Papers

In this paper, we investigate a computational class of random inverse problems that incorporates model uncertainties through random variable parameters nonlinearly in the forward model as well as additive observational uncertainty. Random inverse problems with nonlinear parameter dependencies may arise in engineering, geophysics, image processing or uncertainty quantification. We study a perspective on structural ambiguities due to the non-injectivity of the forward model together with probabilistic ambiguities by assigning mixture model densities with separate components to the parameters, which leads to an in general complex forward model, observation model and posterior. As a result, the mixture-model parameters in the forward model can be viewed as simultaneously describing aspects of both the nonlinear ambiguity and the uncertainty of the inverse problem. The presented solution algorithms are based on Bayesian inversion and Monte Carlo computation for three observation scenarios leading to posterior densities for the input for given output samples or an observed output density. We apply the derived algorithms to analytic 1D and 2D quadratic models, to an epidemiological inverse parameter estimation and to a heat equation for inverse source localization. We numerically demonstrate in which observation scenarios the proposed algorithm can resolve probabilistic ambiguities in the solution and in which they cannot and show the interaction patterns between these two types of ambiguities. The results suggest that the proposed perspective is useful in making residual structural ambiguities visible in highly ambiguous inverse problems, including cases with a finite and an infinite number of solutions.

We consider entire solutions $u: \mathbb{R}^2 \rightarrow \mathbb{R}$ of the Euler-Lagrange equation associated to the variational integral $\int_{\Omega} g(|\nabla u|)\,dx$ with a strictly convex density $g: [0,\infty)\rightarrow \mathbb{R}$ being of linear growth. We show that the condition $\int_{0}^{\infty} t\,g''(t)\,dt < \infty$ implies the Bernstein property, which means that $u$ must be an affine function. If this condition on g is weakened, we still have some partial Bernstein results.

Normally the statistical mechanics of learning treats constraints on weight distributions as restrictions that shrink the space of possible solutions. Therefore, it reduces model capacity. In this paper we would like to take a contrary approach, which, however, is based on the earlier work on distribution-constrained perceptrons. Rather than treating a prescribed weight distribution as a mere restriction, we propose that it defines the intrinsic geometry upon which learning naturally unfolds. We formulate both deep neural networks and variational quantum circuits as gradient flows on a product of Wasserstein manifolds – one classical Wasserstein space for each layer and one quantum Wasserstein space for the circuit parameters. Within this geometry, the capacity reduction, which was previously associated with distributional constraints, appears as the metric structure of the constraint manifold itself. We develop a hierarchical mean-field description for deep networks, extend the framework to the quantum setting using the quantum Wasserstein distance of order 1, and introduce two such practical algorithms, Hierarchical DisCo-SGD and Quantum DisCo, that follow approximate geodesics on the manifold of the product itself. Experiments on teacher-student problems, standard image classification tasks, and small variational quantum classifiers show that respecting these distributional geometries improves generalization, stabilizes training, and reduces the severity of barren plateaus compared with unconstrained and purely norm-based baselines. This approach firstly reframes structural constraints as geometric priors and suggests a route for incorporating biological, spectral, or hardware-derived distributional information into both learning systems, viz., classical and quantum learning.

Sticky channels, which never destroy nor create runs, are some of the simplest types of channels with synchronization errors (such as deletions, insertions, and replications). Despite their simplicity, we know little about the capacity of the sticky channels considered in the literature so far beyond what can be gleaned from purely numerical methods. Towards a broader and systematic exploration of these channels, we initiate the study of additive-noise sticky channels, a basic family of sticky channels that extend each input run by an amount independently sampled from a fixed noise distribution. The capacity of these channels corresponds to the capacity per unit cost of additive-noise memoryless channels over the non-negative integers, and they capture some aspects of the loss of synchronization caused by homopolymer length miscalls in DNA sequencing. We first focus on the setting of Bernoulli additive noise with parameter $p$, and uncover curious behavior of the capacity beyond what numerical methods can tell us. For example, the capacity is constant when $p\in [1/\varphi^2,1/2]$ with $\varphi\approx 1.618$ the golden ratio, achieved by zero-error runlength-constrained coding, and behaves differently when $p\approx 0$ vs. when $p\approx 1$. More generally, we determine the capacity exactly for all $p\leq 1/2$, and when $1/2<p<1$ we give analytical bounds that allow us to characterize the asymptotic behavior of the capacity as $p\to 1$. We also extend our analysis to the setting where input strings have bounded runlengths, motivated by DNA-based data storage. Then, we study additive-noise sticky channels beyond Bernoulli noise. We derive capacity lower bounds for all additive-noise sticky channels whose noise distributions have a given mean and support. These lower bounds allow us to characterize the regime where zero-error runlength-constrained coding is never optimal.

For each $d\geq 1$ we construct a norm-1 Hermitian operator whose Pauli expansion contains $N(d)=\exp(\Omega(d^2))$ terms, each of degree $d$ and magnitude $1/\sqrt{N(d)}$ - the largest magnitude permitted by Parseval's identity. For comparison, if a bounded diagonal operator (or equivalently, a bounded degree-$d$ function on the Boolean cube) has $N(d)$ Pauli coefficients, all of magnitude $\Omega(1/\sqrt{N(d)})$, then $N(d)\leq \exp(\widetilde{O}(d^{1.5}))$. This construction implies the noncommutative Bohnenblust–Hille (BH) constant satisfies $\mathrm{BH}_{M_2}(d)\geq\exp(\Omega(d))$. Together with the upper bounds proved in prior work, this settles the asymptotic growth of $\mathrm{BH}_{M_2}(d)$ as exponential. Our lower bound also asymptotically separates $\mathrm{BH}_{M_2}(d)$ from the (classical) hypercube BH constant $\mathrm{BH}_{\{\pm 1\}}(d)$, which in turn is known to be subexponential: $\mathrm{BH}_{\{\pm 1\}}(d)\leq C^{\sqrt{d \log d}}$. Our Hamiltonians are also unitary and thus quantum Boolean functions in the sense of Montanaro and Osborne (2010). As such they refute the quantum Fourier Entropy-Influence conjecture of Bu et al. (2024), a natural generalization of the classical Fourier Entropy-Influence conjecture due to Friedgut and Kalai (1996).

Distributed statistical analyses often aggregate local posterior or predictive densities by repeated pairwise fusion. When the binary fusion rule is nonassociative, changing the ordered aggregation tree can change the final density and the reported posterior summaries. We study this aggregation-order variation for smooth $f$-divergence balancing in a local density chart around a common reference density. Under square-root-transformed supplied weights and first-order nondegeneracy at each generated merge, every tree in a fixed finite family shares the same first-order coefficient, whereas the first probable tree-dependent term appears at second order. We derive the explicit second-order coefficient, propagate it through every tree in the family by a recursion for the tree-indexed second-order density coefficient, and show that a finite three-state posterior contrast detects the resulting discrepancy. The same coefficient determines density-level and quantity-of-interest diameters, rotation bounds, lower bounds, a limitation of scalar calibration, and a local corrected chart representation. The resulting expansions are uniform over each fixed finite tree family satisfying the stated local conditions.

We study a model of opinion dynamics / social learning / peer-review-based market economics on an evolving network, wherein i) each of the first $N$ agents adopts one of two available opinions arbitrarily, and ii) the $(n+1)$-st agent, for $n\geqslant N$, upon arrival, draws a sample of size $k_{n}$, with replacement, from the past agents, such that the $i$-th agent (for $i\leqslant n$) is included in the sample with probability proportional to the number of times they were previously sampled and agreed with. The $(n+1)$-st agent then decides which opinion to adopt i) based on the proportion of sampled agents conforming to each of the two opinions, and ii) according to a stochastic update rule that involves a memory parameter and a rather general reinforcement function. We study both i) the scenario where $k_{n}=k$ remains fixed with $n$, and ii) the scenario where $k_{n}$ grows at a suitable rate with $n$. This model can be represented as an evolving preferential attachment network wherein each vertex is endowed with one of two possible states, and all edges are directed. It can also be framed as a variant of the celebrated elephant random walk. We study the asymptotics of this stochastic process – in particular, the almost sure convergence, and in case of fixed sample sizes, second order fluctuations, of the relative dominance of each opinion, the influence capital and overall network activity.

Families preserving automorphisms of profinite groups are automorphisms that preserve each of the conjugacy classes of pro-cyclic subgroups. This notion appears in the context of verification of the property that every normal automorphism [i.e., an automorphism that preserves each of the normal closed subgroups] of a certain profinite group such as a nonabelian free profinite group or the absolute Galois group of a $p$-adic local field is an inner automorphism, which was proved by Jarden/Jarden-Ritter. In the present paper, we revisit the notion of a families preserving automorphism from the viewpoint of anabelian geometry. We introduce a natural generalized version of this notion for isomorphisms between general closed subgroups of a profinite group, which we shall refer to as families preserving isomorphisms in the profinite group. With regard to this generalized version, we prove that a large class of profinite groups satisfies a property that families preserving isomorphisms of certain closed subgroups of the profinite groups arise from inner automorphisms of them. For instance, the class includes the absolute Galois groups of Henselian discrete valuation fields of positive residue characteristic or Hilbertian fields. Moreover, as an application of this property, together with delicate considerations on pro-cyclic subgroups of the absolute Galois groups of infinite algebraic extension fields of $\mathbb{Q}$, we obtain a generalized version of the Neukirch-Uchida theorem for $l$-quasi-number fields, where an $l$-quasi-number field is defined to be an algebraic extension field $K$ of $\mathbb{Q}$ whose Galois closure $L$ over $\mathbb{Q}$ satisfies that $[L : \mathbb{Q}]$ is not divisible by $l^{\infty}$. Surprisingly, this result includes the Neukirch-Uchida-type result for the class of subfields of the maximal pro-prime-to-$l$ extension fields of arbitrary number fields.

The classical analysis of the Gaussian mechanism in differential privacy reduces privacy loss for a pair of neighboring datasets to a scalar random variable. While this scalar characterization is sufficient for privacy accounting, each perturbation instance also induces a high-dimensional random triangle formed by the sensitivity vector and the two corresponding noise vectors. In this work, we develop two complementary geometric representations of these random triangles. The first representation maps the normalized squared edge lengths to a simplex. We derive its exact joint density, characterize its elliptical support, and reconstruct the classical privacy loss random variable from the simplex coordinates. The second representation maps the spectral shape of each normalized triangle to a hemisphere. We derive the corresponding density and coordinate mappings, recover the same privacy loss, and characterize an equatorial drift together with band concentration as the dimension increases. These results provide two exact geometric coordinate systems that complement the scalar privacy loss and connect differential privacy with the probabilistic analysis of random shapes.

For Jacobi polynomials with parameters greater than $-1/2$, and with the relative extrema enumerated from the endpoint $x=1$, the normalised modulus at the $k$th extremum of degree $n+1$ is proved to be strictly smaller than that at the $k$th extremum of degree $n$, for $1\leq k\leq n$. This proves the Szegő conjecture, recorded in the 1975 fourth edition of his classic monograph Orthogonal Polynomials, and strengthens it by removing the ordering assumption on the parameters. The proof transforms the Jacobi equation to angular form and compares Prüfer amplitudes at equal phase. Combined with a reduction of de Oliveira Filho and a separate quadratic-transformation argument for the boundary case, the result also settles a question concerning the Lovász theta number of spherical distance graphs in every dimension at least four.

We investigate weighted averages involving Bernoulli polynomials and the Gamma function over regular integers modulo n. Building upon recent identities of Kiuchi and Matsuoka together with our previous transformation formulas for weighted averages over regular integers, we derive explicit asymptotic formulas for a broad class of arithmetic functions. As applications, asymptotic formulas are obtained for the identity, Möbius, divisor and Jordan totient functions.

For a finite simple graph $G$, let $t(G)$ be the largest order of an induced tree and let $g(G)$ be the girth. We prove three consecutive conjectures of DeLaViña's Graffiti.pc program. First, writing $\ell(v)$ for the independence number of the subgraph induced by the neighbourhood of $v$, we prove $t(G) \ge \lfloor g(G)/2 \rfloor - 1 + \max_{v \in V(G)} \ell(v)$. Second, if $\mathrm{Per}(G)$ is the periphery and $f(G) = \max_x d(x, \mathrm{Per}(G))$, we prove $t(G) \ge \frac{2}{3} g(G) + f(G)$, and establish the stronger integral bound $t(G) \ge f(G) + \lceil 2g(G)/3 \rceil$ when $G$ contains a cycle. Third, if $\delta'(G)$ is the second-smallest degree, counted with multiplicity, then every connected non-tree graph satisfies $t(G) \delta'(G) \ge g(G) + 1$. These are Conjectures 141, 142, and 143 of Written on the Wall II. Complete, machine-checked Lean 4 proofs of all three formal statements accompany the manuscript.

In this paper, we present a general formulation to address the problems of covering curves and polygonal chains with triangle, and fitting these curves into triangle. These problems can be formulated as special cases of Bellman's lost-in-a-forest problem (escaping triangular forest) and Moser's worm problem (covered by triangle). We model and reformulate the problem by keeping the curve stationary while allowing the triangle to translate and rotate. Subsequently, we derive the functional minimization formulation with support function constraints to solve. We also prove the equivalence and convergence of the formulas. Finally, we employ numerical methods and present results for covering curves with arbitrary triangles of various angles. We also present some corollaries and variant results, including closed curves and closed polygonal chains.

We consider mixed-integer convex optimization problems in which binary indicators control continuous variables. We introduce the Coordinate Optimality Reformulation (CORe) framework, which augments standard indicator formulations by incorporating coordinate-wise optimality information. The resulting reformulations preserve global optimality while substantially improving branch-and-bound performance, particularly in sparse and structured settings where the coordinate-wise optimality conditions expose exploitable problem structure. We first develop the main components of CORe, including coordinate-wise optimality conditions, closed-form characterizations, and disjunctive reformulations. We then demonstrate the framework across multiple problem families, including quadratic problems and robust single-index models. Computational experiments show that CORe can substantially improve solver performance compared with standard big-$M$ formulations.

Masked prediction learns to infer missing variables from visible context selected by a mask schedule. When does small excess risk guarantee recovery of the true joint distribution? We study this question on finite product spaces under masked-block log loss, with conditionals induced by a single joint distribution. To quantify recovery, we introduce an $\varepsilon$-identifiability modulus measuring the worst-case error among joint distributions with population excess risk at most $\varepsilon$. For data with separated modes pinned down by sufficiently large visible contexts, schedules that always retain such contexts can permit nonvanishing mode-weight errors while incurring exponentially small excess risk. An information decomposition explains why: the loss captures mode-weight mismatch only where the visible context leaves the mode uncertain. We prove two-sided bounds showing that, over a fixed range of mode weights, sensitivity to mode reweighting is governed by residual mode uncertainty averaged over the mask schedule. Assigning schedule mass to low-visibility masks that retain this uncertainty yields recovery bounds within the reweighting family. Beyond this family, positive full-mask probability characterizes uniform control of joint KL divergence by excess risk. Exact calculations and controlled gradient optimization validate these predictions.

Fraczyk and Gelander proved in \cite{FG} that for any simple Lie group $G$ of high rank and for every non-lattice discrete subgroup $\Gamma\leq G$, the injectivity radius of points in $G/\Gamma$ is unbounded, resolving a conjecture of Margulis. In this work we obtain an explicit lower bound on the growth rate of the maximal injectivity radius of points taken from growing balls in $G/\Gamma$. More explicitly, we prove that for any $R>0$, one can embed a ball of radius $c\log^{(4)}R$ in $G/\Gamma$ centered at some point $[g]\in G/\Gamma$ where $g$ is taken from $G_R$ and for some constant $c=c(G,\Gamma)$. In particular, we show that for a general discrete subgroup $\Gamma$, if the injectivity radius growth in $G/\Gamma$ is slower than $\log^{(4)}$, $\Gamma$ must be a lattice. Additionally, we give a new, shorter and simpler proof of the Nevo-Stuck-Zimmer Theorem, saying that every action of a high rank simple group with property $(T)$ is either essentially free or essentially transitive. The results in this paper are obtained using the almost structure of measures from the accompanying paper, together with additional geometric considerations. As a step in the proof, we develop the following characterization for lattices. A discrete subgroup $\Gamma\leq G$ is a lattice if and only if there is a probability measure on $G/\Gamma$ which is sufficiently almost invariant under $G$. More precisely, suppose $\Gamma\leq G$ is a discrete subgroup for which there exists a probability measure $\nu$ on $G/\Gamma$ for which $W_1^{b}(g\nu,\nu)\leq \eps_0$ for some $\eps_0(\Gamma)>0$, then $\Gamma$ is a lattice.

We study the conformally invariant exponential biharmonic Choquard equation in \(\mathbb R^{4}\). Our principal result is the complete spectral resolution of the linearized operator at its conformal bubbles. After stereographic projection, the operator becomes a bounded zeroth-order perturbation of the Paneitz operator, with a compact Riesz component on \(L^{2}(\mathbb S^{4})\). We justify the weak conformal transfer, remove every pole-supported distributional defect, and compute all eigenvalues. The Morse index is one. The kernel is the five-dimensional conformal space. All higher modes satisfy a uniform coercivity estimate. The transverse Hessian of the Adams–Choquard deficit is the same linearized operator. Hence the complete spectrum gives sharp local stability with respect to the Paneitz distance from the conformal extremal manifold. The optimal asymptotic constant is \[ \gamma_{\alpha} =\frac{160-12\alpha-\alpha^{2}}{40(10-\alpha)}, \] and the limiting quotient is minimized precisely by the second spherical harmonics. As a nonlinear preparation, we also prove that every normal distributional finite-mass solution satisfies the hypotheses of Niu's classification theorem and is therefore an explicit translation–dilation bubble.

We describe the image of a quantized BFN Coulomb branch $\mathcal{A}_{G,N}^{\hbar=1}$ under localization (abelianization) map for any $G,N$. In most cases, such as quiver gauge theories without loops, this description works for any flavors, sometimes we need to take generic or formal flavor parameters. The answer is given by a roots and residue condition similar to Ginzburg—Kapranov—Vasserot construction of DAHA arXiv:alg-geom/9512017. As a corollary, for any quantized conical Coulomb branch (and zero or small flavors) we provide a mathematical construction of the sphere trace introduced by Gaiotto and Okazaki arXiv:1911.11126.

In this paper, we develop the theory of several Hom-type anti-algebraic structures in the $\mathbb{Z}_2$-graded setting. We introduce and study Hom-anti-associative superalgebras, Hom-anti-dendriform superalgebras, and Hom-anti-pre-Lie superalgebras, which arise as Hom-type generalizations of the corresponding anti-algebraic structures. Various constructions and examples are provided, together with structural properties and representation-theoretic aspects. In particular, we investigate the role of anti-super-$\mathcal{O}$-operators and anti-Rota-Baxter operators in the construction of Hom-anti-dendriform superalgebras. Furthermore, we introduce the notion of noncommutative Hom-pre-Poisson superalgebras and noncommutative Hom-anti-pre-Poisson superalgebras as Hom-type extensions of noncommutative Poisson-type structures in the graded framework. These structures naturally combine Hom-anti-pre-Lie and Hom-anti-dendriform superalgebras through suitable compatibility conditions. Several relationships between the introduced structures are established, providing a unified framework for studying twisted and graded generalizations of noncommutative Poisson-type algebras.

We identify properties of a Banach space $\mathcal{B}$ of analytic functions on the open unit disk $\mathbb{D}$ in the complex plane ensuring that a multiplication operator $M_\psi: \mathcal{B} \to\mathcal{B}$ is Fredholm if and only if its symbol $\psi$ is bounded away from $0$ near $\partial \mathbb{D}$. The properties we identify are shared by a wide variety of much-studied spaces, including the Hardy spaces $H^p(\mathbb{D})$, weighted Bergman spaces $A^p_\omega(\mathbb{D})$, Hardy-Sobolev spaces $H^2_\beta(\mathbb{D})$, the spaces $S_j^p(\mathbb{D})$ of functions having $j$-th derivative in $H^p(\mathbb{D})$, and the disk algebra $A$. Thus, as a corollary, our work characterizes Fredholm multiplication operators on these spaces. In addition, we describe the closed, finite-codimensional subspaces of $\mathcal{B}$ that are invariant under $M_z: \mathcal{B} \to \mathcal{B}$ in terms of the zeros that the functions in such subspaces have in common. We discuss connections between these subspaces and the problem of characterizing the Fredholm multiplication operators on $\mathcal{B}$, and we prove that $M_z$ restricted to such a subspace is always cyclic, with a polynomial cyclic vector having degree equal to the codimension of the subspace.

Unbiased squared distance covariance and the Hilbert-Schmidt independence criterion (HSIC) are fourth-order U-statistics, yet U-centering evaluates them from pairwise arrays in $O(n^2)$ operations. We show that U-centering is exactly least-squares residualization after fitting additive endpoint effects. This analysis of variance (ANOVA) interpretation explains the zero row sums and the normalization through residual degrees of freedom. The same geometry gives regression representations of dependence and two-sample statistics, exact permutation variances, and relations among pooled and regression mean squares. For arrays indexed by $r$-element subsets and $n\ge2r$, higher-order U-centering removes all lower-order subset effects and projects onto a space of dimension $\binom{n}{r}-\binom{n}{r-1}$. For fixed symmetric square-integrable kernels evaluated on independent and identically distributed (i.i.d.) paired observations, we derive the residual covariance matrix, exact mean-squared errors for estimating the highest-order Hoeffding components, unbiased positive semidefinite (PSD) covariance estimates, and moment formulas under independence and permutation. The normalized residual inner product equals a direct unbiased estimator involving up to $2r$ observations, but can be evaluated in $O(n^r)$ operations for fixed $r$. For four-argument matching kernels, three matrix products yield exactly unbiased estimators of a fourth spectral moment of kernel cross-covariance in $O(n^3)$ arithmetic and $O(n^2)$ storage. A numerical illustration distinguishes dependence structures with equal HSIC and different spectral concentration.

We present an up-to-date overview of approximation methods based on Radial Basis Function (RBF) techniques, which have recently attracted attention across various computational tools and application fields. This review study presents relevant results on RBF techniques, highlighting the associated computational challenges and stability issues that must be addressed when high-performance or parallel computation is required.

Efficient uncertainty quantification for kinetic equations with random inputs is challenging because it requires repeated simulations of high-dimensional models, such as the Boltzmann, Landau, and related collisional equations, whose computational cost can quickly become prohibitive. Multifidelity control variates address this difficulty by coupling a small number of high-fidelity simulations with many evaluations of lower-complexity reduced models. In this work, we analyze the case in which the reduced model is replaced by a neural surrogate rather than evaluated through a classical numerical scheme. We show that the resulting estimator remains unbiased and that the change in the optimal variance induced by the neural approximation is controlled by the error between the exact low-fidelity observable and its neural approximation. This estimate is then combined with residual stability estimates for inhomogeneous Fokker–Planck and Bhatnagar–Gross–Krook surrogates. We also extend the analysis to several control variates and to an asymptotic-preserving (AP) hierarchy containing the limiting Euler observable. In the fluid limit, the optimal hierarchical variance converges to the variance associated with the limiting Euler control, while the contribution of the intermediate kinetic correction vanishes. Numerical tests based on micro–macro neural surrogates illustrate the predicted variance stability and the behavior of the AP hierarchy.

This paper addresses the geometric inverse problem of simultaneously recovering two unknown deterministic source supports in a stochastic parabolic equation, where one source appears in the drift term and the other in the diffusion term. We establish that partial boundary flux measurements alone uniquely determine both supports. Moreover, we prove the existence of minimizers for a perimeter-regularized objective functional. For smooth interfaces, we derive a Hadamard-type boundary representation of the shape derivative. To the best of our knowledge, this is the first such formula for the simultaneous recovery of a drift-source support and a diffusion-source support in an SPDE setting. The derivative exhibits a genuinely stochastic two-channel structure: the adjoint state governs the sensitivity of the drift-source interface, while the martingale component controls the sensitivity of the diffusion-source interface. Based on this formula, we develop a shape-gradient reconstruction method. Numerical experiments demonstrate its effectiveness and its capacity to distinguish between the two source channels under both full and partial boundary observations.

We prove the middle-stair conjecture for every complete bipartite graph. If a parallel chip-firing game on $K_{a,b}$ has configuration $\sigma$ with $2ab-a-b<|\sigma|<2ab$, then its eventual period is $2$. The balanced case $K_{a,a}$ was proved by Ji, Li, and Wang using one-parameter conjugate configurations. We introduce two-parameter conjugates $c^{k,\ell}$, in which the rank shift on one side supplies the additive offset on the other. These conjugates preserve both the total number of chips and the activity. An exact Ferrers-diagram count then produces a nonnegative conjugate with two-round firing coverage on one side. The coverage propagates in alternating two-round waves, giving activity $1/2$; non-clumpiness then forces period $2$.

We show that for every closed, smooth manifold $(M,g)$ of dimension $d$, there exists $c(g)$ such that any nodal domain $\Omega_\lambda$ of a Laplace eigenfunction with eigenvalue $\lambda$ contains a geodesic ball of radius at least $c(g) \lambda^{-1/2} \log\log(\lambda)^{-1/2}$ if $d=3$ and $c(g) \lambda^{-1/2} \log(\lambda)^{-\frac{d-3}{2}}$ if $d >3$. This ball is centered at any point at which the eigenfunction attains its maximum in absolute value within the nodal domain. Furthermore, we show that for any $d \geq 3$, there exist sequences of $\lambda$-nodal domains on $\mathbb{T}^d$ whose inner radius is of order $o(\lambda^{-1/2})$.

This paper presents an autonomous online application designed to support inquiry-based learning in mathematics when in-person instruction is not possible. The application teaches the concept of boundedness of functions through adaptive questioning, requiring active problem solving without teacher intervention. It was tested with 92 Hungarian secondary school students preparing for the advanced-level final examination. Assessment based on Bloom's taxonomy showed that most students achieved at least a medium level of understanding, with many reaching higher cognitive levels. The results suggest that such applications may support learning and complement traditional instruction. Moreover, the study suggests that future AI-based educational systems may benefit from similar inquiry-oriented approaches, emphasizing active thinking and guided problem solving rather than passive answer generation.

Building on work of Kassel–Loday and Zelmanov–Zhang, we show that a Steinberg Lie algebra of the complex group algebra of a (suitable) Bestvina–Brady group is of type $FP_2$ and is not finitely presented. This also allows us to answer the (homological) $0$-$1$-$2$ question for Lie algebras.

We compute the murmuration density for the family of quadratic characters over function fields. We show that the expectation of this density coincides with a correction term arising in the traces of high powers of the Frobenius class in this family, for which we determine large-genus asymptotics. This correction term admits a description in terms of the stable homology of the moduli space of hyperelliptic curves of genus $g$ with one marked Weierstrass point. We further identify this correction as a lower-order term in the one-level density and we obtain a non-vanishing result at the central point for quadratic $L$-functions over function fields, via an alternative approach to using the ratios conjecture. We also compute the murmuration density for the thin family of primitive cubic characters in the Kummer setting.

Scenario-tree model predictive control (MPC) represents future information by a rooted tree and optimizes a nonanticipative policy over that tree. Numerical methods for solving the resulting nonlinear program typically compute their search directions through a sequence of branched linear-quadratic regulator (LQR) subproblems. The standard tree Riccati recursion requires linear work but has a dependency chain proportional to tree height. We present an algebraically exact parallel solver based on rake-compress tree contraction. After independent local control condensation, its two operations act on node and edge data that represent conditional quadratic functions. A rake eliminates a leaf and its parent edge, adding their reduced contribution to the parent-node data. A compress eliminates a unary node and replaces its two adjacent edges by one edge, using the same conditional-value composition as parallel Riccati methods on a chain. Together they contract an arbitrary rooted tree to its root; reversing the contraction recovers every Riccati coefficient, state, control, and multiplier. Given a reusable topology plan, a solve with $N$ nodes and fixed state and control dimensions has $O(N)$ arithmetic work and storage and $O(\log N)$ span, independently of tree height, balance, and maximum out-degree. The formulation allows positive-semidefinite dual regularization, including the unregularized case, and an exact linear-size lifting covers the standard scenario-MPC convention of one control per information node. We prove the contraction identities and equivalence to the Karush-Kuhn-Tucker (KKT) system. Three MIT-licensed JAX packages implement the bidirectional contraction, the dual-regularized LQR solver, and a user-facing primal-dual interior-point solver for tree-structured optimal control.