Papers

We study the master equation for a second-order mean field game on Carnot tori, which means the generic player can move periodically only along admissible trajectories given by the family of vector fields generating the Carnot group. As examples of sub-Riemannian manifolds, Carnot groups represent a type of non-commutative groups characterized by stratified Lie algebra structures. In order to obtain the well-posedness of the master equation, we analyze the properties of its solution by investigating a degenerate mean field game system for which there exists an equivalent characterization with the master equation. The main part of this paper lies in leveraging the regularity properties of solutions to two classes of linear degenerate parabolic equations and a class of linear degenerate coupled systems to derive the existence of solutions to the master equation. The research in this paper is motivated by \cite{19CDLL,24MMM}.

Han's conjecture predicts that a finite-dimensional algebra with eventually vanishing Hochschild homology has finite global dimension. In the tau-Hochschild framework of Cibils, Lanzilotta, Marcos and Solotar, it splits into persistence (Gap A) and survival (Gap B), and a Gap-A failure is already a counterexample. We prove a protected corner theorem bounding Ext at a surviving vertex, settling the Liu-Morin extension conjecture beyond the monomial and special biserial cases. We then establish a trichotomy for Gap-A failures on three strongly connected vertices: the all-infinite case is impossible, and the two-infinite case is completely classified as a mutual dumbbell.

We study the optimal Bohnenblust-Hille constants $ K_{ m , M } $ for complex $ m $-homogeneous polynomials in any number of variables whose monomials with nonzero coefficient involve at most $M$ variables. For every fixed $M$, we prove that these constants satisfy \[ K_{ m , M } \leq A_{ M }^{ M / m } \cdot m^{ ( M^2 - 1 ) / ( 2m ) } \, , \quad \text{for} \ m \geq M \, , \] where $ A_{ M } \geq 1 $ depends only on $ M $. In particular, \[ K_{ m , M } \to 1 \quad \text{as} \ m \to \infty \, , \] and so the corresponding Bohnenblust-Hille constants are asymptotically contractive. The proof exploits the homogeneous structure through a decomposition according to exact monomial-support levels, partitions of the set of variables, multilinear Bohnenblust-Hille estimates, and interpolation with Parseval's identity.

We develop a hierarchical order-resolved relaxation model for a fixed D3Q125 discrete-velocity kinetic formulation. Conventional adaptive collision models often use one scalar rarefaction or nonequilibrium indicator for all retained moment orders, thereby coupling distinct kinetic sectors. Here, a shared macroscopic-gradient background is combined separately with second-, third-, and fourth-order thermodynamic nonequilibrium indicators to define effective measures K2, K3, and K4, each driving its own log-Gaussian relaxation spectrum. Pure-order perturbation tests verify selective activation, with nonmatching sectors remaining at roundoff level. Homogeneous mixed-order, amplitude, and composition tests show lower residual nonequilibrium than a common-sensor model while preserving positive populations. In a smooth periodic compression wave at the stated reference discretization and in the TNE-only sensor limit, the peak total nonequilibrium intensity is reduced by 6.565%, with reductions throughout the domain and in all retained moment sectors. Additional timestep, transport-discretization, relaxation-spectrum, uniform-boost, long-time, and shear-wave studies show that the sign of the hierarchical correction is robust over the tested configurations, while its magnitude depends on timestep, transport scheme, sensor frame, and relaxation spectrum. Quantitative boost checks identify the laboratory-frame raw-Hermite origin of the frame sensitivity without detecting an evident collision-path inconsistency over the tested range. The periodic benchmarks preserve the principal global invariants to floating-point accuracy and remain positive. These results establish the mechanism, selectivity, and numerical behavior of order-resolved activation on a fixed velocity set; independent kinetic-reference validation is still required before claiming universal accuracy improvement.

This paper studies an optimal consumption–investment problem for competitive agents in an \(N\)-player game and its associated mean field game. Each agent invests in an individual risky asset subject to idiosyncratic noise, common noise and downward jump risk, and the interaction among agents is induced by relative performance concerns in both consumption and terminal wealth. In the mean field limit, we characterize a deterministic mean field equilibrium in analytical form by using the stochastic maximum principle. Numerical experiments are presented to illustrate the resulting equilibrium and its financial implications. Finally, based on the obtained mean field equilibrium, we construct an approximate Nash equilibrium for the \(N\)-player game. This model is motivated by \cite{Merton1971} and \cite{Lacker2020}.

We develop a filtered geometric framework for essential surfaces in knot exteriors. Let $\gamma$ be a unit-thickness representative of a knot type $K$ with $\mathrm{Len}(\gamma)\le\Lambda$, and let $F$ be a properly embedded essential surface in the exterior of a fixed-radius tube about $\gamma$, with $\mathrm{Area}(F)\le\Delta$ and relative thickness at least $\tau$, formulated through Federer reach and controlled boundary collars. We prove that this bounded-geometry pair space contains only finitely many pair-isotopy classes, and that equality of explicitly bounded canonical layered codes at resolution $\varepsilon\le c\min\{1,\tau\}$ implies ambient pair-isotopy. In a fixed exterior $E$, the surface systems visible in a geometric window form finite subcomplexes ${ES}_{\Delta,\tau}(E)$ that exhaust the essential-surface complex; isometries act levelwise and $C^{1,1}$ self-diffeomorphisms act with controlled reindexing. For a fixed two-sided surface, compressing disks are filtered in the same way, giving finite geometric witnesses for compressibility and weak reducibility and recovering the index-one characterization in Bachman's topological index theory. On the peripheral torus, every visible numerical boundary slope lies in an explicit writhe window, so that $|r|\le C_{\mathrm{BS}}\Lambda^{4/3}+w(\Delta,\tau)$. Together with finite Reidemeister certificates, the faithful codes give two independent finite recognition mechanisms, one for the knot and one for the carried essential-surface type. The framework is a smooth, triangulation-free analogue of the finiteness philosophy of normal surface theory; it does not assert that a knot exterior has only finitely many essential surfaces without geometric bounds.

We introduce a self-exciting point process with power-law intensity dynamics that admits a finite-dimensional Markovian state representation. The model is constructed to preserve the local jump and slope update structure of power-law Hawkes processes, while replacing global history dependence with a nonlinear Markov chain governing the intensity dynamics. Within a general state-space framework, we establish irreducibility, aperiodicity, and the T-chain property under mild regularity conditions on the inter-arrival time distribution. Under an explicit stability condition, we further prove that the latent state process is positive Harris recurrent, ensuring the existence of a unique invariant distribution. Simulation results based on the local Whittle estimator show that the proposed Markovian intensity model exhibits long-memory behavior near the boundary of the stability region.

In this paper $S$ is a closed symmetric linear relation in a Krein space $\mathfrak H$ with adjoint $S^*$, finite and equal defect numbers $d$, and a boundary mapping $\mathsf{b}: S^* \rightarrow \mathbb{C}^{2d}$ with Gram matrix $\mathsf Q$. We introduce a class $\mathbb A_{S,\mathsf{b}}$ of self-adjoint extensions of $S$ in a Krein space $\widetilde{\mathfrak H}$ containing $\mathfrak H$ as a Krein subspace of finite codimension, together with a class $\mathbb{P}_{\mathsf Q}$ of $d \times 2d$ matrix polynomials. Both classes are equipped with natural equivalence relations. Given $\mathcal{P}(z) \in \mathbb{P}_{\mathsf Q}$, we consider a boundary eigenvalue problem defined by the condition $\mathcal{P}(z)\mathsf{b}(\{f,g\})=0$, $\{f,g\}\in S^*$, and look for its linearizations. By a linearization we mean a linear relation $\widetilde{A} \in \mathbb{A}_{S, \mathsf b}$ such that the Shtraus extension $T_{\widetilde A}(z)$ of $S$ determined by $\widetilde A$ coincides, for all $z \in \overline{\mathbb C}$, with the linear relation $\big\{\{f,g\}\in S^*\,:\,\mathcal P(z)\mathsf b(\{f,g\})=0\big\}$. We prove that this correspondence defines a bijection between the equivalence classes in $\mathbb A_{S, \mathsf b}$ and those in $\mathbb P_{\mathsf Q}$. Moreover, we provide a condition under which the resulting linearizations are spectrally equivalent to the boundary eigenvalue problem: their regular and spectral points coincide, and for each eigenvalue in $\overline{\mathbb C}$ there is a bijection between their Jordan chains.

We classify the finite-extension permutation spectra occurring in an explicit family of mock automorphisms over finite fields. Every finite nonempty divisor-closed subset of $\N_{>0}$ occurs; in particular, this disproves Maubach and Willems conjecture that every mock automorphism permutes infinitely many finite extensions. Every finite-spectrum map satisfying \[ \Jac(F)=I_n,\qquad F|_{\F_q^n}=\id \] has total degree at least $p(q+1)$. For every $n\geq2$, this bound is attained by a map with spectrum $\{1\}$. In dimension one, the same bound and exact spectrum $\{1\}$ are attained over every odd field and over $\F_2$; for even $q>2$ we give a nonexceptional construction with an effective finite bound for its spectrum. If $\delta_p$ denotes the least degree of a nonlinear reduced permutation polynomial over $\mathbb F_p$, then the least degree $\mu_p$ of a nonexceptional one-variable mock automorphism satisfies \[ \mu_2=6,\qquad \mu_3=12,\qquad \mu_p=p\delta_p\quad(p\ge5). \] We also record a cofinite-spectrum criterion and isolate the surviving prime-to-$p$ question. Every nontrivial counterexample constructed here has geometric generic degree divisible by $p$.

We formulate a group chunk theorem in the context of sheaves on sites which generalizes many similar results in model theory and algebraic geometry. Secondly, we develop an algebro-geometric analogue of Hrushovski's method of producing a group chunk from germs of definable functions on stationary types. The use of the model-theoretic tools of canonical bases and elimination of imaginaries is replaced with the use of Hilbert schemes to study “canonical” families of rational morphisms, allowing us to extend the previously known results over more general base schemes. The proofs of these results involve some technical work which may be of independent interest. First, we study partial morphisms in arbitrary categories, and show that a presheaf of partial magmas on a small category admits a universal morphism to a group. On the model theory side, we show that type-definable sets of $M^{eq}$ can be interpreted as sheaf quotients of type-definable sets. On the algebraic geometry side, we develop a theory of rational morphisms and families of rational morphisms of schemes over an arbitrary base.

Stochastic reaction networks often involve components at low copy number, where individual production and degradation events generate substantial fluctuations. Yan et al.\ conjectured in 2019 that for networks with linear degradation and arbitrary cross-regulatory production rates, feedback cannot suppress the stationary fluctuations for each component below the fluctuations of its constant-rate counterpart. Their formulation allows random burst sizes $K_i$: the unit-birth case has $K_i\equiv1$, the biologically important burst model takes $K_i$ to be geometrically distributed, but more general positive integer-valued burst laws are also permitted. The conjecture was recently proved for unit births, $K_i\equiv1$. We show here that the conjecture is false in general by constructing a two-component network with bounded production rates and burst sizes in $\{1,2\}$ for which both stationary Fano factors lie below their common constant-rate baseline. We then prove the conjecture for positive geometric bursts, the canonical burst model in stochastic gene expression. For arbitrary regulatory architecture and nonlinear cross-regulatory production rates, we prove an exact weighted tradeoff that rules out simultaneous suppression of every component below its geometric-burst baseline. We also prove a complementary structural impossibility result for arbitrary positive integer-valued burst laws with finite second moments: if the activating and inhibiting interactions have a globally consistent sign structure, in the sense that every cycle of the regulatory interaction graph contains an even number of negative interactions, then every component individually satisfies $F_{X_i}\ge B_i$, where $F_{X_i}$ is its stationary Fano factor and $B_i$ is its constant-rate burst baseline.

High-dimensional screening is commonly built on covariance, correlation, or least-squares measures. These summary measures can be unstable or even undefined, when predictors are sparse or have heavy-tailed distributions. Building on our recent work on extended codifference and the idea of Ball-covariance, we develop Ball-codifference for marginal screening in statistical modeling with heavy-tailed predictors and responses. The proposed statistic combines the rank-type geometry of random balls with the codifference as a dependency measure constructed based on the characteristic function, so it can be computed without requiring well-defined finite first or second moments. We define Ball-codifference and its normalized screening utility, and formulate a sure independence screening procedure. Large-sample normality follows from a bounded V-statistic and functional-delta-method argument under standard nondegeneracy and regularity conditions. Simulation studies under Gaussian and sub-Gaussian stable designs show that codifference-weighted Ball screening gives competitive or improved recovery of highly associated predictors, especially when tail heaviness is pronounced. Also, our data example illustrates that our variable screening method significantly improves prediction accuracy in linear regression.

Many instances of sequential sampling, including audit and inspection scheduling, representative sampling, and treatment assignment, require selections to be distributed evenly without becoming easy to anticipate or exploit. We study a family of sequential sampling rules that adaptively bias sampling probabilities in order to achieve faster convergence of the empirical distribution to a desired target law, while keeping the resulting samples as unpredictable as possible. The resulting self-balancing sampler is simple to implement, arises naturally among a class of Markovian samplers sharing a certain invariance property, and admits a stochastic mirror-descent interpretation. Our main results show that (i) this self-balancing sampler converges at the fastest possible $O(n^{-1})$ rate with explicit dependence on biasing parameters, beating the standard $O(n^{-1/2})$ rate of IID sampling, (ii) it is the unique solution to a natural entropy-regularized optimization problem which balances the convergence rate of the empirical law and the unpredictability of the samples, and (iii) in the weak-biasing regime, the properly centered counts process converges to an Ornstein-Uhlenbeck process in the diffusive limit. Together, these results support a practical framework for reducing repeated selections and long gaps in coverage without making future selections overly predictable.

We give a bounded-coefficient probabilistic construction of a transcendental entire function $f$ of order two such that every nonempty open subset of the complex plane contains a zero of $f^{(n)}$ for all sufficiently large $n$. This gives an affirmative answer to the transcendental form of Erdős Problem~906. Thus every fixed disk is zero-free for only finitely many successive derivatives. The function satisfies $|f(z)|\leq\sqrt2\exp(|z|^2)$ and is a counterexample to a theorem of Boas and Reddy as printed.

In this paper, we study the supersaturation problems of eventown. Given a family $\mathcal{A}$ of subsets of an $n$ element set, let op$(\mathcal{A})$ denote the number of distinct pairs $A,B\in \mathcal{A}$ for which $|A\cap B|$ is odd. We give extremal eventown constructions and show that for fixed $s\le2^{\lfloor \frac{n}{2} \rfloor}-2$, there exists a collection of $2^{\lfloor\frac{n}{2}\rfloor}+s$ even-sized subsets of an $n$ element set that contains exactly $s\cdot 2^{\lfloor \frac{n}{2} \rfloor-1}$ pairwise intersections of odd size. This extends the range of $s$ in a conjecture proposed by O'Neill from $2^{\lfloor \frac{n}{2} \rfloor}-2^{\lfloor \frac{n}{4} \rfloor}$ to $2^{\lfloor \frac{n}{2} \rfloor}-2$. We also give a construction using symmetric designs to prove that when $k$ is even and $4k-1$ is a prime power, there exists a collection of $2^{\lfloor\frac{4k-1}{2}\rfloor}+s$ even-sized subsets of a $4k-1$ element set $\mathcal{A}_s$ with $op(\mathcal{A}_s)=s \cdot 2^{{\lfloor\frac{4k-1}{2}\rfloor}-1}$, $1\leq s\leq4k-1$.

Below-threshold scattering is studied for the three-dimensional focusing energy-critical Schrödinger equation in the cylindrically symmetric class. For soliton-like compact critical elements, two linked rigidity entrances are isolated: a best fixed-axis window condition for the axial center and a low-frequency tail smallness condition. A shifted localized virial estimate and a finite-mass virial argument after axial zero-momentum normalization exclude these entrances, yielding a conditional cylindrical threshold scattering criterion via compactness reduction.

For every closed surface $\Sigma_g$ of genus $g\geq6$, we prove that there exists a countable union of positive-codimension algebraic subsets of Teichmüller space such that, for every hyperbolic metric $d$ outside this exceptional set, the number of distinct primitive closed-geodesic lengths at most $L$ is bounded below by $\tau(d)e^{\delta(g)L}$, where the explicit constant $\delta(g)$ satisfies $\delta(g)>\frac12$.

Self-orthogonal codes have attracted considerable attention owing to their applications in quantum error-correcting codes, linear complementary dual codes, and a variety of other fields. In this paper, we construct new families of self-orthogonal codes and self-orthogonal minimal codes by establishing criteria that characterize the self-orthogonality of certain linear codes. To this end, we first establish several new criteria for linear codes arising from the defining-set construction to be self-orthogonal. More specifically, for $q > 3$, we show that the code $\mathcal{C}_D$ is self-orthogonal whenever the defining set $D$ is $G$-invariant, where $G \subseteq \mathbb{F}_q^*$ and $|G| > 2$. For $q = 2, 3$, we characterize the self-orthogonality of $\mathcal{C}_D$ using certain character sums. By combining these criteria with partial difference sets, we construct several new families of self-orthogonal codes, which lead to optimal or almost optimal quantum codes. Secondly, via the action of a multiplicative subgroup $G \subseteq \mathbb{F}_q^*$ with $|G| > 2$ on the columns of a projective linear code, we construct self-orthogonal codes with flexible parameters. From their augmented codes, we derive quantum codes. By choosing suitable projective codes, we obtain optimal quantum codes with high parametric flexibility. Thirdly, we employ the characteristic function method to construct linear codes and establish criterion for their self-orthogonality. Based on this, we construct several classes of self-orthogonal minimal codes that violate the Ashikhmin-Barg condition, using vectorial dual-bent functions and $s$-plateaued functions.

Imprecise probability generalizes standard probability theory by replacing a single distribution with a convex set of possible distributions. We show that this generalization requires no change to the standard BDD compilation and weighted model counting pipeline used by discrete probabilistic languages. An imprecise coin flip is simply a BDD variable whose weight is left free rather than fixed. We introduce Imp, a Haskell-embedded DSL for imprecise probabilistic programming. A graded monad, indexed by finite sets of named sources of epistemic uncertainty, restores the commutativity that the standard convex powerset monad lacks, and GHC's type system enforces this at compile time. Weighted model counting is parametric in the semiring, so the same compiled BDD supports exact, differentiable, and interval-bounded inference.

Let $X\sim Bin(N,p)$ and let $c=Nr$ be a prescribed centre with $r\ne p$. The finite formula for $E|X-c|$ is classical, and the existence of a complete large-deviation expansion in powers of $N^{-1}$ follows from Timashev's saddle-point expansion for binomial tails. We give the coefficients in closed form for the stop-loss tail produced by the absolute deviation, with the lattice defect of $Nr$ retained explicitly. The construction combines three standard ingredients: Bernoulli polynomials for the logarithm of successive binomial masses, the exponential Bell recursion, and Eulerian polynomials for weighted geometric sums. The $k$-th coefficient has denominator exactly $(1-\rho)^{2k+2}$, where $\rho$ is the exponential tilt; hence the effective expansion parameter is $1/(N(1-\rho)^2)$, the lattice weighted analogue of the parameter in the incomplete gamma expansion. The case $r=\tfrac12$ gives the fold at the origin of a biased $\pm1$ random walk, and its parity split is identified with the lattice defect of the fold.

Kahane (arXiv:2607.11225) proves that for a fence poset $P$ of size $n$, $n!\,\Omega(P;t)=\sum_{\sigma\in S_n} t^{\mathrm{bl}(\sigma)}$, with $\mathrm{bl}(\sigma)$ a canonical block count, and conjectures (Conjecture 4.4) that the identity extends to the cell poset of any skew shape, with $\mathrm{bl}$ replaced by a constructively-defined extension $\widetilde{\mathrm{bl}}$. We prove the conjecture for every $2\times n$ rectangle – the first genuinely two-dimensional case, where blocks are no longer path-intervals and the statistic is a correlated function of both rows. We introduce a two-parameter growth-state generating function $W_n(\alpha,\beta;t)$ refining $\sum_\sigma t^{\widetilde{\mathrm{bl}}(\sigma)}$ by the rank $\alpha$ of the bottom-row minimum and the rank $\beta$ of the rightmost top cell's block-root, and prove the closed form $W_n(\alpha,\beta;t)=(\alpha-1)!\,B(n,\beta-2)\,S_n(\alpha,\beta;t)$ ($n\ge 1$), where $B(n,k)$ is the Catalan triangle and $S_n$ a rising-factorial skeleton. The proof is an induction on $n$ through Kahane's column-insertion transition, whose load-bearing structural fact – the block-root below a newly inserted top cell is always the global bottom-row minimum – reduces the induction to a Catalan-triangle computation. Summing over $(\alpha,\beta)$ telescopes to $C_n\,t^{\overline{n}}(t+1)^{\overline{n}}$, which by MacMahon's product formula equals $(2n)!\,\Omega(R_{2,n};t)$. Every load-bearing step is a complete general-$n$ argument; the transition table and the inductive step are cross-checked by exhaustive enumeration (through $n=8$ and $n=9$) via an independent from-scratch implementation. General skew shapes and $m>2$ rows remain open.

Gardam's counterexample to the Kaplansky unit conjecture is a unit of $\mathbb{F}_2[P]$, $P$ the Promislow group, with support of size $21$; Gardam asked whether $21$ is least possible. Over $\mathbb{F}_2$ the unit equation is a parity condition on a pair of supports, searchable over word balls. We upgrade two statements to machine-checked certificates: no support pair of sizes $\ge2$ lies in the radius-$3$ ball (a DRAT proof of a ball form of the Craven–Pappas theorem), and every nontrivial unit with supports in the radius-$4$ ball has total support at least $42$, confirming Gardam's expectation in ball-limited form. The relatives $H_4=F(3,4)$ and the Nielsen–Soelberg groups are swept with certificates; $G_3$ is Gardam's amalgam $S$, whose unit we localize into its radius-$4$ ball. $H_4$ resists the twisted-unitary ansatz through radius $6$ for every non-identity dihedral twist. An effective localization principle makes the least support of a nontrivial unit of $\mathbb{F}_2[P]$ computable in principle.

Let \[ {\mathcal P}_2={\mathbb C}[x,q,p,z] \] carry the canonical Poisson bracket determined by $ \{p,x\}=\{z,q\}=1 $ and by the vanishing of the other brackets between distinct generators. Here and throughout, “rank two” means two canonical pairs in the standard indexing of the canonical Poisson algebras; thus there are four polynomial generators and the Poisson tensor has geometric rank four. We give explicit polynomials \[ R,T,D,S\in{\mathbb Q}[x,q,p,z] \] satisfying \[ \{D,R\}=1,\qquad \{S,T\}=1, \qquad \{R,S\}=\{R,T\}=\{D,S\}=\{D,T\}=0, \] while \[ R=x(2-3xq). \] Consequently, the assignment $ (x,q,p,z)\mapsto(R,T,D,S) $ defines a Poisson endomorphism of ${\mathcal P}_2$ that is not an automorphism. This disproves the Poisson Conjecture for two canonical pairs, and hence for every number of canonical pairs at least two. The associated polynomial map of ${\mathbb A}^4$ preserves the canonical symplectic form, has Jacobian determinant one, and has an explicit fiber consisting of exactly three points. The proof uses a polynomial source coordinate system in which the symplectic identity reduces to three displayed coefficient identities. A separate appendix uses the same four output polynomials and their Hamiltonian duals to construct an explicit nonautomorphic endomorphism of the fourth Weyl algebra.

Noncommutative and nonassociative geometry has attracted much attention in both mathematics and physics communities. A prominent example is a stringy $R$-space over an underlying manifold, which exhibits both noncommutative and nonassociative structures and is known to be locally nongeometric. However, previous studies have mainly described $R$-space through algebraic relations of local string position coordinates, instead of a precise global geometric definition. In this paper, we propose a rigorous and systematic mathematical formulation of nongeometric $R$-space within the framework of descent theory. The central idea is to regard nongeometric $R$-space as carrying only local data of a quasitriangular quasi-Hopf algebra on each local coordinate domain of the underlying manifold, which are then glued coherently together to produce a global geometric object. We show that the appropriate global structure is not an ordinary space or a globally defined algebra, but a bicategory-valued stack. This provides a global and geometric definition of $R$-space while still preserving its locally nongeometric intrinsic character.

We develop a real Laplace method for integrals $$ I(z)=\int_N a(x)e^{-z f(x)}\operatorname{dvol}_g(x),\qquad z\to\infty, $$ near a compact, possibly stratified, minimum set of the phase. The basic input is an adapted normal-cell disintegration, defined up to null sets, together with piecewise anisotropic dilations and homogeneous limiting models. The framework simultaneously permits finite $C^2$ stratifications; Borel base refinements and $C^1$ adapted charts; measurable base-dependent cells, relative domains, and incident-stratum assignments; different fiber dimensions and weights on different pieces; a continuous phase and a merely measurable amplitude; and limiting models controlled either by ambient coercivity or directly by cellwise exponential tightness. The cells may converge only in weighted measure, or pointwise under an integrable base majorant. To the best of our knowledge, no existing theorem for real Laplace integrals accommodates this full combination of geometric and analytic assumptions. The tradeoff is that the adapted disintegration and limiting models are supplied hypotheses to be verified in each application.

Let $\mathbb{K}=\mathbb{Q}(\sqrt{-d})\text{ or }\mathbb{Q}(\sqrt{d})$, where $d$ is a positive square-free integer, and denote by $\mathcal{O}_\mathbb{K}$ the ring of integers of $\mathbb{K}$. I investigate the solution of the equation $x_1^{3}-x_2^{2}x_1+1=0$ where $x_1\in \mathbb{K}$ and $x_2\in\mathcal{O}_\mathbb{K}$. The case $\mathbb{K}=\mathbb{Q}(\sqrt{d})$ for $d\equiv 2,3\pmod{4}$ faces infinite units that require a separate treatment. Using the arithmetic of the quadratic integer rings $\mathbb{Z}[\sqrt{d}]]$, together with norm arguments, divisibility properties, and the explicit structure of its unit group, I prove that the equation has exactly two solutions, namely $(x_1,x_2)=(-1,0)~\text{ and }~(1,\sqrt{2})$ for $\mathbb{K}=\mathbb{Q}(\sqrt{2})$ and one solution $(-1,0)$ for $\mathbb{K}=\mathbb{Q}(\sqrt{d})$ As an application, I consider the family of elliptic curves $C_m:Y^{2}=X^{3}-m^{2}X+1,~ m\in\mathcal{O}_\mathbb{K},$ and deduce that, for every $m\neq0,\sqrt{2}$ the Mordell–Weil group $C_m(\mathbb{K})$ contains no rational point of order two.

This paper investigates the problem of optimal piecewise linear approximation of a smooth curve, in which the area of the region enclosed between the graph of the original function and the constructed interpolating polyline is minimized. This problem arises in technological processes such as laser and plasma cutting of metals, milling, additive manufacturing, and in the preparation of control programs for CNC equipment. For a quadratic parabola, the exact value of the minimum area is established, and it is shown that the optimal partition of the interval is uniform. For a cubic function, it is demonstrated that the optimal interpolation nodes are distributed non-uniformly; in the case of two internal partition points, their exact coordinates and the corresponding minimum area are obtained. The results can be applied in CAD systems for optimizing tool paths based on the criterion of minimum deviation area.

Let $G$ be a finitely generated group with a surjective homomorphism $\pi\colon G\to\Z$. The directional growth spectrum $I(\beta)$ is the exponential rate of the number of elements of length at most $n$ lying over $\lfloor\beta n\rfloor$; its maximum is the growth rate. Wherever $I$ has been computed—free, hyperbolic and relatively hyperbolic groups, free abelian groups—it is strictly concave and real-analytic, as Perron–Frobenius theory dictates. We compute $I$ in closed form for the lamplighter groups $F\wr\Z$, $|F|=r+1$, with the standard generators, and obtain a different picture: $I$ is affine on $[-\beta^{*},\beta^{*}]$ and strictly concave beyond it, with a second-order transition at $\beta^{*}$. Elements conditioned to the affine phase backtrack macroscopically, with lamp density independent of $\beta$. The series $\sum_x s^{|x|}y^{\pi(x)}$ is rational, and the transition is an exchange of dominant singularities, the inner one independent of $y$. The peak of $I$ is $\log\omega$ with $\omega^{2}=\omega+r$, the central value $\log\rho$ with $\rho^{3}=\rho+r$; the base group is therefore co-amenable yet grows strictly slower, by an amount unbounded in $r$. For $r=1$ these constants are the golden ratio and the plastic number.

We study the depth required by ReLU networks to exactly represent piecewise linear functions, focusing specifically on the maximum function. This problem has recently received significant attention in both the ML and TCS literature. We prove that $\max_n(x)=\max\{x_1,\ldots,x_n\}$ is exactly representable with two hidden layers for every $n\leq 12$. Previously, this was only known up to $n\leq5$ [Bakaev, Brunck, Hertrich, Stade, Yehudayoff, STOC'26]. We obtain our constructions through an exact computer-assisted search within a space of candidate solutions: After a symmetry reduction, we obtain a finite system of linear equations over $\mathbb{Q}$ such that any solution yields a valid representation of the maximum function. The resulting constructions have a structured first hidden layer, which enables recursive substitution into deeper networks. This yields an exact ReLU representation of $\max_n$ with at most $\lceil \log_6(n/2) \rceil+1$ hidden layers. Consequently, every continuous piecewise-linear function on $\mathbb{R}^d$ admits an exact representation with at most $\lceil\log_6((d+1)/2)\rceil+1$ hidden layers; in particular, two hidden layers suffice for $d\leq 11$. Again, these results improve upon [Bakaev, Brunck, Hertrich, Stade, Yehudayoff, STOC'26], who proved analogous logarithmic bounds with base three.

Scalar metrics are often used to evaluate clusterings against known classes, but they can obscure a fundamental trade-off: clusterings should be informative about class labels while avoiding unnecessary fragmentation. Here we describe normalized scores of cluster homogeneity and parsimony that quantify this trade-off. These scores build on the information bottleneck principle, modified to not reward lossy compression. We show by example and mathematical proof that our definitions of these scores have the intuitive property of varying monotonically under cluster refinement in contrast to related proposals. Extending the information-theoretic framework beyond Shannon entropies, we furthermore derive set-matching and pair-based counterparts of the homogeneity and parsimony scores. These unify commonly used evaluation criteria and show that, in the pair-based setting, the homogeneity-parsimony trade-off recovers the receiver operating characteristic of binary classifiers. We demonstrate the framework's utility for feature selection and algorithm comparison, illustrating how considering scores jointly can clarify clustering operating points and identify Pareto-optimal solutions.