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# Spatially-resolved Brillouin spectroscopy reveals biomechanical abnormalities in mild to advanced keratoconus in vivo

## Abstract

Mounting evidence connects the biomechanical properties of tissues to the development of eye diseases such as keratoconus, a disease in which the cornea thins and bulges into a conical shape. However, measuring biomechanical changes in vivo with sufficient sensitivity for disease detection has proven challenging. Here, we demonstrate the diagnostic potential of Brillouin light-scattering microscopy, a modality that measures longitudinal mechanical modulus in tissues with high measurement sensitivity and spatial resolution. We have performed a study of 85 human subjects (93 eyes), consisting of 47 healthy volunteers and 38 keratoconus patients at differing stages of disease, ranging from stage I to stage IV. The Brillouin data in vivo reveal increasing biomechanical inhomogeneity in the cornea with keratoconus progression and biomechanical asymmetry between the left and right eyes at the onset of keratoconus. The receiver operating characteristic analysis of the stage-I patient data indicates that mean Brillouin shift of the cone performs better than corneal thickness and maximum curvature respectively. In conjunction with morphological patterns, Brillouin microscopy may add value for diagnosis of keratoconus and potentially for screening subjects at risk of complications prior to laser eye surgeries.

## Introduction

Growing evidence indicates that the biomechanical properties of ocular tissues can be diagnostic targets due to their association with various eye diseases and refractive errors1. The cornea is a prototypical example, requiring adequate mechanical stiffness to maintain the structure necessary for good vision in the presence of external mechanical stresses such as in-plane tension and intraocular pressure (IOP)2. The mechanical properties of the cornea stem from the intricate lattice of macromolecules, including collagen fibers and proteoglycans, making up the corneal stroma3,4. Disintegration of this structure alters the biomechanical properties of the stroma and shifts the overall mechanical homeostasis, potentially leading to vision-impairing morphological changes5,6.

Keratoconus (KC) is a bilateral, noninflammatory disease in which the normal prolate cornea locally thins and bulges into a conical shape7. Its etiology is not fully understood, but numerous experimental studies have suggested the importance of biomechanics. Disrupted collagen orientation8,9,10 and reduced mechanical moduli2 have been observed in KC explants and corneas that developed ectasia after laser-assisted in situ keratomileusis (LASIK). In addition, genetic and molecular studies have linked KC to disintegration of the collagen extracellular matrix11,12,13.

Current KC diagnosis is primarily focused on detecting abnormalities in corneal thickness and curvature. Advances in pachymetry and topography have greatly improved the diagnosis and treatment monitoring of KC. However, these morphological changes are thought to be secondary to biomechanical degeneration. Early-stage or ‘subclinical’ KC prior to the appearance of definitive morphological abnormality is considered a major risk factor for post refractive-surgery ectasia. The unmet clinical need for earlier KC detection and screening has motivated the development of increasingly sophisticated morphology-based metrics, specific genetic and molecular markers, and other diagnostic approaches such as measuring corneal biomechanics1,14.

Recently, two commercial instruments: the ocular response analyzer (ORA, Reichert Technologies Inc., USA) and the corneal visualization Scheimpflug technology (Corvis ST, OCULUS Optikgeräte GmbH, Germany), became available, which use a single air-puff to induce millimeter-scale deformation of the cornea and extract various viscoelastic parameters15. Numerous clinical studies have demonstrated the potential of these instruments for detecting moderate to severe KC16,17,18,19,20,21,22,23. Vinciguerra et al. have developed a biomechanical index based on the corneal thickness profile and several Corvis deformation parameters, which has shown potential for diagnosis of early-stage KC24,25. Recent ORA and Corvis studies have made progress towards screening for subclinical KC26,27,28. A drawback of the air-puff technique is that the corneal deformation is inherently dependent on additional factors such as IOP and the anatomical geometry of the eye29. Additionally, this technique can only measure the overall corneal stiffness, so is less sensitive to changes in a specific region of the cornea. Other promising techniques are under development, such as optical coherence elastography30,31,32,33 using various external stimuli34,35,36,37,38,39. However, the safety and feasibility of these techniques in vivo has yet to be tested.

We have previously demonstrated Brillouin light-scattering microscopy, which makes it possible to measure longitudinal mechanical modulus in tissues40. Unlike air-puff-based methods, the technology does not involve corneal deformation and can interrogate a specific region and depth in corneal tissue with optical resolution41. We have established the safety of this technology in humans and reported Brillouin elasticity maps of advanced KC patients’ corneas, providing in vivo evidence of spatial heterogeneity in their biomechanical properties42.

In this paper, we describe the largest clinical Brillouin studies to date, focusing on the early stages of KC. To detect subtle biomechanical changes in mild and moderate KC, we improved the measurement sensitivity and repeatability of Brillouin systems. A careful analysis of Brillouin frequency shifts measured in vivo led to several findings about previously unknown, biomechanical features in KC corneas and normal corneas. We present results that point to promising diagnostic metrics based on spatially-resolved biomechanical measurements.

## Results

### Development of in vivo Brillouin microscopes

We have built two Brillouin microscopes with nearly identical optical designs. One tabletop system was used for studies at MGH in Boston (Fig. 1A). The other rack-mounted system was built at MGH and shipped to Zürich for studies at IROC (Fig. 1B). Compared to our previous instruments42, the new systems have several hardware and software upgrades to improve measurement sensitivity, stability and eye tracking, as well as mechanical robustness. Each system consists of a light source, human interface, spectrometer, and computer (Fig. 1C). The light source is a single-frequency tunable laser with its output spectrum locked to a near-infrared wavelength of ~780 nm and filtered using an etalon43. The laser light is coupled via a polarization-maintaining fiber to the human interface, in which polarization optics route the laser beam to the eye and direct backscattered light to a single-mode fiber. The spectrometer employs two-stage VIPA etalons and apodization filters to achieve a free-spectral range (FSR) of 16 GHz, a resolution of ~120 MHz, and an extinction efficiency of -65 dB. The optical power on the cornea is 3–5 mW (Fig. 1D), which is several times lower than the maximum permissible exposure level according to American National Standard Institutes (ANSI) guidelines (Supplementary Information). With this optical power applied to corneal tissues, an electron multiplying charge-coupled device (EMCCD) camera recorded a total of ~6,000 Brillouin photons per second. The Brillouin frequency shift, which is half the frequency difference between the Stokes and Anti-Stokes spectral peaks, is determined by curve fitting the recorded Brillouin spectra (Figs 1E and S2). With an EMCCD integration time of 0.2 s, we obtained a shot-noise-limited sensitivity of ~±8 MHz in Brillouin frequency shift measurement (Fig. S4). The spatial resolution was measured to be ~5 and ~35 µm in the transverse and axial directions, respectively (Fig. S5). The system employs temperature sensors and temperature-calibrated reference materials to attain short- and long-term repeatability of ~±10 MHz (Fig. S7). For corneal scans, an operator adjusts the human interface using a manual joystick to locate the focus at a desired location in front of the cornea while monitoring an eye-tracking video camera based on pupil detection (Fig. 1F). At each location, the operator starts a motorized axial scan to obtain a depth profile and calculate the mean Brillouin shift over the stroma (Fig. 1G). A typical 2D Brillouin elasticity map constructed from the mean Brillouin shifts measured at various scan locations using standard spatial interpolation is shown in Fig. 1H. Please refer to Materials and Methods and Supplementary Information for more technical details of the systems and data acquisition.

### Brillouin analysis of normal corneas in vivo

To establish a baseline for the normal population, we scanned healthy subjects with corneal thickness in a normal range of 495 to 600 µm, no irregular astigmatism, and no history of ophthalmic pathology or surgery. For each subject, a mean central Brillouin shift was obtained from 5 axial scans in a central region within a 2-mm radius from the pupil center. The measured values from 47 healthy subjects (age: 39 ± 13 y/o, 51% female, one eye per subject with left/right chosen at random) ranged from 5.69 to 5.76 GHz with a standard deviation (SD) of 15 MHz. When analyzed as a function of age (Fig. 2A), the Brillouin shift values tend to increase with age at a nominal slope of ~3 MHz per decade, but the correlation was not statistically significant. 37 of the 47 subjects had both of their eyes scanned. The mean Brillouin shifts were nearly identical within the instrument’s resolution for the left and right eyes: 5.721 ± 0.017 versus 5.723 ± 0.017 GHz (Fig. 2B). The difference between the left and right eyes of each individual subject makes a narrow, Gaussian-like distribution centered at 1.8 MHz (mean) (Fig. 2C). The bilateral symmetry between the two eyes of healthy subjects is remarkable and in contrast with the large interpersonal variability of ± 15 MHz.

### Brillouin measurements of keratoconus patients at different stages of disease

To investigate KC, subjects were scanned who have been diagnosed with KC by corneal specialists. The patients were grouped into 4 stages of severity according to the Amsler-Krumeich classification44 using quantitative criteria, namely refractive error, best spectacle-corrected visual acuity, minimum corneal thickness, and maximum curvature (K-max). For each cornea, pachymetry and topography images identified a “cone” region, which we define as the area within a 1 mm radius from the thinnest point. Brillouin scans were conducted at 20 to 40 lateral locations to generate a Brillouin map of the cornea. A few representative Brillouin maps are shown in Fig. 3, comparison to Brillouin maps of normal corneas (Fig. 3A). For advanced KC corneas in stage III or IV, Brillouin maps showed considerable non-uniformity compared to normal corneas (Fig. 3C). The cones, defined herein by the thinnest points, are approximately coincident with the bulging areas of abnormal curvature and surface elevation, and these areas are characterized by low Brillouin frequency shift values, distinctly different by as much as 100–200 MHz compared to healthy corneas. Stage-I corneas are characterized by mild morphological anomalies, and the thinnest points do not always overlap with the locations of maximum curvature or posterior elevation. Compared to stage-III-IV cases, state-I corneas exhibit much more uniform Brillouin maps, and the regions of lower Brillouin values are less pronounced (Fig. 3B).

The locations of minimum thickness, i.e. cone centers, are distributed predominantly in the center-temporal region (Fig. 4A). The XY coordinate in mm of the center of the distribution was (0.45, -0.85) for stages I and II (n = 34, 36 ± 11 y/o, 47% female) and (0.48, -1.2) for stages III and IV (n = 12, 46 ± 14 y/o, 75% female). The data suggest that the cone location within an individual may not change much as KC progresses. From the Brillouin maps, we calculated the mean Brillouin frequency shifts in the cone regions and plotted the data as a function of minimum corneal thickness (Fig. 4B) and maximum anterior sagittal curvature, K max (Fig. 4C). Weak to moderate trends of decreasing Brillouin frequency shifts with increasing KC severity were measured, with coefficients of determination for linear regression being R2 = 0.40 for corneal thickness and R2 = 0.28 for K-max. The graph in Fig. 4D shows the distributions of Brillouin frequency shift values in cone regions of KC patients in comparison to Brillouin values measured in central regions of normal patients (within a 1-mm radius from the pupil center). While the reduction of Brillouin values is evident for advanced KC, no statistically significant difference was observed between stage-I-II KC and normal corneas. The mean Brillouin values of the stage-I and -II groups are lower than the healthy group by 3 and 7 MHz, respectively. Nonetheless, the observed null correlation results from large interpersonal variability within each group. Notably, the subject-to-subject variation in stage I and II patients is greater than the intrinsic interpersonal difference among healthy subjects.

### Analysis of regional biomechanical variation in keratoconus corneas

The observation of large interpersonal variability in both normal and KC corneas prompted us to seek metrics that are less subject to individual variations. We hypothesized that most of the variability stems from natural personal differences in the composition, collagen organization, and/or hydration level of corneal tissues. We further reasoned that these factors would likely affect Brillouin values uniformly across the entire cornea and therefore can be canceled out in differential metrics, such as regional difference within the cornea. From Brillouin maps, we found that, whereas normal corneas exhibit relatively uniform Brillouin values across their about-4-mm-wide scanned areas (Fig. 5A), the Brillouin shifts in KC corneas have a clear tendency to increase linearly with the distance from the cone in all stages, including stages I and II (Fig. 5B,C). The slope of regional variation increased with the severity of KC (Fig. S10). We compared the mean Brillouin frequency shift in the cone region versus outside-cone region—defined by all scanned points outside a 3-mm radius from the thinnest point—for each cornea. For healthy subjects, central and peripheral Brillouin values are not different (Fig. 5D, n = 16, 39 ± 10 y/o, 25% female). By contrast, highly significant differences (p < 0.001) were found between cone and outside-cone regions for all stages of KC (Fig. 5E,F). Figure 5G shows regional variations or the differences of outside- and inside-cone values for all groups. All KC groups exhibit higher regional differences than the normal group. However, the statistical significance is moderate (p < 0.01) except for stage IV, with considerable overlaps of Brillouin values with the normal group.

### Analysis of bilateral symmetry in KC corneas

KC is known to be bilateral, but often asymmetric. The high degree of left-to-right symmetry in central Brillouin values in healthy subjects motivated us to analyze bilateral symmetry for KC corneas. Representative Brillouin maps from a patient with stage-I KC in both corneas are shown in Fig. 6A. The thickness and posterior elevation maps share symmetric, but not identical, features. Asymmetry in the Brillouin maps is apparent although subtle. To quantify the asymmetry, we calculated the difference in Brillouin frequency shifts between left (OS) and right (OD) eyes in two corresponding locations: near the pupil center and in the cone. The asymmetry in Brillouin values in the central regions of KC corneas was moderately higher (p < 0.01, n = 4, 38 ± 5 y/o, 50% female) than in normal corneas (Fig. 6B). Remarkably, the Brillouin asymmetry between the left and right cone regions, which are identified using the thickness maps, is significantly larger than for the normal group (Fig. 6B). The mean bilateral difference in the stage-I cones was 24 MHz, well separated from the normal population that is predominantly below 12 MHz. Despite the relatively small sample size, the data are compelling and reveal that bilateral symmetry may be a promising metric for diagnosis of early-stage KC. The narrow distribution of the data suggests that the compounding factors that are responsible for the subject-to-subject variability in Brillouin values are equally present in both eyes and, therefore, cancelled out in this differential metric.

## Discussion

The magnitude of the optical frequency shift that we measure is proportional to the velocity of longitudinal acoustic waves in the tissue, which is in turn related to the square root of longitudinal modulus of elasticity45. For viscoelastic materials such as tissues, longitudinal modulus is nearly equal to bulk modulus, or the material’s resistance to volume change. Because most biological tissues are ~70% water by volume, they are nearly incompressible and exhibit high longitudinal modulus, in the range of 2.5 to 3.5 GPa46, producing Brillouin frequencies of 5.69 to 5.77 GHz in corneas42 and up to 6.1 GHz in crystalline lenses47 (at an optical wavelength of 780 nm). The bulk modulus of aqueous humor, which is nearly the same as that of water, is 2.2 GPa, for which we measured a Brillouin shift of 5.25 GHz (Fig. 1G). The higher longitudinal-modulus of corneal tissue compared to water derives from the contribution of the other constituent matter such as ions and proteins, that make the tissue less compressible. Brillouin spectroscopy has been shown to be sensitive to small differences in molecular composition, concentration (or water content)46, and photochemical crosslinking48,49. Brillouin measurement may be less sensitive to processes such as pure polymeric rearrangement or changes that are predominantly entropic, but relatively more sensitive to processes involving changes in molecular concentration and composition.

In contrast to bulk modulus, shear or Young’s modulus characterizes a material’s response to stress under free boundary conditions. Since soft matter is deformed relatively easily at constant volume, its shear modulus is orders-of-magnitude smaller than its bulk modulus. The shear modulus measured by mechanical methods at low deformation frequencies (<100 Hz) is typically ~250 kPa for normal corneal tissue. Although longitudinal and shear moduli are in principle two independent quantities and can have vastly different magnitudes, empirical data obtained with polymers, hydrogels, and various tissues including corneas have shown reasonably high, quantitative correlation between Brillouin frequency shift and quasi-static shear modulus41,42,50. This measurement result suggests that in practice for corneal tissues, natural or pathologic processes that cause shear modulus to increase (or decrease) are likely to cause Brillouin shifts to increase (or decrease).

The Brillouin data from normal adult corneas in vivo showed a modest age-dependence with a slope of ~3 MHz/decade. Unexpectedly, we observed relatively large subject-to-subject differences (±15 MHz SD) even among age-matched subjects (Fig. 2A). Given the interpersonal variability, the rather small age dependence is surprising, as corneal surgeons often experience less soft and pliable corneas in old patients compared to young. Numerous ex vivo mechanical measurements have reported increasing trends of Shear and Young’s moduli with age51,52,53. About 12% increase of Young’s modulus per decade has been estimated53. Employing an often-used empirical log-log conversion factor of 60:1, the slope of 12% per decade in Young’s modulus corresponds to, approximately, 0.2% per decade in longitudinal modulus, or 12 MHz/decade in Brillouin frequency (Supplementary Information). Previously, electron microscopy has revealed thickening of collagen fibrils54 and decrease of their inter-fibrillar spacing55 with age. Such changes in microstructure56, as well as age-related increase of glycation-mediated collagen crosslinking57, are thought to elevate corneal stiffness over age56. The apparent discrepancy between our measurement and previous ex vivo measurements (as well as surgeons’ perception by in vivo palpation) has yet to be resolved. It is interesting to note that in vivo measurements using ORA found no increase or even small decline of corneal hysteresis and corneal resistance factor with age21,58,59,60.

We suspect that the interpersonal variability stems from natural differences in the physiological conditions of the cornea. For example, Brillouin shift is sensitive to the water content in hydrogels and, therefore, hydration level of corneal tissues61. The hydration level is known to differ individually and fluctuate throughout the day62. This hydration variation may contribute to the measured interpersonal variability. The Brillouin shift of tissue is also sensitive to body temperature, as expected from a temperature dependence of 7.4 MHz/°C for water (Fig. S6). It is worth noting that ocular surface temperature was found to be negatively correlated with age63, which may contribute to the little-to-no age dependence we observed in our study. In future, it would be preferable to record body temperature and apply a correction factor to the measured Brillouin values. Since these physiological factors affect globally the entire cornea, we had expected smaller variability for differential metrics. Indeed, we measured substantially smaller “intra-personal” differences between left and right eyes (Fig. 2B) and between the central and peripheral regions of each cornea (Fig. 5D).

Variability in corneal biomechanics across individuals can introduce significant errors in applanation-tonometry IOP readings64. Hence, individual differences in Brillouin value may be used to apply a “biomechanics correction” to improve the accuracy of IOP measurements. Individual differences in Brillouin value may also provide useful input for refractive treatments: it is known that corneal hydration affects the excimer laser ablation rate in LASIK surgeries65,66, and personal biomechanical differences are thought to explain the variability in refractive outcomes following cataract and astigmatic correction keratotomy (AK) surgeries67. It may be possible to individually tailor ablation parameters or AK nomograms based on Brillouin measurements made prior to treatment.

In the present study of KC patients, we confirmed our previous finding that the cones in advanced KC corneas are clearly visible in Brillouin maps as regions of significantly reduced Brillouin shift (Fig. 4B)42. However, the correspondence between Brillouin maps and pachymetry and topography is less clear in early-stage KC corneas, and Brillouin shifts correlate weakly with morphological parameters such as thickness and curvature (Fig. 4B,C). We found no statistical difference between stage-I-II KC patients and healthy subjects (Fig. 4D). The Brillouin values within stage-I and II groups each exhibited larger inter-subject variation than the normal group. The large variations might originate from the physiological factors discussed above and, possibly, additional factors associated with KC pathogenesis.

The diagnosis of KC in this study was performed by corneal experts at IROC using proprietary criteria based on corneal pachymetry and topography maps. We calculated receiver operating characteristic (ROC) curves for two morphological parameters—minimum corneal thickness and maximum sagittal curvature—and two Brillouin parameters—mean Brillouin shift of the cone and regional Brillouin difference—using the data from stage-I patients (n = 25) data (Fig. S11). The area under the curve (AUC) values of the Brillouin parameters were 0.92 for mean Brillouin and 0.85 for regional difference, which are higher or comparable to 0.87 for maximum curvature and 0.77 for minimum thickness. This result supports that the Brillouin parameters may add value for diagnosis of early-stage KC. Future studies with larger patient numbers and more accurate, comprehensive corneal mapping is warranted to validate this finding.

The distinction between early-stage KC and normal groups was more pronounced in the analysis of regional differences between cone and outside-cone regions. In most stage-I and -II subjects and all stage-III and IV subjects, the cone regions had lower Brillouin shifts than the peripheral regions. This supports the long-held hypothesis that the degeneration of corneal tissues occurs focally at the cones14. Given moderate statistical significance (p < 0.01) differentiating stage-I from normal values, the regional-difference parameter has potential to be a useful metric to characterize early-stage KC. However, the two-point comparison alone is unlikely to be diagnostic of KC, considering the overlap of data between different groups (Fig. 5G). With more accurate, comprehensive mapping of the cornea in future, more sophisticated metrics based on specific spatial patterns in Brillouin maps may be developed. Nevertheless, the regional differences could still be of significant importance for clinical practice. One potential application could be personalized CXL treatment planning and outcome efficacy monitoring68.

Another promising metric we have found is the difference in Brillouin shifts between individual left and right eyes. The bilateral asymmetry measured in stage-I KC patients is clearly separated from that of normal corneas, and reflects the clinical experience that in many keratoconus patients one eye presents a more advanced stage of disease relative to the other69. More investigation of this metric in a larger-scale study is warranted, which may include patients with only one eye in stage-I KC and a subclinical or KC suspect contralateral eye. The bilateral asymmetry indices of healthy subjects form a normal distribution with width limited by the system sensitivity. With enhanced Brillouin signals through longer data acquisition time, the distribution in the healthy population may turn out to be narrower. The bilateral asymmetry index may prove useful for identifying patients at an earlier stage or those at risk of developing KC. If so, this index could be used to develop new diagnostic criteria in combination with, or even beyond the current morphology-based gold standard. It is worthwhile to note that it is the bilateral asymmetry measured at the thinnest point that showed the sharp distinction. The asymmetry index measured at the pupil center showed only moderate statistical difference from the normal population (Fig. 5B). This consistently supports the hypothesis of focal weakening in the development of KC14.

An interesting future project would be to compare Brillouin measurements to those obtained from air-puff-based instruments, given the recent success of these methods25,26,27,28. Though these methods are essentially measuring different physical properties of the tissue, correlation may be found between data acquired with these methods, such as corneal hysteresis or the Corvis ST biomechanical index, and certain Brillouin metrics, such as regional difference.

Besides corneas, other ocular tissues such as the crystalline lens and sclera are thought to present a high degree of symmetry in their healthy states. In fact, in our previous study of crystalline lenses in vivo, we have measured large interpersonal variability, as much as ± 60 MHz in age-matched groups, as opposed to small differences between left and right lenses within the measurement uncertainty of ± 10 MHz47. Brillouin-based bilateral asymmetry measurement has potential to be a general, useful technique for the detection of subtle pathological anomalies and drug-induced changes in the biomechanical properties of the lens and sclera.

The current Brillouin systems require a data acquisition time of >0.2 s per depth point and ~12 s for a single axial scan. A higher data acquisition speed is desirable for clinical use. The total number of Brillouin scattered photons that were collected and analyzed at the spectrometer was about 3 × 10−13 of the number of laser photons incident on the corneal surface. The theoretical maximum collection efficiency of Brillouin light scattering in an on-axis confocal configuration is estimated to be $$\frac{2{\pi }^{2}}{{\lambda }^{3}}\frac{{({n}_{s}^{2}-1)}^{2}}{K}{k}_{B}T$$, where kBT = 4.23 × 10−21 J at the cornea temperature, ns is refractive index, K is bulk modulus, and λ is optical wavelength (Supplementary Information). For corneal tissues, the efficiency is ~5 × 10−11. There is room to increase the Brillouin signal by as much as 10-fold by reducing optical loss in the detection path.

In conclusion, these clinical results have demonstrated the potential of Brillouin light scattering spectroscopy for corneal examinations, particularly for detection and treatment monitoring of KC patients. The most promising metric for KC-discrimination are based on focal weakening, a finding made possible by harnessing the intrinsic ability of Brillouin technology to measure local biomechanical properties of tissues with high spatial resolution and sensitivity. Our results suggest that spatially-resolving Brillouin spectroscopy may also prove useful in other applications, including screening subjects at risk of ectasia prior to refractive keratectomy surgeries70,71,72,73,74, optimizing corneal incision parameters in cataract and astigmatic correction keratotomy surgeries by taking individual biomechanics into account67, and treatment-parameter optimization and outcome evaluation for customized CXL treatment68. Brillouin measurements may also be used to improve the accuracy of basic examinations such as intraocular pressure measurement64.

## Methods

### Study design

The objective of this study was to assess the feasibility of using Brillouin spectroscopy for clinical characterization of human corneal biomechanics. Specifically, to evaluate the potential of Brillouin spectroscopy for use in detection and treatment of keratoconus. For this purpose, we examined the interpersonal variability, age-dependence, regional heterogeneity, and bilateral asymmetry in Brillouin values among normal corneas, keratoconus corneas of different severity, and corneas with history of collagen crosslinking treatment.

The general inclusion criteria permitted subjects with clear-enough cornea and media to permit imaging, and who fulfilled the specific study group inclusion/exclusion criteria. Excluded from this study were monocular volunteers. The normal cornea group (n = 47, 39 ± 13 y/o, 51% female) included healthy subjects with normal appearing corneas consistent with the general inclusion/exclusion criteria, with less than ± 3 diopters refractive error, normal corneal thickness (495 to 600 µm), normal corneal topography (no asymmetric or irregular astigmatism, no skewed axis), no corneal pathology, and no history of eye diseases, except presbyopia and/or cataract. The keratoconus group (n = 46, 39 ± 13 y/o, 43% female) included subjects with irregular corneas determined by distorted keratometry mires or irregularities in Scheimpflug photography and slit-lamp biomicroscopic signs such as Vogt’s striae or Fleischer’s ring or corneal scarring consistent with keratoconus, aged from 15–60 y/o. This group included patients diagnosed with KC stage I (n = 25, 36 ± 11 y/o, 46% female), II (n = 9, 45 ± 13 y/o, 44% female), III (n = 4, 47 ± 12 y/o, 40% female) and IV (n = 8, 45 ± 15 y/o, 25% female).

Sample sizes for each group were chosen in advance based on a power analysis. We used preliminary data to estimate the sample size required to establish a correlation between corneal stiffness and progression of the disease and subject age, with >90% power.

Studies were performed at two locations: at the Institute for Refractive and Ophthalmic Surgery (IROC) in Zürich, Switzerland, and at Massachusetts Eye and Ear Infirmary (MEEI) and Massachusetts General Hospital (MGH) in Boston, USA, following approval from the Institutional Review Board (IRB) of Partners HealthCare, the Partners Human Research Committee (PHRC) and the Institutional Review Board of IROC, Zürich. Informed consent was obtained from every patient before imaging. All methods were performed in accordance with the relevant guidelines and regulations. The clinical trial “A Study to Test the Potential of Brillouin Microscopy for Biomechanical Properties Measurements in Human Cornea” is registered at www.clinicaltrials.gov (National Clinical Trial Identifier: NCT03220529).

### Brillouin data acquisition

During Brillouin measurement, the subject sits with their chin and forehead resting in the human interface headrest, directing their gaze towards a fixation target. For healthy subjects with normal corneas, 5 axial scans are taken within a 2 mm zone in the central cornea. To create a Brillouin map of the cornea, 20–40 axial scans are taken at different locations laterally across the cornea, usually within the central 6 mm zone. Light power was calibrated to be 5 mW on the corneal surface. Each axial scan comprises 40 points separated by a step size of 30 μm, spanning air to cornea to aqueous humor. The EMCCD exposure time for each step is typically 300 ms, resulting in a total axial scan time of about 12 s. Following each axial scan, temperature-corrected calibration spectra are taken using known reference materials (polystyrene and water) to determine the dispersion rate (in GHz/pixel) of the spectrum pattern acquired from the cornea, and thus the corresponding Brillouin frequency shift (Fig. S3). From the calibrated axial scan data, points acquired within the corneal stroma are extracted and averaged to yield a single value for each lateral scan location (Fig. 1G). Linear interpolation between these lateral measurement points is then used to create the final 2D Brillouin map (Fig. 1H). Data processing was conducted using MATLAB (Mathworks, Inc.).

### Corneal topographies

The corneal topographies were generated using a commercial Scheimpflug camera (Pentacam HR 70700, OCULUS, Wetzlar, Germany) with thickness mapping and posterior elevation characterization. Patients were asked not to use contact lenses in the two weeks prior to each examination.

### Data analysis

To process a single axial scan, we first determine the rate of spectral dispersion using the included calibration spectrum (see Fig. S3). With this information, we can then determine the Brillouin frequency shift at each of the (typically) 40 scan points spanning air to cornea to aqueous humor. We then segment this Brillouin shift versus axial depth data, based on Brillouin value and other characteristic signal features, in order to identify and extract only points within the corneal stroma. Finally, we average the Brillouin shift measured across all stroma points to obtain a single Brillouin value for the axial scan.

To create a Brillouin map of the cornea, we use standard linear interpolation to connect Brillouin values from 20–40 axial scans at different lateral locations. To analyze the spatial variation in Brillouin values across the cornea, we define various regions of interest (see Fig. S9 for a graphical illustration of these zones). The central region is defined as the zone within a 1-mm radius around the pupil center; the peripheral region is the area with a distance >3 mm from the pupil center. In keratoconic corneas, the ‘cone region’ is defined as the area within a 1 mm radius of the thinnest point (the ‘cone center’) identified using corneal topography, and the ‘outside-cone region’ refers to the area >3 mm from the cone center.

### Statistical analysis

Custom MATLAB software was used for data processing. Statistical analysis was carried out using the Statistics toolbox in MATLAB. To examine the significance of differences observed in the mean Brillouin values of different groups of subjects (such as comparing normal versus KC stage I) we used unpaired, two-tailed t-tests. To evaluate the significance of differences observed within a single cornea (such as comparing one region to another) we used paired, two-tailed t-tests. Data were considered significant if p-values were less than 0.05 (95% confidence intervals).

## Data Availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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## Acknowledgements

We acknowledge Dr. Giuliano Scarcelli and Dr. Sebastien Besner for their contributions during the development of the Brillouin systems, Dr. Valéry Wittwer, Dr. Michael Wolff and Dr. Aaron Wegner for assistance with the age-dependence study, and Dr. Dominik Beck for helping coordinate the IROC trial. The project was funded by the U.S. National Institutes of Health (R01EY025454, P41EB015903) and the Harvard Catalyst Incubator program (UL1-RR025758).

## Author information

P.S., T.S. and S.H.Y. designed the study, P.S., A.M.E. and B.T. constructed the Brillouin systems, P.S., T.G.S., T.K., R.P. and T.S. recruited and classified patients, P.S. and A.M.E. performed Brillouin measurements, P.S., A.M.E., T.G.S., T.S. and S.H.Y. analyzed the data. P.S., A.M.E., T.S. and S.H.Y. wrote the manuscript with input from all authors.

### Competing Interests

T.G.S., B.T., R.P., T.K. and T.S. declare no competing interests. S.H.Y. holds issued and pending patents and P.S. and A.M.E. have a pending patent related to the technology. S.H.Y. is the scientific founder of Intelon Optics, Inc., for which T.Ss. serves on the Medical Advisory Board, and P.S. provides consulting services.

Correspondence to Theo Seiler or Seok-Hyun Yun.


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Home OALib Journal OALib PrePrints Submit Ranking News My Lib FAQ About Us Follow Us+
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 Journal of Biological Sciences , 2001, Abstract: In this study, punching force, stress, and energy of Yalova 12 and Banko onion varieties were examined. These varieties are grown in Bursa region and having an important place for exporting in our country. The effects of storage (right after harvest and after one month of storage) and the position of onion (radial and axial) along with the variety of onion were examined for punching the onions. According to the results, the variety had no effects on punching, but the position of onion was found important statistically. It was found that axial punching requires more force, stress and energy than radial punching.
 Mathematics , 2014, Abstract: We study the relative position of three subspaces in a separable infinite-dimensional Hilbert space. In the finite-dimensional case, Brenner described the general position of three subspaces completely. We extend it to a certain class of three subspaces in an infinite-dimensional Hilbert space. We also give a partial result which gives a condition on a system to have a (dense) decomposition containing a pentagon.
 Mathematics , 2004, Abstract: The relative position of one subfactor of a factor has been proved quite rich since the work of Jones. We shall show that the theory of relative position of several subspaces of a separable infinite-dimensional Hilbert space is also rich. In finite-dimensonal case, Gelfand and Ponomarev gave a complete classification of indecomposable systems of four subspaces. We construct exotic examples of indecomposable systems of four subspaces in infinite-dimensional Hilbert spaces. We extend their Coxeter functors and defect using Fredholm index. There exist close connections with strongly irreducible operators and transitive lattices.
 Physics , 2012, DOI: 10.1051/0004-6361/201118274 Abstract: Context. We detected in 2009 a giant, close-by planet orbiting {\beta} Pic, a young star surrounded with a disk, extensively studied for more than 20 years. We showed that if located on an inclined orbit, the planet could explain several peculiarities of {\beta} Pictoris system. However, the available data did not permit to measure the inclination of {\beta} Pic b with respect to the disk, and in particular to establish in which component of the disk - the main, extended disk or the inner inclined component/disk-, the planet was located. Comparison between the observed planet position and the disk orientation measured on previous imaging data was not an option because of potential biases in the measurements. Aims. Our aim is to measure precisely the planet location with respect to the dust disk using a single high resolution image, and correcting for systematics or errors that degrades the precision of the disk and planet relative position measurements. Methods. We gathered new NaCo data at Ks band, with a set-up optimized to derive simultaneously the orientation(s) of the disk(s) and that of the planet. Results. We show that the projected position of {\beta} Pic b is above the midplane of the main disk. With the current data and knowledge on the system, this implies that {\beta} Pic b cannot be located in the main disk. The data rather suggest the planet being located in the inclined component.
 American Journal of Plant Sciences (AJPS) , 2017, DOI: 10.4236/ajps.2017.89135 Abstract: Okra is a plant rich in nutrients and is very well consumed in Côte d’Ivoire. Despite its many benefits, the production of this vegetable is still weak in our country. The reasons for this include the inadequate selection of varieties, the high cost of inputs and the poverty of the soil for its cultivation. One of the alternatives for sustained production is to solve the problem of soil fertility. In the case of our work, the aim is to improve the yield of okra. To achieve this goal, experiments were undertaken to evaluate the impact of activated charcoal on morphological parameters of a local okra variety. For this purpose, the charcoal used was activated in three different times (activation time equal to 0 days, 15 days and 30 days). The experimental device used is a split-plot with three repetitions, each comprising 12 elementary plots. The various charcoals were buried the same day. Then, the seedling was done with two positions including outside position and inside position. Observations were made on 360 plants. An analysis of the variances was carried out on the morphological parameters. Fruit mass is the variable most influenced by activated charcoal. Thus, the greatest value of the mass was obtained with the charcoal CA0, with outside position.
 Physics , 2005, DOI: 10.1103/PhysRevA.71.062101 Abstract: We revisit the back-action of emitted photons on the motion of the relative position of two cold atoms. We show that photon recoil resulting from the spontaneous emission can induce the localization of the relative position of the two atoms through the entanglement between the spatial motion of individual atoms and their emitted photons. The result provides a more realistic model for the analysis of the environment-induced localization of a macroscopic object.
 Statistics , 2014, DOI: 10.1016/j.sigpro.2015.02.023 Abstract: Localization is a fundamental challenge for any wireless network of nodes, in particular when the nodes are mobile. We present an extension of the classical Multidimensional scaling (MDS) for an anchorless network of mobile nodes, wherein the solutions to the time-varying relative node positions are shown to lie in the derivatives of the time-varying inter-nodal pairwise distances. Moreover, we show that the relative position of a mobile node at each time instance is only dependent on the initial relative position, relative velocity and a common rotation matrix of the respective node, which are estimated using MDS-like and least squares estimators. Simulations are conducted to evaluate the performance of the proposed solutions and the results are presented.
 Ambiente e água : An Interdisciplinary Journal of Applied Science , 2010, Abstract: This editorial describes the process of periodic evaluation by CAPES, based on commercial indices proposed by companies such as the ISI FI by Thomson Reuters and SCImago by Scopus and public indices as proposed by SciELO and other freely calculated indices based on Google Scholar and investigates the impact position of the journal Ambiente & Agua compared with journals from the SciELO base. Results show that Ambiente & Agua is in the right path in the search for quality. Its editorial board strongly recommends that Brazilian public scientific agencies consider tools that are cost free and have large coverage for impact evaluation to allow the impact assessment of a range of emerging journals in a common and wide basis for all scientific journals.
 Mathematics , 2010, DOI: 10.1007/s00222-011-0349-0 Abstract: We introduce the notion of Artin motives and cohomological motives over a scheme X. Given a cohomological motive M over X, we construct its punctual weight zero part $\omega^0_X(M)$ as the universal Artin motive mapping to M. We use this to define a motive E_X over X which is an invariant of the singularities of X. The first half of the paper is devoted to the study of the functors $\omega^0_X$ and the computation of the motives E_X. In the second half of the paper, we develop the application to locally symmetric varieties. Specifically, let Y be a locally symmetric variety and denote by p:W-->Z the projection of its reductive Borel-Serre compactification W onto its Baily-Borel Satake compactification Z. We show that $Rp_*(\Q_W)$ is naturally isomorphic to the Betti realization of the motive E_Z, where Z is viewed as a scheme. In particular, the direct image of E_Z along the projection of Z to Spec(C) gives a motive whose Betti realization is naturally isomorphic to the cohomology of W.
 Physics , 2001, Abstract: Nucleon recoil polarization in electromagnetic reactions with two-nucleon emission is discussed for both (${\vec e},e'{\vec N}N$) and (${\vec \gamma},{\vec N}N$). Numerical results are given for exclusive two-nucleon knockout reactions from $^{16}$O in a theoretical model where final-state interactions, one-body and two-body currents, and the effect of correlations in the initial pair wave function are included.
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## Thursday, March 4, 2010

### Outfield / Utility Projections

Here are my projections for OF (including UTIL).  Ordered by overall value in the traditional 5x5 categories listed in RED.  Additional players will be added as time permits.

(note: HBP are included in BB for these projections)
(click to enlarge image)

### Shortstop Projections

Here are my projections for SS.  Ordered by overall value in the traditional 5x5 categories listed in RED.  Additional players will be added as time permits.

(note: HBP are included in BB for these projections)

### 3rd Base Projections

Here are my projections for 3B.  Ordered by overall value in the traditional 5x5 categories listed in RED.  Additional players will be added as time permits.

(note: HBP are included in BB for these projections)

### 2nd Base Projections

Here are my projections for 2B.  Ordered by overall value in the traditional 5x5 categories listed in RED.  Additional players will be added as time permits.

(note: HBP are included in BB for these projections)


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# K11a56

## Contents

 (Knotscape image) See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a56's page at Knotilus! Visit K11a56's page at the original Knot Atlas!

### Knot presentations

 Planar diagram presentation X4251 X8493 X16,6,17,5 X2837 X18,9,19,10 X20,11,21,12 X22,13,1,14 X6,16,7,15 X14,17,15,18 X12,19,13,20 X10,21,11,22 Gauss code 1, -4, 2, -1, 3, -8, 4, -2, 5, -11, 6, -10, 7, -9, 8, -3, 9, -5, 10, -6, 11, -7 Dowker-Thistlethwaite code 4 8 16 2 18 20 22 6 14 12 10
A Braid Representative

### Three dimensional invariants

 Symmetry type Reversible Unknotting number 1 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a56/ThurstonBennequinNumber Hyperbolic Volume 14.5406 A-Polynomial See Data:K11a56/A-polynomial

### Four dimensional invariants

 Smooth 4 genus Missing Topological 4 genus Missing Concordance genus 3 Rasmussen s-Invariant 0

### Polynomial invariants

 Alexander polynomial −2t3 + 11t2−25t + 33−25t−1 + 11t−2−2t−3 Conway polynomial −2z6−z4 + z2 + 1 2nd Alexander ideal (db, data sources) {1} Determinant and Signature { 109, 0 } Jones polynomial −q5 + 3q4−7q3 + 12q2−15q + 18−17q−1 + 15q−2−11q−3 + 6q−4−3q−5 + q−6 HOMFLY-PT polynomial (db, data sources) −a2z6−z6 + a4z4−3a2z4 + 2z4a−2−z4 + 2a4z2−5a2z2 + 3z2a−2−z2a−4 + 2z2 + a4−3a2 + a−2−a−4 + 3 Kauffman polynomial (db, data sources) a2z10 + z10 + 3a3z9 + 6az9 + 3z9a−1 + 4a4z8 + 7a2z8 + 5z8a−2 + 8z8 + 3a5z7−a3z7−4az7 + 5z7a−1 + 5z7a−3 + a6z6−9a4z6−17a2z6−3z6a−2 + 3z6a−4−13z6−9a5z5−10a3z5−8az5−15z5a−1−7z5a−3 + z5a−5−3a6z4 + 5a4z4 + 13a2z4−4z4a−2−5z4a−4 + 6z4 + 8a5z3 + 12a3z3 + 10az3 + 11z3a−1 + 3z3a−3−2z3a−5 + 2a6z2−a4z2−8a2z2 + 4z2a−2 + 3z2a−4−4z2−2a5z−5a3z−5az−3za−1 + za−5 + a4 + 3a2−a−2−a−4 + 3 The A2 invariant Data:K11a56/QuantumInvariant/A2/1,0 The G2 invariant Data:K11a56/QuantumInvariant/G2/1,0

### "Similar" Knots (within the Atlas)

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {}

### Vassiliev invariants

 V2 and V3: (1, 2)
V2,1 through V6,9:
 V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 4 16 8 $\frac{62}{3}$ $\frac{34}{3}$ 64 $\frac{256}{3}$ $-\frac{128}{3}$ 48 $\frac{32}{3}$ 128 $\frac{248}{3}$ $\frac{136}{3}$ $\frac{11311}{30}$ $\frac{338}{15}$ $\frac{7262}{45}$ $\frac{689}{18}$ $\frac{271}{30}$

V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.

### Khovanov Homology

The coefficients of the monomials trqj are shown, along with their alternating sums χ (fixed j, alternation over r). The squares with yellow highlighting are those on the "critical diagonals", where j−2r = s + 1 or j−2r = s−1, where s = 0 is the signature of K11a56. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-6-5-4-3-2-1012345χ
11           1-1
9          2 2
7         51 -4
5        72  5
3       85   -3
1      107    3
-1     89     1
-3    79      -2
-5   48       4
-7  27        -5
-9 14         3
-11 2          -2
-131           1
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ i = −1 i = 1 r = −6 ${\mathbb Z}$ r = −5 ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = −4 ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ r = −3 ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ r = −2 ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ r = −1 ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ r = 0 ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{10}$ r = 1 ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ r = 2 ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ r = 3 ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ r = 4 ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ r = 5 ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory. See A Sample KnotTheory Session.

Read me first: Modifying Knot Pages.

See/edit the Hoste-Thistlethwaite Knot Page master template (intermediate).

See/edit the Hoste-Thistlethwaite_Splice_Base (expert).

Back to the top.


text:
## Random evolutions processes induced by discrete time Markov chains.(English)Zbl 0946.47029

Let $$X(n)$$ be a stationary Markov chain with $$N-$$ states and $${T_i,1 \leq i \leq N}$$ be bounded operators on a Banach space $$B$$. Define the backward random evolution $$R(0)=I, R(n)=T_{X(0)} \cdots T_{X(n)}$$, $$1 \leq n$$ and $$\widetilde{R(n)}$$ by $$(\widetilde{R(n)} \widetilde{f})_j=E_J [R(n)f_{X(n)}]$$, $$\widetilde{f} \in B^N$$. Then $$(\widetilde{R(n)} \widetilde{f})$$ is a solution of the following equations $u_j(n+1)= \sum_{k=1}^Np_{jk}T_ju_k(n).$ A corresponding forward version defined by $$S(0)=I, S(n)=T_{X(n)} \cdots T_{X(0)}$$, $$1 \leq n$$ also enjoys similar propriety. And the reversed chain $$\widehat{X}(n)$$ defined by the invariant measure also induces backward and forward random evolutions $$\widehat{R}(n)$$ and $$\widehat{S}(n)$$. Similar results hold too. It is then extended to a jump backward random evolution defined by $R(0)=I, R(n)=T_{X(0)}C_{X(0)X(1)}T_{X(1)} \cdots C_{X(n-1)X(n)}T_{X(n)}, 1 \leq n ,$ and its forward version, where $${C_{kl}}, 1 \leq k$$, $$l \leq N$$, are bounded linear operators on the Banach space $$B$$. At the end of this paper, the result is extended to the case where the Markov chain $$X(n)$$ is nonstationary.

### MSC:

 47D07 Markov semigroups and applications to diffusion processes 60J25 Continuous-time Markov processes on general state spaces 60F05 Central limit and other weak theorems

### Keywords:

Random evolution; Markov chain; semigroup of operators
Full Text:


text:
# Interval of convergence and sum of power series

1. May 16, 2015

### nuuskur

1. The problem statement, all variables and given/known data
Consider $\sum\limits_{n=0}^{\infty} \frac{n+1}{(2n)!}(x+1)^{2n+1}$. Find the interval of convergence and sum of the power series.

2. Relevant equations

3. The attempt at a solution
According to the textbook: given the power series $\sum a_n(x-c)^n$ the radius of convergence $R:=\frac{1}{\lim\limits_{n\to\infty}\frac{a_{n+1}}{a_n}}$. It doesn't say anything about a series $\sum a_n(x-c)^{f(n)}$ where $f(n)= 2n +1$, for example. Can I use the said formula to calculate $R$ in this case? $a_n =\frac{n+1}{(2n)!}$ If so then:
$$R:=\frac{1}{\lim\limits_{n\to\infty} \frac{(n+2)(2n)!}{(2n+2)!(n+1)}} = \frac{1}{0}???$$

2. May 16, 2015

### HallsofIvy

Staff Emeritus
So your text book expects you to know that $a^{2n+ 1}= (a)(a^2)^n[/tex]. Yes, that is still a power series. Second, instead of relying on memorized formulas, you need to think about where those formulas come from and what they mean. Here you have [itex]a_n= \frac{n+1}{(2n)!}(x+ 1)^{2n+ 1}$ and $a_{n+1}= \frac{n+ 2}{(2n+2)!}(x+ 1)^{2n+ 3}$
Using the ratio test (which is where your formula comes from), we look at $$\frac{a_{n+1}}{a_n}= \frac{n+2}{(2n+2)!}(x+ 1)^{2n+3}\frac{2n!}{n+1}\frac{1}{x+ 1)^{2n+ 1}}= \frac{n+ 2}{n+1}\frac{(2n)!}{(2n+2)!}\frac{(x+ 1)^{2n+ 3}}{ (x+ 1)^{2n+ 1}}$$

So what is the limit of $\frac{n+2}{n+1}$, the limit of $\frac{(2n)!}{(2n+ 2)!}$, and the limit of $\frac{(x+ 1)^{2n+3}}{(x+ 1)^{2n+1}}$ as x goes to infinity?

(As for your limit that goes to "1/0", it is helpful to use the fact that
$$\frac{1}{\lim_{n\to \infty} a_n}= \lim_{n\to\infty}\frac{1}{a_n}$$
Especially when "$a_n$" is itself a fraction which can be "inverted".
$$\frac{1}{\lim_{n\to\infty}\frac{(n+2)(2n)!}{(2n+2)!(n+ 1)}}= \lim_{n\to\infty}\frac{(2n+2)!(n+1)}{(2n)!(n+ 1)}$$

I presume you know that $\frac{(2n+ 2)!}{(2n)!}= \frac{(2n)!(2n+1)(2n+2)}{(2n)!}$.)

3. May 16, 2015

### pasmith

By the ratio test, the series converges absolutely when $$\lim_{n \to \infty} \frac{|a_{n+1}||(x - c)^{f(n+1)}|}{|a_n||(x - c)^{f(n)}|} < 1.$$ Here this yields $$\lim_{n \to \infty} \left . \frac{(n+2)|x + 1|^{2n+3}}{(2n + 2)!} \right/ \frac{(n+1)|x + 1|^{2n+1}}{(2n)!} = \lim_{n \to \infty} \frac{n+2}{(n+1)(2n+1)(2n+2)}|x+1|^2 = 0.$$ What does that tell you about the values of $|x + 1|$ for which the series converges absolutely?

Don't forget the next part: Find the sum of the power series.

4. May 16, 2015

### wabbit

To apply the formula you use, you can start by setting $y=(1+x)^2$ and analysing the series in y.

5. May 16, 2015

### nuuskur

In response to #3:
From what I see the limit is independent of x, the limit is 0 regardless. Does the series converge in all of $\mathbb{R}$?

#4 - I don't understand why you would use that substitution, there is $(x+1)^{2n+1}$ which is $((x+1)^2)^{n+\frac{1}{2}} = y^{n+\frac{1}{2}}$. Elaborate on your idea a little more, please.

6. May 16, 2015

### wabbit

Well, did you try working out the series in term of $y$ ? Note that the term $y^{1\over2}=x+1$ is a common factor to all terms.

7. May 16, 2015

### nuuskur

Oh do you mean $(x+1)\sum \frac{n+1}{(2n)!}(x+1)^{2n}$?

8. May 16, 2015

### wabbit

Yes but then do that substitution : the series in y is exactly in the form required to apply your theorem - and of course the original series converges for a value $x=x_0$ if and only if the new series converges for $y=y_0=(x_0+1)^2$ .

9. May 16, 2015

### nuuskur

I still get the odd result of $\frac{1}{0}$. I'm not exactly certain how these things work, but it looks like the series is convergent in all of $\mathbb{R}$

If substituting $y:=(x+1)^2$
$y\sum_{n=0}^\infty \frac{n+1}{(2n)!}y^n$. Assuming it's true that the series converges everywhere, then integrating I would get $y^2\sum_{n=0}^\infty \frac{1}{(2n)!}y^n$ which is ALMOST the thing I would want it to be. Problem is with the $(2n)!$ if it only were $n!$.

Is something like this allowed? Let $2n=: m$ then substituting: $y^2\sqrt{y}\sum_{m=0}^\infty \frac{1}{m!}y^m$ and that would be useable, but I'm not convinced this doesn't break some rules.

10. May 16, 2015

### wabbit

The $1\over0$ result does mean "infinite radius of convergence" here, or more rigorously "the series converges everywhere". (That is something you can easily see for yourself if you go back to your textbook and go through the proof of that statement, or you can prove it directly.)

But I'm afraid the substitution was only useful for the purpose of applying your convergence criterion. To find the sum, it's actually easier to use $z=1+x$.

You cannot do that "m=2n" substitution - well, you can, but the power of y is not m then and the sum is over even values only, no good.

What you can do is try some transformation to find a series of the form $\sum \frac{z^{2n}}{(2n)!}$ or something closely related - presumably you already know the sum of that last series.

11. May 16, 2015

### nuuskur

Hmm, this substitution would yield $z\sum_{n=0}^\infty \frac{n+1}{(2n)!}z^{2n}$. If there was no $n+1$ it would be the hyperbolic cosine. Trouble is I don't know how to deal with the $n+1$. No doubt there is an explanation how to deal with this in the textbook, but the text is so abstract, it's difficult to understand "just like that".

12. May 16, 2015

### wabbit

Yes that $n+1$ is troublesome, but you're on the right track.

The easiest thing now is probably to write $\frac{n+1}{(2n)!}=\frac{n}{(2n)!}+\frac{1}{(2n)!}$.

Last edited: May 16, 2015
13. May 16, 2015

### nuuskur

Putting the $z$ back inside for a moment.
$\frac{1}{2}\sum\frac{2(n+1)}{(2n)!}z^{2n+1}$. Integrating with respect to $z$. $\frac{z^2}{2}\sum \frac{1}{(2n)!}z^{2n}$ (pulling $z^2$ back out). This would be the integral of the sum.

Another question: The series does converge Absolutely, however when we analyze a finite convergence radius, we also check the end points. If the series absolutely converges Everywhere does it therefore also converge everywhere? (it feels logical, there are no critical points to consider)

14. May 16, 2015

### wabbit

Yes good move, just be careful not to drop a term somewhere when you finalize the calculation:)

Indeed there is no issue with end points, if a series converges absolutely it converges - this doesn't even have to do with radius of convergence or anything, it's a pointwise statement.

Last edited: May 16, 2015
15. May 16, 2015

### nuuskur

If $\int S(z) = \frac{z^2}{2}\cosh z$ Then $S(z) = z\cosh z + \frac{z^2}{2}\sinh z$ and substituting back for $z= x+1$
$S(x) = (x+1)\cosh (x+1) + \frac{(x+1)^2}{2}\sinh (x+1)$

Side note:

I think it would look better if I did the integration properly, in my earlier "quick n dirty" version I should have also included the constant of integration. This way I can avoid it.
$$\int_0^z\frac{1}{2}\sum_{n=0}^\infty\frac{2(n+1)}{(2n)!}t^{2n+1}\mathrm{d}t$$

Last edited: May 16, 2015
16. May 16, 2015

### wabbit

Right, and if you want to be complete you can add a mention of a theorem that lets you swap the sum and integral here, $\sum\int=\int\sum$ , either using the fact that this is always true for power series inside their radius of convergence, or just referring to the general absolute convergence rule.


text:
Author Topic: TT2 Q4 Question for step4  (Read 1476 times)

hanyu Qi

• Jr. Member
• Posts: 12
• Karma: 4
TT2 Q4 Question for step4
« on: December 02, 2018, 01:36:30 PM »
Hello everyone, I am wondering why the range of $\theta$ is $[0,\pi]$ instead of $[\pi,0]$.

Then the integral estimation would be $|\int_{\gamma_{\epsilon}} f(z) \text{d}z| \leq \int_{\pi}^{0} |f(z)| \text{d}z = \frac{-\pi \epsilon}{\sqrt{\epsilon} (1-{\epsilon}^2)}$ goes to 0 as $\epsilon$ close to 0+.

« Last Edit: December 02, 2018, 01:38:14 PM by Alex Qi »

hanyu Qi

• Jr. Member
• Posts: 12
• Karma: 4
Re: TT2 Q4 Question for step4
« Reply #1 on: December 02, 2018, 01:41:20 PM »
Never mind. I think I know why.

In the answer, f(z) is integral over $-\gamma_{\epsilon}$. I guess this is why its range is $[0,\pi]$


text:
Recently, I have been reading some algebra. This has been immensely enjoyable; I had forgotten how much I used to like algebra. The material here is by no means advanced, but relies on some basic definitions from group theory.

Suppose I have two groups H and K. Can we combine them together? I learnt some years ago now that there is a straightforward way of forming a ‘sum’ of these two objects: One takes the cartesian product H$\times$K, and bestows it with a group operation in the most basic and obvious way possible: (h,k)$\ast$(h’,k’) = (hh’,kk’). The group that results is known as the direct product of H and K.

The same can be done with vector spaces: Given two vector spaces V and W, I can form the direct sum of them, which is thought of as the vector space consisting of ordered pairs of the form (v,w) with v in V and w in W. It is easy to guess what the rules for addition and scalar multiplication must be. However, soon after learning these two definitions, I began to realise that at least a small amount of wool was obscuring my eyes: Suppose I have a vector space X and two subspaces Y and Z such that every vector x in X can be written uniquely in the form x = y + z, for y in Y and z in Z. I would then be invited to observe how this means that X was the direct sum of Y and Z. This didn’t sit well with me. It clearly wasn’t quite the same thing: There was no cartesian product; there was no way that x = y + z was the same as x = (y,z).

I soon came across the same disparity in group theory as well:One may have a group G that can be written as G = HK for two normal subgroups H and K (this means, as in the vector space example, that every g in G can be expressed uniquely as g = hk for h in H and k in K). In what sense is G the direct product of H and K?

### Internality

There are really two different kinds of direct product, as you have now seen. For me, only when trying to fully understand the more complicated semidirect product of groups did the difference become stark enough for me to really take notice. I’ll now stick with groups rather than switching between groups and vector spaces.

Let’s return to the two groups H and K, which I started with. They are completely separate groups with separate identity elements and group operations etc. Let’s denote their direct product by G. The first mathematical observation I’d like to make is that the group G contains copies of both H and K as normal subgroups: The subgroup H’ of G consisting of elements of the form (h,1) is isomorphic to H and the normality is obvious: For x,h in H and k in K, we have

$(x,k)(h,1)(x,k)^{-1} = (x,k)(h,1)(x^{-1},k^{-1}) = (xhx^{-1},1).$

We define K‘ analogously. Secondly, it is trivial to observe that (by construction) every element g of G can be expressed in the form (h,k) = (h,1)(1,k) for h in H and k in K. We write this fact as G = H’K’. Thirdly, the only element of G which belongs to both of H’ and K’ is the identity element (1,1).

The key conceptual point about direct products is that the copies of H and K inside G stay separate. They do not interact with each other at all. The structure of G is determined by H interacting with itself, in the form of the normal subgroup H’, and, simultaneously but independently, K interacting with itself, in the form of the normal subgroup K’.

The construction we have just described is called the external direct product of H and K. The three observations which we made are the core of what a direct product of groups is, because it is from these observations that one may see both external and internal direct products, which we now go on to discuss, in the same light.

Let us suppose now that we start with a (completely different) group G, and normal subgroups H’ and K’. Suppose further that every element g of G can be written as g = hk for h in H’ and k in K’. Lastly, suppose that the only element of G which belongs to both H’ and K’ is the identity element. In this case, we say that G is the internal direct product of H’ and K’. Totally painless. If you ask me, this is the more natural concept.

### Progeny

I was introduced to these concepts in the order I have exposited them here: The external direct product first, and then the internal direct product. The external direct product seemed obvious and natural at first: A robust and basic way to build new groups from old. However, since then, my point of view has completely changed. I think that the other way around is the more natural order.

The internal  direct product captures the following situation: You are given a group G, that, on closer inspection is built out of two groups HK, in such a way that they do not interact with each other – it is easy to show that every element g of G can be written uniquely in the form g=hk, and that the elements of H and K commute with each other (this is even easier to see by looking at the external direct product) . The summands H and K really are like separate dimensions of G – think of the two ‘coordinates’ of $\mathbb{R}^2$ under addition. Now, suppose instead that you are given two separate groups H and K. The concept of the external direct product answers the following question: Is there a group which contains both H and K (or rather isomorphic copies thereof) as subgroups  and which is the internal direct product of said subgroups? The answer is yes and the external direct product is exactly this group (‘up to isomorphism’, obviously). The interpretation I am driving home here is that the external direct product is not nice and natural but rather a contrived and somewhat cumbersome construction used to recreate and achieve the more natural scenario of the internal direct product.

In the next post we will start to think about what happens when there is some interaction between the two subgroups which make up a group in this way and thus explore the concept of the semi-direct product.


text: Asymptotic existence of tight orthogonal main effect plans Our main result is showing the asymptotic existence of tight $\OMEP$s. More precisely, for each fixed number $k$ of rows, and with the exception of $\OMEP$s of the form $2 \times 2 \times \cdots 2 \times 2s\specdiv 4s$ with $s$ odd and with more than three rows, there are only a finite number of tight $\OMEP$ parameters for which the tight $\OMEP$ does not exist. Categories:62K99, 05B15


text:
# Problem with node shape (diamond)

I am drawing a flow chart. Diamond-shaped nodes don't come out right. My minimal working example is below. There are no problems with ShareLaTeX, but it doesn't work on my computer.

\usepackage{tikz}

\usetikzlibrary{shapes.geometric, arrows, fit}

\tikzstyle{decision} = [diamond, minimum width=2cm, minimum height=1cm, text centered, text width=2cm, draw=black, inner sep=1pt, fill=green!10]
\tikzstyle{startstop} = [rectangle, rounded corners, minimum width=3cm, minimum height=1cm,text centered, draw=black, fill=red!30]
\tikzstyle{io} = [trapezium, trapezium left angle=70, trapezium right angle=110, minimum width=2cm, minimum height=1cm, text centered, draw=black, fill=blue!30]
\tikzstyle{process} = [rectangle, minimum width=3cm, minimum height=1cm, text centered,text width=5cm, draw=black, fill=orange!30]
\tikzstyle{endloop} = [ellipse, minimum width=2cm, minimum height=1cm, text centered, text width=3cm, draw=black, inner sep=1pt]
\tikzstyle{input} = [coordinate]
\tikzstyle{output} = [coordinate]
\tikzstyle{arrow} = [thick,->,>=stealth]

\begin{document}

\begin{tikzpicture} [node distance=2cm]

\node [input, name=input] {};
\node(dec1)[decision, below of=input]{$Round <= Limit$};

\end{tikzpicture}

\end{document}

Output on my computer:

Output from ShareLaTeX:

• Welcome to TeX.SX! You can have a look at our starter guide to familiarise yourself further with our format. A tip: If you indent lines by 4 spaces, they'll be marked as a code sample. You can also highlight the code and click the "code" button (with "{}" on it). – ChrisS Jun 27 '18 at 5:51
• Sorry and thanks. Error message for Texstudio is : Undefined control sequence. ...ision, below of=input]{$Round <= Limit$}; – Aaron Jun 27 '18 at 5:56
• see my answer on question tex.stackexchange.com/questions/300759. – Zarko Jun 27 '18 at 6:05

• i can't reproduce your problem, after completing your mwe, it works fine
• you use one old definition (with obsolete tizstyle syntax) of nodes style for flow charts. i suggest to renew it as is used in mwe below (copied from my here):

\documentclass[border=3mm,
tikz]{standalone}
\usetikzlibrary{arrows.meta,
calc, chains,
quotes,
positioning,
shapes.geometric}

\begin{document}
%---------------------------------------------------------------%
\begin{tikzpicture}[
node distance = 8mm and 16mm,
start chain = A going below,
base/.style = {draw, minimum width=32mm, minimum height=8mm,
align=center, on chain=A},
startstop/.style = {base, rectangle, rounded corners, fill=red!30},
process/.style = {base, rectangle, fill=orange!30},
io/.style = {base, trapezium,
trapezium left angle=70, trapezium right angle=110,
fill=blue!30},
decision/.style = {base, diamond, fill=green!30},
every edge quotes/.style = {auto=right}]
]
\node (dec1) [decision] {$Round <= Limit$};
\end{tikzpicture}
\end{document}

• in the case, that you like to have flowchart's nodes style in preamble (for more flowcharts, than you can do this on the following way:

\documentclass[border=3mm,
tikz]{standalone}
\usetikzlibrary{arrows.meta,
calc, chains,
quotes,
positioning,
shapes.geometric}

\tikzset{FlowChart/.style={     % <---
start chain = A going below,
base/.style = {draw, minimum width=32mm, minimum height=8mm,
align=center, on chain=A},
startstop/.style = {base, rectangle, rounded corners, fill=red!30},
process/.style = {base, rectangle, fill=orange!30},
io/.style = {base, trapezium,
trapezium left angle=70, trapezium right angle=110,
fill=blue!30},
decision/.style = {base, diamond, fill=green!30},
every edge quotes/.style = {auto=right}
}% end of FlowChart styles
}

\begin{document}
%---------------------------------------------------------------%
\begin{tikzpicture}[FlowChart,  % <---
node distance = 8mm and 16mm,
]
\node(dec1)[decision]{$Round <= Limit$};
\end{tikzpicture}
\end{document}

• note: i don't see any befits for declaring nodes as coordinates (for input and output). for this purposes you can simply use \coordinate, if you really need it.

• an example of code for relatively simple flowchart you can find here.

• above document examples also works with any other document class, not only with standalone. for example:

\documentclass[ pdftex, oneside,
Einseitiger Druck. 12pt,
Schriftgroesse parskip=half,
footheight = 16pt,  abstracton, DIV=calc,
Satzspiegel berechnen BCOR=8mm, headinclude=false, footinclude=false, listof=totoc, toc=bibliography,
]{scrreprt}
\usepackage{tikz}
\usetikzlibrary{arrows.meta,
calc, chains,
quotes,
positioning,
shapes.geometric}

\tikzset{FlowChart/.style={     % <---
start chain = A going below,
base/.style = {draw, minimum width=32mm, minimum height=8mm,
align=center, on chain=A},
startstop/.style = {base, rectangle, rounded corners, fill=red!30},
process/.style = {base, rectangle, fill=orange!30},
io/.style = {base, trapezium,
trapezium left angle=70, trapezium right angle=110,
fill=blue!30},
decision/.style = {base, diamond, fill=green!30},
every edge quotes/.style = {auto=right}
}% end of FlowChart styles
}

\begin{document}
%---------------------------------------------------------------%
\begin{tikzpicture}[FlowChart,  % <---
node distance = 8mm and 16mm,
]
\node(dec1)[decision]{$Round <= Limit$};
\end{tikzpicture}
\end{document}

gives the same result as both mwe before.

• hmm, if I start a new project, your example works. But when I add exact the same think at my actual project: Undefined control sequence. \node (dec1) [decision] {$Round <= Limit$}; – Aaron Jun 27 '18 at 6:57
• @Aaron, this means that your actual project is corrupted (probably it has missing tikz libraries, etc. without knowing its preamble, i can't say what is going wrong in your document (sorry, my crystal ball is on vacation :-( ). – Zarko Jun 27 '18 at 7:15
• I searched a little bit and I think it is because I have another class.{\documentclass[% pdftex, oneside, % Einseitiger Druck. 12pt, % Schriftgroesse parskip=half, %topmargin = 10pt, headheight = 33pt, headsepline, footsepline, %footheight = 16pt, abstracton, DIV=calc, % Satzspiegel berechnen BCOR=8mm, headinclude=false, footinclude=false, listof=totoc, toc=bibliography, ]{scrreprt} – Aaron Jun 27 '18 at 7:20
• I think your example is working because of the {standalone} environment. – Aaron Jun 27 '18 at 7:22
• no, it also work with any other document class. you only need to add package tikz, which is in my mwe included in option of standalone. please, edit your question and add your code in comment to your code snippet. – Zarko Jun 27 '18 at 7:33


text:
# LU Decomposition of PSD Matrix + Diagonal Matrix

If I have a psd, symmetric matrix $$\mathbf{A}$$ and I need to do LU decomps on $$\mathbf{B_i}= \mathbf{A} + \mathbf{D_i}$$ (where $$\mathbf{D_i}$$ is a diagonal psd matrix, where $$\mathbf{D_i}$$ changes randomly). Is there an easy way to go from the LU of $$\mathbf{A}$$ + some function of $$\mathbf{D_i}$$ to get the LU of $$\mathbf{B_i}$$?

Any references in the literature would be greatly appreciated.

## 1 Answer

Use a Cholesky decomposition or an LDL decomposition instead of LU.

Judging from answers to:

there's no good way to update any of those decompositions. You'd do no worse to recompute the decomposition from scratch each time unless the rank of the diagonal matrix is really low, in which case, maybe you could do something similar to Sherman-Morrison(-Woodbury)-type updates.


text:
digplanet beta 1: Athena
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In differential geometry, the first fundamental form is the inner product on the tangent space of a surface in three-dimensional Euclidean space which is induced canonically from the dot product of R3. It permits the calculation of curvature and metric properties of a surface such as length and area in a manner consistent with the ambient space. The first fundamental form is denoted by the Roman numeral I,

$\!\mathrm{I}(x,y)= \langle x,y \rangle.$

Let X(uv) be a parametric surface. Then the inner product of two tangent vectors is

\begin{align} & {} \quad \mathrm{I}(aX_u+bX_v,cX_u+dX_v) \\ & = ac \langle X_u,X_u \rangle + (ad+bc) \langle X_u,X_v \rangle + bd \langle X_v,X_v \rangle \\ & = Eac + F(ad+bc) + Gbd, \end{align}

where E, F, and G are the coefficients of the first fundamental form.

The first fundamental form may be represented as a symmetric matrix.

$\!\mathrm{I}(x,y) = x^T \begin{pmatrix} E & F \\ F & G \end{pmatrix}y$

Further notation

When the first fundamental form is written with only one argument, it denotes the inner product of that vector with itself.

$\!\mathrm{I}(v)= \langle v,v \rangle = |v|^2$

The first fundamental form is often written in the modern notation of the metric tensor. The coefficients may then be written as $g_{ij}$:

$\left(g_{ij}\right) = \begin{pmatrix}g_{11} & g_{12} \\g_{21} & g_{22}\end{pmatrix} =\begin{pmatrix}E & F \\F & G\end{pmatrix}$

The components of this tensor are calculated as the scalar product of tangent vectors X1 and X2:

$g_{ij} = X_i \cdot X_j$

for i, j = 1, 2. See example below.

Calculating lengths and areas

The first fundamental form completely describes the metric properties of a surface. Thus, it enables one to calculate the lengths of curves on the surface and the areas of regions on the surface. The line element ds may be expressed in terms of the coefficients of the first fundamental form as

$ds^2 = Edu^2+2Fdudv+Gdv^2 \,$.

The classical area element given by $dA = |X_u \times X_v| \ du\, dv$ can be expressed in terms of the first fundamental form with the assistance of Lagrange's identity,

$dA = |X_u \times X_v| \ du\, dv= \sqrt{ \langle X_u,X_u \rangle \langle X_v,X_v \rangle - \langle X_u,X_v \rangle^2 } \ du\, dv = \sqrt{EG-F^2} \, du\, dv.$

Example

The unit sphere in R3 may be parametrized as

$X(u,v) = \begin{pmatrix} \cos u \sin v \\ \sin u \sin v \\ \cos v \end{pmatrix},\ (u,v) \in [0,2\pi) \times [0,\pi).$

Differentiating $X(u,v)$ with respect to u and v yields

$X_u = \begin{pmatrix} -\sin u \sin v \\ \cos u \sin v \\ 0 \end{pmatrix},\ X_v = \begin{pmatrix} \cos u \cos v \\ \sin u \cos v \\ -\sin v \end{pmatrix}.$

The coefficients of the first fundamental form may be found by taking the dot product of the partial derivatives.

$E = X_u \cdot X_u = \sin^2 v$
$F = X_u \cdot X_v = 0$
$G = X_v \cdot X_v = 1$

Length of a curve on the sphere

The equator of the sphere is a parametrized curve given by $(u(t),v(t))=(t,\frac{\pi}{2})$ with t ranging from 0 to $2\pi$. The line element may be used to calculate the length of this curve.

$\int_0^{2\pi} \sqrt{ E\left(\frac{du}{dt}\right)^2 + 2F\frac{du}{dt}\frac{dv}{dt} + G\left(\frac{dv}{dt}\right)^2 } \,dt = \int_0^{2\pi} \sin v \,dt = 2\pi \sin \frac{\pi}{2} = 2\pi$

Area of a region on the sphere

The area element may be used to calculate the area of the sphere.

$\int_0^{\pi} \int_0^{2\pi} \sqrt{ EG-F^2 } \ du\, dv = \int_0^{\pi} \int_0^{2\pi} \sin v \, du\, dv = 2\pi \left[-\cos v\right]_0^{\pi} = 4\pi$

Gaussian curvature

The Gaussian curvature of a surface is given by

$K = \frac{\det \mathit{II}}{\det I} = \frac{ LN-M^2}{EG-F^2 },$

where L, M, and N are the coefficients of the second fundamental form.

Theorema egregium of Gauss states that the Gaussian curvature of a surface can be expressed solely in terms of the first fundamental form and its derivatives, so that K is in fact an intrinsic invariant of the surface. An explicit expression for the Gaussian curvature in terms of the first fundamental form is provided by the Brioschi formula.

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