The integration of quantum computers within classical High-Performance Computing (HPC) infrastructures is receiving increasing attention, with the former expected to serve as accelerators for specific computational tasks. However, combining HPC and quantum computers presents significant technical challenges, including resource allocation. This paper presents a novel malleability-based approach, alongside a workflow-based strategy, to optimize resource utilization in hybrid HPC-quantum workloads. With both these approaches, we can release classical resources when computations are offloaded to the quantum computer and reallocate them once quantum processing is complete. Our experiments with a hybrid HPC-quantum use case show the benefits of dynamic allocation, highlighting the potential of those solutions.

Circadian rhythms in living organisms are temporal orders emerging from biochemical circuits driven out of equilibrium. Here, considering the KaiABC system, a minimal model in the synthetic biology, we study how the oscillation emerges from the circuit made of three Kai proteins and ATP alone. The phase diagram constructed in terms of KaiC and KaiA concentrations reveals a narrowly bounded oscillatory phase, which naturally explains arrhythmia upon protein over-expression. As dictated by the cost-precision trade-offs of the thermodynamic uncertainty relations, the presence of intrinsic noise, amplified in small systems, demands higher free energy cost to achieve greater rhythmic precision. The cost-minimizing condition within the oscillatory phase is found to generate $\sim$21-hr rhythm, which is entrained to 24-hr environmental signals as long as the forcing amplitude is greater than $\sim 10$ \% of the metabolic rate. An optimal level of intrinsic noise can also induce oscillations even beyond the Hopf bifurcation, effectively expanding the oscillatory phase. Our study clarifies how the physical factors, such as regulatory mechanism, energy cost, and stochastic noise contribute to the operation of biological clocks.

Consider costly and time-consuming tasks that add up to the success of a project, and must be fitted into a given time-frame. This is an instance of the classic budgeted maximization (knapsack) problem, which admits an FPTAS. Now assume an agent is performing these tasks on behalf of a principal, who is the one to reap the rewards if the project succeeds. The principal must design a contract to incentivize the agent. Is there still an approximation scheme? In this work we lay the foundations for an algorithm-to-contract framework, which transforms algorithms for combinatorial problems to handle contract design problems subject to the same combinatorial constraints. Our approach diverges from previous works in avoiding the assumption of demand oracle access. As an example, for budgeted maximization, we show how to "lift" the classic FPTAS to the best-possible (approximately-IC) FPTAS for the contract problem. We establish this through our local-to-global framework, in which the local step is to approximately solve a two-sided strengthened variant of the demand problem. The global step then utilizes the local one to find the approximately optimal contract. We apply our framework to a host of combinatorial constraints: multi-dimensional budgets, budgeted matroid, and budgeted matching constraints. In all cases we essentially match the best purely algorithmic approximation. Separately, we also develop a method for multi-agent contract settings. Our method yields the first approximation schemes for multi-agent contract settings that go beyond additive reward functions.

We study a twisted version of Fraser, Lam, and Le's higher boundary measurement map, using face weights instead of edge weights, thereby providing Laurent polynomial expansions, in Plücker coordinates, for twisted web immanants for Grassmannians. In some small cases, Fraser, Lam, and Le observe a phenomenon they call "web duality'', where web immanants coincide with web invariants, and they conjecture that this duality corresponds to transposing the standard Young tableaux that index basis webs. We show that this duality continues to hold for a large set of $\text{SL}_3$ and $\text{SL}_4$ webs. Combining this with our twisted higher boundary measurement map, we recover and extend formulas of Elkin-Musiker-Wright for twists of certain cluster variables. We also provide evidence supporting conjectures of Fomin-Pylyavskyy as well as one by Cheung-Dechant-He-Heyes-Hirst-Li concerning classification of cluster variables of low Plücker degree in $\mathbb{C}[\text{Gr}(3,n)]$.

In high-energy heavy-ion collisions, a nearly perfect fluid is formed, known as the strongly coupled quark-gluon plasma (QGP). After a short thermalization period, the evolution of this medium can be described by the equations of relativistic hydrodynamics. As the system expands and cools, the QGP undergoes a transition into hadronic matter, marking the onset of quark confinement. Direct photons offer insights into an essential stage of evolution, spanning from the onset of thermalization to the suppression of thermal photon production, which occurs within the hadronic phase. This paper builds upon and extends a previously published solution of relativistic hydrodynamics, incorporating an equation of state that falls within the same class as that predicted by lattice QCD. Based on this solution, a completely analytic model is constructed to describe thermal photon production, accounting for the quark-hadron transition. The model is tested against PHENIX measurements of non-prompt direct photon spectra in Au+Au collisions at $\sqrt{s_{NN}} = 200$ GeV. Good agreement is observed between the model predictions and the experimental data, enabling the investigation of the centrality dependence of the initial temperature. These results provide a benchmark for future theoretical and experimental studies of thermal radiation in heavy-ion collisions.

We set up a bootstrap approach to compute the first stringy correction to the supergravity regime of the correlation function of five 20' operators in $\mathcal{N}=4$ super Yang-Mills. We use factorization of Mellin amplitudes, supersymmetric constraints and protected observables to refine our ansatz, leaving only a single undetermined coefficient. We identify non-protected CFT data that is sensitive to the residual ambiguity of our ansatz, and we verify the compatibility of our result with the anticipated behaviour of the Mellin amplitude in the flat-space limit. As a byproduct of our analysis, we also compute the first stringy correction to the four-point correlators of three 20' operators and either one R-symmetry current or one stress tensor.

We design a debiased parametric bootstrap framework for statistical inference from differentially private data. Existing usage of the parametric bootstrap on privatized data ignored or avoided handling possible biases introduced by the privacy mechanism, such as by clamping, a technique employed by the majority of privacy mechanisms. Ignoring these biases leads to under-coverage of confidence intervals and miscalibrated type I errors of hypothesis tests, due to the inconsistency of parameter estimates based on the privatized data. We propose using the indirect inference method to estimate the parameter values consistently, and we use the improved estimator in parametric bootstrap for inference. To implement the indirect estimator, we present a novel simulation-based, adaptive approach along with the theory that establishes the consistency of the corresponding parametric bootstrap estimates, confidence intervals, and hypothesis tests. In particular, we prove that our adaptive indirect estimator achieves the minimum asymptotic variance among all ``well-behaved'' consistent estimators based on the released summary statistic. Our simulation studies show that our framework produces confidence intervals with well-calibrated coverage and performs hypothesis testing with the correct type I error, giving state-of-the-art performance for inference in several settings.

Efficient, low-power, and highly integrated optoelectronic devices remain a critical yet challenging goal.Here, we introduce the extended Kerker effect paradigm that synergizes Kerker's condition with quasi-bound states in the continuum (q-BICs) to overcome these limitations. By engineering dual-mode dispersion, we achieve a high efficiency beam deflector using a membrane metasurface, simultaneously realizing robust parameter tolerance and narrow-linewidth resonances-two typically conflicting properties.Our experiment demonstrates an absolute beam deflection efficiency exceeding 92%, with exceptional spectral and spatial selectivity, including a 4 GHz linewidth, a 2.8o divergence angle, and a quality factor of 114. Additionally, it enables 94% transmission intensity modulation at a pump intensity as low as 0.5 W/cm2 in experiments. The extended Kerker effect provides a scalable platform for energy-efficient and integrable optoelectronic devices, paving the way for transformative advancements in next-generation wireless communications and LiDAR.

We prove a Schwarz-Jack lemma for holomorphic functions on the unit disk with the property that their maximum modulus on each circle about the origin is attained at a point on the positive real axis. With the help of this result, we establish monotonicity and convexity properties of conformal maps of circularly symmetric and bi-circularly symmetric domains. As an application, we give a new proof of Crouzeix's theorem that the numerical range of any $2\times 2$ matrix is a $2$-spectral set for the matrix. Unlike other proofs, our approach does not depend on the explicit formula for the conformal mapping of an ellipse onto the unit disk.

Neutral triple gauge couplings (nTGCs) first arise from the dimension-8 operators of the Standard Model Effective Field Theory (SMEFT), rather than the dimension-4 SM Lagrangian and dimension-6 SMEFT operators, opening up a unique window for probing new physics at the dimension-8 level. In this work, we formulate the nTGC form factors of $ZZV^*$ ($V\!\!=\!Z,γ$) that are compatible with the spontaneous breaking of the SU(2)$\otimes$U(1) electroweak gauge symmetry and consistently match the dimension-8 nTGC operators in the broken phase. We study the sensitivities for probing both the $ZZV^*$ form factors and the corresponding new physics scales through $ZZ$ production (with visible/invisible fermionic $Z$ decays) at high energy $e^+e^-$ colliders including CEPC, FCC-ee, ILC and CLIC. In particular, we identify the dimension-8 operator that contributes to the pure triple $Z$ boson coupling $ZZZ^*$ alone, but not the mixed $ZZγ^*$ coupling. We further study the correlations between probes of the $ZZZ^*$ and $ZZγ^*$ couplings. Using machine learning, we show that angular distributions of the final-state fermions can play key roles in suppressing the SM backgrounds. The sensitivities can be further improved by using polarized $e^\mp$ beams. We demonstrate that machine learning is advantageous for handling the 4-body final states from $ZZ$ decays and improves significantly the sensitivity reaches of probes of nTGCs in $e^+e^-$ collisions. We find that nTGC new physics scales can be probed up to the multi-TeV scale at the proposed $e^+e^-$ colliders.

The aim of this article is to give a complete proof of results of Harish-Chandra linking the irreducibility of parabolic induction of a supercuspidal representation of a p-adic group to the analytic behavior of the mu-function of Harish-Chandra and to show that the proof remains valid in the case of a central extension.M

Shuttling of spin qubits between different locations is a key element in many prospective semiconductor systems for quantum information processing, but the shuttled qubits should be protected from decoherence created by time- and space-dependent noises. Since the paths of different spin qubits are interrelated, the noises acting on the shuttled spins exhibit complex and unusual correlations. We appraise the role of these correlations using the concept of trajectories on random sheets, and demonstrate that they can drastically affect efficiency of the coherence protection. These correlations can also be exploited to enhance the shuttling fidelity, and we show that by encoding logical qubit in a state of two consequtively shuttled entangled spins, high fidelity can be achieved even for very slow shuttling. We identify the conditions favoring this encoding, and quantify improvement in the shuttling fidelity in comparison with the single-spin shuttling.

The multiconfiguration self-consistent field (MCSCF) method is revisited with a specific focus on two-electron systems for simplicity. The wave function is represented as a linear combination of Slater determinants. Both the orbitals and the coefficients of this configuration interaction expansion are optimized according to the variational principle within the Lagrangian formalism, using a Newton optimization scheme. This reduces the MCSCF problem to solving a particular differential Newton system, which can be discretized with multiwavelets and solved iteratively.

A method for the numerical simulation of signed probability distributions for the case of tossing $1/n$-th of a coin is presented and illustrated by examples.

To date, realization of a continuous-wave active atomic clock has been elusive primarily due to parasitic heating from spontaneous emission while repumping the atoms. Here, we propose a solution to this problem by replacing the random emission with coupling to an auxiliary cavity, making repumping a fully collective process. While it is known that collective two-level models do not possess a generic lasing threshold, we show this restriction is overcome with multi-level atoms since collective pumping and decay can be performed on distinct transitions. Using relevant atomic parameters, we find this system is capable of producing an $\mathcal{O}$(100 $μ$Hz)-linewidth continuous-wave superradiant laser. Our principal result is the potential for an operating regime with cavity length vibration sensitivity below $\mathcal{O}(10^{-14} / g)$, including a locus of parameter values where it completely vanishes even at steady-state.

We state necessary and sufficient conditions for the existence of $T$-discrete exponential attractors for semigroups in complete metric spaces. These conditions are formulated in terms of a covering condition for iterates of the absorbing set under the time evolution of the semigroup and imply the existence and finite-dimensionality of the global attractor. We then review, generalize and compare existing construction methods for exponential attractors and show that they all imply the covering condition. Furthermore, we relate the results and concept of $T$-discrete exponential attractors to the classical notion of exponential attractors.

We develop parallel algorithms for simulating zeroth-order (aka gradient-free) Metropolis Markov chains based on the Picard map. For Random Walk Metropolis Markov chains targeting log-concave distributions $π$ on $\mathbb{R}^d$, our algorithm generates samples close to $π$ in $\mathcal{O}(\sqrt{d})$ parallel iterations with $\mathcal{O}(\sqrt{d})$ processors, therefore speeding up the convergence of the corresponding sequential implementation by a factor $\sqrt{d}$. Furthermore, a modification of our algorithm generates samples from an approximate measure $ π_r$ in $\mathcal{O}(1)$ parallel iterations and $\mathcal{O}(d)$ processors. We empirically assess the performance of the proposed algorithms in high-dimensional regression problems, an epidemic model where the gradient is unavailable and a real-word application in precision medicine. Our algorithms are straightforward to implement and may constitute a useful tool for practitioners seeking to sample from a prescribed distribution $π$ using only point-wise evaluations of $\logπ$ and parallel computing.

Large language models (LLMs) are increasingly used to generate labels from radiology reports to enable large-scale AI evaluation. However, label noise from LLMs can introduce bias into performance estimates, especially under varying disease prevalence and model quality. This study quantifies how LLM labeling errors impact downstream diagnostic model evaluation. We developed a simulation framework to assess how LLM label errors affect observed model performance. A synthetic dataset of 10,000 cases was generated across different prevalence levels. LLM sensitivity and specificity were varied independently between 90% and 100%. We simulated diagnostic models with true sensitivity and specificity ranging from 90% to 100%. Observed performance was computed using LLM-generated labels as the reference. We derived analytical performance bounds and ran 5,000 Monte Carlo trials per condition to estimate empirical uncertainty. Observed performance was highly sensitive to LLM label quality, with bias strongly influenced by disease prevalence. In low-prevalence settings, small reductions in LLM specificity led to substantial underestimation of sensitivity. For example, at 10% prevalence, an LLM with 95% specificity yielded an observed sensitivity of ~53% despite a perfect model. In high-prevalence scenarios, reduced LLM sensitivity caused underestimation of model specificity. Monte Carlo simulations consistently revealed downward bias, with observed performance often falling below true values even when within theoretical bounds. LLM-generated labels can introduce systematic, prevalence-dependent bias into model evaluation. Specificity is more critical in low-prevalence tasks, while sensitivity dominates in high-prevalence settings. These findings highlight the importance of prevalence-aware prompt design and error characterization when using LLMs for post-deployment model assessment in clinical AI.

Let $p$ be a prime number. For a field $F$ containing a root of unity of order $p$, let $H^\bullet(F)=H^\bullet(F,\mathbb{F}_p)$ be the mod-$p$ Galois cohomology graded $\mathbb{F}_p$-algebra of $F$. By the Norm Residue Theorem, $H^\bullet(F)$ is a purely quadratic graded-commutative algebra, and is therefore determined by the cup product $\cup\colon H^1(F)\times H^1(F)\to H^2(F)$. We prove that the class of all Galois cohomology algebras $H^\bullet(F)$ is cofinal in the class of all purely quadratic graded-commutative $\mathbb{F}_p$-algebras $A_\bullet$, in the following sense: For every $A_\bullet$ there exists $F$ such that the bilinear map $A_1\times A_1\to A_2$, which determines $A_\bullet$, embeds in the cup product bilinear map $\cup\colon H^1(F)\times H^1(F)\to H^2(F)$. We further provide examples of $\mathbb{F}_p$-bilinear maps which are not realizable by fields $F$ in this way. These are related to recent results by Snopce-Zalesskii and Blumer-Quadrelli-Weigel on the Galois theory of pro-$p$ right-angled Artin groups, as well as to a conjecture by Marshall on the possible axiomatization of quadratic form theory of fields.

The speed-up of parameter estimation is an active field of research in gravitational-wave data analysis. In this paper we present GP15, a deep-learning method that merges residual networks and normalizing flows into a general-purpose, image-based estimator of binary black hole (BBH) parameters. Building on our early work, we map BBH spectrograms from the Advanced LIGO and Advanced Virgo detectors to color channels in an RGB image amenable to be processed with residual networks. GP15 is trained on simulated data for BBH mergers obtained with the \texttt{IMRPhenomXPHM} waveform approximant and tested for all three-detector events from the GWTC-3 and GWTC-2.1 catalogs reported by the LIGO-Virgo-KAGRA (LVK) collaboration. Overall, our model yields good agreement with the LVK results over most parameters. Our simple model can produce large amounts of posterior samples in the order of a second, complementing existing approaches with normalizing flows based on time or frequency representation of gravitational-wave data. We also discuss current shortcomings of our model and possible improvements for future extensions (e.g. including noise conditioning from the detectors' PSD or splitting the parameter space into intrinsic and extrinsic subspaces).

One approach to analyzing entanglement in a gauge theory is embedding it into a factorized theory with edge modes on the entangling boundary. For topological quantum field theories (TQFT), this naturally leads to factorizing a TQFT by adding local edge modes associated with the corresponding CFT. In this work, we instead construct a minimal set of edge modes compatible with the topological invariance of Chern-Simons theory. This leads us to propose a minimal factorization map. These minimal edge modes can be interpreted as the degrees of freedom of a particle on a quantum group. Of particular interest is three-dimensional gravity as a Chern-Simons theory with gauge group SL$(2,\mathbb{R}) \times$ SL$(2,\mathbb{R})$. Our minimal factorization proposal uniquely gives rise to quantum group edge modes factorizing the bulk state space of 3d gravity. This agrees with earlier proposals that relate the Bekenstein-Hawking entropy in 3d gravity to topological entanglement entropy.

We study the CP-violating (CPV) neutral triple gauge couplings (nTGCs) that can be realized via dimension-8 operators in the Standard Model Effective Field Theory (SMEFT). We present a new formulation of the CPV nTGC form factors that is compatible with spontaneous breaking of the electroweak gauge symmetry, and show how these CPV form factors can be matched consistently with the corresponding dimension-8 CPV nTGC operators in the broken phase. We then study probes of the CPV nTGCs at future high-energy $e^+e^-$ colliders with centre-of-mass energies $\sqrt{s}=(0.25, 0.5, 1, 3, 5)$TeV respectively, demonstrating that the $e^{\mp}$ beam polarizations can help to improve the sensitivities of probes of the nTGCs. We estimate that the sensitivity reaches for probing the new physics scales of nTGCs can range from ${O}(\rm{TeV})$ at a 250GeV $e^+e^-$ collider to ${O}(10\,\rm{TeV})$ at an $e^+e^-$ collider of energy $(3-5)$TeV, and that the sensitivities to the nTGC form factors vary from ${O}(10^{-4})$ to ${O}(10^{-6}-10^{-8})$ for the $e^+e^-$ collision energy from 250GeV to $(3-5)$TeV.

We propose a minimal extension to the Standard Model by introducing a Lorentz Invariance Violation (LIV) term into the Z boson's dispersion relation, expressed as $p_μp^μ= M_Z^2 + δ_{LIV} (p_μn^μ)^2$, where $δ_{LIV}$ defines the violation scale and $n^μ$ is a unit Lorentz vector specifying the direction. This modification alters the Z boson propagator and decay rate, impacting the Drell-Yan process cross-section at high-energy colliders. Observable effects are most pronounced near the resonance region at high rapidities ($|Y| > 4$), potentially shifting the perceived Z boson mass and inducing sidereal-time modulations for spacelike and lightlike LIV due to Earth's rotation. We outline a targeted search strategy for ATLAS and CMS, achieving sensitivity to LIV signatures down to $|δ_{LIV}| \approx 10^{-8}$ (or $10^{-9}$ optimistically), offering new insights into historical and future collider data. Our model predicts systematic shifts in weak boson masses at higher collision energies, relevant to past Tevatron and LHC discrepancies, though current data are now consistent.

It is known that for a monomial ideal $I$, the number of minimal generators, $μ(I^n)$, eventually follows a polynomial pattern for increasing $n$. In general, little is known about the power at which this pattern emerges. Even less is known about the exact form of the minimal generators after this power. Let $s\ge μ(I)(d^2-1)+1$, where $d$ is a constant bounded above by the maximal $x$- or $y$-degree appearing in the set $\mathsf{G}(I)$ of minimal generators of $I$. We show that every higher power $I^{s+\ell}$ for any $\ell \ge 0$ can be constructed from certain subideals of $I^s$. This provides an explicit description of~$\mathsf{G}(I^{s+\ell})$ in terms of $\mathsf{G}(I^s)$. Given $\mathsf{G}(I^s)$, this construction significantly reduces computational complexity in determining larger powers of~$I$. This further enables us to explicitly compute $μ(I^n)$ for all $n\ge s$ in terms of a linear polynomial in $n$. We include runtime measurements for the attached implementation in SageMath.

A class $\mathcal{G}$ of graphs is hereditary if it is closed under taking induced subgraphs. We investigate the edge-add class, $\mathcal{G}^{\mathrm{add}}$, consisting of graphs that can be made members of $\mathcal{G}$ by adding at most one edge. While it is known that the operations of vertex deletion and edge deletion preserve the finiteness of forbidden induced subgraphs for classes with finite exclusions, the behavior of edge addition on classes with infinite exclusions remains largely unexplored. We characterize the edge-add class of chordal graphs by their forbidden induced subgraphs and extend the result to a general finiteness theorem: for any fixed $p\ge0$, the set of forbidden induced subgraphs for $p$-edge-add chordal graphs that are not cycles is finite. In contrast, we show that this phenomenon does not extend to perfect graphs. Furthermore, we provide explicit structural bounds proving that edge addition preserves finiteness for base classes with finitely many exclusions. We conclude by providing the complete structural characterizations and explicit minimal obstruction lists for the edge-add classes of split and threshold graphs, and generalize these results to $(p,q)$-edge split graphs.

The remarkable success of foundation models has been driven by scaling laws, demonstrating that model performance improves predictably with increased training data and model size. However, this scaling trajectory faces two critical challenges: the depletion of high-quality public data, and the prohibitive computational power required for larger models, which have been monopolized by tech giants. These two bottlenecks pose significant obstacles to the further development of AI. In this position paper, we argue that leveraging massive distributed edge devices can break through these barriers. We reveal the vast untapped potential of data and computational resources on massive edge devices, and review recent technical advancements in distributed/federated learning that make this new paradigm viable. Our analysis suggests that by collaborating on edge devices, everyone can participate in training large language models with small edge devices. This paradigm shift towards distributed training on edge has the potential to democratize AI development and foster a more inclusive AI community.

The generation of pure spin current, spin angular momentum transport without charge flow, is crucial for developing energy-efficient spintronic devices with minimal Joule heating. Here, we introduce the intrinsic nonlinear pure spin Hall effect (NPSHE), where both linear and second-order charge Hall currents vanish. We show intrinsic second-order spin angular momentum transport in metals and insulators through a detailed analysis of the quantum geometric origin of different spin current contributions. Our comprehensive symmetry analysis identifies 39 magnetic point groups that support NPSHE, providing a foundation for material design and experimental realization. We predict significant nonlinear pure spin Hall currents in Kramers-Weyl metals even at room temperature, positioning them as potential candidates for NPSHE-based spin-torque devices. Our work lays a practical pathway for realizing charge-free angular momentum transport for the development of next-generation, energy-efficient spintronic devices.

We consider a generalized Derrida-Retaux model on a Galton-Watson tree with a geometric offspring distribution. For a class of recursive systems, including the Derrida-Retaux model with either a geometric or exponential initial distribution, we characterize the critical curve using an involution-type equation and prove that the free energy satisfies the Derrida-Retaux conjecture.

In this paper, we study Bass-Serre theory from the perspectives of $C^*$-algebras and topological dynamics. In particular, we investigate the actions of fundamental groups of graphs of groups on their Bass-Serre trees and the associated boundaries, through which we identify new families of $C^*$-simple groups including certain tubular groups, fundamental groups of certain graphs of groups with one vertex group acylindrically hyperbolic and outer automorphism groups $\operatorname{Out}(BS(p, q))$ of Baumslag-Solitar groups. In addition, we study $n$-dimensional Generalized Baumslag-Solitar ($\text{GBS}_n$) groups. We first recover a result by Minasyan and Valiunas on the characterization of $C^*$-simplicity for $\text{GBS}_1$ groups and identify new $C^*$-simple $\text{GBS}_n$ groups including the Leary-Minasyan group. These $C^*$-simple groups also provide new examples of $C^*$-selfless groups and highly transitive groups. Moreover, we demonstrate that natural boundary actions of these $C^*$-simple fundamental groups of graphs of groups give rise to the new purely infinite crossed product $C^*$-algebras.

In the whole set of solutions of the Dirac equation there is a different class referred to as anomalous. Corresponding anomalous particles are independent of conventional ones. The concept of anomalous particles is applicable to Dirac insulators, where electrons obey the Dirac like wave equation. Positively charged antielectrons, which are not holes, can exist in the Dirac insulator. In this material one can create the electron-antielectron pair keeping the valence band completely filled. The anomalous subsystem, associated with the electron-antielectron vacuum, is an inner property of the Dirac insulator. Anamalous particles in the Dirac insulator can be identified in experiments with electric current.