Papers

This paper develops a risk sensitive (RS) Kalman filtering framework for discrete-time linear stochastic singular systems operating under communication constraints imposed by a round-robin protocol. Due to limited network bandwidth, only a subset of the available measurements can be transmitted at each sampling instant, resulting in a periodically varying measurement structure. By employing the Weierstrass canonical form (WCF), the singular system is transformed into an equivalent augmented state space model, yielding a round-robin induced periodic system (RRIPS). A recursive risk sensitive Kalman filter (RSKF) is then developed for the RRIPS through a Bayesian formulation and the minimization of an exponential quadratic cost function, from which the recursive filtering equations are obtained for the original singular system. To enhance robustness against modeling uncertainties and disturbances, an adaptive RS mechanism is introduced in which the risk parameter is adjusted online according to the available covariance information. This adaptive strategy guarantees the positive definiteness of the predicted covariance matrix while adjusting the degree of risk sensitivity to the prevailing estimation uncertainty. Furthermore, sufficient conditions ensuring the filter stability are established using the observability and controllability concepts of periodic systems. The proposed framework reduces to the standard KF for singular systems when the RS parameter vanishes and recovers the standard RSKF when the singular matrix reduces to the identity matrix. Finally, numerical results are presented to demonstrate the effectiveness, robustness, and improved estimation performance of the proposed approach in comparison with the standard KF.

In this article, we first discuss how triplewise independent random variables (rvs) are connected to a complete bipartite graph. Using the connection, we construct a sequence of triplewise independent rvs. We next consider a variance gamma (VG) approximation of sums of such triplewise independent rvs. Using Stein's method and the generalized zero-bias transformation, we obtain our bounds. Related limit theorems are also discussed.

We study the obstacle problem for the parabolic fractional $p-$Laplace equation \[\partial_t u+(-\Delta_p)^su = 0\] in the degenerate range $2<p<2/(1-s)$. We prove that viscosity solutions are locally Lipschitz continuous in space and Hölder continuous in time. If, in addition, $p>1/(1-s)$, the time regularity improves to Lipschitz continuity.

Lewy's classical theorem asserts that a one-to-one planar harmonic mapping has nonvanishing Jacobian. We prove a three-dimensional bounded-distortion analogue: if \[ f:\Omega\subset \mathbb R^3\to \mathbb R^3 \] is nonconstant, sense-preserving, quasiregular, and harmonic componentwise, then \(J_f>0\) throughout \(\Omega\). Thus harmonic quasiconformal mappings between domains in three-space are local harmonic diffeomorphisms. The new point is the Lewy-type differential conclusion \(J_f\neq0\), not merely topological local invertibility, which is already known for sufficiently smooth quasiregular mappings. The proof is by blow-up. A hypothetical zero of \(J_f\) produces a nonconstant homogeneous harmonic polynomial quasiregular mapping \(P:\mathbb R^3\to\mathbb R^3\) of degree \(m>1\). We exclude such homogeneous blow-ups by a second-order trace identity for \(J_P|_{S^2}\): after normalizing the first jet at a positive minimum, the identity gives a negative spherical trace, contradicting the maximum principle. We also derive an affine Liouville theorem for entire harmonic quasiregular mappings in \(\mathbb R^3\).

We propose a Rockafellian relaxation of the Walrasian equilibrium problem for an exchange economy that may not admit one. Market clearing is slackened by a non-negative variable $v$ whose norm is penalized; the relaxation is well posed throughout. As the penalty grows, the residual converges to a vector $v^*_\infty$ of minimum norm in the feasible range of excess demand, measuring the distance to the nearest equilibrium-admitting economy. A stressed Shapley–Shubik example recovers the analytical infeasibility floor to machine

We propose several approaches for bounding the minim\-um distances of the family of quasi-cyclic LDPC codes in the 5G NR standard. In particular, we show that the high-rate [9984, 8448] and the low-rate [25344, 8448] BG1 5G LDPC codes have minimum distances in the ranges {8..14} and {22..57}, respectively. Also we propose a new early termination approach based on circulant modular reduction, which significantly lowers syndrome calculation complexity for the LDPC decoder.

We establish quantitative unique continuation estimates for solutions of the complex Ginzburg-Landau equation in the framework of two-sphere and one-cylinder inequalities. While prior work [Dou et al. SIAM J. Control Optim. (2023)] relied on Carleman estimates, the present study develops a novel logarithmic convexity approach to prove the quantitative unique continuation property. We derive an explicit quantitative unique continuation constant and fully characterize its dependence on the parameters.

Every square matrix is uniquely the sum of a symmetric matrix and a skew-symmetric matrix. We extend this familiar fact to higher-order tensors: every cubic $k$-tensor is uniquely the sum of an $m$-piecewise symmetric tensor and a $(k-m)$-piecewise skew-symmetric tensor, for each choice of $m\leq k$. We study these tensor spaces via the combinatorics of word descents and compositions. Under Schur–Weyl duality, the two spaces naturally decompose into descent representations of the symmetric group. In fact, we prove a variant of Solomon's classical decomposition of its group algebra that explains this decomposition. Our motivation stems from signature tensors in stochastic analysis, algebraic geometry and data science: more specifically, we show that our tensor space decompositions determine the vanishing ideals of the signatures of piecewise linear paths with a fixed number of segments.

We study complex $m$-Hessian equations on bounded hyperconvex domains with right-hand side in $L^1(\Omega)$. The main contribution of this paper is a strong stability result for weak solutions in the Hessian energy sense. More precisely, if $f_j \to f$ in $L^1(\Omega)$ and $u_j, u$ are the corresponding solutions, then \[ \int_\Omega |u_j-u|\, H_m(u_j) \longrightarrow 0. \] This provides convergence in the natural energy topology associated to the Hessian operator, which is significantly stronger than convergence in capacity. For completeness, we also recall the existence of solutions and stability in capacity, which follow from known results in the literature.

This paper develops a systematic method for determining particular solutions of the $k$th-order linear nonhomogeneous recurrence relation $$a_n + c_1 a_{n-1} + \cdots + c_k a_{n-k} = \sum_{j=1}^J p_j(n){r_j}^n$$ with $n \geq k$, $c_k \neq 0$, $r_j \neq 0$. Here each $p_j(n)$ is a polynomial. The main result is the following: for the characteristic polynomial $c(t)=t^k+c_1t^{k-1}+\cdots+c_k$, if $s_j$ denotes the multiplicity of $r_j$ as a root of $c(t)$ ($s_j=0$ when $r_j$ is not a root), then there exists a particular solution of the form $q_n=\sum_{j=1}^J b_j(n)n^{s_j}r_j^n$, where each $b_j(n)$ is a polynomial of the same degree as $p_j(n)$. This result parallels the method of undetermined coefficients for linear ODEs with constant coefficients and yields a systematic procedure for determining the form of particular solutions.

This paper studies the nontrivial orientable $S^2$-bundle $\Sigma \ltimes S^2$ over a closed surface $\Sigma$ of genus $g \geq 1$. We have three main results as follows. We construct the relative Dax invariants for pointed embeddings of $\Sigma$ into $\Sigma \ltimes S^2$, which satisfies isotopy invariance, additivity, and naturality. For some embedded surfaces in $\Sigma \ltimes S^2$ with a fixed geometric dual, we establish a complete classification up to isotopy. We give an alternative construction of a surjective homomorphism $\hat{\Phi}: {\rm MCG}(\Sigma \ltimes S^2) \to \mathbb{Z}^\infty$.

Quantifying how much of the variation in a response is explained by its conditional mean is central to many statistical problems. In this paper, we develop a unified framework for this problem, centered on a normalized conditional mean discrepancy that coincides with the classical Sobol' index for univariate responses and its natural trace-based extension for multivariate responses. We propose a simple estimator of this measure based on nearest-neighbor graphs that can be computed in near-linear time. We establish its consistency and derive its rate of convergence. Further, under the null hypothesis of conditional mean independence, a studentized version of the statistic is asymptotically standard normal. This leads to a computationally efficient test for conditional mean independence that attains the correct asymptotic level and is universally consistent, without requiring bootstrap calibration or sample splitting. Building on the same estimator, we develop a model-free sequential variable-screening procedure that selects a sufficient set with high probability and an exponential error bound. We also discuss extensions of the framework to quantifying interaction effects through higher-order Sobol' indices. Simulations and real-data experiments demonstrate strong power, effective variable screening, and substantial computational gains over existing methods.

We consider the set of matchings of a graph and a local change operation, called a flip, between them. In the combinatorial setting, the base graphs are either complete graphs or complete bipartite graphs, and in the geometric setting, the graphs are embedded on point sets in the plane, with the requirement that edges must be drawn as straight lines and must not cross. For base graphs with an even number of vertices, we consider perfect matchings, i.e., all vertices are matched, and for base graphs with an odd number of vertices, we consider almost-perfect matchings, i.e., all but one vertex of the graph are matched. A 2-flip between two perfect matchings exchanges two edges, and a 1-flip between two almost-perfect matchings exchanges one edge. The corresponding flip graph has the set of perfect or almost-perfect matchings as vertices, with pairs of them connected by an edge if they differ in a 2-flip or 1-flip, respectively. In this work, we provide a comprehensive picture of Hamiltonicity properties of these flip graphs. We prove that the flip graphs in the combinatorial setting are Hamilton-connected, i.e., they admit a Hamilton path between any two vertices, or, if the flip graphs are bipartite, we prove that they are Hamilton-laceable, i.e., they admit a Hamilton path between any two vertices from different partition classes. In the geometric setting, we prove that any path in them misses exponentially many vertices, in particular, they have no Hamilton paths or cycles. For points in convex position and almost-perfect matchings under 1-flips, we complement this by constructing a cycle in the flip graph that visits almost all vertices.

We give a six-dimensional simply connected nilpotent Lie group together with a left-invariant Riemannian metric admitting a nonconstant closed geodesic. The example gives a positive answer to Böhm–Lafuente's question asking whether a homogeneous Riemannian manifold diffeomorphic to Euclidean space can have a closed geodesic.

We consider an inverse source problem for the $k$-Hessian equation \begin{equation*} \sigma_k(D^2u)=f(x) \end{equation*} on the $k$-admissible branch in a smooth uniformly convex domain in $\mathbb R^n$, where $2\le k\le n$. We prove that the nonlinear Dirichlet-to-Neumann map determines the positive smooth source from its values on the restricted large-data rays $t\phi_E|_{\partial\Omega}$, where $E\in Gr(k-1,n)$, $\phi_E(x)=|P_Ex|^2/2$, and $t$ is sufficiently large. The Hessian of each boundary profile has exactly $k-1$ large directions and is flat on $V=E^\perp$. Thus, the leading profile lies on a rank $k-1$ face of the $k$-Hessian structure, and the first source-dependent correction is governed by the missing directions. More precisely, this correction solves fiberwise Poisson equations on the affine sections $\Omega\cap(y+V)$ of dimension $q=n-k+1$. We prove that these sectionwise solutions patch smoothly through glancing points where the sections collapse, and we obtain boundary normal derivative asymptotics by local barriers. The boundary flux of the correction gives the section integrals $\int_{\Omega\cap(y+V)}f\,d\mathcal H^q$. Varying $E$ yields the affine $q$-plane Radon transform of the zero extension of $f$. We give an explicit reconstruction formula through the Fourier slice identity, and the injectivity of the affine Radon transform gives uniqueness. The endpoint $k=n$ recovers the Monge–Ampère chord/X-ray geometry, while the range $n\ge3$ and $2\le k<n$ gives inverse source results for genuinely non-determinant Hessian equations.

Let $P_n=\mathrm{Cay}(S_n,\{r_2,\ldots,r_n\})$ be the pancake graph, with prefix reversals acting on the right. Conjugating Zaks' suffix-reversal permutation Gray code by the full reversal gives a distinguished Hamiltonian cycle $Z_n$ in $P_n$. We determine the stabilizer of this particular cycle under the left regular action of $S_n$. If $\rho=r_{n-1}r_n=[n,1,2,\ldots,n-1]$, then, for every $n\ge3$, $\mathrm{Stab}_{L(S_n)}(Z_n)=\langle L_\rho,L_{r_n}\rangle\cong D_n$, where $D_n$ denotes the dihedral group of order $2n$. The inclusion $\supseteq$ follows from the recursive block decomposition $W_n=(W_{n-1}r_n)^{n-1}W_{n-1}$ and from the palindromy $W_n^R=W_n$. The reverse inclusion follows from a general cyclic-order rigidity lemma: if a Hamiltonian cycle on a finite group is invariant under $L_a$, with $\mathrm{ord}(a)\ge3$, then every left translation preserving the same cycle conjugates $a$ to $a$ or $a^{-1}$. For $n\ge5$, Deng-Zhang's automorphism theorem gives the same stabilizer inside $\mathrm{Aut}(P_n)$; the exceptional ranks are handled separately. We also compute the compression factor of $Z_n$: it is $n$ for $n\ge4$ and $6$ for $n=3$.

We prove uniqueness for an inverse boundary problem for the prescribed Gaussian curvature equation $\det D^2u=K(x)(1+|\nabla u|^2)^2$ on a smooth bounded simply connected planar domain. We compare smooth positive curvatures on a common open class of Dirichlet values for which the convex problems are well posed. Equality of the nonlinear Dirichlet-to-Neumann maps, together with the prescribed first boundary jet of the curvature, implies equality of the curvatures throughout the domain. The first linearized equation has conductivity $\Sigma=K(D^2u)^{-1}$. The planar anisotropic conductivity theorem determines this tensor up to a boundary fixing diffeomorphism. We prove that the Hessian and gradient structure forces this diffeomorphism to be the identity. The difference between the pulled-back background gradient and the original gradient satisfies a local elliptic system whose components have the same scalar principal part and zero boundary Cauchy data. A first-order Carleman estimate gives unique continuation for this system. The argument uses the nonlinear boundary response at one admissible Dirichlet value and its first derivative there.

We consider the initial-boundary value problem for a variant of the Keller-Segel chemotaxis system with doubly degenerate diffusion, i.e. we study \[ \left\{ \begin{array}{ll} u_t = \nabla \cdot (uv\nabla u) - \nabla \cdot (u^2v\nabla v),\\ v_t = \Delta v + u - v,\\ (uv\nabla u-u^2v\nabla v)\cdot\nu=\nabla v\cdot\nu=0,\\ u(x,0)=u_0(x),\quad v(x,0)=v_0(x), \end{array} \right. \] in a smoothly bounded domain $\Omega\subset\mathbb{R}^2$. Crucially, we only assume the sufficiently regular initial data to be nonnegative, but allow those functions to be zero at non-trivial parts of the domain. We show that, despite possibly starting from a degenerate state, the system admits global solutions in a framework of generalized energy solutions, whenever the initial mass is below the threshold number $m_0=4\pi$. Moreover, in a radial setting the threshold number can be increased to $m_0=8\pi$.

Understanding the complex relationships between atmospheric pollutant emissions and their multifaceted determinants presents a dual challenge: driving factors operate at fundamentally different temporal resolutions, from continuously monitored meteorological variables to annually reported socio-economic indicators, and their interconnections evolve dynamically over time. To address these challenges, we propose a Partially Functional Dynamic Structural Equation Model (PFDSEM) that coherently integrates functional covariates (e.g., high-frequency meteorological data) and scalar predictors (e.g., economic and demographic indicators) within a unified dynamic structural framework. The model captures non-stationary temporal dependencies and inter-variable correlations via a Conditional Autoregressive (CAR) structure combined with a Linear Model of Coregionalization (LMC), while functional covariates are incorporated through basis expansion with Bayesian P-spline smoothing. Bayesian inference via Markov Chain Monte Carlo provides full uncertainty quantification. Comprehensive simulation studies confirm accurate parameter recovery and robustness to prior specifications under diverse conditions. Applying the PFDSEM to pollutant emissions data from 30 Chinese provinces (2015–2020), we identify temporally dynamic and province-specific associations between ten categories of socio-environmental factors and ten major air pollutants, including CO$_2$. The results reveal substantial cross-province heterogeneity in the strength and direction of these associations and pronounced seasonal patterns in meteorological effects, offering a quantitative evidence base for designing temporally adaptive and regionally tailored environmental policies.

The Riemann Hypothesis (RH) asserts that every nontrivial zero of the Riemann zeta-function has real part equal to $1/2$. A zero-density theorem provides evidence towards RH by bounding the number of zeros of the zeta-function with real part greater than $1/2$. In 2024, Larry Guth and James Maynard announced a new zero-density theorem which, for a key location in the critical strip, strengthens previous work of Ingham and is the first such improvement in over 80 years. This expository paper places this remarkable achievement in the context of the rich history of zero-density theorems and explores its implications on the distribution of primes in short intervals.

Let $M_n$ be a nonnegative supermartingale with $M_0=1$, and let $T_b$ be the first time $M_n$ crosses $b>1$. Ville's inequality gives $P(T_b<\infty)\leq 1/b$. We show that, at every finite time horizon $N$, $$ 1=bP(T_b\leq N)+V_{b,N}+D_{b,N}+R_{b,N}, $$ where $V_{b,N}$ is the expected overshoot at crossing, $D_{b,N}$ is the cumulative predictable decrease before crossing, and $R_{b,N}$ is the expected value of paths that up to the time horizon $N$ remained below the boundary $b$. The identity follows from telescoping and can be made horizon free without additional integrability assumption. It gives a formula for the slack in Ville's inequality and thus the conservatism of the threshold test $M_n\geq b$. For likelihood ratio martingales, the same terms also describe power: $R_{b,N}$ is the probability, under the alternative, of not crossing by $N$, whereas $bP(T_b\leq N)+V_{b,N}$ is the power. The identity also allows us to verify horizon-free guarantees from a finite computation: $P(T_b<\infty)$ lies between $P(T_b\leq N)$ and $P(T_b\leq N)+R_{b,N}/b$, so a proposed threshold test $M_n\geq b_\alpha$ can be shown to be of level $\alpha\in(0,1)$ for every time horizon based on computations at the single horizon $N$. We illustrate how Ville's identity yields sharper thresholds for Bernoulli and exponential likelihood-ratio tests, and for a conformal test of exchangeability, demonstrating Monte Carlo calibration under a composite, nonparametric null. Lastly, we also establish a continuous-time analogue of Ville's identity.

We establish the existence of time-periodic solutions to the Boltzmann equation for the full range of soft potentials and prove their global stability. The proof relies on refined energy estimates combining Besov and Sobolev regularity with velocity-weighted energy methods. The analysis is complicated by the lack of a spectral gap in the soft potential regime together with the absence of time integrability of the periodic forcing over $\mathbb{R}^+$. This provides a framework for the study of time-periodic behavior for the Boltzmann equation.

We consider the 2D incompressible Euler equations on a periodic channel $\mathbb{T}\times (0,1)$ with inflow-outflow boundary condition $u=(0,1)$ on $\mathbb{T} \times \{0,1 \}$. We also impose the incoming vorticity boundary condition $\omega =\eta $ on $\mathbb{T}\times \{ 0 \}$, where $\eta$ is prescribed. We show that the problem is globally well-posed in Gevrey spaces (for any value of the Gevrey exponent $s>1$) as long as $\eta$ remains Gevrey. This proves that the inflow-outflow velocity boundary condition determines the solution locally in time if and only if the solution is considered in an analytic class. In particular, leaving the analytic class, nonuniquness of solutions occurs already in any Gevrey class, by prescribing $\eta$. Hence, prescribing an analytic inflow-outflow velocity leads to precisely one analytic and continuum $s$-Gevrey solutions for every $s>1$. Furthermore, the result implies that if $\eta$ is analytic, then the unique global solution can lose analyticity in space for all $t>0$, but remain $s$-Gevrey regular for all $s$.

We obtain an infinite-series representation for the arc length of a supercircle in terms of the scale parameter $a$ and the shape parameter $n$. The resulting expression is constructed by means of generalized binomial coefficients and Gauss hypergeometric functions, distinguishing two regimes associated with the value of $n$. We also analyze the absolute convergence of the resulting series. We verify the consistency of the formulation from limiting cases and particular configurations of the family of supercircles: when $n\to0^+$ and $n\to\infty$, the length converges to the value $8a$, corresponding to the limiting rectilinear geometries, whereas for $n=1$ we recover the perimeter of the rhombus with diagonals of length $2a$. In addition, as a validation against supercircles with exact arc length, the formulation reproduces with high numerical precision the arc length of the parabolic star, the astroid, and the circle. Finally, by specializing the circular case $n=2$ and normalizing the length by the diameter $2a$, we obtain a series representation, in terms of hypergeometric functions, for the constant $\pi$.

In this paper, for complete manifolds with a modified Ricci curvature bounded from below, we can successfully set up Cheng-type eigenvalue comparison theorems for the first Dirichlet eigenvalues of the Witten-Laplacian and the weighted $p$-Laplacian on geodesic balls of these manifolds.

Dynamic Wireless Power Transfer (DWPT) could eliminate plug-in charging in cities, but optimal urban deployment is complex. This paper develops a mixed-integer programming model that optimizes DWPT location under signalized intersection dynamics – acceleration, deceleration, and queue-position-dependent dwell time – through probabilistic signal patterns and saturation headway-based modeling. A case study of Kawagoe City, Japan, shows that electrifying 1.5% of the road network is sufficient to sustain continuous urban EV operation without plug-in charging for the baseline scenario, and at most 2.9% suffices across all tested assumptions. Monte Carlo simulations of continuous trip chains averaging approximately 600 km and reaching up to approximately 800 km confirm that optimized 12 kWh-battery deployments sustain operation in all simulated runs, revealing an infrastructure-battery tradeoff corresponding to roughly 1.7-3.0 tonnes CO2e of avoided battery manufacturing emissions per vehicle relative to a conventional 40 kWh urban EV. These findings position DWPT deployment as an environmentally efficient pathway for sustainable urban mobility when deployed optimally.

In recent years, both Appell series and truncated Appell series have garnered significant research interest among scholars. In this paper, we investigate the congruence properties of truncated Appell series and truncated $q$-Appell series. By employing various combinatorial identities and the `creative microscoping' method introduced by Guo and Zudilin, we provide $q$-analogues for several known congruences of truncated Appell series $F_1$ and $F_2$. Meanwhile, we establish a new $q$-supercongruence related to the truncated Appell series $F_4$ and propose a related conjecture, which provides a $q$-analogue for two conjectures by Apagodu and Zeilberger.

For $n\geq 3$, let \[ \Gamma(Q)=\langle E_{ij}(q_{ij}):i\neq j\rangle \] be the subgroup of $\operatorname{SL}(n,\mathbb R)$ generated by elementary matrices with nonzero rational parameters $q_{ij}$. We prove that $\Gamma(Q)$ is always $S$-arithmetic, extending classical integral-parameter results to arbitrary rational parameters. Our method is effective. We also prove strong small-generation results for finite-index subgroups of $S$-arithmetic groups. In particular, $\operatorname{SL}\left(n,\mathbb Z\left[\frac1N\right]\right)$, $n\geq3$, and a broad class of groups $\operatorname{SL}(2,\mathcal O_{F,S})$ have arbitrarily small two-generated finite-index subgroups. We then study the rank-one family \[ \Gamma_q=\left\langle\begin{pmatrix}1&1\\0&1 \end{pmatrix}, \begin{pmatrix} 1&0\\q&1 \end{pmatrix}\right\rangle,\qquad q=\tfrac{s}{t}\in\mathbb Q. \] For $q\neq0,\pm3$, we prove that \[ \Gamma_q=\Gamma_1^{(t)}(s) \] if and only if its upper-triangular subgroup strictly contains \[\left\langle\begin{pmatrix}1&1\\0&1\end{pmatrix}\right\rangle.\] We then apply the rank-one criterion to constructions arising from non-freeness, verifying it for $q=\tfrac{s}{t}\in(-4,4)$ with $1\leq |s|\leq21$, and obtaining infinite families from indefinite binary quadratic forms and Pell-type equations.

Linear Gaussian Bayesian networks, equivalently linear Gaussian structural equation models, recur across statistics, control, and communications; in the vector-valued setting that motivates this work, their nodes are vectors and their edges are matrices. Every quantity of interest is a function of sub-blocks of the joint covariance, which is itself a classical, differentiable map (the K-recursion) from the local edge and innovation parameters. Yet the resulting inference and estimation tasks are usually derived and implemented separately, per task and per topology. Taking this covariance chart as a single backend, we build on it a unified, differentiable covariance calculus in which each task reduces to a few linear-algebra primitives on the one covariance, and automatic differentiation returns every gradient in a single backward sweep, over arbitrary vector-valued directed acyclic graphs and parametrizations, including tied and structured ones. The calculus covers conditioning, conditional-independence testing through mutual information, maximum-likelihood estimation with hidden nodes, and the Slepian–Bangs Fisher information with the local identifiability and Cramér–Rao reliability it induces. It is validated on a linear Gaussian state-space model and a skip-connected (non-chain) extension against the Kalman recursions, d-separation, and the Cramér–Rao bound.

There is an interesting open question: for $n$-D ($n\ge 1$) semilinear Euler-Poisson-Darboux equation $\partial_t^2u-\Delta u+\frac{\mu}{t}\partial_tu=|u|^p$, where $t\ge 1$, $p>1$ and $\mu>0$, the global small data weak solution $u$ will exist when $p>p_{crit}(n,\mu)=\max\{p_s(n+\mu), p_f(n)\}$ with the Strauss exponent $p_{s}(n+\mu)=\frac{n+\mu+1+\sqrt{(n+\mu)^2+10(n+\mu)-7}}{2(n+\mu-1)}$ and the Fujita exponent $p_f(n)=1+\frac{2}{n}$. The blowup of weak solution $u$ has been shown when $1<p\le p_{crit}(n,\mu)$ meanwhile this open question has been solved for $n=1,2$. In the present paper, we focus on this open question for $n=3$ and establish the global existence of small data solution $u$ for $\mu\geq\frac{14}{5}$ (equivalent to $p_{crit}(3,\mu)=p_f(3)=\frac53$) and $p>\max\{\frac53, 1+\frac{2}{\mu}\}$.