Papers

We consider the linear operator for metamaterial modeling $$L_{c,d }=(\chi_{\RE^{3}\backslash\overline \Omega}+c)\nabla\!\cdot (\chi_{\RE^{3}\backslash\overline \Omega}+d)\nabla\,,$$ where the material parameters $c>0$ and $d$ are constant on each connected component of the bounded domain $\Omega$, $d$ is negative on at least one component of $\Omega$ and transmission boundary conditions hold at the interfaces. It turns out that $L_{c,d }$ has, as in the sign-definite case, an absolutely continuous spectrum equal to $[0,+\infty)$; however, contrary to the latter, it has a non-empty point spectrum; in particular, we find conditions on the parameters so that zero is a threshold eigenvalue. In the case $\Omega$ is connected, we study how such a threshold eigenvalue behaves under $\varepsilon$-small analytic perturbations of the parameters. We show that, depending on the position of the number $\frac12\,\frac{1+d(\ve)}{1-d(\ve)}$ relative to the nearest eigenvalue of the Neumann-Poincaré operator of the interface, the zero eigenvalue splits into two branches (analytic in $\varepsilon^{1/2}$), consisting either of two branches of resonances, or of one branch of negative eigenvalues and one of resonances.

Let $G$ be a group and let $A$ be a set with at least two elements. A lazy cellular automaton over $A^G$ acts as the identity except at the cells where a fixed pattern occurs, where it replaces the central symbol by a fixed different symbol. Let $M(G;A)$ be the submonoid of $\mathrm{CA}(G;A)$ generated by the lazy and the invertible cellular automata. We study when $M(G;A) = \mathrm{CA}(G;A)$. Since no lazy cellular automaton is surjective, every surjective element of $M(G;A)$ is invertible; hence $M(G;A) \neq \mathrm{CA}(G;A)$ whenever $A$ is infinite or $G$ has an element of infinite order. Moreover, if $|A| = 2$ and $G$ has a quotient isomorphic to the Klein four-group $C_2 \times C_2$, then $M(G;A) \neq \mathrm{CA}(G;A)$. For locally finite $G$, and in particular for finite $G$, these are the only obstructions: we prove that $M(G;A) = \mathrm{CA}(G;A)$ if and only if $3 \leq |A| < \infty$, or $|A| = 2$ and $G$ has no quotient isomorphic to $C_2 \times C_2$.

In the past two decades, the Bayesian approach to inverse problems in infinite dimensional spaces has seen significant developments in theory, Markov chain Monte Carlo (MCMC) methods, and in applications. The Bayesian approach in finite dimension produces posterior distributions of finite dimensional random variables. In infinite dimensions, the desired objects are random fields and their laws as measures on Hilbert or Banach spaces. There are many textbooks and tutorials available that explain Bayesian methods and MCMC in finite dimensions. To enter the field of Bayesian inverse problems in infinite dimensions, a newcomer has to engage with a quite diverse and extensive literature. The purpose of this article is to provide a basic path to the subject that is easily accessible to a reader with general knowledge in probability and mathematical analysis. We start with a swift introduction to the finite dimensional case. The infinite dimensional case is introduced for Gaussian priors on Hilbert spaces and Gaussian-type likelihood functions. In this situation, mere continuity of the forward map suffices to prove that the posterior measure is well-defined. We also give some new and simple stability results. Two application examples are elaborated, one finite dimensional, one infinite dimensional.

We rigorously derive a dynamic membrane theory for nonlinear viscoelastic solids with Kelvin-Voigt rheology from the corresponding three-dimensional bulk theory, in the limit of vanishing thickness. The bulk model accounts for inertia, a nonsimple elastic energy depending on the second deformation gradient, and a frame-indifferent dissipation. Orientation preservation is ensured by the energy, while self-interpenetration of the limiting membrane is prevented by a uniform averaged invertibility condition. Our analysis relies on a hyperbolic minimizing movements scheme with two time-scales, which recasts the evolution into a hierarchy of incremental, sequential, and time-delayed problems. For fixed thickness, we prove the existence of solutions to the scheme and their convergence to a weak solution of the continuous-time problem as the time-scales vanish. We then identify the membrane limit at each level of the hierarchy, proving the $\Gamma$-convergence of the incremental functionals and the convergence of solutions to the sequential and time-delayed problems. Since the constant in the Korn inequality is not uniform in the thickness, these limits rely on estimates for the time derivative of the Cauchy-Green tensor rather than for the velocity gradient. Finally, as the outer time-scale vanishes, the membrane time-delayed solutions converge to a measure-valued solution of the membrane continuous-time problem.

We prove that every Lie group can be realized by permutations of a countable set, answering a classical question of Ulam. The key deformation construction originated in an argument produced by OpenAI's GPT 6 Astra during extended mathematical discussions.

We introduce the leap group of a graph, the subgroup of the symmetric group on the vertex set generated by graph derangements. We summarize historical results on graph derangements, which we call hops, and prove additional existence and complexity results. The leap groups of various canonical graphs are computed, and their behavior under basic graph operations is analyzed. We offer partial results on which groups can appear as leap groups, and study hops and leaps for infinite and directed graphs.

In this paper, we study a new variant of the ants walk: a model of reinforced random walks for the phenomenon of a colony of ants being able to find the shortest path between their nest and a source of food. In this model, the nest and the source of food are two distinguished nodes in a finite graph. The ants perform successive random walks on this graph, started at the nest and stopped when they first hit the food source; the transition probabilities of each random walk depends on the trajectories of the previous ants, through some edge-weights that represent the amount pheromones deposited on each edge. In our new variant of the model, the ant (1) forgets loops made on the way to the food as soon as they are made, and (2) comes back to the nest following this loop-erased version of its forward trajectory, and deposits pheromone on the edges it crosses by doing so (i.e. adds one to these edges' weights). We say that the ants "find the shortest path" if, almost surely as the number of ants tends to infinity, the proportion of ants that travel along a graph-geodesic path from the nest to the food tends to one and the proportion of ants that travel along the remaining edges tends to zero. We are able to prove that this new variant of the model finds the shortest path when the graph distance between the nest and the source is at most 2. The strength of our result is that there is no other assumption on the shape of the graph. We prove this result using stochastic approximation theory, the electrical network method for random walks on weighted graphs, and the theory of weighted uniform spanning trees (WUSTs). In particular, the link with WUSTs is unique to the model studied in this paper due to the way the ant erases loops on the outward journey.

Automating AI research creates a positive feedback loop between software capability and research effort. We study a stochastic realisation of this loop in which independent, positive, multiplicative improvements arrive at a rate proportional to a power of current capability. Within this model, randomness leaves the deterministic explosion threshold unchanged but replaces the blow-up date by an explicit distribution. For equal-sized improvements the explosion time is a geometrically weighted sum of independent exponentials; its $q$-series representation gives its transforms, moments and tail behaviour, including a two-term small-time asymptotic. For Lomax-distributed improvement sizes the explosion time is exactly Gamma distributed. Self-similarity implies that, with parameters known, the relative uncertainty about the remaining time does not vanish as explosion approaches. We formulate a fixed-sample test between constant and geometrically increasing idea-arrival rates. If the baseline rate and the alternative log-rate increment $\ell$ are known, the minimum sample size at error level $\alpha$ is asymptotic to $(12z_\alpha^2)^{1/3}\ell^{-2/3}$, where $z_\alpha$ is the upper-$\alpha$ standard normal quantile. Uncertainty about the growth exponent near its critical value produces a $1/t$ tail with an explicit constant, and hence an infinite conditional mean waiting time, although explosion is certain given a positive exponent. Separate extensions show that a fixed deployment delay prevents explosion in the specified delayed-feedback models, that power-law diversion of effort shifts the growth threshold, and that in a deterministic oversight race a fixed allocation share does not change the asymptotic regime, except at a critical equality of exponents. Numerical illustrations, which are not empirical forecasts, show how timing and detectability depend on the granularity of improvements.

The Sachdev–Ye–Kitaev (SYK) model is a quantum many-body model of Majorana fermions with all-to-all random interactions. It has attracted wide interest in condensed matter and high-energy physics as a solvable model of non-Fermi liquids and holography. Fix an even integer $q\geq4$, and let $Z_N(\beta)$ be the partition function of the Gaussian $q$-body SYK model. We prove that for all sufficiently small inverse temperatures $\beta > 0$, there exists $\rho=\rho(\beta)>0$ such that \[ N^{q/2 -1}(\log Z_N(\beta)-\mathbb{E}[\log Z_N(\beta)]) \Longrightarrow \mathcal{N}(0,\rho^2). \] In addition to the mathematical results, we discuss how the proofs were developed and refined through interaction between human researchers and AI, and we describe the subsequent formalization of the main arguments in Lean 4. This work thus serves as a case study of a research workflow that integrates human insight, AI-assisted exploration, iterative proof refinement, and formal verification.

This paper deals with the optimal control problem for the general mean-field stochastic differential equations, in which the coefficients depend on both the state process as well as on its law. We focus on the connection between the general maximum principle and the dynamic programming principle for such a control problem. We establish the relationship between the value function of the mean-field optimal control problem and the adjoint equations in the maximum principle. We also give a general mean-field stochastic verification theorem. Some examples are solved explicitly to illustrate our results.

We prove that, for every $c>0$, the image of the generalized Markov-Krein transform on probability measures with finite logarithmic moment is closed under classical additive convolution and under the Mellin convolution. This resolves the positivity conjecture for the additive high-temperature convolution, a one-parameter interpolation between classical and free convolution arising in random matrix theory and representation theory in the regime $N\to\infty, N\beta_N/2\to c$. Our proof relies on a convolution theorem for universally prestarlike functions.

Let $T$ be a bounded linear operator on a complex Banach space $X$. We prove that the algebraic decomposition \[ X=\K(T)\oplus \Hq(T) \] forces both summands to be closed and hence characterizes generalized Drazin invertibility. This improves the classical characterizations of Mbekhta~\resultcite{Theorem~1.6}{MbekhtaKato}, Schmoeger~\resultcite{Theorem~4}{Schmoeger}, and Dajić and Koliha~\resultcite{Theorem~6.7}{DajicKoliha}, by removing their closedness assumptions. We also show that $T$ is generalized Drazin invertible if and only if \[ \dim\bigl(\K(T)\cap \Hq(T)\bigr)<\infty \quad\text{and}\quad \codim\bigl(\K(T)+\Hq(T)\bigr)<\infty. \] For ordinary Drazin invertibility, we obtain analogous finite-defect characterizations involving the hyperkernel and the hyperrange. We also give a characterization using pointwise nilpotence on an invariant summand. Finally, we consider arbitrary invariant algebraic decompositions and give a counterexample when the inverse of the restriction to one summand is not assumed bounded.

We construct smooth functions on the flat torus $\mathbb{T}^{n\ge5}$ that change sign and have negative mean but cannot be realized as the scalar curvature of any metric in the given conformal class.

In this paper, we give a constructive proof that for any integer n>7 (resp. n>6), the numbers 1,2,...,n can be arranged in a circle (resp. in a line) such that the sum of the reciprocals of all adjacent sums equals 1. The above conjecture was originally posed by Zhi-Wei Sun. Partitioning the integers according to n mod 12 into nine residue classes, we prove a recursive proposition and inductively construct new examples. Main lemmas and propositions in introduction are proved in Section 2. The rest of the cases are handled by explicit constructions in appendix.

In this paper, we propose an adaptive variance-reduced stochastic sequential (quadratic constrained) quadratic programming (AVR-SS(QC)QP) method for nonlinear constrained optimization. The framework integrates SPIDER-based variance reduction, stochastic quasi-Newton curvature information, merit-function globalization, and adaptive switching between stochastic SQP and SQCQP models. The switching mechanism is governed by a relative constraint model curvature indicator that measures the second-order constraint information neglected by the linearized SQP model. A smoothed hysteresis rule ac?tivates the SQCQP model only when the relative constraint curvature is significant. An exact L1 merit function with Armijo backtracking globalizes the resulting directions, while a coupled stochastic Lyapunov analysis controls the interaction between gradient-estimation error and optimization progress. Under standard smoothness, sampling, regularity, and second-order conditions, we establish almost-sure consistency of the variance-reduced gradient estimator and global convergence of the KKT residual. We further show that, near regular KKT points, the curvature indicator vanishes and the method eventually selects the computationally cheaper SQP model. Numerical experiments on a sparse principal component analysis problem illustrate the practical be?havior of the proposed framework.

In this paper, we introduce a concept of modified curvature measures for h-convex domains in hyperbolic space and prove a shifted version of the generalized Alexandrov theorem, which characterizes geodesic balls in hyperbolic space via modified curvature measures. This result generalizes the classical work of Kohlmann (1998). We further investigate a volume-preserving shifted $\alpha$-Gauss curvature flow for strictly h-convex hypersurfaces in hyperbolic space and prove that for any $\alpha>0$ the solution exists for all time and converges smoothly to a geodesic sphere. The theory of modified curvature measures developed in the first part serves as a key ingredient in characterizing the Hausdorff limit shape of the flow.

We study a one-dimensional single-excitation model of a scalar photon field coupled to a continuum of two-level atoms with quadratic density. The resulting matrix Hamiltonian has an unbounded spatially varying coupling and is not semibounded. We prove that it is essentially self-adjoint and determine its spectrum explicitly. The continuous spectrum is the half-line $(-\infty,\Omega]$, where $\Omega$ is the atomic resonance frequency, while the spectrum above $\Omega$ consists of a strictly increasing sequence of simple eigenvalues. These eigenvalues are characterized by the interlaced zeros of the Airy function and its derivative and satisfy \begin{align*} \omega_n = \left(\frac{3\pi g n}{4}\right)^{1/2} + \frac{\Omega}{4} + O(n^{-1/2}). \end{align*} The analysis is based on a nonlinear spectral reduction that yields a confining massless relativistic oscillator above $\Omega$ and an inverted oscillator below $\Omega$.

In this work, we develop low-cost non-iterative Sylvester solvers for dynamical low-rank approximation of reaction-diffusion equations and incorporate them into a basis-update & Galerkin mass-lumped finite element method (\mbox{BUG-MLFEM}). By incorporating mass lumping via Gauss-Lobatto quadrature, the spatial discretization yields a matrix differential equation equipped with diagonal positive-definite mass matrices. We establish the weighted dynamical low-rank approximation (WDLRA) within the Galerkin framework of the MLFEM system, proving the theoretical equivalence between the BUG integrator and MLFEM on low-rank subspaces, which guarantees the discrete well-posedness of the implicit BUG steps. To overcome the severe computational bottleneck of standard $O(m^3+n^3)$ Sylvester solvers, we develop tailored, low-cost non-iterative solvers that exploit the inherent sparse-by-small algebraic structure. Under Dirichlet and Neumann boundary conditions (BCs), the large coefficient matrices are banded, enabling efficient diagonalization of the small $r \times r$ factor matrix and reducing the per-step computational complexity to $O((m+n)r^2)$ and storage to $O((m+n)r)$. For periodic BCs, we combine this diagonalization strategy with a rank-2 Sherman-Morrison-Woodbury update to handle wrap-around corner entries at negligible additional cost. Numerical experiments demonstrate first-order temporal and $(k+1)$th-order spatial convergence of the proposed BUG-MLFEM in the discrete $L^2$ norm, where $k$ denotes the polynomial degree of the finite element space, together with significant computational speedups over existing low-rank integrators.

The stability zone of Novikov's problem of planar sections of triply periodic surfaces are labeled by indivisible triples of integers, which we call the soul of the zone. Which triples may occur is not known. We show that, if a Morse function $f$ on $\mathbb T^3$ admits a translation $\tau(x)=x+v$ with $f\circ\tau=-f$, so that the sets $\{f<0\}$ and $\{f>0\}$ are congruent, then in the generic genus-$3$ situation every topological number $L$ satisfies $\langle L,2v\rangle\equiv1\pmod2$. The mechanism is that the electron/hole type of a null-homologous closed trajectory is invariant under the period lattice and is reversed by $\tau$. For $f=\cos2\pi x+\cos2\pi y+\cos2\pi z$ and the $\mu$-cube, this means that the sum of the components of each soul is odd.

We prove an analogue of Min-Oo's conjecture for sufficiently small spherical caps in the unit round sphere. For every dimension $n\ge3$, there exists $r_*(n)>0$ such that the following holds for every round cap of radius $0<r\le r_*(n)$: any smooth compact connected oriented Riemannian fill-in with the same intrinsic boundary metric and with scalar curvature and outward boundary mean curvature at least those of the cap is isometric to the cap.

The $\mathbb{Z}_{p^s}$-additive codes are subgroups of $\mathbb{Z}_{p^s}^n$. A $\mathbb{Z}_{p^s}$-linear generalized Hadamard (GH) code is a GH code over $\mathbb{Z}_p$ which is the Gray map image of a $\mathbb{Z}_{p^s}$-additive code. A recursive construction gives such codes $H_p^{t_1,\dots,t_s}$, of type $(n;t_1,\dots,t_s)$ and length $p^t$, and a partial classification of them is known. In this paper, we complete the classification of these codes for every prime $p$. The main tool is the derived code, obtained by puncturing the code outside a block of coordinates determined by its kernel. For a nonlinear code $H_p^{t_1,\dots,t_s}$ with $t_1\geq2$, the derived code is a $\mathbb{Z}_{p^{s-1}}$-linear GH code of the same family, although the mechanism for odd $p$ differs from the binary one. As a consequence, for $t\geq3$, there are $\mathcal P(t+1)-\mathcal P(t)$ nonequivalent such codes of length $p^t$ if $p$ is odd, and $\mathcal P(t+1)-\mathcal P(t)-1$ if $p=2$, where $\mathcal P(n)$ is the number of partitions of $n$. In particular, this completes the classification of the $\mathbb{Z}_{2^s}$-linear Hadamard codes. We also prove a conjecture on the codes with $t_1\geq2$ and $t_s=0$.

Given complex separable Hilbert spaces $\mathcal{H}_i$ and bounded operators $T_i$ on $\mathcal{H}_i$, for $i=1,\ldots,n$, we consider the tensor product space $\mathfrak{H}:=\mathcal{H}_1\otimes\cdots\otimes\mathcal{H}_n$ and a tuple $(\tilde{T}_1,\ldots,\tilde{T}_n)$ on $\mathfrak{H}$ where \[\tilde{T}_i=I_{\mathcal{H}_1}\otimes\cdots\otimes I_{\mathcal{H}_{i-1}}\otimes T_i\otimes I_{\mathcal{H}_{i+1}}\otimes\cdots\otimes I_{\mathcal{H}_n}.\] A closed jointly $(\tilde{T}_1,\ldots,\tilde{T}_n)$-invariant subspace $\mathfrak{M}$ of $\mathfrak{H}$ is said to be doubly commuting if the tuple $(\tilde{T}_1|_{\mathfrak{M}},\ldots,\tilde{T}_n|_{\mathfrak{M}})$ is doubly commuting. In this paper, we show that if the hereditary polynomials in each $T_i$ are dense in $\mathcal{B}(\mathcal{H}_i)$ with respect to the weak operator topology for $i=1,\ldots,n$, then doubly commuting jointly $(\tilde{T}_1,\ldots,\tilde{T}_n)$-invariant subspaces $\mathfrak{M}$ of $\mathfrak{H}$ are characterized in the form \[\mathcal{M}_1\otimes\cdots\otimes\mathcal{M}_n,\] where $\mathcal{M}_i\subseteq\mathcal{H}_i$ is $T_i$-invariant, for each $i=1,\ldots,n$. As applications, we provide a unified framework for understanding and characterizing doubly commuting quotient modules of weighted Bergman spaces over the open unit polydisc, doubly commuting quotient modules of tensor products of complete Nevanlinna-Pick spaces, and doubly commuting invariant and co-invariant subspaces of certain operators acting on tensor products of de Branges-Rovnyak spaces $H(b)$ associated with both extreme and nonextreme points $b$. This recovers, through a different approach, the known characterizations of doubly commuting quotient modules of the Hardy space over the open unit polydisc and doubly commuting submodules of Jordan blocks of Hardy space over the open unit polydisc.

Finite convexly ordered binary branching $C$-relations have been viewed as a notable exception to the broad applicability of the partite construction in structural Ramsey theory. The absence of a proof of their Ramsey property by the partite construction was noted by Bodirsky in his 2015 survey and pointed out again by Hubička and Konečný in their 2026 survey. We overcome this limitation by giving proofs based on the partite construction of the known Ramsey properties of the class $\mathcal{CT}$ of finite convexly ordered treeable meet semilattices and of the classes of finite convexly ordered $C$-relations with arbitrary branching and with binary branching. Our proofs do not rely on Milliken's tree theorem. The key is to expand $\mathcal{CT}$ by level predicates and a predecessor function so that the locally finite subclass method applies. We then transfer the resulting Ramsey witnesses to the corresponding reduct classes and to the classes of convexly ordered $C$-relations.

The continuous-time mean-variance portfolio selection problem is investigated in this paper through a rigorous integration of backward stochastic differential equation (BSDE) and Kantorovich duality theory. We propose a novel analytical framework for addressing feedback control in McKean-Vlasov stochastic systems. By solving three coupled systems of ordinary differential equations (ODEs), we derive an explicit, closed-loop optimal feedback investment strategy.

We compute the local and global Denef–Loeser topological zeta functions of the determinant on the space of symmetric matrices. It verifies the strong monodromy conjecture of the symmetric determinant function. The computation relates to the geometry of complete quadrics and wonderful embeddings.

In this paper, we improve the upper bound for the tensor rank of the $n \times n$ determinant tensor to $\mathcal{B}_n-2^{n-1}+n+1$ by finding a new tensor decomposition of the tensor, where $\mathcal{B}_n$ denotes the $n$-th Bell number. The main idea is to replace a part of the tensor decomposition suggested by Houston–Goucher–Johnston with a tensor that is essentially the same as the $n \times n$ permanent tensor, and to use Glynn's formula. This upper bound gives an improved upper bound for the Waring rank of the $n \times n$ determinant polynomial for $n \geq 17$ as well.

We study coding multiplicity for self-similar sets on the real line. When the similarity dimension exceeds one, almost every point with respect to the natural self-similar measure has infinitely many codings. We focus on the case when the similarity dimension is at most one. Our analysis proceeds through the recurrence properties and the size of the critical set, defined as the union of the pairwise intersections of the first-level pieces. Our main structural result shows that when the critical set is finite, unboundedness of coding multiplicity is equivalent to periodic recurrence to the critical set. Moreover, every commensurable system with a finite critical set has uniformly bounded coding multiplicity. This conclusion is sharp in the incommensurable setting. We construct a one-parameter family of four-map systems such that for Lebesgue almost every parameter, the critical set is a singleton and the open set condition fails, while the attractor contains a point with infinitely many codings. More generally, for transversal families, the set of parameters admitting an infinite coding fibre is dense among those for which the strong separation condition fails. We also obtain a quantitative lower bound for the Hausdorff dimension of this parameter set. Such parameters can form a meagre set. We continue the study of the geometric size of branching, obtaining dimension formulae for typical coding fibres and almost-sure upper bounds on the dimension of the critical set in transversal families. In contrast, we construct an exponentially separated system with no dimension drop whose critical set has the same Hausdorff dimension as the attractor and whose coding map has a fibre of continuum cardinality. Although we work in the one-dimensional setting, several results and arguments extend to higher dimensions.

In 1994, Friedman, Joichi and Stanton conjectured that coefficients of the power-series $q+(1-q)\prod_{i=n}^{2n-1}\frac{1}{1-q^{i}}$ are nonnegative for $n\geq3$ and $n$ being odd. Let $\qbinom nm$ denote the Gaussian polynomial. In 2003, Prellberg and Stanton conjectured that coefficients of the power-series $(1-q)\qbinom nm\prod_{i=0}^{m-1}\frac{1}{1-q^{i+m}}$ are nonnegative for $0<m<n$, except when $n$ is even and $m\in\{2,n-2\}$. The latter conjecture implies the former. In this paper, we prove a stronger result concerning coefficientwise nonnegativity of another power series, which implies the conjecture of Prellberg and Stanton. The proof uses a Möbius correction from Gaussian polynomials, weight-preserving injections on partitions, explicit linearly independent vectors in the kernel of a derivation operator and the KOH identity.

Time-inconsistent stochastic control admits a natural game-theoretic interpretation in which successive decision dates are represented by different selves. In the Markovian framework of Björk-Khapko-Murgoci, the corresponding continuous-time equilibrium HJB system can be motivated by discrete subgame-perfect games, but the general discrete-to-continuous passage is not automatic. We establish a quantitative convergence theorem for a smooth uniformly elliptic class in which the control may enter both drift and diffusion. Using the exact sample-and-hold transition law on each time step, we construct, near a verified continuous equilibrium branch, a unique regular discrete Markov subgame-perfect equilibrium and prove first-order convergence of the continuation objects in an anisotropic $C^{2+\beta}$ topology and of the equilibrium policies in $C^0$. The key stability mechanism is structural: the discrete first-order condition forces the differential feedback correction in the frozen linearization to be nilpotent, which yields mesh-uniform block estimates through a two-norm dyadic Schauder argument despite the second-order coupling caused by controlled volatility. We also formulate a conditional transfer principle for structured polynomial-growth classes, with error-space stability stated as a separate hypothesis.

In 1984, Bressoud and Subbarao established an interesting weighted partition identity connecting the partition function and the divisor function. Recently, the first author, jointly with Bhoria, Eyyunni, and Maji, extended the work of Bressoud and Subbarao and obtained the following analogous weighted partition identity. For any $n \in \mathbb{N}$, they obtained \begin{equation*} p^{(2)}(n) = \sum_{\pi \in \mathcal{D}(n)} (-1)^{ \# (\pi) }s(\pi) \left( \ell(\pi) - s(\pi) \right). \end{equation*} In the present paper, we provide a combinatorial proof of the above identity and extend the result for $p^{(k)}(n)$, where $p^{(k)}(n)$ denotes the number of partitions of $n$ into exactly $k$ distinct part sizes.