Silicon photonic integrated circuits critically depend on compact on-chip light sources, for which nanowire (NW) lasers are an attractive solution. However, their practical implementation is often limited by broad emission linewidths and poor frequency stability resulting from weak optical feedback. Here, we integrate individual GaAsSb NWs by transfer-printing onto silicon-on-insulator (SOI) racetrack resonators to realize optical feedback at silicon-transparent wavelengths. Finite-difference-time-domain simulations reveal efficient coupling between the hybrid NW-waveguide mode and the fundamental TE resonator mode, with calculated cavity Q-factors exceeding 10$^4$. Experimentally, we observe feedback-induced lasing emission at a low threshold (P$_{th}$) of 8.6 $\pm$ 1.8 $μ$J/cm$^2$. Compared to identical NW lasers without SOI resonator, the linewidth is reduced by more than a factor of four at 3P$_{th}$ and remains stable below 1.8 meV up to 5P$_{th}$. Our results demonstrate NW-based light sources on SOI and show that tailored resonator designs enable improved linewidth control and frequency stabilization.

We study Hamiltonian and symplectic tensor structures in the T-product algebra. We define T-Hamiltonian and T-symplectic tensors and characterize them through their Fourier-domain slices. For T-Hamiltonian tensors we establish the standard block form and the spectral symmetry of T-eigenvalues, while for T-symplectic tensors we derive the inverse and exponential-map properties. Our main result is a constructive T-Williamson normal form for tensors whose Fourier-domain slices are real symmetric positive-definite matrices. We also show that, under the Hermitian symplectic convention adopted here, this decomposition does not extend directly to arbitrary Hermitian positive-definite Fourier-domain slices, and we derive a real-valued recovery criterion under Fourier conjugate symmetry. Numerical experiments verify the construction, exhibit runtime trends consistent with the slice-wise complexity $O(pn^3)$, and illustrate the framework on a Fourier-domain encoding of covariance-matrix families arising in continuous-variable quantum dynamics.

Compression based on asymmetric numeral systems (ANS) combines high encoding and decoding speeds with a compression ratio close to Shannon entropy, while forward modeling of the information source makes it possible to obtain an estimated compressed message size that is less than the entropy. This paper proposes combining these modeling and adaptive coding methods. In addition to ensuring high data processing speeds and compression ratios, this approach enables one to implement the adaptive ANS, which has long remained an important scientific and practical problem.

The Riemann-Roch formula is a cornerstone in the classical theory of algebraic curves. Here we present a novel approach to its proof, by answering a question posed in 2007 by Matthew Baker and Serguei Norine.

The intersection set of Bayesian and nonparametric statistics was almost empty until about 1973, but now is growing at a healthy rate. This chapter, for the {\it Highly Structured Stochastic Systems} book (Oxford University Press, 2003) gives an overview of various theoretical and applied research themes inside this field, partly complementing and extending recent reviews of Dey, M{ü}ller and Sinha (1998) and Walker, Damien, Laud and Smith (1999). The intention is not to be complete or exhaustive, but rather to touch on research areas of interest, partly by example.

We formulate a framework of polynomial diagrams, which are a generalisation of power diagrams (PDs) and anisotropic power diagrams (APDs) allowing for boundaries between cells to be algebraic curves of a prescribed degree. We show that they arise naturally from rephrasing PDs (APDs) as first-degree (second-degree) instances of linear parametrised minimisation diagrams. We also develop an efficient GPU-accelerated framework for fitting polynomial diagrams to image data using Legendre polynomials and by maximising a regularised concave objective function adapted from classical logistic regression literature. A largely self-contained analysis of the optimisation algorithm is also provided, including identification of scale and gauge invariances and the limiting objective function as the regularisation parameter vanishes. We apply the algorithm to fit polynomial diagrams to electron backscatter diffraction images of steel.

In this work a loop quantum corrected model is obtained for spherically symmetric space-times in the vacuum. This effective model is derived by the use of the path integral method, previously employed in several models of Loop Quantum Cosmology. Our principal aim is to find explicit corrections corresponding to inverse triad and holonomy effects that commonly arise from the loop quantization procedure. These corrections modify the Hamiltonian constraint of the classical theory, adding quantum parameters that represent the length of the holonomies considered during quantization. The semiclassical theory yielded reduces to the classical case when small values of such length are taken to be small. Solutions to the effective dynamics of a simplified version of the complete corrected theory are then found and used to describe an effective geometry with inverse triad corrections. This modified space-time represents a black hole with a curvature singularity in its interior which, contrary to its classical counterpart, does not lead to null geodesic incompleteness. For the case of holonomy corrections, preliminary arguments are given in favor of a potential singularity resolution.

Cluster counting dN/dx is a promising method to enhance particle identification for gaseous detectors, especially in next-generation collider experiments like the FCC-ee, where good pion-kaon separation over a broad momentum range is essential. However, its implementation in large-scale systems has been limited by the challenging requirements for high-resolution signal amplification and readout. This paper presents a 24-channel ultra-low-noise preamplifier board designed for drift tube detectors to enable dN/dx measurements. The three-stage amplification topology employs SiGe transistors and integrates dedicated noise-minimization techniques, achieving a charge gain of 21.11 mV/fC from 0.3 fC to 50 fC, a bandwidth of 542 MHz, and a voltage gain of 47.8 dB. The measured voltage noise density is 0.35 nV/sqrt(Hz), surpassing most of the state-of-the-art preamplifiers for gaseous and silicon detectors. Validation tests conducted on the sMDT chambers at the CERN Proton Synchrotron test beam facility demonstrate that the proposed design meets the stringent preamplifier requirements for implementing the dN/dx method in drift-tube detector systems, achieving an equivalent noise charge of 0.14 fC and a signal-to-noise ratio of 73 when operated with a He:iC4H10 (90:10) gas mixture. The design also shows promise for broader application in other gaseous or semiconductor detectors.

We extend the stellar-mass gas-phase metallicity relation (MZR) at $z = 1.1-3.4$ down to the extremely low-mass regime using 183 galaxies with $\log(M_*/M_\odot) = 6.3-10.2$, based on deep JWST/NIRISS slitless spectroscopy from the NGDEEP program. The derived MZR is in excellent agreement with our previous result from 50 galaxies in the GLASS-JWST sample, underscoring the robustness and universality of this relation. Together, these datasets constitute the largest sample of dwarf galaxies yet obtained with NIRISS. The observed MZR slope, $β\simeq0.24\pm0.03$, remains constant across nearly four orders of magnitude in stellar mass. Analytical modeling of the metal-loading factor of outflows ($ζ_\textrm{out}$) indicates that, at $M_*\lesssim10^8M_\odot$, $ζ_\textrm{out}$ becomes progressively less dominant than the gas fraction ($μ_\textrm{gas}$) in regulating the MZR slope. Using this enlarged NIRISS sample, we further test the existence of the fundamental metallicity relation (FMR). We find no robust evidence for an additional SFR dependence beyond the MZR, nor any reduction in metallicity scatter when SFR is included. Examination of systematic uncertainties in \oh, $M_*$ and SFR suggests that the MZR slope ($β\sim0.22$) is robust, and that different assumptions about the strong-line calibrations or star-formation history (SFH) of the galaxies change the slope by less than 1-$σ$. At the current depth of the NIRISS data, evidence for an FMR among high-redshift dwarf galaxies remains inconclusive, highlighting the need for larger samples, and deeper observations.

This paper proposes a novel, nonparametric, interpoint distance-based measure to investigate whether there exist any groups in a set of given data, and if so then, how many groups are prevailing in total. It is a cluster accuracy index useful for arbitrary-dimensional data set, in association with any clustering algorithm having the number of groups specified as a priori. We perform univariate, nonparametric, multiple statistical tests of hypotheses, where as many dependent tests as the sample size are carried out using the interpoint distances. They possess $p$-values to be combined to reach a decision, which is taken in a step-wise process for a possible number of clusters. It reduces the unnecessary computations compared with the other accuracy measures from the literature. Data study establishes the proposed index's efficiency and superiority.

We prove the almost sure weak convergence of a stochastic proximal point method for minimizing a convex integral function in the general nonlinear context of complete geodesic metric spaces of nonpositive curvature (so-called Hadamard spaces), solving a problem of M. Bačák. This method, formulated in the context of a mild growth condition on the function which generalizes Lipschitz continuity, was previously only considered in the context of strong metric regularity conditions or in the context of locally compact spaces. The proof is a combination of a weak almost sure convergence theorem for stochastic processes in Hadamard spaces which confine to a stochastic variant of quasi-Fejér monotonicity, due to previous work of the author, together with a new argument for proving the almost sure convergence of the mean function values of the process towards the minimal value.

In this paper, we consider the subcritical branching random walk in a random environment. We assume the branching and the step jump are independent; and the branching is in random envirenment, i.e., the particles in generation $n$ produce children according the probability measure $F_n\in \mathcal{P}\left(\N_0\right)$, and the $F_n$, $n=1,2,\cdots, $ are i.i.d under the $P_E$. ``subcritical" means that $ a:=\E[X_1]\in (-\infty,0)$, where $X_1:=\log \overline{F}_1$ and $\overline{F}_1$ is the mean of $F_1$.

We study the theoretical constraints on the scalar sector of the Alternative Left-Right Model (ALRM), an $E_6$-motivated extension of the Standard Model based on the gauge group $\mathrm{SU}(3)_c \otimes \mathrm{SU}(2)_L \otimes \mathrm{SU}(2)_{R'} \otimes \mathrm{U}(1)_{B-L}$, supplemented by a global $\mathrm{U}(1)_S$ symmetry. We derive the complete set of tree-level perturbative unitarity constraints on the model, resulting in 14 independent conditions on the quartic scalar couplings. When combined with the boundedness-from-below conditions and the requirement of positive-definite scalar mass-squared eigenvalues, these constraints are found to be complementary, with their simultaneous imposition yielding significantly more stringent restrictions on the parameter space than either set alone. We then perform a one loop renormalization group analysis, evolving the model parameters from the electroweak scale up to a high energy cut-off scale, and requiring that the vacuum stability, the unitarity, and the perturbativity conditions are preserved throughout. The renormalisation group evolution is found to restrict the allowed parameter space considerably beyond the tree-level bounds, with the constraints on the quartic couplings becoming more stringent as the cut off scale is raised. Consequently, the physical scalar masses in the model acquire upper bounds. For the right-hand symmetry breaking scale, $v_R = 10$ TeV and requiring theoretical consistency up to $10^{16}$ GeV, we obtain $m_{H_1^\pm} \lesssim 6.5$ TeV, $m_{H_2^\pm} \lesssim 1.5$ TeV, and $m_{H_1^0} \simeq m_{A_1} \lesssim 1.3$ TeV, with all bounds scaling with $v_R$. These findings offer a predictive and falsifiable framework for searches of the extended Higgs sector of the ALRM at the current and future collider experiments.

It is shown that the product of two nonscalar conjugacy classes of the special linear group SL$(n,K)$ contains matrices of arbitrary trace if $n \ge 4$ and $K$ is an abitrary field or $n=3$ and $K$ is finite.

We report the first rigorous evaluation of the physical mass of electroweak vortex rings, establishing precise values of 18.01 and 26.80 TeV for solutions characterized by different winding numbers. Analysis of the internal structure reveals that repulsive interactions shape the geometry of these configurations, while complex current distributions lead to a neutral analogue of Ampere's circuital law, suggesting a corresponding self-stabilizing pinch mechanism. These findings set the energy scales for the potential observation of such configurations at future colliders and offer a framework for understanding topological structures in the Standard Model.

We study field-controlled spin-valley transport in monolayer MoS$_2$ through a single electrostatic barrier and a uniform off-resonant elliptically polarized irradiation. Starting from the massive Dirac Hamiltonian with intrinsic spin-orbit coupling, we use a high-frequency Floquet expansion to obtain an effective static model with a laser-renormalized mass (gap) term. We solve the scattering problem by spinor matching and derive the exact analytic expression for the transmission. The numerical results show that the drive tunes both the spin-valley-dependent propagation threshold inside the barrier and the Fabry-Pérot phase, creating controllable pass/stop bands. By varying both the laser intensity (amplitude) and the polarization shape, we show that the same junction can be switched between broadband valley filtering and resonance-selective operation, and the valley contrast remains visible in the Landauer conductance. Our findings establish an efficient route for realizing optically reconfigurable valleytronic and spintronic functionalities in MoS$_2$.

We present a quantum circuit implementation of the quantum hashing algorithm (quantum fingerprinting) for a quantum device with restrictions on the application of two-qubit gates by a qubit connectivity graph. We present an optimization technique for the shallow circuit for quantum hashing in the case of a cactus as a qubit connectivity graph. The algorithm has $O(n^3)$ complexity to build the circuit, where $n$ is the number of qubits and $m$ is the number of connections (edges) in the graph. It is improvement compared to the existing exponential-time algorithm in the case of arbitrary graphs. The algorithm uses solution for the shortest non-simple 1-covering path problem as a subroutine. We present an $O(n^3)$-time solution for this graph-theory problem in the case of a cactus. This result can be interesting independently. The algorithm also used for improving of the quantum circuit for Quantum Fourier Transform.

Software defect detection is a critical task in software engineering. However, no prior studies have specifically addressed defect detection in bioinformatics software. Given that the performance of defect detection tasks is primarily influenced by both models and datasets, our experiments controlled for model-related factors and confirmed the limitations of existing datasets in bioinformatics software. To address this issue, we introduce BioDefect, the first dataset specifically designed for defect detection in bioinformatics software, aiming to overcome the limitations of existing datasets in this context. Unlike prior datasets, BioDefect includes complete source code repositories, preserving the actual contextual information of defective code, thereby more accurately reflecting real-world defect scenarios in bioinformatics software. Additionally, BioDefect mitigates issues related to label inconsistency and data leakage, ensuring high data quality and experimental reliability. To evaluate the effectiveness of BioDefect, we conduct a systematic assessment on nine language models (LMs), including DeepSeek-R1. The results demonstrate that BioDefect significantly enhances defect detection performance for bioinformatics software. Compared to existing datasets, BioDefect achieves an average F1-score improvement of 29.61% to 38.04% across all models, highlighting its superior advantages. This study fills a critical research gap in bioinformatics software defect detection, laying a foundation for future studies in this field and offering new insights for improving bioinformatics software quality assurance.

The phase-change material antimony sulfide (Sb$_2$S$_3$) relies on rapid and reversible phase transitions between crystalline and amorphous states, which are critical for their performance in data storage and photonics applications. In this work, a machine learning force field is developed based on the moment tensor potential approach, allowing us to understand the atomistic origin of the structural evolution and crystallization kinetics in Sb$_2$S$_3$ for the first time, by enabling large-scale molecular dynamics simulations (up to 7680 atoms for 40 ns). Sb$_2$S$_3$ shows anisotropic growth rates with the [100] facet exhibiting the fastest growth due to the strong Sb-S covalent bonding along its quasi-1D ribbon-like structure of its crystalline phase. The activation energy for crystal growth is found to be 0.55-0.57 eV, whereas that for diffusion is around 1.16-1.56 eV. The lower activation energy for crystal growth indicates that its heterogeneous crystallization is interface controlled rather than diffusion limited, unlike GST and GeTe with atomic attachment at the solid-liquid interface being energetically favoured over long range atomic transport. These findings provide key insights into the structural, thermodynamic, and kinetic properties of Sb$_2$S$_3$, paving the way for optimizing its functionality including switching speed, reliability, and energy efficiency.

Understanding the competition between first-layer lateral expansion and second-layer nucleation is essential for layer-controlled graphene growth via chemical vapor deposition (CVD). Building on our previous phase diagram framework based on the dimensionless parameters $α$ and $Γ$, we develop an enhanced model incorporating two previously neglected effects: thermal-expansion-induced substrate strain and chemical desorption of carbon monomers via reverse dehydrogenation. First-principles calculations are employed to determine the strain-dependent diffusion and attachment barriers on both exposed and graphene-covered Cu(111) surfaces. By mapping the multi-step CVD process into an effective quasi-physical vapor deposition, we construct a generalized phase diagram characterized by the coupled effects of $α$, $Γ$, and a newly introduced desorption parameter $Z$. Our results show that tensile strain expands the bilayer graphene (BLG) growth window for critical nucleus sizes $i^*>1$. In contrast, chemical desorption suppresses BLG formation in the high-$Γ$ regime via $Z$-dependent monomer depletion. This unified framework provides a predictive guide for the rational synthesis of high-quality bilayer graphene by linking macroscopic growth parameters to microscopic layer-selection mechanisms.

Spin qubits in silicon-MOS (SiMOS) quantum dots have recently demonstrated compatibility with existing industry standard CMOS fabrication techniques. These devices have routinely achieved single- and two-qubit gate fidelities above 99% and demonstrated highly entangled two-qubit Bell states in isolated double quantum dot (DQD) unit cells, however coupling between unit cells has remained challenging. In this work, we present a two unit cell, four-qubit SiMOS processor with universal controllability and fully parallelised state initialisation and readout. We use this processor to generate maximally entangled three-qubit states, including the Greenberger-Horne-Zeilinger (GHZ) state, and certify multipartite entanglement through violation of the classical Mermin-witness bound. By using a fully symmetric dynamically decoupled gate sequence to create our entangled states, we are able to preserve the lifetime of the entanglement beyond $T_2^*$, to a time limited instead by $T_2^\textrm{Hahn}$. These demonstrations pave a road to the scalable operation of larger SiMOS processors, and achieving high purity, long-lived multi-qubit entangled states in them.

We present an analysis of the non-thermal properties of the merging galaxy cluster SPT-CL J2012-5649/Abell~3667 ($z = 0.0556$, $M_{500} = 7.16 \times 10^{14}\ M_\odot$) using the MINOT non-thermal emission modelling framework. The predicted hadronic gamma-ray flux from $pp$ interactions in the $1$--$300\ \mathrm{GeV}$ band is $2.82 \times 10^{-11}\ \mathrm{cm^{-2}\ s^{-1}}$ within $R_{500}$, rising to $1.15 \times 10^{-10}\ \mathrm{cm^{-2}\ s^{-1}}$ at the truncation radius ($3.7\,R_{500}$), in order-of-magnitude agreement with the Fermi-LAT reported flux of $1.3 \times 10^{-10}\ \mathrm{cm^{-2}\ s^{-1}}$. Approximately $76\%$ of the predicted hadronic flux originates from beyond $R_{500}$. The IC contribution from cosmic-ray electrons is subdominant relative to the hadronic $π^0$-decay gamma-ray component by a factor of ${\sim}20$ in the $1$--$300\ \mathrm{GeV}$ energy band, and therefore does not contribute significantly to the observable signal. Although the expected hadronic flux is in approximate agreement with the observed Fermi-LAT flux level in the $1$--$300\ \mathrm{GeV}$ band, the observed spectral index ($Γ= -3.61 \pm 0.33$) is in tension with the hadronic prediction ($Γ\approx -2.4$ to $-2.6$).

This study presents the first experimental implementation of deep reinforcement learning (DRL) for the active real-time suppression of flow-induced vibrations in simultaneously vibrating tandem cylinders using rotary actuation, considering fully actuated and underactuated configurations. In the fully actuated case, where both cylinders are independently controlled, the DRL agent discovers a high-frequency, phase-locked bang-bang control strategy that suppresses the vibrations of both cylinders by more than 95\%. Analysis of the training dynamics reveals a physically interpretable learning process in which the agent first identifies the optimal phase relationship between the actuators before refining the actuation frequency. In the underactuated configuration, where only the upstream cylinder is actuated, equally weighted rewards produce ineffective control, suppressing vibrations only in the actuated cylinder. Introducing asymmetric reward weighting enables the DRL agent to discover a low-frequency lock-on strategy that achieves 70\% and 90\% vibration suppression in the upstream and downstream cylinders, respectively. For staggered arrangements with lateral offset, conventional training fails to converge, requiring a curriculum learning approach. The resulting two-stage curriculum identifies a statically biased bi-harmonic rotational control signal capable of suppressing vibrations in both cylinders. The success of the underactuated control strategy highlights its potential to reduce energy consumption and hardware complexity in multi-body flow control systems.

The generalized quantum Stein's lemma characterizes the optimal asymptotic exponent of the type-II error in quantum hypothesis testing for an independent and identically distributed (IID) null hypothesis against a composite alternative hypothesis. Classically, a probabilistic mixture of IID sources arises as a natural generalization of IID sources, and, in the non-composite setting, the optimal type-II error exponent in hypothesis testing for such classical mixed sources is known to be characterized concisely by the worst-case component of the mixture. In this work, we extend these foundational results to composite quantum hypothesis testing where the null hypothesis is a mixed source, i.e., a probabilistic mixture of IID quantum states, and the alternative hypothesis is composite as in the generalized quantum Stein's lemma. When the type-I error vanishes asymptotically, we characterize the optimal type-II error exponent of this composite quantum hypothesis testing problem in terms of the worst-case component of the mixture, by developing techniques for the non-commutative quantum setting inspired by the classical information-spectrum analysis. We also show that the analogous characterization does not hold in general for a fixed nonzero type-I error threshold, by providing a counterexample beyond the vanishing type-I error regime. These results clarify the applicability of the generalized quantum Stein's lemma to highly non-IID null hypotheses arising from arbitrary finite probabilistic mixtures of IID quantum states.

A pseudo-Euclidean Novikov superalgebra $A$ is a Novikov superalgebra endowed with a non-degenerate symmetric bilinear form $\langle,\rangle$ such that all left multiplication operators are $\langle,\rangle$-antisymmetric. In this case, the associated Lie superalgebra $(A^{-},$\langle,\rangle$)$ is a flat pseudo-Euclidean Lie superalgebra. In this paper, we investigate the structure of pseudo-Euclidean Novikov superalgebras. In particular, we introduce a distinguished subclass, called Milnor superalgebras, and prove that any pseudo-Euclidean Novikov superalgebra whose two-sided ideal is non-degenerate belongs to this class. We provide a method for constructing pseudo-Euclidean Novikov superalgebras. We also introduce a double extension procedure for pseudo-Euclidean Novikov superalgebras and show that every such superalgebra with a degenerate two-sided ideal can be obtained via this method. Furthermore, we establish that any pseudo-Euclidean Novikov superalgebra is either a Milnor superalgebra or can be obtained by a sequence of double extensions starting from a Milnor superalgebra. As an application, we provide a complete classification of pseudo-Euclidean Novikov superalgebras of total dimension at most four.

In this paper, we derive the multi-peakon dynamical system of a class of Camassa-Holm-type equations with quadratic nonlinearities. We also consider the analytical properties for the Cauchy problem. Firstly, we establish local well-posedness of solutions in Besov spaces and then provide the blow-up criteria. Subsequently, we impose appropriate sufficient conditions on the initial data to guaranty that the corresponding solution either exists globally or blows up in a finite time. Finally, we prove the ill-posedness in the Besov space $B_{2,\infty}^{3/2}$ by utilizing the non-traveling wave solutions.

Computing posterior distributions in large-scale Bayesian linear inverse problems is challenging due to the high dimensionality of the parameter space. In this work, we develop a data-informed framework that shifts the computational focus from the parameter space to the data space. We rigorously characterize an intrinsically low-dimensional data space, establish its isometric embedding into the parameter space, and show that the prior-to-posterior update is confined to a data-informed subspace. This perspective allows posterior inference to be carried out in a reduced data-informed subspace. Based on this formulation, we propose a quotient-space Golub--Kahan bidiagonalization method to construct data-informed Krylov subspaces, and integrate empirical Bayesian inference into the iterative framework, enabling simultaneous hyperparameter estimation and posterior approximation in a matrix-free manner. Numerical experiments on representative problems support the theoretical framework and demonstrate the effectiveness of the resulting method.

We introduce a new propositional logic, called very weak subintuitionistic logic $\mathbf{VF}$, by adapting the relational semantics of Fitting, Marek, and Truszczyński for the pure logic of necessitation $\mathbf{N}$ to the propositional setting. We prove that $\mathbf{VF}$ and its closed negative extensions are sound and complete with respect to this semantics, and that they have the disjunction property and the finite frame property. We also prove that $\mathbf{VF}$ is strictly weaker than the weak subintuitionistic logic $\mathbf{WF}$ of Shirmohammadzadeh Maleki and de Jongh. Finally, we study modal companions of $\mathbf{VF}$ and its closed negative extensions via Corsi's modified Gödel translation.

We establish a quantum Fisher information (QFI) duality for distributed quantum sensor networks with local phase encoding. For any $N$-qubit probe state, where $N$ denotes the number of sensors, $F_Q(\boldsymbol{w}^\top \boldsymbolθ) + F_Q(\boldsymbol{v}^\top \boldsymbolθ) \leq N$ for all unit orthogonal sensing directions $\boldsymbol{w}$ and $\boldsymbol{v}$, with equality for all equatorial states when $N=2$ and for Greenberger--Horne--Zeilinger (GHZ) states when $N\geq 2$. Heisenberg-limited precision for direction $\boldsymbol{w}$, $F_Q(\boldsymbol{w}^\top \boldsymbolθ)=N$, saturates the bound and simultaneously forces zero QFI for all other independent directions. This can be interpreted as the condition for parameter privacy in distributed quantum sensing: attaining Heisenberg-limited precision for the sensing target renders all alternative privacy-intrusive estimations impossible.

By using Brownian motion and stochastic calculus, we establish a second main theorem for holomorphic curves into a projective subvariety $V\subset\mathbb P^n(\mathbb C)$ with an arbitrary family $\mathcal Q$ of $q$ hypersurfaces $Q_1,\ldots,Q_q$ concerning its distributive constant $Δ_{\mathcal Q,V}$. In our result, the counting functions are truncated to level $H_V(d)-1$, where $d=lcd(°Q_1,\ldots,°Q_d)$ and $H_V(d)$ is the Hilbert function of $V$. As an application of the second main theorem, we give a uniqueness theorem for holomorphic curves from $\mathbb C$ into $V$ sharing an arbitrary family of hypersurfaces regardless of multiplicity.