizing cones. For $n+1 \leq 7$, Bronsard & Novack characterized all minimizing $(1,2)$-clusters as standard lenses, whose exterior interface is planar. For $n+1 \in [8,2700]$, the authors together with Bronsard showed the existence of a locally minimizing $(1,2)$-cluster whose exterior interface blows down to some (unknown, possibly non-unique) singular area-minimizing hypercone. For $n+1=8$, this was shown independently by Novaga, Paolini & Tortorelli. Here we develop a refined construction using the Hardt-Simon foliation that realizes prescribed cones. For a singular area-minimizing hypercone $C$ that has an isolated singularity or is cylindrical, we show that if $C$ satisfies an explicit energy bound, then there is a locally minimizing $(1,2)$-cluster whose exterior interface is asymptotic to $C$ with quantitative rates. In fact, if $C$ is an area minimizing Lawson cone satisfying this energy bound, we produce a countably infinite family of distinct locally minimizing clusters asymptotic to $C$, distinguished by their prescribed asymptotic decay to leading order. We verify this energy bound for the generalized Simons cones $C_{k,k}$ in every even ambient dimension $n+1 = 2k+2\geq 8$, and for the cylindrical cone $C_{3,3}\times\mathbb{R}$ in $\mathbb{R}^9$, where $C_{3,3}$ is the Simons cone, therefore answering the cone realization problem in these cases. This in particular removes the upper bound of 2700 on the ambient dimension when $n+1$ is even in our preceding work.