Analytic Matrices Research Articles

On the periodicity of singular vectors and the holomorphic block-circulant SVD on the unit circumference

We investigate the singular value decomposition of a rectangular matrix that is analytic on the complex unit circumference, which occurs, e.g., with the matrix of transfer functions representing a broadband multiple-input multiple-output channel. Our analysis is based on the Puiseux series expansion of the eigenvalue decomposition of analytic para-Hermitian matrices on the complex unit circumference. We study the case in which the rectangular matrix does not admit a full analytic singular value factorization, either due to partly multiplexed systems or to sign ambiguity. We show how to find an SVD factorization in the ring of Puiseux series where each singular value and the associated singular vectors present the same period and multiplexing structure, and we prove that it is always possible to find an analytic pseudo-circulant factorization, meaning that any arbitrary arrangements of multiplexed systems can be converted into a parallel form. In particular, one can show that the sign ambiguity can be overcome by allowing non-real holomorphic singular values.

Invertible bases and root vectors for analytic matrix-valued functions

We revisit the concept of a minimal basis through the lens of the theory of modules over a commutative ring $R$. We first review the conditions for the existence of a basis for submodules of $R^n$ where $R$ is a Bézout domain. Then, we define the concept of invertible basis of a submodule of $R^n,$ and when $R$ is an elementary divisor domain, we link it to the Main Theorem of G. D. Forney Jr. [SIAM J. Control, 13:493-520, 1975]. Over an elementary divisor domain, the submodules admitting an invertible basis are precisely the free pure submodules of $R^n$. As an application, we let $\Omega \subseteq \mathbb{C}$ be either a connected compact set or a connected open set, and we specialize to $R=\mathcal{A}(\Omega)$, the ring of functions that are analytic on $\Omega$. We show that, for any matrix $A(z) \in \mathcal{A}(\Omega)^{m \times n}$, $\ker A(z) \cap \mathcal{A}(\Omega)^n$ is a free $\mathcal{A}(\Omega)$-module and admits an invertible basis, or equivalently a basis that is full rank upon evaluation at any $\lambda \in \Omega$. Finally, given $\lambda \in \Omega$, we use invertible bases to define and study maximal sets of root vectors at $\lambda$ for $A(z)$. This in particular allows us to define eigenvectors also for analytic matrices that do not have full column rank.

Capacity and Control: Superintendent-School Board Relations in Locally Controlled Districts

Purpose: In the wake of school board protests nationwide, it is essential for educational leaders, policymakers, and researchers to understand locally controlled educational governance. The stability of educational districts depends on superintendents navigating relationships with their school boards, yet little research exists that addresses board–superintendent relationships in locally controlled districts. Using theories of local will and capacity, and community power relationships, I examine what factors shape the extent of local control enacted by school boards, and how these factors affect superintendent–school board relations. Research Methods: The qualitative case study design is bounded as one multi-district union superintendent and three rural school boards. Data collection included semistructured interviews, ethnographic observations, and document collection to understand the relational dynamics of local control. Data analysis included in vivo and a priori coding, and the development of analytic matrices. Findings: All boards demonstrated local will and some local capacity, which explained the dimensions of enactment of local control. Rural board capacity—tenure, expertise, and residency of board members—influenced the extent of local control. Cross-case analysis revealed a relationship between board capacity, community capacity, and board–superintendent relationships. These patterns are theorized as local control school board–superintendent relationships typology, based on board capacity and local community capacity. The relationships include role contestation, confusion, collaboration, and dependence. Implications for Research and Practice: The study expands the theory of local capacity to include community and board dimensions and describes the complexities of superintendent–board relationships. The study concludes with recommendations for practice, policy, and research on locally controlled school boards.

Trajectories of HIV management among virally suppressed and unsuppressed female sex workers in the Dominican Republic: A comparative qualitative analysis

ABSTRACT Purpose Despite suboptimal HIV outcomes among female sex workers (FSW), limited research has been conducted on factors that impact viral suppression among this population. Examining narratives of HIV management, we examined how experiences of diagnosis, treatment initiation, and ongoing care behaviours shaped viral suppression outcomes over time. Methods We conducted 20 in-depth interviews with FSW in Santo Domingo, Dominican Republic. Using narrative and thematic qualitative approaches, we developed analytic summaries and matrices to compare trajectories of managing HIV between suppressed and unsuppressed participants. Results Regardless of suppression status, participants described similar narratives of overcoming initial challenges to HIV management through personal resilience and social support. Unsuppressed participants identified more delays in initiating antiretroviral therapy and more lapses in adherence due to less active acceptance of their HIV status and more persistent experiences of economic hardship and HIV stigma. Conclusions We found that individual, interpersonal and structural factors, including stigma and economic precarity, differentiated trajectories towards viral suppression among FSW indicating the importance of multilevel interventions. Improved access to mental health services and social support could promote greater early acceptance of HIV status and progress towards viral suppression among FSW.

Accurate cosmic microwave background covariance matrices: Exact calculation and approximations

Context. A reliable estimation of cosmological parameters from pseudo-power spectrum estimators requires accurate covariance matrices. Aims. We focus on the analytical calculation of covariance matrices. We consider the case of observations of the cosmic microwave background (CMB) in temperature and polarization on a small footprint such as in the South Pole Telescope third-generation (SPT-3G) experiment, which observes 4% of the sky. Power spectra evaluated on small footprints are expected to have strong correlations between modes, and these need to be accurately modeled. Methods. We present for the first time an algorithm that allows an efficient (but computationally expensive) exact calculation of analytic covariance matrices. Using it as our reference, we tested the accuracy of existing fast approximations of the covariance matrix. Furthermore, we propose a new approximation that is designed to be more precise. Finally, we derived the covariance matrices for mask-corrected power spectra estimated by the PolSpice code. In particular, in the case of a small sky fraction, we included the effect of the apodization of the large-scale modes. Results. We find that when the power spectrum is binned in wide bandpowers, current approximations of the covariance matrix are correct up to the 5% level on the SPT-3G small sky footprint. Our new approximation improves the previous approximations and reaches a precision of 1% for the wide bandpowers. It is generally more than four times more accurate than current approaches. Conclusions. While we considered the specific case of the CMB, our results are applicable to any other cosmological probe that requires the calculation of pseudo-power spectrum covariance matrices.

Most numbers are not normal

AbstractWe show, from a topological viewpoint, that most numbers are not normal in a strong sense. More precisely, the set of numbers $x \in (0,1]$ with the following property is comeager: for all integers $b\ge 2$ and $k\ge 1$ , the sequence of vectors made by the frequencies of all possibile strings of length k in the b-adic representation of x has a maximal subset of accumulation points, and each of them is the limit of a subsequence with an index set of nonzero asymptotic density. This extends and provides a streamlined proof of the main result given by Olsen (2004) in this Journal. We provide analogues in the context of analytic P-ideals and regular matrices.

"I Didn't Sign Up for This": Perspectives from Persons Living with Dementia and Care Partners on Challenges, Supports, and Opportunities to Add Geriatric Neuropalliative Care to Dementia Specialty Care.

In the United States, dementia specialty centers affiliated with centers of excellence for research hold promise as locations to develop innovative, holistic care in care systems otherwise siloed by discipline or payer. We conducted foundational research to inform development of patient-and family-centered palliative care interventions for dementia specialty centers. We interviewed persons living with dementia (PLWD), current, and former care partners (CP) recruited from a specialty dementia clinic and purposively selected for variation across disease syndrome and stage. A framework method of thematic analysis included coding, analytic matrices, and pattern mapping. 40 participants included 9 PLWD, 16 current CPs, and 15 former CPs of decedents; 48% impacted by Alzheimer's disease dementia. While help from family, support groups and adult day centers, paid caregiving, and sensitive clinical care were invaluable to PLWD, CPs, or both, these supports were insufficient to navigate the extensive challenges. Disease-oriented sources of distress included symptoms, functional impairment and falls, uncertainty and loss, and inaccessible care. Social and relational challenges included constrained personal and professional opportunities. The obligation and toll of giving or receiving caregiving were challenging. Clinical care challenges for PLWD and/or CPs included care fragmentation, insufficient guidance to inform planning and need for expert interdisciplinary clinical care at home. Findings highlight the breadth and gravity of gaps, which surpass the disciplinary focus of either behavioral neurology or palliative care alone. Results can inform the development of novel interventions to add principles of geriatrics and neuropalliative care to dementia care.

Analytic Gaussian covariance matrices for galaxy N -point correlation functions

We derive analytic covariance matrices for the $N$-Point Correlation Functions (NPCFs) of galaxies in the Gaussian limit. Our results are given for arbitrary $N$ and projected onto the isotropic basis functions of Cahn & Slepian (2020), recently shown to facilitate efficient NPCF estimation. A numerical implementation of the 4PCF covariance is compared to the sample covariance obtained from a set of lognormal simulations, Quijote dark matter halo catalogues, and MultiDark-Patchy galaxy mocks, with the latter including realistic survey geometry. The analytic formalism gives reasonable predictions for the covariances estimated from mock simulations with a periodic-box geometry. Furthermore, fitting for an effective volume and number density by maximizing a likelihood based on Kullback-Leibler divergence is shown to partially compensate for the effects of a non-uniform window function.

Invariants of Linear Control Systems with Analytic Matrices and the Linearizability Problem

The paper continues the authors’ study of the linearizability problem for nonlinear control systems. In the recent work (Sklyar, Syst Control Lett 134:104572, 2019), conditions on mappability of a nonlinear control system to a preassigned linear system with analytic matrices were obtained. In the present paper, we solve more general problem on linearizability conditions without indicating a target linear system. To this end, we give a description of invariants for linear nonautonomous single-input controllable systems with analytic matrices, which allow classifying such systems up to transformations of coordinates. This study leads to one problem from the theory of linear ordinary differential equations with meromorphic coefficients. As a result, we obtain a criterion for mappability of nonlinear control systems to linear control systems with analytic matrices.

Cosmological constraints from BOSS with analytic covariance matrices

We use analytic covariance matrices to carry out a full-shape analysis of the galaxy power spectrum multipoles from the Baryon Oscillation Spectroscopic Survey (BOSS). We obtain parameter estimates that agree well with those based on the sample covariance from two thousand galaxy mock catalogs, thus validating the analytic approach and providing substantial reduction in computational cost. We also highlight a number of additional advantages of analytic covariances. First, the analysis does not suffer from sampling noise, which biases the constraints and typically requires inflating parameter error bars. Second, it allows us to study convergence of the cosmological constraints when recomputing the analytic covariances to match the best-fit power spectrum, which can be done at a negligible computational cost, unlike when using mock catalogs. These effects reduce the systematic error budget of cosmological constraints, which suggests that the analytic approach may be an important tool for upcoming high-precision galaxy redshift surveys such as DESI and Euclid. Finally, we study the impact of various ingredients in the power spectrum covariance matrix and show that the non-Gaussian part, which includes the regular trispectrum and super-sample covariance, has a marginal effect ($\lesssim 10 \%$) on the cosmological parameter error bars. We also suggest improvements to analytic covariances that are commonly used in Fisher forecasts.

An efficient way to incorporate the geometric phase in the time-dependent wave packet calculations in a diabatic representation.

We present a new approach to incorporate the geometric phase in the time-dependent wave packet calculations based on the analytic diabatic potential energy matrices for two-state systems connecting via a conical intersection. The approach only requires information on the location of the conical intersection and the adiabatic potential energy surface of the ground electronic state and merely takes the same computational cost as a diabatic calculation. Demonstrations of the benchmark H + H2/HD reactions show that the new approach can accurately include the geometric phase in dynamics calculation and can be easily extended to the cold regime where the GP effects become more pronounced. Due to its simplicity and numerical efficiency, the new approach has the potential to extend the dynamics study of the geometric effects to a wide range of reaction systems.

Lyapunov behavior and dynamical localization for quasi-periodic CMV matrices

In this article, we prove Lyapunov behavior and dynamical localization for analytic quasi-periodic CMV matrices in the regime of positive Lyapunov exponents. The localization results of Wang and Damanik [28] are extended.

Cognitive testing of the PHQ-9 for depression screening among pregnant and postpartum women in Kenya

BackgroundAfrican women face high rates of depression, particularly during pregnancy or postpartum or after a recent HIV diagnosis. The Patient Health Questionnaire-9 (PHQ-9) depression screening tool has been quantitatively validated and extensively used to identify depression and link individuals to care. However, qualitative work is necessary to identify important opportunities to improve PHQ-9 question comprehension and performance among Kenyan women.MethodsWe administered the Kiswahili or English PHQ-9 (based on preference) to 29 pregnant and postpartum women in Thika, Kenya. Following administration, we conducted cognitive interviews with a purposive sample of 20 women. We used analytic memos and data matrices to identify themes around scale acceptability, comprehension, and decision and response processes.ResultsMost participants preferred to answer the PHQ-9 in Kiswahili (N = 15; 52%). Among the 20 interview participants, 12 (60%) had scores ≥5, indicating depressive symptoms. Overall, participants found the scale acceptable as an interviewer-administered tool. Participants reported few problems related to comprehension but had difficulty answering items not relevant to their lives (e.g., “watching television”) and double-barreled items (e.g., “poor appetite or overeating”). They were hesitant to endorse items related to “duties as a wife and mother” and suicidal ideation. Most participants had difficulty distinguishing between response options of “several days” and “more than half the days”.ConclusionsWe detected several problems related to PHQ-9 comprehension, decision processes, and response processes. We provide recommended changes to instructions and item wording to improve PHQ-9 validity among Kenyan women.

Observations on the Computation of Eigenvalue and Eigenvector Jacobians

Many scientific and engineering problems benefit from analytic expressions for eigenvalue and eigenvector derivatives with respect to the elements of the parent matrix. While there exists extensive literature on the calculation of these derivatives, which take the form of Jacobian matrices, there are a variety of deficiencies that have yet to be addressed—including the need for both left and right eigenvectors, limitations on the matrix structure, and issues with complex eigenvalues and eigenvectors. This work addresses these deficiencies by proposing a new analytic solution for the eigenvalue and eigenvector derivatives. The resulting analytic Jacobian matrices are numerically efficient to compute and are valid for the general complex case. It is further shown that this new general result collapses to previously known relations for the special cases of real symmetric matrices and real diagonal matrices. Finally, the new Jacobian expressions are validated using forward finite differencing and performance is compared with another technique.

On mappability of control systems to linear systems with analytic matrices

The paper deals with the problem of analytic linearizability for nonlinear non-autonomous control systems, i.e., the problem of mapping of a nonlinear non-autonomous control system to a linear system with analytic matrices by a transformation of coordinates. We introduce a concept of a driftless form of a non-autonomous system, which allows simplifying our analysis. We give linearizability conditions and find invariants for nonlinear systems that are mappable to a preassigned linear system.

Trauma and Relationship Strain: Oral Histories With World Trade Center Disaster Responders.

Existing models of couple functioning after trauma are primarily based on the experiences of returning military veterans. In this study, we conducted thematic analysis of a purposive sample of 49 oral histories of responders to the 9/11/01 terrorist attacks to understand how they navigated life with their spouses after the response experience. Use of multiple coders and analytic matrices increased analytic rigor. In the sample, 34.7% disclosed a posttraumatic stress disorder (PTSD) diagnosis and another 22.7% mentioned experiencing at least one trauma symptom. Most responders had not sought mental health intervention, relying instead on their spouses' caregiving. Responders reported limited disclosure to their spouses about the details of their 9/11/01 response work, which may have helped them cope emotionally with repeated 9/11/01 clean-up duties. Shared values regarding the common good and patriotism were important for maintaining an intimate relationship after 9/11/01, and helping partners understand and feel understood by each other.

Analytic expressions of the Transmission, Reflection, and source function for the community radiative transfer model

The phase matrix in scalar radiative transfer is symmetric that grants reciprocity principle in radiative transfer. The reciprocity principle is so useful that one may compute a single transmission matrix and a single reflection matrix for a homogeneous medium regardless of in upward or downward direction. The symmetric phase matrix is also important as one only needs to solve for a real eigensolution. An eigensolution is often used in a radiative transfer solver because of its high computational efficiency. However, the phase matrix in vectorized radiative transfer is generally not symmetric which challenges the reciprocity principle and forces us to deal with a complex eigensolution that requires a major effort in computational coding tangent-linear and adjoint models. This paper introduces an approach to retain the reciprocity principle in radiative transfer and applies a Taylor expansion of analytic transmittance and reflection matrices for a base optical depth together with a doubling-adding method beyond the base in the vectorized Community Radiative Transfer Model (CRTM). The value of the base optical depth depends on the maximum absolute value of the phase matrix elements. In comparison with other forward radiative transfer models, the extended vectorized CRTM agrees well with those models. The computational efficiency among the CRTM and those models is comparable. The tangent-linear and adjoint modules of the vectorized CRTM can be used for assimilating microwave, infrared, visible and ultraviolet sensor radiances.

Characterization of the consistent completion of analytic hierarchy process comparison matrices using graph theory

AbstractDecision‐making is frequently affected by uncertainty and/or incomplete information, which turn decision‐making into a complex task. It is often the case that some of the actors involved in decision‐making are not sufficiently familiar with all of the issues to make the appropriate decisions. In this paper, we are concerned about missing information. Specifically, we deal with the problem of consistently completing an analytic hierarchy process comparison matrix and make use of graph theory to characterize such a completion. The characterization includes the degree of freedom of the set of solutions and a linear manifold and, in particular, characterizes the uniqueness of the solution, a result already known in the literature, for which we provide a completely independent proof. Additionally, in the case of nonuniqueness, we reduce the problem to the solution of nonsingular linear systems. In addition to obtaining the priority vector, our investigation also focuses on building the complete pairwise comparison matrix, a crucial step in the necessary process (between synthetic consistency and personal judgement) with the experts. The performance of the obtained results is confirmed.

Improved potential energy surfaces of thioanisole and the effect of upper surface variations on the product distribution upon photodissociation

Electronically nonadiabatic photodissociation can be investigated in detail by experiments. This provides an opportunity to validate the current theory of electronically nonadiabatic processes, but is great challenge because the time scale of photodissociation processes can be long compared to the current capability for accurate direct dynamics simulations. To circumvent this difficulty, we have been using analytic diabatic potential energy matrices, such that the simulation time scale can be extended to the nanosecond region without neglecting the effects of external electron correlation. In previous work, we have developed full-dimensional three-state potential energy matrices for thioanisole photodissociation. In the current work, we extend this work in two main ways: (i) we improve the treatment of the initial torsional potential, and (ii) we shift the potential energy surfaces to investigate the quantitative effect on the dissociation lifetimes and product branching ratios of changing the energy and location of the S1–S2 conical intersection.

The Effects of WTO and Preferential Trade Agreements on Trade Flows: Count Data Models With Dual Endogenous Binary Treatments

The World Trade Organization (WTO) and preferential trade agreements (PTA) are key factors in international trade liberalization. However, their relative effects on trade flows are controversial (ranging from 16 percent to 277 percent), depending on whether the endogeneity of these two variables is taken into account. Surprisingly, there is no paper estimating the endogenous trade effects of WTO and PTAs simultaneously, thus the existing findings about these variables might be questioned. To fill the gap of the trade literature, we thus extend the econometric model of Terza (1998) and empirical works of Egger et al. (2011) by proposing a count data models with dual endogenous binary regressors. A two-step quasi-maximum likelihood approach to estimation and inference is introduced. Since we derive the analytic gradient and hessian matrices of the log likelihood function of the proposed model, we can easily conduct the Monte Carlo simulations where the data observation is over 10,000 and the parameters to be estimated are over several hundreds, as is typically found in the empirical studies of international economics. Indeed, the simulation results are promising under various configurations considered in this paper. We then apply our approach to the data of Egger et al. (2011) who only investigate the trade effects of PTA. Our results show that both WTO and PTA membership have huge impacts on bilateral trade flows, increasing the value by 926 percent and 218 percent respectively. On the contrary, their partial trade effects are much smaller (13 percent and 74 percent) if the endogeneity is totally ignored.