Papers

Ranking data arise in scientific and machine learning applications, including recommendation systems, information retrieval, voting, marketing, and AI preference ranking from human feedback. Existing statistical work has primarily focused on inference tasks such as preference estimation, rank aggregation, and ranking prediction. However, generating realistic synthetic rankings from an observed population is important for privacy-preserving data sharing, benchmark construction, simulation, and uncertainty quantification. This task is challenging because rankings are high-dimensional combinatorial objects with non-Euclidean dependence structures, while ranking populations often exhibit substantial preference heterogeneity. We propose a framework for population-level generative modeling through a latent preference simplex embedding. It estimates a low-dimensional latent preference simplex through a likelihood-based ranking model, leverages flow matching to learn the population distribution of latent preferences, and generates new rankings through the fitted probabilistic ranking model. We show that ranking generation admits an oracle reduction to latent distribution learning and derive finite-sample generative guarantees that clarify how the number of items, ranking length, and latent dimension affect accuracy. Experiments on synthetic and real datasets demonstrate improved population-level fidelity and provide a statistically interpretable representation of preference heterogeneity.

Tarski's high school algebra problem asks whether every true identity concerning addition, multiplication, and exponentiation of positive integers follows from a list of 11 elementary identities. Surprisingly, Wilkie showed that the following identity is valid over the positive integers and yet does not follow from Tarski's axioms: \begin{align*} &\left((1+x)^y + (1+x+x^2)^y\right)^x \cdot \left((1+x^3)^x + (1+x^2+x^4)^x\right)^y = \\ &\left((1+x)^x + (1+x+x^2)^x\right)^y \cdot \left((1+x^3)^y + (1+x^2+x^4)^y\right)^x. \end{align*} Gurevič gave an algebra on 59 elements that satisfies Tarski's axioms but not Wilkie's identity, and over the years several authors whittled down the size of such a countermodel, culminating in a countermodel of size 12 due to Burris and Yeats. On the other hand, Zhang proved that there is no countermodel with fewer than 11 elements. Using SAT, we prove that the smallest countermodels are of size 12, as conjectured by Burris and Yeats. Moreover, we show that there are exactly 8,957,952 countermodels on 12 elements up to isomorphism and provide a simple classification of them. Our SAT approach outperforms dedicated tools for finding countermodels in equational theories, namely Mace4 and SEM. Furthermore, using autoformalization, we prove the correctness of our main result in Lean.

The finite-generator corona theorem for bounded slice regular functions on the quaternionic unit ball was recently established by Colombo, Pozzi, Sabadini, and Wick. In the present paper, we extend the quaternionic \(H^\infty\)-corona theorem to countably many generators and obtain quantitative estimates that are independent of the cardinality of the generating family. We also establish the corresponding \(H^p\)-corona theorem for the full range \(1\leq p<\infty\), with quantitative norm estimates for both finite and countable families of generators. In the Hilbert-space setting, we prove a quaternionic Leech factorization theorem for Hardy-space multipliers and derive, as a consequence, a Toeplitz corona characterization. Our approach is based on a fixed-slice \(2\times2\) complex matrix realization of the slice regular product, together with operator-valued corona and factorization techniques.

We study constrained statistical learning over infinite-dimensional hypothesis classes in the fully nonconvex setting, and establish universal PACC learnability of the solutions of dual algorithms: Probably Approximately Correct on Constraints, guaranteeing optimality and constraint satisfaction at once. This strengthens near-PACC results, whose feasibility residual no amount of data can remove. Optimality is caught between generalization, governed by Rademacher complexity and favoring small classes, and strong Lagrangian duality, which rests on Lyapunov convexity for vector measures and needs decomposability, a demand pulling the other way. We reconcile the two by posing the population problem over a universal RKHS $\mathcal{H}_K$, dense in a decomposable envelope, and learning over norm balls of growing radius. This yields the Tikhonov complexity $\mathfrak{T}^{\varepsilon}_{n}$, the least RKHS norm reaching an $\varepsilon$-optimal Lagrangian level set; we prove it finite, obtain exact learnability of the optimal value, and make the sample threshold explicit and polynomial in $1/\varepsilon$ under a source condition. Feasibility is harder: absent convexity the Lagrangian may not attain its infimum, and dual information pins down only an averaged constraint-risk vector, not the risks of any returned predictor. We introduce the closure-realization gap $\varepsilon^\star_\infty$, an index of how well $\mathcal{H}_K$ retrieves feasible solutions from dualization; it is a property of the problem, not of a modeling choice. Learnability is exact when $\varepsilon^\star_\infty=0$, in particular under dual differentiability, and near-PACC with residual exactly $\varepsilon^\star_\infty$ otherwise. Finally, no distribution-free threshold exists already in the unconstrained specialization, so universality is the canonical frame for dual algorithms over large hypothesis classes.

Spatial transport can create apparent biological heterogeneity even when organoids are intrinsically identical. We develop a transport-to-phenotype null model for diffusively dosed liver-cancer organoid arrays. The model couples bulk diffusion and clearance to partially accessible adsorption, reversible surface residence, productive internalization, and intracellular state dynamics. Matched asymptotics reduce the perforated-domain problem to a Green-function system, while renewal resolvents describe desorption, re-adsorption, and residence-time effects. Across $2000$ random ten-organoid arrays with localized dosing, the predicted transport-only maturation coefficient of variation has median $0.623$; one-factor design changes move this median between $0.27$ and $0.86$. After matching array-mean exposure, distributed dosing reduces the baseline spread approximately fivefold. The analysis also shows that, in a conservative reflecting chamber, desorption changes uptake timing and allocation but not total eventual uptake; reductions in total uptake require a competing loss channel. Residence laws with equal means can nevertheless produce different transient phenotypes. The spatial reduction is verified against finite-element solutions of the full PDE, and the time reconstruction against numerical Laplace inversion. Finally, a large-batch theorem shows that increasing batch size averages independent process variation but not shared line or batch effects. The framework provides a geometry-specific null against which measured organoid heterogeneity can be assessed.

These are detailed expository notes on the staggered optical-lattice hard-core Bose gas of Aizenman, Lieb, Seiringer, Solovej and Yngvason. They restate and expand the arguments and their principal inputs in a common notation, and describe the central and orbitwise decompositions of thermodynamic-limit equilibrium states. The purpose is to make compressed or delegated steps explicit, not to introduce a new method, strengthen the estimates, or otherwise improve the cited results.

Let $A$ be a unital $C^*$-algebra and $I$ be a symmetrically normed ideal of $A$. I introduce and study lexicographic functional calculus (LFC), a kind of multivariate functional calculus for tuples $(a_1,\ldots,a_m)$ of noncommuting self-adjoint elements of $A$ ”acting in lexicographic order,” i.e., from left to right, with an element $b_i \in I$ ”inserted” between the action of $a_i$ and $a_{i+1}$ for each $i=1,\ldots,m-1$. The reason for its introduction is an application to ”functional calculus calculus,” the differential calculus of maps induced by single-variate (continuous) functional calculus. Specifically, I prove that if $f\colon\mathbb{R}\to\mathbb{C}$ is sufficiently regular and $a\in A_{\mathrm{sa}}:=\{c\in A:c^*=c\}$, then $f_{a,I}(b):=f(a+b)-f(a)\in I$ for all $b\in I_{\mathrm{sa}}:=I\cap A_{\mathrm{sa}}$, the map $f_{a,I}\colon I_{\mathrm{sa}}\to I$ is Fréchet $C^k$, and the $k^{\text{th}}$ Fréchet derivative of $f_{a,I}$ may be written in terms of LFC applied to the $k^{\text{th}}$ divided difference of $f$, a function of $k+1$ variables. This result recovers or vastly generalizes nearly all comparable results in the literature. For example, it simultaneously recovers the following three highly related results on the regularity of the function $f_A\colon A_{\mathrm{sa}}\to A$ defined by $a\mapsto f(a)$: (1) If $A$ is commutative and $f\in C^k(\mathbb{R})$, then $f_A$ is Fréchet $C^k$; (2) if $A$ is finite dimensional and $f\in C^k(\mathbb{R})$, then $f_A$ is Fréchet $C^k$; and (3) if $f\colon\mathbb{R}\to\mathbb{C}$ is ”slightly better than $C^k$,” e.g., belongs to the homogeneous Besov space $\dot{B}_1^{k,\infty}(\mathbb{R})$, then $f_A$ is Fréchet $C^k$ no matter the choice of $A$. Before LFC, there was no unifying framework for results (1)–(3); in particular, there was no single result of which they were all corollaries.

In this article we consider polynomials orthogonal with respect to the inner product $$\langle P,Q \rangle_S = \int_{\mathbb{R}}P(x)Q(x) \mathrm{e}^{-V(x)} \, \mathrm{d}x + \lambda\int_{\mathbb{R}}P^\prime(x)Q^\prime(x) \mathrm{e}^{-V(x)} \, \mathrm{d}x$$ where $V$ is a polynomial of even degree (at least four) and positive leading coefficient, and $\lambda > 0$ a constant. The above inner product is a special case of the so-called Sobolev inner product. We show how one can represent the associated Sobolev orthogonal polynomials as a species of Type I multiple-orthogonal polynomial. This correspondence makes use of the WKB asymptotics of a certain second order ODE. From this we may write a Riemann–Hilbert problem for the Sobolev orthogonal polynomials, from which one can deduce a Christoffel–Darboux-type formula for the projection kernel.

We investigate the analyticity of the free energy of the Ising model in the presence of a non-zero external magnetic field, at high temperature, and at low temperature. Using the Fernandez–Procacci convergence criterion for cluster expansions, together with generating-function techniques and graph-theoretical methods, we derive improved convergence conditions in all three regimes. In particular, the generating-function approach yields sharper estimates for polymers and contours in the strong-field and low-temperature regimes, while a new high-temperature expansion based on Veblen's theorem provides a substantially larger analyticity region than the classical results in the literature.

Throughout the last decade, Gaussian universality has been widely studied for high-dimensional estimation problems. Most of the literature focuses on i.i.d. sensing matrices or accounts for special forms of dependence, such as block dependence or other specific row/column dependencies. More general simultaneous row and column mixing has not yet been fully studied. In this paper, we focus on that setting. We prove a Gaussian universality theorem for the lasso in the sparse regime, where the non- Gaussian covariates have linearly dependent rows and columns. To the best of our knowledge, our setting permits a broader simultaneous row and column dependence structure than those treated in much of the prior universality literature. Numerical illustrations for various sparse profiles support the universality claims of this paper.

We study rigidity phenomena in closed Finsler manifolds without conjugate points under assumptions on the $S$-curvature. We prove that a closed $C^\omega$ Finsler manifold with constant $S$-curvature, continuous Green bundles, and admitting a hyperbolic closed geodesic must be Riemannian. In the $C^\infty$ setting, the same conclusion holds under the additional assumption that the geodesic flow is transitive. As a consequence, we obtain rigidity results for Finsler manifolds with uniform visibility universal covering. Our approach is based on the analysis of the Cartan vector field as a Jacobi field and its interaction with the geometry of Green bundles.

Price forecasts are evaluated in EUR/MWh of error, while storage earns euros; their link depends on the decision rule. We separate the optimal value of a signal V(S|C) from the revenue J(g,S) achieved by an implemented policy. For a price-taking asset with daily throughput bound L and a conditionally sub-Gaussian price law of scale s, the optimal gain over climatology is at most L s sqrt(2 I(S;Pi|C)). This is a square-root envelope, not a prediction for an individual policy. On 939 French day-ahead days the tested bound sits at least two orders of magnitude above the reference uplift. Along a nested Gaussian garbling family, however, plug-in policy revenue is non-monotone even though signal information is monotone: at the residual-noise scale the best Gaussian-shrinkage rule achieves -46% of the uplift, below climatology. The mechanism is ordinal. On a relaxation, optimal schedules depend on interval ordering and profitable-pair tests. A constructed rank-only bid captures 90% of the uplift; exchanging the cheapest and dearest entries of an otherwise exact price vector cuts total revenue by 53%. Within the designed sweep, rank agreement has in-sample R^2 = 0.99, against 0.34 for the information upper bound. Climatology already earns 78% of perfect-foresight revenue. Information quantity alone therefore does not value a forecast-driven policy; realised schedules and ordering must be evaluated explicitly.

Let $L =-\Delta_{\mathbb{H}^n} +V$ be the Schödinger operator on the Heisenberg group $\mathbb{H}^n$, where $\Delta_{\mathbb{H}^n}$ is the sub-Laplacian, and $V$ is a nonnegative potential belonging to the reverse Hölder class $RH_q(\mathbb{H}^n)$ for some $q > Q/2$, where $Q:=2n+2$ is the homogeneous dimension of $\mathbb{H}^n$. In this paper, motivated by the work of De León-Contreras and Torrea \cite{DT}, we introduce the Lipschitz spaces $\Lambda_L^\alpha (\mathbb{H}^n)$, $0< \alpha <2$, adapted to $L$ via a pointwise second-order difference condition involving the critical radius function $\rho$ related to $V$, and also introduce another type of Lipschitz spaces $\Gamma^{\alpha/2}_L(\mathbb{H}^n)$, $0< \alpha <\infty$, adapted to $L$ in terms of the heat semigroup $e^{-tL}$. We show that for $0< \alpha <2-(Q/q)$, $\Lambda_{L}^\alpha (\mathbb{H}^n) =\Gamma_L^{\alpha/2} (\mathbb{H}^n)$ with equivalent norms. Applications of $\Gamma^{\alpha/2}_L(\mathbb{H}^n)$ to the regularity of the fractional powers of the operator $L$ are also given.

In a preceding paper, we used the graph-theoretic linear framework to show how transient properties of continuous-time Markov processes – splitting probabilities and the moments of first-passage time (FPT) distributions – could be expressed as rational algebraic functions of the transition rates, by using spanning forests of the underlying graph. This contrasts with the related rational formulas for steady-state (s.s.) probabilities, which use only spanning trees. The biophysicist Terrell Hill sketched a procedure for calculating mean FPTs and splitting probabilities in terms of the s.s. probabilities of a modified Markov process, thereby converting calculations using ensembles of trajectories to those using a single trajectory. Similarly, Mark Kac showed that the mean recurrence time to a state of a Markov process could be expressed in terms of the s.s. probability of that state. Here, we explore further the relationships between transient and s.s. properties, forests and trees, and ensemble and single-trajectory calculations. We formalise Hill's procedure by introducing a Hill operator, $H_u[G]$, on a graph, $G$, and use it to calculate all moments of the conditional and unconditional FPTs from $u$ as rational functions of the s.s. probabilities of $H_u[G]$, which arise from trees and "exchange factors" which arise from forests. We then combine this with an unravelling operator, $U_v[G]$, to calculate all moments of the recurrence time distribution as rational functions of the s.s. probabilities of $G$ and related exchange factors. Surprisingly, the Hill operator turns out to be a left-inverse to the unravelling operator, suggesting that the algebra of operators on linear framework graphs may be of broader interest. Our results integrate and generalise previously disparate findings into a common repertoire of rational algebraic formulas for FPTs of Markov processes.

We study rigidity phenomena for holomorphic foliations associated with irreducible cuspidal cubics. If $C\subset\PP^2$ is such a cubic, we denote by $p$ its cusp and by $q$ its unique smooth inflection point (or flex), where the tangent line has contact of order three with the cubic. Our first main result is an intrinsic global classification through degree seven: if a foliation $\F$ of degree at most seven leaves $C$ invariant, has singular set $\{p,q\}$, the germ $\F_p$ admits a nonconstant holomorphic first integral with $C_p$ as its unique separatrix, and the reduction of $\F_q$ contains no saddle-nodes, then $\F$ has degree two and is projectively equivalent to the standard cuspidal foliation \[ \dd(x^2+y^3)=0. \] In particular it admits a rational first integral. The proof combines polynomial normal forms adapted to the invariant cusp, a global contact constraint at the smooth flex, and weighted local equations arising from the holomorphic first integral. These ingredients eliminate all cases up to degree seven; only the extremal degree-seven case requires a final exact algebraic computation over $\mathbb Q$. Our second main result concerns degree-two deformations. Finally, for $n\geq3$ we introduce the cuspidal linear pull-back locus $\mathcal C_n^{\mathrm{cusp}}$, prove that it is a proper closed irreducible algebraic subset of the classical linear pull-back component of $\operatorname{Fol}_2(\PP^n)$, and prove rigidity of the base foliation for deformations which remain in that pull-back locus. This leads to an intrinsic pull-back recognition problem and, more broadly, to the study of geometrically distinguished irreducible algebraic subloci inside irreducible components of spaces of projective foliations.

This paper studies the uplink performance of OFDM-based cell-free massive MIMO systems in the presence of hardware impairments affecting both low-cost access points and wireless fronthaul transceivers. We consider a centralized architecture with functional split option~8, where the sampled baseband signals are forwarded from the access points to the central processing unit through wireless fronthaul links operating at mmWave or sub-THz frequencies. A unified analytical framework is developed to characterize the aggregate impact of hardware distortions across the access and fronthaul links. In particular, closed-form expressions are derived for channel estimation and uplink spectral efficiency under hardware-impaired transmission and reception. To mitigate the resulting performance degradation, distortion-aware linear receive combining is considered for the wireless fronthaul. Moreover, a max-min fair resource allocation framework is developed to jointly optimize the uplink transmit powers, fronthaul transmit powers, and fronthaul time expansion factor.

The proof of Proposition 1 in the paper [J. Math. 2022, (2022) #7783128] contains a gap, but its statement is correct. Here, we fix this gap.

We establish the global convergence of the (non-homogeneous) Fitness Centrality algorithm for general graphs, deriving an explicit convergence bound for the corresponding fixed-point iteration. Furthermore, we show how the convergence can be dramatically improved by Anderson acceleration and by switching to Newton's method once a sufficiently good approximation to the fixed point has been found. The efficacy of this strategy is illustrated by numerical experiments on different types of graphs.

Let $T$ be a ternary $\Gamma$–semiring and let $\mathsf{SMod}_{T,\Gamma}$ be the category of ternary $\Gamma$–semimodules with the intrinsic five–ary action $T\times \Gamma \times M \times \Gamma \times T\to M$.This paper develops a congruence–first calculus in $\mathsf{SMod}_{T,\Gamma}$.Kernels and quotients are taken in the sense of congruences, not in the sense of cosets of subobjects, because coset quotients generally fail without subtraction. We prove that every morphism admits a canonical factorization through its kernel congruence, establish congruence versions of the three isomorphism theorems, and describe coequalizers as quotients by generated congruences. Exactness is formulated in the regular/Barr-exact sense: kernels are kernel pairs and exactness is expressed by equality of kernels with regular images. Finally, we record a projective generation criterion and a concrete failure of a classical diagram lemma in this Barr-exact but non-abelian setting.

In this paper, we generalize the classical addition theorems for the Jacobi elliptic functions, giving numerous implicit and explicit formulas. We first present determinant identities involving several of the Jacobi functions. Then, we prove explicit addition formulas for a single function. Novel 4-,5-,6- and 7-term addition formulas are also given alongside a general $n$-term addition formula.

We establish a Kneser-based anticoncentration estimate for uniform subset sums in composite cyclic groups. The estimate contains a periodic loss and is weaker than its prime-modulus counterpart. Nevertheless, together with known small- and large-set results, it proves that, for every fixed $t\geq2$ such that $\mathbb{Z}_t$ is strongly sequenceable and every sufficiently large prime $p$, every subset of $\mathbb{Z}_{tp}\setminus\{0\}$ has a valid ordering, thus establishing the analogue of Graham's rearrangement conjecture for this family of composite cyclic groups. We then identify the structural source of this loss. An inverse theorem shows that failure of the stabilizer-free growth underlying prime-type anticoncentration forces almost all of the set into a proper subgroup or one of its cosets. We exploit this structure by reverse absorption. Iterating the resulting dichotomy between non-periodic anticoncentration and structured concentration proves that every subset of \[ \mathbb{Z}_k\setminus\{0\}, \qquad k=\prod_{i=1}^{s}p_i^{e_i}, \qquad \sum_{i=1}^{s}e_i\leq L, \qquad p_1<\cdots<p_s\leq\gamma p_1, \] admits a valid ordering whenever $L$ and $\gamma$ are fixed and the primes $p_i$ are sufficiently large.

Tumor progression is an inherently multiphysical phenomenon in which interstitial fluid dynamics, biochemical transport, and cellular mechanics interact across multiple spatiotemporal scales. Classical mesh-based solvers, although accurate, impose prohibitive computational costs for the repeated evaluations demanded by inverse parameter identification and future patient-specific predictive pipelines. In this work we introduce a Physics-Informed Neural Network (PINN) framework for a tractable chemo-fluidic continuum model of tumor growth that couples an advection-diffusion-reaction (ADR) equation for the tumor volume fraction with a quasi-static Darcy pressure equation for the interstitial fluid pressure. By intentionally decoupling the solid-mechanical equilibrium, we obtain a three-equation system whose gradient structure is stable under automatic differentiation, enabling robust deep-learning optimization. The network simultaneously learns both state variables from physics constraints alone (forward problem) and recovers hidden transport parameters from sparse, noisy synthetic measurements (Data-Assimilation PINN, DA-PINN, inverse problem). We verify the forward solver against a high-resolution finite-difference (FD) reference, achieving a mean absolute error below 0.002. For the inverse problem, starting from an initial permeability estimate of 0.08 (a factor of 4x above the true value of 0.02) with only 5% spatially sparse observations corrupted by 5% Gaussian noise, the DA-PINN recovers the permeability with a relative error below 5%. These results demonstrate that physics-informed deep learning constitutes a viable, computationally efficient route to multiphysics oncology modeling and lays the mathematical groundwork for future integration into clinical data assimilation pipelines.

We give an explicit simple bridgeless cubic graph on 112 vertices with no Petersen coloring, and hence no normal 5-edge-coloring. The graph is identified by the SHA-256 digest in Theorem 1.1. It is assembled from three copies of a four-pole L and a claw six-pole C; in turn, L is assembled from four copies of a four-pole F and one copy of C, where F is obtained from the Petersen graph by deleting the endpoints of one edge. We give direct SAT formulations for Petersen colorings and normal 5-edge-colorings. CaDiCaL 3.0.1 returned UNSAT for both formulas, and drat-trim verified the resulting DRAT proofs. The ancillary archive contains the construction, an explicit relabeling, the encoders, certificates, hashes, and verification programs. Combined with a theorem of Ma, Mattiolo, Steffen, and Wolf, the counterexample also implies that infinitely many connected simple bridgeless cubic graphs have no Petersen coloring. We also give a separately verified, nonisomorphic $D_3$-symmetric 112-vertex counterexample. We do not address whether 112 is minimum.

In this paper, we compute the canonical and Kobayashi-Nomizu connections, together with their curvature, on Lorentzian four-dimensional nilpotent Lie groups endowed with a product structure. We also classify the algebraic Ricci solitons associated with these connections.

We prove a colorful volume Helly theorem for convex sets in $\mathbb R^2$: There is a constant $V>0$ such that if $\mathfrak F_1,\mathfrak F_2,\mathfrak F_3,\mathfrak F_4$ are finite families of convex sets in $\mathbb R^2$ and if $|\bigcap_1^4F_i|\ge V$ for every transversal $F_i\in \mathfrak F_i,\; (i=1,\ldots,4)$, then $|\bigcap \mathfrak{F}_i|\ge 1$ for some $i$. Here $|A|$ is the Lebesgue measure of $A\subset \mathbb R^d$. The main ingredient is the following theorem. Let $Q_1,\ldots,Q_4\subset\mathbb R^2$ be convex quadrilaterals of area at most $1$, where of course each $Q_i$ is the intersection of 4 halfplanes. Then for every $Q_i$ there is one of these halfplanes $H_i$, say, such that $|\bigcap_1^4 H_i| \le 4096$.

Chavan and Zwonek recently proved that every $C^1$ Kobayashi distance isometry of the diamond is holomorphic or antiholomorphic. We show that the $C^1$ assumption is superfluous.

We study pointwise convergence rates of purely random partition estimators in nonparametric regression, where the partition – into hyper-rectangles by purely random trees, or into Voronoi cells by prototype rules – is built independently of the responses. Our analysis rests on a single geometric criterion, shape regularity, relating the diameter of a cell to its volume, which is shown by Bettinger, Portier and Saumard (2026) to be necessary and sufficient, up to logarithmic factors, for achieving the minimax rate $n^{-1/(d+2)}$. We show that centered and uniform trees are not shape-regular – their cells' aspect ratio grows exponentially with the number of splits with probability bounded away from zero – explaining the super-logarithmic corrections in their error bounds, whereas Mondrian trees, whose splits adapt to the current cell geometry, are shape-regular in probability and attain the minimax rate. The same analysis applied to Voronoi partitions yields the first pointwise concentration bounds for Proto-NN, resolving an open problem of Györfi and Weiss (2021), and shows that OptiNet achieves the minimax rate with markedly better success probability – even almost surely, for a suitable choice of parameters – thanks to its $\eta$-net construction.

This chapter deals with nonparametric inference for quantiles from a Bayesian perspective, using the Dirichlet process. The posterior distribution for quantiles is characterised, enabling also explicit formulae for posterior mean and variance. Unlike the Bayes estimator for the distribution function, our Bayes estimator for the quantile function is a smooth curve. A Bernstein–von Mises type theorem is given, exhibiting the limiting posterior distribution of the quantile process. Links to kernel-smoothed quantile estimators are provided. As a side product we develop an automatic nonparametric density estimator, free of smoothing parameters, with support exactly matching that of the data range. Nonparametric Bayes estimators are also provided for other quantile-related quantities, including the Lorenz curve and the Gini index, for Doksum's shift curve and for Parzen's comparison distribution in two-sample situations, and finally for the quantile regression function in situations with covariates.

We develop a proof-carrying recurrence–boundary method for affine families of positive-definite Nahm sums. A universal coordinate-contiguous relation provides exact algebraic cells for finite certificates; a tropical face-limit theorem identifies parameter directions along which a higher-rank Nahm sum degenerates to a lower-rank theta or theta–hypergeometric boundary value; and a recurrence–boundary principle recovers the initial sums from sufficiently many independent boundary limits. Machine learning and reinforcement learning are used only for discovery. Evolutionary symbolic search proposes recurrences and asymptotic rays from exact algebraic data, while a $Q$-learning agent searches for short sequences of legal contiguous-cell identities. No learned output is accepted as proof: every successful candidate is replaced by an exact symbolic certificate. As the main application, we prove Shi and Wang's Conjecture~3.8 (arXiv:2607.23257) for the Nahm sums dual to Zagier's twelfth rank-three example. The search finds a second-order recurrence for a one-parameter family and a five-cell certificate for it, together with two asymptotic rays leading to binary and unary theta series. These give two linear equations for the two initial sums. Solving the resulting $2\times2$ system reduces the conjecture to two generalized-eta identities, certified by valence arguments on $\Gamma_1(300)$ and $\Gamma_1(100)$. Consequently the dual of Zagier's twelfth example is modular, and every member of the affine family is an explicit $\mathbb{Z}[q,q^{-1}]$-combination of the two base products. Complete verification artifacts accompany the paper.

Non-Hermitian degeneracies are usually discussed through spectral coalescence, whereas entanglement is a property of eigenvectors and need not be fixed by the eigenvalues alone. We formulate a compact bipartite model that separates these two notions. A Hermitian operator with a degenerate eigenspace is transformed by an invertible non-unitary similarity map. The resulting Hamiltonian is non-Hermitian and retains a non-defective degeneracy at every finite value of the non-Hermiticity parameter. For an exactly solvable two-qubit realization, the right eigenstates evolve continuously from product states to maximally entangled states although the spectrum is unchanged. We distinguish the positive right-state reduced density matrix from the generally non-positive biorthogonal reduction, for which entropy may become complex. The same two-qubit solution gives a closed partial-transpose negativity and a Schmidt-gauged non-local magic. Entanglement grows monotonically with the non-Hermiticity parameter, whereas the non-local magic vanishes for both the product and maximally entangled limits and is largest at an intermediate coupling. In larger bipartite spaces, the Page entropy and Haar-averaged purity provide reference values for eigenstate typicality. These diagnostics separate non-defective degeneracy, exceptional-point sensitivity, Haar-typical entanglement, and non-stabilizer correlations without relying on a proliferation of basis-dependent spectral quantities.