Papers

Write $\mu_t$ for the $t$-th roots of unity and $z^{\pm1}$ for $r$ free reciprocal pairs. We study $\Phi_{t,r}(\beta)=s_{\lambda}(\mu_t,z^{\pm1})$, $\beta=\lambda+\delta$, and the question the companion paper left open after $r=1$: when does it vanish? We factor the evaluation into classical branching followed by a torsion filter, and the shape depends on the parity of $t$: the point lies in the orthogonal group with determinant $(-1)^{t+1}$. For odd $t$ it sits in the identity component: an ordinary restriction $SO_{2R'+1}\downarrow SO_{2m'+1}\times SO_{2r}$, the filter an odd orthogonal character at a principal element of order $h+1$. For even $t$ in the other: a twining, a virtual expansion, and a torsion element regular but not principal; there we prove the filter, with its sign. One description covers both: the filter is nonzero exactly when the shifted point is regular semisimple. Both are minimal-level fusion projections: the even of type $C$, the odd's tensor sector of type $B$. Affine folding accounts for $0,\pm1$; what it does not survives as conjectures. The highest surviving weight is the dominant vertex of the numerator's Newton polytope minus the denominator's — the latter proved, the former conditional on a single-orbit property — and the class there is conjecturally primitive, $\pm$ the generator of the rank-one quotient. For odd $t$ and one $\Lambda$ that numerator is a signed transversal count in $\{0,\pm1\}$ by the equal-rank character formula, leaving one division. We invert it in closed form, along an arithmetic progression; the quotient is $\pm\epsilon_t\det M$ for an explicit $0/{\pm}1$ matrix — an interval matrix up to signs, hence totally unimodular, which settles (L1). The fibre count is a permanent, odd only when $1$, so at a dominant index a multi-hit fibre sums to zero. Two extremal statements remain. What is unproved is measured, in both parities.

Altun and Hancer \cite{AltunHancer2019} proved that every continuous \(P\)-contractive self-map of compact metric space is an almost Picard operator, and asked whether it must in fact be a Picard operator. We give an affirmative answer. The proof follows by combining the existing results with a final compactness argument.

For a centered, variance-one random variable $X$ with finite fourth moment, let $x$ be the vector of square-free degree-$d$ monomials in $n$ independent copies of $X$. At the critical scale $d^2/n \to \lambda \in (0,\infty)$, the normalized squared length of $x$ converges to the lognormal variable $R = \exp(\sqrt{\lambda v}\, Z - \lambda v/2)$, where $v = \mathbb{E} X^4 - 1$ and $Z$ is standard normal. If $\binom{n}{d}/N \to c \in (0,\infty)$, the sample covariance of $N$ independent copies of $x$ has an almost-sure limiting spectral law: the free compound-Poisson law with rate $1/c$ and jump distribution $\operatorname{Law}(cR)$. It reduces to Marchenko-Pastur when $\lambda v = 0$. The proof shows that subtracting the contribution of the sample length leaves vanishing quadratic-form fluctuations, even when $\mathbb{E} X^3 \neq 0$; length and direction need not be independent.

Anytime solvers return usable solutions before they terminate and improve them while time remains. Their progress is commonly summarized by a primal integral, a final gap, or convergence curves. Each summary leaves consequential choices open: how to score a run before its first feasible solution, whether poor incumbents are truncated, and whether the reference value is updated after the experiment or version-frozen beforehand. These choices become visible when runs produce no valid incumbent or improve a published best-known value. We study a normalized signed primal integral built from a bounded relative gap. It assigns an intrinsic worst value to an empty run, requires no acceptance threshold, and assigns negative instantaneous gaps to incumbents that beat a frozen reference. We compare the smooth difference-over-sum kernel with a signed version of Berthold's max-normalized gap and use the former as a working default. We also distinguish analysis-time from version-frozen reference policies and recommend reading the score with the raw final gap, mean convergence curve, and target-attainment curve. We evaluate these choices on a five-arm routing campaign and two model-fidelity ladders. The two signed kernels preserve every panel ordering in this study, whereas a common acceptance threshold compresses the distances between arms and reverses one panel ordering. Updating 54 of the 212 reference values changes score levels and removes all negative scores, while leaving the observed panel orderings unchanged. An exploratory screen also identifies 12 panel comparisons in which similar integral scores conceal materially different endpoints or attainment rates. The implementation, frozen inputs, and generators are openly released.

We consider the family ${\mathcal{C}}_d^{\Pi_{d-1}}$ of all $d$-dimensional copulas whose $(d-1)$-dimensional marginals are all equal to the $(d-1)$-dimensional product copula $\Pi_{d-1}$ and tackle the natural question, `how far away' from the $d$-dimensional product copula $\Pi_d$ elements in ${\mathcal{C}}_d^{\Pi_{d-1}}$ can be. We provide definitive answers for both, the uniform metric $d_\infty$ as well as the stronger, conditioning-based metric $D_1$. The established results clearly indicate that the family ${\mathcal{C}}_d^{\Pi_{d-1}}$ is larger than one might expect.

Stochastic mean-field control (MFC) provides a fundamental framework for coordinating large populations of interacting agents under uncertainty, with a wide range of applications. Existing numerical and deep-learning methods solve one MFC problem instance at a time and must be re-optimized whenever the task changes. In this work, we formulate stochastic MFC as an operator-learning problem and develop, to the best of our knowledge, the first mesh-free, self-supervised neural operator for stochastic MFC. The main challenge is that the diffusion term in the controlled Fokker–Planck equation precludes deterministic transport-map representations. We address this challenge by combining the probability-flow ODE with an invertible normalizing-flow-based transformer, which recasts the dynamics as a deterministic continuity equation and enables closed-form score evaluation through the exact inverse and analytical log-determinant of the normalizing flow, with $\mathcal{O}(d)$ cost per particle for networks of fixed size. Through transformer-based in-context learning, task prompts, represented by compact distribution parameters or raw particle clouds, condition the transport map, enabling a single pretrained operator to solve unseen tasks in one forward pass. The resulting Normalizing Flow Invertible Solution Transformer (NFIST) is trained end-to-end by minimizing the stochastic control objective directly, requiring no precomputed numerical solutions for training. We further prove the consistency of the proposed operator-learning formulation with task-by-task optimization. Numerical experiments on stochastic optimal control, Schrödinger bridge, systemic-risk control, and obstacle-avoiding path planning demonstrate effective zero-shot generalization while substantially reducing the computational cost of solving large families of stochastic MFC problems.

Adding a uniform external magnetic field to a mean-field spin-glass model usually requires a new analysis specific to the model. The disordered system we consider corresponds to the Gaussian binary branching random walk (BRW) - in the spirit of Derrida and Spohn [21] and studied from the statistical-physics point of view by Jagannath [22] - and we prove that this is not the case : a single elementary observation - that the resulting Hamiltonian is still a BRW, now with independent, but non-identically distributed, displacements - allows us to use the available results for general BRW. Combining classical and recent results on general BRW (Biggins [3], Chauvin and Rouault [13], Mallein [25]), one obtains an essentially complete picture of the model in an external magnetic field : the ground state, the free energy, the one-step replica symmetry breaking (1-RSB) transition, and the limiting genealogical overlap distribution with Poisson-Dirichlet statistics for the Gibbs weights. We then prove a strong concentration result for the magnetization under the Gibbs measure at low temperature, giving its explicit optimal value. It turns out that this one-replica statement is not enough to control the classical Ising overlap between two independently sampled configurations : an elementary counterexample (Remark 4.1) shows that concentration of each replica's magnetization does not, by itself, determine their joint correlation. We address this by developing a two-replica large-deviation argument - resting on the classical method of types and the subadditivity of Shannon entropy, and taking the form of a uniform Chernoff bound over the joint empirical type of a pair of configurations, matched against a two-replica concentration estimate - which we use to obtain the distribution of a second (Ising) hypercube-type overlap.

In this paper, we study Levine equivalence in aura topological spaces. We show that the aura topology is determined by the reachability preorder induced by the scope function and is therefore an Alexandrov topology. We prove that the aura-Levine hull has the explicit representation $K_{\mathfrak a}(A)=\uparrow_{\mathfrak a}A=S_{\mathfrak a}^{\infty}(A)$, and hence two subsets are aura-Levine equivalent if and only if they have the same eventual forward spread. We also describe the quotient structure of the equivalence classes, characterize separation properties through the reachability preorder, classify scope functions that induce the same Levine equivalence, and examine preservation under aura-continuous mappings. For finite aura spaces, we provide an explicit procedure for deciding aura-Levine equivalence using the reflexive-transitive closure of the scope relation.

In this article, we consider an energy-critical quintic wave equation on a bounded domain $\Omega\subset\mathbb{R}^3$ with nonlinear and nonlocal Kelvin–Voigt damping of the form $-\|\nabla u_t\|_{L^2(\Omega)}^\alpha\Delta u_t$, where $\alpha\in\mathbb{R}_+=[0,\infty)$. Under suitable hypotheses on the quintic source term, we establish the well-posedness of the problem and investigate its long-time dynamics in the natural energy space $\mathcal H=H_0^1(\Omega)\times L^2(\Omega)$. For every $\alpha\in\mathbb{R}_+$, we show that the associated dynamical system $(\mathcal H,S^\alpha(t))$ is gradient and dissipative, and we prove a stabilization estimate that yields asymptotic smoothness and, consequently, the existence of a compact global attractor $\mathcal A_\alpha$. The same estimate provides an upper bound for the Kolmogorov $\varepsilon$-entropy of $\mathcal A_\alpha$ and, in the limiting case $\alpha=0$, reduces to a quasi-stability inequality, which implies that $\mathcal A_0$ has finite fractal dimension. Furthermore, we prove that the family $\{\mathcal A_\alpha\}_{\alpha\in\mathbb{R}_+}$ is uniformly bounded in the higher-regularity space $\mathcal H_1=(H^2(\Omega)\cap H_0^1(\Omega))\times H_0^1(\Omega)$. Finally, we establish the upper semicontinuity of $\{\mathcal A_\alpha\}_{\alpha\in\mathbb{R}_+}$ at $\alpha=0$, showing that the attractors associated with the nonlinear and nonlocal Kelvin–Voigt damping converge to the global attractor of the limiting problem with classical linear Kelvin–Voigt damping.

We classify pairs consisting of a right square pyramid and a right square prism, each having rational base side length and rational height, for which the two solids have equal surface area and equal volume. We prove that, up to scaling by a common positive rational factor, there is a unique such pair. The problem reduces to determining the rational points on a genus 2 bielliptic curve whose Jacobian has rank 2. We accomplish this using a quadratic Chabauty computation followed by the Mordell–Weil sieve. The same analysis gives an analogous classification for right circular cones and right circular cylinders.

The inverse Born series provides a general perturbative framework for representing nonlinear inverse maps between infinite-dimensional function spaces and has found numerous applications in inverse problems governed by partial differential equations and integral equations. Motivated by its operator-theoretic structure, we develop a systematic framework for constructing neural operators that approximate the operator expansions arising in the inverse Born series. Our approach combines the analytical representation of the inverse Born series with the expressive power of neural operators, yielding data-driven approximations of the nonlinear inverse map while preserving the underlying operator structure. The proposed framework is applicable to a broad class of inverse problems and is presented in a general functional-analytic setting. To demonstrate its practical performance, we consider two representative examples: inverse scattering and the Calderón (electrical impedance tomography) problem. Numerical experiments show that the constructed neural operators accurately approximate the inverse Born expansions and produce high-quality reconstructions across a range of test cases. These results indicate that the proposed methodology provides an effective and computationally efficient approach for learning nonlinear inverse operators and suggests a promising direction for integrating classical operator expansions with modern neural operator architectures.

We complete Banach's isometric conjecture over the complex field. More precisely, if \(X\) is a complex normed space and, for some \(2\leqslant n<\dim_{\C}X\), all its \(n\)-dimensional complex subspaces are isometric as metric spaces, then the norm is induced by a Hermitian inner product. We also prove the quaternionic counterpart. The central geometric argument first treats real star bodies without convexity or central symmetry; applied to circled complex or quaternionic bodies, it shows that mutually real-linearly equivalent hyperplane sections force the ambient body to be a Hermitian ellipsoid. The proof adapts the bundle-degree mechanism introduced by Lu and Yang for the real case. Finally, we obtain extensions to absolutely homogeneous functions, graded Fréchet spaces, metrisable locally convex spaces, and compatible translation-invariant metrics.

The ruling polynomial is a Legendrian invariant that is closely related to the augmentation variety of the Legendrian. We characterize all graded and ungraded ruling polynomials, and construct Legendrian links realizing each possible polynomial. The graded augmentation varieties of the Legendrians we construct all have trivial cluster algebra structures. Finally, we construct Legendrian knots admitting $k$ exact Lagrangian fillings with $\chi(L)=n$ that are pairwise smoothly non-isotopic for $n\leq 1$, and $k\geq 0.$

We give an alternate expression for the Kloosterman sums associated with the long Weyl element of \(GL(n)\) that avoides the use of Plücker coordinates. The expression is instead based on the use of the well-known Cauchy-Binet formula from linear algebra. From this expression, we prove some important properties of these Kloosterman sums.

In the digital twin paradigm, online parameter updating is essential. Unlike a static model, a digital twin must continuously adapt to the evolving dynamics of the system it represents. Adaptive observers, which jointly estimate states and parameters from online data, are therefore an increasingly important tool. In this work, we formulate a mode-wise adaptive observer for a class of autonomous switched nonlinear systems with switched unknown parameters. The main challenges are twofold: removing the disturbance that switching injects into the parameter-error dynamics and guaranteeing persistence of excitation over a finite-time window without relying on an input signal. To address them, we assign a dedicated adaptive observer to each mode, active only on its corresponding interval, which directly removes the zero-input disturbance caused by switching. We then introduce a finite-window persistence-of-excitation condition together with a minimum dwell-time condition, under which the parameter estimation error is contractive and the state estimation error is bounded within the active interval of each mode. The performance of the proposed approach is illustrated with an academic example and with a practical example of evolutionary therapies.

Estimating the difference of two Stein's score functions is a fundamental problem in generative modeling. In particular, score differences arise naturally in transfer learning, where the score difference provides the mechanism for adapting a pre-trained model to a new target distribution, and in diffusion model-based post-training methods such as discriminator guidance. Existing estimators for score differences in these settings either lack of statistical consistency or are difficult to scale up in high-dimensions. We propose a statistically consistent and scalable estimator for score differences based on Sobolev regularization, which plays a crucial role in ensuring consistency and stablizing the training in the small-sample regime. Mathematically, we establish a convergence rate of $O(n^{-\frac{s-1}{d+2s-2}})$ where $d$ is the dimension and $s$ denotes the smoothness of the underlying densities, and provide a minimax lower bound of $\tilde{\Omega}(n^{-\frac{2(s-1)}{d+2s}})$ (in mean-squared error). Empirically, our estimator exhibits significantly improved stability in small-sample regimes compared to existing methods. We demonstrate its effectiveness on real-world tasks, including transfer learning for ECG signal generation, where it substantially outperforms non-regularized score difference estimators in downstream classification performance.

We prove a limit multiplicity conjecture of Hee Oh for principal congruence covers of geometrically finite hyperbolic manifolds, with an explicit power-saving rate. The rate is governed by the return of Patterson–Sullivan shadows through a fixed compact core, giving a geometric interpretation to the exceptional eigenvalue density. For convex-cocompact groups, a packing argument for enlarged shadows gives a stronger rate.

We establish exact second-order coding rate regions at the two corner points of the capacity region of the two-user Gaussian multiple-access channel. For any average error probability $\varepsilon\in(0,1)$, we characterize the $n^{-1/2}$-scale fluctuations of achievable rates around each corner point, proving a converse that matches the previously known achievability bound. The proof first extracts a rectangular subcode that preserves the independence of the two transmitted codewords. The Polyanskiy–Verdú good-code output distribution theorem then yields log-determinant constraints that induce a spectral decomposition of a trimmed codebook into diffuse and exceptional subspaces. An entropic Brascamp–Lieb projection inequality controls the codeword inner product on the diffuse subspace, while variance-inflated Gaussian output distributions handle the low-dimensional exceptional subspace. Combining these two treatments yields the joint Gaussian limit required for the matching second-order converse.

Let $p_k(n)$ be the number of $k$-polymatroids on $[n]$. We show that for every fixed $k \geq 1$, we have \[ \left\lfloor \frac{k}{2} \right\rfloor \cdot \binom{n}{\lfloor n/2 \rfloor} \cdot (1+o(1)) \le \log_2 p_k(n) \le k \cdot \binom{n}{\lfloor n/2 \rfloor} \cdot (1+o(1)). \] We also show that for $k \geq 2$, almost all $k$-polymatroids are (i) connected, (ii) proper, and (iii) not linearly representable over any field.

For continuous viscosity solutions $u$ of fully nonlinear uniformly elliptic equations $F(D^2 u)=0$, the work of Nadirashvili–Tkachev–Vlăduţ shows that $u$ need not be $C^2$ in dimensions $n\ge 5$. It is therefore natural to ask how large the set $\operatorname{Sing}(u)$, consisting of points at which $u$ has no $C^2$ neighborhood, can be. Under an additional $C^1$ assumption on $F$, Armstrong–Silvestre–Smart proved that $\operatorname{Sing}(u)$ has Hausdorff dimension at most $n-\varepsilon$ for some $\varepsilon >0$. In this paper, we show that, in dimensions $n\ge 5$, the Hausdorff dimension of $\operatorname{Sing}(u)$ cannot be bounded away from $n$ under uniform ellipticity alone. In fact, we prove the stronger statement: $\operatorname{Sing}(u)$ can be any compact nowhere dense set in $\mathbb{R}^5$ modulo a countable set. Consequently, in every dimension $n\geq 5$, the Hausdorff dimension of $\operatorname{Sing}(u)$ can be any number in $[n-5, n]$; moreover, $\operatorname{Sing}(u)$ can even have positive Lebesgue measure. In contrast, for every dimension $n$ and every uniformly elliptic operator $F$, we prove that the set $\Sigma_{2,0}(u)$ of points at which $u$ is not twice differentiable has Hausdorff dimension at most $n-\varepsilon$ for some $\varepsilon>0$. In particular, this quantitatively strengthens Trudinger's theorem that $u$ is twice differentiable almost everywhere. More generally, we prove for every $\alpha\in [0,1)$ that the set $\Sigma_{2,\alpha}(u)$ of points at which $u$ has no $C^{2,\alpha}$ expansion has Hausdorff dimension at most $n-\varepsilon(1-\alpha)$.

We develop a real twistor-space formulation of four-dimensional CFT Wightman correlators. We write the conformal generators and little-group constraints in ambitwistor space. The quadric condition $Z\cdot W$=0 arises naturally from the little-group constraints. We then solve for both the parity-even and parity-odd correlators in the ambitwistor variables. We further establish the connection between the ambitwistor correlators and the ones in the helicity-basis Grassmannian via a half-Fourier transform, and also develop the Penrose transform to recover the corresponding position-space correlators. We notice that for correlators with multiple tensor structures, the Penrose transform yields the full space of conformally invariant structures, but regularizing the associated Schwinger integrals selects the physical correlator. Finally, starting from the covariant Grassmannian formulation, we construct massive twistors that make the little-group symmetries manifest and develop their Penrose transform, which again reproduces the correct position-space correlators.

Via the circle method, we deduce that all sufficiently large positive integers are sums of two primes and six positive Fibonacci numbers, and are also sums of two primes and seven Lucas numbers. We also prove that each sufficiently large positive integer can be written as a sum of two primes, three Fibonacci numbers, and three Lucas numbers.

This paper presents new theoretical results on generalizing the Jaccard distance for lattices and real valuations. We demonstrate that when the valuation is strictly positive, monotone, and modular, the Jaccard distance satisfies the triangle inequality on arbitrary lattices, effectively generalizing earlier results that depended heavily on distributivity. Moving to relatively complemented distributive lattices (which safely drop the requirement for the global bounds found in Boolean algebras), we prove the triangle inequality holds as long as the valuation is positive, monotone, supermodular, and $\log$-submodular. Additionally, we adapt the symmetric-difference Jaccard formulation for submodular valuations to sectionally complemented distributive lattices. Shifting to necessary conditions, we prove that supermodularity is a strict requirement for the standard generalized Jaccard distance to operate as a valid metric. Finally, we map the practical value of relaxing these structural constraints to computational fields like quantum information theory, formal concept analysis, and machine learning, closing with a brief look at open mathematical problems.

Near maximum distance separable (NMDS) codes of dimension 3 and length n over the finite field with q elements are equivalent to (n,3)-arcs in PG(2,q). For every odd prime power q we construct, by adding five suitable points to a conic of PG(2,q), a family of [q+6,3,q+3] NMDS codes and determine their weight distributions completely; three distinct weight enumerators occur, governed by two explicit quadratic-character conditions on the parameters. For every odd prime power q, no code of our family is monomially equivalent to a code of the recent [q+6,3,q+3] NMDS family of Fan, Wang and Xu, even where the weight enumerators of the two families coincide: the separating invariant is a triple of geometric data attached to the underlying arc. Extending the configuration by a sixth point on a distinguished external line, we further obtain [q+7,3,q+4] NMDS codes, together with their weight distributions, for every odd prime power q >= 11 (admissible parameters exist for no q <= 9); the existence proof combines exact and Weil-type character sum estimates with a finite computer verification. Our proofs are purely geometric and rest on a simple counting identity for the trisecant lines of a point set obtained by extending a conic. All the codes constructed are optimal locally recoverable codes with locality 2.

Stanley's tree-isomorphism conjecture asks whether the chromatic symmetric function distinguishes nonisomorphic trees. We give a reconstruction criterion that permits repeated leaf-component orders. For a proper tree, collapse each leaf component to its center and record its order as a vertex weight. We prove that the chromatic symmetric function reconstructs the tree whenever every nonleaf vertex of this weighted core has a weight that occurs nowhere else in the core. Repetitions among core leaves are unrestricted. The proof uses only the leading star-basis partition and the coefficients immediately above it. As a consequence, the conjecture holds for an infinite class of diameter-six trees not covered by the condition that all leaf-component orders are distinct. We also give a canonical integer-partition model for arbitrary diameter-six trees and an exact cut-partition implementation intended for further work. The unrestricted diameter-six case remains open.

Trust assessment is a fundamental component of cooperative and connected vehicle systems. However, existing approaches operate primarily at the level of individual vehicles, making it difficult to reason about trust evolution across road segments. In this paper, we propose a spatio-temporal trust-field framework that aggregates microscopic vehicle-level trust into a continuous representation over space and time. The trust field is formally defined on road segments. We conducted simulation-based experiments using synthetic trajectories generated under controlled conditions, enabling analysis of trust-field behavior in simple road scenarios. Beyond theoretical modeling, we study an implication of the trust-field concept: reconstructing the full trust field from sparse roadside-unit (RSU) measurements. We compare (i) a coordinate-based deep learning baseline that learns a generic trust field from sparse samples and (ii) a field-informed deep learning method that treats trust as a latent quantity carried by vehicles and enforces measurement consistency through the aggregation mechanism. The field-informed approach more accurately recovers trajectory-aligned low-trust patterns and yields improved reconstruction error.

We prove unique ergodicity of the projective process associated with the two dimensional Navier–Stokes equation in vorticity form, with additive diagonal noise acting on every nonzero real Fourier phase and satisfying two sided power law bounds. Consequently, the exact Furstenberg–Khasminskii formula for the top Lyapunov exponent holds. The main new ingredient is a compact dense mechanism for asymptotic generalized coupling in the absence of a Foiaş–Prodi type high-low mode decomposition for the projective dynamics. Using the dense range of the Malliavin derivative and compactness of the state derivative, we construct finite rank perturbations of the Wiener path that compensate, to first order, for perturbations of the initial condition, leaving a residual whose logarithmic growth has negative stationary mean and hence contracts locally at an exponential rate, with a cost controlled by a triangular scheme of blockwise Ramer transformations.

A well-known conjecture of Chern, do Carmo, and Kobayashi asserts that, for $n,m \geq 1$, the scalar curvature of a closed, minimally immersed $n$-submanifold of $\mathbb{S}^{n+m}$ with second fundamental form of constant length takes values in a discrete set. This property holds in every codimension when $n \in \{1,2\}$. We disprove this conjecture for all $n \geq 3$ with $m \geq 4$, and for even $n \geq 4$ with $m \geq 3$.

Conformal symmetry strongly constrains correlation functions. In momentum space, the conformal Ward identities are solved for three-point functions by integrals of a product of three Bessel functions ("triple-$K$ integrals"); more generally, integrals of this type ("multiple-$K$ integrals"), serve as the building blocks of a wide class of higher-point conformal correlators. These integrals belong to the class of generalised hypergeometric functions, and while their series representations are known in principle, none converges throughout the entire physical region of kinematic space selected by momentum conservation. In this work, we construct series representations adapted to exactly this physical domain. Using the method of brackets, which turns the evaluation of definite integrals into solving a linear system of algebraic equations, we derive various Lauricella-type series representations for multiple-$K$ integrals. For triple-$K$ integrals, conventionally expressed in terms of the Appell $F_4$ function whose series converges only outside the triangle-inequality region, we find instead a compact, two-branch series that converges throughout the entire physical region and for arbitrary scaling dimensions. We then extend this construction to general multiple-$K$ integrals: by introducing a new set of kinematic variables, we build an iterative series representation that converges for all physical kinematic configurations.

In this paper, we provide estimators for quantum Rényi and Tsallis entropies with nearly optimal sample complexity. Specifically, for order $\alpha$, dimension $d$, and additive error $\varepsilon$, 1. For $0 < \alpha < 1$, the sample complexity is $O(d^{1+1/\alpha}/\varepsilon^{1/\alpha} + d^{1/\alpha-1}/\varepsilon^{2})$ for Rényi entropy and $O(d^{1+1/\alpha}/\varepsilon^{1/\alpha} + d^{2-2\alpha}/\varepsilon^2)$ for Tsallis entropy. In particular, for $0 < \alpha \leq 1/2$, the sample complexity for both entropies is $O(d^{1+1/\alpha}/\varepsilon^{1/\alpha})$. 2. For non-integer $\alpha > 1$, the sample complexity is $O(d^2/\varepsilon^{1/\alpha} + d^{1-1/\alpha}/\varepsilon^2)$ for Rényi entropy. Our upper bounds improve the quantum Rényi entropy estimators due to Acharya, Issa, Shende, and Wagner (2017) and the quantum Tsallis entropy estimators due to Chen, Liu, and Wang (2026), and match the lower bounds recently established by Wang (2026).