Operator Method Research Articles

Coexisting metabolic dysfunction-associated steatotic liver disease exacerbates in-hospital outcomes in patients with heat stroke

PurposeThis study aimed to investigate the impact of coexisting metabolic dysfunction-associated steatotic liver disease (MASLD) on in-hospital mortality and organ injury markers in patients with heat stroke (HS).ApproachHS patients were retrospectively identified between July 1, 2022 and September 30, 2023 at West China Hospital, Sichuan University. Baseline characteristics, such as demographics, initial vital signs, and organ functional indicators were collected. Outcome events included organ injury and in-hospital mortality. The Least Absolute Shrinkage and Selection Operator (Lasso) method was employed to identify the optimal predictors for in-hospital mortality in HS patients. Subsequently, multivariable logistic regression analysis was performed to assess the relationship between the presence of MASLD and in-hospital mortality as well as organ function indicators.FindingsA total of 112 patients were included in the study, in which 27 (24.1%) had coexisting MASLD. Compared to those without MASLD, patients with MASLD had higher levels of various organ injury markers such as aspartate aminotransferase, urea nitrogen, serum cystatin C, creatinine, uric acid, myoglobin, creatine kinase and its isoenzymes upon admission (P < 0.05). The multivariable Logistic regression analysis indicated that the presence of MASLD is an independent risk factor for in-hospital mortality in HS patients.ConclusionThis study firstly indicated that coexisting MASLD may exacerbate organ injury in HS patients and serve as an independent risk factor for in-hospital mortality.

Predicting thyroid involvement in primary Sjögren’s syndrome: development and validation of a predictive nomogram

IntroductionPatients with Primary Sjögren’s syndrome (pSS) are at a higher risk of thyroid disorders than the general population. This retrospective analysis of 202 patients with pSS was conducted to uncover risk factors associated with thyroid involvement and to create a predictive model for this condition.MethodsWe analyzed 202 patients with pSS from Guang’anmen Hospital, China Academy of Chinese Medical Sciences, with 105 cases of thyroid involvement and 97 without. The Least Absolute Shrinkage and Selection Operator method was used to identify key variables for our risk model. These variables were then subjected to multivariate logistic regression to develop the model. The accuracy of the model was assessed through the C-index, receiver operating characteristic curves, calibration plots, and decision curve analysis, with internal validation via bootstrapping.ResultsHigh-sensitivity C-reactive protein (HCRP), pulmonary disease, pharyngeal dryness, forgetfulness, night sweats, hyperuricemia, nasal dryness, anxiety, Ro52, and aspartate aminotransferase (AST) were incorporated into the nomogram. The model showed robust discrimination and calibration abilities. Decision curve analysis indicated the clinical utility of our nomogram in intervening on the probability thresholds of thyroid disease.ConclusionBy integrating HCRP, pulmonary disease, pharyngeal dryness, forgetfulness, night sweats, hyperuricemia, nasal dryness, anxiety, Ro52, and AST, our thyroid risk nomogram can predict the risk of thyroid involvement in patients with pSS, aiding in more informed treatment strategies.

Numerical version of complex scaling Green operator method

Abstract A new treatment of scattering problems is formulated by improving the complex scaling Green operator (CSGO) method, which can calculate the transition or scattering matrix from the spectral representation of the complex-scaled Green function and the asymptotic solution expressed in terms of the analytic function. In the present formulation, the CSGO method is extended to handle the general scattering problem in the charged particle systems by employing the Coulomb wave function numerically calculated under the complex-scaled asymptotic Hamiltonian. We call the new method the CSGO-numerical (CSGON) method, and the CSGON method is applied to the scattering problem of the 4He + proton system to check its validity. The method of the continuum level density (CLD), which is another method to solve the scattering problem, is also applied to the same system. The numerical precision and the applicability of the CSGON method is discussed by comparing with the CLD and CSGO methods.

Simulating the Landau-Zener sweep in deeply sub-Ohmic environments.

With the goal to study dissipative Landau-Zener (LZ) sweeps in realistic solid-state qubits, we utilize novel methods from non-Markovian open quantum system dynamics that enable reliable long-time simulations for sub-Ohmic environments. In particular, we combine a novel representation of the dynamical propagator, the uniform time evolving matrix product operator method, with a stochastic realization of finite temperature fluctuations. The latter greatly reduces the computational cost for the matrix product operator approach, enabling convergence in the experimentally relevant deeply sub-Ohmic regime. Our method allows the exact simulation of dynamical protocols with long operation times, such as the LZ sweep, in challenging parameter regimes that are realized in current experimental platforms.

Novel insights of disulfidptosis-mediated immune microenvironment regulation in atherosclerosis based on bioinformatics analyses

Atherosclerosis (AS) is the leading cause of coronary heart disease, which is the primary cause of death worldwide. Recent studies have identified disulfidptosis as a new type of cell death that may be involved in onset and development of many diseases. However, the role of disulfidptosis in AS is not clear. In this study, bioinformatics analysis and experiments in vivo and in vitro were performed to evaluate the potential relationship between disulfidptosis and AS. AS-related sequencing data were obtained from the Gene Expression Omnibus (GEO). Bioinformatics techniques were used to evaluate differentially expressed genes (DEGs) associated with disulfidptosis-related AS. Hub genes were screened using least absolute shrinkage and selection operator (LASSO) and random forests (RF) methods. In addition, we established a foam cell model in vitro and an AS mouse model in vivo to verify the expressions of hub genes. In addition, we constructed a diagnostic nomogram with hub genes to predict progression of AS. Finally, the consensus clustering method was used to establish two different subtypes, and associations between subtypes and immunity were explored. As the results, 9 disulfidptosis-related AS DEGs were identified from GSE28829 and GSE43292 datasets. Evaluation of DEGs using LASSO and RF methods resulted in identification of 4 hub genes (CAPZB, DSTN, MYL6, PDLIM1), which were analyzed for diagnostic value using ROC curve analysis and verified in vitro and in vivo. Furthermore, a nomogram including hub genes was established that accurately predicted the occurrence of AS. The consensus clustering algorithm was used to separate patients with early atherosclerotic plaques and patients with advanced atherosclerotic plaques into two disulfidptosis subtypes. Cluster B displayed higher levels of infiltrating immune cells, which indicated that patients in cluster B may have a positive immune response for progression of AS. In summary, disulfidptosis-related genes including CAPZB, DSTN, MYL6, and PDLIM1 may be diagnostic markers and therapeutic targets for AS. In addition, these genes are closely related to immune cells, which may inform immunotherapy for AS.

Investigation of risk signatures associated with anoikis in thyroid cancer through integrated transcriptome and Mendelian randomization analysis

BackgroundAnoikis is intricately associated with the malignant progression of cancer. Thyroid cancer (THCA) is the most common endocrine tumor, metastasis is closely related to treatment response and prognosis of THCA. Hence, it is imperative to comprehensively identify predictive prognostic genes and novel molecular targets for effective THCA therapy.MethodsDifferential expression analysis and weighted gene co-expression network analysis (WGCNA) were utilized to mine differentially expressed anoikis-related (DE-ARGs). Then, the prognostic genes were identified and a risk signature was constructed for THCA using univariate Cox analysis and least absolute shrinkage and selection operator (LASSO) method. Furthermore, the associations between risk signature and immune infiltration, immunotherapy, as well as potential mechanisms of action were determined using multiple R packages and Wilcoxon test. Finally, Mendelian randomized (MR) analysis was conducted to investigate the causal relationship between the prognostic genes and THCA.ResultsIn total, six prognostic genes (LRRC75A, METTL7B, ADRA1B, TPD52L1, TNFRSF10C, and CXCL8) related to anoikis were identified, and the corresponding risk signature were constructed to assess the survival time of THCA patients. Immunocorrelation analysis demonstrated the anoikis-relevant risk signature could be used to evaluate immunotherapy effects in THCA patients, and the infiltration of immune cells was correlated with the degree of risk in THCA patients. According to two-sample MR analysis, there was the significant causal relationship between CXCL8 and THCA (odds ratio [OR] > 1 & p< 0.05), and the increase of its gene expression would lead to an increased risk of THCA. Furthermore, real-time quantitative polymerase chain reaction (RT-qPCR) confirmed the upregulated expression patterns of these prognostic genes in THCA tissues.ConclusionIn conclusion, we constructed the risk signature related to anoikis for THCA, which might have important clinical significance for improving the quality of life and treatment effect of THCA patients.

Radiomics analysis in differentiating osteosarcoma and chondrosarcoma based on T2-weighted imaging and contrast-enhanced T1-weighted imaging

This study was performed to investigate the diagnostic value of radiomics models constructed by fat suppressed T2-weighted imaging (T2WI-FS) and contrast-enhanced T1-weighted imaging (CET1) based on magnetic resonance imaging (MRI) for differentiation of osteosarcoma (OS) and chondrosarcoma (CS). In this retrospective cohort study, we included all inpatients with pathologically confirmed OS or CS from Second Xiangya Hospital of Central South University (Hunan, China) as of October 2020. Demographic and imaging variables were extracted from electronic medical records and compared between OS and CS group. Totals of 530 radiomics features were extracted from CET1 and T2WI-FS sequences based on MRI. The least absolute shrinkage and selection operator (LASSO) method was used for screening and dimensionality reduction of the radiomics model. Multivariate logistic regression analysis was performed to construct the radiomics model, and receiver operating characteristic curve (ROC) was generated to evaluate the diagnostic accuracy of the radiomics model. The training cohort and validation cohort included 87 and 29 patients, respectively. 8 CET1 features and 15 T2WI-FS features were screened based on the radiomics features. In the training group, the area under the receiver-operator characteristic curve (AUC) value for CET1 and T2WI-FS sequences in the radiomics model was 0.894 (95% CI 0.817–0.970) and 0.970 (95% CI 0.940–0.999), respectively. In the validation group, the AUC value for CET1 and T2WI-FS sequences in the radiomics model was 0.821 (95% CI 0.642–1.000) and 0.899 (95% CI 0.785–1.000), respectively. In this study, we developed a radiomics model based on T2WI-FS and CET1 sequences to differentiate between OS and CS. This model exhibits good performance and can help clinicians make decisions and optimize the use of healthcare resources.

Modeling and upgrade of disaster-resilient interdependent networks using machine learning

Recent global emergencies emphasize the critical role of reliable communication networks. As dependence on critical infrastructures grows, the focus shifts from isolated failures to designing networks capable of withstanding disasters, taking into account their interdependence with infrastructures like the power grid. This paper investigates the problem of the disaster resilient upgrade of interdependent networks, focusing on enhancing network resilience during emergencies and ensuring a service-level agreement. We analyze how the interdependency between the networks affects the disaster resilience and propose heuristic methods for network operators to improve resilience against disasters. Furthermore, to address the challenge of hidden interdependencies, we present a novel approach using graph neural networks for predicting interdependency between networks based on historical data of failures. Using simulations with real networks and earthquake data, we demonstrate that limiting the number of interdependent edges per node significantly affects resilience. We show that if sufficient data is available graph neural networks can learn the connection between failures and interdependencies, and capable of predicting interdependencies. Additionally, we show that selecting appropriate upgrade methods can reduce network upgrade costs by up to 20%.

Macro-Driven Stock Market Volatility Prediction: Insights from a New Hybrid Machine Learning Approach

This study comprehensively investigates stock market volatility based on over one hundred monthly macroeconomic variables, applying machine learning models. Methodological contribution integrating the random forest (RF) with the least absolute shrinkage and selection operator methods (LASSO). Importantly, the RF-LASSO model can robustly achieve the best forecasting performance under different circumstances. In addition, we focus on model explanation from different perspectives based on permutation importance and shapley additive explanation (SHAP) methods. This study illuminates novel insights into the realm of stock market volatility, harnessing the transformative potential of machine learning methodologies.

Can we skip invasive biopsy of sentinel lymph nodes? A preliminary investigation to predict sentinel lymph node status using PET/CT-based radiomics

BackgroundSentinel lymph node (SLN) biopsy (SLNB) is considered the gold standard for detecting SLN metastases in patients with invasive ductal breast cancer (IDC). However, SLNB is invasive and associated with several complications. Thus, this study aimed to evaluate the diagnostic performance of a non-invasive radiomics analysis utilizing 2-deoxy-2-[18F]fluoro-d-glucose positron emission tomography/computed tomography (18F-FDG-PET/CT) for assessing SLN metastasis in IDC patients.MethodsThis retrospective study included 132 patients with biopsy-confirmed IDC, who underwent 18F-FDG PET/CT scans prior to mastectomy or breast-conserving surgery with SLNB. Tumor resection or SLNB was conducted within one-week post-scan. Clinical data and metabolic parameters were analyzed to identify independent SLN metastasis predictors. Radiomic features were extracted from each PET volume of interest (VOI) and CT-VOI. Feature selection involved univariate and multivariate logistic regression analysis, and the least absolute shrinkage and selection operator (LASSO) method. Three models were developed to predict SLN status using the random forest (RF), decision tree (DT), and k-Nearest Neighbors (KNN) classifiers. Model performance was assessed using the area under the receiver operating characteristic curve (AUC).ResultsThe study included 91 cases (32 SLN-positive and 59 SLN-negative patients) in the training cohort and 41 cases (29 SLN-positive and 12 SLN-negative patients) in the validation cohort. Multivariate logistic regression analysis identified Ki 67 and TLG as independent predictors of SLN status. Five PET-derived features, three CT-derived features, and two clinical variables were selected for model development. The AUC values of the RF, KNN, and DT models for the training cohort were 0.887, 0.849, and 0.824, respectively, and for the validation cohort were 0.856, 0.830, and 0.819, respectively. The RF model demonstrated the highest accuracy for the preoperative prediction of SLN metastasis in IDC patients.ConclusionThe PET-CT radiomics approach may offer robust and non-invasive predictors for SLN status in IDC patients, potentially aiding in the planning of personalized treatment strategies for IDC patients.

Generalized Langevin dynamics for single beads in linear elastic networks

We derive generalized Langevin equations (GLEs) for single beads in linear elastic networks. In particular, the derivations of the GLEs are conducted without employing normal modes, resulting in two distinct representations in terms of resistance and mobility kernels. The fluctuation-dissipation relations are also confirmed for both GLEs. Subsequently, we demonstrate that these two representations are interconnected via Laplace transforms. Furthermore, another GLE is derived by utilizing a projection operator method, and it is shown that the equation obtained through the projection scheme is consistent with the GLE with the resistance kernel. As simple examples, the general theory is applied to the Rouse model and the ring polymer, where the GLEs with the resistance and mobility kernels are explicitly derived for arbitrary positions of the tagged bead in these models. Finally, the GLE with the mobility kernel is also derived for the elastic network with hydrodynamic interactions under the pre-averaging approximation. ©2024 American Physical Society 2024 American Physical Society

Grassmann time-evolving matrix product operators: An efficient numerical approach for fermionic path integral simulations.

Developing numerical exact solvers for open quantum systems is a challenging task due to the non-perturbative and non-Markovian nature when coupling to structured environments. The Feynman-Vernon influence functional approach is a powerful analytical tool to study the dynamics of open quantum systems. Numerical treatments of the influence functional including the quasi-adiabatic propagator technique and the tensor-network-based time-evolving matrix product operator method have proven to be efficient in studying open quantum systems with bosonic environments. However, the numerical implementation of the fermionic path integral suffers from the Grassmann algebra involved. In this work, we present a detailed introduction to the Grassmann time-evolving matrix product operator method for fermionic open quantum systems. In particular, we introduce the concepts of Grassmann tensor, signed matrix product operator, and Grassmann matrix product state to handle the Grassmann path integral. Using the single-orbital Anderson impurity model as an example, we review the numerical benchmarks for structured fermionic environments for real-time nonequilibrium dynamics, real-time and imaginary-time equilibration dynamics, and its application as an impurity solver. These benchmarks show that our method is a robust and promising numerical approach to study strong coupling physics and non-Markovian dynamics. It can also serve as an alternative impurity solver to study strongly correlated quantum matter with dynamical mean-field theory.

Predicting Invasiveness in Lepidic Pattern Adenocarcinoma of Lung: Analysis of Visual Semantic and Radiomic Features.

To differentiate invasive lepidic predominant adenocarcinoma (iLPA) from adenocarcinoma in situ (AIS)/minimally invasive adenocarcinoma (MIA) of lung utilizing visual semantic and computer-aided detection (CAD)-based texture features on subjects initially diagnosed as AIS or MIA with CT-guided biopsy. From 2011 to 2017, all patients with CT-guided biopsy results of AIS or MIA who subsequently underwent resection were identified. CT scan before the biopsy was used to assess visual semantic and CAD texture features, totaling 23 semantic and 95 CAD-based quantitative texture variables. The least absolute shrinkage and selection operator (LASSO) method or forward selection was used to select the most predictive feature and combination of semantic and texture features for detection of invasive lung adenocarcinoma. Among the 33 core needle-biopsied patients with AIS/MIA pathology, 24 (72.7%) had invasive LPA and 9 (27.3%) had AIS/MIA on resection. On CT, visual semantic features included 21 (63.6%) part-solid, 5 (15.2%) pure ground glass, and 7 (21.2%) solid nodules. LASSO selected seven variables for the model, but all were not statistically significant. "Volume" was found to be statistically significant when assessing the correlation between independent variables using the backward selection technique. The LASSO selected "tumor_Perc95", "nodule surround", "small cyst-like spaces", and "volume" when assessing the correlation between independent variables. Lung biopsy results showing noninvasive LPA underestimate invasiveness. Although statistically non-significant, some semantic features showed potential for predicting invasiveness, with septal stretching absent in all noninvasive cases, and solid consistency present in a significant portion of invasive cases.

Inverted oscillator: quantum discrete spectrum

Abstract We use the invariant operator method to investigate the quantum characteristics of the inverted oscillator. We introduce a unitary transformation that maps a time-dependent general Hermitian linear invariant operator to a time-independent one, which describes a harmonic oscillator with a unit mass and a constant frequency. Our investigation includes three distinguished cases: negative frequency, zero frequency, and positive frequency. Our most interesting result concerns the last case. For the case of negative frequency, we propose a new inner product by the introduction of the new metric operator η. Coherent state for this case will be constructed.

Advanced Model with a Trajectory Tracking Stabilisation System and Feasible Solution of the Optimal Control Problem

In this study, we consider the extended optimal control problem and search for a control function in the class of feasible functions for a real control object. Unlike the classical optimal control problem, the control function should depend on the state, not time. Therefore, the control synthesis problem for the initial-state domain should be solved, instead of the optimal control problem with one initial state. Alternatively, an optimal trajectory motion stabilisation system may be constructed. Both approaches—control and trajectory motion stabilisation system syntheses—cannot be applied to real-time control, as the task is too complex. The minimum threshold of quality criteria is searched for in the space of mathematical expression codes. Among other problems, the search space is difficult to define and the gradient is hard to determine. Therefore, the advanced control object model is used to obtain a feasible control function. The advanced model is firstly obtained before solving the optimal control problem and it already includes a trajectory motion stabilisation system; in particular, this stabilisation system is synthesised in advance at the control system design stage. When the optimal control problem appears, it is solved in real time in the classical statement, and a control function is searched for as a function of time. The advanced control object model also uses the reference model to generate the optimal trajectory. The search for the optimal control function is performed in real time and considers the synthesised stabilisation system of motion along a determined trajectory. Machine learning control via symbolic regression, namely, the network operator method, is used to directly solve the control synthesis problem. An example solution of the optimal control problem, with an advanced model moving in the environment with obstacles for a group of two mobile robots, is presented. The obtained solution is a control function for a reference model that generates a trajectory from a class of trajectories stabilised with the object’s control system.

Development of a prognostic model to predict 90-day mortality in hospitalised cancer patients (PROMISE tool): a prospective observational study

Prognostic factors for ambulatory oncology patients have been described, including Eastern Cooperative Oncology Group (ECOG), tumor stage and malnutrition. However, there is no firm evidence on which variables best predict mortality in hospitalized patients receiving active systemic treatment. Our main goal was to develop a predictive model for 90-day mortality upon admission. Between 2020 and 2022, we prospectively collected data from three sites for cancer patients with hospitalizations. Those with metastatic disease receiving systemic therapy in the 6 months before unplanned admission were eligible to this study. The least absolute shrinkage and selection operator (LASSO) method was used to select the most relevant factors to predict 90-day mortality at admission. A multivariable logistic regression was fitted to create the PROgnostic Score for Hospitalized Cancer Patients (PROMISE) score. The score was developed in a single-center training cohort and externally validated. Of 1658 hospitalized patients, 1009 met eligibility criteria. Baseline demographics, patient and disease characteristics were similar across cohorts. Lung cancer was the most common tumor type in both cohorts. Factors associated with higher 90-day mortality included worse ECOG, stable/progressive disease, low levels of albumin, increased absolute neutrophil count, and high lactate dehydrogenase. The c-index after bootstrap correction was 0.79 (95% CI, 0.75-0.82) and 0.74 (95% CI, 0.68-0.80) in the training and validation cohorts, respectively. A web tool (https://promise.vhio.net/) was developed to facilitate the clinical deployment of the model. The PROMISE tool demonstrated high performance for identifying metastatic cancer patients who are alive 90 days after an unplanned hospitalization. This will facilitate healthcare providers with rational clinical decisions and care planning after discharge. Merck S.L.U., Spain.

Feature selection with annealing for forecasting financial time series

Stock market and cryptocurrency forecasting is very important to investors as they aspire to achieve even the slightest improvement to their buy-or-hold strategies so that they may increase profitability. However, obtaining accurate and reliable predictions is challenging, noting that accuracy does not equate to reliability, especially when financial time-series forecasting is applied owing to its complex and chaotic tendencies. To mitigate this complexity, this study provides a comprehensive method for forecasting financial time series based on tactical input–output feature mapping techniques using machine learning (ML) models. During the prediction process, selecting the relevant indicators is vital to obtaining the desired results. In the financial field, limited attention has been paid to this problem with ML solutions. We investigate the use of feature selection with annealing (FSA) for the first time in this field, and we apply the least absolute shrinkage and selection operator (Lasso) method to select the features from more than 1000 candidates obtained from 26 technical classifiers with different periods and lags. Boruta (BOR) feature selection, a wrapper method, is used as a baseline for comparison. Logistic regression (LR), extreme gradient boosting (XGBoost), and long short-term memory are then applied to the selected features for forecasting purposes using 10 different financial datasets containing cryptocurrencies and stocks. The dependent variables consisted of daily logarithmic returns and trends. The mean-squared error for regression, area under the receiver operating characteristic curve, and classification accuracy were used to evaluate model performance, and the statistical significance of the forecasting results was tested using paired t-tests. Experiments indicate that the FSA algorithm increased the performance of ML models, regardless of problem type. The FSA hybrid models showed better performance and outperformed the other BOR models on seven of the 10 datasets for regression and classification. FSA-based models also outperformed Lasso-based models on six of the 10 datasets for regression and four of the 10 datasets for classification. None of the hybrid BOR models outperformed the hybrid FSA models. Lasso-based models, excluding the LR type, were comparable to the best models for six of the 10 datasets for classification. Detailed experimental analysis indicates that the proposed methodology can forecast returns and their movements efficiently and accurately, providing the field with a useful tool for investors.

Infinite Grassmann time-evolving matrix product operator method for zero-temperature equilibrium quantum impurity problems

Infinite Grassmann time-evolving matrix product operator method for zero-temperature equilibrium quantum impurity problems

Recent advances in peritoneal dialysis catheter placement: the association between method of insertion, operator, and prior abdominal surgery and outcomes.

Peritoneal dialysis (PD) is associated with comparable outcomes to hemodialysis and is much less expensive to provide in most parts of the world. A well functioning PD catheter is required, but complications are frequent, and while there are guidelines for the placement and care of a PD access, they are based on low-quality evidence. Data reporting the outcomes of percutaneous, as compared to surgical pathways for placement of catheters and the role of operator type in determining outcomes are examined. The impact of prior abdominal procedures on patient selection and the prognostic importance of the presence of adhesions at the time of catheter insertion are reviewed. There are conflicting data about the relative merits of percutaneous as compared to surgical placement of catheters that may relate to differences in patient selection, operator experience, or operator volume. Prior abdominal procedures are not associated with worse PD catheter outcomes and likely should not influence patient selection, except in exceptional circumstances. Finally, the presence of adhesions is associated with a higher risk of complications. However, even in the presence of adhesions, the termination of PD therapy and need for re-intervention occurs in <20% of patients.

Reply to the Comment on “A New Boson Expansion Theory Utilizing a Norm Operator”

Abstract The comment is based on misunderstandings and, as a result, does not evaluate the norm operator method. We refute the comment. The structure of the norm operator composed of all the excitation modes determines whether the form of a practical boson expansion is a small parameter expansion. KT-3 is chimerical. It should be taken seriously that the small parameter expansions do not hold with truncating the contributions of all phonon excitation modes that are not adopted as boson excitations. The formulation using the boson expansion of the projection operator on to the boson vacuum does not allow the limitation of the phonon excitation number beforehand.