Papers

We study the semiconcavity property of viscosity solutions to Hamilton–Jacobi equations with Neumann boundary conditions. Unlike the state-constraint case, minimizing trajectories associated with the Neumann problem may fail to be $C^1$, so the classical approach based on the regularity of minimizers is no longer available. To overcome this difficulty, we introduce a comparison argument between the constrained action associated with the Skorokhod problem and the unconstrained action, avoiding any use of higher regularity of reflected minimizing trajectories. Under a structural decomposition assumption on the Hamiltonian at the boundary, we establish the estimate \[u(x+h,t+\sigma)+u(x-h,t-\sigma)-2u(x,t)\leq C(|h|+\sigma)^{\frac{3}{2}}.\] An explicit example shows that the power $3/2$ in this estimate cannot be improved.

The minimum-weight perfect matching (MWPM) decoder is a standard decoding strategy for surface codes, but its performance can degrade considerably under biased noise. In this work, a modified family of surface codes, termed the XYZ planar code, is introduced, and MWPM is extended to posterior MWPM (pMWPM), which exploits the information provided by mixed stabilizers with almost no increase in decoding complexity. The XYZ–pMWPM and XZZX code–decoder combinations are compared under homogeneous independent and identically distributed (IID) Pauli noise and processor-derived independent nonidentically distributed (INID) noise. Under homogeneous biased Pauli noise, pMWPM substantially increases the threshold of the XYZ planar code relative to conventional MWPM. At finite code distances, particularly for small codes under strong bias, XYZ–pMWPM achieves logical error rates comparable to or lower than those of XZZX. Under INID noise, the pseudothreshold ordering observed under the corresponding homogeneous IID references is not always preserved; in some hardware–distance settings, XYZ–pMWPM achieves pseudothresholds comparable to or higher than those of XZZX. Within XZZX, qubit-specific weighting consistently yields higher INID pseudothresholds than average-rate weighting for the hardware instances considered. It is further found that the IID–INID pseudothreshold shift depends strongly on the physical-error scalarization and the construction of the homogeneous IID reference. These results show that pMWPM efficiently exploits the mixed-stabilizer structure of the XYZ planar code and provides competitive finite-distance decoding under biased and spatially nonuniform noise.

In geophysical flows such as large-scale ocean dynamics, the vertical viscosity is often much smaller than the horizontal viscosity. This anisotropy makes it natural to ask whether solutions of the full anisotropic compressible Navier–Stokes equations converge, as the vertical viscosity coefficient $\varepsilon \to 0$, to solutions of a horizontally dissipative limit system, and whether this limit can be justified globally in time. Prior work has answered this question locally in time or in the incompressible setting. We resolve this problem for the three-dimensional compressible Navier–Stokes equations in the upper half-space with the Navier slip boundary condition. This paper establishes two main results for small perturbations of a constant equilibrium state. First, we prove the existence of a unique global-in-time solution whose conormal Sobolev norm remains uniformly bounded for all $t \ge 0$ and all $\varepsilon \in (0,1)$. Second, we justify the global vanishing vertical viscosity limit. More precisely, we show that the solutions converge to a global solution of the horizontally dissipative compressible Navier–Stokes system. This provides the first rigorous justification of the anisotropic viscosity limit for compressible flows on an infinite time interval.

We study a Schrödinger equation in the upper half-space with a nonlinear Neumann boundary interaction driven by the Bessel operator $\Ba$, $a>-1$. The problem arises naturally as an extension formulation for a nonlocal NLS with memory and can also be interpreted as a Schrödinger evolution with a nonlinear singular source concentrated on a codimension-one interface. We first develop a complete linear theory for the associated inhomogeneous problem with nonhomogeneous Neumann data. A central ingredient is a new Duhamel representation formula that separates bulk and boundary dynamics and identifies the precise role of the boundary propagator. Using this formula, we establish sharp Strichartz estimates adapted to the geometry of the half-space and the singular structure induced by the Bessel operator. The analysis reveals a basic dichotomy between the regimes $a\ge 0$ and $-1<a<0$: in the former, bulk and boundary exhibit a unified dispersive behavior, whereas in the latter the dispersive structure becomes anomalous, requiring weighted estimates and distinct functional frameworks for the bulk and boundary contributions. As an application of the linear theory, we prove well-posedness results for the nonlinear problem in the $L_a^2$-mass critical and subcritical regimes. For $a\ge 0$, we obtain global well-posedness for sufficiently small critical data, together with local and global results in the subcritical case. In the anomalous range $-1<a<0$, we establish existence and uniqueness on arbitrary finite time intervals for sufficiently small critical data, as well as local well-posedness in the subcritical regime. The results provide a unified dispersive framework for Schrödinger equations with nonlinear boundary interactions and singular extension structures associated with nonlocal-in-time dynamics.

We introduce the problem of entropy equivalence testing for probability distributions, a relaxation of the well-studied closeness testing problem, where the distribution testing algorithm is now only required to distinguish, given samples from two unknown distributions $p,q$ and a parameter $\varepsilon \in(0,1/2]$, between $p=q$ and $|H(p)-H(q)| \geq \varepsilon$ (where $H$ denotes the Shannon entropy). We provide a time- and sample-efficient algorithm for this task, showing that the optimal sample complexity for this task can be significantly lower than that of closeness testing. As an application, we leverage this result to provide the first non-trivial testing algorithm for (standard) closeness of low-degree Bayesian networks, which significantly improves on either the sample or time complexity of a baseline based on full learning.

Recent studies on binomials of the form $F_r(x) = x^r(1 + \chi(x))$ over $\mathbb{F}_{p^n}$ have shown that these functions can exhibit very low boomerang uniformity. In this paper, we focus on the specific behavior of such binomials in characteristic $3$, where instances of extremely low boomerang uniformity-namely $0$ or $1$-seem to arise more frequently than in other characteristics. First, we provide a systematic analysis of Almost Perfect Nonlinear (APN) power functions in characteristic $3$. We present an explicit parametrization of APN exponents arising from the construction of Zha and Wang and demonstrate through numerical results for $n \le 13$ that this generalized framework accounts for several previously known and sporadic APN instances. Building on this classification, we identify and rigorously prove two classes of binomials $F_r$ that are locally-PN and possess the minimum possible boomerang uniformity of $0$. These classes involve exponents derived from the aforementioned APN construction and the differentially 4-uniform exponent $r = 2 \cdot 3^{\frac{n-1}{2}} + 1$. Furthermore, we analyze the binomial $F_r$ with $r = 3^n - 3$, proving that it is locally-APN with boomerang uniformity $1$ when $n\ge 5$ is odd, and completely determine its boomerang spectrum through the evaluation of character sums. Our results clarify and extend existing studies on the cryptographic properties of binomials, providing a systematic characterization of several classes of binomials with very low boomerang uniformity in characteristic $3$.

We consider the stochastic heat equation on the integer lattice $\mathbb{Z}^d$ in dimension $d \geq 3$ and with small coupling constant. We show uniqueness of global solutions within the class of positive functions that are stationary in time and whose asymptotic growth in space is subexponential. Our proof relies on a factorization formula for the point-to-point partition function in the associated polymer model.

We study the functional code $C_d(\mathcal{X})$, introduced by G. Lachaud in 1996, in the case where $\mathcal{X}$ is a rank $n$ degenerate Hermitian variety $P\mathcal{U}_{n-1}$ in $\mathbb{P}^n(\mathbb{F}_{q^2})$ and $d\leq q$. We establish an upper bound for the maximum number of $\mathbb{F}_{q^2}$-rational points in the intersection of $P\mathcal{U}_{n-1}$ with an $\mathbb{F}_{q^2}$-hypersurface of degree at most $q$ in $\mathbb{P}^n$. Using this bound, we determine the parameters of the codes $C_d(P\mathcal{U}_{n-1})$ in the cases $n=2,3,4$. We also characterize the hypersurfaces that correspond to the minimum distance of these codes in the cases $n=2,3,4$.

We present three versions of the classic two-pile game one-or-one-or-one-of-both generalized to the multi-pile context. In each case, we explore the resulting $\mathcal{P}$-positions. In the first version, there is a simple pattern. In the other two versions, we find partial solutions in each case through two experimental routes. First by limiting the number of piles, then by limiting the number of tokens per pile.

We develop an asymptotic statistical theory for parameter estimation from a class of non-i.i.d. periodic binary event-detection processes subject to nonparalyzable dead time and gating, which we call "dead-time event detection" (DED) processes. Such processes arise in single-photon lidar, fluorescence lifetime imaging, X-ray astronomy, and particle or radiation flux measurements in nuclear physics, where each detection renders the radiation/particle detector inactive for a recovery interval. Our theory quantifies how dead time and gating affect the fundamental lower bounds of estimation and identifies practical estimators that attain these bounds. First, we identify a sufficient statistic, showing in particular that activation counts can carry statistically useful information discarded by conventional histogramming hardware. We then prove local asymptotic normality and derive the corresponding Fisher-information rate, thereby obtaining fundamental lower bounds for estimation from DED processes. We prove that the maximum likelihood estimator (MLE), widely used in DED applications, attains these lower bounds. Since computing the MLE typically requires solving a nonconvex optimization problem, we also propose Le Cam one-step estimators, which attain the same asymptotic bounds with only a single local correction rather than iterative optimization. We illustrate the validity of our asymptotic theory and the practical usefulness of one-step estimators through the example of single-photon lidar in both simulations and real-data experiments.

Let $(M, g, f, \tau)$ be a complete Ricci shrinker satisfying $\textrm{Ric}+\nabla^2f=\frac{g}{2\tau}$ and let $R$ denote its scalar curvature. For a confined function $V$ on $M$, we obtain a lower bound for the lowest eigenvalue of the Schrödinger operator $-\Delta+\frac{R}{4}+V$, expressed in terms of an integral quantity involving $V$ and the shrinker entropy, and the equality case is characterized by the potential functions. We further generalize this estimate to complete Riemannian manifolds via Perelman's $\mu$-functional. We also study the drifted Schrödinger operator $-\Delta_f+V$ on smooth metric measure spaces. In particular, on Ricci shrinkers, we derive a lower bound for its lowest eigenvalue, with equality if and only if $V$ is affine.

In this work, we establish a relationship between the sum of irreducible character degrees and the number of twisted involutions associated with the automorphisms of a finite group. We develop algorithmic frameworks for evaluating these quantities in the context of inner automorphisms and the symmetric group $\mathfrak{S}_n$. As an application, we provide a criterion for identifying groups that possess complex (non-real) irreducible representations and explore the structural consequences arising from these results.

Scaling recommendation models is a central challenge in recommender systems. Recently, RankMixer has emerged as an effective solution, operating on a unified token representation and alternating between token mixing and per-token feedforward networks (P-FFNs) to achieve scalable performance. However, RankMixer suffers from embedding collapse, where learned representations have low effective rank, limiting expressivity and underutilizing the expanded representation space. Through empirical analysis and theoretical insights, we identify rigid token mixing and P-FFN modules as the primary causes of this phenomenon, jointly inducing a damped oscillatory trajectory in effective-rank evolution across layers. To address it, we propose RankElastor, a novel architecture that produces spectrum-robust representations with provable collapse mitigation. RankElastor introduces two components: (i) parameterized full mixing, which enables expressive token mixing with improved spectral robustness; and (ii) GLU-improved P-FFNs, which stabilize representation spectra through GLU-style FFN modules. Extensive experiments on large-scale industrial datasets demonstrate that RankElastor consistently improves recommendation performance, mitigates embedding collapse, and exhibits robust scaling behavior. Code is available at this GitHub repository: https://github.com/vasile-paskardlgm/RankElastor

We study the question of global attraction, in the energy norm, for finite energy solutions to classical particle coupled to a scalar wave field. The attraction could take place to either the set of the stationary solutions in the case of a confining potential or to the soliton manifold in the case of zero potential. By energy conservation argument we establish that in both cases there is no global attraction.

Curl-measure fields are $p$-integrable vector fields whose distributional curl is a vector-valued Radon measure with finite total variation. They were introduced in arXiv:2509.26465, where, for $p= \infty$, the existence of tangential traces for bounded Lipschitz domains was established, together with the tangential property of the trace. In this paper, we show that the same tangential property holds for domains that are sets of finite perimeter.

A graph-theoretic analogue of Iwasawa theory, initiated by Gonet and Vallières, has attracted considerable interest in the study of Iwasawa invariants. On the other hand, for a pair of prime numbers $(r,\ell)$, one obtains a graph, called the supersingular $\ell$-isogeny graph (SIG), whose adjacency matrix has eigenvalues given by the $\ell$-th Fourier coefficients of the weight 2 Eisenstein series and newforms of level $r$. In this paper, we fix prime numbers $r$ and $p$, and let $\ell$ vary over infinitely many primes. We then investigate the distribution of the Iwasawa $\lambda$-invariants of the constant $\mathbf{Z}_p$-towers over the SIGs, thereby revealing connections among graph theory, Iwasawa theory, elliptic curves, and the Galois representations attached to newforms. At the end of this paper, we propose a conjecture concerning the Galois orbits of newforms.

We develop new conservative discontinuous Galerkin (DG) methods for nonlinear wave problems, focusing on the generalized Korteweg-de Vries (gKdV) equation and the coupled Hirota-Satsuma KdV (HS-KdV) system. The proposed methods preserve mass through the single-valued structure of numerical traces, while energy and Hamiltonian conservation are enforced by implicitly determining penalty parameters in the numerical traces through auxiliary conservation constraints. In our previous work [11], we developed a conservative DG method for the gKdV equation; however, that formulation involves the time derivative of the jump of the approximate solution, which complicates extensions beyond second-order temporal accuracy. Our new formulation overcomes this limitation by introducing a redesigned trace configuration that eliminates the derivative-of-jump term. This novel enhancement seamlessly paves the way for higher-order time discretizations and requires solving fewer nonlinear systems per time step than the previous approach. For the coupled HS-KdV system, we present the first conservative DG method that preserves all three invariants of the exact solution. Numerical results demonstrate the accuracy and expected convergence behavior of the proposed methods, as well as long-time stability and strong conservation properties for both the gKdV equation and HS-KdV system.

We prove that dichotomies given by growth rates that are either faster or slower than exponential either do not occur or are inconsequential in the setting of skew-products with compact base. A similar conclusion is obtained for the nonuniform exponential behavior. To achieve this, we study families of translated linear nonautonomous differential equations for which we prove propagation results. We also study translations of growth rates under a comparison criteria.

The recent proof of the unramified Geometric Langlands Conjecture has attracted a lot of publicity, so this seems like a good time to address the title question. In one line, the Geometric Langlands correspondence is an algebraic spectral theorem for a certain class of differential equations called automorphic sheaves: it asserts they can be decomposed into monochromatic objects, which diagonalize the action of natural symmetries (Hecke operators), and it describes the corresponding colors or frequencies (Langlands parameters). The statement is very technical and esoteric sounding, the proof takes thousands of pages, and there are relatively few easily stated immediate consequences. So what's the deal? In this brief survey I will present the subject informally as a blueprint for a master plan for the study of nonabelian symmetry, touching on some of the main motivations, connections and structures that have emerged.

Several machine learning models are defined for inputs of any size, such as graphs with different numbers of nodes and point clouds containing varying numbers of points. The universality properties of such any-dimensional models remain poorly understood, as universality is traditionally studied for models accepting inputs of a fixed size, defined on a compact subset of their domain. In sharp contrast, any-dimensional models can be viewed as sequences of functions defined on growing-sized inputs, and it is not clear in which sense they can be universal. We develop a systematic approach to establish any-dimensional universality, by identifying any-dimensional functions with a unique function taking inputs in a suitable infinite-dimensional limit space containing inputs of all finite sizes as well as their limits. Using the symmetries of these inputs and relations between inputs of different sizes, we show that this limit space admits a natural topology with rich families of compact sets on which any-dimensional universality can be established. We illustrate our approach by showing that several existing architectures fail to be universal, and we propose simple modifications that restore universality.

We develop techniques for solving the relative isoperimetric problem on polygonal domains in $\mathbb{R}^2$, with special attention paid to corners. As an application, we solve the relative isoperimetric problem for a square with a square corner removed.

Individual fairness, the notion that "similar individuals should be treated similarly," provides a strong and flexible fairness guarantee for algorithmic decision makers. However, a barrier to implementing individual fairness in practice is the difficulty of learning the similarity metric over individuals. In this work, we present an algorithm for learning a Mahalanobis similarity metric from triplet queries of the form "is individual $i$ more similar to individual $j$ or $k$?" We work in the standard Bradley-Terry model for pairwise comparisons. Our algorithm consists of a spectral initialization step followed by gradient descent. We provide extensive theoretical guarantees on our algorithm, showing that it converges quickly to the ground truth metric despite the non-convexity of the loss in our model. Because our focus is on fairness, we also show that individual fairness with respect to an estimated metric is sufficient to achieve similar fairness with respect to the true metric. We also discuss potential applications of our work to AI model tuning. Finally, we present experimental results that demonstrate the convergence of our algorithm and the fairness performance of downstream fair predictors trained on our estimated metric.

We prove a functional extension of an exponential inequality originally proposed by Bin Zhao and proved by Xiaosheng Mou. The main result asserts that if $\alpha_1\leq \cdots\leq \alpha_n$ and $\sum_{k=1}^n \alpha_k=0$, then \[ \sum_{k=1}^n \phi(k\alpha_k)\geq 0 \] for every odd function $\phi$ that is increasing and convex on $[0,\infty)$. The proof is based on a truncated-sum comparison and the stop-loss characterization of the increasing convex order. As consequences, we recover the original exponential inequality and obtain polynomial and integral variants.

This paper is devoted to studying the generalized Ostrovsky equation \begin{eqnarray*} u_{t}-\beta\partial_{x}^{3}u-\gamma\partial_{x}^{-1}u+\frac{1}{k+1}(u^{k+1})_{x}=0,k\geq5 \end{eqnarray*} with $\beta<0,\gamma>0$. Firstly, by using the density theorem in the mixed Lebesgue spaces, we prove that $X_{s,b}\hookrightarrow C(\mathbb{R};H^{s}(\mathbb{R})) \hookrightarrow C(\mathbb{R};L_{x}^{\infty})$ with $s>1/2,b>1/2.$ Secondly, we present a new proof of the convergence problem of linear Ostrovsky equation, which is slightly different from the proof of Theorem 1.1 (Convergence problem of Ostrovsky equation with rough data and random data, Indiana Univ. Math. J. 71(2022), 1897-1921.) Thirdly, we investigate the pointwise convergence problem of the generalized Ostrovsky equation. Fourthly, for the solution $u$ to the Cauchy problem for the generalized Ostrovsky equation, we prove that $u=u_{1}+u_{2},t\in[-\delta,\delta]$, and $u_{2}$ possesses better regularity than $u$, where $u_{1}$ is the linear part of $u$ and $u_{2}$ is the nonlinear integral part. Fifthly, we investigate the nonlinear smoothing and the uniform convergence problem of the generalized Ostrovsky equation. Finally, when data $f$ belongs to $H^{s}(\mathbb{R})(s>\frac{1}{2}-\frac{2}{k+1},k\geq6)$ and $\lim\limits_{|x|\rightarrow{\infty}}f=0$ and $\mathscr{F}_{x}(U(t)f)\in L^{1}(\mathbb{R}),$ for $t\in [-\delta,\delta],$ we prove that $\lim\limits_{|x|\rightarrow{\infty}}u=0$. The key ingredients are high-low frequency technique, maximal function estimates related to low frequency and some Strichartz estimates which can be proved with the aid of the Stein complex interpolation Theorem.

This paper presents some qualitative properties of positive solutions to the strongly coupled system \[ \begin{cases} \displaystyle - \Delta u + \tau u = \frac{2 p}{p + q} \left( I_\alpha \ast |v|^q \right) |u|^{p - 2} u &\text{in} ~ \mathbb{R}^N, \\ \\ \displaystyle - \Delta v + \eta v = \frac{2 q}{p + q} \left( I_\alpha \ast |u|^p \right) |v|^{q - 2} v &\text{in} ~ \mathbb{R}^N, \end{cases} \] with $\tau, \eta > 0$, $N \in \mathbb{N}$, $0 < \alpha < N$, \[ \max \left\{1, \frac{2 \alpha}{N}\right\} < p, q < 2^* \quad \text{and} \quad \frac{2 (N + \alpha)}{N} < p + q < 2_\alpha^*, \] where $I_\alpha$ denotes the Riesz potential, \[ 2^* := \begin{cases} \infty, &\text{if} ~ N \in \{1, 2\}, \\ \frac{2 N}{N - 2}, &\text{if} ~ N \geq 3, \end{cases} \quad \text{and} \quad 2_\alpha^* := \begin{cases} \infty, &\text{if} ~ N \in \{1, 2\}, \\ \frac{2 (N + \alpha)}{N - 2}, &\text{if} ~ N \geq 3. \end{cases} \] More precisely, by means of the moving planes method, we prove that positive solutions to this system are radially symmetric and strictly radially decreasing when $p, q \geq 2$, and we obtain a classification result for positive ground states in the case $p = q$ and $\tau = \eta$.

Polar codes are the first error-correcting code proven to achieve channel capacity based on infinite code length. The Successive Cancellation List Flip (SCLF) decoding algorithm was proposed by flipping an erroneous bit during the next decoding attempt. To identify the erroneous bits, the Log-Likelihood Ratio (LLR) is used to indicate the reliability of each decision bit. To improve the accuracy of the erroneous bit prediction, we propose deep-learning-aided (DL-aided) SCLF decoding algorithms. We first offer a stacked LSTM network that contains new features to train our models, which are able to improve the accuracy of the prediction of positions of erroneous bits. Then we separately train the stacked LSTM models to predict the position of both the first and second erroneous bits and whether to continue flipping. As a result, the DL-aided SCLF decoding algorithms based on the proposed stacked LSTM \mbox{flip-1} model, stacked LSTM \mbox{flip-2} model, and the stacked LSTM \mbox{continue-flipping} check (CFC) model are able to provide a better performance at a lower number of average decoding attempts when compared to other state-of-the-art decoding algorithms.

The Permutation Equivalence Problem (PEP) for linear complementary dual (LCD) codes reduces to Graph Isomorphism (GI) through the orthogonal projector onto the code, but this requires a trivial hull. We ask whether the reduction extends when the standard inner product is replaced by a nondegenerate symmetric bilinear form \(M\). We make precise what it means for \(M\) to yield a valid reduction to GI and prove that, for length \(n \geq 3\) and every field size, this holds if and only if \(M\) is permutation-invariant. The reduction then reaches, beyond LCD codes, a class of codes with one-dimensional hull: exactly those whose hull is spanned by a vector with nonzero coordinate sum. We count these codes exactly; for odd \(q\), asymptotically the method covers a fraction \(\prod_{i \geq 2}(1+q^{-i})^{-1}\) of all codes, \(1 + q^{-1}\) times the fraction of LCD codes. A polynomial-time reduction algorithm is given.

We generalize the stabilizer formalism for entanglement-assisted quantum error-correcting codes with noisy ebits (EAQECCs-Ne) from the binary case to the general $q$-ary case, where $q$ is a prime power. By leveraging the structure of the generalized Pauli group over $\mathbb{F}_q$ and symplectic geometry over $\mathbb{F}_q^{2n}$, we establish a unified framework for constructing EAQECCs-Ne for qudit systems. Equivalent formulations in terms of symplectic geometry over $\mathbb{F}_q$ and additive codes over $\mathbb{F}_q^{2n}$ are derived. We further construct several families of $q$-ary EAQECCs with noise ebits and analyze their performance compared to optimal stabilizer codes. Our results demonstrate that under certain noise conditions, the proposed EAQECCs-Ne can outperform standard stabilizer codes with equivalent error-correcting capability, offering a promising approach for fault-tolerant quantum computation in high-dimensional quantum systems.

Locally standard $T$-pseudomanifolds were introduced by the authors in a previous work. They are topological stratified pseudomanifolds equipped with torus actions. Their equivariant homeomorphism types are classified by characteristic data under the homotopy equivalence condition. In this paper, we show that this condition can be removed when the dimension is at most three. We also characterize those of dimension at most three that are topological manifolds, in terms of their orbit spaces.

A nonlinear time-delay model is proposed to describe the interaction dynamics between criminal and non-criminal populations, combining social influence mechanisms, saturation effects represented by a Holling type II functional response, and time-dependent law-enforcement actions. The delay accounts for the latency between exposure to criminal behavior and behavioral response, introducing memory effects that naturally lead to a delay differential equations framework. Fundamental analytical properties, including positivity, global existence, and invariance of the feasible region, are established to ensure the mathematical consistency of the population interpretation. In the autonomous setting, explicit threshold conditions governing the stability of the criminal-free equilibrium and the emergence of coexistence states are derived, while the delay is shown to induce stability switches and oscillatory regimes through characteristic root crossings. In the non-autonomous case, topological degree arguments guaranty the existence of strictly positive periodic solutions, indicating that long-term dynamics depend primarily on the averaged law enforcement intensity measures rather than on short-term fluctuations. These results identify time delay as a key structural mechanism underlying recurrent patterns and complex temporal behavior in crime dynamics.