Time-dependent Operator Research Articles

Projection-based model reduction with dynamically transformed modes

We propose a new model reduction framework for problems that exhibit transport phenomena. As in the moving finite element method (MFEM), our method employs time-dependent transformation operators and, especially, generalizes MFEM to arbitrary basis functions. The new framework is suitable to obtain a low-dimensional approximation with small errors even in situations where classical model order reduction techniques require much higher dimensions for a similar approximation quality. Analogously to the MFEM framework, the reduced model is designed to minimize the residual, which is also the basis for ana posteriorierror bound. Moreover, since the dependence of the transformation operators on the reduced state is nonlinear, the resulting reduced order model is obtained by projecting the original evolution equation onto a nonlinear manifold. Furthermore, for a special case, we show a connection between our approach and the method of freezing, which is also known as symmetry reduction. Besides the construction of the reduced order model, we also analyze the problem of finding optimal basis functions based on given data of the full order solution. Especially, we show that the corresponding minimization problem has a solution and reduces to the proper orthogonal decomposition of transformed data in a special case. Finally, we demonstrate the effectiveness of our method with several analytical and numerical examples.

Shortcuts to Adiabaticity in Driven Open Quantum Systems: Balanced Gain and Loss and Non-Markovian Evolution

A universal scheme is introduced to speed up the dynamics of a driven open quantum system along a prescribed trajectory of interest. This framework generalizes counterdiabatic driving to open quantum processes. Shortcuts to adiabaticity designed in this fashion can be implemented in two alternative physical scenarios: one characterized by the presence of balanced gain and loss, the other involves non-Markovian dynamics with time-dependent Lindblad operators. As an illustration, we engineer superadiabatic cooling, heating, and isothermal strokes for a two-level system, and provide a protocol for the fast thermalization of a quantum oscillator.

Fractional Order of Evolution Inclusion Coupled with a Time and State Dependent Maximal Monotone Operator

This paper is devoted to the study of evolution problems involving fractional flow and time and state dependent maximal monotone operator which is absolutely continuous in variation with respect to the Vladimirov’s pseudo distance. In a first part, we solve a second order problem and give an application to sweeping process. In a second part, we study a class of fractional order problem driven by a time and state dependent maximal monotone operator with a Lipschitz perturbation in a separable Hilbert space. In the last part, we establish a Filippov theorem and a relaxation variant for fractional differential inclusion in a separable Banach space. In every part, some variants and applications are presented.

Maximal regularity for the damped wave equations

We consider the problem of maximal regularity for non-autonomous Cauchy problems ¨ u(t) + B(t) ˙ u(t) + A(t)u(t) = f (t) (t ∈ [0, τ ]), u(0) = u 0 , ˙ u(0) = u 1. Here, the time dependent operator A(t) is bounded from V to V ′ and B(t) is associated with a family of sesquilinear forms with domain V. We prove maximal L p-regularity results and other regularity properties for the solutions of the previous problems under minimal regularity assumptions on the operators.

Evaluation of lymph node ratio and lymph node yield as prognosticators of locoregional recurrence in p16‐associated oropharyngeal squamous cell carcinoma

The prognostic value of lymph node yield (LNY) and lymph node ratio (LNR), or the ratio of number of metastatic LNs to total number dissected, has not been well established in p16-associated oropharyngeal squamous cell carcinoma (OPSCC). This retrospective cohort study evaluated locoregional disease-free survival (LRDFS) in 82 patients with p16+ OPSCC who underwent neck dissection at a single institution from 2009 to 2017. LNR and LNY cutoffs were estimated using time-dependent receiver operator characteristic (ROC) curves. Prognostic significance of these cutoffs was compared with Eighth Edition AJCC Nodal Staging. An increased LNR ≥ 0.129 was associated with worse 2-year LRDFS (66.9% vs 96.8%, P = .005). LNY was not significantly associated with LRDFS (P = .304). An LNR-based risk model was a better prognosticator than Eighth Edition AJCC Nodal Staging (Harrell's C, 0.9065 vs 0.7668). LNR has good prognostic utility in predicting LRDFS in p16+ OPSCC, but further evaluation of this measure is warranted.

On a Time and State Dependent Maximal Monotone Operator Coupled with a Sweeping Process with Perturbations

In this paper, we state, in separable Hilbert spaces, the existence of absolutely continuous solutions for a couple of evolution problems governed by time and state dependent maximal monotone operator and closed convex sweeping process, with perturbations.

Path integral method for quantum dissipative systems with dynamical friction: Applications to quantum dots/zero-dimensional nanocrystals

Abstract In this study, a path integral approach for an isolated dynamical system which contains a moving body and its surrounding coupled to each other by a fractional dynamical friction is constructed. The dissipative system is characterized by a total Lagrangian L total = L − E d holding the energy term E d = ∫ 0 t f ( x ˙ , x , τ ) d τ which is dissipated by a dynamical fractional friction force f ( x ˙ , x , τ ) . Such a position- and time-dependent friction is in fact motivated from the Brownian motion. Our methodology aims to substitute the standard action by S = ∫ 0 t ( L ( x ˙ , x , τ ) − ∫ 0 τ f ( x ˙ , x , τ ) d μ α ( ξ ) ) d τ where [ μ α ] = − α , 0 α ≤ 1 with the particular choice d μ α ( τ ) = d τ Λ α ( τ ) for some scalar function Λ α ( ξ ) . A modified time-dependent Schrodinger equation is obtained which is characterized by a time-dependent Hamiltonian operator which is suitable to describe for quantum systems characterized by a time-dependent mass. Applications to quantum dots/zero-dimensional nanocrystals in which the size of particles is close to the exciton Bohr radius of the material are studied and explored in details by taking into account both the electron and the hole within the quantum dots besides the energy gap of the bulk. For the case of Cadmium selenide, we have obtained a decrease in the density of state in agreement with recent experimental results. Several consequences were discussed in some details.

Neutrophil-lymphocyte score: A novel prognostic scoring system that utilized the dynamic change of neutrophil, lymphocyte, and albumin and its comparison to other indices.

3048 Background: Indications for immune checkpoint inhibitor (ICI) in cancer care are expanding rapidly. There is increasing need for accurate decision tool to better guide treatment. We have constructed a new prognostic scoring system, neutrophil-lymphocyte score (NRS), based on the nonlinear dynamic change of neutrophil to lymphocyte ratio (NLR) in relation to survival over the first cycle of ICI treatment. We compared this novel system to existing indices such as NLR, lymphocyte to monocyte ratio (LMR), platelet to lymphocyte ratio (PLR), Advanced Lung Cancer Inflammation Index (ALI), and Systemic Immune-inflammation Index (SII). Methods: This is a retrospective analysis of 837 patients at Ohio State University from 2011-18. Neutrophil (ANC), lymphocyte (ALC), platelet (plt), monocyte (AMC), albumin (alb), and body mass index (BMI) were collected at baseline. Repeat labs were collected at cycle 2. NLR = ANC/ALC, ALI = BMI x alb / NLR, LMR = ALC/AMC, SII = platelet x NLR, PLR = plt/ALC. NLR Ratio = baseline NLR / repeat NLR. Based on the association between NLR and the overall survival, we assigned 1 point (p) for baseline NLR < 0.7, 6p for 0.7 to < 2, 5p for 2 to < 3, 4 p for 3 to < 4, 3 for 4 to 5, 2p for 5 to < 9, and 1p for ≥9. We also assigned 1p for NLR ratio < 0.6, 2p for 0.6 to < 0.8, 3p for 0.8 to < 1.2, 5p for 1.25 to < 1.4, 3p for 1.4 to < 1.6, and 2p for ≥1.6. NLS = sum of these 2 scores . NLS_A = NLS*alb. Time-dependent receiver operator characteristic (ROC) curves with integrated time-dependent area under the curve (TD AUC) values were used to evaluate the predictive accuracy of each index for survival. Results: For baseline and repeat values, all indices were statistically significant (P < 0.001) in predicting survival. Baseline integrated TD AUC were: ALI 0.704, NLR 0.692, SII 0.663, LMR 0.645, and PLR 0.612. All of the repeat indices at cycle 2 had higher prognostic value than their baseline counterparts. Integrated TD AUC for indices at cycle 2 were: ALI 0.740 (with baseline BMI), NLR 0.729, SII 0.694, LMR 0.671, and PLR 0.652. NLS_A was a composite score based on the dynamic change of NLR from cycle 1 to 2 and the treatment alb with integrated TD-AUC at 0.754. Conclusions: Indices constructed from ANC, ALC, AMC, Plt, alb, and BMI can be obtained inexpensively and provide great prognostic value for pts on ICI. We have constructed a novel scoring system (NLS_A) and demonstrated its improvement over the current prognostic indices. Studies with a larger cohort are needed to further improve and validate this system.

HBcrAg and pg RNA and the therapeutic effect in HBeAg-positive patients receiving anti-viral therapy, baseline serum HBV-RNA is a powerful predictor of response.

We used HBV core antigen (HbcrAg), pre-genomic RNA (pg RNA) and other biomarkers to evaluate the therapeutic effect in HBV infected patients receiving anti-viral therapy. 127HBeAg-positive patients were enrolled: 35 patients received nucleotide therapy, 14 patients received interferon and 78 patients received combination therapy with both. HBcrAg, pg RNA and other biomarkers were detected at different time points, we defined the decreased titre of HBcrAg and HBeAg from baseline to 6 and baseline to 12months as ∆HBcrAg and ∆HBeAg, which were used to predict HBeAg seroconversion. Furthermore, we used the time-dependent receiver operator curve of different markers to analyse HBeAg seroconversion. For HBeAg seroconversion: at 6months, 0.75 log10 U/mL of ∆HBcrAg and 1.47 log10 PEI U/mL of ∆HBeAg showed maximum predictive value in receiver operator curve analysis (Youden's index values for area under the curve of 0.687 and 0.646, respectively). At 12months, 2.05 log10 U/mL of ∆HBcrAg and 1.92 log10 PEI U/mL of ∆HBeAg showed improved prediction (maximum Youden's index values, with areas under the curve of 0.688 and 0.698, respectively).pg RNA was a better predictor of outcome due and the concentrations of 6.20 log10 I U/mL of pg RNA and 8.0 log10 U/mL of HBcrAg were cut-off values for response in a Kaplan-Meier curve analysis. Our results may be used to identify the pg RNA concentration in patients at baseline and ∆HBcrAg during therapy who are likely to achieve HBeAg seroconversion according to the cut-off value at different time points, thus helping to evaluate the therapeutic effect.

The Cauchy Problem for the Iterated Klein–Gordon Equation with the Bessel Operator

An analogue of the Cauchy problem for an iterated multidimensional Klein–Gordon equation with a time-dependent Bessel operator is investigated. Applying the generalized Erdelyi–Kober operator of fractional order, we reduce the formulated problem to the Cauchy problem for the polywave equation. Applying a spherical mean method, we construct an explicit formula to solve this problem for the polywave equation; then, basing on this solution, we find an integral representation of the solution of the formulated problem. The obtained formula allows one to immediately discern the character of the dependence of the solution on the initial functions and, in particular, to establish conditions for the smoothness of the classical solution. The paper will be useful for specialists engaged in the resolving of problems of higher spin theory.

Time-dependent rational extensions of the parametric oscillator: quantum invariants and the factorization method

New families of time-dependent potentials related to the parametric oscillator are introduced. This is achieved by introducing some general time-dependent operators that factorize the appropriate constant of motion (quantum invariant) of the parametric oscillator, leading to new families of quantum invariants that are almost-isospectral to the initial one. Then, the respective time-dependent Hamiltonians are constructed, and the solutions of the Schrödinger equation are determined from the intertwining relationships and by finding the appropriate time-dependent complex-phases of the Lewis–Riesenfeld approach. To illustrate the results, the parameters of the new potentials are fixed such that a family of time-dependent rational extensions of the parametric oscillator is obtained. Moreover, the rational extensions of the harmonic oscillator are recovered in the appropriate limit.

Integrated Analysis of the Functions and Prognostic Values of RNA Binding Proteins in Lung Squamous Cell Carcinoma.

Lung cancer is the leading cause of cancer-related deaths worldwide. Dysregulation of RNA binding proteins (RBPs) has been found in a variety of cancers and is related to oncogenesis and progression. However, the functions of RBPs in lung squamous cell carcinoma (LUSC) remain unclear. In this study, we obtained gene expression data and corresponding clinical information for LUSC from The Cancer Genome Atlas (TCGA) database, identified aberrantly expressed RBPs between tumors and normal tissue, and conducted a series of bioinformatics analyses to explore the expression and prognostic value of these RBPs. A total of 300 aberrantly expressed RBPs were obtained, comprising 59 downregulated and 241 upregulated RBPs. Functional enrichment analysis indicated that the differentially expressed RBPs were mainly associated with mRNA metabolic processes, RNA processing, RNA modification, regulation of translation, the TGF-beta signaling pathway, and the Toll-like receptor signaling pathway. Nine RBP genes (A1CF, EIF2B5, LSM1, LSM7, MBNL2, RSRC1, TRMU, TTF2, and ZCCHC5) were identified as prognosis-associated hub genes by univariate, least absolute shrinkage and selection operator (LASSO), Kaplan–Meier survival, and multivariate Cox regression analyses, and were used to construct the prognostic model. Further analysis demonstrated that high risk scores for patients were significantly related to poor overall survival according to the model. The area under the time-dependent receiver operator characteristic curve of the prognostic model was 0.712 at 3 years and 0.696 at 5 years. We also developed a nomogram based on nine RBP genes, with internal validation in the TCGA cohort, which showed a favorable predictive efficacy for prognosis in LUSC. Our results provide novel insights into the pathogenesis of LUSC. The nine-RBP gene signature showed predictive value for LUSC prognosis, with potential applications in clinical decision-making and individualized treatment.

BV Continuous Solutions of an Evolution Inclusion with Maximal Monotone Operator and Nonconvex-Valued Perturbation. Existence Theorem

An evolution inclusion with the right-hand side containing a time-dependent maximal monotone operator and a multivalued mapping with closed nonconvex values is studied in a separable Hilbert space. The dependence of the maximal monotone operator on time is described with the help of the distance between maximal monotone operators in the sense of Vladimirov. This distance as a function of time has bounded variation with an upper bound given by a nonatomic positive Radon measure. By a solution of the inclusion one means a continuous function of bounded variation whose differential measure (Stieltjes measure) is absolutely continuous with respect to the positive Radon measure above, and the values of the density of this differential measure with respect to the Radon measure belong to the right-hand side of the inclusion almost everywhere. Under the traditional assumptions on the perturbation (measurability, Lipschitzianity in the phase variable in the Hausdorff metric, linear growth condition), the existence of solutions is proven and some properties of the solution set are established.

Development and validation of a RNA binding protein-associated prognostic model for lung adenocarcinoma.

RNA binding proteins (RBPs) dysregulation have been reported in various malignant tumors and associated with the occurrence and development of cancer. However, the role of RBPs in lung adenocarcinoma (LUAD) is poorly understood. We downloaded the RNA sequencing data of LUAD from the Cancer Genome Atlas (TCGA) database and determined the differently expressed RBPs between normal and cancer tissues. The study then systemically investigated the expression and prognostic value of these RBPs by a series of bioinformatics analysis. A total of 223 differently expressed RBPs were identified, including 101 up-regulated and 122 down-regulated RBPs. Eight RBPs (IGF2BP1, IFIT1B, PABPC1, TLR8, GAPDH, PIWIL4, RNPC3, and ZC3H12C) were identified as prognosis related hub gene and used to construct a prognostic model. Further analysis indicated that the patients in the high-risk subgroup had poor overall survival(OS) compared to those in low-risk subgroup based on the model. The area under the curve of the time-dependent receiver operator characteristic curve of the prognostic model are 0.775 in TCGA cohort and 0.814 in GSE31210 cohort, confirming a good prognostic model. We also established a nomogram based on eight RBPs mRNA and internal validation in the TCGA cohort, which displayed a favorable discriminating ability for lung adenocarcinoma.

Identification of a long non-coding RNA signature for predicting prognosis and biomarkers in lung adenocarcinoma.

Long non-coding RNAs (lncRNAs) have a number of functions in various cellular processes and are potential prognostic factors for lung adenocarcinoma (LUAD). A gene risk model could provide novel evidence to improve the prediction of overall outcomes and provide more potential biomarkers. The present study aimed improve a previously published method of gene signature construction to make it more robust and accurate. The lncRNA expression profiles from 594 patients with LUAD were obtained from The Cancer Genome Atlas (TCGA) database and samples were divided into high- and low-risk groups based on median risk scores calculated using a prognosis-related risk score formula. Univariate Cox regression, least absolute shrinkage and selection operator algorithm and multivariate Cox regression were performed to construct a gene signature based on the differentially expressed lncRNAs in patients with LUAD. The robustness and accuracy of the present model was assessed using area under the calculated curves (AUC) and Kaplan-Meier (K-M) survival analysis of the high- and low-risk cohorts. Potential biomarkers associated with survival status were then identified using K-M survival analysis and potential biomarker functions were predicted using enrichment analysis of co-expressed mRNAs. The gene signature constructed contained 44 lncRNAs. The AUCs for 3- and 5-year survival with the model were 0.836 and 0.818, respectively, of a time-dependent receiver operator characteristic curve. Moreover, lncRNAs AC124804.1 and MIR34AHG were identified using K-M survival analysis and the potential function of these two lncRNAs was predicted using Gene Ontology and Kyoto Encyclopedia of Genes and Genomes functional enrichment. The present lncRNA model provides novel insight which may improve prediction of prognosis for patients with LUAD and identify potentially novel biomarkers for the diagnosis.

Ladder Invariants and Coherent States for Time-Dependent Non-Hermitian Hamiltonians

We extend the Dodonov–Malkin–Man’ko–Trifonov (DMMT) invariant method (Malkin et al. Phys. Rev. D 2, 1371 1, J. Math. Phys. 14, 576 2; Dodonov et al. Int. J. Theor. Phys. 14, 37 3; Dodonov and Man’ko Phys. Rev. A 20, 550 4) to time-dependent pseudo-fermionic systems by introducing ladder invariant operators (time-dependent integrals of motion) which play the role of time-dependent pseudo-fermionic operators and constructing time-dependent pseudo-fermionic coherent states (PFCS) for such systems. As illustrative example, we study in details the time-dependent parity-time-symmetric two-level system under synchronous combined modulations. We explicitly determine time-dependent pseudo-fermionic operators and construct time-dependent PFCS for this physical system. We show that our approach can be extended to time-dependent pseudo-bosonic systems.

Information geometry aspects of minimum entropy production paths from quantum mechanical evolutions.

We present an information geometric analysis of entropic speeds and entropy production rates in geodesic evolution on manifolds of parametrized quantum states. These pure states emerge as outputs of suitable su(2;C) time-dependent Hamiltonian operators used to describe distinct types of analog quantum search schemes. The Riemannian metrization on the manifold is specified by the Fisher information evaluated along the parametrized squared probability amplitudes obtained from analysis of the temporal quantum mechanical evolution of a spin-1/2 particle in an external time-dependent magnetic field that specifies the su(2;C) Hamiltonian model. We employ a minimum action method to transfer a quantum system from an initial state to a final state on the manifold in a finite temporal interval. Furthermore, we demonstrate that the minimizing (optimum) path is the shortest (geodesic) path between the two states and, in particular, minimizes also the total entropy production that occurs during the transfer. Finally, by evaluating the entropic speed and the total entropy production along the optimum transfer paths in a number of physical scenarios of interest in analog quantum search problems, we show in a clear quantitative manner that to a faster transfer there corresponds necessarily a higher entropy production rate. Thus we conclude that lower entropic efficiency values appear to accompany higher entropic speed values in quantum transfer processes.

Evolution Inclusions Governed by Time-Dependent Maximal Monotone Operators with a Full Domain

In this paper, we study the existence of solutions for evolution inclusions governed by time-dependent maximal monotone operators with a full domain. Without assumptions concerning time-regularity on the time-dependent maximal monotone operators, and by using the Moreau-Yosida regularization technique, we establish the existence of solutions in Hilbert spaces. The theoretical result is applied to prove the existence of solutions for semicoercive sweeping processes with velocity constraint.

Volumetric parameters derived from FLT-PET performed at completion of treatment predict efficacy of Carbon-ion Radiotherapy in patients with locally recurrent Nasopharyngeal Carcinoma.

The purpose of this study was to investigate the role of 3'-deoxy-3'-[18F]fluorothymidine (FLT)-PET for predicting the outcome of patients with locally recurrent nasopharyngeal carcinoma (LR-NPC) treated by carbon-ion radiotherapy (CIRT). Patients received FLT-PET/CT scan one-week prior to or after completion of CIRT were enrolled in the study. All patients were from prospective trials or treated using a standardized protocol. Time-dependent receiver operator characteristics (ROC) were used to determine the optimal cutoff values for FLT-PET parameters. Univariable and multivariable analyses of local progression-free survival (LPFS) were performed using Cox regression, to examine the prognostic value of FLT-PET parameters, including SUVmax, metabolic tumor volume (MTV) and total lesion thymidine (TLT). A total of 41 patients were enrolled. Elevated MTV and TLT were significantly associated with worse LPFS, in both univariable and multivariable analyses. ROC analysis revealed that both an MTV value higher than 8.6 and a TLT value higher than 14.9 were predictive of increased risk of developing local recurrence, the adjusted HRs were 5.59 (p=0.009) and 7.76 (p=0.002), respectively. In conclusion, FLT-PET was found to be a promising prognostic tool for LR-NPC patients and might play a role in the treatment guidance.

Time-dependent PT -symmetric quantum mechanics in generic non-Hermitian systems

$\mathcal{PT}$-symmetric quantum mechanics has been considered an important theoretical framework for understanding physical phenomena in $\mathcal{PT}$-symmetric systems, with a number of $\mathcal{PT}$-symmetry related applications. This line of research was made possible by the introduction of a time-independent metric operator to redefine the inner product of a Hilbert space. To treat the dynamics of generic non-Hermitian systems under equal footing, we advocate in this work the use of a time-dependent metric operator for the inner-product between time-evolving states. This treatment makes it possible to always interpret the dynamics of arbitrary (finite-dimensional) non-Hermitian systems in the framework of time-dependent $\mathcal{PT}$-symmetric quantum mechanics, with unitary time evolution, real eigenvalues of an energy observable, and quantum measurement postulate all restored. Our work sheds new lights on generic non-Hermitian systems and spontaneous $\mathcal{PT}$-symmetry breaking in particular. We also illustrate possible applications of our formulation with well-known examples in quantum thermodynamics.