Papers

In this paper, we introduce the notion of adaptive strong monotonicity and establish a linear convergence of the recently proposed Generalized Proximal Point Algorithm (GPPA) for computing approximate solutions to nonmonotone inclusion problems in Hilbert spaces. We also investigate the Inexact Generalized Proximal Point Algorithm (IGPPA) in the presence of nondiminishing errors.

Let $K$ be an imaginary quadratic field of discriminant~$-d<-11$, and for $\ell\geq1$, let $\psi$ be a Hecke character of $K$ with trivial finite conductor and infinite type~$(2\ell,0)$. In this work we prove that for any prime $p>3$, the quotient $$ \frac{\langle\theta_{\psi},\theta_{\psi}\rangle}{\Omega_{K}^{4\ell}}, $$ is $\lambda$-integral for a prime ideal $\lambda$ over $p$, where $\langle\theta_{\psi},\theta_{\psi}\rangle$ is the Petersson norm of the theta function $\theta_{\psi}=\theta_{\psi}(\tau)$ attached to $\psi$, and $\Omega_{K}$ denotes the Chowla–Selberg period attached to~$K$, and the product $$ \prod_{i=1}^{h_{K}}\frac{\langle\theta_{\psi_{i}},\theta_{\psi_{i}}\rangle}{\Omega_{K}^{4\ell}} $$ is rational, where the product is over all $h_{K}$ of the underlying Hecke characters.

Symmetry is central to modern machine learning and physics: invariances and equivariances improve sample efficiency, robustness, and out-of-distribution generalization, while symmetry principles guide scientific modeling. Yet for stochastic dynamical systems the relevant continuous symmetries are rarely known, and symmetry discovery for SDEs has remained essentially unexplored. We introduce LieStoNet, an end-to-end, template-free framework for discovering Lie-point symmetries of SDEs directly from spatiotemporal trajectories, without prespecifying symmetry groups, templates, or canonical coordinates. Building on the seminal SDE Lie-symmetry theory of Gaeta and Quintero (1999), which formalizes Lie-point SDE symmetries and their relation to Fokker-Planck symmetries, LieStoNet learns neural surrogates for drift and diffusion from increments, then learns projectable generators by enforcing the SDE determining equations, separately regularizing for closure under Lie brackets, adherence to the Lie algebra axioms (bilinearity, antisymmetry, Jacobi), and a non-redundant independent basis. The surrogate also defines an associated Fokker-Planck equation, enabling optional discovery of its Lie-point symmetries in parallel. Across multiple canonical SDEs with known analytic symmetries, LieStoNet recovers generators consistent with the ground-truth symmetry algebra, providing interpretable symmetry discovery for noisy dynamics. Code is available at \href{https://github.com/sumit-sinha-seas/LieStoNet_Final.git}{this link}.

We consider the Riemann problem for a 3x3 system of conservation laws with generalized Chaplygin pressure $p(\rho,v)=-\frac{A(v)}{\rho^\alpha}$, $0<\alpha\leq 1$, where the pressure depends on an additional transported variable. We analyze the system's wave structure and classify the Riemann solutions. Whenever classical solutions consisting of shocks, rarefaction waves, and contact discontinuities fail to exist, singular solutions arise. We verify that these satisfy the conservation laws in the distributional sense within the classical Dirac delta framework, and compare them with Nedeljkov's shadow-wave construction, giving two complementary descriptions of the same singular solution. We further study admissibility via the Dafermos maximum entropy dissipation principle, with several examples showing how it selects the physically relevant solution. Lax-Friedrichs simulations illustrate the Riemann wave patterns and provide a comparison with the analytical results. To construct viscous profiles for the isolated overcompressive $\delta$-shock, we assume $\alpha\in\mathbb{Q}\cap(0,1]$ and apply the Dafermos regularization together with a spherical blow-up. Working in three directional charts, we construct the reduced singular concatenation consisting of the left outer orbit, the middle orbit on the blown-up boundary, and the right outer orbit. We then prove that, for sufficiently small positive viscosity, this singular concatenation perturbs to a heteroclinic orbit. Consequently, the isolated overcompressive $\delta$-shock is realized as the zero-viscosity limit of a family of self-similar Dafermos viscous profiles.

The planar Pompeiu problem, originating in 1929, and the associated Schiffer conjecture are long-standing rigidity questions linking rigid-motion integral transforms and Fourier zero sets to overdetermined Neumann eigenvalue problems. We construct a bounded simply connected noncircular domain $\Omega\subset\mathbb{R}^2$ with real-analytic Jordan boundary and a nonconstant function $u$ such that $(\Delta+k^2)u=0$ in $\Omega$, $u=1,\partial_\nu u=0$ on $\partial\Omega $ for some $k\in(31.967007261,31.967007293)$. Thus $u$ is a Neumann eigenfunction which is constant on the boundary, and $\Omega$ is a counterexample to Schiffer's conjecture. Green's identity also gives $\widehat{\mathbf 1_\Omega}(k\omega)=0$ $(\omega\in\mathbb S^1)$, so $\Omega$ fails the Pompeiu property and is also a counterexample to the planar Pompeiu conjecture for bounded simply connected Lipschitz domains. We obtain the domain as $\Omega=\phi(\mathbb{D})$, where $\phi$ is a ten-fold symmetric conformal map close to an explicitly listed polynomial of degree $301$. On the unit disc, the analytic problem becomes a cubic operator equation on real coefficient spaces, $F(g,p)=g+|p|^2(1+Kg)=0$, where $K$, expressed in a disk-polynomial basis, is an explicit inverse of the Laplacian on the range compatible with zero Dirichlet and Neumann traces, and $p=k\phi'$. Positivity of the disk-polynomial linearisation coefficients, sharp bounds for $K$, and monotone control of the infinite tails establish an a posteriori contraction near the listed polynomial in a weighted coefficient algebra, and hence an exact zero of $F$.

Problem F16 of Croft, Falconer, and Guy asks for the least worst-case length needed to join prescribed pairs of points by pairwise disjoint planar arcs, when the distance within each pair is at most one. The book suggests that the answer for two pairs is $\sqrt2$. We prove this exactly. The lower bound is a crossing argument in a square. The upper bound follows from a sharp ellipse lemma. As a consequence, the value proposed in the book for three pairs, $(\sqrt3+1)/2$, cannot be correct under the literal formulation, because the constants are nondecreasing in the number of pairs.

We prove that smooth cubic threefolds are symplectically irrational. This recovers the classical irrationality theorem of Clemens-Griffiths. The obstruction is given by the formal monodromy of the quantum connection around the irregular singularity.

Let $k$ be an algebraically closed field of characteristic $p\geq 5$, and let $\mathrm{Ver}_p^+$ be the even part of the Verlinde fusion category $\mathrm{Ver}_p$, the semisimplification of $\mathrm{Rep}_k(\mathbb Z/p)$. Let $\mathfrak g$ be a linearly reductive Lie algebra in $\mathrm{Ver}_p^+$, i.e., one whose finite-dimensional representations are semisimple. A basic class of examples is obtained by semisimplifying a simple Lie algebra over $k$ equipped with the action of $\mathbb Z/p$ by a principal unipotent element, when $p$ exceeds its Coxeter number. We prove that $\mathfrak g$ is invariantless, i.e., that the unit object is not a summand of $\mathfrak g$. For odd $m$ with $3\leq m\leq p-2$, set $\mathfrak{g}_m:=\operatorname{Hom}_{\mathrm{Ver}_p^+}(L_m,\mathfrak g)$ and $E_{\mathfrak g}:=\bigoplus_{3\leq m\leq p-2,\ m\ {\rm odd}}\mathfrak g_m^{(1)}[m]$, where $(1)$ denotes Frobenius twist. Our main result is an isomorphism of graded algebras $H^\bullet_{\mathrm{CE}}(\mathfrak g)\cong\bigwedge^\bullet E_{\mathfrak g}^*$. We also identify this algebra with the de Rham cohomology $H^\bullet_{\mathrm{dR}}(G)$ of the group scheme $G=\exp(\mathfrak g)$ and show that the induced graded Hopf algebra structure agrees with the standard one on the exterior algebra. Moreover, if $V$ is a simple $\mathfrak g$-module on which $\mathfrak g$ acts nontrivially, then $H^\bullet_{\mathrm{CE}}(\mathfrak g,V)=0$. Hence for every finite-dimensional $\mathfrak g$-module $V$ one has $H^\bullet_{\mathrm{CE}}(\mathfrak g,V)\cong\bigwedge^\bullet E_{\mathfrak g}^*\otimes V^{\mathfrak g}$. This recovers the theorem of Borel and Chevalley on the cohomology of complex semisimple Lie algebras and its analogue in sufficiently large positive characteristic. We also prove similar results for relative cohomology.

A set of vertices $S$ of a graph $G$ is locating-dominating if $S$ is dominating and, for each pair of distinct vertices not in $S$, their neighborhoods in $S$ are distinct. We present results on the minimum density of such sets in the infinite hexagonal grid with a finite number of rows $k$, also known as the hexagonal strip of width $k$, which we denote by $H_k$. For each $k\geq 2$, we present either an optimal solution or a quasi-optimal solution for $H_k$ that is within $1.3\%$ of the optimum. We describe an exact exponential-time algorithm for fixed k, which we implemented to find optimal solutions for $k \leq 5$. As the infinite grid $H_{k}$ always admits a periodic optimal solution, to deal with larger values of $k$, we present an integer linear program that finds an optimal periodic solution for $H_{k}$ for each fixed period. This program yields high-quality feasible solutions for $H_7$ and $H_8$, which we then combine with an optimal solution for $H_3$ to obtain quasi-optimal solutions for all $k\geq 6$. All these solutions admit a very short description.

We study the dependence of the top Lyapunov exponent of a mixed Markov quasi-periodic cocycle on the transition probabilities. The transition matrix is assumed primitive, the Markov extension on the torus is assumed non-resonant along directed cycles, and the top Lyapunov exponent is assumed simple. We first prove the result in the irreducible case by a forward Markov transfer operator and contraction estimates uniform over the initial Markov state. The reducible case is then obtained, as in the Bernoulli argument of Bezerra-Sánchez-Tall, by decomposing along measurable invariant sections into restricted and quotient bundle cocycles and inducting on the fiber dimension. A measurable invariant section need not admit a continuous trivialization, even when the original matrices cocycle are continuous. We therefore formulate the irreducible analytic case for essentially bounded measurable bundle cocycles with essentially bounded inverses.

Put $q=r-1$, $t=k-1$, and $D=tq+1$. For an edge $e$ of a linear $C^r_{1,k}$-free $r$-uniform hypergraph, define \[ \delta_H(e)=\sum_{v\in e}\frac{1}{d_H(v)}-\frac{r}{D}. \] The defect satisfies $\delta_H(e)\ge 0$. At equality, every vertex of $e$ has degree $D$, and the petal trace at $e$ is a disjoint union of $t$ affine planes of order $q$. Equivalently, restoring the base line gives $t$ projective planes of order $q$ with common line $e$. For growing crowns, the equality structure is stable in the following sense. If $q_j\to\infty$, $2\le t_j\le q_j$, and $e_j$ is an edge of a finite linear $C_{1,t_j+1}^{q_j+1}$-free hypergraph satisfying \[ \frac{t_j^2}{q_j}\to0, \qquad t_j^3\delta_{H_j}(e_j)\to0, \] then, for every fixed $0<\theta<1$, \[ \frac{ |\{f\ne e_j:f\cap e_j\ne\varnothing,\ \delta_{H_j}(f)\ge\theta/t_j^2\}| }{(q_j+1)(t_jq_j)} \to1. \] Thus an edge close to equality is adjacent almost entirely to edges with defect of order at least $t_j^{-2}$. A uniform form gives absolute constants $Q_0,\varepsilon_0,c_0>0$ such that, whenever $q\ge Q_0t^2$ and $\delta_H(e)<\varepsilon_0/t^3$, at least $\tfrac12(q+1)tq$ neighbors of $e$ have defect at least $1/(100t^2)$. Consequently, \[ c^{\mathrm{lin}}_{q+1,t+1}\le t-\frac{c_0}{t}, \] where $c^{\mathrm{lin}}_{r,k}$ denotes the asymptotic linear Turán coefficient for $C^r_{1,k}$.

Let $F/\mathbb{Q}_p$ be a finite extension and let $\breve{G}$ be a connected reductive group with connected center satisfying $p>h_{\breve{G}}$ (the Coxeter number). We write down explicit polynomial equations for the reduced Emerton-Gee stacks (the Borel and the twisted Borel versions that jointly cover the entire $\mathcal{X}_{\breve{G},\mathrm{red}}^{\mathrm{EG}}$), by introducing the mod $p$ Weil-Deligne stacks capturing derived structures of $(\varphi, \Gamma)$-modules in reduced families. This allows us to algorithmically compute the set of irreducible components of $\mathcal{X}_{\breve{G},\mathrm{red}}^{\mathrm{EG}}$, thereby establishing its equidimensionality, and the existence of crystalline lifts by showing the total number of irreducible components equals the number of (mod $p$) crystalline (or potentially semistable) components. The last step of the arguments is to interpolate the rigid analytic Weil-Deligne stacks and the mod $p$ Weil-Deligne stacks to deduce the number of potentially semistable components. An initial implementation of the algorithms based on Gröbner basis establishes the existence of crystalline lifts in the $\breve{G}=\mathrm{F}_4$ case for all $F/\mathbb{Q}_p$, and in the remaining $\breve{G}=\mathrm{E}_6, \mathrm{E}_7$, and $\mathrm{E}_8$ cases for all but finitely many $F/\mathbb{Q}_p$.

We study vector-linear function computation over three-layer networks with a fixed target function and a fixed source-access pattern. We develop a support-constrained row-space framework that represents a linear computing code by a global row space. This space must contain the target row space and be generated by rows satisfying the local support-constraints of the network. We prove that this representation is equivalent to the existence of a linear computing code. For any prescribed global row space, we give a necessary and sufficient condition for its realization and determine the minimum uniform communication load at the middle nodes. The condition is expressed in terms of the ranks of the local subspaces supported on the source-access sets. It separates the exact local realization problem from the outer problem of designing the global row space and yields a variational characterization of the linear computing capacity. We then apply the framework to MDS targets over cyclic networks. We identify when the target row space alone is sufficient and when auxiliary rows are required. We determine the capacity in the dense regime and in the sparse regime whenever the cut-set bound is integral. For the remaining sparse parameters, we give a general linear construction whose achievable rate equals the integer part of the cut-set bound.

Quadratic Gradient (QG) is a Newton-type optimization framework that bridges first-order gradient descent and second-order optimization by incorporating curvature information into gradient updates. Simplified Quadratic Gradient (SQG) reduces the complexity of QG construction while preserving its optimization capability, whereas Quasi-Quadratic Gradient (QQG) extends the quadratic gradient principle to quasi-Newton methods such as BFGS. In this paper, we propose **Generalized Quadratic Gradient (GQG)**, a unified framework that extends the quadratic gradient principle to a broader class of Newton-type optimization algorithms. By abstracting the common structure of existing quadratic gradient methods, we show that the fundamental requirement of quadratic gradient construction is not limited to specific Hessian approximations, such as constant Hessian matrices, diagonal Hessian approximations, or BFGS-based Hessian surrogates. Instead, it can be generalized to any positive-definite curvature matrix satisfying the stationary condition of a local quadratic model. Based on this perspective, we investigate the construction of generalized quadratic gradients using various positive-definite Hessian surrogates beyond BFGS, providing a broader foundation for developing curvature-aware optimization algorithms.

We prove that every holomorphic differential form of non-top degree on a locally symmetric variety with Baily-Borel boundary of codimension greater than 1 is square-integrable with respect to the invariant metric. This generalizes a theorem of Weissauer in the Siegel modular case.

The paper develops the theory of quandle algebras. We show that over integral domains of characteristic other than 2, quandle algebras of ordered commutative quandles have no nontrivial idempotents. We also compute the automorphism groups of quandle algebras of trivial quandles and dihedral quandles of odd orders over arbitrary commutative rings, with partial results for dihedral quandles of even orders. Finally, a computer search provides the first examples of quandle algebras of commutator width 2.

Drifting objectives compare a target and model distribution through a vector field observed noisily at finitely many locations. We ask what distributional conclusion such a frozen measurement system warrants. For integrable antisymmetric interactions and absolutely continuous laws in a declared finite density basis, the unnormalized sampled numerator satisfies $\operatorname{vec}(V_X)=Mc$, where $c$ is an antisymmetric mismatch and $M$ is probe-dependent. This identity yields an a posteriori total-variation (TV) upper confidence bound accounting for held-out field noise, estimated-operator error, and externally validated $L^1$ residual radii around normalized density approximants in the span; a nonpositive observability margin returns the trivial TV bound and abstains. The audit recomputes this numerator from held-out samples; a normalized drift statistic requires a separate joint numerator–denominator analysis. For Gaussian-RBF interactions, a global envelope supports distribution-free and empirical-Bernstein radii without truncation, with companion bounds for the Laplace similarity in the original drifting objective. We characterize random-probe observability by a population Gram matrix, identify rank and symmetry degeneracies, and prove large-bandwidth collapse toward mean matching. Synthetic studies exercise Gaussian and Laplace numerators, separately prespecified bounded-vector and variance-adaptive radii, Monte Carlo-calibrated operators, nonzero residual radii around normalized finite-basis approximants, outward-rounded observability bounds, and designed abstention. A joint basis-size/dimension stress path extends evaluation through $m=8$. The result is a conditional diagnostic for a finite density class, or for normalized finite-basis density approximants with external residual radii, not a universal guarantee from small training drift.

Elimination geometry studies the exact nonnegative defect left after an auxiliary variable is optimized out. Part~II audits this certified-elimination interface through three theorem families. For integrability, local gaps form a repeated partial-elimination tower precisely under nonnegativity, touching, and zero elementary-square circulation. Hodge projection constructs nearest finite-space repairs and separates curvature, certification, and global period taxes. For representation and price, a descent gate characterizes surjective refinements and rules out strictly convex separable, duplication-invariant weight penalties. Dummy and target-calling lifts show that unrestricted lift design is vacuous, whereas the Fidelity-coherence frontier isolates irreducible cost. A modal Gram block classifies unconstrained quadratic lift-coupling germs and yields aligned Gaussian risk ledgers. Bernstein spectral functions preserve graph-Laplacian structure. Graph-universal Markov resolvents exist exactly up to first-order fractional qualification, and universal polynomial spectral penalties reduce to nonnegative Laplacian multiples. This linear limit does not preclude Part~I's nonlinear CDF repair and sharp CRPS theory. For converses, finite-order target-jet preservation uniquely requires subtraction of the corresponding defect jet, up to flat terms; approximate jets yield quasi-descent and stationary limits. Fixed-marginal coupling obeys a dependence-benefit frontier. Reversing certified KL in entropy-regularized control eliminates dimension-free guarantees, while a fixed measurement preserves all Umegaki defects of a faithful Gibbs family exactly when that family commutes.

We consider the following generalization of Troesch's classical two-point-BVP in order to take into account charge separation, which is required by physical considerations: $w''=L\sinh(Lw), w(0)=W_0<0<W_1=w(1)$. In contrast with the classical case, the solution of this generalized equation possesses boundary layer at both ends. We determine an explicit, approximate solution and verify its precision in several ways. The analytic solution hinges on a result concerning a class of overdetermined first order BVPs. Presenting this matter constitutes the second objective of the article.

We discuss entire solutions $u \in C^2(\mathbb{R}^2)$ of the equation ${\rm div}(\nabla f(\nabla u)) = 0$ with strictly convex density $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ of subquadratic growth and prove that $u$ is an affine function provided that at least one partial derivative is bounded from one side. Further results concern the behaviour of non-affine entire solutions.

The safety and reliability of intelligent-transportation-systems (ITS) can be greatly enhanced through adding redundancy in the information-sharing network of the vehicles. In this paper, we first model the mixed traffic of human-driven and autonomous vehicles as a distributed system observability problem over a network of communicating vehicles. We clearly show that a strongly-connected network with minimum n links (with n as network size) is sufficient for the observability of mixed traffic network. Then, we present graph-theoretic results on adding redundancy to the changing network of vehicles to make it resilient to the failure of certain number of vehicles/sensors or their data-sharing links. Finally, we employ distributed observer design to validate our results over a simple mixed traffic example.

Let G be a torsion-free group whose real group algebra R[G] has no zero divisors, and let u -> u^{*theta} be an l2-isometric anti-involution of R[G] (a composition of the inversion involution with a ring automorphism and a sign character). We prove a compactness theorem: for every finite *theta-closed support window W the constant mu*(W) = min{ ||w^{*theta}w||_2 : ||w||_2 = 1, supp(w) in W } is strictly positive, and every real theta-unitary element (u^{*theta}u = 1) supported in W satisfies ||u||_2 <= mu*(W)^{-1/2}. In particular the integer theta-unitary elements supported in W form a finite, effectively enumerable set: the a priori infinite "height" direction of the unit search collapses to a single real constant. For the Promislow (Hantzsche-Wendt) group P and the window B(4) that hosts Gardam's counterexample to the unit conjecture over F_2, we combine this with exact SAT-certified height ladders (heights <= 31 per stratum, DRAT-certified master cell at larger heights), a radius-free depth-tail theorem, and a mirror symmetry between the +- strata, reducing the vanishing of all nontrivial theta-unitary units u = +-1 (mod 2) with supp(u) in B(4) to a single certified lower bound on mu*(B(4)). We report numerical estimates mu*(B(4)) ~ 1.4e-3, well above the required threshold, and prove three structural results about the remaining certification problem: no linear (Cauchy-Schwarz) dual certificate exists, because P carries theta-anti-unitary trivial elements; and, numerically, the level-2 sum-of-squares relaxation is boundary-pinned with an explicit slope, both in the direct and in the ideal-multiplier formulation – the obstruction being a spurious pseudo-moment that no measure can realize. We also contrast the mechanism with Z[D_infinity], where torsion produces zero divisors, mu* = 0, and genuinely unbounded unipotent families of twisted unitaries.

The Mandelbrot and Julia sets, generated by the quadratic iteration $z_{n+1}=z_n^2+c$, are foundational objects in complex dynamics. We study the three-variable principal-value complex-power iteration $z_{n+1}=z_n^x+c$, where $z_0,c,x\in\mathbb{C}$. The triples producing all-time well-defined and bounded orbits form a locus $\mathcal{B}\subset\mathbb{C}^3$. Fixing two coordinates yields three natural families of coordinate fibers, denoted $M(z_0,x)$, $J(c,x)$, and $E(z_0,c)$. For $x=2$, $M(0,2)$ is the classical Mandelbrot set, $J(c,2)$ is the classical filled Julia set, and $\partial J(c,2)$ is the classical Julia set. We focus on the Exponent Set, or E-Set, obtained by fixing $(z_0,c)$ and varying the complex exponent $x$. We prove three groups of structural results. First, for explicit parameter families, including pure-power real and unit-circle cases and an additive example with nonzero real and imaginary parts, no finite universal escape radius exists: for every prescribed radius, one can choose an exponent whose bounded orbit makes a finite excursion beyond that radius. Second, we construct a boundary point at which an extended-valued escape-time function is discontinuous for one strict threshold. Third, we prove a vertical boundedness asymmetry in which the principal-argument convention enters explicitly.

Starting with some historical considerations, we examine the various attempts to model old age that fall within the realm of mathematical physics. For example, we study the mathematical theory of declining functions, which, since Laurent Schwartz's work on distributions, are generally defined on what is called a "Schwartz space." We explain how this essentially analytic formalization applies to the representation of the decline of vitality (or of a number of vital functions). We then show that there exist many other possible formalizations that utilize the resources of algebra, in particular the theory of "inclines" of Cao, Kim, and Roush and that of relational structures, on which an algebraic notion of "age," in Cameron's sense, can be defined. But we maintain, in conclusion, that a more positive representation of old age, consistent, moreover, with what a number of cultural sources from Antiquity to the 20th century tell us, should mobilize another type of formalization, capable of selecting, within the overall debate between the living being and its environment, persistent structures that do not decline amidst the decline of others.

Let $P_1(n)$ and $P_2(n)$ be the largest and second-largest distinct prime factors of $n$. Alladi and Johnson asked whether there is a bounded function $f$ on the primes such that the averages $\frac{1}{x}\sum_{2\le n\le x}f(P_1(n))$ and $\frac{1}{x}\sum_{2\le n\le x}f(P_2(n))$ converge to different limits, where $f(P_2(n))=0$ when $n$ is a prime power. We show that this cannot happen. In fact, whenever the first average converges, the second also converges to the same limit.

For the classic bridge and torch problem with crossing times $\left\{1,\ldots,n\right\}$, we derive a closed-form expression for the optimal crossing time $T\left(n\right)$ using a recurrence relation derived from the problem's optimal substructure, thus obtaining \[T\left(n\right)=\frac{n^2}{4}+3n-5+\frac{\left(-1\right)^n-1}{8}\] which holds for all $n\ge 2$. We generalize the problem to a bridge of capacity 3 and obtain the optimal crossing time \[T_3\left(n\right)=\frac{n^{2}}{6}+2n-\frac{181}{36}+\frac{\left(-1\right)^{n}}{4}-\frac{2}{9}\cos\left(\frac{2n\pi}{3}\right)\] which holds for all $n\ge 7$. As such, we obtain a new sequence A392834 in the On-Line Encyclopedia of Integer Sequences. Lastly, we also explore this problem for star graphs, and see how we can recover some classic identities involving the sum of floor functions.

A certified exhaustive computation shows that every simple cubic bipartite graph on at most 58 vertices contains a cycle of length 4, 8, or 16. Consequently, any cubic bipartite counterexample to the Erdos-Gyarfas conjecture has at least 60 vertices, improving the established published lower bound for this class from 30 to 60. The proof begins with a Moore-bound observation: below 62 vertices, a cubic bipartite graph avoiding 4- and 8-cycles must contain a 6-cycle. Viewing the graph as the Levi graph of a linear symmetric v3-configuration turns this 6-cycle into a Berge triangle. Up to symmetry, only two rooted extensions are possible. A complete restricted-growth search on at most 29 points exhausts both search trees. The computation is checked by two separately implemented exact procedures using different C16 oracles and by a static witness certificate. Source code, certificates, and reproduction instructions are archived with the paper.

For a positive integer $n$, let $\mathcal{M}_n$ be the set of distinct multinomial coefficient values $n!/(p_1!\cdots p_t!)$, where $(p_1,\ldots,p_t)$ ranges over the integer partitions of $n$. We study two complementary aspects of the additive structure of $\mathcal{M}_n$: the maximum cardinality $s(n)$ of a Sidon subset and the multiplicities of unordered pair sums in the full set. A product embedding, strongly Sidon subsets, and estimates for prime partitions give \[ \liminf_{n\to\infty}\frac{\log(s(n)/n)\log\log n}{\sqrt{\log n}}\geq\frac{\pi}{\sqrt{3}}. \] Long arithmetic progressions and separated copies yield a complementary lower bound $2/(3\sqrt{3})$ for the normalized Sidon defect. Writing $r_n(t)$ for the number of unordered representations $t=x+y$ with $x,y\in\mathcal{M}_n$, we study the collision excess $C(n)$, the number $D(n)$ of multiply represented sums, the maximum multiplicity $\mu(n)$, and the cumulative profile $T(n,k)$. We prove \[ \liminf\frac{C(n)}{n^2\log n}\geq\frac18,\qquad \liminf\frac{D(n)}{n^{3/2}\sqrt{\log n}}\geq\frac{4}{3\sqrt{3}},\qquad \liminf\frac{\mu(n)}{\sqrt{n\log n}}\geq\frac12. \] We also obtain a scaled lower envelope for the full profile, an additive-energy bound, stabilization with respect to the number of variables, and an explicit family of trinomial collisions. Exact Sidon values through $n=16$, the lower bound $s(17)\geq89$, and pair-sum statistics through $n=20$ are reported with reproducible computational material.

The fundamental groups of compact 3-manifolds are known to be residually finite. Feng Luo conjectured that a stronger statement is true, by only allowing finite groups of the form $\mathrm{PGL}(2,R)$, where $R$ is a finite commutative ring. In earlier work, this conjecture was disproven in full generality. The conjecture arose in the context of orientable connected compact 3-manifolds which are geometrizable. By constructing explicit faithful linear representations using rings with nilpotent elements, we demonstrate that the conjecture holds for six of the eight Thurston model geometries, namely all but $\mathbb{S}^3$ and $\widetilde{\mathrm{SL}_2}$. In the case of $\mathbb{S}^3$, the conjecture holds if we replace $\mathrm{PGL}(2,R)$ with $\mathrm{GL}(2,R)$. A spherical counterexample for the projective variant is the Poincaré homology sphere $\Sigma(2,3,5)$. In the case of $\widetilde{\mathrm{SL}_2}$, the conjecture fails to hold for both the projective and non-projective variants; a counterexample is provided by the Brieskorn sphere $\Sigma(2,3,7)$.

We consider the largest eigenvalues of submatrices of the classical Gaussian random matrices with a finite rank spike in the critical BBP regime. We show that they converge to a multilevel interlacing particle system whose lowest level is the Airy$_\beta$ point process. Our construction and scaling limits extend in a natural way to non-classical $\beta >0$.