Papers

This paper explores topological phase transitions in the momentum space of the $\kappa$-deformed $\phi^4$ lattice model within the mean-field approximation. We show that the double-well structure of the momentum space gives rise to degenerate momentum states at high energies and that this change is marked by a phase transition. We investigate this transition using both numerical and analytical methods, with the analytical approach based primarily on singularity theory. Our numerical analysis shows that the energy at which the topological phase transition occurs is related to the noncommutative parameter by the simple power law $E \sim 1/\alpha^{4}$. By mapping the constant-energy-level surface to the $A_{3}$ singularity, we obtain the same result analytically. We also present results unique to noncommutative systems, showing that the fraction of particles in degenerate momentum states has a nonzero commutative limit. As a nonperturbative effect, this result provides a testable prediction for the existence of noncommutative spacetime.

Let $p = 4q + 1$ be a prime integer, where $q$ is an odd prime. Let $L = \mathbb{Q}(\zeta_p)^+$ denote the maximal totally real subfield of the $p$-th cyclotomic field $\mathbb{Q}(\zeta_p)$. In this paper, we prove that if $U_L^+=U_L^2$ (i.e., every totally positive unit is a square in $L$), then for any non-trivial subfield $K \subseteq L$ and any non-trivial totally positive unit $\varepsilon \in U_K^+$, the absolute trace satisfies the sharp lower bound $\Tr_{K/\mathbb{Q}}(\varepsilon) \ge p - 4$. As an application, we show that a spherical fusion category $\C$ with global dimension $\dim(\mathcal{C})=p=4q+1$ is pointed, where $2$ is a primitive root modulo $q$, or $p=4q+1<2^{22}$ and $p\neq 29$.

This paper investigates how bistatic delay information improves the observability and localization performance of 3D moving targets in distributed Wi-Fi sensing. Motivated by emerging IEEE 802.11bf WLAN sensing, we consider one transmitter, multiple distributed receivers, and a single target with unknown position and velocity. We compare Doppler-only sensing with delay-Doppler sensing, in which delay-capable receivers provide additional bistatic path-length information for target localization. We derive the Fisher information matrix (FIM) and the position equivalent FIM (EFIM) for both sensing formulations. The resulting observability analysis shows that Doppler-only localization generically requires six receivers, whereas delay-Doppler localization can become locally identifiable with three receivers under non-degenerate geometry. We further study mixed receiver architectures in which only a subset of receivers provides delay estimates. Simulations with Wi-Fi-like bandwidths confirm the transition and quantify the roles of bandwidth, Doppler noise, receiver geometry, and delay-capable receiver availability.

We prove that every holomorphic isometric embedding, with respect to the Bergman metrics, from a complex unit ball of maximal dimension into irreducible classical bounded symmetric domain of rank at least two that is not biholomorphic to a Type IV domain is equivalent to Mok's embedding. This completely resolves a problem of Mok.

Let $C_n^k$ denote the $k$th power of the cycle $C_n$. The equivalence covering number $\operatorname{eq}(G)$, studied by Alon, is the minimum number of equivalence subgraphs needed to cover all edges of $G$. King conjectured that $\operatorname{eq}(C_n^k)=\Omega(k)$ for sufficiently large $n$ relative to $k$. We study $\operatorname{eq}(C_n^k)$ and prove that, for every $n \geq 2k + 2$, we have $1 + \lceil\log_2(k+1)\rceil \leq \operatorname{eq}(C_n^k) \leq 1 + 2\lceil\log_2(k+1)\rceil$. We further show that, whenever $n\ge(k+1)\max\{41,5\lceil\log_2(k+1)\rceil\}$, \begin{equation*} \operatorname{eq}(C_n^k)\in\left\{1+\lceil\log_2(k+1)\rceil,\,2+\lceil\log_2(k+1)\rceil\right\}. \end{equation*} Moreover, for every integer $n\ge \max\{2k+2,k^2(k+2)\}$, the following exact values hold in the indicated cases: \begin{equation*} \operatorname{eq}(C_n^k)=\begin{cases} 1+\lceil\log_2(k+1)\rceil, & k+1\neq2^{\lceil\log_2(k+1)\rceil} \text{ or }(k+1)\mid n,\\[3pt] 2+\lceil\log_2(k+1)\rceil, & k+1=2^{\lceil\log_2(k+1)\rceil} \text{ and }2\nmid n. \end{cases} \end{equation*}

We establish the weak type \((1,1)\) endpoint estimate for operator-valued rough homogeneous singular integrals with kernels satisfying the classical \(L\log L\) integrability and cancellation conditions. This extends Seeger's endpoint theorem [{J. Amer. Math. Soc.} 9 (1996)] to the noncommutative setting. A key ingredient is an off-diagonal analysis that controls interactions between different kernel scales and avoids an additional logarithmic loss.

Codes in the Levenshtein metric that can correct multiple edits (including deletions, insertions, and substitutions) have attracted a lot of attention recently owing to their application in DNA-based storage and racetrack memories. In this work, we present new techniques for designing explicit binary codes of length $n$ that correct $t$ edits with at most $(2t+1)\log n+O_t(\log\log n)$ bits of redundancy. We first propose to use a polynomial hash to construct a systematic code that corrects $t$ edits with $(4t+2)\log n+O_t(\log\log n)$ bits of redundancy. Then we observe that there are many polynomial hashes like this, some are good, some are bad. So, we propose the second technique, called entropy recycling, to use the indices of good hashes to carry information. As a consequence, we obtain a partial-systematic code of length $n$ that corrects $t$ edits with at most $(2t+1)\log n+O_t(\log\log n)$ bits of redundancy. Finally, we provide deterministic polynomial-time encoding and decoding algorithms for our code.

A random geometric graph is a random graph model whose vertex set is a binomial point process, where two vertices are connected by an edge if each lies within the ball of radius $r$ centered at the other. In this paper, we investigate the asymptotic fluctuations of the threshold radius required for the maximum degree of the graph to reach a target value. Assuming this target value grows with the number of vertices $n$ at an order of at least $\log n$ but strictly less than $(\log n)^d$, we show that these fluctuations follow a Gumbel distribution. We also investigate the spatial configurations of vertices with extremely large degrees, and show that the point process marked by their degrees converges in distribution to a Poisson point process.

We consider the recovery of functions of the weighted Wiener classes $W_s^\mu$ from perturbed Fourier–Hermite coefficients in the metric of $L_\infty(\mathbb R)$. For the summation methods of the class $\mathcal S^\theta$ we find error estimates on $W_s^\mu$. The estimates take into account the behaviour of the Hermite functions both in the oscillatory region and near the turning point: the maxima of functions with different indices are attained at different points, and the supremum of a sum over a block of indices is substantially smaller than the sum of the suprema. Owing to this, in particular, we establish a common lower bound for all finite diagonal methods; for $s\ne4/3$ and $p\ne4/3$ it matches the upper bound and gives the exact order of the error in this class, while at the critical points the gap between the bounds does not exceed $(\log(e/\delta))^{1/4}$ up to a constant. If, in addition, $s>4/3$ or $p<4/3$, then this order is also optimal among all recovery operators that use the whole sequence of data.

We construct smooth irreducible plane curves whose complete amoeba contour is a smooth embedded curve equal to the amoeba boundary, although the curves are not torically equivalent to simple Harnack curves. Finite monomial coverings yield explicit infinite families with higher logarithmic multiplicity and nonmaximal normalized amoeba area. We classify the Newton polygons admitting such covering constructions through their exponent lattices, give an exact connectedness criterion in terms of monodromy, and treat primitive polygons by a separate phase-sensitive construction. Exact triangular and trapezoidal families are complemented by positive-genus investigations exhibiting bounded amoeba-complement components. These results show that a perfectly exposed amoeba contour does not determine the real structure, logarithmic sheet number, or Harnack type of the underlying curve.

We construct smooth plane curves whose logarithmic contours are smooth and embedded but strictly larger than their amoeba boundaries. The examples include connected irreducible curves, disconnected unions of nonlinear components with two-dimensional amoebas, nested families with arbitrarily many components, and a uniform construction realizing every positive genus. Exact critical-locus calculations distinguish the boundary-producing folds from compact or noncompact contour branches lying in the amoeba interior. These constructions show that smoothness of the complete contour does not force contour–boundary equality and that internal critical-value branches persist under monomial transformations, controlled perturbations, and finite monomial coverings.21 pages, 14 figures

This paper is a continuation of Part~I \cite{QTWPartI}, where blow-up analysis and compactness results for the CR Nirenberg problem were established. We prove some existence and degree results for the CR Nirenberg problem on the standard sphere. More precisely, we first prove existence results under antipodal and, more generally, finite-group symmetries. Without symmetry, we derive a Leray–Schauder degree formula under natural flatness conditions, yielding existence results for positive prescribed curvatures with certain homogeneous expansions at critical points. We also obtain Aubin-type inequalities under a vanishing first-moment condition, uniformly near the critical exponent.

We study the Cauchy problem in $\R^{3}$ for the repulsive compressible Navier-Stokes-Riesz system with Riesz exponent $0<s<1$ and viscosity $0<\varepsilon\leq1$, where the Riesz interaction $\nabla(-\Delta)^{-s}(\rho-\bar{\rho})$ is a generalization of the Coulomb interaction for electrons. For small perturbations of a constant equilibrium, with the solenoidal component of the initial velocity of order $\mathcal{O}(\varepsilon)$, we prove the global existence and uniqueness of smooth solutions. We derive time-decay estimates in $L^{2}$ norms and $L^{\infty}$ norms that capture both uniform-in-$\varepsilon$ dispersive behavior and viscosity-dependent dissipation. We further establish a global-in-time inviscid limit to the irrotational global solution of the compressible Euler-Riesz system whose initial data consist of the same density and the curl-free component of the velocity, with an explicit convergence rate in $W^{k,p}$ norms. The proof combines viscosity-adapted dispersive estimates, normal-form analysis and nonlinear energy estimates with control of both negative and positive Sobolev norms.

An elementary space is a pair $(X, \mathcal A)$, where $\mathcal A$ is an algebra of subsets of $X$. Every elementary space $(X, \mathcal A)$ is naturally a topological space when equipped with the topology $\tau_{\mathcal A}$ generated by $\mathcal A$. Elementary mappings on elementary spaces are defined in a manner analogous to measurable mappings on measurable spaces. In addition, the notion of an extension of an elementary space is introduced. Let $M(X, \mathcal A)$, denote the ring of all real-valued elementary functions on $(X, \mathcal A)$. In this article, we study the ring $M(X, \mathcal A)$ and extensions of an elementary space. We show that, for any elementary space $(X, \mathcal A)$, there exists a unique compact elementary extension $(\beta_{\mathcal A}X, \bar{\mathcal A})$ of $X$, analogous to the Stone-$\check C$ech compactification, which contains every other extension of $X$. This extension has the property that every elementary map from $X$ into any compact topological space extends continuously to $\beta_{\mathcal A}X$. Next, we show that $\beta_{\mathcal A}X$ contains a realcompact extension $(\upsilon_{\mathcal A}X, \bar{\mathcal A}_R)$ of $X$, such that every elementary map from $X$ into any realcompact topological space extends continuously to $\upsilon_{\mathcal A}X$. Moreover, $\upsilon_{\mathcal A}X$ is the smallest realcompact extension of $X$ and $(X, \mathcal A)$ is realcompact if and only if $X=\upsilon_{\mathcal A}X$. Also, we see that, every metric space equipped with the $\sigma$-algebra of all Borel sets is realcompact. Finally, we observe that the maximal ring of quotients of $M(X, \mathcal A)$ is the direct limit of locally constant (continuous) functions on dense open subsets of $(X, \tau_{\mathcal A})$. Some questions raised in the literature are also answered.

We provide a simplified and more human-friendly version of the proof of the Deligne–Drinfeld conjecture recently given by OpenAI. No originality is claimed.

We study the initial boundary value problem for the compressible Navier-Stokes-Poisson equations without damping in the three-dimensional half-space. The velocity satisfies the no-slip boundary condition, and the electric potential satisfies the homogeneous Neumann boundary condition. For small perturbations of a constant state in $H^3$, with a small initial electric field in $L^2$, we establish global existence and uniqueness of strong solutions. The uniform estimates combine density dissipation, tangential and normal derivative estimates, and a modified energy that accounts for the electric boundary contribution through the Helmholtz projection. We also prove algebraic decay under an additional low-frequency assumption on the initial data. If the density perturbation, velocity and electric field have finite tangential negative Sobolev norms of order $-s$, where $1/2\leq s<1$, then the solution decays at the rate $(1+t)^{-s/2}$ in the full energy norm. The decay estimate follows from the formulation in terms of the momentum $\rho u$, uniform negative-order estimates and interpolation with the dissipation.

We construct a positive-definite symmetric integral $12\times12$ matrix $A$ such that, for every solution $Q$ of Nahm's equations $1-Q=Q^A$ with $Q_i\ne0,1$ for all $i$, the associated element $\sum_i[Q_i]$ is torsion in the Bloch group of $\mathbb{Q}(Q_1,\ldots,Q_{12})$. However, the Nahm sum $F_{A,B,C}(\tau)$ is nonmodular for every $B\in\mathbb{Q}^{12}$ and $C\in\mathbb{Q}$. Thus the torsion condition for all solutions in Nahm's conjecture is not sufficient for the existence of a modular Nahm sum.

We study the homogenization of an optimal control problem governed by a Navier-Stokes-Cahn-Hilliard system in a periodically perforated three-dimensional domain. The obstacles are Lipschitz sets of size $\varepsilon^\alpha$, $\alpha>3$, where $\varepsilon$ is the period. The fluid satisfies the no-slip condition on the outer boundary and the free-slip condition on the obstacles, the phase-field equation contains a source term, and the convective term carries a pore-scale Reynolds number that tends to zero as $\varepsilon\to0$. The control is a distributed body force, and the cost functional tracks the velocity in $L^2$ and the order parameter in $H^1$. Since uniqueness of weak solutions of the state system is not known for fixed $\varepsilon$, the control-to-state relation is multivalued, and we pose the microscopic problems over control-state pairs. We prove the existence of microscopic optimal pairs and the sequential $\Gamma$-convergence of the cost functionals to the cost functional of a control problem governed by the unsteady Stokes-Cahn-Hilliard system on the full domain. As a consequence, the optimal values converge, every cluster point of microscopic optimal pairs is optimal for the homogenized problem, and the optimal controls converge strongly along the corresponding subsequences. The $\liminf$ inequality follows from the homogenization of the state system, including the strong convergence of the gradients of the order parameter. Only the recovery sequence requires uniqueness of weak solutions of the homogenized system, which we prove when the outer boundary is of class $C^{2,1}$.

We prove an explicit lower spectral bound for independent random vectors with prescribed group multiplicities. Let $v_{ij}\in\mathbb C^d$, $1\le i\le l$, $1\le j\le k_i$, be independent random vectors with finite support and $\mathbb E v_{ij}v_{ij}^*=A_i$, where $\sum_{i=1}^l A_i=I_d$ and $\operatorname{Tr}(A_i)\ge\epsilon$. Set $k=\max_i k_i$, $n=\max_i\operatorname{rank}(A_i)$, and $K=\sum_{i=1}^l k_i$. If $n>(6+4\sqrt2)k$ and $2\sqrt{nk}-k<\epsilon\le n/2$, then, with positive probability, \[ \lambda_K\left(\sum_{i=1}^l\sum_{j=1}^{k_i}v_{ij}v_{ij}^*\right) \ge n-(\sqrt{k}+\sqrt{n-\epsilon})^2. \] As a consequence, we obtain a restricted invertibility result selecting exactly $k_i$ vectors from the $i$th group of a Parseval frame. The proof combines interlacing families, a reciprocal transformation of mixed characteristic polynomials, and a multivariate barrier estimate that accommodates repeated differentiation in the same variable.

When can combining source and target data yield much better performance than using either alone? From a minimax viewpoint, writing $R(n,n_T)$ for the minimax risk from $n$ source and $n_T$ target observations, we ask when $R(n,n_T)$ decays strictly faster than both $R(n,0)$ and $R(0,n_T)$. We call this the Synergistic Learning Phenomenon (SLP) and quantify it by the Synergistic Advantage Ratio $\mathrm{SAR}=\min(R(n,0),R(0,n_T))/R(n,n_T)$. We study domain adaptation with a known link: source regression $f$, target regression $g=\mathcal{A}f$ for a known operator $\mathcal{A}$, and possibly different designs. Three links—identity, integration, and differentiation—and the density ratio $\rho=h_T/h_S$ determine whether synergy is possible, at what rate, and in which sample-size window. For the identity link (covariate shift) with a Beta$(a,1)$ source and uniform target, a known necessary condition for SLP is $a\geq2+1/(2\beta)$, where $\beta$ is the smoothness of $f$ (Result I). An integral link $g(x)=\int_0^xf(t)\,dt$ can yield the joint rate $(nn_T)^{-1/2}$ even for a bounded ratio (Result II), and under a Beta$(a,1)$ source has a phase diagram with cuts at $a=2$ and $a=3$: integral SLP for $1<a<2$, an extra level-split rate $n_T^{-1}$ for $2<a<3$, and covariate-shift-type spatial SLP for $a>3$ (Result III). A derivative link $g=\dot f$ synergizes only under a stronger singularity (Result IV), giving a sharp integration–differentiation asymmetry under a bounded ratio. Across all three links, synergy arises from role separation: each dataset resolves a distinct bottleneck associated with the link and designs. We characterize the sample-size windows for synergy and how SAR varies with $n$ and $n_T$. Simulations support the predicted phase structure, and a mirrored integration–differentiation analysis of real vehicle and smartphone signals illustrates the asymmetry.

Projected Walk on Spheres (PWoS) solves surface PDEs through ambient sampling and closest-point projection, but tightly folded embeddings can force short steps or ambiguous projections despite simple intrinsic geometry. We introduce Walk-on-Ambient-Patch (WoAP), which treats the embedding as a solver variable. Edge-isometry constraints and tangent-point repulsion separate intrinsically distant regions, optionally in higher codimension; patch states track residual projection branches without introducing PDE boundaries. We analyze the resulting geometry–sampling tradeoff: additional dimensions provide separation freedom but reduce uniform tangent progress, while strong tangent-biased proposals incur large importance-weight moments. Folded-surface experiments demonstrate reduced projection conflicts, a separate hard-constraint pilot verifies metric preservation, and an internal fixed-RMSE comparison favors higher dimensions at stricter accuracy targets. Together, these results motivate choosing the embedding and projection state jointly with the stochastic solver.

Let \(E(X)\) count the even integers between \(4\) and \(X\) that are not sums of two primes. We combine Pintz's explicit formula for the major arcs with the exponential-sum estimate of Maynard, Pandey, and Radziwiłł to prove \(E(X)\ll_\varepsilon X^{7/12+\varepsilon}\) whenever all Dirichlet \(L\)-functions have a common fixed zero-free half-plane \(\Rea s>\theta_0\), with \(\theta_0<1\). The uniform \(7/8\) theorem reported in the OpenAI mathematics release supplies this input. Under the stronger hypothesis that every nontrivial Dirichlet zero has real part at most \(\theta\), where \(1/2\leq\theta\leq2/3\), we prove \[ E(X)\ll_{\theta,\varepsilon}X^{g(\theta)+\varepsilon}, \qquad g(\theta)=\frac{\theta(5-4\theta)}{2(2-\theta)}. \] Thus every fixed \(\theta<2/3\) gives an exponent strictly below \(7/12\). The latter argument uses a classical fixed-modulus zero-density estimate and is independent of the new exponential-sum theorem.

Model averaging emerges as an attractive ensemble technique that strategically synthesizes all available information from potential candidate models to enhance estimation accuracy. Despite of having been widely practiced in various areas in biostatistics and economics, its application to causal inference is rather limited. We propose to estimate the average treatment effect (ATE) by forming the weighted combination of the robust rank-based estimators obtained from a series of candidate models, where the model weights, selected via the growing window cross-validation, dynamically adjust the importance of candidate models. The proposed robust rank-based model averaging (RRMA) is not only robust to handle potential outliers and heavy-tailed error distributions but also capable to reduce the risk of model mis-specification and enhance the flexibility and the accuracy of estimation. Theoretically, we establish the desirable asymptotic properties of RRMA, including the asymptotic optimality, consistency and asymptotic distribution. Moreover, motivated by the asymptotic distribution of RRMA, an easy-to-implement confidence interval is also developed for the ATE. Simulation experiments and an empirical application demonstrate the superiority of RRMA over existing competitors in estimation accuracy, coverage probabilities and widths of confidence intervals.

Using the geometric realization spectral sequence and the Serre spectral sequence, we compute $K_1(\operatorname{Mat}_*)$, the first K-theory group of the proto-exact category of pointed matroids, answering a question of Jun, Sistko and Wright.

We give the topological classification of some simplest univariate complex fewnomials, and find tropical fans of the fewnomial strata with each topological type. Specifically, for the support $ A=\{d,d+1,d+2,d+3,d+4\},\quad d\geq3, $ we classify the topological equivalence classes of polynomials having two critical values of ramification type $(2,2)$. There is one class for odd $d$ and two classes for even $d$. We describe the branches at infinity of the corresponding one-dimensional stratum using Newton polytopes of certain complete intersection representations of the sought strata, and determine their tropicalizations. The combinatorial types of the obtained tropical polynomials distinguishes all three kinds of topological equivalence classes. These computations suggest a correspondence between topological equivalence classes of polynomials and combinatorial types of tropicalizations of their branches at infinity, as well as a possible periodicity phenomenon under translations of the support.

We study the mean-squared error (MSE) rate-distortion function (RDF) of finite Gaussian mixture models (GMMs) in $\mathbb{R}^d$, for which neither the RDF nor an optimal test channel is known in closed form. It can be sandwiched between the genie-aided conditional RDF $R_{\mathbf{X} | T}(D)$, which can be characterized in closed-form for a GMM, and $R_{\mathbf{X} | T}(D)+I(\mathbf{X};T)$, where $T$ denotes the latent component label. We show that the upper bound can be achieved operationally by the proposed Gibbs-relabelling method (GRM), which conveys a label from the posterior $P_{T|\mathbf{X}}$ through channel simulation at rate $I(\mathbf{X};T)$, followed by vector quantization of the resulting exactly Gaussian conditional sources by reverse water-filling. Using the Shannon lower bound (SLB), we prove that GRM attains the SLB, and hence the RDF, of a general finite mixture if and only if every component attains its own SLB. For a GMM this holds throughout the fully active interval $D/d<\min_{k,m}\lambda_{k,m}$, where $\lambda_{k,m}$ are the eigenvalues of the component covariances, and for scalar GMMs the condition is also necessary. Since GRM cannot operate below the label cost $I(\mathbf{X};T)$, we propose a complementary stochastic scalar quantization (SSQ) scheme for low rates, which simulates an $M$-ary reconstruction index. We show that at low rates SSQ with $M\geq 2$ matches the Shannon RDF to first order: $R(D_{\max}-\delta)=\delta/(2\tau^2)+o(\delta)$, where $\tau^2$ is a directional sub-Gaussian proxy of the mixture, and quantify the price of a finite codebook through a strict rate gap at high rates and a distortion floor whose joint scaling in $M$ and number of mixture components is characterized. Numerical results illustrate both regimes and the finite-blocklength cost of channel simulation.

Nonlinear Constrained Optimisation (NCO) requires simultaneous satisfaction of the Karush-Kuhn-Tucker (KKT) conditions. Existing methods enforce these conditions through projections, penalties, barriers, or sequential approximations, potentially introducing computational complexity and convergence difficulties. For nonconvex problems, a KKT point need not be a local optimum. This paper reformulates NCO as stabilisation of a unified KKT sliding residual manifold and develops SM-KKT, a projection-free solver. Stationarity, equality feasibility, and regularised Fischer-Burmeister complementarity are integrated into a unified residual system, with complementarity encoding inequality feasibility, dual feasibility, and complementary slackness. The manifold's zero set is shown to be equivalent to the KKT conditions. The required Jacobian saddle-point inertia is characterised and shown, under stated assumptions, to be equivalent to the simultaneous satisfaction of the Linear Independence Constraint Qualification (LICQ) and Second-Order Sufficient Condition (SOSC). This characterisation establishes local stability of the structured correction algorithm under these conditions and distinguishes nondegenerate locally optimal KKT solutions from points failing second-order optimality. SM-KKT computes structured primal–dual corrections and applies a discrete sliding reaching law to drive the KKT residual into a prescribed boundary layer in finite time. Four nonlinear benchmarks show recovery of reference constrained optima with KKT residuals below $10^{-6}$, earlier boundary-layer reaching, and favourable KKT accuracy compared with projected primal-dual, augmented-Lagrangian, quadratic-penalty, and log-barrier baselines.

We study federated learning from continuously arriving data: each client samples from its own distribution, these distributions drift over time, and the server must track the minimizer of the clients' weighted average loss. Clients are not alike as they collect data at different rates, with different noise, and change at different speeds. The main question is how much history should each one keep? A single shared window must serve every client at once, and so makes a fast-changing client carry the sampling noise of a slow, nearly stationary one. We give each client a memory of its own, sized to that client's noise, arrival rate, and drift, and refresh it by restarted accelerated method that starts from a common point. The tracking error is then bounded by \[ R\left(\sum_m w_m^{3/2}\,\sigma_m\sqrt{\gamma_m/\lambda_m}\right)^{2/3}, \] for clients with weight $w_m$, noise $\sigma_m$, arrival rate $\lambda_m$, and drift rate $\gamma_m$ on a domain of diameter $R$. This is never worse than what any common window can achieve, and strictly better whenever noise and drift are spread unevenly across clients. A lower bound that puts different groups of clients on different drift clocks shows this quantity is the right one. The two bounds agree up to a constant whenever the clients occupy boundedly many refresh scales, and in general differ only by the sixth root of the number of scales. Experiments show the gain comes from the memory rather than from the solver: it persists across accelerated, stochastic-gradient, and one-step methods, and against a separately tuned common-memory baseline.

For a graph $G$, let $\nu(G)$ denote the maximum number of pairwise edge-disjoint triangles and let $\tau(G)$ denote the minimum number of edges meeting every triangle. Tuza's conjecture asserts that $\tau(G)\le 2\nu(G)$ for every graph $G$. We first note that Tuza's conjecture is equivalent to the following statement: for every fixed $c\ge 0$, if $G$ is a sufficiently large $n$-vertex graph with $e(G)\ge n^2/4-cn$, then $\tau(G)\le 2\nu(G)$. In this paper, we prove this statement for every $c<2$.

We study the variation of hull dimensions of symplectic linear codes under equivalence transformations preserving the symplectic distance. For a symplectic linear code $C$, we introduce an even-valued invariant $\Delta(C)$, which depends only on the equivalence class of $C$. This invariant can be expressed as the generic rank of an associated space of alternating matrices and, equivalently, as twice a corresponding matroid matching number. If $C$ has dimension $k$, we prove that the minimum symplectic hull dimension among all codes equivalent to $C$ is $k-\Delta(C)$. Consequently, $C$ is equivalent to a symplectic LCD code if and only if $\Delta(C)=k$. We further show that every symplectic MDS code of even dimension is monomially equivalent to a symplectic LCD code. For a symplectic self-orthogonal code $C$, we determine all attainable hull dimensions: for every integer $r$ with $0\le r\le \Delta(C)/2$, there exists a monomially equivalent code whose symplectic hull has dimension $k-2r$. We also refine a standard form for the generator matrix of a symplectic self-orthogonal code and derive the corresponding parity-check matrix and the Gram matrix of the dual code. In the self-dual case, we characterize $\Delta(C)$ through minors of an associated symmetric matrix. Furthermore, we relate hull variation to graph matchings, establish a bound for $\Delta(C)$, and give graph constructions producing symplectic self-dual MDS, optimal, and best-known codes.