Pulsar glitches are usually regarded as stochastic, independent events triggered by sudden angular momentum transfer from the neutron star's superfluid interior to its crust. However, dense glitching episodes in the Crab pulsar suggest that some temporally proximate small glitches may instead form parts of broader dynamical episodes. Here we reanalyse more than five decades of Crab timing data by grouping nearby glitches into glitch clusters. In this clustered sequence, adjacent waiting times are consistent with preferred temporal organization around $\sim 3.5$ yr, and every-other cluster intervals indicate a longer-timescale component near $\sim 7$ yr. Cluster size correlates more strongly with preceding than with subsequent waiting times, with the clearest signal arising from the longer pre-history of the system. These results suggest that clustering primarily regularizes the temporal structure of the Crab glitch record and support a picture in which Crab glitches are better interpreted as temporally coupled, history-dependent collective events rather than as fully independent stochastic occurrences.

The stochastic cross-dock door design problem entails determining the number of doors and their nominal capacities under uncertainty. The inbound flow of commodities from origin nodes is assigned to the entry doors consolidated in the platform, and the outbound flow is assigned to the exit doors to be delivered to the destination nodes. This problem combines three high computational difficulties, namely, NP-hard quadratic combinatorics, uncertainty in the main parameters, and ambiguity in their probability distribution. Distributionally robust optimization is considered to deal with these uncertainties. A two-stage mixed binary quadratic model is presented, where the first stage decisions are related to the design of the platform and the second stage ones are related to the assignment of the commodity flow to the doors in the members of the ambiguity set. The goal is to minimize the highest total cost in the ambiguity set, subject to the constraint system for each of those members. In addition to the risk-neutral version, a risk-averse formulation is presented. Given the difficulty of this problem, a min-max matheuristic scheme based on a scenario cluster decomposition is proposed for obtaining lower and upper bounds. A computational study is conducted to compare the solutions provided by the straightforward use of the state-of-the-art solvers CPLEX and Gurobi, as well as to validate the proposed matheuristic scheme.

For $p\in [1,\infty]$, we define a smooth manifold structure on the set $AC_{L^p}([a,b],N)$ of absolutely continuous functions $γ\colon [a,b]\to N$ with $L^p$-derivatives for all real numbers $a<b$ and each smooth manifold $N$ modeled on a sequentially complete locally convex topological vector space, such that $N$ admits a local addition. Smoothness of natural mappings between spaces of absolutely continuous functions is discussed, like superposition operators $AC_{L^p}([a,b],N_1)\to AC_{L^p}([a,b],N_2)$, $η\mapsto f\circ η$, for a smooth map $f\colon N_1\to N_2$. For $1\leq p <\infty$ and $r\in \mathbb{N}$ we show that the right half-Lie groups $\text{Diff}_K^r(\mathbb{R})$ and $\text{Diff}^r(M)$ are $L^p$-semiregular. Here $K$ is a compact subset of $\mathbb{R}$ and $M$ is a compact smooth manifold. An $L^p$-semiregular half-Lie group $G$ admits an evolution map $\text{Evol}:L^p([0,1],T_e G)\to AC_{L^p}([0,1],G)$, where $e$ is the neutral element of $G$. For the preceding examples, the evolution map $\text{Evol}$ is continuous.

Given a subset $A$ of $\mathbb{R}^n$, we define \begin{align*} \mathrm{conv}_k(A) := \left\{ λ_1 s_1 + \cdots + λ_k s_k : λ_i \in [0,1], \sum_{i=1}^k λ_i = 1 , s_i \in A \right\} \end{align*} to be the set of vectors in $\mathbb{R}^n$ that can be written as a $k$-fold convex combination of vectors in $A$. Let $γ_n$ denote the standard Gaussian measure on $\mathbb{R}^n$. We show that for every $\varepsilon > 0$, there exists a subset $A$ of $\mathbb{R}^n$ with Gaussian measure $γ_n(A) \geq 1- \varepsilon$ such that for all $k = O_\varepsilon(\sqrt{\log \log(n)})$, $\mathrm{conv}_k(A)$ contains no convex set $K$ of Gaussian measure $γ_n(K) \geq \varepsilon$. This result acts as a complement to the recent affirmative resolution of Talagrand's convexity conjecture by Hua, Song, and Tudose, which states that a universal dilation of the threefold Minkowski sum $A+A+A$ of a large set $A$ guarantees a large convex subset. Our approach utilises concentration properties of random copulas and the application of optimal transport techniques to the empirical coordinate measures of vectors in high dimensions.

This is the third work in a series on the (local in time) well-posedness of the initial boundary value problem (IBVP) for the vacuum Einstein equations in general relativity with geometric boundary conditions. Here we study the conformal-mean curvature boundary conditions, consisting of the conformal class of the boundary metric and mean curvature of the boundary. We prove that at metrics of uniformly bounded geometry to all orders, the linearized problem has a solution space with dense range in $C^{\infty}$ and establish a Holmgren-type uniqueness theorem valid for general smooth linearized solutions. These results require the addition of an arbitrary corner angle term at the intersection of the Cauchy surface and the timelike boundary.

We prove that for two-component maps in dimension two, rank-one convexity is equivalent to quasiconvexity. The essential tool for the proof is a fixed-point argument for a suitable set-valued map going from one component to the other that preserves decomposition directions within the $(H_n)$-condition formalism. The existence of a fixed point ensures that, in addition to keeping decomposition directions, joint volume fractions are respected as well, leading to the fundamental fact that every two-dimensional, two-component gradient can be reached by lamination. When maps have more than two components, fixed points exist for every combination of two components, but they do not match in general. Higher dimension would require further insight on how to organize and deal with triangulations for piece-wise affine maps.

We present a quantum preconditioning framework for solving linear systems arising from a finite difference discretization of the Poisson equation. It is based on the combination of the Schrödingerization technique \cite{JLY22b,JLYPRL24} and the BPX multilevel preconditioner in order to achieve near-optimal complexity. The Schrödingerization technique transforms linear partial and ordinary differential equations into Schrödinger-type systems with unitary evolution in one higher dimension, making them suitable for quantum simulation. A key contribution is a structure-aware construction of the block-encoding for the symmetrically preconditioned matrix $A_S = S^\top A S$, where $A$ is the stiffness matrix and $S$ encodes the BPX preconditioner in factored form. By establishing a novel commuting identity, we avoid the unfavorable normalization scaling that would otherwise arise from naive multiplication of block-encodings. This yields an exact block-encoding of $A_S$ with normalization $\mathcal{O}(d^2(L+1))$, where $d$ is the spatial dimension and $L$ is the number of levels. Combined with the Schrödingerization-based Hamiltonian simulation, the overall quantum algorithm achieves a query complexity of $\mathcal{O}\big(\mathrm{poly}(d)\varepsilon^{-1} \mathrm{polylog}(\varepsilon^{-1}) \big)$ for estimating linear functionals of the solution to a given tolerance $\varepsilon$.

Ehler and Gröchenig defined spherical $t$-design curves to be curves whose associated line integrals exactly average all degree at most $t$ polynomials. These authors posed the question of finding spherical $t$-design curves $γ_t$ on $S^d$ of asymptotically optimal arc length $\ell(γ_t)\asymp t^{d-1}$ as $t\to\infty$. This work investigates analogues of this question for $\textit{$\varepsilon_t$-approximate}$ and $\textit{weighted $t$-design curves}$, proving existence of such curves on $S^d$ achieving this asymptotic arc length for odd $d\in\Bbb N_+$ in the approximate setting (where $\varepsilon_t\asymp1/t$ as $t\to\infty$) and all $d\in\Bbb N_+$ in the weighted setting (where these curves have weight functions which are strictly positive at all but finitely many points). Formulas for such weighted $t$-design curves for $d\in\{2,3\}$ are presented.

Let X be a proper smooth variety over an algebraically closed field k of characteristic p>0. This thesis studies the p-primary component of the Brauer group of X.

A family of sets is $s$-intersecting if every pair of its sets has at least $s$ elements in common. It is an $s$-star if all its members have some $s$ elements in common. A family of sets is called $s$-EKR if all its $s$-intersecting subfamilies have size at most that of some $s$-star. For example, the classic 1961 Erdős-Ko-Rado theorem states essentially that the family of $r$-sized subsets of $\{1,2,\ldots,n\}$ is $s$-EKR when $n$ is a large enough function of $r$ and $s$, and the 1967 Hilton-Milner theorem provides the near-star structure of the largest non-star intersecting family of such sets. Two important conjectures along these lines followed: by Chvátal in 1974, that every subset-closed family of sets is 1-EKR, and by Holroyd and Talbot in 2005, that, for every graph, the family of all its $r$-sized independent sets is 1-EKR when every maximal independent set has size at least $2r$. In this paper we present similar 1-EKR results for families of length-$r$ paths in graphs, specifically for sun graphs, which are cycles with pendant edges attached in a uniform way, and theta graphs, which are collections of pairwise internally disjoint paths sharing the same two endpoints. We also prove $s$-EKR results for such paths in suns, and give a Hilton-Milner type result for them as well. A set is a transversal of a family of sets if it intersects each member of the family, and the transversal number of the family is the size of its smallest transversal. For example, stars have transversal number 1, and the Hilton-Milner family has transversal number 2. We conclude the paper with some transversal results involving what we call triangular families, including a few results for projective planes.

The best-worst method is a well-known distance based multi-criteria decision-making method used for computing the weights of decision criteria. This article provides a comprehensive analytical examination of the taxicab distance based model of the method, with the objectives of investigating the uniqueness of these solutions, and performing a rigorous consistency analysis. To achieve this, an optimal modification based optimization problem, equivalent to the original one, is first formulated. This reformulated problem is then solved analytically, and the optimal weight sets are derived from its solutions. Contrary to the prevailing understanding derived from numerical experiments with the taxicab model, our analytical framework proves that the model can, in fact, lead to multiple optimal weight sets, and we formally establish the conditions for this occurrence. A mixed-integer linear programming model is then employed to compute the consistency index. A decision-maker-aided selection strategy is also proposed for addressing non-uniqueness of optimal weight sets. In addition, threshold values of the consistency ratio to assess the admissibility of given preferences are also established. This framework provides a solid mathematical foundation that enhances the understanding of the model and eliminates the requirement for optimization software. By significantly improving the model's computational efficiency, it enables implementation in large-scale, dynamic real-world applications such as electricity market bidding strategies and portfolio rebalancing under market volatility. The effectiveness of the proposed framework is demonstrated through numerical examples, and its practical applicability is illustrated via a smartphone selection problem.

The integration of photonic components into surface ion traps provides a scalable approach for trapped-ion quantum computing, sensing, and metrology, enabling compact systems with enhanced stability and precision. However, the introduction of optical apertures in the trap electrodes can distort the trapping electric field. This can lead to excess micromotion (EMM) and ion displacement which degrade the performance of quantum logic operations and optical clocks. In this work, we systematically investigate the electric field distortion in a surface ion trap with integrated waveguides and grating couplers using Finite Element Method (FEM) simulations. We analyze methods to reduce these distortions by exploiting symmetries and transparent conductive oxide materials.

Acceleration ethics addresses the tension between innovation and safety in artificial intelligence. The acceleration argument is that risks raised by innovation should be answered with still more innovating. This paper summarizes the theoretical position, and then shows how acceleration ethics works in a real case. To begin, the paper summarizes acceleration ethics as composed of five elements: innovation solves innovation problems, innovation is intrinsically valuable, the unknown is encouraging, governance is decentralized, ethics is embedded. Subsequently, the paper illustrates the acceleration framework with a use-case, a generative artificial intelligence language tool developed by the Canadian telecommunications company Telus. While the purity of theoretical positions is blurred by real-world ambiguities, the Telus experience indicates that acceleration AI ethics is a way of maximizing social responsibility through innovation, as opposed to sacrificing social responsibility for innovation, or sacrificing innovation for social responsibility.

The paper establishes results following two interconnected directions. 1. Let $G$ be a Roelcke precompact closed subgroup of the group $\mathrm{Sym}(ω)$ of permutations of the natural numbers. Let $\mathrm{Aut}(G)$ denote the group of continuous automorphisms of $G$. Then $\mathrm{Inn}(G)$ is closed in $\mathrm{Aut}(G)$, where $\mathrm{Aut}(G)$ carries the topology of pointwise convergence for its (faithful) action on the cosets of open subgroups. Under the stronger hypothesis that~$G$ is oligomorphic, $\+ N_G/G$ is profinite, where $\+ N_G$ denotes the normaliser of~$G$ in $\mathrm{Sym}(ω)$, and the topological group $\mathrm{Out}(G)= \mathrm{Aut}(G)/\mathrm{Inn}(G)$ is totally disconnected, locally compact. 2a. We provide a general method to show smoothness of the isomorphism relation for appropriate Borel classes of oligomorphic groups. We apply it to two such classes: the oligomorphic groups with no algebraicity, and the oligomorphic groups with finitely many {essential} subgroups up to conjugacy. 2b. Using this method we also show that if $G$ is in such a Borel class, then $\mathrm{Aut}(G)$ is topologically isomorphic to an oligomorphic group, and $\mathrm{Out}(G)$ is profinite.

Suppose $μ$ and $ν$ are probability measures on $\mathbb{R}$ satisfying $μ\leq_{cx} ν$. Let $a$ and $b$ be convex functions on $\mathbb{R}$ with $a \geq b \geq 0$. We are interested in finding $$\sup_{\mathbf{M}} \sup_τ \mathbb{E}^{\mathbf{M}} \left[ a(X) I_{ \{ τ= 1 \} } + b(Y) I_{ \{ τ= 2 \} } \right] $$ where the first supremum is taken over consistent models $\mathbf{M}$ (i.e., filtered probability spaces $(Ω, \mathbf{F}, \mathbb{F}, \mathbb{P})$ such that $Z=(z,Z_1,Z_2)=(\int_{\mathbb{R}} x μ(dx) = \int_{\mathbb{R}} y ν(dy), X, Y)$ is a $(\mathbb{F},\mathbb{P})$ martingale, where $X$ has law $μ$ and $Y$ has law $ν$ under $\mathbb{P}$) and $τ$ in the second supremum is a $(\mathbb{F},\mathbb{P})$-stopping time taking values in $\{1,2\}$. Our contributions are first to characterise and simplify the dual problem, and second to completely solve the problem under some structural assumptions on the measures $μ$ and $ν$ (namely that $μ$ and $ν$ are absolutely continuous probability measures that satisfy the Dispersion Assumption). A key finding is that the canonical set-up in which the filtration is that generated by $Z$ is not rich enough to define an optimal model and additional randomisation is required. This holds even though the marginal laws $μ$ and $ν$ are atom-free. The problem has an interpretation of finding the robust, or model-free, no-arbitrage bound on the price of a Bermudan option with two possible exercise dates, given the prices of co-maturing European options.

Piecewise-Deterministic Markov Processes (PDMPs) hold significant promise for sampling from complex probability distributions. However, their practical implementation is hindered by the need to compute model-specific bounds. Conversely, while Hamiltonian Monte Carlo (HMC) offers a generally efficient approach to sampling, its inability to adaptively tune step sizes impedes its performance when sampling complex distributions like funnels. To address these limitations, we introduce three innovative concepts: (a) a Metropolis-adjusted approximation for PDMP simulation that eliminates the need for explicit bounds without compromising the invariant measure, (b) an adaptive step size mechanism compatible with the Metropolis correction, and (c) a No U-Turn Sampler (NUTS)-inspired scheme for dynamically selecting path lengths in PDMPs. These three ideas can be seamlessly integrated into a single, `doubly-adaptive' PDMP sampler with favourable robustness and efficiency properties.

In our series of papers, we prove that smooth Fano threefolds in positive characteristic lift to the ring of Witt vectors. Moreover, we show that they satisfy Akizuki-Nakano vanishing, $E_1$-degeneration of the Hodge to de Rham spectral sequence, and torsion-freeness of Crystalline cohomologies. In this paper, we establish these results for the case when $|-K_X|$ is very ample and the Picard group is generated by $ω_X$.

We observe for the first time in silico the transition to a linear regime in the primary damage production in tungsten. As the critical plasma-facing material in fusion reactors, radiation damage in tungsten has been studied extensively in experiments and simulations. Irradiation experiments routinely produce recoils in the MeV range while full atomistic modelling has been limited to a few hundred keV. Here we bridge these scales with extremely large-scale and accurate machine-learning-driven molecular dynamics simulations with recoil energies up to 2 MeV in systems up to one billion atoms. We reveal four regimes of primary damage as a function of damage energy, with a transition to a high-energy regime that deviates from all previous models. Curiously, the start of the high-energy regime coincides with the highest possible recoil energy to tungsten atoms from fusion-emitted neutrons (300 keV).

It is well-known that the Sobolev spaces $W^{k,p}(\mathbb R^d)$ are vector lattices with respect to the pointwise almost everywhere order if $k \in \{0,1\}$, but not if $k \ge 2$. In this note, we consider negative $k$ and show that the span of the positive cone in $W^{k,p}(\mathbb R^d)$ is a vector lattice in this case. We also prove a related abstract result: if $(T(t))_{t \in [0,\infty)}$ is a positive $C_0$-semigroup on a Banach lattice $X$ with order continuous norm, then the span of the cone $X_{-1,+}$ in the extrapolation space $X_{-1}$ is a vector lattice. This complements results obtained by Bátkai, Jacob, Wintermayr, and Voigt in the context of perturbation theory and provides additional context for the theory of infinite-dimensional positive systems.

Classical results of Cauchy and Dehn imply that the 1-skeleton of a convex simplicial polyhedron $P$ is rigid i.e. every continuous motion of the vertices of $P$ in $\mathbb R^3$ which preserves its edge lengths results in a polyhedron which is congruent to $P$. This result was extended to convex smplicial polytopes in $\mathbb R^d$ for all $d\geq 3$ by Whiteley, and to generic realisations of 1-skeletons of simplicial $(d-1)$-manifolds in $\mathbb R^{d}$ by Kalai for $d\geq 4$ and Fogelsanger for $d\geq 3$. We will generalise Kalai's result by showing that, for all $d\geq 4$ and any fixed $1\leq k\leq d-3$, every generic realisation of the $k$-skeleton of a simplicial $(d-1)$-manifold in $\mathbb R^{d}$ is volume rigid, i.e. every continuous motion of its vertices in $\mathbb R^d$ which preserves the volumes of its $k$-faces results in a congruent realisation. In addition, we conjecture that our result remains true for $k=d-2$ and verify this conjecture when $d=4,5,6$.

Previous work of Chan--Church--Grochow and Baker--Wang shows that the set of spanning trees in a plane graph $G$ is naturally a torsor for the Jacobian group of $G$. Informally, this means that the set of spanning trees of $G$ naturally forms a group, except that there is no distinguished identity element. We generalize this fact to graphs embedded on orientable surfaces of arbitrary genus, which can be identified with ribbon graphs. In this generalization, the set of spanning trees of $G$ is replaced by the set of spanning quasi-trees of the ribbon graph, and the Jacobian group of $G$ is replaced by the Jacobian group of the associated regular orthogonal matroid $M$ (along with an associated regular representation of $M$). Our proof shows, more generally, that the family of "BBY torsors" constructed by Backman--Baker--Yuen and later generalized by Ding admit natural generalizations to (regular representations of) regular orthogonal matroids. In addition to shedding light on the role of planarity in the earlier work mentioned above, our results represent one of the first substantial applications of orthogonal matroids (also called "even delta-matroids" or "Lagrangian orthogonal matroids") to a natural combinatorial problem about graphs.

In this paper we consider sequences $u_j:B_2\subseteq M\to N$ of stationary harmonic maps between smooth Riemannian manifolds with uniformly bounded energy $E[u_j]\equiv \int |\nabla u_j|^2\leq Λ$ . After passing to a subsequence it is known one can limit $u_j\to u:B_1\to N$ with the associated defect measure $|\nabla u_j|^2 dv_g \to |\nabla u|^2dv_g+ν$, where $ν= e(x)\, H^{m-2}_S$ is an $m-2$ rectifiable measure \cite{lin_stat}. For a.e. $x\in S=\operatorname{supp}(ν)$ one can produce a finite number of bubble maps $b_j:S^2\to N$ by blowing up the sequence $u_j$ near $x$. We prove the energy identity in this paper. Namely, we have at a.e. $x\in S$ that $e(x)=\sum_j E[b_j]$ for a complete set of such bubbles. That is, the energy density of the defect measure $ν$ is precisely the sum of the energies of the bubbling maps.

We give a dynamical, relatively elementary proof of an "absorption theorem" which is closely related to a well-known result due to Matui. The construction is in the spirit of an earlier joint work of the author and S. Robert. In an appendix we explain how to use this result to correct the dynamical proof given by Melleray--Robert of a classification theorem for orbit equivalence of minimal ample groups due to Giordano, Putnam and Skau (the original argument had a gap).

There has been a surge of recent interest in the Wigner's Friend paradox, sparking several novel thought experiments and no-go theorems. The main narrative has been that Wigner's Friend highlights a counterintuitive feature that is unique to quantum theory, and which is closely related to the quantum measurement problem. Here, we challenge this view. We argue that the gist of the Wigner's Friend paradox can be reproduced without assuming quantum physics, and that it underlies a much broader class of enigmas in the foundations of physics and philosophy. To show this, we first consider several recently proposed Extended Wigner's Friend scenarios, and demonstrate that some of their implications for the absoluteness of observations can be reproduced by classical thought experiments that involve the duplication of agents. Crucially, some of these classical scenarios are technologically much easier to implement than their quantum counterparts. Then, we argue that the essential structural ingredient of all these scenarios is a feature that we call "Restriction A": that a physical theory cannot give us a probabilistic description of the observations of all agents. Finally, we argue that this difficulty is at the core of other puzzles in the foundations of physics and philosophy, and demonstrate this explicitly for cosmology's Boltzmann brain problem. Our analysis suggests that Wigner's Friend should be studied in a larger context, addressing a frontier of human knowledge beyond quantum foundations: to obtain reliable predictions for experiments in which these predictions can be privately but not intersubjectively verified.

In this paper, we are concerned with the low Mach number limit for the compressible Navier-Stokes equation with a stationary force and ill-prepared initial data in the three-dimensional whole space. The convergence result of the stationary solutions toward the corresponding incompressible flow is obtained when the stationary force is small enough. Under the assumption that the initial perturbation around the stationary solution is small enough, the convergence result of the perturbation toward the corresponding perturbation around the stationary incompressible flow is obtained globally in time. The proof relies crucially on the Strichartz type estimate for the linearized semigroup around the motionless state which reflects not only its dispersive property but also dissipative properties of the linearized operator.

This paper develops a Geometric Analysis-based Route Safety Assessment (GARSA) framework to enhance the safety of marine vehicles navigating in restricted waters. Utilizing line and point geometric elements to define waterway boundaries, the framework enables the construction of a dynamic width characterization function to quantify spatial safety within complex restricted navigation spaces. An iterative method is developed to calculate the dynamic width characterization function, enabling an abstracted spatial property representation of waterways. Based on this, a navigational safety index that accounts for the overall route safety as well as the spatial constraints of local segments is created, enabling the selection of the safest waterway to pass through. Case studies in Hamburg Port and hypothetical waterways show that GARSA identifies spatial safety differences among routes. For example, GARSA assigned safety assessment values of 0.6 and 135 to two intra-port transit routes of 7,258 m and 7,845 m, respectively, with the higher value indicating a safer route. Overall, GARSA provides a quantitative basis for safety-oriented route decision-making of marine vehicles in restricted waters.

One of the fundamental results in graph minor theory is that for every planar graph~$H$, there is a minimum integer~$f(H)$ such that graphs with no minor isomorphic to~$H$ have treewidth at most~$f(H)$. The best known bound for an arbitrary planar $H$ is ${O(|V(H)|^9\operatorname{poly~log} |V(H)|)}$. We show that if $H$ is the disjoint union of cycles, then $f(H)$ is $O(|V(H)|\log^2 |V(H)|)$, which is a $\log|V(H)|$ factor away being optimal.

Given a smooth, bounded domain $Ω\subset\mathbb{R}^N$, we establish the existence of two non-trivial, non-negative solutions to the semilinear degenerate elliptic equation \begin{align*} \left. \begin{array}{l} -Δ_λu=μg(z)|u|^{r-1}u+h(z)|u|^{s-1}u \;\text{in}\; Ω u\in H^{1,λ}_0(Ω) \end{array}\right\} \end{align*} where $Δ_λ=Δ_x+|x|^{2λ}Δ_y$ denotes the Grushin Laplacian Operator, $z=(x,y)\inΩ$, $N=n+m;\, n,\, m\geq 1$, $λ>0$, $0\leq r<1<s<2^*_λ-1$ and $μ$ is a positive parameter. The functions $g$ and $h$ may change sign and $2^*_λ=\frac{2Q}{Q-2}$ is the critical Sobolev exponent associated with the homogeneous dimension $Q=n+(1+λ)m$ of $Δ_λ$. In the critical case $s=2^*_λ-1$, we further show that the problem admits at least two non-trivial, non-negative solutions under the additional assumptions $g\geq 0$ and $h\equiv 1$.

It is a fundamental question why quantum mechanics employs complex numbers rather than solely real numbers. In this work, we conduct the first analysis of imaginarity quantification in neutrino flavor and spin-flavor oscillations. As quantum systems in coherent superposition, neutrinos are ideal candidates for quantifying imaginarity within the resource theoretic framework, using measures such as the $\ell_1$-norm and the relative entropy of imaginarity. We show that in the case of two-flavor mixing, these measures of imaginarity are nonzero. The measures of imaginarity reach their extreme values when the probabilistic features of quantum theory are fully maximized, i.e., both the transitional and survival probabilities are approximately equal. Our study reveals that the imaginarity, as a resource, can be harnessed not solely from the presence of a complex phase in the mixing matrix but also from the intrinsic quantum dynamics of time evolution itself. We further extend our analysis to explore the dynamics of three-flavor neutrino mixing, incorporating the effects of a nonzero $CP$ phase.

Ongoing advancements in microbiome profiling have provided unprecedented insights into the molecular dynamics of microbial communities, sparking a surge of interest in uncovering the microbiome's critical role in human health. Identifying microbial features linked to clinical outcomes, however, remains challenging due to the high-dimensional, sparse, and compositional nature of microbiome data. Additionally, many microbial taxa, although classified as distinct, may share functional roles, complicating traditional variable selection methods. To overcome these obstacles, we introduce Bayesian Regression with Agglomerated Compositional Effects (BRACE), a novel approach using a spike-and-cluster prior combining Bernoulli activity indicators, an Ewens exchangeable partition prior on the finite active set, and a projection-based constrained Gaussian prior on cluster effects to perform data-adaptive clustering and variable selection. The methodological innovation of our work lies in how we combine the Ewens partition prior with a projection-based constrained Gaussian on the cluster atoms to enforce the sum-to-zero constraint. BRACE groups microbial taxa with similar effects on the outcome, yielding more interpretable models while enabling effective dimension reduction. Through comprehensive simulations and a real-world application examining the influence of oral microbiome composition on insulin resistance, we demonstrate BRACE's superior performance over existing methods, particularly in identifying key features with shared effects on outcomes.