Pair Of Modes Research Articles

Examination of genomic profile of true single circulating tumor cells (sCTC) from hepatocellular carcinoma and its effect on mutational features.

e15039 Background: Hepatocellular carcinoma (HCC) is the fifth leading cause of cancer-related deaths worldwide. Despite numerous NGS studies unveiling diverse mutational patterns through tissue biopsies, the absence of tissue for biopsy and distinct biomarkers complicates both diagnosis and treatment. The utilization of sCTC genomics holds promise in furnishing clinically relevant and actionable insights to tailor personalized diagnostics and therapy for HCC. We present a comparative analysis of sCTC and paired ctDNA obtained from primary HCC patients for better Tx possibilities. Methods: Genomic profile from 24 sCTCs/CTC clusters (19 sCTCs, 79%; 5 CTC clusters, 21 %) revealed distinct mutational patterns compared to paired ctDNA. Live sCTCs/CTC clusters were isolated from peripheral blood of treatment naïve 12 Hepatobiliary Cancer with advanced and metastatic disease patients using OncoRadar technology. Whole genomes from sCTCs/CTC clusters were amplified, target enriched with hybridization capture using a comprehensive 1080 gene panel-OncoIndx to obtain sequencing libraries, and sequenced on Illumina NextSeq2000 platform in paired end mode (depth 500x). The raw sequence alignment and variant calling was performed using iCare software. Paired ctDNA samples were sequenced the same way, except at the depth of 5500x. Results: Pathogenic (P)/likely pathogenic (LP) variants were detected from all sCTCs, while ctDNA was negative for 40% samples. sCTCs showed at least twice as many P variants compared to paired ctDNA. ATM (50%), PIK3CA (42%), ARID1A /1B (42%), TP53/RB1 (34%), and IDH1 (34%) were among the most frequently mutated genes, while 42% sCTCs showed amplified CCND1/2 variants. In comparison, paired ctDNA had no distinct mutational pattern. Genomic profile of sCTC showed concordance with paired ctDNA for at least one gene mutation when compared at patient level. At pathway level, all sCTCs were mutated for at least one HRR pathway component, while 60% sCTCs had at least one mutation in receptor tyrosine kinase (RTK) pathway (FGFR1, ERBB, ALK, ROS1). sCTC population showed a strong cooccurrence of HRR mutation and epigenetic regulatory pathway mutations (OR =4, CI=0.4-60). At individual mutation level, 30 % concordance was observed in sCTC population, suggesting high mutational heterogeneity. Besides actionable targets, 50% sCTC were enriched with resistance conferring variants such as STK11, NF1, STAT5B, and RB1. Conclusions: The genomic profile of sCTCs in primary HCC pts revealed a complex mutational landscape, offering complementarity to ctDNA for clinical insights. sCTCs exhibited more actionable targets and greater heterogeneity compared to paired ctDNA, making sCTC genomics valuable for pts ineligible for tissue biopsy or lacking clinically relevant information from ctDNA.

Study of single live circulating tumor cells capture and their genomic profile and enriched mutations of PIK3CA and HRR pathway in patients with breast cancer.

e13161 Background: Capturing live single circulating tumor cells (sCTC) for intra-cellular compressive genomic profiling and concordance with ctDNA is highly challenging. Utilization of single sCTC genomics holds promise for detecting clinically relevant, actionable insights to tailor personalized therapy, especially when No Mutations Detected (NMDs) using CfDNA. Despite numerous studies providing valuable insights on mutational landscapes from breast tumors, predicting cancer progression, metastasis, and relapse continue to pose major challenges. ctDNA is suboptimal for 30-40% population with disease relapse/progression after a primary therapy. We show comprehensive genetic profile (CGP) of live sCTCs and paired ctDNA from a diverse breast cancer population. Methods: 26 live sCTCs (2 CTC clusters) were isolated from 12 patients. Live sCTCs were isolated from 12 Breast Cancer patients using OncoRadar live sCTC capture method. Whole genomes were amplified using multiple displacement amplification, target enriched with hybridization capture using OncoIndx (CGP 1080 gene panel) to obtain sequencing libraries, and sequenced on Illumina platform in paired end mode (depth 500x). The raw sequence alignment and variant calling was performed using iCare software. Paired ctDNA samples were processed the same way, except for sequencing depth (5500x). Results: Compared to paired ctNDA, genomic profiling of sCTCs revealed complex mutational patterns. Whereas sCTCs showed least two pathogenic (P) or likely pathogenic (LP) variants, paired ctDNA was negative for 25% cases (3/12). 92% sCTCs showed mutations in the PIK3CA-MTOR pathway with PIK3CA being the most prevalent (73 %). A significant association was observed between co-occurrence of HER2 amplification and PIK3CA mutations (P =0.04, OR=10). Similarly, sCTCs showed a strong correlation between HRR pathway and PIK3CA pathway mutations (r=0.76, P =0.003, OR=3.43). Genomic profile of sCTCs and paired ctDNA showed concordance with at least one gene mutation when compared at individual patient level. Additionally, at individual mutation level, PIK3CA exon 10 mutations were most frequent (61.5%) along with BRCA2 pathway variants, followed by IDH1/2(54%), ARID1A (50%), NRAS (46%), FGFR2 (42%) and TP53 (42%). Interestingly, 70% sCTCs showed the presence of therapy resistance conferring variants (NF1, ARID1A, ESR1, RB1 and PTEN), compared to only 25 % observed in paired ctDNA. Conclusions: The genomic profile of sCTCs represents true tumor heterogeneity and can be combined with ctDNA for better clinical outcome. sCTCs exhibited high mutational heterogeneity compared to paired ctDNA. Therapy resistance signatures were more prevalent in sCTCs and may offer valuable clinical insights for patients unable to provide tissue biopsy or show negative ctDNA.

Topological phases and edge modes of an uneven ladder

Abstract We investigate the topological properties of a two-chain quantum ladder with uneven legs, i.e., the two chains differ in their periods by a factor of 2. Such an uneven ladder presents rich band structures classified by the closure of either direct or indirect bandgaps. It also provides opportunities to explore fundamental concepts concerning band topology and edge modes, including the difference of intracellular and intercellular Zak phases, and the role of the inversion symmetry (IS). We calculate the Zak phases of the two kinds and find excellent agreement with the dipole moment and extra charge accumulation. We also find that configurations with IS feature a pair of degenerate two-side edge modes emerging as the closure of the direct bandgap, while configurations without IS feature one-side edge modes emerging as not only the closure of both direct and indirect bandgaps but also within the band continuum. Furthermore, by projecting to the two sublattices, we find that the effective Bloch Hamiltonian corresponds to that of a generalized Su–Schrieffer–Heeger model or the Rice–Mele model whose hopping amplitudes depend on the quasimomentum. In this way, the topological phases can be efficiently extracted through winding numbers. We propose that uneven ladders can be realized by spin-dependent optical lattices and their rich topological characteristics can be examined by near future experiments.

Optical vectorial-mode parity Hall effect: a case study with cylindrical vector beams

The vectorial optical field (VOF) assumes a pivotal role in light-matter interactions. Beyond its inherent polarization topology, the VOF also encompasses an intrinsic degree of freedom associated with parity (even or odd), corresponding to a pair of degenerate orthogonal modes. However, previous research has not delved into the simultaneous manipulation of both even and odd parities. In this study, we introduce and validate the previously unexplored parity Hall effect for vectorial modes using a metasurface design. Our focus lies on a cylindrical vector beam (CVB) as a representative case. Through the tailored metasurface, we effectively separate two degenerate CVBs with distinct parities in divergent directions, akin to the observed spin states split in the spin Hall effect. Additionally, we provide experimental evidence showcasing the capabilities of this effect in multi-order CVB demultiplexing and parity-demultiplexed CVB-encoded holography. This effect unveils promising opportunities for various applications, including optical communication and imaging.

Fluid–structure–surface interaction of a flexibly mounted pitching and plunging flat plate in proximity to the free surface

This experimental study investigates the fluid–structure–surface interactions of a flexibly mounted rigid plate in axial flow, focusing on flow-induced vibration (FIV) response and vortex dynamics of the system within a reduced velocity range of $U^*=0.29\unicode{x2013}8.73$ , corresponding to a Reynolds number range of $Re=518\unicode{x2013}15\,331$ . The plate, with one and two degrees of freedom (DoFs) for pitching and plunging oscillations, is examined at various submerged heights near the free surface. Results show that the plate exhibits divergence instability at low reduced velocities in both 1DoF and 2DoF systems. As the flow velocity surpasses a critical reduced velocity, periodic limit-cycle oscillations (LCOs) occur, increasing in amplitude until a second critical reduced velocity is reached. Beyond this point, LCOs are suppressed, and the plate experiences an increased static divergence angle with further flow velocity increase. The proximity to the free surface significantly influences the FIV response, with decreasing submerged heights leading to reduced LCO amplitudes and a shift of instabilities to higher reduced velocities. Vortex dynamics are analysed using time-resolved volumetric particle tracking velocimetry and hydrogen bubble flow visualisation. The analysis reveals disruptions in the symmetric flow field near the free surface, causing elongation and fragmentation of vortices in the wake of the plate, as well as vortex coupling. Proper orthogonal decomposition (POD) identifies dominant coherent structures, including leading-edge and trailing-edge vortices, captured in the first and second paired modes. On the other hand, higher POD modes capture the interaction of vortices in the wake and near the free surface.

Damage assessment in L-shaped bends by nonlinear feature guided waves

L-shaped bends are crucial lightweight structural components in aerospace manufacturing, renowned for their high specific strength and stiffness. However, the geometric configuration of these bends makes them susceptible to stress concentration in the bending zone. Effective technique for damage assessment of the bending region of L-shaped bends is highly demanded. This paper proposes a nondestructive testing method using nonlinear ultrasonic feature guided waves (FGWs) for early damage assessment of L-shaped bends. Semi-analytical finite element (SAFE) method and perfectly matched layer (PML) technique are employed to analyze FGW propagation in L-shaped bends. Three-dimensional finite element models are utilized to investigate the FGWs propagation in the bending region of L-shaped bends in a numerical perspective. Cumulative second harmonic generation (SHG) of FGWs is successfully observed by selecting suitable mode pairs based on synchronism and non-zero power flux conditions, which provides a promising alternative for early damage assessment in L-shaped bends. Experimental findings are consistent with the numerical ones. Subsequently, thermal damage and plastic deformation with varying gradients were introduced into the bending region of L-shaped bends. The use of SHG of selected FGW mode has effectively assessed the damage with an improved sensitivity, compared to conventional linear acoustic one. This investigation substantiates the potential of nonlinear FGWs in detecting early damage in L-shaped bends, offering promise for future industrial testing applications.

Ultra-high Q resonances based on zero group-velocity modes accompanied by bound states in the continuum in 2D photonic crystal slabs

Optical resonators made of 2D photonic crystal (PhC) slabs provide efficient ways to manipulate light at the nanoscale through small group-velocity modes with low radiation losses. The resonant modes in periodic photonic lattices are predominantly limited by nonleaky guided modes at the boundary of the Brillouin zone below the light cone. Here, we propose a mechanism for ultra-high Q resonators based on the bound states in the continuum (BICs) above the light cone that have zero-group velocity (ZGV) at an arbitrary Bloch wavevector. By means of the mode expansion method, the construction and evolution of avoided crossings and Friedrich-Wintgen BICs are theoretically investigated at the same time. By tuning geometric parameters of the PhC slab, the coalescence of eigenfrequencies for a pair of BIC and ZGV modes is achieved, indicating that the waveguide modes are confined longitudinally by small group-velocity propagation and transversely by BICs. Using this mechanism, we engineer ultra-high Q nanoscale resonators that can significantly suppress the radiative losses, despite the operating frequencies above the light cone and the momenta at the generic k point. Our work suggests that the designed devices possess potential applications in low-threshold lasers and enhanced nonlinear effects.

Coherent Imaging with Photonic Lanterns

Photonic lanterns (PLs) are tapered waveguides that gradually transition from a multimode fiber geometry to a bundle of single-mode fibers (SMFs). They can efficiently couple multimode telescope light into a multimode fiber entrance at the focal plane and convert it into multiple single-mode beams. Thus, each SMF samples its unique mode (lantern principal mode) of the telescope light in the pupil, analogous to subapertures in aperture masking interferometry (AMI). Coherent imaging with PLs can be enabled by the interference of SMF outputs and applying phase modulation, which can be achieved using a photonic chip beam combiner at the backend (e.g., the ABCD beam combiner). In this study, we investigate the potential of coherent imaging by the interference of SMF outputs of a PL with a single telescope. We demonstrate that the visibilities that can be measured from a PL are mutual intensities incident on the pupil weighted by the cross correlation of a pair of lantern modes. From numerically simulated lantern principal modes of a 6-port PL, we find that interferometric observables using a PL behave similarly to separated-aperture visibilities for simple models on small angular scales (<λ/D) but with greater sensitivity to symmetries and capability to break phase angle degeneracies. Furthermore, we present simulated observations with wave front errors (WFEs) and compare them to AMI. Despite the redundancy caused by extended lantern principal modes, spatial filtering offers stability to WFEs. Our simulated observations suggest that PLs may offer significant benefits in the photon-noise-limited regime and in resolving small angular scales at the low-contrast regime.

Evidence of the fractional quantum spin Hall effect in moiré MoTe2.

Quantum spin Hall (QSH) insulators are two-dimensional electronic materials that have a bulk band gap similar to an ordinary insulator but have topologically protected pairs of edge modes of opposite chiralities1-6. So far, experimental studies have found only integer QSH insulators with counter-propagating up-spins and down-spins at each edge leading to a quantized conductance G0 = e2/h (with e and h denoting the electron charge and Planck's constant, respectively)7-14. Here we report transport evidence of a fractional QSH insulator in 2.1° twisted bilayer MoTe2, which supports spin-Sz conservation and flat spin-contrasting Chern bands15,16. At filling factor ν = 3 of the moiré valence bands, each edge contributes a conductance with zero anomalous Hall conductivity. The state is probably a time-reversal pair of the even-denominator 3/2-fractional Chern insulators. Furthermore, at ν = 2, 4 and 6, we observe a single, double and triple QSH insulator with each edge contributing a conductance G0, 2G0 and 3G0, respectively. Our results open up the possibility of realizing time-reversal symmetric non-abelian anyons and other unexpected topological phases in highly tunable moiré materials17-19.

Miniaturized Wideband Circularly Polarized Triangular Patch Antennas based on Characteristic Mode Analysis

A miniaturized, wideband circularly polarized (CP) antenna based on coupled triangular patches is presented. Initially, two identical triangular patches with shorting pins are placed close to each other in a perpendicular orientation. Hence, a pair of orthogonal modes can be produced based on the coupled resonators. Under the characteristic mode analysis (CMA), it can be found that the 90∘ phase difference is achieved by modulating gap distance and shorting pins numbers. Both the shape of triangle patches and shorting pins contribute to the miniaturization. To further improve the AR bandwidth, a third patch is added to form a new mode. Thanks to the triple modes produced by the three patch elements, two AR minima are constructed to broaden the AR bandwidth. With this compact arrangement and shorting pins, a miniaturized wideband CP patch antenna with a 5.2% AR bandwidth is successfully implemented. The overall size of the antenna is merely 0.34λ0×0.33λ0×0.046λ0.

Gene-gene interaction network analysis indicates CNTN2 is a candidate gene for idiopathic generalized epilepsy.

Twin and family studies have established the genetic contribution to idiopathic generalized epilepsy (IGE). The genetic architecture of IGE is generally complex and heterogeneous, and the majority of the genetic burden in IGE remains unsolved. We hypothesize that gene-gene interactions contribute to the complex inheritance of IGE. CNTN2 (OMIM* 615,400) variants have been identified in cases with familial adult myoclonic epilepsy and other epilepsies. To explore the gene-gene interaction network in IGE, we took the CNTN2 gene as an example and investigated its co-occurrent genetic variants in IGE cases. We performed whole-exome sequencing in 114 unrelated IGE cases and 296 healthy controls. Variants were qualified with sequencing quality, minor allele frequency, in silico prediction, genetic phenotype, and recurrent case numbers. The STRING_TOP25 gene interaction network analysis was introduced with the bait gene CNTN2 (denoted as A). The gene-gene interaction pair mode was presumed to be A + c, A + d, A + e, with a leading gene A, or A + B + f, A + B + g, A + B + h, with a double-gene A + B, or other combinations. We compared the number of gene interaction pairs between the case and control groups. We identified three pairs in the case group, CNTN2 + PTPN18, CNTN2 + CNTN1 + ANK2 + ANK3 + SNTG2, and CNTN2 + PTPRZ1, while we did not discover any pairs in the control group. The number of gene interaction pairs in the case group was much more than in the control group (p = 0.021). Taking together the genetic bioinformatics, reported epilepsy cases, and statistical evidence in the study, we supposed CNTN2 as a candidate pathogenic gene for IGE. The gene interaction network analysis might help screen candidate genes for IGE or other complex genetic disorders.

A Novel Dual-Layer 2-bit MMW Transmitarray Based on Two Pairs of Orthogonal Modes

A Novel Dual-Layer 2-bit MMW Transmitarray Based on Two Pairs of Orthogonal Modes

Propagation characteristics of circumferential feature guided waves propagation in a revolved topological waveguide

In our previous research, propagation characteristics and combined harmonic generation of feature guided waves (FGWs) in welded joints of plate structure, were analyzed theoretically and observed numerically. Circumferential ultrasonic guided waves find extensive applications in assessing rotating structures, highlighting the importance of investigating the propagation characteristics of circumferential feature guided waves (CFGWs) in such revolved topological waveguides. In this study, propagation characteristics of CFGW in revolved topological waveguides are investigated. Semi-analytical finite element (SAFE) method in conjunction with a perfectly matched layer (PML) within a cylindrical coordinate system is used to explore the modal and dispersion characteristics of CFGWs. Second-harmonic generation (SHG) of CFGWs is further analyzed and discussed both in theoretical and numerical manner. Cumulative SHG of selected CFGW mode pairs, propagation in the welded joint of the revolved topological waveguide, is observed successfully. The obtained results demonstrate that the energy trapping effect of CFGWs in the welded joint of revolved topological waveguide amplifies the generation of cumulative second harmonics, thereby enhancing the measurability of SHG of CFGW. This investigation reveals the potential of damage assessment in revolved topological waveguides by CFGWs.

Bi-global stability of supersonic backward-facing step flow

Supersonic backward-facing step (BFS) flow is numerically studied using direct numerical simulation (DNS) and global stability analysis (GSA) with a free stream Mach number of 2.16 and a Reynolds number of 7.938 × 105 based on the flat-plate length L and free stream conditions. Two-dimensional BFS flow becomes unstable to three-dimensional perturbations as the step height h exceeds a certain value, while no two-dimensionally unstable mode is found. Global instability occurs with the fragmentation of the primary separation vortex downstream of the step. Two stationary modes and one oscillatory unstable mode are obtained at a supercritical ratio of L/h = 32.14, among which the two stationary modes originate from the coalescence of a pair of conjugate modes. The most unstable mode manifests itself as streamwise streaks in the reattached boundary layer, which is similar to that in shock-induced separated flow, although the flow separation mechanisms are different. Without introducing any external disturbances, the DNS captures the preferred perturbations and produces a growth rate in agreement with the GSA prediction in the linear growth stage. In the quasi-steady stage, the secondary separation vortex breaks up into several small bubbles, and the number of streamwise streaks is doubled. A low-frequency unsteadiness that may be associated with the oscillatory mode is also present.

Noise-induced collective actuation in active solids.

Collective actuation describes the spontaneous synchronized oscillations taking place in active solids when the elasto-active feedback, which generically couples the reorientation of the active forces and the elastic stress, is large enough. In the absence of noise, collective actuation takes the form of a strong condensation of the dynamics on a specific pair of modes and their generalized harmonics. Here we report experiments conducted with centimetric active elastic structures, where collective oscillation takes place along the single lowest energy mode of the system, gapped from the other modes because of the system's geometry. Combining the numerical and theoretical analysis of an agent-based model, we demonstrate that this form of collective actuation is noise-induced. The effect of the noise is first analyzed in a single-particle toy model that reveals the interplay between the noise and the specific structure of the phase space. We then show that in the continuous limit, any finite amount of noise turns this new form of transition to collective actuation into a bona fide supercritical Hopf bifurcation.

An enhanced wideband tracking method for characteristic modes

Abstract An enhanced wideband tracking method for characteristic modes (CMs) is investigated in this paper. The method consists of three stages, and its core tracking stage (CTS) is based on a classical eigenvector correlation-based algorithm. To decrease the tracking time and eliminate the crossing avoidance (CRA), we append a commonly used eigenvalue filter (EF) as the preprocessing stage and a novel postprocessing stage to the CTS. The proposed postprocessing stage can identify all CRA mode pairs by analyzing their trajectory and correlation characteristics. Subsequently, it can predict corresponding CRA frequencies and correct problematic qualities rapidly. Considering potential variations in eigenvector numbers at consecutive frequency samples caused by the EF, a new execution condition for the adaptive frequency adjustment in the CTS is introduced. Finally, CMs of a conductor plate and a fractal structure are investigated to demonstrate the performance of the proposed method, and the obtained results are discussed.

A Thin‐Layer, Li+ Compensation, Moisture‐Tolerant Interfacial Modification of Cu Substrate Toward the Anode‐Less, Energy/Power Dense Li Metallic Batteries

AbstractThe unregulated metallic deposition and continuous cracking of the fragile solid electrolyte interphase are considered the critical barriers that compromise the cyclability of lithium metal batteries (LMB), especially under low N/P ratio (&lt;3) pairing modes. Herein, an ultra‐thin (5 µm), lightweight (0.25 mg cm−2), and moisture‐proof interfacial layer composed of the high‐entropy alloys (denoted as HEAs) and interweaved carbon nanotubes (CNTs) scaffold is constructed to modify the current collector, moreover, the thermally‐induced Li22Si5 alloy blended with the hydrophobic ethylene‐vinyl acetate copolymer (EVA) is infiltrated into the scaffold pores as the moisture‐proof cation reservoir. The HEA@CNT/Li22Si5@EVA interfacial layer not only maximizes the Li‐utilization degree with minimal voltage divergence in symmetric cells but also compensates for irreversible Li depletion in the pouch‐format anode‐less models. As the HEA@CNT/Li22Si5@EVA‐Cu substrate paired with the LiNi0.8Mn0.1Co0.1O2 cathode in a 200 mAh prototype, the phase evolution of oxide cathode and efficient Li utilization at the anode substrate can be real‐time monitored by the transmission‐mode operando X‐ray diffraction. This interfacial layer strategy affords multifunctionality to enable the LMB prototyping without excessive Li abuse. Consequently, cycling endurance and the balanced energy densities (420.1 Wh kg−1) are obtained on the whole cell.

Cluster synchronization in a semiconductor laser

Cluster synchronization is a general phenomenon in a network of non-locally coupled oscillators. Here, we show that cluster synchronization occurs in semiconductor lasers, where the beat notes between the pairs of adjacent longitudinal modes of the laser cavity constitute a collection of coupled phase oscillators. Non-local coupling arises from the standing-wave nature of the cavity with finite mirror reflectivities, which we can actively control. Varying the coupling, we can bring the laser into a state of cluster synchronization where the two beat note families oscillate at two distinct collective frequencies. Using a coherent beat note detection technique, we show that the beat notes within the two families are synchronized in the opposite configurations—in-phase and antiphase.

Witnessing edge modes in trimerized circuit quantum electrodynamic lattice

We propose a scheme to investigate and witness edge modes of general one-dimensional photonic trimers in a circuit quantum electrodynamic lattice. These in-gap edge modes are strictly and analytically solved and the criteria for their emergence are indicated respectively. Moreover, the energy spectrum of the system shows two different regimes characterized by a discrepancy in the number of edge modes. Specifically, while there are always a couple of edge modes only present at one single boundary in both the regimes, one of the regimes also shows another pair of edge modes localized on the opposite boundary. Furthermore, these edge modes are witnessed with the aid of continuous-time quantum walks and average photon number measurements. Our scheme provides a comprehensive method for studying the edge modes of matter.

Real-space decomposition of p-wave Kitaev chain

We propose an extended Bogoliubov transformation in real space for spinless fermions, based on which a class of Kitaev chains of length 2N with zero chemical potential can be mapped to two independent Kitaev chains of length N. It provides an alternative way to investigate a complicated system from the result of relatively simple systems. We demonstrate the implications of this decomposition by a Su–Schrieffer–Heeger Kitaev model, which supports rich quantum phases. The features of the system, including the groundstate topology and nonequilibrium dynamics, can be revealed directly from that of sub-Kitaev chains. Based on this connection, two types of Bardeen–Cooper–Schrieffer (BCS)-pair order parameters are introduced to characterize the phase diagram, showing the ingredient of two different BCS pairing modes. Analytical analysis and numerical simulations show that the real-space decomposition for the ground state still holds true approximately in presence of finite chemical potential in the gapful regions.