arXiv Math @arxiv_math@qoto.org

We study a class of infinite-horizon impulse control problems with execution delay in discrete time. Using probabilistic methods, particularly the notion of the Snell envelope of processes, we construct an optimal strategy among all admissible strategies for both risk-neutral and risk-sensitive utility functions. Furthermore, we establish the existence of bounded $ε$-optimal strategies. This framework provides a robust approach to handling execution delays in discrete-time stochastic systems.

We show that the energy critical Wave Maps equation from $\mathbb{R}^{2+1}$ to $\mathbb{S}^2$ and restricted to the co-rotational setting with co-rotation index $k = 2$ admits finite time blow up solutions of finite energy on $(0, t_0]\times \mathbb{R}^2$, $t_0>0$, and concentrating two concentric bubble profiles at the frequency scales $λ_1(t) = e^{α(t)},\,α(t)\sim \big|\log t\big|^{β+1}$, as well as $λ_2(t) = t^{-1}\cdot \big|\log t\big|^β$. The parameter $β>\frac32$ can be chosen arbitrarily. This shows that soliton resolution scenarios with finite time blow up and $N = 2$ collapsing profiles, i. e. bubble trees, do occur for this equation.

Solutions to nonlinear integro-differential systems are regular outside a negligible closed subset whose Hausdorff dimension can be explicitly bounded from above. This subset can be characterized using quantitative, universal energy thresholds for nonlocal excess functionals. The analysis is carried out via the use of nonlinear potentials and allows to derive fine properties of solutions under sharp assumptions on data and kernel coefficients.

The $S$-gap shifts have a dynamically and combinatorially rich structure. Dynamical properties of the $S$-gap shift can be related to the properties of the set $S$. This interplay is particularly interesting when $S$ is not syndetic such as when $S$ is the set of prime numbers or when $S=\{2^n\}$. It is a well known result that the entropy of the $S$-gap shift is given by $h(X) = \log λ$, where $λ>0$ is the unique solution to the equation $\sum_{n \in S} λ^{-(n+1)}=1$. Fix a point $w$ of the full shift $\{1,2, \dots, k\}^\mathbb{Z}$. We introduce the $(S,w)$-gap shift which is a generalization of the $S$-gap shift consisting of sequences in $\{0,1, \dots, k\}^\mathbb{Z}$ in which any two $0$'s are separated by a word $u$ appearing in $w$ such that $|u|\in S$. We extend the formula for the entropy of the $S$-gap shift to a formula describing the entropy of this new class of shift spaces. Additionally we investigate the dynamical properties including irreducibility and mixing of this generalization of the $S$-gap shift.

Motivated by emerging urban applications in commercial, public sector, and humanitarian logistics, we revisit continuous $p$-hub location problems in which several facilities must be located in a continuous space such that the expected minimum Manhattan travel distance from a random service provider to a random customer through exactly one hub facility is minimized. In this paper, we begin by deriving closed-form results for a one-dimensional case and two-dimensional cases with up to two hubs. Subsequently, a simulation-based approximation method is proposed for more complex two-dimensional scenarios with more than two hubs. Moreover, an extended problem with multiple service providers is analyzed to reflect real-life service settings. Finally, we apply our model and approximation method using publicly available data as a case study to optimize the deployment of public-access automated external defibrillators in Virginia Beach.

We study the multiplicity of the singularity of mean curvature flow with bounded mean curvature and Morse index. For $3\leq n\leq 6$, we show that either the mean curvature or the Morse index blows up at the first singular time for a closed smooth embedded mean curvature flow in $\mathbb{R}^{n+1}$.

We consider a one-dimensional symmetric Levy process that has local time. In the first part, we construct a self-adjoint extension of the generator of the process so that the constructed operator corresponds to the generator with the delta potential. Using the constructed operator, we extend the Feynman-Kac formula to the case of delta function-type potentials and prove a limit theorem for an operator semigroup corresponding to this formula. In the second part, we construct a one-parameter family of distributions that attract the sample paths of the process to a given point. We show that this family weakly converges to the distribution of a Feller process and prove a limit theorem for the distribution of a point where an attracted sample path comes.

Let $A$ be an associative algebra over an algebraically closed field $K$ of characteristic 0. A decomposition $A=A_1\oplus\cdots \oplus A_r$ of $A$ into a direct sum of $r$ vector subspaces is called a \textsl{regular decomposition} if, for every $n$ and every $1\le i_j\le r$, there exist $a_{i_j}\in A_{i_j}$ such that $a_{i_1}\cdots a_{i_n}\ne 0$, and moreover, for every $1\le i,j\le r$ there exists a constant $β(i,j)\in K^*$ such that $a_ia_j=β(i,j)a_ja_i$ for every $a_i\in A_i$, $A_j\in A_j$. We work with decompositions determined by gradings on $A$ by a finite abelian group $G$. In this case, the function $β\colon G\times G\to K^*$ ought to be a bicharacter. A regular decomposition is {minimal} whenever for every $g$, $h\in G$, the equalities $β(x,g)=β(x,h)$ for every $x\in G$ imply $g=h$. In this paper we describe the finite dimensional algebras $A$ admitting a $G$-grading such that the corresponding regular decomposition is minimal. Moreover we compute the graded codimension sequence of these algebras. It turns out that the graded PI exponent of every finite dimensional $G$-graded algebra with regular grading such that its regular decomposition is minimal, coincides with its ordinary (ungraded) PI exponent.

In this work, we present novel randomized compression algorithms for flat rank-structured matrices with shared bases, known as uniform Block Low-Rank (BLR) matrices. Our main contribution is a technique called tagging, which improves upon the efficiency of basis matrix computation while preserving accuracy compared to alternative methods. Tagging operates on the matrix using matrix-vector products of the matrix and its adjoint, making it particularly advantageous in scenarios where accessing individual matrix entries is computationally expensive or infeasible. Flat rank-structured formats use subblock sizes that asymptotically scale with the matrix size to ensure competitive complexities for linear algebraic operations, making alternative methods prohibitively expensive in such scenarios. In contrast, tagging reconstructs basis matrices using a constant number of matrix-vector products followed by linear post-processing, with the constants determined by the rank parameter and the problem's underlying geometric properties. We provide a detailed analysis of the asymptotic complexity of tagging, demonstrating its ability to significantly reduce computational costs without sacrificing accuracy. We also establish a theoretical connection between the optimal construction of tagging matrices and projective varieties in algebraic geometry, suggesting a hybrid numeric-symbolic avenue of future work. To validate our approach, we apply tagging to compress uniform BLR matrices arising from the discretization of integral and partial differential equations. Empirical results show that tagging outperforms alternative compression techniques, significantly reducing both the number of required matrix-vector products and overall computational time. These findings highlight the practicality and scalability of tagging as an efficient method for flat rank-structured matrices in scientific computing.

Continuous actions of semigroups over a topological space are discussed. We focus on semigroups that can be embedded into a group, and study the problem of defining a "natural extension," that is, defining a corresponding group action by homeomorphisms over an appropriate space that serves as an invertible extension of the original semigroup action. Both feasibility and infeasibility results are shown for surjective continuous semigroup actions, characterizing the existence of such an extension in terms of algebraic properties of the semigroup. We finish by briefly studying topological dynamical properties of the natural extension in the amenable case.

We study distribution of orbits sampled at polynomial times for uniquely ergodic topological dynamical systems $(X, T)$. First, we prove that if there exists an increasing sequence $(q_n)$ for which the rigidity condition \[ \max_{t<q_{n+1}^{4/5}}\sup_{x\in X}d(x, T^{tq_n}x)=o(1) \] is satisfied, then all square orbits $(T^{n^2}x)$ are equidistributed (with respect to the only invariant measure). We show that this rigidity condition might hold for weakly mixing systems, and so as a consequence we obtain first examples of weakly mixing systems where such an equidistribution holds. We also show that for integers $C>1$ a much weaker rigidity condition \[ \max_{t<q_n^{C-1}}\sup\limits_{x\in X}d\left(x, T^{tq_n}x\right)=o(1) \] implies density of all orbits $(T^{n^C}x)$ in totally uniquely ergodic systems, as long as the sequence $(ω(q_n))$ is bounded.

In the theory of species, differential as well as integral operators are known to arise in a natural way. In this paper, we shall prove that they precisely fit together in the algebraic framework of integro-differential rings, which are themselves an abstraction of classical calculus (incorporating its Fundamental Theorem). The results comprise (set) species as well as linear species. Localization of (set) species leads to the more general structure of modified integro-differential rings, previously employed in the algebraic treatment of Volterra integral equations. Furthermore, the ring homomorphism from species to power series via taking generating series is shown to be a (modified) integro-differential ring homomorphism. As an application, a topology and further algebraic operations are imported to virtual species from the general theory of integro-differential rings.

This paper presents an embedding-based approach for solving switched optimal control problems (SOCPs) with dwell time constraints. At first, an embedded optimal control problem (EOCP) is defined by replacing the discrete switching signal with a continuous embedded variable that can take intermediate values between the discrete modes. While embedding enables solutions of SOCPs via conventional techniques, optimal solutions of EOCPs often involve nonexistent modes and thus may not be feasible for the SOCP. In the modified EOCP (MEOCP), a concave function is added to the cost function to enforce a bang-bang solution in the embedded variable, which results in feasible solutions for the SOCP. However, the MEOCP cannot guarantee the satisfaction of dwell-time constraints. In this paper, a MEOCP is combined with a filter layer to remove switching times that violate the dwell time constraint. Insertion gradients are used to minimize the effect of the filter on the optimal cost.

In this work, we discuss several results concerning Serrin's problem in convex cones in Riemannian manifolds. First, we present a rigidity result for an overdetermined problem in a class of warped products with Ricci curvature bounded below. As a consequence, we obtain a rigidity result for Einstein warped products. Next, we derive a soap bubble result and a Heintze-Karcher inequality that characterize the intersection of geodesic balls with cones in these spaces. Finally, we analyze the analogous ovedetermined problem for the drift Laplacian, where the ambient space is a cone in the Euclidean space.