Papers

For an elliptic curve $E/\mathbb{Q}$ of rank two with complex multiplication, Coates, Liang and Sujatha gave a criterion for the vanishing of $Sha(E/\mathbb{Q})[p^\infty]$ at a good ordinary prime $p$ and applied it to five such curves for $p < 30{,}000$. We prove a cyclotomic criterion of the same kind: outside an explicit set of primes, the normalised second Taylor coefficient of the Mazur-Tate-Teitelbaum $p$-adic $L$-function at the central point is a $p$-adic unit if and only if the cyclotomic $p$-adic regulator is a unit and $Sha(E/\mathbb{Q})[p^\infty] = 0$, and, by the theory of Bannai and Kobayashi, if and only if an explicit combination of three critical Hecke $L$-values of weight $2p - 1$ has valuation exactly two. Following the algorithm of Stein and Wuthrich, we compute the regulator for the same five curves at every good ordinary prime below $30{,}000$: it is a unit at all but three of the $8{,}050$ primes outside the excluded set. The one case that the criterion of Coates, Liang and Sujatha left open, $p = 577$ for $y^2 = x^3 + 34x$, is settled by the new criterion. A Lean 4 formalisation of the first equivalence, assuming stated results from the literature, is provided.

Let $R=L_{\F_2}(1,2)$. We prove that $\GL_r(R)$ is integrally acyclic for every $r\geq1$ and that the canonical map $\St_r(R)\to\GL_r(R)$ is an isomorphism for every $r\geq3$. The unit group $R^\times$ is finitely presented, and we describe an explicit finite presentation. The homology calculation uses leaf coordinates, simultaneous extensions of ordered frames, and finite-field actions on stabilizers. The Steinberg argument lifts relations from a simply connected frame complex and refines coordinates. We also state separate criteria for the two arguments over rings of characteristic two.

We present an iteration algorithm for vacuum and Einstein scalar-field equations in double-null gauge, which transform the non-linear PDE into systems of ODE. The numerical realization combines characteristic constraint solves, LGL spectral elements, pole-free spherical operators, Galerkin projection, and independent first-order residual and consistency checks.

We generalise Yin's motivic integration with section to the Hensel-minimal framework, with and without volume forms, building on Stout and Vermeulen's work on a Hrushovski-Kazhdan style motivic integral for $1$-h-minimal valued fields. In particular, we get a bijective motivic integral \emphà la} Hrushovski-Kazhdan between certain Grothendieck groups of definable sets on the valued field and the coarse Grothendieck groups on $\mathrm{RV}$ in the Denef-Pas language, with angular component. In the second half of the article, we build a Cluckers-Loeser-like framework and show that our constructions indeed give back Cluckers-Loeser motivic integral. Thus Cluckers-Loeser motivic integration derives from an isomorphism of Hrushovski-Kazhdan motivic integration, but in the Denef-Pas language with a section of $\mathrm{RV}$.

Entropy conditions are widely used in empirical-process analyses to establish asymptotic properties of M-estimators. In regression problems with responses taking values in a general metric space, a commonly imposed condition requires the entropy integral associated with a shrinking ball centered at the target object to remain uniformly bounded as the ball radius tends to zero. We show that this condition generally fails in the quadratic Wasserstein space of univariate probability distributions supported on a compact interval. Specifically, when the target object is a strictly increasing and absolutely continuous distribution function whose derivative is bounded away from zero and infinity, the corresponding entropy integral diverges as the ball radius tends to zero. The same divergence persists even when the ambient space is restricted to the class of distribution functions satisfying fixed uniform two-sided Lipschitz bounds. These results indicate that different asymptotic analyses are required for the quadratic Wasserstein space.

Let $M^n$ be a closed oriented manifold of even dimension $n=2m\ge2$, equipped with a smooth metric $g_1$ and a Riemannian metric $g_2$. We compute the Wodzicki residue of $D_{g_1}^2D_{g_2}^{-n}$ on the common exterior bundle, where $D_{g_r}=d+\delta_{g_r}$. This is the noncommutative integral of $D_{g_1}^2$ relative to the reference operator $D_{g_2}$. Its local density involves the curvatures of the two metrics and the difference of their Levi–Civita connections. Integration by parts gives a closed-manifold formula without explicit derivatives of the connection difference. For compact manifolds with boundary and $n\ge4$, we assume that both metrics are Riemannian and satisfy $g_r=h_r(x_n)^{-1}g^{\partial M}+dx_n^2$ near the boundary, with $h_r(0)=1$. We compute the noncommutative residues of two factorizations involving even and odd powers of $D_{g_2}$. Their interior contributions coincide, whereas their boundary terms are explicit multiples of the integral of $K_{g_1}-K_{g_2}$, with the coefficient for the even factorization twice that for the odd one. Here $K_{g_r}$ is the trace of the second fundamental form with respect to the inward unit normal. The calculation uses a second-order residue formula and direct boundary symbol expansions. When the metrics coincide, the formulas reduce to the Hodge–de Rham Kastler–Kalau–Walze identity.

Given a matched pair of Hopf algebras, the double cross product construction yields a new Hopf algebra, the prototypical example of which is the generalized quantum double arising from a (Hopf) skew pairing. In this paper, we specifically investigate double cross products with projections in the framework of Radford's biproduct theory, and establish that such an algebraic structure is equivalent to a simplified version of (Yetter-Drinfeld) relative Rota-Baxter operators on Hopf algebras recently introduced by Sciandra, with the corresponding braided Hopf algebras lying in the Yetter-Drinfeld module category under coadjoint coactions. We further examine conditions under which this algebraic structure induces a matched pair of actions on a single Hopf algebra, which is in turn equivalent to the notion of braiding operators on Hopf algebras that yield solutions of the braid equation. In addition, we show that such relative Rota-Baxter operators are morphisms between certain Doi-Hopf modules.

For integers $t\geq -1$, let $\theta_t$ be the largest real root of the Shanks polynomial $g_t(X)=X^3-tX^2-(t+3)X-1$ and put $K_t=\mathbb{Q}(\theta_t)$. We classify the points on $E_n:y^2=x^3-n^2x$ over $K_t$ with abscissa $\theta_t-1$ when $n$ is a positive integer and $E_n(\mathbb{Q})$ has rank zero. We determine the four pairs $(d,t)$ with $d>0$ rational for which $(\theta_t-1)/d^2$ is an abscissa on $E_3(K_t)$. We also exclude the abscissa $\theta_t-1$ on $E_5(K_t)$ and $E_6(K_t)$ and prove that only finitely many parameters $t$ admit this abscissa for each fixed positive integer $n$.

We present an energy-stable discontinuous Galerkin (DG) spatial discretization of the intrinsic beam equations with Kelvin-Voigt damping. We reformulate the governing equations by splitting the total sectional resultants into elastic and viscous parts, thereby exposing a mixed system without mixed space-time derivatives. For homogeneous cantilever boundary data, the continuous system obeys an energy dissipation identity. With characteristic upwinding and alternating auxiliary traces the DG discretization is energy stable at the semidiscrete level for the stated homogeneous split boundary closure. The method is implemented in the Julia-based simulation framework Trixi.jl. A physically compatible manufactured solution with all state and viscous components active assesses convergence, while a rotating-beam test verifies the analytic constant-spin branch and the actuated energy balance.

Consider a random geometric graph with vertices given by a Poisson point process, and whose edges depend on independent marks corresponding to the vertices and pairs of vertices. In this paper, we study two related questions on this general model: the number of edge crossings in a projection of this graph, and its graph-theoretical planarity. We focus on models with heavy-tailed mark distributions, in particular with polynomial tails with arbitrary exponents. We show that the asymptotic behaviour varies significantly depending on this exponent.

We study the stochastic first-order oracle complexity for constrained or regularized convex-concave min-max optimization and stochastic monotone variational inequalities. We focus on the case when suboptimality is measured in terms of the gradient mapping, also known as, forward-backward or natural residual, an optimality notion that generalizes the gradient norm for unconstrained problems. In this setting, under standard unbiased oracle access with now-standard variance assumptions, the best-known complexity for making the norm of the gradient mapping less than $\varepsilon$ is $\widetilde{O}(\varepsilon^{-4})$, compared to the near-optimal $\widetilde{O}(\varepsilon^{-2})$ that is established in the unconstrained case. We bridge this gap to improve the gradient mapping complexity for constrained convex-concave min-max problems to $\widetilde{O}(\varepsilon^{-2})$. We then extend to prove the same complexity for problems without the bounded variance, by using the Blum-Gladyshev assumption.

Tutte's $3$-flow conjecture asserts that every $4$-edge-connected graph admits a nowhere-zero $3$-flow. In 2013, Lovász, Thomassen, Wu, and Zhang proved that every odd-$7$-edge-connected graph admits a nowhere-zero $3$-flow; consequently, the conjecture holds for $4$-edge-connected graphs with no edge-cut of size $5$. We consider the complementary situation in which $5$-edge-cuts are allowed. We show that, when edge-cuts of size $5$ are permitted, Tutte's $3$-flow conjecture holds for graphs with no edge-cut of any size from $6$ to $k$, where $k$ is an absolute constant. In fact, $k=40$ suffices and we prove a stronger version in which only nontrivial edge-cuts of those sizes are forbidden. A graph is called essentially $t$-edge-connected if deleting any set of at most $t-1$ edges leaves at most one nontrivial component. Motivated by Jaeger's weak $3$-flow conjecture, we prove an analogous result for essential edge connectivity: every $4$-edge-connected, essentially $41$-edge-connected graph admits a nowhere-zero $3$-flow.

We study the local continuation of a steady vortex sheet under a prescribed deformation $g_s= e^{2\sigma_s}g_{\mathrm{e}}$ of a conformal metric on the unit disk. Representing the moving sheet by a normal graph, we formulate the streamline and Bernoulli conditions as a nonlinear residual whose two components have different Sobolev orders. If the two one-sided tangential velocities of the reference sheet do not vanish simultaneously, the derivative of the unnormalized residual is Fredholm of index two. Fixing the mean normal displacement and the total circulation gives an index-zero problem; when the normalization differential restricted to the unnormalized kernel is onto $\mathbb{R}^2$, we obtain a parameter-dependent Lyapunov-Schmidt reduction. This yields a locally unique normalized branch in the nondegenerate case and explicit first- and second-order necessary conditions in the degenerate case. For a rotationally symmetric reference metric and a concentric circular sheet, the linearized problem separates into Fourier blocks and only finitely many modes can be resonant. In the constant-curvature disk family, every mode $k\geq2$ is nonresonant, whereas the first mode produces a two-dimensional reduced problem. The two first-mode kernel profiles are normal traces of ambient Killing fields, but they do not generate fixed-boundary symmetries or, by themselves, a nonconcentric branch.

Recently, Kang, Kim, and Lee \cite{Kang2026} developed a unified moment-trace framework for symmetric partition statistics using complete Bell polynomials and their inversion formula. In this paper, we apply this framework to crank statistics for $t$-core partitions and overpartitions. For $t\in\{5,7,11,17,19\}$, we show that the normalized even crank moment generating functions for $t$-core partitions admit partition-trace representations in terms of the functions $D^{(t)}_{2s}(\tau)$, together with suitable Bernoulli-number shifts. We also establish inverse trace formulas that recover $D^{(t)}_{2s}(\tau)$ from the corresponding normalized even crank moments. For overpartitions, we obtain analogous trace and inverse-trace identities for the normalized even moments associated with the first and second residual crank generating functions. As applications, we use complete Bell polynomials and their inversion formula to obtain explicit expressions for the $t$-core partition numbers and overpartitions number in terms of sums involving divisor function.

Let $v$ be an indeterminate. We give a presentation of the mirabolic quantum Schur algebra $\mathcal{MS}_v(2,d)$ in terms of generators and defining relations. More precisely, we determine the kernel of the natural epimorphism from Rosso's mirabolic quantum $\mathfrak{sl}_2$, $\mathbf U^{\mathrm{mir}}_v(2)=\langle e,f,k^{\pm 1},\ell\rangle$, onto $\mathcal{MS}_v(2,d)$. Setting $P_d(k)=\prod_{r=0}^{d}(k-v^{2r-d})$ and $Q_d(k)=\prod_{r=1}^{d}(k-v^{2r-d})$, we prove $\mathcal{MS}_v(2,d)\cong \mathbf U^{\mathrm{mir}}_v(2)/ \left\langle P_d(k),(\ell-1)Q_d(k)\right\rangle$. We further determine the split Wedderburn decomposition of this quotient and provide an equivalent presentation in terms of weight idempotents.

We study the problem of perturbing a given matrix X belonging to a subspace S of linearly structured real matrices of order n to its nearest counterpart Y=X+D, where D belongs to S and Y possesses an eigenvalue with prescribed lower bounds on its algebraic multiplicity and geometric multiplicity. The proposed framework takes care of both cases where the target eigenvalue is known and where it is treated as an unknown decision variable to be determined jointly with the structured perturbation. We establish necessary and sufficient conditions for the existence of such matrices by expressing the feasibility constraints through Jordan-chain relations in structural coordinates. This allows us to formulate the problem as a nonlinear constrained nested optimization that minimizes the Frobenius norm of the perturbation. To solve this, we implemented a two-level strategy, utilizing MATLAB's fmincon for continuous inner optimization via a multi-start Sequential Quadratic Programming approach, while the outer level solves a discrete optimization problem. Comprehensive numerical experiments on various classes of structured matrices, including comparisons with existing methods, demonstrate the effectiveness and accuracy of the proposed approach.

We investigate the properties of matrices obtained from a geometric transformation of the first $N$ rows of Pascal's triangle. For $N > 2$, their congruence properties form a Pascal tiling, that is, a perfect alternation between entries congruent to $0 \pmod{N}$ and the others, if and only if $N$ is prime. This result yields an alternative proof of the classical congruence $L_N-1\equiv 0 \pmod{N}$ for prime $N$, where $L_N$ denotes the $N$th Lucas number. Within the framework of the Pascal tiling theorem, this congruence can be expressed as a sum of entries lying along a diagonal of one of the matrices under consideration; when $N$ is prime, each of these entries is congruent to $0 \pmod{N}$. By contrast, for Fibonacci pseudoprimes, the sum remains congruent to $0 \pmod{N}$ while at least one of its terms is not. Finally, these results are interpreted in terms of decompositions of binomial coefficients and extended to multinomial coefficients, leading to a study of the associated symmetries. This perspective highlights the case where $N$ is a prime power and clarifies the conditions under which a Pascal tiling arises.

Turán's Tetrahedron Problem asks to determine the Turán density of the complete hypergraph $K_4^{(3)}$ (tetrahedron). This problem, posed by Turán in 1941, is one of the most famous problems in extremal combinatorics and its solution would attract \$500 prize from Erd\H{o}s. In the 1980s, Erd\H{o}s and S\'os asked to determine Tur\'an densities of $K_4^{(3)-}$ (broken tetrahedron) and $K_4^{(3)}$ (tetrahedron) when edges are constrained to be uniformly distributed in the host hypergraph. The presumably easier case of the broken tetrahedron was solved by Glebov, Kr\'al' and Volec [Israel J. Math. 211 (2016), 349-366] and Reiher, R\"odl and Schacht [J. Eur. Math. Soc. 20 (2018), 1139-1159]. We solve the tetrahedron case by proving that the uniform Tur\'an density of $K_4^{(3)}$ is equal to 1/2; this confirms that Rödl's lower bound construction from 1986 is optimal.

Let \(R_p=\|x_p\|^2/p\Rightarrow \nu\) and \(p/N\to c\in(0,\infty)\). We prove that the radial quadratic-form condition (RQC), requiring energy in deterministic subspaces to follow the parent radius, is equivalent to the following spectral property: for one fixed \(\alpha\in(0,1)\), every deterministic rank-\(\lfloor \alpha p\rfloor\) projected covariance has empirical spectral distribution converging to \(\mu_{c\alpha,\nu}\). The radius law \(\nu\) may have arbitrary tails; no moment assumptions, conditional isotropy, or independence between radius and direction are required. We also show that the radius law can be recovered from the projected spectral limits when their common limit has finite second moment and subspaces of vanishing relative dimension carry vanishing normalized energy, again without moment assumptions on the original vectors. A finite-sample inverse estimate bounds quadratic-form defects by bounded tests of expected projected spectra. The converse argument combines angular symmetrization and fantope rigidity with a determinant comparison between radial column deletion and spectral trimming. A cancellation of the deleted radial factors makes the comparison error depend only on the deleted fraction, rather than on the magnitudes of the deleted radii.

We study cohomological Hall algebras of curves and their actions on the homology of Quot schemes. We introduce the virtual homology of Quot schemes and show that it is preserved by both creation and annihilation actions. We prove that the torsion CoHA is isomorphic to a shuffle algebra and, equivalently, to a braided symmetric algebra associated with a Yang-Baxter operator. We use this description to determine the ideal of tautological relations for punctual Quot schemes and obtain a new basis for their cohomology rings. Finally, we introduce a universal way to double the torsion CoHA and show that it acts naturally on the virtual homology of Quot schemes of arbitrary type.

We present a compact implicit high-resolution finite volume scheme for one-dimensional multicomponent transport problems with nonlinear Langmuir sorption. The method combines a second order compact implicit discretization with WENO reconstruction and a time limiter for the treatment of steep gradients and discontinuities. A special attention is given to the Courant number used in the limiter, where local characteristic Courant numbers based on the eigenvalues of the inverse retardation matrix are considered. The compact implicit schemes are compared with analogous explicit first and second order methods. Numerical experiments show second order convergence for smooth solutions and demonstrate that the implicit approach can be used with significantly larger Courant numbers. The method is also applied to a three-component displacement chromatography problem with sharp concentration fronts.

We prove that no strongly regular graph with parameters $(266, 45, 0, 9)$ exists. The proof is formalized in Lean 4 and Mathlib without external infeasibility certificates or assumed classification theorems. A hypothetical graph gives a rank-$12$ integral Gram lattice with an integral centroid. A Lorentzian change of form, a marked $D_7$ gluing, and an explicit rank-six complement produce a positive-definite even unimodular lattice of rank $24$, together with the original indexed family of $220$ vectors. Harmonic theta identities and a root-isolation inequality force the root system $A_{11} \perp D_7 \perp E_6$. First and second moments then exclude the possible complements: the final case reduces to an impossible binary projection identity $4x + 4y - 2z = 50$. A type-$A$ subcase is closed by a separate classification-free proof of the known nonexistence of a quasi-symmetric $2$-$(56, 12, 9)$ design with intersections $0, 3$. That argument constructs a Krein graph and forces a Steiner $3$-$(12, 4, 1)$ design, contradicting its replication equation. The formal theorem depends only on the three standard Lean axioms and has also been checked independently with nanoda. The archived formalization is release v2.0.0.

We characterize the uniform theta graph $\Theta(t,2)$ by the spectrum of its normalized adjacency matrix, or equivalently, by the spectrum of its normalized Laplacian matrix. We also investigate the periodicity of Grover walks on nonregular graphs, which is closely related to the eigenvalues of the normalized adjacency matrix and those of the time evolution matrix of the Grover walk. We show that the Dutch windmill graph $D_n^{(t)}$ is $2n$-periodic and that the uniform theta graph $\Theta(t,n)$ is $(2n+2)$-periodic. Furthermore, we completely determine the connected $6$-periodic graphs and prove that they are precisely $D_3^{(t)}$ with $t \geq 2$ and $\Theta(t,2)$ with $t \geq 1$.

Let $J$ be the Jordan subalgebra of the free associative algebra $\Q\langle X\rangle$ generated by $X$ and $1$ under $a\cdot b=\tfrac12(ab+ba)$. We give a polynomial criterion for an associative polynomial to belong to $J$. For every $n\ge1$, Robbins' recursion gives a family of operators $T_\alpha\in\Q[S_n]$, indexed by compositions of $n$ into parts $1$ and $2$. We prove that right multiplication by $U_n=\sum_\alpha T_\alpha^\dagger T_\alpha$ has image $\SJ(n)$, the space of multilinear Jordan elements. If its characteristic polynomial is $t^{e_n}q_n(t)$, where $q_n(0)\ne0$, then $\Pi_n=I-q_n(U_n)/q_n(0)$ is an idempotent and $f\in\SJ(n)$ if and only if $fq_n(U_n)=0$, or equivalently $f\Pi_n=f$. The same operators give the criterion for arbitrary homogeneous polynomials of degree $n$. We also prove $\SJ(n)\cong\bigoplus_{\lambda\vdash n}r_\lambda S^\lambda$, where $r_\lambda=\rank\rho_\lambda(U_n)$. Thus $\dim\SJ(n)=\sum_\lambda d_\lambda r_\lambda$ and $e_n=\sum_\lambda d_\lambda(d_\lambda-r_\lambda)$, with $d_\lambda=\dim S^\lambda$. We give a recursion for the corresponding row spaces and use it to compute the multiplicities over $\Q$ through degree $12$. Exact block characteristic polynomials through degree $7$ and direct computations in small degrees illustrate the criterion.

This paper develops a finite element method for wave-structure interaction problems arising in the hydroelastic modelling of submerged elastic plates. The approach is formulated for both two-dimensional settings and three-dimensional channels. For the fluid part of the problem, we use a symmetric discontinuous Galerkin scheme that captures the discontinuity in the potential across the plate. To impose the Sommerfeld radiation conditions, we take inspiration from the Dirichlet-to-Neumann map approach and use the analytic solution in the semi-infinite domains to construct appropriate non-local boundary conditions. For the plate, we use a continuous/discontinuous Galerkin method to resolve the 4$^{\rm th}$ order operator without using continuously differentiable finite elements. We prove well-posedness of the method, showing that the resulting sesquilinear form is bounded and satisfies the Gårding inequality, leading to a unique and stable solution. We implement our formulation as an open-source tool using the Gridap package in Julia. We validate our method against a hypersingular boundary integral method, finding excellent agreement. The flexibility of the methodology is demonstrated through simulations of a two-dimensional slanted plate and three-dimensional plate geometries including centred rectangular plates, offset rectangular plates, and annulus-shaped plates. The finite element method presented here can be readily applied to model wave energy conversion of piezoelectric bimorphs. Furthermore, the method could easily be extended to consider wave scattering from fixed or rigid structures or from variable bottom topography, and is not limited to hydroelastic applications.

To mitigate the scalability bottleneck in the radio access network (RAN) in federated edge learning (FEEL), over-the-air federated learning (AirFL) exploits waveform superposition over multiple-access channels (MACs) for analog model aggregation. However, coherent AirFL typically relies on stringent PHY-layer conditions such as accurate channel state information (CSI), tight time/frequency synchronization, and frequent transceiver calibration for signal alignment. However, these requirements, if not impossible to be met, incur substantial communication and computation overhead. In this paper, we propose a non-coherent AirFL (NCAirFL) protocol over a broadband single-antenna MAC, leveraging binary dithering, unbiased non-coherent detection, and long-term error feedback to waive the need for instantaneous CSI. For NCAirFL with general smooth non-convex objectives and a constant learning rate, we establish a convergence bound achieving the convergence rate in the same order of $\mathcal{O}(1/\sqrt{T})$ as communication-ideal FedAvg, where $T$ is the total number of communication rounds. To further improve communication efficiency under data and wireless resource heterogeneity, we also derive a lower bound on the expected single-round objective decrease in the global loss conditioned on device scheduling, building upon which a surrogate objective function is obtained for jointly optimal device selection and power control. Experimental results on MNIST and CIFAR-10 corroborate that NCAirFL achieves learning performance close to FedAvg in practical settings, with the proposed device scheduling policy substantially accelerating convergence.

Let \(N_M(\lambda)\) be the spectral counting function of the sub-Laplacian on the compact Heisenberg manifold \(M=\Gamma\backslash\mathbb H_d\), where $\Gamma$ is a lattice subgroup of the Heisenberg group $\mathbb H_d$. In 2016, Strichartz \cite[J. Geom. Anal.]{Str16} proved the Weyl law with remainder \(R_M(\lambda)=N_M(\lambda)-A_d\operatorname{vol}(M)\lambda^{d+1} = O_M(\lambda^d\log\lambda)\), and conjectured the optimal remainder to be \(O_M(\lambda^d)\). In this work, we establish a new upper bound and the first two-sided lower bounds $$ R_M(\lambda)=O_M\!\left(\lambda^d(\log\lambda)^{2/3}\right), \qquad R_M(\lambda)=\Omega_{M,\pm}\!\left(\lambda^d\log\log\lambda\right). $$ As a result, this implies that the sharp polynomial order is $d$, and disproves Strichartz's conjecture.

We consider the one-dimensional random growth surface introduced by Adhikari and Chatterjee [AC24], where the height function is defined recursively via the heights at two neighboring sites and an independent noise term at each space-time point. In the subcritical regime, assuming uniformly bounded eighth moments of the noise variables and appropriate regularity of the driving function, we establish convergence in law of the suitably rescaled height function to the solution of the additive stochastic heat equation. Our approach employs a direct recursive renormalization scheme. A central component is a graph representation of the renormalization terms, which enables the derivation of the high-moment bounds essential to the renormalization procedure.

A knot invariant is a fingerprint shared by equivalent knots: unequal values certify inequivalence, while equal values may conceal different knots. Under two rooted random-diagram models, we prove that the incomplete normalized Alexander polynomial separates random pairs but collides in a growing database. If $D_n$ and $D_n'$ are independent $n$-crossing diagrams, then $\Pr\{\Delta_{K(D_n)}=\Delta_{K(D_n')}\}=O(n^{-1/2})$, and every fixed normalized Alexander polynomial is exponentially rare. With exponentially high probability, the diagram shadow contains linearly many disjoint opened-trefoil slots. Conditional on the shadow and exterior crossing signs, their indicators are independent Bernoulli$(1/4)$ variables whose sum is a binomial coordinate in the $3$-adic determinant valuation. Consequently, $\sup_{a\geq 1}\Pr\{\det K(D_n)=a\}=O(n^{-1/2})$. For a discrete invariant $I$, let $\alpha_n(I)=\Pr\{I(D_n)=I(D_n')\}$. Its collision entropy is $H_2(I(D_n))=-\log\alpha_n(I)$. An independent sample of size $M$ has $\binom{M}{2}\alpha_n(I)$ colliding pairs on average and birthday scale $\alpha_n(I)^{-1/2}$. If $M_n^2\alpha_n(I)\to\infty$, a repeated value occurs with probability tending to one even when $\alpha_n(I)\to0$. For the determinant, $M=o(n^{1/4})$ suffices for collision freedom with high probability; a matching lower bound is open. We also give exact finite-population formulas for fixed censuses, calculate the expected cost of invariant cascades, and relate collision probability to the frequency of calls to a complete equivalence procedure. On a balanced pair-classification benchmark, the normalized-Alexander rule has balanced accuracy $1-O(n^{-1/2})$ but cannot distinguish inequivalent pairs in one fiber.

We prove that continuous $2$-convex solutions of the $\sigma_{2}$ equation with a density bounded above and below by positive constants belong to $W^{2,1}_{\mathrm{loc}}$ in every dimension. If, in addition, the density is continuous, the solutions belong to $W^{2,p}_{\mathrm{loc}}$ for every $1<p<\infty$.