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In this paper, we have investigated the projective action of the Lie group SL(3,R) on the homogeneous space ℝ핡². Following the Erlangen approach, the action of the subgroups of SL(3,R) on the projective lines has been examined. Projective properties, in particular, plane duality and cross-ratio, are discussed for the SL(3,R) action. [ABSTRACT FROM AUTHOR]

This paper aims to enhance the existing Automatic Meter Reading (AMR) technologies for utilities in the public services sector, such as water, electricity, and gas, by allowing users to regularly upload images of their meters, which are then automatically processed by machines for digit recognition. We propose an end-to-end AMR approach designed explicitly for unconstrained environments, offering practical solutions to common failures encountered during the automatic recognition process, such as image blur, perspective distortion, partial reflection, poor lighting, missing digits, and intermediate digit states, to reduce the failure rate of automatic meter readings. The system's first stage involves checking the quality of the user-uploaded images through the SVM method and requesting re-uploads for images unsuitable for digit extraction and recognition. The second stage employs deep learning models for digit localization and recognition, automatically detecting and correcting issues such as missing and intermediate digits to enhance the accuracy of automatic meter readings. Our research established a gas meter training dataset comprising 52,000 images, extensively annotated across various degrees, to train the deep learning models for high-precision digit recognition. Experimental results demonstrate that, with the simple SVM model, an accuracy of 87.03% is achieved for the classification of blurry image types. In addition, meter digit recognition (including intermediate digit states) can reach 97.6% (mAP), and the detection and correction of missing digits can be as high as 63.64%, showcasing the practical application value of the system developed in this study. [ABSTRACT FROM AUTHOR]

Mobile and handheld electronic products such as smartphones and dashboard cameras are frequently used to record occurrences of traffic accidents nowadays. Most existing studies require at least one entire lane marking as a reference to calculate a certain amount of driving distances and are not adapted to a targeted vehicle with a rapidly changing speed in a short time. Therefore, existing approaches cannot provide the desired accuracy of speed estimation in real-life scenarios. In this study, we obtain dynamic time and space data from recorded footage using dashboard cameras and then apply photogrammetry and cross-ratio methods to estimate the vehicle speed. In addition, the proposed method is applied to other cars' speed estimation and ego-speed estimation even though both cars are moving. The experimental results show that given the frame rate on the recorded footage, our proposed method only needs one object in each frame to estimate the vehicle speed even at a rapidly changing speed. Our proposed method shows that the difference between the estimated speed and the reference speed by the Global Positioning System (GPS) is smaller than 1 km/h when only one car is on the move, and smaller than 3 km/h when both cars are on the move. Nevertheless, the difference between the estimated speed and the reference speed is between 1.46 km/h and 5.28 km/h when one car moves by AUPD method. To our knowledge, there is no existing method that estimated the front vehicle speed when both cars are on the move using dashboard cameras. [ABSTRACT FROM AUTHOR]

Hyperbolic fillings of metric spaces are a well-known tool for proving results on extending quasi-Moebius maps between boundaries of Gromov hyperbolic spaces to quasi-isometries between the spaces. For a hyperbolic filling Y of the boundary of a Gromov hyperbolic space X, one has a quasi-Moebius identification between the boundaries ∂ Y and ∂ X . For CAT(-1) spaces, and more generally boundary continuous Gromov hyperbolic spaces, one can refine the quasi-Moebius structure on the boundary to a Moebius structure. It is then natural to ask whether there exists a functorial hyperbolic filling of the boundary by a boundary continuous Gromov hyperbolic space with an identification between boundaries which is not just quasi-Moebius, but in fact Moebius. The filling should be functorial in the sense that a Moebius homeomorphism between boundaries should induce an isometry between there fillings. We give a positive answer to this question for a large class of boundaries satisfying one crucial hypothesis, the antipodal property. This gives a class of compact spaces called quasi-metric antipodal spaces. For any such space Z, we give a functorial construction of a boundary continuous Gromov hyperbolic space M (Z) together with a Moebius identification of its boundary with Z. The space M (Z) is maximal amongst all fillings of Z. These spaces M (Z) give in fact all examples of a natural class of spaces called maximal Gromov hyperbolic spaces. We prove an equivalence of categories between quasi-metric antipodal spaces and maximal Gromov hyperbolic spaces. This is part of a more general equivalence we prove between the larger categories of certain spaces called antipodal spaces and maximal Gromov product spaces. We prove that the injective hull of a Gromov product space X is isometric to the maximal Gromov product space M (Z) , where Z is the boundary of X. We also show that a Gromov product space is injective if and only if it is maximal. [ABSTRACT FROM AUTHOR]

This paper aims to enhance the existing Automatic Meter Reading (AMR) technologies for utilities in the public services sector, such as water, electricity, and gas, by allowing users to regularly upload images of their meters, which are then automatically processed by machines for digit recognition. We propose an end-to-end AMR approach designed explicitly for unconstrained environments, offering practical solutions to common failures encountered during the automatic recognition process, such as image blur, perspective distortion, partial reflection, poor lighting, missing digits, and intermediate digit states, to reduce the failure rate of automatic meter readings. The system’s first stage involves checking the quality of the user-uploaded images through the SVM method and requesting re-uploads for images unsuitable for digit extraction and recognition. The second stage employs deep learning models for digit localization and recognition, automatically detecting and correcting issues such as missing and intermediate digits to enhance the accuracy of automatic meter readings. Our research established a gas meter training dataset comprising 52,000 images, extensively annotated across various degrees, to train the deep learning models for high-precision digit recognition. Experimental results demonstrate that, with the simple SVM model, an accuracy of 87.03% is achieved for the classification of blurry image types. In addition, meter digit recognition (including intermediate digit states) can reach 97.6% (mAP), and the detection and correction of missing digits can be as high as 63.64%, showcasing the practical application value of the system developed in this study.

The cross-ratio (CR)-based method exploits the invariance property of CRs in projective transformation to determine a screen point corresponding to the pupil center. However, this point is essentially the intersection of the eyeball optical axis (OA) and the screen, rather than the actual point-of-regard (POR). In addition, the premise of CR calculation is that the corneal reflection points of four on-screen light sources are coplanar with the 3D pupil center, but they are only assumed to be coplanar. To solve these issues, this paper proposes an improved CR-based gaze estimation method using weighted average and polynomial compensation. Under the configuration of a single camera and two light sources, the 3D corneal center and the normal vector of virtual pupil plane are first estimated using the eyeball imaging model, and then four reference planes parallel to the virtual pupil plane are determined based on the geometric model of pupil and screen corner points. The screen point corresponding to the intersection of each reference plane and the line connecting the camera optical center and the imaging pupil center is calculated using the conventional CR-based method. Thus, the point where the OA intersects the screen is determined by the weighted average of these four points. Finally, a polynomial is learned to compensate it to the POR. The experimental results show that the gaze accuracy can reach 1.33° when the more accurate eye is selected, and it would be improved by 24% by calculating the joint POR, which is competitive with the state-of-the-art methods using more complex systems. On the basis of simplifying the system configuration of CR-based methods, the proposed method avoids the non-coplanarity of 3D pupil center and corneal reflection plane, and improves the gaze estimation performance.

Cross-ratio is an important local measure of the strength of dependence among correlated failure times. If a covariate is available, it may be of scientific interest to understand how the cross-ratio varies with the covariate as well as time components. Motivated by the Tremin study, where the dependence between age at a marker event reflecting early lengthening of menstrual cycles and age at menopause may be affected by age at menarche, we propose a proportional cross-ratio model through a baseline cross-ratio function and a multiplicative covariate effect. Assuming a parametric model for the baseline cross-ratio, we generalize the pseudo-partial likelihood approach of Hu et al. (Biometrika 98:341-354, 2011) to the joint estimation of the baseline cross-ratio and the covariate effect. We show that the proposed parameter estimator is consistent and asymptotically normal. The performance of the proposed technique in finite samples is examined using simulation studies. In addition, the proposed method is applied to the Tremin study for the dependence between age at a marker event and age at menopause adjusting for age at menarche. The method is also applied to the Australian twin data for the estimation of zygosity effect on cross-ratio for age at appendicitis between twin pairs.

This paper aims to perform metric measurements of narrow street façades using single image captured by smartphone's camera. Since tight area accompanied by narrow street limits object to camera distance, object lines perpendicular to façade do not appear in image and consequently their vanishing point (VP) is hard to detect. Accordingly, semi-automated MATLAB® application was designed depending only on two orthogonal VPs. Novelty of work comes from using smartphone as a cost and time efficient tool for measurements, depending only on two VPs, and applying image line refinement approach exploiting detected VPs. Three single images were captured by three different smartphones. Then, undistorted single images were formed after calibrating cameras. Image lines for horizontal and vertical object lines were extracted semi-automatically. Two VPs were detected applying two models: Model-I solves for vanishing points' Cartesian coordinates, whereas Model-II solves for angle coordinate peaks of histogram. Image line refinement approach was applied before applying cross-ratio using one horizontal and one vertical reference lines to calculate object lengths of 46 check lines (horizontal and vertical). Proposed models provided reliable and comparable results. Applying line refinement approach improved solution with best overall accuracy of 0.010 m and 0.011 m for Model-I and Model-II, respectively. [ABSTRACT FROM AUTHOR]

This paper aims to perform metric measurements of narrow street façades using single image captured by smartphone’s camera. Since tight area accompanied by narrow street limits object to camera distance, object lines perpendicular to façade do not appear in image and consequently their vanishing point (VP) is hard to detect. Accordingly, semi-automated MATLAB® application was designed depending only on two orthogonal VPs. Novelty of work comes from using smartphone as a cost and time efficient tool for measurements, depending only on two VPs, and applying image line refinement approach exploiting detected VPs. Three single images were captured by three different smartphones. Then, undistorted single images were formed after calibrating cameras. Image lines for horizontal and vertical object lines were extracted semi-automatically. Two VPs were detected applying two models: Model-I solves for vanishing points’ Cartesian coordinates, whereas Model-II solves for angle coordinate peaks of histogram. Image line refinement approach was applied before applying cross-ratio using one horizontal and one vertical reference lines to calculate object lengths of 46 check lines (horizontal and vertical). Proposed models provided reliable and comparable results. Applying line refinement approach improved solution with best overall accuracy of 0.010 m and 0.011 m for Model-I and Model-II, respectively.

Medical studies often collect bivariate survival data including time to the same type of disease in twins or time to two different diseases from the same individuals. Cross-ratio, defined as the ratio of conditional hazard functions for one event given the other, is often used as a measure of dependency between the two survival outcomes. Statistical methods have also been proposed to estimate the effect of covariates on cross-ratio in order to identify common factors influencing both survival outcomes. There are currently three estimation approaches for cross-ratio, namely, a parametric approach using the Clayton copula, a semi-parametric two-stage approach proposed by Shih and Louis, and a nonparametric pseudo-partial likelihood approach proposed by Hu et al. In this paper, we compare the three estimation approaches on estimating covariates' effect on the cross-ratio in simulation studies. We found that the nonparametric pseudo-partial likelihood estimation approach performed well and that the method was also robust under various model assumptions. Data from a longitudinal cohort of elderly African Americans were used to illustrate the three estimation approaches. [ABSTRACT FROM AUTHOR]

In recent years, exploration of local characteristics have been an effective feature extraction technique for managing multimedia repository for image retrieval. These class of approaches accelerate the image retrieval. However, with the increase of online intrusion, it is also essential to provide confidentiality for the private online data with effective functionality and significant retrieval accuracy. In this paper, we have solved the protuberant bottleneck problem of privacy-preserving in CBIR using random projection technique for feature encryption with effective retrieval. For this, we formulated highly discriminative interest points based features that are rotation and translation invariant and are constructed using k -means clustering and silhouette concept. Moreover, we have devised locally likely arrangement hashing technique that can incorporate fast, accurate, and secure matching over encrypted features. Based on the experiment, it substantially outperforms the recent state-of-the-art techniques in terms of retrieval accuracy and time overhead on various standard benchmarks, namely, Corel, Olivia, ALOI, and MPEG-7. The proposed method on average provided retrieval accuracy of 83.10%, 81.30%, 81.62%, 73.01% for Corel, Olivia, ALOI and MPEG-7 plaintexted feature, respectively while retrieval accuracy of 79.90% for Corel, 79.10% for Olivia, 79.97% for ALOI, and 67.34% for MPEG-7 using the encrypted feature. [ABSTRACT FROM AUTHOR]

The hyperbolic Möbius transformations, which have been defined and proved to be hyperbolic conformal in Golberg and Luna-Elizarrarás (Math Methods Appl Sci 2020, ), are generalizations of Möbius transformations in complex space C (i) and hyperbolic space D to multidimensional hyperbolic space D n . In this paper, we study the hyperbolic Möbius transformation in bicomplex space B C isomorphic to D 2 in detail, present a conjugacy classification according to the number of fixed points in S L (2 , B C) , and detailedly prove that the cross-ratio is invariant under hyperbolic Möbius transformations. Furthermore, the present paper generalizes the classical results, which have closed relation with fixed points and cross-ratios, to B C and may give new energy for the development of hyperbolic Möbius groups. [ABSTRACT FROM AUTHOR]

In this work, we derived some new algebraic results on continued fraction expansion of the ratio of the two consecutive balcobalancing numbers, circulant matrices and spectral norms, Pythagorean triples, characteristic polynomials and eigenvalues of the nth power of the companion matrices, Cassini and Catalan identities, cross-ratios and Heisenberg groups related to balcobalancing numbers, Lucas-balcobalancing numbers and balcobalancers. [ABSTRACT FROM AUTHOR]

Drought is a major natural hazard that impacts agriculture, the environment, and socio-economic conditions. In 2018 and 2019, Europe experienced a severe drought due to below average precipitation and high temperatures. Drought stress affects the moisture content and structure of agricultural crops and can result in lower yields. Synthetic Aperture Radar (SAR) observations are sensitive to the dielectric and geometric characteristics of crops and underlying soils. This study uses data from ESA's Sentinel-1 SAR satellite to investigate the influence of drought stress on major arable crops of the Netherlands, its regional variability and the impact of water management decisions on crop development. Sentinel-1 VV, VH and VH/VV backscatter data are used to quantify the variability in the spatio-temporal dynamics of agricultural crop parcels in response to drought. Results show that VV and VH backscatter values are 1 to 2 dB lower for crop parcels during the 2018 drought compared to values in 2017. In addition, the growth season indicated by the cross-ratio (CR, VH/VV) for maize and onion is shorter during the drought year. Differences due to irrigation restrictions are observed in backscatter response from maize parcels. Lower CR values in 2019 indicate the impact of drought on the start of the growing season. Results demonstrate that Sentinel-1 can detect changes in the seasonal cycle of arable crops in response to agricultural drought. [ABSTRACT FROM AUTHOR]

A satellite navigation system makes it simple to find and navigate to a specific position. Although a carrier measurement is required to establish a precise position due to the characteristics of the carrier observation, it is difficult to determine a robust position in a poor signal reception environment such as urban areas. Various studies are being carried out to overcome this problem, with cycle slips being the most important factor. With only a single frequency, it is very challenging to detect cycle slips in multiple satellite channels at the same time. A geometry-based technique is proposed in this study as a technical solution for detecting simultaneous cycle slips for multiple channels utilizing only a single-frequency receiver. The method could detect a half-wavelength size of cycle slip for each channel through the geometry information. [ABSTRACT FROM AUTHOR]

This article relates the Grassmannian complexes of geometric configurations to the tangent to the Bloch-Suslin complex and to the tangent to Goncharov’s motivic complex. By means of morphisms, we bring the geometry of configurations in tangent groups, $T\mathcal{B}_2(F)$ and $T\mathcal{B}_3(F)$ that produce commutative diagrams. To show the commutativity of diagrams, we use combinatorial techniques that include permutations in symmetric group S6. We also create analogues of the Siegel’s cross-ratio identity for the truncated polynomial ring F[ε]ν.

Drought is a major natural hazard that impacts agriculture, the environment, and socio-economic conditions. In 2018 and 2019, Europe experienced a severe drought due to below average precipitation and high temperatures. Drought stress affects the moisture content and structure of agricultural crops and can result in lower yields. Synthetic Aperture Radar (SAR) observations are sensitive to the dielectric and geometric characteristics of crops and underlying soils. This study uses data from ESA’s Sentinel-1 SAR satellite to investigate the influence of drought stress on major arable crops of the Netherlands, its regional variability and the impact of water management decisions on crop development. Sentinel-1 VV, VH and VH/VV backscatter data are used to quantify the variability in the spatio-temporal dynamics of agricultural crop parcels in response to drought. Results show that VV and VH backscatter values are 1 to 2 dB lower for crop parcels during the 2018 drought compared to values in 2017. In addition, the growth season indicated by the cross-ratio (CR, VH/VV) for maize and onion is shorter during the drought year. Differences due to irrigation restrictions are observed in backscatter response from maize parcels. Lower CR values in 2019 indicate the impact of drought on the start of the growing season. Results demonstrate that Sentinel-1 can detect changes in the seasonal cycle of arable crops in response to agricultural drought.

A satellite navigation system makes it simple to find and navigate to a specific position. Although a carrier measurement is required to establish a precise position due to the characteristics of the carrier observation, it is difficult to determine a robust position in a poor signal reception environment such as urban areas. Various studies are being carried out to overcome this problem, with cycle slips being the most important factor. With only a single frequency, it is very challenging to detect cycle slips in multiple satellite channels at the same time. A geometry-based technique is proposed in this study as a technical solution for detecting simultaneous cycle slips for multiple channels utilizing only a single-frequency receiver. The method could detect a half-wavelength size of cycle slip for each channel through the geometry information.

In this paper, we carry the obtained results for a certain class of Moufang-Klingenberg planes co-ordinatized by the local alternative ring of dual numbers, over to a class of Moufang-Klingenberg planes coordinatized by the plural algebra of order m which is a local alternative ring. So, we show that its collineation group is transitive on quadrangles and therefore that coordinatization of the planes is independent of the choice of the coordinatization quadrangle. Moreover, by a conjugacy class we discuss the definition of cross-ratio of four distinct points on the projective lines for the planes. [ABSTRACT FROM AUTHOR]

Consider two orientation preserving circle homeomorphisms with different type of singularities: critical map fc with unique cubic critical point xc and fb with one break point xb. Suppose their rotation numbers coincide and irrational i.e. p(fc) = p(fb) =: p and p is irrational number. J. C. Yoccoz proved that C³ critical map with cubic critical point and irrational rotation number pc := p(fc) is topologically conjugated to linear rotation fp(x) = x + p (mod1). Grachek and Swiatek showed that the unique fc- invariant probability measure pc is singular with respect to Lebesgue measure `. A. Dzhalilov and K. Khanin proved that the circle map fb ΞC2+"(S¹ / {xb}), " > 0; with one break point xb and irrational rotation number pb := p(fb) topologically conjugated to fp and the conjugacy pb between fb and fp is singular function. We prove that the conjugation h between fc and fb is singular function on the circle i.e. h(x) is continuous on S¹ and h0(x) = 0 a.e. with respect to Lebesque measure `. The distortions of cross-ratio for critical circle maps are also investigated. [ABSTRACT FROM AUTHOR]

This research is concerned with the study of the projective plane over a finite field Fq. The main purpose is finding partitions of the projective line PG( 1,q3) and the projective plane PG(2,q2 ) q= 11, in addition to embedding PG(1, q) into PG(1, q3 ) and PG(2,q ) into PG(2,q2 ). Clearly, the orbits of PG( 1,q3)q = 11 are found, along with the cross-ratio for each orbit. As for PG( 2,q2), 13 partitions were found on PG(2,q2 ) each partition being classified in terms of the degree of its arc, length, its own code, as well as its error correcting. The last main aim is to classify the group actions on PG(2,11 ). [ABSTRACT FROM AUTHOR]

• A binary step-response serial coded target (BSSCT) is proposed, the binary step period in the image has a strong robustness to external noise interference, and the capacity can be greatly expanded by periodically encoding corner elements of coded targets. • A robust decoding algorithm based on multi-radius neighborhood search is proposed, the robust decoding of the dot-dispersing coded target using cross-ratio invariance needs to ensure P2-invariant stability. • The BSSCT coding and decoding method based on step period has a high recognition rate and location accuracy with different illumination environments, noise environments, and measurement angles. In close-range photogrammetry, it is difficult to meet the measurement requirements of large scenes in actual engineering due to the limited capacity of coded targets. To expand the capacity of the coded target, we propose a binary step-response serial-coded target (BSSCT). The BSSCT introduces periodic binary wave information as an additional feature in the dot-dispersing coded target. Also, a robust recognition algorithm for the BSSCT is developed, the P2-Invariant, and the step period is used for decoding. The capacity of the coded target reaches 7 magnitudes without increasing the auxiliary points. Under different lighting conditions and viewing angles, the measurement experiments show that the BSSCT outperforms other state-of-the-art coded targets. Our BSSCT is a promising standard method for large-field system calibration and object measurement. [ABSTRACT FROM AUTHOR]

Using Mizar [1], in the context of a real vector space, we introduce the concept of affine ratio of three aligned points (see [5]). It is also equivalent to the notion of "Mesure algèbrique"1, to the opposite of the notion of Teilverhältnis2 or to the opposite of the ordered length-ratio [9]. In the second part, we introduce the classic notion of "cross-ratio" of 4 points aligned in a real vector space. Finally, we show that if the real vector space is the real line, the notion corresponds to the classical notion3 [9]: The cross-ratio of a quadruple of distinct points on the real line with coordinates x1, x2, x3, x4 is given by: (x 1 , x 2 ; x 3 , x 4) = x 3 - x 1 x 3 - x 2 . x 4 - x 2 x 4 - x 1 $$({x_1},{x_2};{x_3},{x_4}) = {{{x_3} - {x_1}} \over {{x_3} - {x_2}}}.{{{x_4} - {x_2}} \over {{x_4} - {x_1}}}$$ In the Mizar Mathematical Library, the vector spaces were first defined by Kusak, Leonczuk and Muzalewski in the article [6], while the actual real vector space was defined by Trybulec [10] and the complex vector space was defined by Endou [4]. Nakasho and Shidama have developed a solution to explore the notions introduced by different authors4 [7]. The definitions can be directly linked in the HTMLized version of the Mizar library5. The study of the cross-ratio will continue within the framework of the Klein- Beltrami model [2], [3]. For a generalized cross-ratio, see Papadopoulos [8]. [ABSTRACT FROM AUTHOR]

There is a fundamental relationship between projective geometry and the perspective imaging geometry of a pinhole camera. Projective scales have been used to measure within images from the beginnings of photogrammetry, mostly the cross-ratio on a straight line. However, there are also projective frames in the plane with interesting connections to affine and projective geometry in three dimensional space that can be utilized for photogrammetry. This article introduces an invariant on the projective plane, describes its relation to affine geometry, and how to use it to reduce the complexity of projective transformations. It describes how the invariant can be use to measure on projectively distorted planes in images and shows applications to this in 3D reconstruction. The article follows two central ideas. One is to measure coordinates in an image relatively to each other to gain as much invariance of the viewport as possible. The other is to use the remaining variance to determine the 3D structure of the scene and to locate the camera centers. For this, the images are projected onto a common plane in the scene. 3D structure not on the plane occludes different parts of the plane in the images. From this, the position of the cameras and the 3D structure are obtained. [ABSTRACT FROM AUTHOR]

This paper defines the pressure metric on the Moduli space of Margulis spacetimes without cusps and shows that it is positive definite on the constant entropy sections. It also demonstrates an identity regarding the variation of the cross-ratios. [ABSTRACT FROM AUTHOR]

Cross-ratio invariant is used widely in projective transformations for eye gaze estimation. Establishing a virtual plane projection is an important step to use this property. Most of traditional cross-ratio approaches only used fixed parameters to calculate the gazing point. This paper proposes gazing point dependent eye gazing estimation approach. Our contributions are three-folded. First, we model a dynamic virtual plane projection, which is tangent to the cornea of pupil, to estimate the position of the gazing point. Second, we introduce a two stage approach consisting of rough-to-precise framework for gazing point estimation based on the gazing point dependent virtual plane projection. Third, a heuristic strategy which contains off-line and on-line parameter learning for gazing point estimation is proposed. The experiment results show that our approach can significantly improve the gazing estimation performance with an average accuracy of 0.70°. [ABSTRACT FROM AUTHOR]

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In [11], the second author considered the positive integer solutions of the Pell equation x -- Dy² = 1 for some specific values of D including D = k² -- 2 for an integer k ≥ 2. In this paper, we are able to give the nth integer solution (xn,yn) of x² -- (k² -- 2)y² = 1 by a different method and then we set an integer sequence Wn = pWn-i -- qWn-2 with parameters p = k² -- 2 and q = 1 and derive some algebraic relations on it. [ABSTRACT FROM AUTHOR]

We solve the classification problem for integrable lattices of the form u, = f( u ,..., u ) under the additional assumption of invariance with respect to the group of linear-fractional transformations. The obtained list contains five equations, including three new ones. Difference Miura-type substitutions are found, which relate these equations to known polynomial lattices. We also present some classification results for generic lattices. [ABSTRACT FROM AUTHOR]

There is tremendous scientific and medical interest in the use of biomarkers to better facilitate medical decision making. In this article, we present a simple framework for assessing the predictive ability of a biomarker. The methodology requires use of techniques from a subfield of survival analysis termed semi-competing risks; results are presented to make the article self-contained. As we show in the article, one natural interpretation of semi-competing risks model is in terms of modifying the classical risk set approach to survival analysis that is more germane to medical decision making. A crucial parameter for evaluating biomarkers is the predictive hazard ratio, which is different from the usual hazard ratio from Cox regression models for right-censored data. This quantity will be defined; its estimation, inference, and adjustment for covariates will be discussed. Aspects of causal inference related to these procedures will also be described. The methodology is illustrated with an evaluation of serum albumin in terms of predicting death in patients with primary biliary cirrhosis. [ABSTRACT FROM AUTHOR]