Papers

Let $K_N^{(r)}$ denote the $N$-vertex complete $r$-uniform hypergraph. For an $r$-uniform hypergraph $H$ and an integer $k\geq2$, the $k$-color Ramsey number $R(H,k)$ is the least integer $N$ such that every $k$-edge-coloring of $K_N^{(r)}$ contains a monochromatic copy of $H$. When $k\mid\esize(H)$, the zero-sum Ramsey number $R(H,\mathbb Z_k)$ is the least integer $N$ such that every edge-labeling of $K_N^{(r)}$ by elements of $\mathbb Z_k$ contains a copy of $H$ whose edge labels sum to $0$ in $\mathbb Z_k$. We settle two conjectures and a problem concerning these two Ramsey numbers. First, Caro and Provstgaard proposed exact values for the zero-sum Ramsey numbers over $\mathbb Z_2$ of delta-systems with an even number of edges. We determine these numbers and thereby prove their conjecture. Second, for a forest $F$ with $m$ edges, let $tF$ denote the disjoint union of $t$ copies of $F$. Caro conjectured that $R(tF,\mathbb Z_{mt})=R(tF,2)$ for all sufficiently large $t$. We show that this conjecture does not hold for double stars. Caro also asked whether there exists a tree $T$ with $m$ edges such that $R(T,\mathbb Z_m)>R(T,2)$. We answer this question affirmatively by constructing an infinite family of such trees.

Given an additive character $\psi$ of an arbitrary finite field $\mathbb{F}_{p^n}$ and elements $a,b\in \mathbb{F}_{p^n}$, $b\neq 0$, we prove a bound on bilinear Kloosterman sums $\sum_{x\in B_1}\sum_{y\in B_2}\psi(axy+bx^{-1}y^{-1})$, where $B_1, B_2$ denote boxes in $\mathbb{F}_{p^n}$. Here, a box is a coordinate parallelepiped obtained by restricting the coefficients of field elements, with respect to a fixed basis of $\mathbb{F}_{p^n}$ over $\mathbb{F}_p$, to intervals in $\mathbb{F}_p$. Our estimates are nontrivial when $|B_1||B_2|>p^{n/2+\epsilon}$ and hence in a range not accessible by the Weil bound. We also consider the random walk on $\mathbb{F}_{p^n}$ defined by $S_k=S_0+W_1+\cdots+W_k$, where $W_i=aX_iY_i+b(X_iY_i)^{-1}$ and $X_i,Y_i$ are independent uniformly distributed random variables on $B_1$ and $B_2$ respectively. We show that the nontrivial Fourier coefficients of $S_k$ decay exponentially. Consequently, every nonzero $\mathbb{F}_p$-linear projection of $S_k$, as well as the full distribution of $S_k$ converge to the uniform distribution on $\mathbb{F}_p$ and $\mathbb{F}_{p^n}$, respectively. We also obtain bounds on the rate at which the entropy of $S_k$ converges to its maximal value.

In this paper, we introduce a new framework for computing certain localized $\mathrm{v}$-numbers of a class of ideals called coordinate-saturated ideals, which includes certain classes of Lovász-Saks-Schrijver (LSS) ideals and (generalized) binomial edge ideals associated with graphs. For a forest graph $G$, we derive an explicit formula for the localized $\mathrm{v}$-number of the LSS ideal $L_G^{\mathbb{K}}(d)$, denoted by $\mathrm{v}_{\mathfrak{p}_{\emptyset}(G)}(L_G^{\mathbb{K}}(d))$, for all $d \geq 2$, where $\mathbb{K}$ is an algebraically closed field. As a consequence, we prove that $\mathrm{v}(L_G^{\mathbb{K}}(d)) \leq \mathrm{reg}(R/L_G^{\mathbb{K}}(d)),$ where $\mathrm{v}(L_G^{\mathbb{K}}(d))$ and $\mathrm{reg}(R/L_G^{\mathbb{K}}(d))$ denote the $\mathrm{v}$-number of $L_G^{\mathbb{K}}(d)$ and the Castelnuovo-Mumford regularity of $R/L_G^{\mathbb{K}}(d)$, respectively. Also, we give an upper bound for $\mathrm{v}_{I_{K_{n}}}(L_{G}^{\mathbb{R}}(2))$, where $\mathbb{R}$ is field of real numbers. Furthermore, we provide combinatorial descriptions of the localized $\mathrm{v}$-number of parity binomial edge ideals, denoted by $\mathrm{v}_{\mathfrak{p}^{+}(G)}(\mathcal{I}_{G})$, and, as an application, show that $\mathrm{v}(\mathcal{I}_G) \leq \mathrm{reg}(R/\mathcal{I}_G)$ for several classes of non-bipartite graphs. Finally, we prove that $\mathrm{v}(J_G^k)\leq \mathrm{reg}(R/J_G^k)$ for all powers of binomial edge ideals of closed graphs $G$.

We prove a conjecture of Datar-Mete-Song \cite{DMS} characterizing $J$-slope semi-stability by the minimal $J$-slope. More precisely, for a semi-stable pair of Kähler classes $(\alpha,\beta)$ on a compact Kähler manifold $X$, every big and nef birational test class has slope at least the topological $J$-slope, whereas an unstable pair admits a test class with strictly smaller slope. We also introduce the $J$-null locus of a semi-stable pair and prove that it is an analytic subset of $X$ if $X$ is a compact Kähler surface or a compact toric Kähler manifold. In the toric invariant case, we show that Murakami's \cite{Murakami} weak solution to the $J$-equation is smooth and Kähler on the dense big torus $(\mathbb{C}^*)^n$ of $X$.

Decentralized Federated Learning (DFL) has emerged as an optimal privacy-preserving solution; however, it remains vulnerable to opportunistic behaviors due to the absence of a central coordinator. While Evolutionary Game Theory (EGT) serves as a powerful framework for analyzing such behaviors, existing studies often assume that agents possess perfect rationality and maintain static strategies. To address these limitations, this paper proposes a novel EGT framework designed to analyze strategic evolution and enhance overall system performance. The primary contributions of this work are threefold: First, we model peer-to-peer (P2P) interactions on a lattice network structure under the assumption of bounded rationality. Second, we formulate a comprehensive payoff matrix incorporating training costs, communication overhead, and cooperative rewards, while tailoring a strategy update rule that captures spatial propagation dynamics. Third, we integrate a reputation-based reward-and-punishment mechanism to effectively deter free-riding behaviors. Simulation results demonstrate that the framework significantly outperforms the baseline. Specifically, it increases average accuracy from approximately 70% to 82%, elevates cooperation frequency to approach 100% (compared to below 5% in the baseline), and drops accuracy variance from around 0.40 to 0.002, thereby accelerating uniform convergence and ensuring system stability.

This paper is concerned with generalized hyperbolic circle packings on compact bordered surfaces, endowed with finite polygonal cellular decompositions. We investigate the problem of realizing generalized hyperbolic circle packings with prescribed geodesic curvatures at boundary vertices, prescribed total geodesic curvatures at interior vertices, and prescribed discrete Gaussian curvatures at the centers of dual circles. We give a necessary and sufficient condition for the existence of such generalized hyperbolic circle packings and show their uniqueness. We also establish a discrete Schwarz–Pick lemma in this setting, including comparison results for vertex curvatures, generalized circle arc lengths, distances, and areas, together with the corresponding rigidity statements.

We establish a sharp integral nonexistence criterion for the Lane–Emden system of inequalities \[ -\Delta u\ge v^p,\qquad -\Delta v\ge u^q, \qquad p,q>0,\quad pq>1, \] on arbitrary infinite, connected, locally finite weighted graphs. In the asymmetric case $p\ne q$, set $P=\max\{p,q\}$. If, for some root $o\in V$, \[ \sum_{n=2}^{\infty} \frac{n^{2pq+2P-1}}{\mu(B(o,n))^{pq-1}}=\infty, \] then every nonnegative solution $(u,v)$ satisfies $u\equiv v\equiv0$. The proof combines flow decomposition of the finite Green current with nonlinear testing. In the symmetric case $p=q>1$, the Liouville problem reduces, via the sum $u+v$, to the scalar criterion \[ \sum_{n=2}^{\infty} \frac{n^{2p-1}}{\mu(B(o,n))^{p-1}}=\infty. \] Weighted half-line examples show that the critical logarithmic endpoint in the asymmetric result is sharp.

For time-fractional gradient flows, natural dissipation statements are often integrated or memory-modified rather than pointwise differential inequalities for the original energy. We construct a smooth, compactly supported, globally semiconvex energy and an absolutely continuous solution on a finite time interval of a time-fractional gradient flow along which the original energy is strictly increasing on an explicit subinterval. The construction incorporates the initial layer directly into the trajectory. After transformation, the curve and its fractional derivative are polynomial, the curve is a regular immersion, and the Hessian remains bounded below up to the anchor.

The Fomin-Greene Duality Theorem for finite posets and Wei's Duality Theorem for demimatroids each relate sets of extremal invariants via seemingly similar dualities. This paper describes precisely how these two dualities relate. It is shown that the chain and antichain numbers each satisfy a Wei-type duality with antichain- and chain-deletion numbers, respectively, and that, for each poset with at least two elements, its chain and antichain demimatroids are not related by any composition of standard demimatroid duality operations. Their upper Wei-number numbers do however determine each other via Fomin-Greene duality. The two dualities thereby coincide for Wei numbers but not for the underlying rank functions.

We investigate how peer and institutional incentives jointly shape the evolution of cooperation, social welfare, and enforcement efficiency in social dilemmas. In a Prisoners Dilemma with four strategies, unconditional cooperators (C), defectors (D), social punishers (SP), and social rewarders (SR), we allow decentralised peer incentives and centralised institutional incentives to act simultaneously, with the institution able to reward or punish any subset of strategies. In infinite well-mixed populations, we analyse the resulting four-strategy replicator dynamics, and in structured populations we use agent-based simulations on square lattices to study spatial effects and network reciprocity. Intervention schemes are evaluated by equilibrium states and evolutionary flow for infinite well-mixed populations, by cooperation levels and social welfare for structured populations, defined as aggregate population payoff net of institutional cost. We find that peer punishment most strongly promotes cooperation, whereas peer reward is more beneficial for social welfare. Institutionally rewarding peer incentive strategies substantially improves both cooperation and welfare, while subsidising unconditional cooperators has little impact. Under institutional punishment, directly penalising defectors is the only consistently effective policy; punishing peer incentive strategies dismantles decentralised incentives, reduces cooperation, and harms social welfare, showing that maximising cooperation does not necessarily optimise overall societal benefit. Our findings provide design principles for institutions seeking to balance cooperation promotion with welfare maximisation.

We study the Wasserstein gradient flow of the squared Maximum Mean Discrepancy (MMD) generated by the nonsmooth energy kernels $K(z)=-|z|^q$, $0<q<2$. In dimensions $d\ge2$, the corresponding energies are not displacement semiconvex, so standard Wasserstein-gradient-flow theory does not apply. When $d+q-2>0$, we prove global well-posedness on $\mathbb{R}^d$ for probability densities in subcritical $L^p$ spaces, with targets in the same integrability class and with finite moments. We also include the one-dimensional Coulomb endpoint $d=q=1$. For the associated $N$-particle system, we prove global noncollision and fixed-$N$ convergence to the collision-free critical set, a particle-to-continuum criticality principle, and a modulated-energy mean-field estimate that yields convergence of the particle dynamics to the continuum flow as $N\to\infty$ on every finite time interval. We also construct collision-free saddle equilibria, showing that deterministic particle trajectories need not approach global empirical minimizers. For $1\le q<2$, every continuum solution in our class has a narrowly relatively compact orbit, every $\omega$-limit point is Lagrangian critical, and the orbit approaches the Lagrangian critical set. For $0<q<1$, the same conclusions hold under uniform-in-time moment and subcritical $L^p$ bounds. We prove that an absolutely continuous Lagrangian critical point equals the target when the source and target have finite moments of order $q$, except when $0<q<1$ and $d\in\{1,3\}$. Under the preceding uniform bounds, rigidity gives convergence of the continuum flow to the target throughout the rigid part of the well-posedness range. Finally, we show that no initial-data-independent multiplicative MMD decay modulus exists on $\mathbb{R}^d$, and that global Polyak–Łojasiewicz inequalities fail in several whole-space and periodic Riesz/Coulomb regimes.

We establish a Fenchel-Willmore-Chen inequality for smooth metric measure spaces with a lower bound on the 1-Bakry-Emery k-Ricci curvature. This extends previous results and yields a unified proof. In addition, a Sobolev and isoperimetric inequality for nonnegative 1-weighted Ricci curvature is proven.

Consider $V_*$, the subgroup of R. Thompson's group $V$ which stabilises $0^\omega$ under the natural action upon the Cantor set, $\{0, 1\}^\omega$. Let $G_*$ be any subgroup of a finitely generated group $G$. We show that $G_*$ is a context-free subgroup of $G$ if and only if $G_*$ is a pullback of $V_*$ under a homomorphism $G \rightarrow V$. In particular, this shows the existence of a hardest context-free membership problem. As a consequence, we prove that a group $G$ embeds into $V$ if and only if it is the transition group of a finite union of context-free automata, or equivalently, if there exist finitely many context-free subgroups of $G$ whose cores intersect trivially.

This paper contains a new version of a final identification theorem for the `generic' groups of finite Morley rank, developing further the results of [4] and [5]. It is prepared for use in the Hrushovski Programme of classification of infinite simple groups of finite Morley rank.

We allow the exponent $\alpha$ that defines the positive Bergman operator on the upper half-plane to differ from the exponent $\gamma$ that defines the underlying weighted Lebesgue spaces. In this setting, we prove a sharp weak-type off-diagonal weighted inequality, with explicit and best-possible dependence on the Békollé–Bonami characteristic of the weight. We then deduce strong-type off-diagonal inequalities for the same operator. The key new tool is an off-diagonal extrapolation theorem, adapted to allow a change in the power of the distance to the boundary along the way.

Let $S^n$ be the unit round sphere with its intrinsic angular metric, normalized so that $\operatorname{diam}S^n=\pi$. For finite homogeneous metric spaces $X$, put \[ \delta_n=\inf_X d_{GH}(X,S^n). \] The main open problem is whether $\inf_{n\ge2}\delta_n>0$. Gelander's theorem gives $\delta_n>0$ in each fixed dimension, but not uniformly. An abstract cross-polytope construction gives the universal upper bound $\delta_n\le\pi/4$. In the opposite direction, ChatGPT combines the passage from small Gromov–Hausdorff error to an approximate finite action on the sphere, logarithmic stability of approximate inner-product-preserving maps due to Cuesta, operator-norm stability of almost representations, and Green's width theorem for finite transitive sets. This gives the quantitative bound \[ \delta_n\ge \frac{c}{(1+\log(n+1))^2} \] for all sufficiently large $n$.

One-way spatial marching methods separate upstream- from downstream-propagating disturbances using a rational approximation of a spectral projector. Among existing one-way Navier–Stokes (OWNS) formulations, the recursive variant, OWNS-R, is the most economical, evaluating the approximation as a product of $N$ resolvent factors. This product amplifies rounding errors multiplicatively, imposing a flow-dependent upper limit on the approximation order that cannot be known in advance. We reformulate the same approximation as an additive partial-fraction sum (OWNS-Summation, OWNS-S). In exact arithmetic, the recursive and summation evaluations are equivalent when supplied with identical weights and poles; in floating-point arithmetic, they are not. Each of the $N$ resolvent solves acts on the same input state and contributes independently to a weighted sum, preventing multiplicative error amplification. The approximation order $N$ therefore becomes a pure convergence parameter, and the solves can run in parallel. Using the same auxiliary poles as OWNS-R, OWNS-S remains accurate in every configuration tested, showing that the recursive evaluation, rather than the poles, is the dominant source of instability. A paired greedy parameter-selection procedure is also introduced, with candidates drawn from analytic estimates of the upstream and downstream spectral regions, avoiding eigen-decomposition during the numerical march. OWNS-S is validated on incompressible, hypersonic and transonic boundary layers. In the transonic case, the disturbance spectrum reorganises from a subsonic to a supersonic topology during the march. The one-way computation proceeds continuously through this transition at a streamwise resolution unattainable by the parabolised stability equations.

In this paper, we consider stochastic optimal control problems with infinite-horizon joint chance constraints. By means of an appropriate state augmentation, we reformulate the original problem as a constrained Markov decision process, in which both the cost and the constraint function exhibit an additive structure. We then prove that this formulation enjoys strong duality, thereby enabling us to reformulate the problem as an equivalent unconstrained one in the Lagrange dual framework. We propose a dual-ascent algorithm to solve the resulting problem and show that it converges to a deterministic Markov policy defined over the augmented state space that is both optimal and feasible. To accommodate continuous state-input spaces, we propose a dedicated learning algorithm to approximate the value function in an offline training setting, thereby significantly reducing the computational complexity of the online control phase. We then test our approach on a numerical example and demonstrate its effectiveness compared to online predictive control methods in terms of performance and computational complexity.

Let \(g\) be a monic polynomial of degree \(r<n\), and let \(C_g(n)\) be the coefficient-vector code formed by multiples \(ug\) with \(\deg(ug)<n\). We study the coefficient-space MDS locus \(M_{n,r}\). The companion construction identifies coefficient space with the moduli of cyclic matrix-vector pairs, and the remainder-orbit map embeds it as a smooth complete intersection in the standard big cell of \(\operatorname{Gr}(r,n)\). We prove that every normalized maximal Plucker coordinate pulls back, up to sign, to a power of the constant coefficient \(A_0\) times a Schur polynomial \(S_\kappa(g)=s_\kappa(\Lambda_g)\), where \(\kappa\subseteq (n-r)^{r-1}\). Hence the universal MDS polynomial is \[D_{n,r}=A_0\prod_{\kappa\subseteq (n-r)^{r-1}}S_\kappa.\] This description yields a flat non-MDS boundary over \(\mathbb{Z}\), explicit degree and finite-field estimates, and a length filtration governed by sparse multiples. It also gives bad-characteristic criteria on root-multiplicity strata and density-one results on the irreducible stratum. Finally, for \(r\ge 3\) and \(N\ge r+3\), every nonempty first-failure layer over an algebraically closed field has a dense open non-GRS locus.

A quasi-metric measure space (qm-space) is a set with a directed distance whose symmetrization is a complete separable metric, together with a Borel probability measure of full support. Motivated by the problem of determining the pyramid limits of beta measures on forward Funk balls, we construct a compact metric space of pyramids of qm-spaces. The associated-pyramid map from the concentration-distance space of qm-spaces into this compact space is a $1$-Lipschitz topological embedding with dense image. As a secondary result, we prove that the box-distance space of qm-spaces is complete and separable.

We prove the logarithmic fractional-moment exponent for the mass of a shrinking ball under the critical two-dimensional stochastic heat flow. The proof builds on the geometric block decomposition of Gu and Tsai and introduces a fractional-product change of measure that preserves block independence and allows the fractional moments to be analyzed.

We prove a conjecture of Nieraeth asserting that if the Hardy–Littlewood maximal operator $M$ is bounded on an $s$-concave Banach function space $X$ for some $s\in (1,\infty)$, then $M$ is also bounded on the associate space $X'$. The proof is based on a new argument establishing the Fefferman–Stein inequality on $X$.

The present work presents a real Paley-Wiener theorem within the space of multi-anisotropic ultradifferentiable functions. We also show new properties of this space.

In this paper, we study a class of nonlinear Schrödinger equations (NLS) in a unified way. This class includes two important examples: the intermediate NLS (INLS) and the Calogero–Moser derivative NLS (CM-DNLS). Our main results are twofold. First, we prove small-data global well-posedness in $L^2(\mathbb{R})$ for a broad class of equations. This includes both focusing and defocusing CM-DNLS and the INLS for arbitrary choices of its parameters. Second, we prove small-data scattering in $L^2(\mathbb{R})$ under an additional assumption. This result covers both focusing and defocusing CM-DNLS and the INLS for specific choices of its parameters. Both the formulation of the problem, including the notion of solution, and the proofs rely crucially on a linear theory for Schrödinger equations with rough time-dependent potentials. This theory is also of independent interest, since we allow potentials so rough that the standard Duhamel formulation may not make sense. Our approach is perturbative and is built on the bilinear Strichartz estimate proved by Ozawa and Tsutsumi in 1998. In particular, our argument does not rely on integrability.

In recent work, the authors defined a sequence $(a_n)_{n\ge 0}$ to be $m$-convolutive exactly if \begin{align*} \sum_{n\ge 0} a_{mn} q^n = \left(\sum_{n\ge 0} a_n q^n\right)^m \end{align*} for a specific positive integer $m$ and provided proofs of the $2$- and $3$-convolutivity of a small number of sequences arising from primitive eta-products. Since the completion of that work, the authors have discovered many new instances of convolutive eta-products. The main focus of this work is to unify all but one of these instances in a parametric way.

Total quadratic ball discrepancy measures the deviation of codeword counts from uniformity over all Hamming balls. We prove that, whenever a Hamming space and cardinality admit a perfect code, the discrepancy minimizers among all subsets of that cardinality are precisely the perfect codes. This extends Barg's binary minimizing result and characterizes equality over arbitrary finite alphabets. At parameter sets that satisfy the sphere-packing condition and Lloyd's integral-root condition for correcting one or two errors, we give an explicit lower bound whose attainment is equivalent to perfect tiling: the discrepancy that a perfect code with these parameters would have. For one error, these include parameters at which the existence of a perfect code is open. For alphabets of size at least four, the discrepancy is an affine function of an energy with a strictly completely monotonic potential; for these alphabets, known universal energy bounds also give these lower bounds, and, when a perfect code exists, known universal optimality results also give the characterization of the minimizers. Over the ternary alphabet, complete monotonicity fails, and we prove the ternary bounds directly at all Hamming parameters and at the Golay parameters, the only nontrivial ternary cases.

In this paper, we introduce the lower and upper decay indicators and the associated decay character on the Heisenberg group. These notions are used to derive decay estimates for the linear fractional diffusion equation and to characterize the decay of initial data in several function spaces. We then study the Cauchy problem $$ \partial_t^2u+\left(-\Delta{\mathrm H}\right)^{\delta_1}u+\left(-\Delta_{\mathrm H}\right)^{\delta_2}\partial_tu=0, \quad \delta_1\in\left[0,\frac{\delta_2}{2}\right], $$ and establish decay estimates for solutions and their derivatives in homogeneous fractional Sobolev spaces in terms of the decay characters of the initial data. These results recover known estimates and extend them to new classes of data. We also investigate the corresponding semilinear problem with nonlinearity $|u|^p$. Global existence and decay are proved for $$ p>1+\frac{2\omega\delta_1}{Q-2\omega\delta_2}, \quad \omega=\frac{Q}{Q+\min{r_{\mathrm H}(u_0),r_{\mathrm H}(u_1)-2\delta_2}+2\delta_2}, $$ together with the corresponding critical case. Finally, by constructing test functions adapted to the nonlocal fractional sub-Laplacians, we establish blow-up results and identify the critical exponent $$ p=1+\frac{2m\delta_1}{Q+m\gamma-2m\delta_2} $$ for initial data in $\dot H_m^{-\gamma}(\mathbf H_n)$, where $m\in(1,2]$ and $\gamma\in\left[0,Q-\frac{Q}{m}\right)$.

Prefix caching reuses the key–value (KV) states of shared prompt prefixes and can substantially reduce the time to first token (TTFT) of large language model (LLM) inference. In a wireless edge network, however, prefix states are distributed across geographically separated GPU nodes. A nearby node offers a short radio path but may provide little reuse, whereas a more distant node may cache a longer matching prefix but incur additional communication and queueing delay. Moreover, persistent prefixes and active-request KV states compete for the same GPU memory, so aggressive caching can reduce inference concurrency and create queueing hotspots. This paper develops a stochastic-geometry and queueing framework for this spatial communication–caching–computation tradeoff. We represent the prompt workload by a prefix forest and define an ancestor-closed cache profile that may contain multiple reusable prefixes. Edge GPU nodes form a Poisson point process and are independently marked by cache profile, yielding analytically tractable spatial tiers. We derive the profile-association probability, conditional serving-distance distribution, token-level computation-offloading ratio, and TTFT coverage probability under a load-aware association policy. A fixed-point formulation captures the coupling between spatial association and multi-server GPU queues, while an outer optimization selects the cache-profile distribution subject to static-memory and stability constraints. Analytical and Monte Carlo results agree closely. The results show that the latency-optimal node need not be the nearest node, that TTFT can be non-monotonic in cached-prefix depth because of GPU-memory coupling.

This paper studies survival of certain bird species (Zebra Finches) under different food availability conditions. A competing risks model is studied under improved adaptive type-II progressive censoring scheme (IAT-II PCS) referring to this phenomena. Two independent competing causes of failure are considered where lifetime of these failures are assumed to follow Gompertz distribution with unknown scale and shape parameters. Maximum likelihood estimators (MLEs) of the unknown parameters are derived. It is established that they exist uniquely. Asymptotic confidence intervals (ACIs) are also constructed using asymptotic normality property of the MLE. Bayes estimates are obtained with respect to both non-informative and informative priors under different loss functions. Highest posterior density (HPD) credible intervals are calculated. A Monte Carlo simulation study is conducted to compare the performance of the proposed estimates. Three optimality criteria are studied to obtain the optimal censoring scheme. Finally, a real life data set is analyzed for further illustrations.

Let $D$ be a digraph. In this note, an arc coloring of $D$ is an assignment of colors to the arcs of $D$ such that no two arcs with a common tail receive the same color and no two arcs with a common head receive the same color. Under such a coloring, each vertex $v$ is associated with an out-color set and an in-color set, consisting of the colors assigned to the arcs with tail $v$ and to the arcs with head $v$, respectively. An arc coloring of $D$ is vertex-distinguishing if any two distinct vertices have different out-color sets and different in-color sets. The minimum number of colors required for a vertex-distinguishing arc coloring of $D$ is called the vertex-distinguishing chromatic index of $D$, denoted $\chi_{vd}^{\prime}(D)$. In 2016, Li, Bai, He, and Sun conjectured that $\chi_{vd}^{\prime}(D)=k(D)$ for any digraph $D$ with at most one source and at most one sink, where $k(D)$ is a natural lower bound determined by the outdegree and indegree sequences of $D$. We confirm this conjecture.