Papers

This paper is concerned with propagation phenomena of spatially periodic combustion reaction-diffusion equations in exterior domains. It is known that there is a pulsating front connecting 0 and 1 with positive speed in $\mathbb{R}^N$ for any direction $e\in\mathbb{S}^{N-1}$. We first prove that there exists an entire solution originating from a pulsating front in the exterior domain. Then, we prove that the entire solution propagates completely. Additionally, by constructing appropriate super- and sub-solutions, we establish that the entire solution is a transition front connecting 0 and 1, and that it is trapped between two translates of the pulsating front as $t\rightarrow +\infty$. Finally, under a suitable assumption, we show that the entire solution converges to the same pulsating front as $t\rightarrow +\infty$, as well as the uniqueness of such entire solutions.

We study the structure and regularity of (infinitesimally) invariant measures of the solutions to stochastic differential equations $dX_t = b(X_t)\,dt + dZ_t$, where $(Z_t)_{t\geq 0}$ is a Lévy process. We show, in particular, that the invariant measure has to satisfy a Volterra-type convolution equation; since we can obtain the kernels explicitly, we are able to apply regularity methods from harmonic analysis. As an application, we get a very short proof – in any dimension – of a classic result due to Sato and Yamazato on the form of the invariant measure of a generalized Ornstein–Uhlenbeck process.

This paper presents a dual-algebraic framework for control barrier functions (CBFs) that guarantees deterministic execution using exclusively elementary arithmetic. We develop this deterministic approach to solve a fundamental bottleneck in safety-critical control: pointwise minima compose intersecting safe sets, but generate non-smooth boundaries where standard Lie derivatives fail. Existing mathematical workarounds inject approximation bias, probabilistic non-determinism, or combinatorial execution delays that impede hard real-time hardware certification. By embedding the system state and vector field into the dual-number ring, our method extracts both the composite barrier value and its exact directional derivative in a single evaluation. The standard floating-point minimum deterministically isolates a single vertex of the Clarke generalized gradient for the quadratic-program solver. We prove this selected vertex constitutes a Clarke subgradient and the resulting simultaneous-enforcement safety filter guarantees forward invariance. The arithmetic overhead remains a fixed constant factor, independent of state dimension and constraint count. We extend this framework to finite $\min$/$\max$ Boolean compositions, for which enforcement of the routed constraint of each $\delta$-active clause guarantees forward invariance, and to systems of higher relative degree, for which a bivariate truncated-dual evaluation extracts the control coupling without symbolic differentiation. Three numerical examples illustrate the computational scaling.

Physics informed neural networks (PINNs) incorporate Maxwell's equations, initial conditions, boundary conditions, and measurement data directly into the learning process, transforming the solution of partial differential equations into a constrained optimization problem. As such, PINNs are attracting increased attention in computational electromagnetics as an alternative to traditional time-domain solvers. This paper presents a PINN formulation for the time-domain Maxwell's equations incorporating split-field perfectly matched layers (PMLs). A key advantage of the split-field PML formulation is that the same governing equations can be applied in both the physical and PML regions, simplifying the PINN formulation and loss construction. The proposed approach is validated using one-dimensional and two-dimensional Gaussian pulse problems with PML. The PINN solutions show good agreement with analytical and finite-difference time-domain (FDTD) reference solutions, demonstrating the feasibility of combining PINNs with split-field PMLs for open-domain time-domain electromagnetic simulations.

We prove a finite-scale coarse-grained decomposition for the three-dimensional incompressible Navier-Stokes equations near the Caffarelli-Kohn-Nirenberg local regularity framework. The first part is a local resolution lemma for the scale-critical quantity: for every spatial filter length ell > 0, Psi(r) <= 4 Psi^ell(r) + 4 Omega^ell(r), where Psi^ell(r) is the corresponding coarse-grained velocity-pressure quantity and Omega^ell(r) is the explicitly defined subfilter residual. Thus a CKN-bad scale is either visible at the resolved level or is carried by unresolved velocity-pressure oscillation. The second part is an exact fixed-chain depletion theorem for the combined pressure-flux work distribution G^ell = Pi^ell + div(P^ell U^ell), Pi^ell = -R^ell : grad U^ell, which is the signed work density appearing in the localized resolved-energy balance. For finite-dimensional active test families with common endpoint traces, we obtain a constructive active-work extraction and a weighted telescoping inequality: forward combined work and resolved dissipation are paid by the initial localized kinetic energy, explicit localization leakage, and negative combined work/backscatter.

A graph $G$ is $\mathcal F$-saturated if $G$ contains no member of $\mathcal F$, but the addition of any non-edge creates a copy of a member of $\mathcal F$. For $m\ge 1$, let $(m+1)K_3$ denote the vertex-disjoint union of $(m+1)$ triangles. In this paper, we study $(m+1)K_3$-saturated graphs. We construct a family of $(m+1)K_3$-saturated graphs which gives the uniform upper bound $\operatorname{sat}((m+1)K_3,n)=O(n^{3/2})$ for all $m\ge 1$ and $n\ge 3m+3$. For general $(m+1)K_p$-saturated graphs, Faudree et al. determined $\operatorname{sat}((m+1)K_p, n)$ for sufficiently large $n$. In the case of $p=3$, we lower Faudree's threshold from $n \ge 12m + 3$ to $n\ge 9m+5$ and prove that $\operatorname{sat}((m+1)K_3,n)=n+6m-1.$ We also provide two structural restrictions on the components obtained after deleting all vertices of degree $|V(G)|-1$ from an $(m+1)K_3$-saturated graph.

The performance of gradient methods depends critically on step-size selection. Spectral step-size design involves constructing candidates from finite history and ordering them during iteration. We address both through a finite-moment spectral framework. For candidate construction, we relate Huang–Dai–Liu determinant pencils, limited-memory steepest-descent Krylov–Ritz extraction, pseudo-memory moment realization, and Gu–Du recovery of spectral nodes and component weights. The framework distinguishes generalized-eigenvalue recoverability, finite-window implementability, and component-weight interpretability. Under explicit positivity, rank-one, regularity, and exact-recovery assumptions, these properties become compatible in a common finite-moment regime, with the long, short, and mean LMSD constructions as representatives. For ordering, we identify spectral rebound: a step targeting a lower spectral scale may reactivate previously reduced high-frequency components. Idealized one-target, block, and recomputed memory-\(m\) models yield phase-count recurrences motivating a high-frequency-first strategy with spectral refreshes. We propose a rebound-aware spectral gradient method (RASG) that ranks Ritz candidates by frequency and estimated strength, and sets phase duration via a Ritz-uncertainty estimator. Two variants, RASG-KT and RASG-LMSD, use a Kato–Temple-inspired estimator and a long/short LMSD discrepancy. For strictly convex quadratics, global \(R\)-linear convergence holds under an admissibility safeguard. Experiments on ill-conditioned quadratic problems and two large-scale convex problems illustrate the behavior.

We prove a residual-work compatibility theory for positive-cone viscoelastic Reynolds states. In Oldroyd–B, the entropy cancellation eliminates incompressible transport and upper-convected stretching against polymeric stress work. After quotienting pressure tensors and spatial means, positive pressure-free residual work can be paid only by the conformation residual through the entropy-dual lever (G=I-A^{-1}). This yields the closed residual-work cone (P+(\alpha/2)\int_{\mathbb T^d}G:S,dx\le 0), exact windowed tests, and the least-cost Hilbert-space repair; constrained closures pay through the projected lever. Strong residual-data limits preserve this cone, while weak–weak limits require the product-defect measure carried by (G:S). The augmented topology is sharp, as shown by localized residual packets. For FENE-P, the lever (G_b(C)=((b-d)/(b-\operatorname{tr}C))I-C^{-1}) makes the finite-extensibility boundary a genuine residual-work boundary.

Let $C$ be the middle-third Cantor set and $\mu$ be the Cantor-Lebesgue measure on $C$. A conjecture of Velani states that $\mu(W_2(\tau))=0$ if $\tau>1$ and $\mu(W_2(\tau))=1$ if $0<\tau\leq 1$, where $W_2(\tau)=\left\{x\in[0,1]: \|2^nx\|<n^{-\tau}\ {\rm for\ infinitely\ many }\ n\in\mathbb{N} \right\}$. We prove that the conjecture holds for $\tau>\frac{1}{\gamma}-\frac{1-\gamma}{3-\gamma}\,(\approx 1.429)$ and $0<\tau<\frac{\gamma}{12}\,(\approx 0.052)$, where $\gamma=\frac{\log2}{\log3}$ is the Hausdorff dimension of $C$. This improves the known results on both the null part ($\tau>\frac{1}{\gamma}-\frac{0.078(1-\gamma)}{\gamma(2-\gamma)}\approx 1.552$, due to Allen, Baker, Chow, and Yu (2023)) and the full measure part ($0<\tau\leq 0.01$, due to Baker (2025)). Our key innovation is to establish the estimate \[\sum_{n=1}^{N}|\widehat{\mu}(h2^n)|^2\ll N^{1-\gamma}\] and its consequences: \[ \sum_{n=1}^{N}|\widehat{\mu}(h2^n)|\ll N^{1-\frac{\gamma}{2}},\quad \sum_{n=1}^{N}n^{- \sigma}|\widehat{\mu}(h2^n)|\ll_{\sigma} N^{1-\frac{\gamma}{2}-\sigma},\] where $0<\sigma<1-\frac{\gamma}{2}$, and all estimates are uniform in $h\in\mathbb{Z}\setminus\{0\}$. For the full measure part, our approach also generalizes to self-similar measures on a class of missing-digit sets.

We establish a connection between the part-whole principle and the quantity $\text{ded }\kappa$ – a generalized cardinal characteristic related to the number of Dedekind cuts of a linear order. As consequences, we improve a result of Mancosu and Massas on generalized probability functions and propose some questions.

A subset $S$ of a group $G$ is conjugacy distinguished if the union of all conjugates of $S$ is closed in the profinite topology on $G$. We prove that if $M = \mathbb{H}^3/\Gamma$ is a hyperbolic $3$-manifold of finite volume, $g \in \Gamma$, and $H$ is an abelian subgroup of $\Gamma$, then the coset $gH$ is conjugacy distinguished in $\Gamma$. A subset $S \subset G$ is conjugacy distinguished from a class of subgroups if, for every $K$ in the class that is disjoint from the union of conjugates of $S$, there exists a homomorphism $\varphi \colon G \rightarrow F$, where $F$ is a finite group, such that $\varphi(K)$ is disjoint from the union of conjugates of $\varphi(S)$. In previous work, we proved that if $M = \mathbb{H}^3/\Gamma$ is a hyperbolic $3$-manifold of finite volume, then a coset of a maximal parabolic subgroup with cusp $C$ is conjugacy distinguished from the class of maximal parabolic subgroups of $\Gamma$ with cusps distinct from $C$. We extend this result by proving that a coset of a loxodromic subgroup is conjugacy distinguished from the class of maximal parabolic subgroups of $\Gamma$.

Size-selective harvesting is often justified by the intuition that larger individuals have higher value and should therefore be harvested above a critical size. We study a controlled size-structured transport model in which a scalar environmental variable, generated by the population itself, modifies the vital rates \(g(E,l)\) and \(\mu(E,l)\). The first result is a corrected stationary closure theory: crowding-suppressed growth increases residence density at the inflow, so the closure derivative is not determined by pointwise profile monotonicity. Instead, $\Phi'(E)=\mathsf A(E)-\mathsf C(E)$, an integrated balance between residence-time amplification and cumulative survival loss. The second result is an exact stationary adjoint reduction. The nonlocal switching correction has rank one, $S=S_{\rm red}-\frac{A}{1-B}\psi$, and its zero-discount feedback gain satisfies the identity $B(0)=\Phi'(E^*)$. Thus the same scalar governs stationary closure sensitivity and threshold fragility. We also establish finite-horizon well-posedness and compactified optimal-control existence in a spatial-\(BV\) policy class. Numerical certification in a density-dependent von Bertalanffy model shows when minimum-size harvesting persists and when vital-rate feedback creates multiple-switch harvest windows.

Information measures acquire operational meaning through coding theorems. The quantum conditional mutual information (QCMI) is nonnegative due to strong subadditivity, yet a direct connection with channel coding has remained elusive. In this work, we propose a quantum communication task-conditional quantum communication-that fills this gap. We show that the optimal rate for establishing quantum correlation between two parties, assisted by a third system, is given by half the QCMI. This result naturally extends the classical key generation capacity of Csiszár and Ahlswede to the quantum domain. We place our model within the family tree of quantum protocols and compute the conditional capacity for several example channels. Our results provide new insights for code design in reliable quantum information processing.

We prove endpoint continuation criteria for stress-diffusion-free incompressible viscoelastic flows by working modulo pressure. In two space dimensions, the pressure-free part of any smooth spectral isotropic stress reduces to a single active deviatoric channel (q_1(a,|Y|^2)Y), where (C=aI+Y) and (\operatorname{tr}Y=0). This scalar quotient structure allows a weighted active-deviatoric energy to cancel the top-order coupling between polymer stretching and the divergence of the active stress. On compact conformation windows the resulting high-order estimate depends only on an endpoint velocity clock and a logarithmic conformation norm. For Oldroyd–B this gives continuation of strong two-dimensional solutions under (\nabla u\in L^1_tB^0_{\infty,1}), while for FENE-P it gives continuation under (\nabla u\in L^2_tB^0_{\infty,1}). In both models the compact conformation window and logarithmic bound are derived from the velocity clock and the model barriers, rather than imposed as independent hypotheses. The criteria are formulated in integer Sobolev strong-solution classes and do not assert Leray-type weak-solution or critical-space local well-posedness results. We also identify a static operator obstruction showing the functional necessity of the logarithmic threshold for the pressure-free stress map. In three dimensions the quotient contains an additional residual channel (q_2(Y^2)^\circ), so the exact scalar closure is intrinsically two-dimensional. On prescribed compact windows this residual can be absorbed by viscosity; for Oldroyd–B and FENE-P it vanishes because (q_2\equiv0).

We develop a unified methodology that integrates spectral theory, Riemann-Hilbert problems, and inverse scattering theory for the construction and numerical evaluation of transform pairs associated with linear variable-coefficient partial differential equations. The approach combines analytical formulae with numerical methods for ordinary differential equations and Riemann-Hilbert problems, yielding a hybrid analytical-numerical strategy for working with these transforms. Results are presented for transforms arising in the Dirac equation, demonstrating accurate computations, even in the presence of discontinuous coefficients.

We study iterates of a quadratic $f= x^2+1/c\in K[x]$. If the number of factors of $f^n:=f\circ f \circ ... \circ f$ is bounded by a constant independent of $n$, then $f$ is said to be eventually stable. This paper is an extension to $\mathbb{Q}(i)$ of the paper \cite{evstb}, which considered $f$ over $\mathbb{Q}$. The conjecture "if $f^2$ is irreducible, then $f^n$ is irreducible for all $n$" extends to $\mathbb{Q}(i)$, but due to the lack of a linear ordering on $\mathbb{Q}(i)$, an auxiliary function is involved in a specific $n$ to check. The elusive case of $c\equiv 2 \bmod 4$ (as a $\mathbb{Z}$ equivalence class) is shown to be "stable" over $\mathbb{Q}(i)$, offering more evidence for \cite[Conjecture 1]{evstb}. Stability for $c\equiv 1\bmod 2$ (as a $\mathbb{Z}[i]$ equivalence class) is not as fully handled as over $\mathbb{Z}$, however.

Solving large-scale instances of the Traveling Salesman Problem (TSP) exactly is computationally expensive. Researchers often employ graph sparsification methods to improve computational efficiency. Traditional sparsification methods typically rely on fixed heuristics and fail to fully exploit instance-specific structural information. In this paper, we propose Graph Edge Sparsification (GES), a learning-based sparsification approach for Euclidean TSP. By incorporating geometric structural information and combinatorial optimization technology, our proposed method adaptively generates a sparsification graph for different instances, significantly reducing the graph size and accelerating the solving process. Experimental results demonstrate that our sparsification method can prune up to 95% of edges on the MATILDA dataset, while keeping the solution gap within 1% of the optimal value. Moreover, our approach exhibits strong generalization capability on the TSPLIB benchmark.In some large-scale instances, the pruning rate exceeds 99%, while the optimality gap remains below 1%.

We present a new method for structural sequence analysis grounded in Algorithmic Information Theory (AIT). At its core is the Ladderpath approach, which extracts nested and hierarchical relationships among repeated substructures in linguistic sequences – an instantiation of AIT's principle of describing data through minimal generative programs. These structures are then used to define three distance measures: a normalized compression distance (NCD), and two alternative distances derived directly from the Ladderpath representation. Integrated with a $k$-nearest neighbor classifier, these distances achieve strong and consistent performance across in-distribution, out-of-distribution (OOD), and few-shot text classification tasks. In particular, all three methods outperform both gzip-based NCD and BERT under OOD and low-resource settings. These results demonstrate that the structured representations captured by Ladderpath preserve intrinsic properties of sequences and provide a lightweight, interpretable, and training-free alternative for text modeling. This work highlights the potential of AIT-based approaches for structural and domain-agnostic sequence understanding.

We study reverse Araki-type trace inequalities and log-majorizations beyond the exponent $2$. For arbitrary nonnegative nondecreasing weights, we show that $s=2$ is the sharp dimension-free boundary: for every $s>2$, explicit one-parameter $3\times3$ positive definite examples violate the reverse Liu–Cheng trace inequality and the corresponding dual formulation of Shi–Wei–Wang, whereas the reverse inequality remains valid for every $s\geq1$ in dimension $2$. For power weights, a larger region survives and is bounded by a sharp hyperbola. In normalized variables, for $s>2$, \[ A^{r+s}B^s \succ_{\log}A^r (A^{1/2} BA^{1/2})^s \] holds for all positive semidefinite matrices in every finite dimension if and only if $0\leq r\leq s/(s-2)$; beyond this range, even the associated trace inequality fails for $3\times3$ positive definite matrices. Equivalently, for $0<p\leq q$ and $q>2p$, the sharp condition is $0\leq r\leq pq/(q-2p)$. Combined with the known all-$r$ regime $p\leq q\leq2p$, this completes the reverse log-majorization phase diagram.

We develop a rigorous theory of anisotropic mixed fractional Landau inequalities for rotating compressible fluid flows at high Mach numbers, incorporating Coriolis and centrifugal forces. We introduce rotating fractional Sobolev spaces $\mathcal{W}^{\nu,p}_{\alpha,\Omega}(\mathbb{R}^k)$, which encode directional scaling, fractional dissipation and rotational effects. We prove sharp fractional Landau inequalities with explicit dependence on the Mach number, the rotation rate $|\Omega|$ and the anisotropy vector $\alpha$. Key tools are a rotating Littlewood–Paley decomposition, anisotropic maximal estimates with rotational corrections, and commutator estimates for the Coriolis term. As applications, we establish stability bounds for neural operators approximating rotating compressible flows and derive optimal approximation rates of order $N^{-\nu/d_{\alpha,\Omega}}$, where $d_{\alpha,\Omega}=\sum_i\alpha_i^{-1}+\kappa|\Omega|^{2/\nu}$ is the effective anisotropic–rotational dimension. Our results provide a mathematical foundation for analysing high–Mach rotating flows and for designing physically informed neural architectures.

The binding number, denoted by $\mbox{bind}(G)$, of a graph $G$ is defined as the minimum value of $\frac{|N_G(X)|}{|X|}$ taken over any non-empty subset $X$ of $V(G)$ with $N_G(X)\neq V(G)$. A graph $G$ is said to be $r$-binding if $\mbox{bind}(G)\geq r$. The adjacency matrix of a graph $G$ is denoted by $A(G)$. The largest eigenvalue of $A(G)$ is called the spectral radius of $G$. An $H_b$-factor of a graph $G$ is defined as a spanning subgraph $F$ of $G$ such that for any $v\in V(G)$, $d_F(v)$ belongs to the set $\{1,3,5,\ldots,b-1,b\}$, where $b$ is an even integer with $b\geq2$. This note establishes a sufficient condition to guarantee that a connected $\frac{1}{b-1}$-binding graph $G$ of even order contains an $H_b$-factor based on the spectral radius.

How sparse can a nontrivial solution of a discrete Schrödinger equation be? In this note we study Dirichlet solutions on a finite $d$-dimensional lattice box, allowing an arbitrary real potential, and measure sparsity by the number of lattice sites at which the solution is nonzero (assuming it is nonzero at the origin). Our main result is a dimension-reduction principle: the minimal possible support size cannot decrease when the dimension increases. Consequently, any lower bound proved in dimension $d-1$ automatically yields the same lower bound in dimension $d$. As an application, we obtain a nearly sharp lower bound in four dimensions, matching the best-known two-dimensional constructions up to a logarithmic factor.

What we call q-normed (linear) spaces were introduced (under the name pseudonormed spaces) into constructive analysis by D.L. Johns as a means of handling spaces, such as L-infinity, in which not all elements are constructively normable. We prove a number of q-normed-space analogues/generalisations of standard theorems in the constructive analysis of normed linear spaces, and give examples showing that analogues of two of those theorems are essentially nonconstructive.

We will first discuss the question of which integers $n$ have exactly one group of order $n$, namely the cyclic group $\mathbb{Z}/n\mathbb{Z}$. We will see that these are the integers that are relatively prime to the Euler totient function $\phi(n)$. Then we discuss how many groups there are of order $p^3$ for each prime $p$. We end with a couple of interesting results and conjectures pertaining to groups of squarefree order.

A commutative cancellative monoid is atomic if every nonunit factors into atoms, and an integral domain is atomic if its multiplicative monoid of nonzero elements is atomic. Several weakenings of atomicity have been introduced and studied during the past decade, including near atomicity, almost atomicity, and quasi-atomicity. Although nearly atomic monoids that are not atomic were already known, whether there exist nearly atomic integral domains that are not atomic had remained open. We answer this question affirmatively by constructing an explicit nearly atomic integral domain that is not atomic. We also strengthen the classical Anderson–Anderson–Zafrullah characterization of the finite factorization property by proving that an integral domain is an FFD if and only if it is both nearly atomic and IDF. We conclude by showing that near atomicity cannot be weakened to almost atomicity in this characterization, even within the class of IDF domains.

Let $W^T$ map Dirichlet data $f$ to time $T$ state $u^f(T, \cdot)$, we study the $K$-normal operator $C^T_K:= (W^T)^*KW^T$ where $(W^T)^*$ is the standard $L^2(dx)$ adjoint. We prove that it is a locally finite sum of Fourier Integral Operators (FIOs) away from the glancing directions with canonical relation involving collision data at time $T$ between pairs of velocities. However, if the operator is $M$-self-adjoint, then the canonical relation for $C^T_M$ loses the collision data. Thus, when system satisfies certain coupling requirement, we demonstrate off-polarization leakage detection and introduce a new collision rigidity problem. Combine these two parts, we prove an inverse problem of velocity recovery from $C^T_I$ for multi-velocity wave models and variable coefficient isotropic elasticity system, where $I$ is the identity matrix.

In this paper, we show a zero-Hopf bifurcations in a four-dimensional hyperchaotic Qi system. While the hyperchaotic dynamics of this model have been extensively investigated, the existence and bifurcation of zero-Hopf equilibria have not been previously analyzed. We first characterize the equilibrium points and determine the parameter conditions under which the origin becomes a zero-Hopf equilibrium. Using second-order averaging theory, we establish the existence of up to four small amplitude periodic solutions bifurcating from the origin under suitable parameter perturbations. We also provide analytical expressions for the initial conditions of these solutions.

Let us denote with $\mathcal{A}^2_\lambda(\mathbb{C}_+)$ ($\lambda > -1$) a weighted Bergman space over the right half-plane $\mathbb{C}_+$, which admits a unitary representation of $\mathbb{R}$ given by the (imaginary) translations of $\mathbb{C}_+$. We study the von Neumann algebra $\mathfrak{T}\big(\mathcal{A}^2_\lambda(\mathbb{C}_+)\big)$ of bounded translation invariant operators. We prove that every element of $\mathfrak{T}\big(\mathcal{A}^2_\lambda(\mathbb{C}_+)\big)$ is a Toeplit operator $T^{(\lambda)}_A$ for some translation invariant operator $A$ of the ambient $L^2$-space of $\mathcal{A}^2_\lambda(\mathbb{C}_+)$. Furthermore, we prove that this can be achieved through a commutative von Neumann algebra $\mathfrak{A}_\lambda$ that yields an assignment $A \mapsto T^{(\lambda)}_A$ that turns out to be a $*$-isomorphism of $*$-algebras. Our main tool is a Bargmann transform $\mathcal{B}_\lambda$ for which we establish several operator and representation theoretic properties. We also describe the translation invariant subspaces of $\mathcal{A}^2_\lambda(\mathbb{C}_+)$ and obtain formulas for the diagonalizing spectral functions for translation invariant Toeplitz operators whose symbols are translation invariant operators. The latter generalize previously known results for function symbols.

Fix $b\in(0,1)$, let $1\leq k\leq n$, and let $A=(A_{ij})$ be an $n\times n$ random matrix with independent real entries satisfying $$ \sup_{x\in\mathbb{R}}\mathbb{P}\{A_{ij}=x\}\leq b<1 \qquad (1\leq i,j\leq n). $$ We show that there exists $c>0$ such that $$ \mathbb{P}\{\operatorname{rank} A\leq n-k\}\leq \exp(-cnk), \qquad 1\leq k\leq n. $$

Topology-constrained optimization arises in applications such as medical image segmentation, geometric design, and network generation, where global structural correctness matters. Yet Betti numbers and persistent homology are discrete and combinatorial, making them difficult to control directly by gradient-based optimization. We propose a differentiable framework that replaces discrete topological counting with a soft spectral relaxation of the Hodge Laplacian. Candidate simplices are embedded in a fixed ambient complex, while simplex inclusion is represented by continuous activations, producing a smooth spectral path between different hard complexes. The soft states are used only as optimization surrogates; in the hard limit, the ordinary Hodge Laplacian and its homological zero modes are recovered. We establish operator-norm and spectral perturbation bounds connecting hard Betti numbers to soft near-zero modes, and derive differentiable objectives using heat, resolvent, and polynomial low-pass filters. Controlled experiments on Vietoris–Rips point clouds show more spatially distributed gradients, reduced scale-normalized derivative variation near persistence-pairing changes, and geometry-aware update directions in the tested settings. Experiments on graph clique complexes further show controllable shifts in sampled hard Betti regimes and compatibility with standard graph objectives. The framework provides a complementary Hodge-spectral approach to persistent-homology-based topology optimization.