Papers

For Gaussian approximation over high-dimensional rectangles under unrestricted covariance, Chernozhukov et al. (2023b) conjectured that the $n^{-1/4}$ rate, up to logarithmic factors, is near-optimal. We show that, under the coordinatewise subexponential condition with scale $B_n$ and the marginal variance lower bound condition with constant $b$ in Chernozhukov et al. (2023b), the approximation error in dimension $d$ is bounded by \begin{align*} C_b\min\left\{ 1,\, \left(\frac{B_n^2}{n}\right)^{1/3}\{\log(2dn)\}^{7/3} + \frac{B_n}{\sqrt n}\{\log(2dn)\}^{5/2} \right\}. \end{align*} In particular, for bounded $B_n$ and polynomial dimension, the new bound is $n^{-1/3}$ and therefore falsifies the polynomial-dimensional $n^{-1/4}$ near-optimality conjecture. The proof uses a two-stage interpolation and a rank-free matrix-weighted Gaussian surface bound, which may be of independent interest. The initial proof attempt was generated by ChatGPT 5.6 Pro (OpenAI) and subsequently corrected and rewritten by the authors. The machine-checked Lean formalization of the proof can be found at the GitHub repository (https://github.com/WeihanZhang2001/cubic-root-gaussian-approximation-under-unrestricted-covariance).

We investigate the filtration structure of the skein algebra induced by the (quantum) trace map. Utilizing the structure, we prove that relative $\operatorname{SL}(3, \mathbb{C})$ character varieties of punctured surfaces admit log Calabi-Yau compactifications and prove the (weak) geometric P=W conjecture.

The Hörmander-Bernhardsson constant $\mathscr{C}$ is the sharp constant in $|f(0)|\le \mathscr{C} \|f\|_1$ for entire functions of exponential type $\le \pi$. We prove that $\mathscr{C} = 2\pi \theta_*^{-2}$ where $\theta_*$ is the least positive singularity of the regular solution $v$ with $v(0)=0$ of the cosh-Gordon equation $v_{\theta\theta} + v_\theta/\theta = \cosh(v)$. It is known that $\mathscr{C}$ is a scaling limit in $n$ from the analogous problem for polynomials of degree $\le n$. We reformulate the polynomial problem as a Padé approximation problem at infinity. The associated matrix Riemann-Hilbert problem is analyzed by a Deift-Zhou steepest descent whose local parametrix is built from a Painlevé transcendent.

Berestycki and Graham proved large-dilation uniqueness for bounded positive solutions of \[ -\Delta u=f(u)\quad\hbox{in }\kappa\Omega, \qquad u+\alpha\partial_\nu u=0\quad\hbox{on }\partial(\kappa\Omega),\] when $\alpha$ is fixed, and remarked that the dilation threshold should not depend on $\alpha$. We show that it does not, including at the Dirichlet and Neumann endpoints, for possibly unbounded uniformly $C^{2,\gamma}$ domains. The half-space linearizations have a common positive spectral gap over the compactified boundary parameter. The Dirichlet end requires a separate compactness argument because the Robin coefficient diverges there. After rescaling by its inverse, the equation has a harmonic half-space limit. A Liouville lemma rules out a nonzero limiting trace, and the resulting endpoint compactness, together with the half-space gap and localization, yields the uniform uniqueness statement. When $\partial\Omega\neq\varnothing$, we further obtain convergence of the spectral bottom to its half-space value with error $O(\kappa^{-1/2})$. For bounded $C^{4,\gamma}$ domains the boundary layer also has a first mean-curvature correction, with remainder $O(\kappa^{-1-\gamma}+\kappa^{-2})$ on each fixed boundary strip.

We establish local-in-time dispersive and sharp Strichartz estimates for the linear wave equation with Neumann boundary conditions inside a cylindrical convex domain $\Omega \subset \mathbb{R}^3$ featuring a non-empty smooth boundary $\partial\Omega$. The cylindrical geometry dictates that the nonnegative radius of curvature vanishes along the axial direction. By utilizing an explicit spectral analysis based on the zeros of the Airy function derivative, alongside a tailored Airy–Poisson summation formula, we capture the microlocal behavior of multi-reflected waves and caustics. Finally, we apply these sharp Strichartz estimates without loss of derivatives to prove local well-posedness and small-data global well-posedness for the energy-critical quintic nonlinear wave equation (NLW) subject to Neumann boundary conditions.

A classification is given of regular, rotary, and birotary Cayley maps of which the vertex number is coprime to the valency. The classification is involved in constructions of new examples of interesting Cayley maps.

Checkerboard edge-midpoint geometry provides an exact planar Varignon parallelogram for every spatial quadrilateral, allowing a local tangent frame and normal to be recovered directly from nodal positions even when the raw quadrilateral is warped. Building on this property, we develop a position-only thin-shell discretization with no independent director, rotation, or strain variables. The connected edge-midpoint surface is taken as the physical midsurface: the first fundamental form is evaluated on planar B faces, while a W-centered second fundamental form is constructed from variations of neighboring B-face normals, so membrane and bending share the same geometric carrier. A variational-kernel analysis shows that smooth second-order consistency does not eliminate lattice-scale blind modes. After quotienting out the raw checkerboard gauge, the B metric has one physical membrane blind direction and the symmetric W curvature has two curvature blind directions. We introduce a quotient-minimal membrane compatibility coordinate $X_M$ and an objective reference-relative curvature coordinate $X_W^{rel}$, placed consistently in the $O(t)$ membrane and $O(t^3)$ bending sectors. The formulation admits an explicit midpoint quotient, complete flat blind-mode classification, rigid-motion objectivity, reference-state consistency, and an $O(h^2)$ near-isometry approximation result for aligned generalized cylinders. Numerical tests show second-order curvature convergence, targeted removal of the membrane defect, and a sub-percent, refinement-decaying influence of $X_W^{rel}$. Linear and nonlinear shell benchmarks further demonstrate flat bending, curved-shell membrane-bending coupling, thickness sensitivity, large rotation, nonlinear pinching, and localized ovalization within a single position-only framework.

For $N\geq 3$ and a potential phase $\vartheta\in\mathbb{R}$, we study the non-self-adjoint almost Mathieu matrix obtained by multiplying the discrete Laplacian by a complex phase with angle $\varphi\in\mathbb{R}$, $A_N(\varphi,\vartheta)=e^{i\varphi}(S+S^{-1})/2+\operatorname{diag}(\cos(2\pi j/N+\vartheta))_{j\in\mathbb{Z}/N\mathbb{Z}}$, where $S e_j=e_{j+1}$ is the periodic shift on $\mathbb{C}^N$. We derive a Chambers formula and isolate the part $Q_{N,\varphi}$ of the characteristic polynomial that depends only on $N$ and $\varphi$, but not on $\vartheta$ or on a change of boundary conditions for the shift operator. We then show, for every $N$, that the zeros of $Q_{N,\varphi}$ lie on the two perpendicular lines $e^{i\varphi/2}\mathbb{R}\cup e^{i(\varphi/2+\pi/2)}\mathbb{R}$. For even $N$, the same property holds for the matrices $A_N(\varphi,\vartheta)$ with $\vartheta\in 2\pi\mathbb{Z}/N$, and we compute their limiting eigenvalue measure explicitly. For $\varphi\in[-\pi,\pi]$, the eigenvalue distribution approximates elliptic-integral densities with masses $1-|\varphi|/\pi$ and $|\varphi|/\pi$, and maximal radii $2|\cos(\varphi/2)|$ and $2|\sin(\varphi/2)|$, respectively. At $\varphi=\pi/2$, the central polynomial $Q_{N,\varphi}$ factors into positive quartic factors. This proves that the Scottish flag matrix, after Trefethen and Chapman, has its spectrum on the two diagonal lines of the saltire.

For periodic Schrödinger operators $H_V=-\Delta+V$ with bounded real-valued potentials on $\mathbb R^d$ with $d\ge2$, we show that for generic potentials, each endpoint of every spectral gap is attained by a single Bloch band at only finitely many quasimomenta, and has a nondegenerate Hessian at every attaining point. This proves the Spectral Edge Conjecture for periodic Schrödinger operators.

The loop mirror Heisenberg-Virasoro algebra, an embedded subalgebra of the loop Heisenberg-Virasoro algebra, admits a family of interesting truncated subalgebras including those of Takiff type and \(\mathfrak{bms}_3\) type. We give a complete classification of simple Harish-Chandra modules over the loop mirror Heisenberg-Virasoro algebra, whose simple modules fall into three categories, highest weight modules, lowest weight modules, and evaluation modules of the intermediate series. As a by-product, we classify all simple Harish-Chandra modules over the truncated mirror Heisenberg-Virasoro algebras \(\mathcal{L}(n)\) for \(n\geq2\). By virtue of shift operators in the \(d\)-parameter family, we give a more streamlined proof of Theorem 3.3 from the work [Classification of simple $W_n$-modules with finite-dimensional weight spaces, J. Reine Angew. Math., 720 (2016), 199-216] by Y. Billig and V. Futorny, which states the key Billig-Futorny identity. Furthermore, our approach can be extended to the computation of annihilators for uniformly bounded modules over some other Lie (super)algebras.

We leverage Polishchuk's Poisson blow-up criterion in the context of algebro-geometric integrable systems. In more detail, one may associate an integrable system $\tau:\mathfrak{X}\longrightarrow\mathfrak{B}$ to each affine Poisson scheme $\mathfrak{X}$ over $\mathbb{C}$. We prove that the blow-ups of $\mathfrak{X}$ along fibers of $\tau$ are Poisson schemes occurring in a family $\widetilde{\mathfrak{X}\times\mathfrak{B}}\longrightarrow\mathfrak{B}$, where $\widetilde{\mathfrak{X}\times\mathfrak{B}}$ is itself a Poisson scheme. This result is subsequently specialized to the adjoint quotient $\tau:\mathfrak{g}\longrightarrow\mathfrak{g}/\!/G=:\mathfrak{c}$ of a finite-dimensional complex semisimple Lie algebra $\mathfrak{g}$ with integrating algebraic group $G$. We show that the family $\widetilde{\mathfrak{g}\times\mathfrak{c}}\longrightarrow\mathfrak{c}$ is flat, conical, and equipped with a canonical Poisson Hamiltonian $G$-variety structure. We also obtain Poisson-geometric results on the fibers of this family, which are blow-ups of $\mathfrak{g}$ along regular adjoint orbit closures.

We introduce and study completeness for multiplicative $n$-ary Hom-Nambu superalgebras. Because an $n$-ary Hom-Nambu bracket is not necessarily totally super-skew-symmetric, we define its center as the intersection of its positional centers. A multiplicative $n$-ary Hom-Nambu superalgebra is complete when its center is trivial, and every $\alpha^{k+1}$-derivation is inner for all $k \geq 0$. We show that this notion reduces to the usual completeness for multiplicative $n$-Hom-Lie superalgebras. Furthermore, we establish a completeness criterion for surjective brackets with a zero twisting map and provide a complete $n$-ary Hom-Nambu superalgebra that is not an $n$-Hom-Lie superalgebra. We also study the direct sums of complete $n$-Hom-Lie superalgebras and the behavior of centers under twisting. Finally, starting from a multiplicative Hom-Lie superalgebra, we consider recursively induced multiplicative $n$-ary Hom-Nambu superalgebras. When the twisting map is surjective, we prove that triviality of the center is preserved and reflected in this construction. We also observe that every binary $\alpha^k$-derivation satisfies the corresponding relative $\alpha^k$-derivation identity for the induced bracket. Finally, we determine the complete members in the selected low-dimensional Hom-Lie and $3$-Hom-Lie superalgebra classifications.

Recently Dixit, Kumar, and Srivastava investigated what they called Rascoe and non-Rascoe partitions. These are defined to be the set of distinct partitions where the length of the partition is a part of the partition and is not a part respectively. In this note we provide a $q$-series theoretic proof of two identities regarding the generating function for unrestricted Rascoe and non-Rascoe partitions fulfilling a request of Dixit, Kumar, and Srivastava. We also prove a conjecture of Beck relating non-Rascoe partitions and the rank of a partition.

We study extinction and extinguishment in a two-type continuous-state nonlinear branching model driven by Brownian branching noise and spectrally positive stable jumps. The populations are subject to nonlinear self-regulation and a mixed-sign predator–prey interaction: the second promotes the first, whereas the first suppresses the second. Two complementary structures are developed. An exact power–logarithmic cancellation functional removes the interaction drifts and yields stochastic Lyapunov estimates, nonexplosion, and boundary criteria. In the multiplicative regimes, integrating-factor identities and geometric Lévy factorizations express extinction through weighted exposure clocks and reduce the long-time analysis to effective decay rates. These methods yield almost-sure extinction criteria and identify a regime in which finite-time extinction and nonextinction coexist. On nonextinction, both populations remain positive at all finite times and converge jointly to zero, exhibiting joint extinguishment rather than positive persistence.

We prove that for every integer $n\geq 2$ and $8\pi(n-1)<\rho<8\pi n$, the singular Liouville equation $\Delta u+\e^u=\rho\delta_0$ on a rectangular torus $E_{\mathrm{i}b}=\mathbb{C}/(\mathbb Z+\mathrm{i} b\mathbb Z)$ has exactly $n$ solutions, which are all axisymmetric. Together with previous results by Chen-Lin and Lin-Wang, this yields that \begin{itemize} \item $E_{\mathrm{i} b}$ admits no spherical metrics with a conical singularity of angle $2\pi\vartheta$ as long as $\vartheta$ is a positive odd integer. \item For every integer $n\geq 1$, $E_{\mathrm{i} b}$ admits exactly $n$ spherical metrics with a conical singularity of angle $2\pi\vartheta$ for each $\vartheta\in (2n-1, 2n+1)$. \end{itemize} The basic idea is to prove that the linearized equation has only trivial solutions in the space of axisymmetric functions. The previous method of analysing nodal domains via Bol's isoperimetric inequality only works for $\rho\leq 8\pi$. We develop a unified approach for all $\rho\in (0,+\infty)\setminus 8\pi\mathbb{N}_{\geq 1}$ by exploring the deep connection with the monodromy of the classical Lamé equation.

We establish Łojasiewicz–Simon type gradient inequalities for the Yamabe functional on bounded smooth domains near configurations consisting of finitely many concentrating bubbles, with or without a regular component. A weighted decomposition separates the finite-dimensional spectral and bubble parameters from an infinite-dimensional coercive remainder. We obtain quantitative estimates for the parameters, the remainder, and the corresponding energy gaps, in both the regular-plus-bubbling and the pure-bubbling regimes. These inequalities quantify the deviation from such surfaces and are essential for studying the dynamics of related parabolic flows.

Let $\mathcal M$ be a semifinite von Neumann algebra equipped with a semifinite faithful normal trace $\tau$. We establish Jensen's trace inequality in full generality and characterize its equality case, which answers two questions raised in [Kosaki2013] and [HaradaKosaki2008]. As an application, we derive noncommutative Lamperti-type inequalities and their equality conditions. Employing this result, we characterize linear isometries (not necessarily surjective) on a class of $F$-normed noncommutative Orlicz spaces, which provides a noncommutative Lamperti's theorem for linear isometries.

We establish the existence of smooth solutions to the symmetric maximal surface equation with degenerate boundary conditions. Moreover, we prove that these solutions maximize the associated area functionals. This result serves as the Lorentzian analogue of minimal graphs in hyperbolic spaces together with their associated area minimizing problem.

Let $\mu$ be an $n$-AD-regular measure in $\mathbb{R}^d$. Chousionis, Garnett, Le and Tolsa [CGLT] proved that $\mu$ is uniformly $n$-rectifiable if and only if the square function built from the density differences $\Delta_\mu(x,r)=\mu(B(x,r))/r^n-\mu(B(x,2r))/(2r)^n$ satisfies a Carleson condition. In this paper we show that the same characterization holds if the density is first composed with a function $F$ which is bi-Lipschitz on the interval $[c_0^{-1},c_0]$ determined by the AD-regularity constant $c_0$. The main example is $F=\log$, introduced in [Le], for which the square function takes the scale-invariant form $\Delta_\mu^{\log}(x,r) = \log\bigl(\mu(B(x,r))/\mu(B(x,2r))\bigr)+n\log 2$. We give a complete proof, extend the statement to the smooth square functions of [CGLT], where the density is replaced by the convolution of $\mu$ with a Gaussian or a more general radial kernel, discuss what happens when $F$ is not bi-Lipschitz, and treat the case $\mu(\mathbb{R}^d)<\infty$, where the behavior of $F$ near zero enters in only one of the two implications. We also show that the qualitative characterization of $n$-rectifiable measures by Tolsa and Toro [TT], in terms of the same square function at $\mu$-almost every point, holds after composition with any locally bi-Lipschitz $F$. This requires neither AD-regularity nor doubling, and for $F=\log$ the condition $\lim_{r\to0}\Delta_\mu(x,r)=0$ becomes $\lim_{r\to0}\mu(B(x,r))/\mu(B(x,2r))=2^{-n}$.

We establish local Calderón–Zygmund estimates near the natural exponent for very weak solutions to matrix-weighted quasilinear equations\[ - \mathrm{div\,} A_{\mathbb{M}}(x,Du) = - \mathrm{div\,} A_{\mathbb{M}}(x,\mathbf{f}), \qquad A_{\mathbb{M}}(x,\xi)=\mathbb{M}(x)A(x,\mathbb{M}(x)\xi), \] where $A$ has $p$-growth and strong monotonicity, and $\mathbb{M}$ is a measurable positive-definite matrix field. We assume that $\mathbb{M}$ has bounded condition number and that \(\omega:=|\mathbb{M}|^p\in A_p\), without imposing uniform upper or lower bounds on $\mathbb{M}$. This extends the near-natural Calderón–Zygmund theory developed by Adimurthi–Phuc \cite{AP15} to the matrix-degenerate setting: There exists $\delta_0>0$ such that every very weak solution $u \in W^{p-\delta_{0}}_{\omega, \mathrm{loc}}$ satisfies \[ \mathbf{f}\in L^\gamma_{\omega,\mathrm{loc}} \Longrightarrow Du\in L^\gamma_{\omega,\mathrm{loc}} \] for $p-\delta_0\le\gamma\le p+\delta_0$. The proof requires handling the lack of energy estimates below the natural exponent and the use of Lipschitz truncation in the weighted setting. We achieve this through comparison estimates, higher integrability, and weighted analysis techniques.

We prove the transcendental Morse inequality conjecture of Boucksom-Demailly-Păun-Peternell. Among the corollaries of this result, this proves orthogonality of divisorial Zariski decompositions, shows that the pseudoeffective and movable cones are dual, and establishes the differentiability of the volume function on the big cone, on all compact Kähler manifolds.

The tropical Schottky problem asks which principally polarized tropical abelian varieties are isomorphic to tropical Jacobian varieties equipped with their canonical principal polarizations. We prove the tropical Matsusaka criterion and the tropical Matsusaka–Ran criterion, which are the main results of this paper, and show that their hypotheses are equivalent. Together with the tropical Poincaré formula proved by Gross–Shokrieh, these results give a geometric solution to the tropical Schottky problem. In particular, we give an affirmative answer to a question posed by Brannetti–Melo–Viviani. Several tools are developed for the proofs of the two criteria. In particular, we establish Hodge-type inequalities for ample tropical Cartier divisors on tropical abelian varieties and prove a splitting criterion for principally polarized tropical abelian varieties.

We study high-frequency propagation for a viscoelastic wave equation with spatially dependent hereditary memory in relative-history form. The kernel may have the integrable singularity $\mathfrak m(s,x)=s^{p-1}m(s,x)$, $0<p<1$; the regular case $p=1$ is included. Using the half-amplitude propagation distance and corresponding travel time as units, the wavelength $h\ll1$ yields the memory factor $\varepsilon=h^{1-p}$. We construct exact solutions with full two-scale geometric-optics expansions in powers $h^{k+(1-p)\ell}$. Memory modifies the propagation geometry through the instantaneous modulus $\sigma+\varepsilon\int_0^\infty\mathfrak m(s,\cdot)\,d s$, while the kernel singularity contributes $C_p=\Gamma(p)e^{i\pi p/2}$ to the leading transport equation. For $0<p<1$, this produces fractional scales, frequency-dependent attenuation, and a dispersive phase correction; for $p=1$, the fractional hierarchy disappears, attenuation is frequency independent, and the transport phase correction vanishes. We also derive a local damped wave equation whose incoming high-frequency solutions approximate the hereditary solutions with $O(h)$ error in semiclassical $C^k$ norms. Exterior observations for all incident directions and $0<h\ll1$ uniquely recover $\sigma_{\mathfrak m}$ and the full temporal jet of $m$ at $s=0$, which determine the expansion modulo $O(h^\infty)$. Finally, a contraction-semigroup argument gives well-posedness and arbitrary finite-order Sobolev regularity for spatially dependent weakly singular kernels and prescribed full prehistory, with explicit compatibility conditions and estimates uniform in $\varepsilon$. These estimates justify the geometric-optics construction.

Stochastic programming (SP) is a powerful framework for decision-making under uncertainty, but its practical adoption in industry is often hindered by the difficulty in understanding the causal relationships that drive optimal solutions. In the two-stage SP, strategic first-stage decisions are coupled with operational second-stage recourse decisions. When the number of scenarios under consideration is large, understanding the direct link between the uncertainty realization and optimal recourse strategy becomes computationally and cognitively demanding. Common approaches to improve interpretability include trained classification trees or scenario reduction, replacing the large scenario set with a representative subset. This is often achieved through post-hoc clustering (e.g., k-means) based on uncertainty realizations or optimal recourse decisions. While useful, these methods only provide a statistical approximation of the solution space and may fail to reveal the underlying structural properties of the recourse problem that drive optimal first-stage decisions. This work introduces a novel, deterministic approach to explainability using multiparametric programming (mp) within a Benders decomposition framework. We reformulate the recourse subproblem as a multiparametric linear program, generating an explicit map of Critical Regions (CRs), which are polyhedral partitions of the uncertainty space. This allows us to cluster scenarios analytically rather than statistically. We demonstrate this methodology on a supply chain planning under demand uncertainty. Our results show that 100 stochastic scenarios map to exactly six critical region clusters. This mapping allows us to explain optimal capacity planning decisions as a precise trade-off between specific operational modes, providing a fully transparent interpretation of the stochastic solution.

Electrochemical distributed energy resources (DERs), such as electrolyzers and battery energy storage systems, consume large amounts of electricity. Buying that power directly from wholesale markets can sharply reduce operating costs. Accessing these markets, however, requires a minimum bid size that individual small or medium-scale units struggle to meet. Grouping several units to bid as a single participant clears this barrier while letting each unit use its own low-cost renewable generation. The common remedy is to hire a third-party aggregator, but aggregators charge commissions and solve a centralized optimization that often favors certain units over others. A fairer alternative is peer-to-peer (P2P) participation, where units form a self-governing group and bid jointly with no central authority. Because each self-interested unit is unwilling to share private data, coordination is best posed as a game-theoretic distributed optimization problem. Existing P2P methods, however, address only single-round spot markets and ignore the two-stage structure of real wholesale markets, where participants commit a day ahead and continuously adjust in real time. We close this gap with a two-stage receding-horizon generalized Nash equilibrium (GNE) game. Each unit re-optimizes its strategy every five minutes over a rolling one-hour horizon, while the group satisfies both market stages collectively. Units exchange only publicly visible aggregate power, never private cost or production data. We apply the approach to a six-unit fleet on the PJM market. It delivers 37% higher profit, 29% lower electricity cost, and 92% less renewable curtailment than individual participation, with every unit better off.

This work study a differential game of pursuit-evasion involving countably many pursuers $p_{1},p_{2},\cdots,p_{m}$ and a single evader $e$ under point-wise (geometric) constraints in the sequence space $l_{2}.$ The pursuers evolve according to specified differential equations of $1^{st}$ order while the evader follows a $2^{nd}$ order differential equation. The game period is a fixed time interval of length $\theta$ unit of time. The minimum distance between evader and the pursuer at the terminal time denotes the game payoff. The pursuers aim is to minimize the payoff, whereas the evader seeks to maximize it. We obtain the solution of the game, including optimal strategies of the players construction, as well as game value estimation.

R-equivalence is an equivalence relation on the set of colorings of an oriented knot diagram by a quandle. In this paper, we show that a certain subgroup of the inner automorphism group of a quandle acts on the R-equivalence class of a given coloring by the quandle. We also determine the R-equivalence classes of colorings of a diagram of a $2$-bridge knot by a dihedral quandle completely, under a certain condition.

We solve Tingley's problem for Haagerup noncommutative $L_p$-spaces, where $1<p\ne 2<\infty$, which gives an affirmative answer to Mori's problem \cite[Problem~6.3]{Mori}.

We examine pursuit and evasion problems in the space $ \mathbb{R}^{n} $ involving a lone pursuer and evader. Motion of each player is governed by a first-order linear differential equation. The players' control functions are subject to the Gronwall-type inequality. The game's duration is fixed and represented by a positive number $\vartheta $. When the state of a pursuer coincides with that of the evader, we then say the pursuit is completed. On the contrary, evasion is possible when throughout the game there is no agreement between the states of pursuer and evader. The result obtained related to the pursuit problem depends on the attainability domains of the two players. On the other hand, one sufficient condition is given for evasion to be possible.

Given a discrete group $\Gamma$ acting on a compact space $X$, and a stabilizer subgroup $\Lambda\leq \Gamma$ of $X$, we use the Rieffel induction of covariant $(\Lambda, X)$-representations to study classes of representations induced by certain characters on $\Lambda$ and the $\text{C}^*$-algebras they generate. When $X$ is a $\Gamma$-boundary, we obtain new classes of simple, traceless group $\text{C}^*$-algebras. When the $\Gamma$-boundary $X$ is an extreme boundary, we show that the associated group $\text{C}^*$-algebras are also purely infinite, answering part of a question posed by Kalantar and Scarparo on $\text{C}^*$-algebras associated with Thompson's groups. The latter $\text{C}^*$-algebras are shown to be selfless in the sense of Robert.