Papers

We investigate a nonlinear quantum Fokker–Planck equation with self-consistent collision frequency, bulk velocity, and temperature. In contrast to quantum Fokker–Planck equations with prescribed diffusion and friction coefficients, the macroscopic quantities are nonlinear functionals of the distribution function. The equation preserves mass, momentum, and kinetic energy, admits a quantum entropy dissipation structure, and propagates the Pauli admissible range in the fermionic case. Its collision operator is also formally connected to the quantum Landau equation. For the Cauchy problem in the three-dimensional whole space, we prove the global-in-time existence and uniqueness of strong solutions near a global quantum equilibrium. The proof is based on a perturbative macro–micro energy method that combines microscopic coercivity, estimates for nonlinear velocity moments, and a macroscopic dissipation argument. We further establish the propagation of nonnegativity and the fermionic Pauli upper bound. Under an additional negative Sobolev assumption on the initial perturbation, we obtain algebraic decay rates toward equilibrium.

Let $k$ and $n$ be positive integers such that $2 \leq k \leq n+1$. We prove that the number of $k$-chains in a union-closed family with universe $[n]$ and size $m$ is minimized when its member sets are largest possible. We also show that, whenever the minimum is nonzero and $m>n$, there are no other minimizing families.

Let $\mu$ be a probability measure on the unit circle with Verblunsky coefficients $\alpha$. Lukic conjectured that a weighted entropy condition with finitely many critical points is equivalent to a decomposition of $\alpha$ into components localized at those points. We give a counterexample for two critical points of multiplicity three, found by GPT-5.6. The construction uses two phase modes with common power-decay exponent $3/20$. The sequence admits the Lukic's decomposition conditions, but its weighted entropy for the corresponding two-point weight equals $-\infty$.

In this article, we study the transcendence of special values of certain Goss type $L$-series at non-negative integers, which takes values in a function field of characteristic $p$. We show that, for a Drinfeld module $\varphi$ over $K$ and an Artin representation $\rho:G_K\to \operatorname{GL}_n(\overline{\mathbb{F}}_q)$, the twisted special value $L(\varphi^\vee,\rho,k)$ is transcendental over $K$ for every non-negative integer $k$. The proof uses the theory of Artin twists of Drinfeld modules, Taelman's regulators of $t$-modules, and the algebraic independence theorem of Gezmis and Namoijam for tractable coordinates of logarithms. As a consequence, we deduce the transcendence of the special $L$-value $L(\rho,k)$ for every positive integers $k$.

Latent low-dimensional structure in datasets of natural and engineered systems enables their sparse sensing, or full-state reconstruction from historical data and very few carefully chosen localized measurements. Depending on the reconstruction algorithm, sensor locations, and measurement noise, the reconstruction risk curves demonstrate a diversity of patterns including a dramatic peak in error known as double descent in Machine Learning literature. Here we explore those scenarios under a unified Data-Noise Averaging theory. Qualitatively, we formulate sufficient criteria for double descent to emerge through a catastrophic amplification of a pathological signal in reconstruction. Quantitatively, we predict the detailed risk curves at a fraction of computational cost, trace reconstruction instability to individual sensors and their combinations, and provide regularization mechanisms to mitigate the instability. We demonstrate results for both static reconstruction of Sea Surface Temperature patterns and time integration of a reduced order model of a PDE.

In this paper, we address two fundamental challenges in the numerical solution of dynamic opti- mal transport (OT) problems. The first challenge arises when the initial and/or terminal densities approach zero and no positive lower bound is available. In this regime, conventional methods may become unstable or computationally inefficient, since the Lipschitz constant of the objective can scale like the reciprocal of the cube of the density. As a result, near-zero regions may lead to slow convergence or even divergence. The second challenge concerns the substantial memory cost of the dynamic formulation, whose discretization over fine spatial and temporal grids requires storing vari- ables across the entire space-time domain. This storage burden quickly becomes prohibitive as the grid is refined or the spatial dimension increases. To overcome the first difficulty, we reformulate the classical discretized dynamic OT problem so that the resulting objective admits an exact proximal mapping. When combined with a linearized alternating direction method of multipliers (LADMM), this reformulation yields an efficient and robust algorithm that remains stable even in challenging settings where the density may vanish. To reduce the memory burden, we further introduce a dis- tributed formulation in which the optimization variables are partitioned across multiple agents. This design substantially lowers the storage requirement for each agent and can also lead to computational acceleration. We validate the proposed framework through numerical experiments in one- and two- dimensional spatial settings under varying levels of difficulty. The results demonstrate the stability, robustness, and scalability of the proposed method in comparison with conventional approaches.

This paper defines a family of Hurwitz–Lerch type functions whose coefficients are the \(k\)-augmented centered triangular numbers. For this family, we obtain the convergence conditions, a reduction formula, and an Euler-operator form. A Vandermonde-based inversion formula is derived for a class of polynomially weighted Hurwitz–Lerch functions. The family considered here is the quadratic case with geometric factors \(1\), \(2\), and \(4\). The resulting formulas show that three consecutive functions recover the classical Hurwitz–Lerch transcendent and its first two Euler derivatives. We also derive recurrence formulas, ordinary generating functions, finite sums, and special values. The values at \(z=1\) are expressed through Hurwitz zeta functions and Bernoulli polynomials. When \(a=1\), the numerator polynomials of the rational values \(H_k(z,-m,1)\) are written in terms of Eulerian polynomials, while the alternating values \(H_k(-1,-m,a)\) are expressed through Euler polynomials.

We provide closed-form expressions for standard 1-Wasserstein distance ($W_1$) and Sliced 1-Wasserstein distance ($SW_1$) for translated probability measures on $\mathbb{R}^d$. Furthermore, we construct an explicit lower bound for the constant $c_d$ established by Carlier, Figalli, Mérigot, and Wang between $W_1$ and $SW_1$ [1] for translated probability measures on $\mathbb{R}^d$. Sliced Wasserstein distances are computationally efficient and used in image generation and other applications, which raises natural questions in computational efficiency versus theoretical correctness. A lower bound on the constant $c_d$ gives information about the cost of that efficiency.

We prove that a general pencil of plane cubics over $\mathbb C$ has exactly $12$ common flex lines. This answers a question of Ciliberto, Miranda, and Roé. The main result of this paper was obtained using generative AI, particularly ChatGPT 5.5 Pro and the Danus system.

\noindent Given an AKSZ theory $\mathbb{T}$ on a manifold $M$, with target a graded vector space $Y$, we formulate a 1-dimensional theory $\mathbb{t}$ on graphs (the “first quantisation picture for $\mathbb{T}$”), whose partition functions reproduce the Feynman graphs of $\mathbb{T}$. More precisely, the theory $\mathbb{t}$ is itself a 1d AKSZ theory with the target built out of $M$, and involving a coupling to 1d supergravity. It yields a form on the space of metric graphs (with length $T$ of an edge and its de Rham differential $\mathrm{d} T$ interpreted as the zero-modes of the graviton and gravitino, respectively); its integral yields the sum of Feynman graphs of $\mathbb{T}$. We study the theory $\mathbb{t}$ in the BV-BFV formalism; a gauge-fixing of $\mathbb{T}$ corresponds to a gauge-fixing of $\mathbb{t}$. At the classical level, $\mathbb{t}$ assigns to vertices certain Lagrangian submanifolds $L_k$ in Cartesian powers $\Phi^{\times k}$ of the phase space $\Phi$ of $\mathbb{t}$. These submanifolds can be thought of as defining a cyclic $\mathrm{L}_\infty$-algebra in Weinstein's symplectic category (“dequantising” the cohomological vector field on the target %target AKSZ dg structure of $\mathbb{T}$). In the path integral construction of $\mathbb{t}$, Lagrangians $L_k$ determine sewing conditions for fields on the incident edges at a $k$-valent vertex. We give examples of this paradigm, such as when $\mathbb{t}$ on edges is the Witten-Morse supersymmetric quantum mechanics (which corresponds to a particular type of gauge-fixing for $\mathbb{T}$ and $\mathbb{t}$). In the example where $\mathbb{T}$ is the non-abelian Chern–Simons theory with structure Lie algebra $\mathfrak{su}(2)$, we describe the vertex Lagrangian $L_{\mathrm{W}}$ (the “Wigner Lagrangian” ).

In a matrix-free geometric-multigrid FGMRES solver for three-dimensional SIMP topology optimization, a reported converged solve is not always a converged solve. On four of 102 held-out states, the projected residual used for stopping falls below $10^{-6}$ while a recomputed true residual is 1.35 to 49.5 times the tolerance; in an unguarded optimization trajectory, 22 of 40 state solves reach the iteration cap without raising an error. We formulate floor selection as a verified control problem: probe the frozen state at the original floor, use two residual features to choose the first attempted floor, and accept no solution until $||f-Ku||/||f||\le10^{-6}$ is recomputed. The two-feature rule matches 98 of 102 reference classifications; the residual guard detects the four missed escalations, and all 102 selected solves satisfy the tolerance. Relative to always using a $10^{-3}$ floor, the policy preserves the original operator on 24 admissible states and avoids mean compliance and gradient changes of 31.0% and 0.340 on those severe random states, and 0.48% and 0.008 on seven optimized designs, at 2.5 times the mean wall time. In a 12-state subset of the held-out states, eight still require escalation at the conventional floor $10^{-6}$. In a nine-state control with the preconditioner's adaptive components disabled, every failure is visible and no false acceptance occurs, tying the stopping-estimate drift to the iterate-dependent preconditioner. The recomputed residual is the correctness safeguard; the probe and floor ladder govern an implementation-specific cost-fidelity tradeoff.

Persistent homology summarizes the birth and death of topological features, but it does not by itself specify where a feature is located in the underlying complex. Harmonic persistent homology addresses this by assigning canonical harmonic cycle representatives to bars. In earlier work, Basu and Cox showed that harmonic representatives of simple bars maximize the total relative weight placed on essential simplices, the simplices that are forced to appear in representatives of the corresponding class. In this paper we prove that, for generic one-dimensional bars, this preference is stronger than an aggregate maximization statement. Every essential edge has strictly larger coefficient, in absolute value, than every non-essential edge in the harmonic representative, and the absolute values of the coefficients of all essential edges are equal. The key argument is a finite-dimensional variational characterization of the harmonic representative as a minimum-norm chain with prescribed boundary, combined with an elementary graph-theoretic cut argument. We then prove that the result is special to dimension one. In higher dimensions, the analogous coefficient-wise dominance statement fails. We give examples to show that harmonic representatives can place larger coefficients on non-essential higher-dimensional simplices than on essential ones. These results clarify both the power and the limitations of using harmonic representatives to assign geometric significance to simplices in persistent homology.

We classify signatures of bounded-variation paths in finite-dimensional real normed spaces whose logarithms are entire, in the sense of superexponential homogeneous decay. The zero-first-level fibre is trivial; if the first level is $v\neq0$, the signature is $a\e^v a^{-1}$ with $a$ a bounded-variation signature, and every such conjugate is entire. For a given path, $a$ may be chosen from a prefix. For a tree-reduced representative, a gate-selected prefix gives weak path conjugacy to a line. The proof uses a finite-dimensional spectral-growth statement: if $q\geq1$, $A:\C\to M_q(\C)$ is entire, and $\norm{\e^{A(z)}}\leq C\e^{\tau\abs{z}}$, then the $k$th characteristic-polynomial coefficient of $A(z)$ has degree at most $k$ for $1\leq k\leq q$. Universal matrix isospectrality, resonant developments, metric-tree fixed points and compactness, and Stieltjes–Fourier reconstruction establish the bounded-variation modified Lyons–Sidorova conjecture under a hypothesis imposed only on the whole path.

A ternary code is trifferent if every three distinct codewords have a coordinate in which their symbols are pairwise distinct. Let $T(n)$ be the maximum size of a trifferent code of length $n$. The classical Körner–Marton construction gives $T(n)\ge c_0(9/5)^{n/4}$ for an absolute constant $c_0>0$. We prove the polynomial strengthening $T(n)\ge c\sqrt{n}(9/5)^{n/4}$ for an absolute constant $c>0$. Our proof refines the outer-code step in the Körner–Marton concatenation. We encode non separating triples as edges of a $3$-uniform hypergraph, randomly thin its vertex set, and remove high-degree vertices together with all remaining Berge cycles of lengths two and three. The resulting locally sparse hypergraph admits a large independent set by a theorem of Verstraete and Wilson, producing the additional factor $\sqrt n$. Concatenation with the length-four Tetra code then yields the stated lower bound.

The claw-free Schur-positivity conjecture, recorded by Stanley (1998) and credited there to Gasharov, asserts that the chromatic symmetric function of every claw-free graph is Schur-positive. We give a counterexample on 12 vertices: the line graph $G$ of the graph obtained from a 4-cycle by attaching triangles at two opposite vertices and pendant edges at the other two satisfies $[s_{(3,3,3,3)}]X_G = -64$. The coefficient follows from a short computation by hand and is also reproduced by three exact implementations. An exhaustive computation over all 216,777 connected claw-free graphs on at most 11 vertices shows that every one is Schur-positive, so 12 vertices is the minimum order of any counterexample. A complete census of the 1,728,404 connected claw-free graphs on 12 vertices finds exactly two non-Schur-positive isomorphism classes; the other has graph6 code K?`CR@`bAbRB and coefficient $[s_{(3,3,3,3)}] = -40$.

A 6-net is a simplicial triangulation of the $2$-sphere with maximum degree $\leq 6$. Experiments suggest that every $6$-net admits a unique realization as an undented Euclidean polyhedron built from unit equilateral triangles, and a unique realization as an ideal equilateral hyperbolic polyhedron. We call these neoplatonic solids and ideal neoplatonics. A net is prime if every 3-cycle bounds a face. A computer-assisted proof shows that every prime $6$-net with $v \leq 50$ has a unique realization as a convex ideal neoplatonic. Numerical homotopy from this realization yields an approximate Euclidean neoplatonic, and a computer-assisted proof shows that a true Euclidean neoplatonic lies nearby, though we do not prove uniqueness. Using the separating-triangle decomposition, we extend Euclidean existence to all $10{,}412{,}340$ $6$-nets with $v\leq50$, counted up to combinatorial isomorphism.

Let $S=P_{cA_0}Q_{B_0}P_{cA_0}$ be the spatio-spectral concentration operator of bounded sets $cA_0,B_0\subset\mathbb{R}^d$, and let $\Lambda_\varepsilon=\#\{n:\varepsilon<\lambda_n(S)<1-\varepsilon\}$ be its plunge count. For $A_0$ and $B_0$ finite disjoint unions of bounded axis-parallel open boxes, we prove an explicit uniform upper bound on $\Lambda_\varepsilon$, valid for every $d\geq1$, $c>0$, and $0<\varepsilon<1/2$, with all constants written in terms of the side lengths. On the range $\alpha\geq4$, $c\geq2$, and $\alpha^{-c}<\varepsilon<1/2$, it gives $\Lambda_\varepsilon\leq Cc^{d-1}\log(1/\varepsilon)\log\!\bigl(\alpha c/\log(1/\varepsilon)\bigr)$. Kulikov and Dam Larsen previously proved this order on that range for a broader class; the present contribution is an independent proof and an explicit all-parameter estimate for product boxes. The proof uses a telescoping tensorization of $P_{(cA_0)^c}Q_{B_0}P_{cA_0}$ into $d$ elementary tensor operators, with one one-dimensional off-diagonal factor and $d-1$ localization factors. Schatten quasi-norms then multiply across tensor factors, and the single logarithm arises only from the normal direction. For the model cube pair, we also prove that, when $\varepsilon<4^{-d}$, $\Lambda_\varepsilon\geq M_a^d=\Omega((\log c)^d)$. Using an exact trace identity, an explicit cubic minorant, and the sine-kernel determinant asymptotics of Basor and Widom, we further obtain $\operatorname{Tr}((S-S^2)^m)=\beta_m\pi^{-2}\log c+O_m(1)$ for each fixed $m$, where $\beta_m=B(m,m)$, together with a two-sided fixed-depth window estimate of order $\log c$. The lower bound is not matching, and the fixed-order statements are not uniform in $m$.

This paper investigates the convexity structure and contact set of the $Q$-tensor flow with anisotropic Landau–de Gennes elastic energy and a singular entropy potential $f(Q)$. The flow naturally exhibits a coercivity property, yielding the $H^1$ regularity of $f(Q)$ for almost every time. Combining this $H^1$ regularity with spherical averaging and capacity theory, the contact set $$\mathcal{C}(Q)\triangleq \{x\mid f(Q(x))=\infty\}$$ is characterized. In particular, it is shown to be empty in two spatial dimensions and to have Hausdorff dimension at most one in three spatial dimensions.

Sendov's conjecture, first formulated in 1958, asserts that if a complex polynomial $f$ of degree $n \geq 2$ has all its zeros in the closed unit disk ${z \in \mathbb{C} : |z| \leq 1}$, then, for every zero $\lambda_0$ of $f$, there exists a critical point $\zeta$ of $f$ such that $|\zeta-\lambda_0| \leq 1$. The conjecture was previously known to hold for polynomials of degree $n \leq 8$, as well as in several special higher-degree cases. In 2022, using compactness methods, balayage, and the argument principle, Tao~\cite{Tao22} proved that there exists an absolute constant $n_0$ such that Sendov's conjecture holds for all $n \geq n_0$. However, Tao's argument does not provide an explicit admissible value of $n_0$. In the present paper, we make Tao's result effective and prove that one may take $$ n_0 = 10^{200000}. $$

This paper develops a canonical decomposition and reconstruction theory for local-unit-aligned ordered semigroups. Two coherentizations of the positive-idempotent skeleton are introduced. The finer one records the closure forced within local-unit blocks, while the multiplication-coherent quotient yields a join-semilattice of canonical blocks. Components are the fibers of these blocks. For comparable blocks $A\le B$, each positive idempotent $q\in B$ defines a transport homomorphism $x\mapsto xq$ from the component over $A$ to that over $B$. Keeping all such maps gives a resolved-transport family, which replaces the single connecting map used in an ordinary direct system. These canonical data reconstruct multiplication without additional assumptions: two elements are transported to their join component, and their product is the least product of corresponding transported images. They also determine every comparison directed from a lower component to a higher one. The full ambient order is recovered under any of three explicit order-recovery conditions. In particular, every totally ordered local-unit-aligned semigroup is completely reconstructed by its resolved-transport data. A complementary result recovers the order from componentwise order duality when a suitable component-preserving anti-automorphism is available. When each component receiving a proper transition has a least positive idempotent, the resolved family collapses to a single least-target map. Under a natural monotonicity condition, these maps form an ordinary direct system and induce a directed lexicographic order. Examples show both why the additional order conditions are needed and why resolved transports cannot in general be replaced by ordinary transition maps.

In this paper, we prove all gaps in the Simon conjecture for connected closed minimal surfaces immersed in unit spheres. Our proof uses the line-bundle ladder of Lin, Wang, and Xu and a Dolbeault computation to obtain two-sided ordered spectral comparisons on the two-sphere under Gaussian curvature $K_g\geq1$ and $K_g\leq1$, with rigidity in the equality case. Applying these comparisons at consecutive Calabi scales forces the Gaussian curvature to be one of the two endpoint values. We also establish the corresponding rigidity theorem for closed surfaces with parallel mean curvature vector. Finally, we prove a rotationally symmetric higher-dimensional analogue under a Ricci lower bound and a sectional-curvature upper bound.

Reduction replaces a mathematical object with a simpler model that retains the relevant information. We introduce left and right vector fields on small categories as tools for reducing finite acyclic categories while preserving their directed homotopical information. We relate these fields to directed deformation retracts and beat-object reductions, and show that right vector-field reductions preserve the directed sectional category of right directed fibrations and the global sections of functorial databases. We also extend initial scaffolds from posets to acyclic categories. These provide smaller indexing categories that preserve limits and, in particular, globally coherent selections in databases. Finally, we prove that initial scaffolds are preserved by right directed deformation retracts and hence by right vector-field reductions.

In ridepooling services, mobility providers must communicate an expected arrival time before customers decide whether to book. An ambitious commitment attracts higher demand than a conservative one, but is correspondingly harder to fulfill. Provider revenue is therefore governed by an endogenous interplay between the announced commitment and the demand it induces. We study this interplay in a traveling salesman setting in which all customers share a shuttle to a common destination, formalizing it as the Traveling Salesman Problem with Service Commitments (TSP-SC), a two-stage stochastic optimization problem. In the first stage, the provider announces an arrival commitment to all customers interested in the trip, each of whom independently accepts or rejects the offer with a probability decreasing in the committed time. In the second stage, an Orienteering Problem determines a revenue-maximizing subset of accepting customers to be served within the commitment; the remaining customers are rejected. We show that optimal commitments lie in a finite candidate set induced by the tour lengths of customer subsets, and develop two exact algorithms, a general and a faster variant for homogeneous customers, together with an Adaptive Binomial Sampling Heuristic (BSH) for larger instances. In a numerical study on instances with up to 100 customers, the heuristic deviates from the optimal expected revenue by at most 0.32~% on average across all exactly solvable sizes. The results further reveal a striking operational regularity: as the customer base grows, the revenue-maximizing policy converges to a regime in which roughly one in three accepting customers is rejected, a rejection rate that remains stable across instance sizes.

A relaxed shifted matrix-splitting modulus-based iteration method is proposed for solving large-scale sparse linear complementarity problems. The method extends the classical modulus-based matrix-splitting framework by introducing two shift parameters into the matrix splitting while preserving the original coefficient matrix. Different choices of the splitting lead to shifted Jacobi, Gauss-Seidel, SOR, and AOR-type schemes. Sufficient convergence conditions are established for \(P\)-matrices and \(H_+\)-matrices. For structured sparse matrices, practical AOR-type conditions and explicit admissible intervals for the shift parameters are also derived. These intervals are used together with a grid-search procedure to select efficient parameter pairs. Numerical experiments on four large-scale sparse test problems, including a quasi-variational inequality model, show that the proposed method is competitive with several existing modulus-based and projected methods. A direct comparison with the two-parameter MAOR method further illustrates the effect of the proposed shifted splitting.

Let $X^4_m=\{x_0^m+\dots+x_5^m=0\}\subset\mathbb{P}^5$ be the Fermat fourfold of degree $m$. We give a computer-assisted proof of the Hodge conjecture for $X^4_m$ for every odd $m\le 199$: three geometric closure criteria combined with an exhaustive machine census of the Hodge $(2,2)$-orbits, with a completeness proof and re-verifiable certificates. Criteria: (1) if the character multiset splits into two zero-sum triples, the rational Hodge block is transported from a $(1,1)$-substructure of a product of Fermat curves, hence algebraic; (2) algebraicity follows for characters that, after adjoining two vanishing pairs, decompose into an Aoki standard sextuple and a grade-$2$ Hodge quadruple; (3) the exceptional class at $m=33$ has a quasi-decomposable lift to level $66$, whose algebraicity descends along $X^4_{66}\to X^4_{33}$. The census covers the $89$ levels $21\le m\le 199$, $m\ne 23$, classifies all $78{,}299$ Galois-orbit representatives, and isolates thirteen orbits beyond decomposability, quasi-decomposability and Aoki's standard cycles: six close by the $*$-split criterion, seven by the two-pair and level-lifted closures, leaving none. Every orbit carries a machine-checked witness (negative screenings for the terminal ones), and the census is reproduced by an algorithmically independent implementation and brute force through $m=143$. Seven of the thirteen are gap classes outside Aoki's lattice calculus, new to the author's knowledge; for the other six, algebraicity is also derivable from that calculus, the explicit presentations being the new content. An exact Jacobi-sum computation at $p=67$ shows no cycle defined over $\mathbb{Q}(\zeta_{33})$ projects nontrivially onto the exceptional $m=33$ block; more generally, over any finite extension of $\mathbb{Q}(\zeta_{33})$ carrying a certifying cycle, every residue degree above $67$ is divisible by $6$.

In this paper we study autoencoders, a special class of deep neural nets (DNNs) whose performance can be characterized via their fixed points. This perspective naturally raises questions of existence, stability, and basins of attraction of these fixed points. These questions are addressed via the contractive properties of autoencoders, and are closely related to the notion of edge-of-chaos. Edge-of-chaos (EoC) is an important notion in the theory of DNNs. It describes the critical regime separating ordered and chaotic signal propagation through a randomly initialized network. Initialization at or near this critical regime offers several theoretical and practical advantages, including stability of the network w.r.t. perturbations of the input. EoC was previously introduced for broad classes of neural networks using mean-field averaging methods. In this paper we modify the notion of EoC for the study of autoencoders. Specifically, we introduce local and global EoC for autoencoders that control local (small) and global (arbitrary) perturbations of the input respectively. The study of stability of autoencoders falls within the scope of nonlinear problems in Random Matrix Theory (RMT). Our analysis of local EoC is based on spectral techniques of RMT, whereas global EoC is studied by employing Sudakov-Fernique inequality for Gaussian processes.

Motivated by the solution of Descartes famous problem on the chains of tangent circles given by D.Mathews and O.Zymaris, we give an algorithmic solution to the feasibility problem for radii of Steiner 4-chains studied by K.Kirajiev. More precisely, we obtain a complete characterization of the ordered quadruples of positive numbers which coincide with a quadruple of curvatures of the circles forming a Steiner 4-chain. Our approach makes essential use of the numerical invariants of Steiner chains introduced in a recent paper of R.Schwartz and S.Tabachnikov. We compute all invariants of Steiner 4-chains and establish the algebraic relations between those invariants which are used in the proof of the main result. Several related observations and illustrative examples are also given. Keywords: Steiner chain, parent circles, Steiner porism, poristic Steiner chains, Descartes circle theorem, invariant moments of curvatures, algebraic relations between invariants, symmetric Steiner chains, feasibility problem for Steiner chains

We study two planar unit-distance graphs (UDGs) that both look like heptagons but have opposite rigidity yet kindred arithmetic: the braced heptagon {35,1} (19 vertices, 35 edges) and the complete heptagon {21,2} (21 vertices, 42 edges). Neither is straightedge-and-compass constructible – already the regular 7-gon is not, by Gauss-Wantzel – so the natural setting is origametry: the construction of each realization by Huzita-Hatori folds, with the quadratic fold O5 (straightedge-and-compass strength) versus the cubic Beloch fold O6. The complete heptagon is over-braced and globally rigid, with a unique realization whose 42 edges are certified unit exactly over Q(cos 2pi/7), a cyclic cubic field: every vertex is an origami number, reached by one Beloch fold. The braced heptagon is isostatic (minimally rigid) with thousands of non-congruent realizations and a large generic realization count N = 3869504 = 2^6 * 103 * 587; yet we prove that its unit realization is again origami-constructible. In Pegg's explicit construction the pinning angle is a root of an irreducible degree-12 palindromic polynomial whose Chebyshev reduction, an irreducible sextic, factors into quadratics over the heptagon cubic Q(cos 2pi/7); hence the coordinate field is a {2,3}-tower of degree 3*2^k, reached by one trisection followed by quadratics. The primes 103 and 587 therefore belong to the generic complex count, not to the folded coordinates. We solve the governing cubic explicitly by folding (Lill's method and the Beloch fold), place both graphs inside Alperin's field of origami numbers, and read the fold-branch signs of the construction as mountain-valley assignments with an empirical Maekawa-type balance. All claims are checked in exact arithmetic.

We study the Fisher–KPP equation on the half-line under Dirichlet,Neumann, and Robin boundary conditions. For the autonomous logistic equation, we identify bounded stationary profiles converging to $1$ and obtain exponential far-field comparison estimates. We prove local uniform convergence of nontrivial Neumann solutions to $1$. Assuming local uniform convergence of the Robin solution to its stationary profile, we derive asymptotic Neumann–Robin comparison estimates. We then consider small time-periodic Neumann and Robin boundary forcing. Under exponential stability of the homogeneous linearized semigroup, we construct a locally unique small periodic lifted mild solution and obtain a first-order expansion with a uniform $O(\varepsilon^2)$ remainder in $C_0([0,\infty))$.

We develop a refined asymptotic theory for renewal processes with regularly varying inter-arrival distributions in the finite-variance regime. Assuming that $1-F(t)=t^{-\alpha}L(t)$ with $\alpha\in(2,3)$ and $L$ slowly varying, we establish second-order asymptotic expansions for both the renewal convolution and the renewal function. The results reveal a two-scale structure: the equilibrium tail $Q_F(t)$ governs the local renewal correction, while the integrated tail $\rho(t)=\int_t^\infty Q_F(u)\,du$ determines the global deviation from equilibrium. A particular emphasis is placed on the critical threshold $\alpha=2$, where the asymptotic behaviour undergoes a structural transition. In this borderline case, the power-law hierarchy collapses and the dominant correction is governed by an integrated slowly varying tail. Nevertheless, the local renewal correction mechanism persists, and the correction term remains asymptotically proportional to the equilibrium tail. The analysis is based on a systematic use of Laplace-transform techniques and Tauberian transfer principles, which allow us to relate singular behaviour in the Laplace domain to precise time-domain asymptotics. The results provide a unified description of second-order renewal asymptotics across the entire range $2\leq\alpha<3$, and point to possible extensions to renewal equations and to age-dependent branching processes.