Unknown Length Research Articles

Application of cultured epidermal autograft, JACE®, improves survival rate in extensive burns: A propensity score matching study using Tokyo registry data.

Cultured epidermal autograft, JACE®, was introduced into the Japanese national health insurance system in 2009 and has been used in more than 1000 cases of extensive burns. The aim of this study was to investigate whether the use of JACE® contributes to survival rate in extensive burns. In this study, 119 cases were selected from 3990 cases in Tokyo Burn Unit Association registry data from 2009 to 2023, excluding cases with less than 40% total body surface area, cases of deaths within 4 weeks and cases with unknown length of hospital stay. In total, 25 patients treated with JACE® were selected and matched with another 25 patients who did not receive JACE® using propensity score matching. The results showed that patients treated with JACE® had a significantly higher survival rate than did those who were not treated with JACE® at all time points between 6 and 9 weeks post-injury. In addition, there was no significant difference in length of hospital stay between the groups. These results suggest that the use of JACE® in patients with extensive burns contributes to patient survival and does not prolong hospital stay.

“I Can See Your Password”: A Case Study About Cybersecurity Risks in Mid-Air Interactions of Mixed Reality-Based Smart Manufacturing Applications

Abstract This paper aims to present a potential cybersecurity risk existing in mixed reality (MR)-based smart manufacturing applications that decipher digital passwords through a single RGB camera to capture the user’s mid-air gestures. We first created a test bed, which is an MR-based smart factory management system consisting of mid-air gesture-based user interfaces (UIs) on a video see-through MR head-mounted display. To interact with UIs and input information, the user’s hand movements and gestures are tracked by the MR system. We setup the experiment to be the estimation of the password input by users through mid-air hand gestures on a virtual numeric keypad. To achieve this goal, we developed a lightweight machine learning-based hand position tracking and gesture recognition method. This method takes either video streaming or recorded video clips (taken by a single RGB camera in front of the user) as input, where the videos record the users’ hand movements and gestures but not the virtual UIs. With the assumption of the known size, position, and layout of the keypad, the machine learning method estimates the password through hand gesture recognition and finger position detection. The evaluation result indicates the effectiveness of the proposed method, with a high accuracy of 97.03%, 94.06%, and 83.83% for 2-digit, 4-digit, and 6-digit passwords, respectively, using real-time video streaming as input with known length condition. Under the unknown length condition, the proposed method reaches 85.50%, 76.15%, and 77.89% accuracy for 2-digit, 4-digit, and 6-digit passwords, respectively.

The role of efferocytosis and transplant rejection: Strategies in promoting transplantation tolerance using apoptotic cell therapy and/or synthetic particles.

Recently, efforts have been made to recognize the precise reason(s) for transplant failure and the process of rejection utilizing the molecular signature. Most transplant recipients do not appreciate the unknown length of survival of allogeneic grafts with the existing standard of care. Two noteworthy immunological pathways occur during allogeneic transplant rejection. A nonspecific innate immune response predominates in the early stages of the immune reaction, and allogeneic antigens initiate a donor-specific adaptive reaction. Though the adaptive response is the major cause of allograft rejection, earlier pro-inflammatory responses that are part of the innate immune response are also regarded as significant in graft loss. The onset of the innate and adaptive immune response causes chronic and acute transplant rejection. Currently employed immunosuppressive medications have shown little or no influence on chronic rejection and, as a result, on overall long-term transplant survival. Furthermore, long-term pharmaceutical immunosuppression is associated with side effects, toxicity, and an increased risk of developing diseases, both infectious and metabolic. As a result, there is a need for the development of innovative donor-specific immunosuppressive medications to regulate the allorecognition pathways that induce graft loss and to reduce the side effects of immunosuppression. Efferocytosis is an immunomodulatory mechanism with fast and efficient clearance of apoptotic cells (ACs). As such, AC therapy strategies have been suggested to limit transplant-related sequelae. Efferocytosis-based medicines/treatments can also decrease the use of immunosuppressive drugs and have no detrimental side effects. Thus, this review aims to investigate the impact of efferocytosis on transplant rejection/tolerance and identify approaches using AC clearance to increase transplant viability.

Persistence Length of PEGMA Bottle Brushes Determined by Pyrene Excimer Fluorescence.

Seven pyrene-labeled poly(oligo(ethylene glycol) methyl ether methacrylate)s (PyEG5-PEGnMAs) were prepared with n = 0, 3, 4, 5, 7, 9, and 19 ethylene glycol units by copolymerizing a small amount of penta(ethylene glycol) 1-pyrenemethyl ether methacrylate with an EGnMA monomer. The conformation of the PyEG5-PEGnMA polymers evolved from a random coil for PyEG5-PEG0MA or poly(methyl methacrylate) to a polymeric bottle brush (PBB) architecture with increasing side chain length. The fluorescence decays of the PyEG5-PEGnMA samples were fitted according to the fluorescence blob model (FBM) whose parameters were used, in combination with the Kratky-Porod equation, to calculate the persistence length of these polymers. The persistence lengths obtained from the PEF experiments were found to increase with the square of the number (NS) of non-hydrogen atoms in the side chain as expected theoretically. The persistence lengths found with the PyEG5-PEGnMA samples in DMF also matched those found earlier for another series of PEGnMA samples labeled with 1-pyrenebutanol. The good agreement found between the persistence lengths obtained with the PEGnMA samples labeled with two different pyrene derivatives illustrates the robustness of the method and its applicability for measuring the unknown persistence length of polydisperse polymer samples.

Passivity based hierarchical multi-task tracking control for redundant manipulators with uncertainties

Most research so far on hierarchical multi-task control with strict priority requires exact dynamic and kinematic models. However, when a robot picks up tools of unknown lengths, orientations and mass, its kinematics and dynamics are varying and unavailable. There are few controllers that can achieve asymptotic tracking of multi-task in the presence of kinematic and dynamic uncertainties. Thus, in this paper we present a passivity-based adaptive Jacobian hierarchical multi-task controller. New reference velocity and kinematic regressor matrix are defined for the multi-task case with strict priority to guarantee the asymptotic convergence of both tracking error and parameter estimation error. A novel natural adaption law is specially designed for this method to guarantee the correctness of estimated dynamic parameters and increase the rate of convergence. Experiment results are presented to illustrate the performance of the proposed controller.

“I Just Wanted Nothing More Than to Get in a Real Shower”: Patient Experience of the Inpatient Wait for a Heart Transplant

BackgroundPatients waiting for heart transplant may be hospitalized for weeks to months before undergoing transplantation. This high-stress period is further complicated by restrictions of daily privileges including diet, rooming, access to the outdoors, and hygiene (eg, limited in ability to shower). However, there is a paucity of research on the experience of this waiting period. We sought to describe the inpatient experience among patients awaiting heart transplantation and to better understand the needs of inpatients waiting for heart transplant. Methods and ResultsWe conducted in-depth, semistructured phone interviews with a purposeful sample of patients who received a heart transplant in the past 10 years and waited in the hospital for at least 2 weeks before surgery. Using the prior literature, the lived experience of the lead author, and input from qualitative experts, we developed an interview guide. Interviews were recorded, transcribed, and analyzed in an iterative process until theoretical saturation was achieved. A 3-person coding team identified, discussed, and reconciled emergent themes. We conducted interviews with 15 patients. Overarching themes included food, hygiene, relationship with health care professionals, living environment, and stressors. Patients reported that strong bonds were formed between the patients and the staff, and the overwhelming majority only had positive comments about these relationships. However, many expressed negative comments about the experience of the food and limitations in personal hygiene. Other stressors included the unknown length of the waiting period, lack of communication about position on the transplant list, worry about family, and concerns that their life must be saved by the death of another. Many participants described that they would benefit from more interaction with recent heart transplant recipients. ConclusionsHospitals and care units have the opportunity to make small changes that could greatly benefit the experience of waiting for a heart transplant, as well as the experience of hospitalization more generally.

Comparison of radiography and CT for the evaluation of third carpal bone slab fractures in Thoroughbred racehorses.

Slab fractures of the third carpal bone (C3) are a common cause of lameness in Thoroughbred racehorses. Information on fracture morphology is commonly obtained from radiographs or CT. This retrospective, methods comparison aimed to explore the agreement between radiography and CT for imaging C3 slab fractures and discuss the contribution of the latter to clinical case management. Thoroughbred racehorses with a slab or incomplete slab fracture of C3 identified on radiographs that subsequently underwent CT examination were included. Fracture characteristics (location, plane, classification, displacement, comminution) and fracture length as a percentage of the proximodistal length of the bone, termed the proximodistal fracture percentage (PFP) were recorded independently from both modalities and then compared. Across all fractures (n=82) radiographs and CT showed slight agreement on the presence of comminution (Cohen's Kappa (κ) 0.108, P 0.031) and moderate agreement on fracture displacement (K 0.683, P<0.001). Computed tomography identified comminution in 49 (59.8%) and displacement in nine (11.0%) fractures that were not detected by radiographs. Half of the fractures were only seen on flexed dorsoproximal-dorsodistal oblique (DPr-DDiO) radiographs and therefore were of unknown length without additional CT imaging. Incomplete fractures that could be measured on radiographs (n=12) had a median (IQR) PFP of 40% (30%-52%) on radiographs and 53% (38%-59%) on CT, a statistically significant difference (P= 0.026). Radiography and CT showed the poorest agreement when determining the presence of comminution. Additionally, radiography often underestimated the incidence of displacement, and fracture length, and resulted in more fractures being classified as incomplete when compared to CT.

Second gradient continuum model for anisotropic elastic and piezoelectric structures calibrated based on phonon dispersion relations

In this paper, we present a variant of the second gradient continuum model that allows description of the spatial dispersion effects of high-frequency waves in anisotropic crystals. The considered model takes into account the strain and inertia gradient effects and the classical electro-mechanical coupling. The simplified constitutive equations of the model contain seven and four length scale parameters for piezoelectric hexagonal crystals and elastic cubic crystals, respectively. It is shown that all except one length scale parameter can be identified based on the fitting of the continuum model to the lattice dynamics calculations for the dispersion relations of the bulk acoustic phonons. Examples of the identification are presented for hexagonal piezoelectric ZnO and AlN and for cubic diamond and Si crystals. Using a calibrated model, we then obtain the dispersion relations for the high-frequency Bleustein–Gulyaev wave in piezoelectric crystals and for the shear horizontal surface wave in elastic crystals. The last one is predicted by the gradient models only for the high frequencies, while at low frequencies this wave degenerates to the bulk shear wave. Examples of calculations are also provided for the Rayleigh wave propagating through the homogeneous piezoelectric half-space and layered ZnO(AlN)/diamond/Si structures. It is shown that this kind of surface wave can be used to identify the last unknown length scale parameter of the model.

Understory plant communities fail to recover species diversity after excluding deer for nearly 20 years

White-tailed deer ( Odocoileus virginianus) have been overabundant in eastern North America for more than five decades, resulting in depauperate understories and ricocheting effects on higher trophic levels. Even after deer populations are reduced, understory plant communities may fail to recover for an unknown length of time due to persistent legacy effects. We surveyed understory plant communities in six deer exclosures and paired reference plots in northwestern Pennsylvania to determine the degree to which 19 years of deer exclusion was sufficient for recovery of species richness, diversity, percent cover, and understory structural complexity. We observed a 2.3-fold increase in tree cover and a 60% reduction in fern cover in the ground layer, as well as a 114-fold increase in foliage density between 80 and 200 cm above ground level, in exclosures compared to reference plots. However, the exclosures did not permanently support higher overall percent cover, species richness, or diversity in the ground layer, nor did we detect any meaningful divergence in community composition between exclosures and reference plots. We conclude that 19 years of release from chronic over-browsing are sufficient to restore understory structural complexity, but recovery of diversity in the ground layer will require more time or direct intervention.

Mapping of quantitative trait locus reveals PsXI gene encoding xylanase inhibitor as the candidate gene for bruchid (Callosobruchus spp.) resistance in pea (Pisum sativum L.).

Pea (Pisum sativum L.) is an important legume crop for both food and feed. Bruchids (Callosobruchus spp.) are destructive insect pests of pea in the field and during storage. In this study, we identified a major quantitative trait locus (QTL) controlling seed resistance to C. chinensis (L.) and C. maculatus (Fab.) in field pea using F2 populations derived from a cross between PWY19 (resistant) and PHM22 (susceptible). QTL analysis in the two F2 populations grown in different environments consistently identified a single major QTL, qPsBr2.1, controlling the resistance to both bruchid species. qPsBr2.1 was mapped onto linkage group 2 between DNA markers 18339 and PSSR202109 and explained 50.91% to 70.94% of the variation in resistance, depending on the environment and bruchid species. Fine mapping narrowed down qPsBr2.1 to a genomic region of 1.07 Mb on chromosome 2 (chr2LG1). Seven annotated genes were found in this region, including Psat2g026280 (designated as PsXI), which encodes a xylanase inhibitor and was considered as a candidate gene for bruchid resistance. PCR amplification and sequence analysis of PsXI suggested the presence of an insertion of unknown length in an intron of PWY19, which causes variation in the open reading frame (ORF) of PsXI. Moreover, the subcellular localization of PsXI differed between PWY19 and PHM22. These results together suggested that PsXI encoding xylanase inhibitor is responsible for the bruchid resistance of the field pea PWY19.

Hummingbird: Dynamic Path Validation With Hidden Equal-Probability Sampling

Path validation has already been incrementally deployed in the Internet architecture. It secures packet forwarding by enabling end hosts to negotiate specific forwarding paths and enforcing on-path routers to prove their forwarding behaviors along these paths. Most existing path validation solutions target static paths, paying less attention to fully dynamic paths that support flexible routing. In this paper, we present Hummingbird as the first validation solution over fully dynamic paths. It features a hidden equal-probability sampling technique. Gaining efficiency via routers probabilistically sampling packets to validate, we craft the sampling probability such that each router validates a similar amount of packets given an unknown path length. We further hide the state of whether a packet has been sampled and validated using a lightweight, non-cryptographic scheme. This prevents attackers from differentiating and selectively mis-forwarding packets. We validate security and efficiency of Hummingbird through both theoretical proof and experimental evaluation.

Local structure of liquid/vapour interfaces approaching the critical point.

Investigating the structure of fluid interfaces at high temperatures is a particularly delicate task that requires effective ways of discriminating liquid from vapour and identifying the location of the liquid phase boundary, thereby allowing to distinguish intrinsic from capillary fluctuations. Several numerical approaches require introducing a coarse-graining length scale, often heuristically chosen to be the molecular size, to determine the location of the liquid phase boundary. Here, we propose an alternative rationale for choosing this coarse-graining length scale; we require the average position of the local liquid phase dividing surface to match its flat, macroscopic counterpart. We show that this approach provides additional insight into the structure of the liquid/vapour interface, suggesting the presence of another length scale beyond the bulk correlation one that plays an important role in determining the interface structure.

An actinometric method to characterize performance of reflecting UVC reactors used for water treatment

The applicability of chemical actinometry to characterize the fluence in UV reactors with reflections, non-parallel light, and variable water transmittance is limited due to the unknown effective path length or hydraulic shortcuts within the reactor. In this study, the effects of reflection and transmittance on actinometry were examined and a new, optimized and easy method for determining fluence was developed. KI/KIO3 and uridine actinometry experiments were carried out under controlled conditions using a collimated beam apparatus and a completely mixed batch reactor with or without diffuse reflection and compared to biodosimetry results. Whereas optically opaque actinometers such as KI/KIO3 are not directly capable of predicting the fluence of reflecting reactors, the results of uridine actinometry are influenced by reflection and transmission. To precisely predict the fluence rate in UV reactors with uridine, knowledge about the effective optical path length of the light is needed. Here, an existing method to mathematically calculate the optical path length was adopted and optimized for uridine actinometry. Results for average fluence were validated by biodosimetry using MS2 phages under different degrees of reflection and transmission. It could be shown that by modifying the bottom of the reactor with diffusely reflecting polytetrafluoroethylene foil, the fluence rate was increased by a factor of approximately 2.6 and the path length by factor of 2.4. When only half of the bottom was covered with reflective foil, fluence rate increased by a factor of 1.8 and path length by 1.8. Although this new approach cannot replace biodosimetry, to predict the fluence distribution received by microorganisms, it can provide means to characterize more complex reactor designs, validate results of advanced reactor modeling, and quantify fluence for non-parallel irradiation and reflective light, especially for the application of high fluence (e.g., advanced oxidation processes), where biodosimetry may be too sensitive. Further, comparing the fluence obtained with actinometry to the results of biodosimetry might qualitatively indicate hydraulic short cuts or unideal fluence distributions for flow-through reactors.

Automatic orientation of historical terrestrial images in mountainous terrain using the visible horizon

Historical terrestrial images are the only visual sources documenting alpine environments shortly after the end of the Little Ice Age. Despite their unique value, they are largely unused for quantifying environmental changes because of the difficult and time-consuming estimation of the unknown camera parameters. For most images large parts of the captured scenery have vastly changed over time, making automatic feature point matching infeasible. In contrast, the visible image horizon seems to remain stable over time and hence, appears to be a suitable feature for image orientation. Since the focal length is unknown for historical terrestrial images, existing methods, focusing solely on estimating the exterior orientation of recent imagery, can not be applied. Accordingly, it was investigated if the horizon is suitable to estimate both the interior and exterior orientation of historical terrestrial images, with an accuracy comparable to manually oriented images. In a first step, the whole horizon was used to approximate the unknown camera parameters, reducing the potential search space. In the subsequent spatial resection these approximations were further refined using salient points along the horizon. We evaluated our approach using 204 manually oriented reference images. With the proposed method the accuracy of the estimated exterior orientation could be significantly improved compared to previous works. Additionally, the unknown focal length was estimated within 5% of the true focal length for 75% of the images. As historical terrestrial images are commonly used for monoplotting, the accuracy for 2400 manually selected checkpoints was evaluated. This analysis showed that for 63% of the images the same accuracy as with manually oriented images was achieved. For additional 22% the estimated camera parameters were still accurate enough to serve as initial estimates for a subsequent manual orientation. In 15% of the images our method completely failed. Due to the vastly changing scenery and oblique viewing geometry, finding the initial camera parameters, in our experience, is often the most challenging and time consuming step during manual orientation of historical images. Hence, in 85% of the images this initial step can be replaced with our method, leading to a significantly reduced effort for orienting whole collections of historical terrestrial images.

Fast and Accurate Pose Estimation with Unknown Focal Length Using Line Correspondences.

Estimating camera pose is one of the key steps in computer vison, photogrammetry and SLAM (Simultaneous Localization and Mapping). It is mainly calculated based on the 2D-3D correspondences of features, including 2D-3D point and line correspondences. If a zoom lens is equipped, the focal length needs to be estimated simultaneously. In this paper, a new method of fast and accurate pose estimation with unknown focal length using two 2D-3D line correspondences and the camera position is proposed. Our core contribution is to convert the PnL (perspective-n-line) problem with 2D-3D line correspondences into an estimation problem with 3D-3D point correspondences. One 3D line and the camera position in the world frame can define a plane, the 2D line projection of the 3D line and the camera position in the camera frame can define another plane, and actually the two planes are the same plane, which is the key geometric characteristic in this paper's estimation of focal length and pose. We establish the transform between the normal vectors of the two planes with this characteristic, and this transform can be regarded as the camera projection of a 3D point. Then, the pose estimation using 2D-3D line correspondences is converted into pose estimation using 3D-3D point correspondences in intermediate frames, and, lastly, pose estimation can be finished quickly. In addition, using the property whereby the angle between two planes is invariant in both the camera frame and world frame, we can estimate the camera focal length quickly and accurately. Experimental results show that our proposed method has good performance in numerical stability, noise sensitivity and computational speed with synthetic data and real scenarios, and has strong robustness to camera position noise.

Nuclear magnetic resonance as a tool to determine chlorine percentage of chlorinated paraffin mixtures

A new simple method for chlorine percentage calculations (method C), from proton nuclear magnetic resonance (1H NMR) spectroscopy, has been established and applied to an industrial chlorinated paraffin (CP) mixture and 13 single-chain CPs of known carbon chain lengths. Two modified methods (method A and B), originating from the work of Sprengel et al., have been utilized on the same single-chain mixtures. All samples were analysed by 1H NMR and two-dimensional heteronuclear quantum coherence (HSQC) for this purpose. All three methods worked well for medium chlorinated (45–55% Cl) single-chain mixtures of known carbon chain lengths. Method A yielded the best result for mixtures of lower chlorine content (<45% Cl), method C gave better estimations for higher chlorine contents (>55% Cl). Compared to Mohr's titration, method A showed a deviation of 0.7–7.8% (3.6% average), method B 4.1–11.3% (7.0% average) and method C 0.6–11.6% (5.2% average), for all 13 single-chain mixtures. The new method C is the only method that could be applied for determining the chlorine percentage of industrial mixtures of multiple, unknown chain lengths.

Calibrating the Scan Statistic: Finite Sample Performance Versus Asymptotics

Abstract We consider the problem of detecting an elevated mean on an interval with unknown location and length in the univariate Gaussian sequence model. Recent results have shown that using scale-dependent critical values for the scan statistic allows to attain asymptotically optimal detection simultaneously for all signal lengths, thereby improving on the traditional scan, but this procedure has been criticised for losing too much power for short signals. We explain this discrepancy by showing that these asymptotic optimality results will necessarily be too imprecise to discern the performance of scan statistics in a practically relevant way, even in a large sample context. Instead, we propose to assess the performance with a new finite sample criterion. We then present three calibrations for scan statistics that perform well across a range of relevant signal lengths: The first calibration uses a particular adjustment to the critical values and is therefore tailored to the Gaussian case. The second calibration uses a scale-dependent adjustment to the significance levels rather than to the critical values and this adjustment is therefore applicable to arbitrary known null distributions. The third calibration restricts the scan to a particular sparse subset of the scan windows and then applies a weighted Bonferroni adjustment to the corresponding test statistics. This Bonferroni scan is also applicable to arbitrary null distributions and in addition is very simple to implement. We show how to apply these calibrations for scanning in a number of distributional settings: for normal observations with an unknown baseline and a known or unknown constant variance, for observations from a natural exponential family, for potentially heteroscadastic observations from a symmetric density by employing self-normalisation in a novel way, and for exchangeable observations using tests based on permutations, ranks or signs.

Pottery from Tell Khaiber: A Craft Tradition of the First Sealand Dynasty By Daniel Calderbank (Archaeology of Ancient Iraq 1). Ludlow: Moonrise Press 2021. Pp. xii + 282. £ 35. ISBN 978-1-910169-02-5 (cloth).

<i>Pottery from Tell Khaiber: A Craft Tradition of the First Sealand Dynasty</i> By Daniel Calderbank (Archaeology of Ancient Iraq 1). Ludlow: Moonrise Press 2021. Pp. xii + 282. £ 35. ISBN 978-1-910169-02-5 (cloth).

CALCULATION OF A TRAJECTORY FOR THE MOVEMENT OF A LOAD OF SWINGING SPRING

Pendulum oscillations in the vertical plane of a suspended weightless spring, while maintaining the straightness of its axis, are considered. In the literature, this type of pendulum is called a swinging spring. The required trajectory of the load of the swinging spring is modeled using a computer using the values ​​of the mass of the load, the stiffness of the spring and its length without load. In addition, the initial values ​​of the parameters for initiating oscillations of the oscillating spring are used: the initial angle of deviation of the spring axis from the vertical, the initial rate of change of this angle, as well as the initial parameter of the spring elongation and the initial rate of change of the elongation. The calculations were performed using the Lagrange equation of the second kind. Variants of finding conditionally periodic trajectories of movement of a point load of a swinging spring with a movable attachment point are considered.&#x0D; The relevance of the topic is determined by the need to research and improve new technological schemes of mechanical devices, which include springs. In particular, the study of the conditions for separating from chaotic vibrations of mechanical structure elements and determining the rational values ​​of parameters to ensure periodic trajectories of their vibrations.&#x0D; Calculation options are given to obtain periodic trajectories of cargo movement, when the given parameters are:&#x0D; - the length of the spring without load and its stiffness with an unknown value of the mass of the load;&#x0D; - the length of the spring without load and the value of the mass of the load with an unknown spring stiffness;&#x0D; - the value of the mass of the load and the stiffness of the spring with an unknown length of the spring without load.&#x0D; The results can be used as a paradigm for studying nonlinear coupled systems, as well as for calculating options for mechanical devices, when it is necessary to dissociate themselves from chaotic movements of loads, and provide periodic trajectories of their movement.

Periodic Orbits of Nonlinear Ordinary Differential Equations Computed by a Boundary Shape Function Method

In the paper, we determine the period of an n-dimensional nonlinear dynamical system by using a derived formula in an (n + 1)-dimensional augmented space. To form a periodic motion, the periodic conditions in the state space and nonlinear first-order differential equations constitute a special periodic problem within a time interval with an unknown length. Two periodic problems are considered: (a) boundary values are given and (b) boundary values are unknown. By using the shape functions, a boundary shape function method (BSFM) is devised to obtain an initial value problem with the initial values of the new variables given. The unknown terminal values of the new variables and period are determined by two iterative algorithms for the case (a) and one iterative algorithm for the case (b). The periodic solutions obtained from the BSFM satisfy the periodic conditions automatically. For the numerical example, the computed order of convergence displays the merit of the BSFM. For the sake of comparison, the iterative algorithms based on the shooting method for cases (a) and (b) were developed by directly implementing the Poincaré map into the fictitious time-integration method to determine the period. The BSFM is better than the shooting method in terms of convergence speed, accuracy, and stability.