In this article, we study the equation $$ (-Δ_p)^s u = f(u) $$ in a bounded domain $Ω\subset \mathbb{R}^n$, where $n\geq 2$, $p>2$, and $f$ is locally Lipschitz. We establish a strong comparison principle in a fairly general setting and use it to derive symmetry results for positive $C^1$ solutions satisfying Dirichlet boundary conditions. We also show that the $C^1$ regularity assumption is indeed satisfied for $p\in \left[2,\frac{2}{1-s}\right)$.

An ultimate theory of a phase of matter should describe all its universal properties via quantities that are measurable numerically and experimentally. In this work, we present a microscopic universal theory of symmetry-enriched topological quantum spin liquids (TQSLs) in two spatial dimensions, which directly utilizes microscopically measurable quantities to describe the universal properties. This theory applies to generic TQSLs, which can be Abelian or non-Abelian, chiral or non-chiral. The symmetries are also general, which can include both internal and lattice symmetries, unitary and anti-unitary symmetries, and discrete and continuous symmetries. There can be spin-orbit coupling, the microscopic degrees of freedom may transform linearly or projectively under the symmetries, and the symmetries can permute anyons. The input of the theory is some microscopic states with anyons, operators that control the dynamics of anyons, and symmetry actions in the TQSL, and its output is a set of data characterizing the universal properties, whose underlying mathematical structure is a generalization of category theory. Based on this theory, we find an explicit bijective map between the universal data characterizing a TQSL with a symmetry described by a group $G$, where the symmetry actions may include both lattice and internal symmetries, and the corresponding universal data for a TQSL with only an internal symmetry group $G$, and thus establish a precise crystalline equivalence principle. We demonstrate our theory in symmetry-enriched TQSLs realized on quantum processors based on superconducting qubits, trapped ions, and Rydberg atoms, and in each example we verify the Lieb-Schultz-Mattis anomaly matching condition. Our theory provides a solid basis for identifying and manipulating symmetry-enriched TQSLs, which further paves the way for fault-tolerant quantum computation based on these systems.

We formulate a quantization scheme for open electromagnetic systems with arbitrary passive boundary conditions. Rather than specifying reservoirs phenomenologically, the method identifies them from the dissipation geometry of the Maxwell operator. Factoring the imaginary part of the Maxwell operator gives a bosonic realization of the field operator and separates the fluctuation channels into medium-assisted reservoirs from material absorption and boundary-assisted reservoirs from exchange through the open boundary. Depending on the boundary condition, the latter become free-space radiation modes, impedance-load channels, guided port modes, or more general boundary channels. Green-function input-output relations then follow as an application, yielding frequency-dependent scattering and noise kernels without Markov or single-mode assumptions. To illustrate the practical application, we consider a lossy structure with mixed impedance and outgoing boundaries, and photonic integrated circuit configurations with waveguide port boundaries.

We establish several improvements to the main results of [PR19] and [CP21], refining the seminal self-improving method for generalized Poincaré inequalities from [FPW98, MP98]. These results, together with various related applications, stem from a general self-improving property for functions satisfying the local inequality $$\frac{1}{|Q|}\int_Q |f(x)-f_Q|\,dx \le a(Q)$$ for all cubes $Q\subset\mathbb{R}^n$. The functional $a$ is assumed to obey a specific discrete geometric summability condition. By restricting our focus to axis-parallel cubes in $\mathbb{R}^n$, this geometric setting allows us to obtain sharper estimates than those available in more general metric measure spaces.

Source code vulnerability detection remains a long-standing challenge due to the increasing scale, structural complexity, and semantic diversity of modern codebases. Conventional static-analysis or rule-based approaches often fail to capture subtle execution dependencies, while single-modality learning models tend to overlook critical structural information embedded beyond the lexical surface of source code. To improve robustness across heterogeneous code patterns, we propose FusionVul, a joint representation learning framework that integrates sequential syntactic representations extracted by a pretrained Transformer encoder with structural semantics propagated through a graph neural network. The framework further incorporates a cross-attention-based feature fusion network to enable fine-grained cross-modal interaction and employs a sample-aware weighting mechanism to integrate multiple predictive branches. Experimental results on four datasets demonstrate that FusionVul achieves superior F1 scores on datasets with highly dispersed function size distributions and broader vulnerability-type coverage, such as SVulD and DiverseVul, reflecting its capability to capture complex and diverse vulnerability patterns.

There is growing interest in constructing conformal prediction sets that provide approximate or asymptotic conditional coverage guarantees, capturing local data heterogeneity. However, methods like localized conformal prediction (LCP) may face challenges in ensuring reliable prediction sets in regions with sparse calibration data. This paper introduces Enhanced Localized Conformal Prediction (ELCP), a novel approach that incorporates auxiliary data to refine localized prediction sets while preserving finite-sample marginal coverage guarantees. By utilizing a density-ratio-weighted kernel estimator, ELCP seamlessly integrates auxiliary and calibration data, accommodating potential distributional shifts and improving the local reliability of prediction sets. Theoretical analysis confirms that ELCP maintains marginal coverage and enhances asymptotic test-conditional coverage. Simulation results demonstrate its superior local coverage and smaller prediction sets compared to standard LCP, highlighting its effectiveness in settings with limited calibration data but available auxiliary information from related tasks.

Neutral group-IV vacancy (XV, X = Si, Ge, Sn, and Pb) centers in diamond are emerging solid-state spin-photon interfaces because of their favorable spin coherence and inversion symmetry-protected optical transitions. However, stabilizing their neutral charge state typically requires stringent Fermi-level engineering in high purity boron-doped diamond, which poses significant materials-growth challenges. Here, we demonstrate that dimensional confinement in diamane provides an alternative route to charge-state stabilization without intentional doping. Using first-principles calculations, we show that quantum confinement and surface termination cooperatively tune the host band gap and shift the occupied defect states upward from the valence-band edge, thereby enlarging the thermodynamic stability window of the neutral charge state and suppressing valence-band assisted excitation pathways. We further reveal that the thickness and surface termination of diamane enable systematic tuning of the electronic structure, zero-field splitting, and spin-orbit coupling of XV centers while largely preserving their optical transition energies. Among the structures considered, hydrogenated diamane offers the most favorable balance between charge-state stability and magneto-optical performance. More broadly, our findings establish dimensional confinement as a general strategy for engineering the charge, optical, and spin properties of solid-state quantum defects.

We investigate the structural correlations of hard spheres confined within a narrow cylindrical pore in the quasi-one-dimensional regime, where interactions are restricted to nearest neighbors. Using a Laplace-space formulation of the radial distribution function (RDF), we determine the correlation lengths and oscillation frequencies associated with its long-distance decay. In addition to the global RDF, we analyze transverse-resolved RDFs that account for the positions of particle pairs across the pore cross section. While these observables are associated with the same underlying pole spectrum, their residues depend on the transverse configuration and can vanish due to symmetry. As a result, different particle-pair configurations may be governed by different leading poles and display different correlation lengths and oscillation frequencies. In particular, the global RDF does not always reflect the longest-ranged correlations found in transverse-resolved observables. We examine how this behavior depends on density and confinement. In the strong-confinement limit, the system approaches the Tonks-gas behavior at finite pressure, and the differences between the RDFs disappear.

It is an interesting open problem whether every compact Hausdorff space can be realized as the maximal $d$-spectrum of an arithmetic frame. We approach this problem by generalizing the $d$-nucleus to a stably continuous frame. We use Priestley duality to characterize the resulting $\underline d$-nucleus, which allows us to prove that every locally compact Hausdorff space can be realized as the maximal $\underline d$-spectrum of a continuous regular frame. As a corollary, we obtain that every locally Stone space can be realized as the maximal $d$-spectrum of an algebraic regular frame.

In complex quantum many-body systems such as atomic nuclei, the interplay between regular collective motion and irregular intrinsic dynamics gives rise to fluctuations that cannot be fully captured by existing global theoretical models. Nuclear mass, which exhibits smooth trends across the nuclear chart together with localized deviations, provides a sensitive observable for investigating such irregular dynamics. In this work, we employ a variety of neural network architectures, which serve as controlled nonlinear filters within a Hierarchical Residual Decomposition framework to progressively extract and suppress the chaotic many-body signature (characterized by $1/f$ spectral correlations) in nuclear mass residuals. The resulting Physics-Informed Neural Ensemble (PINE) model combines multiple mass models and neural network architectures, enabling a systematic suppression of coherent and chaotic components, after which the remaining fluctuations are analyzed using Fourier-based spectral diagnostics across different mass regions. Our results show that hierarchical neural residual learning efficiently removes the dominant low-frequency correlations and suppresses the quantum-chaotic spectral rigidity, driving the residuals toward the uncorrelated white-noise limit. This systematic suppression provides a quantitative diagnostic of the underlying scale-dependent complexity and many-body correlation structure of nuclear mass deviations.

In multi-hop device-to-device (D2D) networks, the optimization of network-level metrics is particularly difficult due to the tight coupling between network-layer routing and physical-layer resource allocation. Departing from traditional average-performance metrics, this paper addresses the joint optimization of routing paths, transmission power, and bandwidth allocation. We formulate a generalized cost function to minimize the maximum transmission time (i.e., the bottleneck delay) alongside the total energy consumption. To tackle the resulting highly non-convex formulation, we propose a novel block coordinate descent (BCD) framework. At the network layer, we develop two adaptive routing algorithms: a matrix-free Frank-Wolfe (MF-FW) algorithm for fast execution in dense topologies, and a low-rank primal-dual interior-point method (LR-PDIPM) that bypasses dense matrix inversions via the Sherman-Morrison formula for high-precision solutions. At the physical layer, we design a parallel dual ascent algorithm leveraging a time-domain perspective transformation to solve the resource allocation subproblem to global optimality. The proposed BCD framework is proven to converge to an ε-neighborhood of a stationary point. Through comprehensive experiments, the proposed BCD framework establishes its superiority in achieving the optimal delay-energy trade-off. Specifically, the LR-PDIPM variant achieves a maximum 9.14-fold reduction in total energy consumption and up to an order of magnitude improvement in energy efficiency, while maintaining a bounded maximum delay gap (up to 3.78-fold) relative to the best baseline. Meanwhile, the warm-start MF-FW variant identifies near-optimal solutions in mere seconds, serving as a highly practical engineering approach.

Recently, topological MV-algebras have been investigated by several mathematicians. In this paper, we mainly show that for every Hausdorff topological MV-algebra $A$, there exists a natural topological isomorphism $i_A:A\rightarrow A^\bullet$ of $A$ onto a closed subalgebra of the pathwise connected, locally pathwise connected topological MV-algebra $A^\bullet$. Furthermore, we show that there is an extension to a bounded continuous function on the MV-algebra $A^\bullet$ for each continuous real-valued bounded function on a topological MV-algebra $A$. Finally, we prove that if $φ:A_1\rightarrow A_2$ is a continuous homomorphism of topological MV-algebras, then $φ$ admits a natural extension to a continuous homomorphism $φ^\bullet:A_1^\bullet\rightarrow A_2^\bullet$; in addition, if $φ$ is open and onto, then so is $φ^\bullet$.

Shear banding is a well-known and widespread instability in strained solids: under external strain, the deformation localizes along a line in two dimensions or a plane in three dimensions. Developing a proper theoretical description of this phenomenon is key to understanding mechanical failure in solid materials. Very recently, a nonlinear theory extending classical elasticity to include plastic deformations as topological charges was proposed, offering detailed predictions on the nature and consequences of the shear-banding instability. The theory derives a Hessian operator whose lowest eigenvalue vanishes at the onset of instability, and the corresponding critical eigenmode describes the displacement field across the shear band. The resulting soft mode possesses the selected localization scale and subsequently saturates into a finite-width shear band. The aim of this Letter is to examine this theory numerically, establishing the role of topological screening and nonlinear instability as the mechanisms governing shear banding during athermal quasistatic deformation. We show that the displacement profile around the shear band is directly determined by the screening parameter and the nonlinear coefficient, thereby quantitatively verifying the theoretical predictions. Our results demonstrate that shear banding differs fundamentally from fracture: it arises from a nonlinear instability of an elastic field screened by plastic deformations. This establishes topological screening as the essential mechanism governing shear banding in amorphous solids.

In this paper, we revisit the axisymmetric Nirenberg problem for certain prescribed scalar curvature functions whose critical points include the north and south poles and whose order of flatness at both points is $n-2$. It has been known for some time that the solution set is compact when the parameter points $(K_1,K_2,a_1,a_2)\in\mathbb{R}^4$, determined by the first two leading expansions of the prescribed scalar curvature functions at the poles, stay away from a critical three-dimensional hypersurface $Σ^0$, and that loss of compactness occurs near $Σ^0$. We construct, in dimensions $n \geq 4$, two-bump axisymmetric solutions for parameter points sufficiently close to $Σ^0$ from one side, where the relevant side is determined by the next-order flatness terms. This gives a one-sided refinement of the previously known blow-up phenomenon near the critical hypersurface $Σ^0$: two-bump blow-up occurs from this side, whereas the solution set remains compact on the opposite side.

We propose a cascaded Rydberg antiblockade (RAB) regime via a Floquet modulation in four fully connected interacting atoms, which establishes a new synthetic dimension, Dicke-state lattice (DSL), in the space of collective spin excitations. By applying a global periodic driving, we synthesize an effective Hamiltonian that enables perfect state transfer across the five-site DSL with multiple programmable pathways from stepwise nearest-neighbor jumps to a single-step transition. This DSL platform further allows us to simulate a dynamic Su-Schrieffer-Heeger model, where soft quantum control is employed to achieve topologically inspired full RAB $|0000\rangle \to |1111\rangle$ with enhanced robustness against disorder. Moreover, by incorporating the shortcut to adiabaticity technique, we generate high-fidelity entangled twin-Fock and Greenberger-Horne-Zeilinger states on the four atoms within sub-microsecond timescales, outperforming the speed limits of conventional adiabatic protocols. Our work demonstrates a flexible and programmable synthetic dimension for quantum simulation and multipartite entanglement engineering in Rydberg atom arrays, paving the way for the future development of quantum information processing.

In quantum mechanics, the wave function encodes all information about a particle through its probability density and phase. While stationary states are characterized by conserved quantum numbers such as energy and, in central potentials, angular momentum, non-stationary superpositions - particularly in anisotropic or time-dependent fields - generally lack equally universal descriptors of their spatiotemporal dynamics. Here we introduce quantum hodographs: the trajectories traced in time by the expectation value of observable vector quantities. For a free electron in a superposition of three plane waves, all hodographs of the probability current lie on a universal cubic surface with conical singularities. Rational frequency (energy) difference ratios produce non-contractible loops with well-defined winding numbers. In anisotropic harmonic oscillators the Ehrenfest trajectories - hodographs of the expectation value of particle radius-vector - form three-dimensional Lissajous knots, echoing the classical Thomson vortex-atom model. Externally driven quantum systems allow controllable initiation of knotted hodographs. We propose an optical modulation spectroscopy scheme for reconstructing these topological features in trapped ions and single-electron systems. The topological indices of the loops and knots are robust to parameter variations, offering a new tool for characterizing complex quantum dynamics.

We answer two open questions about weakly o-minimal fields posed by Macpherson, Marker, and Steinhorn: whether weakly o-minimal fields have the exchange property and whether they have generic differentiability. We construct an ordered field $(K,+,\cdot,\le)$ and a function $f : K \to K$ such that the expansion $(K,+,\cdot,\le,f)$ has a weakly o-minimal complete theory but $f$ is nowhere differentiable. In an appendix, we prove that algebraic closure has the exchange property in any weakly o-minimal theory of ordered fields.

We propose a lowest-order \(H(\operatorname{div})\)-conforming mixed extended virtual element method for elliptic interface problems on interface-unfitted polygonal meshes. The flux and pressure are approximated by subdomain-wise extended \(H(\operatorname{div})\)-VEM spaces and by piecewise constants, respectively. On cut elements, the computable polynomial projection is defined on the whole background element and then restricted to the two subdomains. Compared with BDM-type polynomial spaces, the mixed VEM space contains a non-polynomial component, which gives rise to additional consistency terms on cut elements. To control these terms, we use an enhanced kernel stabilization on cut elements and an interface normal-flux average in the mixed coupling. A corrected interface-flux penalty and a local divergence ghost penalty are added to obtain cut-position-independent stability without using a volume div-div augmentation. We prove continuity, a discrete inf-sup condition, and an optimal first-order error estimate in a mesh-dependent norm. The constants are independent of the mesh size and of the position of the interface relative to the background mesh, but may depend on the coefficient contrast.

Cloaking and holography -- usually treated as distinct problems -- are two limits of a single operation that we call acoustic disguising, realized here using immersive boundary conditions on a closed surface. Driving the boundary with homogeneous Green's functions suppresses any incident field inside the enclosed volume and cloaks unknown objects broadband; driving it with scattering Green's functions synthesizes a holographic scatterer indistinguishable from a target for arbitrary illuminations. Combining the two, using heterogeneous Green's functions, replaces the scattering signature of one object with that of another, transforming its acoustic identity. We demonstrate the framework in three-dimensional FDTD simulations driven by impulsive Green's functions, complemented by data-driven Green's-function retrieval, establishing a direct route to real-time 3D acoustic cloaking, holography, cloning, and disguising.

We study the $t$-uniform hypergraphicality problem under a compressed representation of the degree sequence. Instead of listing all vertex degrees explicitly, the input consists of pairs $$ (δ_1,n_1),\dots,(δ_k,n_k), $$ meaning that exactly $n_i$ vertices have degree $δ_i$. Thus the parameter $k$ denotes the number of distinct degrees. Although deciding $t$-hypergraphicality is NP-complete for every fixed $t>2$, we prove that the problem is fixed-parameter tractable parameterized by $(k,t)$. Our result shows that tractability extends substantially beyond previously known bounded-range regimes: even degree sequences with large overall degree spread can be handled efficiently when the number of distinct degrees is bounded. Our approach decomposes hyperedges according to their types with respect to the degree classes, yielding a bounded-dimension spectrum representation. Using balancing hinge-flips, we show that every feasible spectrum can be transformed into a realization of the prescribed degree sequence. This leads to an integer programming feasibility formulation with $$ \binom{t+k-1}{k-1} $$ variables. Applying Lenstra's theorem yields an FPT algorithm running in time $$ f(k,t)\cdot \mathrm{poly}(L), $$ where $L$ denotes the encoding length of the compressed input.

Privacy-preserving Personalized Federated Learning (PFL) enables clients to collaboratively train personalized models without exposing raw data, but exchanged model updates remain vulnerable to inference attacks from honest-but-curious servers. Homomorphic Encryption (HE) addresses this by allowing server-side aggregation directly on encrypted updates, with the CKKS scheme being particularly suitable due to its native support for approximate floating-point arithmetic. However, no prior work has examined how to configure CKKS for PFL deployments, leaving practitioners without principled guidance on parameter selection that directly affects privacy, precision, and computational cost. This paper presents pFedCKKS, a generic framework integrating CKKS into PFL, and provides the first systematic parameter selection guide for practitioners. We derive the full CKKS parameter constraints under 128-bit security for the PFL setting, showing the selection problem reduces to choosing just two values: the inner and outer ciphertext prime. Implemented using the Flower framework and TenSEAL library, pFedCKKS is evaluated on the FEMNIST, CelebA and Sentiment140 datasets with FedFinetune, Ditto and FedPer which represents PFL algorithms. Experimental results reveal an empirical trade-off between precision and computational/communication costs. This allows us to draw a concrete guideline for selecting proper CKKS parameters that balance efficiency and accuracy in real-world deployments of pFedCKKS.

The results of new photometric observations of two SU~UMa type dwarf novae are presented. Superhumps were detected on the light curves of these systems during superoutbursts and their periods and amplitudes determined. The classification of objects as dwarf novae of the ER~UMa subclass (for RZ~LMi) and subclass WZ~Sge (for KV~Dra) has been clarified. Orbital period $0^{d}.0586$ for KV Dra was determined.

In this article, a rigorous well-posedness result is established for a moist compressible primitive-equation system with phase changes. The model couples the compressible primitive equations to a bulk microphysics model for vapor water, cloud water, and rain water, including evaporation, condensation, auto-conversion, collection, and sedimentation effects. Global strong well-posedness is proved for initial data sufficiently close to constant equilibria.

The main ingredient of this paper is an edge-local variable-degree projection on a triangle that is uniformly stable in both L2 and H1. We use this two-dimensional operator in two tetrahedral constructions. First, on a reference tetrahedron, we build an H1-stable projection from a high order polynomial space onto a variable-degree space whose degrees are prescribed independently on edges, faces, and in the volume. Since the tetrahedral projection is local and trace-compatible, it also gives an h- and p-uniform stable decomposition, in the weighted energy norm, for conforming hp spaces, and hence a uniform additive Schwarz preconditioner for the conforming Laplace operator. Second, on a uniformly regular mapped tetrahedral mesh with elementwise variable polynomial degrees, the same triangular projection gives the finite-layer edge truncation needed in a p-uniform stable DG-to-CG decomposition for the symmetric IPDG norm. The DG-to-CG decomposition, combined with the conforming splitting, gives the IPDG preconditioner. The constants depend only on reference shapes, the local degree-spread bound within each tetrahedron, the neighbor-degree bound across mesh faces, uniform map-regularity, patch cardinalities, and the coefficient path constants; they are independent of h, of the local polynomial degrees, and of the coefficient contrast.

Modern scientific workflows increasingly span diverse computing architectures, yet executing a single computational model across disparate systems often forces researchers to maintain fragmented, site-specific pipelines. In this paper, we address this challenge within the domain of computational neuroscience by presenting a unified, cloud-based workflow orchestrated via EBRAINS JupyterLab. This workflow enables users to transparently execute spiking neural networks on both von-Neumann supercomputers and neuromorphic hardware. Using a single federated identity, the system dispatches jobs to HPC sites (JUSUF, Galileo100) via PyUNICORE and to the SpiNNaker-1 neuromorphic system via the Neuromorphic Computing Platform Interface. To guarantee cross-site reproducibility and mitigate software version drift, we utilize a zero-installation execution mode that dynamically pulls PMIx-aware Apptainer containers to HPC compute nodes. Furthermore, we demonstrate genuine model-level portability using the NESTML domain-specific language, allowing custom neuron models to be written once and automatically compiled for either the NEST (C++) or sPyNNaker backends. Validated with a balanced random network case study, this work illustrates a practical, end-to-end path for hardware-agnostic workflows while highlighting the critical role of containerization and domain-specific languages in achieving true cross-platform reproducibility.

Bound states in the continuum (BICs) are spatially localized states embedded in the continuous spectrum without hybridizing with extended bulk modes. Recent advances in topological band theory have greatly enriched the understanding of BICs, which gives rise to boundary-localized topological BICs with extremely high robustness against disorders. However, there remains a challenge in realizing corner-localized topological BICs in a three-dimensional system due to the absence of both realistic theoretical models and effective topological characterization schemes. In particular, how to engineer a controllable number of corner-localized topological BIC is still an open question. Here, we propose that the corner-localized topological BICs can emerge in a class of generalized breathing pyrochlore lattice with general inter-cell hoppings. We further show that the number of BICs at each corner can be arbitrarily adjusted by changing the parameters of inter-cell hoppings. Remarkably, although these corner-localized topological BICs are intertwined with a substantial number of bulk modes, we can accurately characterize them through the polarized topological charges, which are nodal points with topological properties in Brillouin zone and are measurable in experiments. We also reveal three types of topological phase transitions of corner-localized BICs, which are associated with the different ways of closing the bulk energy gap and can be intuitively captured by the polarized topological charges. This work not only promotes the theoretical research of corner-localized topological BICs, but also opens an avenue for their experimental observation in the future.

This work formalizes the notions of structure and pattern for three distinct one-dimensional spin-lattice models (finite-range Ising, solid-on-solid, and three-body), using information-theoretic and computation-theoretic methods. We begin by presenting a novel derivation of the Boltzmann distribution for finite one-dimensional spin configurations embedded in infinite ones. We next recast this distribution as a stochastic process, thereby enabling us to analyze each spin-lattice model within the theory of computational mechanics. In this framework, the process's structure is quantified by excess entropy (predictable information) and statistical complexity (stored information), and the process's structure-generating mechanism is specified by its epsilon-machine. To assess compatibility with statistical mechanics, we compare the configurations jointly determined by the information measures and epsilon-machines to typical configurations drawn from the Boltzmann distribution, and we find agreement. We also include a self-contained primer on computational mechanics and provide code implementing the information measures and spin-model distributions.

Combinatorial optimization can be addressed with quantum computing through variational quantum algorithms, but a central challenge in this approach is to design an ansatz expressive enough to explore the feasible subspace of the Hilbert space where the optimal solution lies. Another major challenge is tuning the Lagrange multipliers in penalty terms to enforce feasibility and guarantee solution quality. To address both challenges, we propose the first feasibility-preserving mixed Dicke state ansatz for Hamming weight constrained combinatorial optimization, extending the density matrix formalism to structurally encode equality and inequality constraints directly into the quantum circuit, thereby eliminating the need for penalty terms in the objective function. The proposed framework handles both constraint types, with the pure Dicke state ansatz recovered as a special case corresponding to equality constraints, and generalizes to multiple constraint groups via tensor products of individual pure or mixed Dicke states. We validate the proposed approach in the context of combinatorial portfolio optimization across three experimental scenarios with increasing constraint complexity, using the CMA-ES optimizer and comparing its performance against random search with replacement restricted to the feasible subspace. As the feasible search space grows, the proposed ansatz demonstrates a clear advantage over random search in terms of the number of objective function calls required to identify high-quality solutions. Hardware experiments on IBM NISQ processors confirm that noise mitigation and circuit transpilation optimizations remain open challenges for practical deployment. The framework is general and directly applicable to other combinatorial optimization problems with Hamming weight constraints.

Let $G$ be a graph. The size and the signless Laplacian spectral radius of $G$ are denoted by $e(G)$ and $q(G)$, respectively. A spanning subgraph $F$ of $G$ is called an $H_b$-factor of $G$ if $d_F(v)\in\{1,3,5,\ldots,b-1,b\}$ for every $v\in V(G)$, where $b\geq2$ is an even integer. Lu and Wang obtained a sufficient condition according to the number of odd components in $G-S$ for a connected graph $G$ of even order to have an $H_b$-factor, where $S$ is a subset of $V(G)$ [H. Lu, D. Wang, On Cui-Kano's characterization problem on graph factors, J. Graph Theory 74 (2013) 335--343]. In this paper, motivated by Lu and Wang's above result, we establish a lower bound for the size in an $n$-vertex connected graph $G$ with given minimum degree to guarantee that $G$ has an $H_b$-factor. Further, we show a lower bound for the signless Laplacian spectral radius in an $n$-vertex 2-connected graph $G$ with given minimum degree to ensure that $G$ has an $H_b$-factor.

Control-variate and polynomial-maximization (PMM) estimators are optimized at a single fixed distribution, yet they are increasingly proposed to strengthen hypothesis tests, which decide between two regions of a parameter family. We give a closed-form criterion for when this transfer succeeds. For an H0-centered augmentation of a target moment statistic with null-optimized weight vector K0, the alternative-side expectation equals the target plus K0^T mu_a,H1, where mu_a,H1 is the alternative-side mean of the augmenting basis. Null-variance reduction therefore transfers without bias only under the orthogonality condition K0^T mu_a,H1 = 0; requiring each augmenting function to remain mean-zero is sufficient but not necessary. We instantiate the criterion on the recently proposed Wiedermann-Shi third-order cumulant test for measurement-error independence. A second-order PMM correction is unbiased and lower-variance under the null (relative efficiency >= 1 in all 36 conditions; aggregated mean ARE values 1.23-5.16; Type-I 0.04-0.09), yet provably inconsistent under the alternative: the antisymmetric polynomial auxiliaries acquire nonzero means, attenuating the target by a closed-form factor and costing 7-52 percentage points of power, worst where the test is strongest and worsening under heavy tails. A fourth-order variant reduces variance (ratio 1.127) but fails a nuisance guard (rejection 0.295 versus 0.10). We derive a reusable alternative-consistency acceptance gate for variance-reduced test statistics.