Papers

We introduce strong normality for holomorphic curves and logharmonic mappings, extending classical normality concepts. We establish an extension of the rescaling characterization due to Lohwater and Pommerenke for not strongly-normal maps. In addition, we also study the Bloch mappings, little-Bloch mappings and prove Zalcman-Pang type rescaling results for them. The framework is further extended to strongly $\varphi$-normal mappings, yielding a unified treatment across these settings.

Control barrier functions (CBFs) are a widely applied modular tool to ensure safe operation of nonlinear dynamical control systems. However, for their construction accurate knowledge of the system dynamics is typically needed. This requirement was recently alleviated for relative-degree-one systems using techniques from prescribed performance control (PPC) or funnel control (FC). This article extends the model-free CBF design to nonlinear systems of arbitrary relative degree. Moreover, we show with a simple example that a straightforward extension of existing results for relative-degree-one systems fails. Instead, we utilize novel techniques from funnel control to characterize a subset of the controls satisfying a CBF condition without requiring a dynamic model or state measurement. Finally, we demonstrate the applicability of our results on a seven degrees of freedom robotic manipulator with relative degree two.

The independence polynomial of a graph encapsulates all independent sets of differing sizes, a task classified as NP-hard in theoretical computer science. This article examines the independence polynomial of zero divisor graphs in commutative rings. We demonstrate that the independent sets, represented as a sequence of coefficients of the independence polynomial, exhibit unimodality and log-concavity. Therefore, for the independence polynomial of some zero divisor graphs, the unimodal conjecture is true. Additionally, the characteristics of the zeros of the independence polynomial are delineated, along with their corresponding annular regions on the plane.

Despite the widespread use of PID controllers in engineering practice, designing optimal PID parameters has long been regarded as a challenging problem in both theory and practice, particularly when faced with uncertain nonlinear dynamical systems. Based on the authors' PID control theory established recently for MIMO nonlinear uncertain systems (Zhao and Guo, 2022), which provides a concrete PID parameter set for global stability of PID controlled systems, this paper further proposes a near-optimal PID tuning method, where only input-output (zeroth-order) data on the control performance is available. The tuning method is formulated as a constrained optimization problem and solved by an iterative learning algorithm, referred to as HRS-KW algorithm, that combines a hysteretic random search with the Kiefer-Wolfowitz algorithm, aiming at utilizing the advantages of both global exploration and local gradient acceleration. This method operates without requiring precise structural knowledge of the system dynamics, yet its almost sure convergence to an epsilon-optimal solution for the PID parameters can be guaranteed in theory while ensuring closed-loop system stability. Simulation results illustrate that our HRS-KW algorithm outperforms other related optimization methods, exhibiting better convergence to the prescribed epsilon-optimal performance set.

We study weak solutions of the one-dimensional pressureless Euler-Poisson-alignment system. When smooth solutions develop singularities, distributional weak solutions are not unique. We introduce an entropy-based selection principle via an associated scalar balance law with time-dependent flux and establish global well-posedness for its entropy solutions. The resulting entropic solution yields a uniquely selected weak solution of the Euler-Poisson-alignment system. In the attractive regime, it is compatible with sticky particle dynamics, while in the repulsive regime atomic states may disperse, revealing a fundamental qualitative difference between the two cases.

Let R denote the class of all computable, causal functionals that are rate-independent in the classical sense (invariant under monotone time reparametrizations), and let Pi_n be the Preisach extremum stack of an input sequence u_{0:n}. We prove a characterization theorem establishing that every F in R satisfies Fu = f(Pi_n) for a computable f, and derive two information-theoretic results. First, under any probability measure on u_{0:n}, the equality I(u_{0:n}; Fu) = I(Pi_n; Fu) holds for every F in R and is an immediate corollary of the characterization theorem. Second, the main result: Pi_n is a Shannon-minimal sufficient statistic in the sense that I(u_{0:n}; Pi_n) <= I(u_{0:n}; S) for every random variable S from which all R-queries are computable. The proof uses the finite indicator family of [Frydrych, 2026] to reconstruct Pi_n from any sufficient S. As a corollary, online maintenance of Pi_n suffices for rate-independent estimation: the NNLS estimator of the Preisach measure mu can be assembled from the incremental stack process (Pi_t)_{t=0}^n in O(k * L^2) memory per step, where k = |Pi_t| and L is the grid resolution.

Kan's discovery of dynamical systems with two attractors whose basins of attraction both have full support, featured specific examples of skew product systems of interval diffeomorphisms forced by expanding circle maps. The interval diffeomorphisms are polynomial maps and can hence be considered on the Riemann sphere. We consider the resulting skew product systems of holomorphic maps on the Riemann sphere forced by expanding circle maps, and establish the existence of three attractors whose basins of attraction all have full support and are thus intermingled.

Predictive tasks constitute subprocedures within human–AI interactions (HAIs) involving professionals and AI systems. Complementarity occurs when an HAI outperforms a benchmark defined by its constituent agents. Existing formalizations, however, focus on a single human and a single AI, leaving multi-agent interactions insufficiently characterized. We introduce a tree-based mathematical framework for multi-agent complementarity. Ordered agent-role configurations decorate the leaves of rooted planar binary trees and local aggregation rules generate protocol outputs evaluated against either a best-average or a pointwise-min benchmark. We establish four results. First, under linear pooling, every protocol tree provides a bijective coordinate system for the interior of the simplex of global leaf weights. Arbitrary trees with the same ordered leaves admit output- and complementarity-preserving reparameterizations, realized locally through path-independent Tamari rotations. However, HAI constraints can violate admissibility preservation between equivalent protocols. Second, for regression under squared loss, we characterize complementarity through residual correction and derive optimal linear-pooling weights. We also characterize protocol-indifference loci under identical local parameters. Third, selector-based protocols cannot achieve complementarity relative to the pointwise-min benchmark, independently of task and loss. Fourth, in binary classification, HAIs cannot achieve complementarity under endpoint-monotone losses and the pointwise-min benchmark, with an analogous obstruction for multiclass cross-entropy. Numerical illustrations examine the main claims. We provide a mathematical foundation for analyzing multi-agent predictive subprocedures within HAIs, establishing a first step toward understanding how interaction, aggregation, and real-world constraints affect HAI performance.

Text-to-video (T2V) models trained on large-scale web data can generate undesired content, motivating interventions that reduce harmful outputs without sacrificing visual quality. Activation steering offers an attractive mechanistic alternative to finetuning and prompt filtering, but existing T2V steering methods remain limited, typically applying coarse, non-anticipative interventions that can lead to oversteering and content degradation. To close this gap, we propose Latent Activation Linear-Quadratic Regulator (LA-LQR), a reduced-order optimal control framework for minimally invasive T2V steering. LA-LQR formulates T2V inference as a dynamical system and computes closed-loop feedback interventions that steer activations toward desired feature setpoints while penalizing unnecessary perturbations. To make optimal control feasible for high-dimensional video activations, we project activations onto a low-dimensional, task-relevant subspace derived from contrastive prompt pairs, estimate local linear dynamics in this latent space, and solve a latent LQR problem to obtain timestep- and layer-specific steering signals. We provide theoretical bounds relating latent setpoint tracking to raw activation-space feature control, and empirically validate the fidelity of the reduced latent dynamics. On concept steering and video safety benchmarks, LA-LQR reduces unsafe generations relative to baselines, while preserving prompt fidelity and visual quality.

We obtain weighted centro-affine Bochner formulas on spherical caps associated with smooth strictly convex hypersurfaces. As a consequence, we prove weighted Poincaré inequalities on caps and on intersections of caps for a class of weights depending on the position vector $X$ of the hypersurface. In the unconditional case, we obtain a centro-affine Poincaré inequality with weight $|X|^2$, which is used to prove a Brunn–Minkowski inequality for the $(n+2)$-nd dual quermassintegral. We also establish an $L_0$-Brunn–Minkowski inequality for the $q$-th dual quermassintegral for $q\in(0,n)$, with equality only for dilates, and an $L_p$-Brunn–Minkowski inequality for $q=n+\alpha$ whenever \[ 0<\alpha\le \frac{2p(1-p)}{2-p}, \] which in particular covers the range $q\in(n,n+6-4\sqrt{2}]$ for suitable $p\in(0,1)$. These Brunn–Minkowski inequalities imply weighted centro-affine Poincaré inequalities and uniqueness results for the $L_{p,q}$-Minkowski problem in the unconditional class. Our main contribution is the introduction of a flat logarithmic centro-affine geometry on the positive orthant $(0,\infty)^n$, adapted to the multiplicative structure of the $L_0$-sum. In this geometry, a Bochner formula yields a sharp Poincaré inequality, as well as a new proof of the centro-affine Poincaré inequality with constant $n$ due to Kolesnikov–Milman, for unconditional bodies and unconditional functions.

We study $n$-dimensional annuli for $n\,\in\,\{2,\dots,N\}$ with $N\,<\,\infty$. We choose a non-dimensional setting such that for any fixed $n $ and given number ${\cal A}>0$ the annuli ${\Omega}_{(n),\cal A}$ are defined as space between two concentrical balls with radii ${\cal A}/2$ and ${\cal A}/2 +1$ in ${ R}^{n}$. For these geometries we provide calculated (precise) Poincaré constants. These depend on ${\cal A}$ and the dimension $n$. Additionally we find a direct match of the Poincaré constants for solenoidal vector fields in ${R}^{n}$ and the Poincaré constants for scalar functions in ${ R}^{n+2}$ (all with vanishing Dirichlet traces). This is based on the relation of the first eigenvalues and one eigenfunction of the (scalar) Laplace and the Stokes operator. In addition we consider the limit ${\cal A}\,\to\,0$. In this context problems in domains ${\Omega}_{(n),\sigma}^{*}$ are investigated. These domains enable us to use the Green's function of the Laplacian with vanishing Dirichlet traces to show that the first eigenvalue here tends to the first eigenvalue of the corresponding problem on the open unit ball in ${ R}^{n}$. On the other hand, we take advantage of the so-called small-gap limit for ${\cal A}\to\infty$.

We construct a correspondence with a character equation of discrete series representations of $\mathrm{GSp}_{2n}$ over a $p$-adic field and its inner form for $p>2$ and $n\geq 2$. This is a generalization of the works by Chan and Gan and the local Jacquet-Langlands correspondence. Moreover, our results partially reveal what the local Langlands correspondence of $\mathrm{GSp}_{2n}$ and its inner form over $p$-adic fields should be.

We study decentralized stochastic smooth convex optimization, where $M$ workers minimize an average objective using local stochastic gradients and neighbor-only communication over a fixed gossip network. A central question in this setting is to determine the largest number of workers that can be used under a total budget of $N$ gradient samples while still preserving the centralized $O(1/\sqrt N)$ statistical rate. We introduce an accelerated decentralized method that preserves this rate for up to $\smash{M\lesssim \sqrt{\rho}\,N^{3/4}}$ workers, where $\rho$ is the spectral gap of the gossip network, improving the best prior maximal scaling of $\smash{M\lesssim \rho\sqrt N}$. The method is based on a one-step-delayed stochastic acceleration scheme that enables workers to interleave minibatching with accelerated gossip while controlling residual disagreement, and its guarantee depends only logarithmically on the optimum-local heterogeneity. We also establish a matching lower bound for linear-span decentralized first-order methods, showing that the method is optimal up to logarithmic factors.

A non-zero unital ring $R$ is said to be centrally essential if for every nonzero element $a$ of $R$, there exist non-zero central elements $x$ and $y$ with $ax = y$. In the paper, almost fully prime centrally essential rings are described in terms of ideal extensions, centrally essential Dorroh extensions, and trivial extensions.

Let $P_r$ denote an integer with at most $r$ prime factors, counted with multiplicity. It is known that every sufficiently large integer $N$ satisfying $N \equiv 3 \pmod{24}$ and $5 \nmid N$, can be written in the form $N= x_1^2+x_2^2+x_3^2$ where $x_1,x_2,x_3$ are integers. In this paper, we prove that the above representation in the following two different forms (i) $x_1x_2x_3$ is a $P_{67}$-number; (ii) each $x_i$ is a $P_{27}$-number. This result improves on the previous result of Waibel\cite{Wa}, in which $P_{72}$ was obtained in place of $P_{67}$. The proofs combine the higher-dimensional sieve, a Richert-type weighted sieve method introduced by Cai \cite{Cai} with a Bombieri-Vinogradov type result given by Waibel\cite{Wa}. Applying the same method in a one dimensional sieve setting, we also show that every sufficiently large $N$ not of the form $4^k(8l+7)$ can be written in the form \[ N = x^{2} + y^{2} + (2^{a} z)^{2}, \] where $x,y,a,z$ are non-negative integers and $z$ is a $P_{18}$-number. This improves upon a result of Banerjee \cite{Ban} who obtained $P_{118}$ in place of $P_{18}$.

Let \((A, L)\) be a polarized abelian variety of dimension \(g \geq 1\) over an algebraically closed field of characteristic zero. We prove that every nontrivial line bundle \(P\) in the connected component \(\operatorname{Pic}^0(A)\) of the Picard variety is arithmetically Cohen–Macaulay (ACM) with respect to \(L\). For \(g \geq 2\) and any fixed nontrivial \(P \in \operatorname{Pic}^0(A)\), we construct by induction an infinite sequence of indecomposable ACM vector bundles \(E_r\) of every rank \(r \geq 1\). In addition, this paper studies classification questions for ACM line bundles and shows that, for abelian varieties of dimension at least two, the category of ACM bundles is of wild representation type. This paper settles the existence problem for nontrivial ACM bundles on polarized abelian varieties and supply large explicit families of indecomposable examples

A ring $R$ with a non-zero identity element is said to be centrally essential if for any non-zero element $a\in R$, there exist non-zero central elements $x,y\in R$ such that $ax = y$. We describe centrally essential rings in a large subclass of formal triangular matrix rings and in a subclass of the matrix ring $M_3(R)$ over the ring $R$.

In this paper, we present a GPU-accelerated framework for nonlinear model predictive control (NMPC) based on direct transcription and second-order interior-point methods. Many real-world systems exhibit nonlinear dynamics that cannot be accurately captured by linear models, motivating the use of NMPC. However, NMPC requires the repeated real-time solution of optimal control problems (OCP), which become computationally demanding large-scale nonlinear programs (NLPs) after transcription. Although GPU acceleration has emerged as a promising approach for nonlinear optimization, existing GPU-based NMPC workflows reconstruct structurally identical OCPs at each solve. This introduces substantial overhead even though successive solves differ only through updated system measurements or reference trajectories. To address this limitation, we introduce a parametric interior-point formulation that exploits the fixed structure of transcribed OCPs, enabling reuse of structure-dependent computations (e.g., symbolic factorization in sparse Cholesky) across re-solves. We evaluate the proposed framework on distillation column and 2D heated plate benchmarks against state-of-the-art CPU and GPU configurations. The results show that the framework achieves over an order-of-magnitude speedup in total NMPC run times. These improvements are primarily driven by reduced per-iteration solve times, with GPU execution achieving up to a 94% reduction compared to the baseline. Overall, the results demonstrate the effectiveness of exploiting repeated problem structure in GPU-accelerated NMPC and highlight the potential of the proposed framework to expand the envelope of real-time NMPC applications.

We introduce a bias-corrected weighted-logistic (bcw) formulation for ODE models of gene regulatory networks. Each gene's regulatory function is a single sigmoid $\sigma(\lambda S_i+\lambda b_i)$ of the signed weighted regulator sum $S_i$, with a combinatorial bias $b_i=-\lambda^{-1}\log(2^{m_i}-1)$ depending only on the in-degree $m_i$ and the steepness $\lambda$. We prove this is the unique single-sigmoid correction recovering the critical-point value $1/2^{m_i}$ of the product-of-logistics model and sharing its canonical equilibrium $x_i^*=\kappa_i/(\gamma_i 2^{m_i})$ under self-consistency. The bcw system inherits global $C^\infty$ well-posedness, an explicit Lipschitz bound, a positively invariant box, and strictly positive basal output. We derive an exact algebraic identity for the discrepancy $f^{prod}-f^{bcw}$ as a sum over proper non-empty regulator subsets; a slope ratio $2-2^{1-m_i}$ at the critical point; a curvature mismatch breaking the symmetry $\sigma''=0$ for $m_i\ge2$; a no-go result excluding global equivalence by any state-independent constant; and a mean-field reading of the bias as a zero-order log-sum-exp approximation. At a shared equilibrium the Jacobians satisfy $J^{bcw}=D(J^{prod}+\Gamma)-\Gamma$ with $D=\mathrm{diag}(2-2^{1-m_i})$, giving a stability dichotomy. In the contractive regime $\gamma_i>\kappa_i\lambda m_i/4$ both systems converge globally and exponentially to the shared equilibrium at rate $\alpha=\min_i(\gamma_i-\kappa_i\lambda m_i/4)$. A two-gene self-activating negative-feedback motif illustrates the theory: the bcw form undergoes Hopf bifurcation at $2/3$ of the product threshold and sustains limit cycles where the product is linearly stable, with eigenvalue amplification factor $D=3/2$.

The slant function on a regular surface in three-space, introduced by Koenderink, is the inner product of a fixed light direction with the unit normal; under the standard Lambertian model it gives the brightness of the surface illuminated from that direction. We study the singularities of this function and show that their type is characterised by the differential geometry of the parabolic set and its spherical image under the Gauss map. At every point of the surface, we first describe the light directions along which the slant function has a degenerate singularity; such singularities arise only at the parabolic points. We then determine the precise singularity type at representative parabolic points. At the first-order inflection and first-order vertex of Gauss, and at the cusp of Gauss at which the parabolic set is regular, the finer geometry of the spherical image is reflected in the singularity of the slant function.

We prove a No-Core Principle for stationary actions of countable groups. Namely, if a Borel set intersects almost every orbit in finitely many points and has positive measure, then it is supported, modulo null sets, on the finite-orbit part of the action. This extends to stationary actions a basic regularity phenomenon known for measure-preserving actions. We apply this principle to the geometry of Stationary Random Subgroups. For a finitely generated group, we prove that the Schreier graph of a stationary random subgroup has almost surely 0,1,2, or infinitely many ends. Finally, we contrast this probabilistic regularity with the topological notion of Boomerang subgroups: for every $k\geq 3$, including $k=\aleph_0$, we construct a Boomerang subgroup of $\mathbb{F}_3$ whose Schreier graph has exactly $k$ ends.

We consider the isothermal Euler-Poisson system with linear damping on a periodic domain in the large damping regime. For arbitrarily large smooth initial data with density bounded away from vacuum, we prove the global-in-time existence of smooth solutions. The argument is based on a large-damping bootstrap scheme, a modified density estimate revealing hidden parabolic dissipation, top-order weighted cancellations, and a comparison with an auxiliary drift-diffusion–Poisson system. We further prove exponential relaxation to the homogeneous equilibrium and establish a large-data overdamped limit as the damping coefficient tends to infinity. In the slow time scale, the density converges quantitatively to the large smooth solution of the drift-diffusion–Poisson system with the same initial density. After subtracting a fast initial layer from the rescaled flux, the flux converges quantitatively to the corresponding drift-diffusion flux.

We study the problem of estimating smooth optimal transport (OT) maps between two probability distributions under differential privacy (DP) constraints. Leveraging wavelet-based density estimators and recent stability bounds for smooth OT maps, we propose differentially private estimators that apply to both central and local DP models. Our main estimator achieves near-minimax optimal rates in dimension $d \geq 2$, and we complement it with a quantile-based estimator that attains minimax optimal rates in dimension $d = 1$ under central DP. We further establish matching minimax lower bounds, confirming the near-optimality of our approach. To the best of our knowledge, this constitutes the first differentially private procedure for OT map estimation with minimax optimality guarantees.

This paper investigates the asymptotic behavior of the norm, as a bounded operator in $L^2(\mathbb{R})$, of the $n$-th Calderón commutator $T_{A,n}$ on the graph of a Lipschitz function $A:\mathbb{R}\to\mathbb{R}$. We prove the estimate $\|T_{A,n}\|_{L^2\to L^2} \leq Cn\|A'\|_\infty^n$, thus formalizing a claim by Mateu and Verdera via a symmetrization strategy and the $T1$ theorem. We also show that additional regularity on $A$ yields sublinear growth in $n$. Specifically, for $A$ supported in $[0,1]$, the bound improves to a behavior of the form $\sqrt{n}\|A'\|_\infty^n$ under a Dini condition on $A'$, or if $A'$ belongs to the logarithmic Besov space $B^{1,0}_{1,1}(\mathbb{R})$. This space contains all compactly supported functions in the Sobolev spaces $H^s(\mathbb{R})$ for $0<s<1,$ as well as functions of bounded variation. These refined estimates are established through an alternative framework based on Hörmander-type conditions and interpolation, bypassing the standard $T1$ approach. Counterexamples are provided to demonstrate that the Dini and Sobolev fractional regularity conditions are incomparable.

Let $s(n)=s_{10}(n)$ be the decimal sum-of-digits map. Building on the circle-method framework of Drmota-Mauduit-Rivat for digital restrictions on primes, we make the constants explicit at the points needed to obtain an effective surjectivity statement for digit sums of primes. We exhibit an explicit integer $M < 1.78 \times 10^{32}$ such that every integer $m \ge M$ with gcd$(m,9)=1$ occurs as $s(p)$ for at least one prime $p$. We also prove an explicit lower bound $A_m(10^{2m/9}) \ge C_q(m) 10^{2m/9}/m^{3/2}$, where $C_q(m)$ is explicit, positive above the sufficient threshold, and bounded away from $0$ along each admissible residue class. Existence of a non-numerical threshold follows from the DMR asymptotic theory and was noted by Harman; to the best of our knowledge, this is the first published explicit numerical threshold for this surjectivity statement. The proof combines explicit major-arc estimates, a fully explicit replacement for DMR's implicit prime exponential-sum input, and constant-tracked Type-II minor-arc estimates. As an application, we prove the infinitude of OEIS A070027, the primes whose iterated digit-sum chain remains prime until reaching a one-digit prime. We also record related effective consequences for additive primes and digit-sum additive decompositions.

This paper presents a method of multivariate scattered data interpolation and approximation that produces optimal Lipschitz-continuous approximation, subject to the desired monotonicity constraints. This method relies on tight upper and lower approximations to the data, and is similar in its spirit to the nearest-neighbour approximation but does not suffer from discontinuities. Local Lipschitz interpolation and Lipschitz smoothing are also presented. This approach falls under the umbrella of instance-based approximation with no training phase, and it is suitable for GPU-based parallelisation. A Python GPU-friendly package LipFit which implements the methods discussed is discussed.

We analyze cluster states in an ensemble of Stuart-Landau oscillators with two subpopulations of different frequencies. Our main goal is to compare the descriptions of the system's dynamics obtained via the standard first-order phase approximation and the second-order phase reduction. We demonstrate that the second-order model not only provides quantitative improvements in the description but also reveals new dynamical states not present in the standard Kuramoto theory. In particular, it describes bistability of synchronous states in the minimal setup of two coupled oscillators and the existence of three-cluster states, forbidden in the first-order phase description by the Watanabe-Strogatz theory. The very good agreement between the second-order approximation results and the results of numerical simulations of the original Stuart-Landau network highlights the usefulness of high-order phase-reduction models.

It has been proven that real and regular soliton solutions of the KP equation are classified in terms of the totally nonnegative Grassmannian. It is well known that vertex operators can be used to construct soliton solutions. In this paper, we consider several regular soliton solutions and study their combinations through products of vertex operators. In general, the resulting solutions become singular. Totally nonnegative elements are parametrized by the Le-diagrams introduced by Postnikov. We show that the resulting singular solutions can be parametrized by Go-diagrams, which extend Le-diagrams and arise in the Deodhar decomposition of the Grassmannian.

Suppose that pebbles are distributed on the vertices of a directed graph D. A directed pebbling step u -> v along an arc u -> v removes two pebbles from u and places one pebble on v. We study the stacking number stack(D), the least integer t >= 2 such that every configuration with t pebbles can be transformed by a finite sequence of pebbling steps into a configuration with all pebbles on a single vertex, and the clearing number clear(D), defined analogously by requiring a final configuration with one pebble. Our main result is the formula stack(C_n) = n(2^{n-1}-1)+1 for the directed n-cycle C_n, for n >= 2. We also prove that, for finite simple digraphs with at least two vertices, stack(D) is defined precisely for strongly connected digraphs, and clear(D) is defined precisely for strongly connected digraphs whose directed cycle lengths have greatest common divisor 1.

We establish a small-time large deviation principle for the stochastic Landau–Lifshitz–Baryakhtar equation using the framework of exponential equivalence. This result characterizes the asymptotic behavior of the solution on very short time scales. In particular, it shows that, as the stochastic thermal fluctuations become small, the magnetization remains exponentially concentrated near its initial state, reflecting the short-time stability of the magnetization dynamics. The associated rate function provides a quantitative measure of deviations from the initial state and the resulting short-time stability.