arXiv Math @arxiv_math@qoto.org

This paper investigates multivariable extremum seeking using unit-vector control. By employing the gradient algorithm and a polytopic embedding of the unknown Hessian matrix, we establish sufficient conditions, expressed as linear matrix inequalities, for designing the unit-vector control gain that ensures finite-time stability of the origin of the average closed-loop error system. Notably, these conditions enable the design of non-diagonal control gains, which provide extra degrees of freedom to the solution. The convergence of the actual closed-loop system to a neighborhood of the unknown extremum point is rigorously proven through averaging analysis for systems with discontinuous right-hand sides. Numerical simulations illustrate the efficacy of the proposed extremum seeking control algorithm.

We present an explicit numerical approximation scheme, denoted by $\{X^n\}$, for the effective simulation of solutions $X$ to a multivariate stochastic differential equation (SDE) with a superlinearly growing $κ$-dissipative drift, where $κ>1$, driven by a multiplicative heavy-tailed Lévy process that has a finite $p$-th moment, with $p>0$. We show that for any $q\in (0,p+κ-1)$, the strong $L^q$-convergence $\sup_{t\in[0,T]}\mathbf{E} \|X^n_t-X_t\|^q=\mathcal{O} (h_n^γ)$ holds true, in particular, our numerical scheme preserves the $q$-moments of the solution beyond the order $p$. Additionally, for any $q\in (0,p)$ we establish strong uniform convergence: $\mathbf{E}\sup_{t\in[0,T]} \|X^n_t-X_t\|^q=\mathcal{O} ( h_n^{δ_q^\mathrm{uc}} )$. In both cases we determine the convergence rates $γ$ and $δ_q^\mathrm{uc}$. In the special case of SDEs driven solely by a Brownian motion, our numerical scheme preserves super-exponential moments of the solution. The scheme $\{X^n\}$ is realized as a combination of a well-known Euler method with a Lie--Trotter type splitting technique.

Elliptic integral-differential operators resembling the classical elliptic partial differential equations are defined over a compact d-dimensional p-adic domain together with associated Sobolev spaces relying on coordinate Vladimirov-type Laplacians dating back to an idea of Wilson Zúñiga-Galindo in his previous work. The associated Poisson equations under boundary conditions are solved and their $L_2$-spectra are determined. Under certain finiteness conditions, a Markov semigroup acting on the Sobolev spaces which are also Hilbert spaces can be associated with such an operator and the boundary condition. It is shown that this also has an explicitly given heat kernel as an $L_2$-function, which allows a Green function to be derived from it.

Deng, Hani, and Ma [arXiv:2503.01800] claim to resolve Hilbert's Sixth Problem by deriving the Navier-Stokes-Fourier equations from Newtonian mechanics via an iterated limit: a Boltzmann-Grad limit (\(\varepsilon \to 0\), \(N \varepsilon^{d-1} = α\) fixed) yielding the Boltzmann equation, followed by a hydrodynamic limit (\(α\to \infty\)) to obtain fluid dynamics. Though mathematically rigorous, their approach harbors two critical physical flaws. First, the vanishing volume fraction (\(N \varepsilon^d \to 0\)) confines the system to a dilute gas, incapable of embodying dense fluid properties even as \(α\) scales, rendering the resulting equations a rescaled gas model rather than a true continuum. Second, the Boltzmann equation's reliance on molecular chaos collapses in fluid-like regimes, where recollisions and correlations invalidate its derivation from Newtonian dynamics. These inconsistencies expose a disconnect between the formalism and the physical essence of fluids, failing to capture emergent, density-driven phenomena central to Hilbert's vision. We contend that the Sixth Problem remains open, urging a rethink of classical kinetic theory's limits and the exploration of alternative frameworks to unify microscale mechanics with macroscale fluid behavior.

The core group is an invariant of unoriented virtual links. We introduce a peripheral structure for the core group, in which the longitudes are sensitive to orientations. We show that the combination of the core group and its peripheral structure is equivalent, as a link invariant, to the combination of the $π$-orbifold group and its peripheral structure. Examples show that the peripheral structure of the core group can be used to verify noninvertibility of some knots and links.

Singularly perturbed systems (SPSs) are prevalent in engineering applications, where numerically solving their initial value problems (IVPs) is challenging due to stiffness arising from multiple time scales. Classical explicit methods require impractically small time steps for stability, while implicit methods developed for SPSs are computationally intensive and less efficient for strongly nonlinear systems. This paper introduces a Stabilized Multirate Explicit Scheme (SMES) that stabilizes classical explicit methods without the need for small time steps or implicit formulations. By employing a multirate approach with variable time steps, SMES allows the fast dynamics to rapidly converge to their equilibrium manifold while slow dynamics evolve with larger steps. Analysis shows that SMES achieves numerical stability with significantly reduced computational effort and controlled error. Its effectiveness is illustrated with a numerical example.

We use a complex-valued transformation of the Kuramoto model to develop an operator-description of the linear stability in finite networks of nonlinear oscillators. This mathematical approach offers analytical predictions for the linear stability of $q$-states, which include phase synchronization ($q = 0$) and waves with different spatial frequencies ($|q| > 0$). This approach seamlessly incorporates the presence of time delays (represented by phase-lags in the coupling). With this, we are able to analytically determine the specific combination of connectivity and time delays (phase-lags) that leads to any given $q$-state to be linearly stable. This approach offers a geometric perspective of linear stability in finite networks in terms of the connectivity and delays (phase-lag), and it opens a path to designing and controlling the spatiotemporal dynamics of individual oscillator networks.

In this paper we consider the problem of computing the stationary distribution of nearly completely decomposable Markov processes, a well-established area in the classical theory of Markov processes with broad applications in the design, modeling, analysis and optimization of computer systems. We design general classes of algorithmic solution approaches that exploit forms of mixed-precision computation to significantly reduce computation times and that exploit forms of iterative approximate methods to mitigate the impact of inaccurate computations, further reduce computation times, and ensure convergence. Then we derive a mathematical analysis of our general algorithmic approaches that establishes theoretical results on approximation errors, convergence behaviors, and other algorithmic properties. Numerical experiments demonstrate that our general algorithmic approaches provide significant improvements in computation times over the most efficient existing numerical methods.

We study distributionally robust optimization (DRO) problems with uncertainty sets consisting of high dimensional random vectors that are close in the multivariate Wasserstein distance to a reference random vector. We give conditions under which the images of these sets under scalar-valued aggregation functions are equal to or contained in uncertainty sets of univariate random variables defined via a univariate Wasserstein distance. This allows to rewrite or bound high-dimensional DRO problems with simpler DRO problems over the space of univariate random variables. We generalize the results to uncertainty sets defined via the Bregman-Wasserstein divergence and the max-sliced Wasserstein and Bregman-Wasserstein divergence. The max-sliced divergences allow us to jointly model distributional uncertainty around the reference random vector and uncertainty in the aggregation function. Finally, we derive explicit bounds for worst-case risk measures that belong to the class of signed Choquet integrals.

We consider a history-dependent variational inequality (P) which models the frictionless contact between a viscoelastic body and a rigid obstacle covered by a layer of soft material. The inequality is expressed in terms of the displacement field, is governed by the data f (related to the applied body forces and surface tractions) and, under appropriate assumptions, it has a unique solution, denoted by u. Our aim in this paper is to perform a sensitivity analysis of the inequality (P), including the study of the regularity of the solution operator $f \mapsto u$. To this end, we start by proving the equivalence of (P) with a fixed point problem, denoted by (Q). We then consider an associated optimal control problem, for which we present an existence result. Then, we prove the directional differentiability of the solution operator and show that the directional derivative is characterized by a history-dependent variational inequality with time-dependent constraints. Finally, we prove two well-posedness results in the study of problems (P) and (Q), respectively, and compare the two well-posedness concepts employed.

The study of stochastic variational principles involves the problem of constructing fixed-endpoint and adapted variations of semimartingales. We provide a detailed construction of variations of semimartingales that are not only fixed at deterministic endpoints, but also fixed at first entry times and first exit times for charts in a manifold. We prove a stochastic version of the fundamental lemma of calculus of variations in the context of these variations. Using this framework, we provide a generalization of the stochastic Hamilton-Pontryagin principle in local coordinates to arbitrary noise semimartingales. For the corresponding global form of the stochastic Hamilton-Pontryagin principle, we introduce a novel approach to global variational principles by restricting to semimartingales obtained as solutions of Stratonovich equations.

The Ishango Bone, discovered in 1950 near the Semliki River in present-day Democratic Republic of Congo and dated to over 24,000 years ago, features a series of 16 grouped numerical notches arranged across three columns. The interpretation of these markings has long been debated, ranging from numerical intention to decorative symbolism. This study presents a newly identified structural pattern, uncovering four irregularities in the numerical groupings that, when resolved, reveal a consistent internal logic across all three columns. The analysis identifies repeating sums, dualistic pairing structures, and cross-column symmetry, pointing toward a deliberate and complex mathematical arrangement. These findings suggest the notches may have functioned as a form of reference system, potentially using stone markers for visual or narrative purposes. Hypotheses include links to cosmic cycles, mathematical instruction, or mythological integration. While interpretations remain speculative, this study argues for an interdisciplinary reevaluation of the Ishango Bone's significance in terms of mathematical, cultural, and cosmological understanding. It offers new insights into the symbolic thinking and cognitive capacities of prehistoric societies and challenges conventional assumptions about the origins of numeracy.

In 1969 J. Verhoeff provided the first examples of a decimal error detecting code using a single check digit to provide protection against all single, transposition and adjacent twin errors. The three versions of such a code that he presented are length 3-digit codes with 2 information digits. Existence of a 4-digit code would imply the existence of 10 such disjoint 3-digit codes. This paper presents 3 pairwise disjoint 3-digit codes. The codes developed herein, have the property that the knowledge of the multiset of digits included in a word is sufficient to determine the entire codeword even though their positions were unknown. Thus the codes are permutation-free, and this fulfills his desire to eliminate cyclic errors. Phonetic errors, where 2 digit pairs of the forms X0 and 1X are interchanged, are also eliminated.

In this manuscript, we extend the global gradient estimates for positive solutions to the heat equation under a general compact Finsler $CD(-K,N)$ geometric flow and derive the corresponding Harnack inequality.

We present a mathematical framework for quantifying energy efficiency in intelligent systems by linking energy consumption to information processing capacity. Our proposed watts per intelligence metric integrates algorithmic thermodynamic principles of Landauer with computational models of machine intelligence. By formalising the irreversible energy costs of computation, we derive rigorous lower bounds on energy usage of algorithmic intelligent systems and their adaptability. We introduce theorems that constrain the trade offs between intelligence output and energy expenditure. Our results contribute to design principles for energy efficient intelligent systems.

In 1971, by induction on $n$ and using a two-term linear recurrence relation, Graham and Pollak got a beautiful formula $$\det(D_n)=-(n-1)(-2)^{n-2}$$ on the determinant of distance matrix $D_n$ of a tree $T_n$ on $n$ vertices. The recurrence relations are very crucial when proving this formula by inductive method: in 2006, Yan and Yeh used two-term and three-term recurrence relations; in 2020, Du and Yeh used a homogeneous linear three-term recurrence relation. In this paper, we analyze the subtree structure of the tree and find four-term, five-term, six-term and seven-term homogeneous linear recurrence relations on $\det(D_n)$, as a corollary new proofs of Graham and Pollak's formula can be given.

A red-white coloring of a nontrivial connected graph $G$ is an assignment of red and white colors to the vertices of~$G$. Associated with each vertex $v$ of $G$ of diameter $d$ is a $d$-vector, called the code of $v$, whose $i$th coordinate is the number of red vertices at distance $i$ from $v$. A red-white coloring of $G$ for which distinct vertices have distinct codes is called an ID-coloring of $G$. In 2025, a criterion to determine whether a red-white coloring of a path is an ID-coloring or not was presented by Kono, with the aid of a result shown by Marcelo et al. in 2024. The criterion utilizes the fact that ID-colorings of paths are ``opposite'' of colorings with a certain symmetry. In this paper, we establish a similar criterion that can be applied for cycles whose order is a prime number at least 3. In order to do so, we employ an analogous approaches used for the criterion for paths, i.e., we pay attention to symmetries of given red-white colorings of cycles.

The goal of this paper is to show that Stokes data coming from flat bundles form a locally geometric derived stack locally of finite presentation. This generalizes existing geometricity results on Stokes data in four different directions: our result applies in any dimension, $\infty$-categorical coefficients are allowed, derived structures on moduli spaces are considered and more general spaces than those arising from flat bundles are permitted.

Let $G$ be a finite group. For subgroups $U$ and $V$ let $1_U^G$ and $1_V^G$ be the permutation characters for the action of $G$ on the right cosets of $U$ and $V$, respectively. Let $N$ be a normal subgroup of $G$. Norbert Klingen, in his book, shows that if $1_U^G=1_V^G$, then $1_{NU}^G=1_{NV}^G$. We give a counterexample to an argument in his proof and we give a new proof of this statement.

A status updating system is considered in which multiple processes are sampled and transmitted through a shared channel. Each process has its dedicated server that processes its samples before time stamping them for transmission. Time stamps, however, are prone to errors, and hence the status updates received may not be credible. Our setting models the time stamp error rate as a function of the servers' busy times. Hence, to reduce errors and enhance credibility, servers need to process samples on a relatively prolonged schedule. This, however, deteriorates timeliness, which is captured through the age of information (AoI) metric. An optimization problem is formulated whose goal to characterize the optimal processes' schedule and sampling instances to achieve the optimal trade-off between timeliness and credibility. The problem is first solved for a single process setting, where it is shown that a threshold-based sleep-wake schedule is optimal, in which the server wakes up and is allowed to process newly incoming samples only if the AoI surpasses a certain threshold that depends on the required timeliness-credibility trade-off. Such insights are then extended to the multi-process setting, where two main scheduling and sleep-wake policies, namely round-robin scheduling with threshold-waiting and asymmetric scheduling with zero-waiting, are introduced and analyzed.