We investigate sample-to-sample fluctuations of the shear modulus in ensembles of particle packings near the jamming transition. Unlike the average modulus, which exhibits distinct scaling behaviours depending on the interparticle potential, the fluctuations obey a critical exponent that is independent of the potential. Furthermore, this scaling behaviour has been confirmed in two-dimensional packings, indicating that it holds regardless of spatial dimension. Using this scaling law, we discuss the relationship predicted by heterogeneous-elasticity theory between elastic-modulus fluctuations and the Rayleigh scattering of sound waves across different pressures. Our numerical results provide a useful foundation for developing a unified theoretical description of the jamming critical phenomenon.

We present a dedicated search for gravitational-wave backgrounds with nontrivial peak structures using data from the first three and the initial part of the fourth observing runs of the LIGO-Virgo-KAGRA network. The analysis is motivated by a variety of early-Universe models characterized by signals with multiple peaks. We introduce a model independent parameterization of double-peaked spectra based on the superposition of two normalized broken power laws and perform a Bayesian inference study using the LIGO-Virgo-KAGRA isotropic cross-correlation data. While no statistically significant evidence for a multi-peak background is found, the analysis provides constraints on the inter-peak slopes in correlation with the signal amplitude. These results exhibit LIGO-Virgo-KAGRA's ability to probe signals beyond a single peak structure and establish a foundation for future targeted searches for nontrivial gravitational waves background spectral shapes in future observing runs and the advanced detector era.

The analysis of noise-induced escape in populations of bistable elements is challenging, because nonlinearity, coupling, and noise all play essential roles. We show that the interplay of these three factors yields three qualitatively distinct escape mechanisms depending on coupling strength in populations of diffusively coupled bistable elements. To clarify dominant driving factors of escape dynamics, we develop a model-reduction approach, deriving three effective one-dimensional dynamics: nonlinear mean-field Fokker-Planck equation in the weak-coupling regime, stochastic mean-field dynamics in the strong-coupling regime, and deterministic mean-field dynamics in the intermediate regime. We validate these reduced descriptions by comparing predicted mean escape times with numerical simulations. We identify a distinct dominant driving factor of collective escape in each regime. Notably, the three mechanisms emerge through the interplay of nonlinearity, diffusive coupling, and dynamical noise -- rather than bifurcations of the noise-free system. Our approach serves as a framework applicable to other stochastic nonlinear systems with diffusive coupling, motivating a further search for similar synergistic phenomena.

We study the stability of $d$-dimensional ($d=3,4,5$) de Sitter and Minkowski spacetimes within the framework of semiclassical gravity sourced by a strongly coupled quantum field with a gravity dual. Our stability results are derived from a careful analysis of the $d$-dimensional Lichnerowicz equation with mass-squared $m^2$ and of semiclassical equations involving the dimensionless parameter $γ_d$. For $d=3$, we find that Minkowski spacetime is always unstable against perturbations, whereas de Sitter spacetime becomes stable when a dimensionless parameter $γ_3$ exceeds a critical value. In $d=4$, both de Sitter and Minkowski spacetimes become unstable when the parameter $γ_4$ exceeds its critical value. In contrast, in $d=5$, de Sitter and Minkowski spacetimes remain stable for almost all values of the parameter $γ_5$, except for a regime in which higher-curvature corrections become comparable to the Einstein tensor.

This paper investigates the reconfiguration variant of the Constraint Satisfaction Problem (CSP), referred to as the Reconfiguration CSP (RCSP). Given a CSP instance and two of its solutions, RCSP asks whether one solution can be transformed into the other via a sequence of intermediate solutions, each differing by the assignment of a single variable. RCSP has attracted growing interest in theoretical computer science, and when the variable domain is Boolean, the computational complexity of RCSP exhibits a dichotomy depending on the allowed constraint types. A notable special case is the reconfiguration of graph homomorphisms -- also known as graph recoloring -- which has been studied using topological methods. We propose a novel algebraic approach to RCSP, inspired by techniques used in classical CSP complexity analysis. Unlike traditional methods based on total operations, our framework employs partial operations to capture a reduction involving equality constraints. This perspective facilitates the extension of complexity results from Boolean domains to more general settings, demonstrating the versatility of partial operations in identifying tractable RCSP instances.

Littlewood--Paley theory began with the classic paper of Littlewood and Paley (J.\ E.\ Littlewood, R.\ E.\ A.\ C.\ Paley, {\em Theorems on Fourier Series and Power Series}. J. Lond. Math. Soc. (1), {\bf 6} (1931), 230--33). We discuss this paper and its impact from a historical perspective. We include an outline of the results in the paper and their subsequent significance in relation to developments over the last century, and set them into the context of the current state of the art in harmonic analysis and beyond.

We develop a fully numerical framework to compute and visualize the \emph{hypershadow}\cite{Novo:2024wyn}, the three-dimensional generalization of the black hole shadow in five-dimensional spacetimes. Our method is based on backward ray tracing and allows flexible control over observer position, enabling the reconstruction of the full shadow volume. For visualization, we combine discrete sampling with surface contouring and introduce reflection difference maps on central slices to quantify mirror symmetries. Applying this method to the Schwarzschild-Tangherlini and Myers-Perry geometries, we validate the former's spherical symmetry and systematically discuss the hypershadow's dependence on observer position and black hole spin parameters. We also provide compact quantitative measures for size reduction and global displacement, revealing clear monotonic trends. The framework is readily extendible to other metrics and opens the way to numerical studies of more exotic objects, such as black rings and their prospective toroidal hypershadows.

We present a long-term broad-band temporal and spectral study of a TeV BL Lac source TXS 0518+211 by analyzing nearly 16 years (MJD 54682 -- 60670) of simultaneous optical, UV and X-ray light curves from \textit{Swift}-XRT/UVOT and gamma-ray light curves from \textit{Fermi}-LAT. Based on the availability of simultaneous multi-wavelength data and considering flux level as the depiction of AGN-jet activity we identified 11 epochs (named as Epoch-A to Epoch-K) and investigated temporal as well as spectral variability during these epochs to understand the emission properties in this source. The fractional variability analysis reveals that, in all epochs, X-ray light curve exhibits relatively high degree of variability in compared to the optical, UV and gamma-ray light curves. The flux-flux plots among different bands, in general, show weak to moderate correlation with Spearman correlation coefficient ranging from 0.29 to 0.58. Notably, during Epoch-I, we detect a possible orphan flare exhibiting increase in the X-ray flux level ($\sim$ 2.4 times of the total average flux) but with no corresponding counterpart seen in the optical, UV bands. In contrast, during Epoch-K, we detect a significant decrease in the X-ray flux but no corresponding decrease in optical, UV and gamma-ray bands. Overall, our study reveals several changes in the flux states and complex nature of jet dominated emission processes. We tested one-zone and two-zone leptonic scenarios and for most of the epochs, the latter one provides a better description of the broad-band emission in this TeV BL Lac source.

In this short article, we prove that the Halton sequence, one of the most well-known low-discrepancy sequences, is not quasi-uniform in any dimension $d \ge 2$ with any pairwise relatively prime bases. We further disprove the quasi-uniformity of some Halton-type sequences, including the $p$-dimensional Faure sequence in base $p$, $p \in \mathbb{P}$, which provides an alternative proof of the known results.

We discuss the choice and implementation of inter-valley potentials in the so-called two bands $\mathbf{k}\cdot\mathbf{p}$ model for the opposite $X$, $Y$ or $Z$ valleys of silicon. We focus on the description of valley splittings in Si/SiGe heterostructures for spin qubits, with a particular attention to alloy disorder. We demonstrate that the two bands $\mathbf{k}\cdot\mathbf{p}$ model reproduces the valley splittings of atomistic tight-binding calculations in relevant heterostructures (SiGe spikes, wiggle wells...), yet at a much lower cost. We show that the model also captures the effects of valley-orbit mixing and yields the correct inter-valley dipole matrix elements that characterize manipulation, dephasing and relaxation in spin/valley qubits. We simulate a realistic Si/SiGe spin qubit device as an illustration, and discuss electron-phonon interactions in the two bands $\mathbf{k}\cdot\mathbf{p}$ model. Beyond spin qubits, this model enables efficient simulations of SiGe heterostructure devices where spin and valley physics are relevant.

The nodal structure of bound-state wave functions for one-dimensional quantum systems with quartic energy-momentum dispersion and polynomial potentials is analysed by using the semiclassical approximation and variational approach. For energies of bound states, we derive the quantization condition, obtained by using the complex Wentzel method, where we take into account perturbative (up to the fourth order) and nonperturbative in the Planck constant corrections. The bound-state energies and wave functions for the harmonic and quartic potentials are compared with those found by applying the variational approach utilizing the universal Gaussian basis. It is shown that the classical oscillation theorem, valid for systems with quadratic energy-momentum dispersion, breaks down in the classically forbidden region where wave functions also have nodes, while it still remains valid in the classically allowed region. These results are confirmed in addition via the solutions of the exactly solvable problem of the fourth-order Schrodinger equation with a square well potential.

We develop a phase reduction method for reaction-diffusion systems with a discrete delay. On the basis of the recent developments in the phase reduction theory for infinite-dimensional systems, we introduce a bilinear form tailored to spatially extended systems involving a discrete delay. By solving the adjoint equation associated with the bilinear form, we obtain the phase sensitivity function, which quantifies the shift of the phase in response to a given perturbation. The theory is verified numerically with the use of the Schnakenberg system with a discrete delay in one spatial dimension. We further demonstrate the utility of the theory by optimizing the interaction between a pair of the Schnakenberg systems, with the use of the phase equation, for maximizing the stability of in-phase synchronization. This study serves as a step towards establishing a theory for analyzing oscillatory systems that involve both spatial degrees of freedom and delay.

The General Coordinates Network (GCN) is NASA's time-domain and multimessenger alert system. GCN distributes two data products: automated "Notices" and human-generated "Circulars" that report the observations of high-energy and multimessenger astronomical transients. The flexible and nonstructured format of GCN Circulars, comprising more than 40,500 Circulars accumulated over three decades, makes it challenging to manually extract observational information, such as redshift or observed wave bands. In this work, we employ large language models (LLMs) to facilitate the automated parsing of transient reports. We develop a neural topic modeling pipeline with open-source tools for the automatic clustering and summarization of astrophysical topics in the Circulars archive. Using neural topic modeling and contrastive fine-tuning, we classify Circulars based on their observation wave bands and messengers. Additionally, we separate gravitational-wave event clusters and their electromagnetic counterparts from the Circulars archive. Finally, using the open-source Mistral model, we implement a system to automatically extract gamma-ray burst (GRB) redshift information from the Circulars archive, without the need for any training. Evaluation against the manually curated Neil Gehrels Swift Observatory GRB table shows that our simple system, with the help of prompt-tuning, output parsing, and retrieval augmented generation (RAG), can achieve an accuracy of 97.2% for redshift-containing Circulars. Our neural search-enhanced RAG pipeline accurately retrieved 96.8% of redshift Circulars from the manually curated archive. Our study demonstrates the potential of LLMs to automate and enhance astronomical text mining and provides a foundational work for future advances in transient alert analysis.

Starting from the purely microscopic model, we go beyond conventional mean-field theory and develop a self-consistent microscopic thermodynamic framework for disordered 2D superconductors. It incorporates the fermionic Bogoliubov quasiparticles, bosonic Nambu-Goldstone (NG) quantum and thermal phase fluctuations in the presence of long-range Coulomb interactions, and topological Berezinskii-Kosterlitz-Thouless (BKT) vortex-antivortex fluctuations on an equal footing, to self-consistently treat the superconducting gap and superfluid density. This unified phase-fluctuating description naturally recovers the previously known limiting results: the superconducting gap in the 2D limit can remain robust against long-wavelength NG phase fluctuations at $T=0^+$ due to Coulomb-induced regularization, while the gradual proliferation of BKT fluctuations as the system approaches criticality drives a separation between the global superconducting transition temperature $T_c$ and the gap-closing temperature $T^*$. In contrast to mean-field theory, which predicts 2D superconductivity to be independent of carrier density and non-magnetic disorder (Anderson theorem), the incorporation of phase fluctuations generates a density- and disorder-dependent zero-point gap $Δ(0)$ and consequently $T_c$ and $T^*$. Remarkably, applications to bilayer MoS$_2$ [Nat. Nanotechnol. 14, 1123 (2019)] and disordered InO$_x$ thin films [Nat. Phys. 21, 104 (2025)] quantitatively reproduce key experimental observations in excellent agreement. The framework offers a useful theoretical tool for understanding phase-fluctuation-dominated superconductivity.

The discovery of topological phases has ushered in a new era of condensed matter physics and revealed a variety of natural and artificial materials. They obey the bulk-boundary correspondence (BBC), which guarantees the emergence of boundary states with nonzero topological invariants in the bulk. Widespread attention has been paid to extending topological phases to nonlinear and non-Hermitian systems. However, the BBC and topological invariants of non-Hermitian nonlinear systems remain largely unexplored. Here, we establish a complete BBC and topological characterization of the topological phases in a class of non-Hermitian nonlinear-eigenvalue systems by introducing an auxiliary system. We restore the BBC broken by non-Hermiticity via employing the generalized Brillouin zone on the auxiliary system. Remarkably, we discover that the interplay between non-Hermiticity and nonlinearity creates an exotic complex-band topological phase that coexists with the real-band topological phase. Our results enrich the family of nonlinear topological phases and lay a foundation for exploring novel topological physics in metamaterial systems.

Data is found everywhere, from health and human infrastructure to the surge of sensors and the proliferation of internet-connected devices. To meet this challenge, the data engineering field has expanded significantly in recent years in both research and industry. Traditionally, data engineering, Machine Learning, and AI workloads have been run on large clusters within data center environments, requiring substantial investment in hardware and maintenance. With the rise of the public cloud, it is now possible to run large applications across nodes without owning or maintaining hardware. Serverless functions such as AWS Lambda provide horizontal scaling and precise billing without the hassle of managing traditional cloud infrastructure. However, when processing large datasets, users often rely on external storage options that are significantly slower than direct communication typical of HPC clusters. We introduce Cylon, a high-performance distributed data frame solution that has shown promising results for data processing using Python. We describe how we took inspiration from the FMI library and designed a serverless communicator to tackle communication and performance issues associated with serverless functions. With our design, we demonstrate that the scaling efficiency of AWS Lambda achieves within 6.5% of serverful AWS (EC2) at 64 nodes, based on implementing direct communication via NAT Traversal TCP Hole Punching.

Complex multilayer network datasets have become ubiquitous in various applications, including neuroscience, social sciences, economics, and genetics. Notable examples include brain connectivity networks collected across multiple patients or trade networks between countries collected across multiple goods. Existing statistical approaches to such data typically focus on modeling the structure shared by all networks; some go further by accounting for individual, layer-specific variation. However, real-world multilayer networks often exhibit additional patterns shared only within certain subsets of layers, which can represent treatment and control groups, or patients grouped by a specific trait. Identifying these group-level structures can uncover systematic differences between groups of networks and influence many downstream tasks, such as testing and low-dimensional visualization. To address this gap, we introduce the GroupMultiNeSS model, which enables the simultaneous extraction of shared, group-specific, and individual latent structures from a sample of networks on a shared node set. For this model, we establish identifiability, develop a fitting procedure using convex optimization in combination with a nuclear norm penalty, and prove a guarantee of recovery for the latent positions as long as there is sufficient separation between the shared, group-specific, and individual latent subspaces. We compare the model with MultiNeSS and other models for multiplex networks in various synthetic scenarios and observe an apparent improvement in the modeling accuracy when the group component is accounted for. Experiment with the Parkinson's disease brain connectivity dataset demonstrates the superiority of GroupMultiNeSS in highlighting node-level insights on biological differences between the treatment and control patient groups.

Dipoles in triangular optical ladders constitute a flexible platform for the study of the interplay between geometric frustration and long-range anisotropic interactions, and in particular for the observation of the spontaneous onset of chirality. Frustration magnifies the effect of the dipolar interactions in itinerant polarized dipolar bosons. As a result, the dipole-induced transition between a chiral superfluid and a non-chiral two-component superfluid may be observed for current state-of-the-art temperatures even for the weak inter-site interaction characterizing magnetic atoms in standard optical lattices. On the other hand, pinned spin-$1/2$ dipoles, which we discuss in the context of polar molecules in two rotational states, realize frustrated dipolar XXZ spin models. By controlling the external electric field strength and orientation, these systems can explore a rich ground-state landscape including chiral and nematic phases, as well as intriguing chiral dynamics.

Velocity and temperature distributions are both crucial for modeling compressible wall-bounded turbulent flows. The compressible law of the wall for velocity has been extensively examined through velocity transformations. However, a well-established temperature transformation remains an open issue. We propose new Van Driest type (VD-type) and semi-local type (SL-type) temperature transformation for compressible turbulent channel flow. Our approach is based on an analysis of the momentum and energy balance equations in the overlap layer. It accounts for the influences of mixing length model, the work of the body force, and the turbulent kinetic energy (TKE) flux. The proposed transformations are evaluated using data from direct numerical simulations and wall-resolved large eddy simulations of compressible turbulent channel flow. The SL-type transformation provides better data collapse than the VD-type in the viscous sublayer and buffer layer. With a suitable mixing length model, the SL-type transformed temperature agrees well with the incompressible temperature profile or the extended law of the wall. For the isothermal wall, the integral mean error over the entire boundary layer remains below 2% for most cases, with root mean square value of about 1.7%. The results highlight the importance of mitigating the energy imbalance in the transformation. This work identifies the multi-layer structure of the turbulent TKE flux, which in turn enables approximate models and corresponding simplified yet effective temperature transformations. Applications of the proposed approach in near-wall modeling and inverse transformation, as well as its potential extension to more general configurations, are also discussed.

This paper focuses on decentralized composite optimization over networks without a central coordinator. We propose a novel decentralized symmetric ADMM algorithm that incorporates multiple communication rounds within each iteration, derived from a new constraint formulation that enables information exchange beyond immediate neighbors. While increasing per-iteration communication, our approach significantly reduces the total number of iterations and overall com- munication cost. We further design optimal communication rules that minimize the number of rounds and variables transmitted per iteration. The proposed algorithm is shown to achieve linear convergence under standard and relatively weak assumptions (e.g., metric subregularity). Extensive experiments on regression and classification tasks validate the theoretical results and demonstrate superior performance compared to existing decentralized optimization methods.

Quasiparticle hybridization remains a major challenge to realizing and controlling exotic states of matter in existing quantum simulation platforms. We report the absence of hybridization for compact localized states (CLS) emerging in the chiral spin liquid described by the Yao-Kivelson model. The CLS form due to destructive quantum interference at fine-tuned coupling constants and populate perfectly flat quasiparticle bands on an effective kagome lattice. Using a formalism for general Majorana-hopping Hamiltonians, we derive exact expressions for CLS for various flux configurations and both for the topological and trivial phases of the model. In addition to finite-energy matter fermions with characteristic spin-spin correlations, we construct compact localized Majorana zero modes attached to $π$-flux excitations, which enable non-Abelian braiding of Ising anyons with minimal separation. Our results inform the quantum simulation of topologically ordered states of matter and open avenues for exploring flat-band physics in quantum spin liquids.

We develop pivotal and spherical versions of graded extension theory. We define the corresponding analogues of Brauer-Picard $2$-categorical groups and realize them as fixed points of natural $\mathbb{Z}$ and $\mathbb{Z}/2\mathbb{Z}$ $2$-categorical actions. We classify graded extensions of a pivotal tensor category by monoidal $2$-functors into the pivotal Brauer-Picard $2$-categorical group. A similar statement is proven for spherical (unimodular) tensor categories. We also develop an obstruction theory for determining when pivotal and spherical structures can be extended.

Mixed-state phase transitions have recently attracted growing attention as a new frontier in nonequilibrium quantum matter and quantum information. In this work, we introduce the measurement-dressed imaginary-time evolution (MDITE) as a novel framework to explore mixed-state quantum phases and decoherence-driven criticality. In this setup, alternating imaginary-time evolution and projective measurements generate a competition between coherence-restoring dynamics and decoherence-inducing events. While reminiscent of monitored unitary circuits, MDITE fundamentally differs in that the physics is encoded in decoherent mixed states rather than in quantum trajectories. Using numerical simulations of the one-dimensional transverse-field Ising model and the two-dimensional columnar dimerized Heisenberg model, we demonstrate the existence of this kind of mixed-state phase transitions. Notably, these transitions appear to exhibit critical behavior inconsistent with known universality classes. In addition, we provide a diagrammatic representation of the evolving state, which naturally enables efficient studies of MDITE with quantum Monte Carlo and other many-body numerical methods, thereby extending investigations of mixed-state phase transitions to large-scale and higher-dimensional systems. Our results establish MDITE as a versatile platform for investigating mixed-state criticality and uncover new classes of decoherence-driven nonequilibrium phase transitions.

The Hawking radiation spectrum from a collapsing null shell can be derived via the double copy of a simpler gauge theory calculation. Analyzing the non-abelian Yang-Mills root of this process, we demonstrate that the radiation spectrum is thermal in the color charge eigenvalue $λ$, not energy. Considering the $SU(N_c)$ gauge theory in the large $N_c$ limit, we find the differential spectrum $dN/ dλ$ is a product of the gravitationally familiar Planck-like factor and the color phase space density, modeled here as the Wigner semicircle from random matrix theory. This reveals that apparent energy thermality in gravity is the direct dual of charge thermality in its underlying non-abelian gauge theory.

The detection of primordial gravitational waves would offer a direct evidence of inflation and valuable insights into the dynamics of the early universe. During post-inflation reheating period, when the inflaton coherently oscillates at the bottom of its potential, primordial stochastic gravitational waves may be sourced by its perturbative decay into particles of different spins. Assuming the behavior of the potential near the minimum as a polynomial $V(ϕ)\sim ϕ^k$, where $k\ge 2$, and treating the inflaton as coherently oscillating classical field, we calculate the decay of inflaton into a pair of spin $3/2$ particles accompanied by graviton emission. We numerically study the reheating dynamics and calculate the stochastic gravitational wave spectra. Our analysis shows that the gravitational wave spectra can offer insights into the microscopic physics during inflation.

Thermally activated delayed fluorescence (TADF) emitters are essential for next-generation, high-efficiency organic light-emitting diodes (OLEDs), yet their rational design is hampered by the high computational cost of accurate excited-state predictions. Here, we present a comprehensive benchmark study validating semi-empirical extended tight-binding (xTB) methods -- specifically sTDA-xTB and sTD-DFT-xTB -- for the high-throughput screening of TADF materials. Using an unprecedentedly large dataset of \num{747} experimentally characterized emitters, our framework demonstrates a computational cost reduction of over \qty{99}{\percent} compared to conventional TD-DFT, while maintaining strong internal consistency between methods (Pearson $r \approx \num{0.82}$ for \deltaest), validating their utility for relative molecular ranking. Validation against \num{312} experimental \deltaest values reveals a mean absolute error of approximately \qty{0.17}{\electronvolt}, a discrepancy attributed to the vertical approximation inherent to the HTS protocol, underscoring the methods' role in screening rather than quantitative prediction. Through large-scale data analysis, we statistically validate key design principles, confirming the superior performance of Donor-Acceptor-Donor (D-A-D) architectures and identifying an optimal D-A torsional angle range of \qtyrange{50}{90}{\degree} for efficient TADF. Principal Component Analysis reveals that the complex property space is fundamentally low-dimensional, with three components capturing nearly \qty{90}{\percent} of the variance. This work establishes these semi-empirical methods as powerful, cost-effective tools for accelerating TADF discovery and provides a robust set of data-driven design rules and methodological guidelines for the computational materials science community.

Trigonometric formulas for eigenvalues of $3 \times 3$ matrices that build on Cardano's and Viète's work on algebraic solutions of the cubic are numerically unstable for matrices with repeated eigenvalues. This work presents numerically stable, closed-form evaluation of eigenvalues of real, diagonalizable $3 \times 3$ matrices via four invariants: the trace $I_1$, the deviatoric invariants $J_2$ and $J_3$, and the discriminant $Δ$. We analyze the conditioning of these invariants and derive tight forward error bounds. For $J_2$ we propose an algorithm and prove its accuracy. We benchmark all invariants and the resulting eigenvalue formulas, relating observed forward errors to the derived bounds. In particular, we show that, for the special case of matrices with a well-conditioned eigenbasis, the newly proposed algorithms have errors within the forward stability bounds. Performance benchmarks show that the proposed algorithm is approximately ten times faster than the highly optimized LAPACK library for a challenging test case, while maintaining comparable accuracy.

We establish nonuniqueness of solutions for Cauchy problems of semilinear heat equations with a wide class of nonlinearities. Specifically, we consider \[ \begin{cases} \partial_tu-Δu=f(u), & x\in\mathbb{R}^N,\ t>0,\\ u(x,0)=u_0(x), & x\in\mathbb{R}^N, \end{cases} \] where $N>2$. We assume that the growth rate of $f$ is less than the Joseph-Lundgren exponent for $N>10$ and it satisfies certain assumptions guaranteeing a positive radial singular stationary solution $u^*$. We prove that if $u_0=u^*$, then the problem has at least two positive solutions, namely $u^*$ and $u(t)$ which satisfies $u(t)\in L_{loc}^{\infty}(0,t_0;L^{\infty}(\mathbb{R}^N))$ for some $t_0>0$ and $$ u(t)\to u^*\quad\text{in}\ L^γ_{ul}(\mathbb{R}^N)\quad\text{as}\ t\to 0^+ $$ for $1\le γ<N(p_f-1)/2$, where $p_f:=\lim_{u\to\infty}uf'(u)/f(u)$ is a growth rate of $f$. Hence, nonuniqueness problem can be reduced to the existence problem of a positive radial singular stationary solution. The method of construction of $u(t)$ is based on the monotonicity argument. Transformations of forward self-similar solutions for $f(u)=u^p$ and $e^u$ play a crucial role.

Recently, Ruffini et al. [Phys. Rev. Lett. 134 (2025) 8, 081403] pointed out that the repetitive Penrose process cannot drain the entire extractable energy of a Kerr black hole. In this Letter, we alternatively point out the charge of a Reissner-Nordström black hole cannot drop down to exactly zero via the repetitive electric Penrose process that is terminated after a finite number of iterative steps, indicating a new thermodynamical third-law analog for the repetitive electric Penrose process.

We investigate a class of spatially covariant vector field theories on a flat background, where the Lagrangians are constructed as polynomials of first-order derivatives of the vector field. Because Lorentz and $\mathrm{U}(1)$ invariances are broken, such theories generally propagate three degrees of freedom (DOFs): two transverse modes and one longitudinal mode. We examine the conditions under which the additional longitudinal mode is eliminated so that only two DOFs remain. To this end, we perform a Hamiltonian constraint analysis and identify two necessary and sufficient degeneracy conditions that reduce the number of DOFs from three to two. We find three classes of solutions satisfying these degeneracy conditions, corresponding to distinct types of theories. Type-I theories possess one first-class and two second-class constraints, type-II theories have four second-class constraints, and type-III theories contain two first-class constraints. The Maxwell theory is recovered as a special case of the type-III theories, where Lorentz symmetry is restored.