Geometric Explanation Research Articles

A Generalization of Pythagoras's Theorem and Application to Explanations of Variance Contributions in Linear Models

Many aspects of the geometry of linear statistical models and least squares estimation are well known. Discussions of the geometry may be found in many sources. Some aspects of the geometry relating to the partitioning of variation that can be explained using a little‐known theorem of Pappus and have not been discussed previously are the topic of this report. I discuss, using the theorem, how geometric explanation helps us understand issues relating to contributions of independent variables to explanation of variance in a dependent variable. A particular concern that the theorem helps explain involves nonorthogonal linear models including correlated regressors and analysis of variance.

Effects of group size on aggregation against desiccation in woodlice (Isopoda:Oniscidea)

AbstractAggregation in terrestrial isopods, a behaviour that results in the formation of dense clusters, is readily accepted as a mechanism of resistance to desiccation. Thus, aggregation is considered to be an adaptation to terrestrial life in this fully terrestrial suborder of crustaceans. In the present study ofPorcellio scaberLatreille, a cosmopolitan species, individual water loss is investigated experimentally as a function of the size of the aggregates and, for the first time, over a large range of group sizes (groups of 1, 10, 20, 40, 60, 80 and 100 individuals). From the perspective of an isolated individual, aggregation behaviour is effective in reducing the rate of water loss whatever the group size, and reduces the individual water loss rate by more than half in large groups. However, the water loss rate of an individual follows a power law according to group size. Accordingly, if the addition of individuals to small groups strongly reduces the water losses per individual, adding individuals to large groups only slightly reduces the individual water losses. Thus, the successful reduction of the water loss rate by this aggregation behaviour is confirmed, although only up to a certain limit, particularly if the number of individuals per aggregate exceeds 50–60 under the experimental conditions used in the present study. Moreover, the individual surface area exposed to the air, as a function of group size, follows a similar pattern (i.e. a similar power law). Thus, a geometrical explanation is proposed for the nonlinear water losses in woodlice aggregates. These results are discussed in relation to the group sizes observed both in the laboratory and the field.

Lagrangian Floer Superpotentials and Crepant Resolutions for Toric Orbifolds

We investigate the relationship between the Lagrangian Floer superpotentials for a toric orbifold and its toric crepant resolutions. More specifically, we study an open string version of the crepant resolution conjecture (CRC) which states that the Lagrangian Floer superpotential of a Gorenstein toric orbifold $\mathcal{X}$ and that of its toric crepant resolution $Y$ coincide after analytic continuation of quantum parameters and a change of variables. Relating this conjecture with the closed CRC, we find that the change of variable formula which appears in closed CRC can be explained by relations between open (orbifold) Gromov-Witten invariants. We also discover a geometric explanation (in terms of virtual counting of stable orbi-discs) for the specialization of quantum parameters to roots of unity which appears in Y. Ruan's original CRC ["The cohomology ring of crepant resolutions of orbifolds", Gromov-Witten theory of spin curves and orbifolds, 117-126, Contemp. Math., 403, Amer. Math. Soc., Providence, RI, 2006]. We prove the open CRC for the weighted projective spaces $\mathcal{X}=\mathbb{P}(1,\ldots,1,n)$ using an equality between open and closed orbifold Gromov-Witten invariants. Along the way, we also prove an open mirror theorem for these toric orbifolds.

Doubling of the algebra and neutrino mixing within noncommutative spectral geometry

We study the physical implications of the doubling of the algebra, an essential element in the construction of the noncommutative spectral geometry model, proposed by Connes and his collaborators as offering a geometric explanation for the standard model of the strong and electroweak interactions. Linking the algebra doubling to the deformed Hopf algebra, we build Bogoliubov transformations and show the emergence of neutrino mixing.

An extension of the Levi-Weckesser method to the stabilization of the inverted pendulum under gravity

Sufficient conditions are given for the stability of the upper equilibrium of the mathematical pendulum (inverted pendulum) when the suspension point is vibrating vertically with high frequency. The equation of the motion is of the form $$ \ddot{\theta}-\frac{1}{l}\bigl(g+a(t)\bigr) \theta=0, $$ where l,g are constants and a is a periodic step function. M. Levi and W. Weckesser gave a simple geometrical explanation for the stability effect provided that the frequency is so high that the gravity g can be neglected. They also obtained a lower estimate for the stabilizing frequency. This method is improved and extended to the arbitrary inverted pendulum not assuming even symmetricity between the upward and downward phases in the vibration of the suspension point.

Constraints on noncommutative spectral action from Gravity Probe B and torsion balance experiments

Noncommutative spectral geometry offers a purely geometric explanation for the standard model of strong and electroweak interactions, including a geometric explanation for the origin of the Higgs field. Within this framework, the gravitational, the electroweak and the strong forces are all described as purely gravitational forces on a unified noncommutative space-time. In this study, we infer a constraint on one of the three free parameters of the model, namely the one characterising the coupling constants at unification, by linearising the field equations in the limit of weak gravitational fields generated by a rotating gravitational source, and by making use of recent experimental data. In particular, using data obtained by Gravity Probe B, we set a lower bound on the Weyl term appearing in the noncommutative spectral action, namely β≳10−6m−1. This constraint becomes stronger once we use results from torsion balance experiments, leading to β≳104m−1. The latter is much stronger than any constraint imposed so far to curvature squared terms.

Geometric explanation of conic-section interference fringes in a Michelson interferometer

A simple geometric method based on wave-front analysis is developed to provide a concise explanation for the various interference fringes in a Michelson interferometer without the compensator plate. In view of the fact that a homocentric pencil of rays from a point light source becomes astigmatic as it passes obliquely through a glass plate, the wave-front deviation from a spherical one is obtained by calculating the astigmatic focal distance of the central ray. If the compensator plate is removed, the two central rays along the interfering paths have different astigmatic focal distances (AFDs), therefore, the optical path length difference can never be compensated with the movement of the reflected mirrors. The wave-front difference or the optical path difference is determined to the accuracy of second order by comparing the two pairs of principal curvature radii, and the trivial point-by-point calculation of the optical path length difference (OPD) is avoided. Numerical results support the theoretical analysis with great accuracy.

Efficient, uninformative sampling of limb darkening coefficients for two-parameter laws

Stellar limb darkening affects a wide range of astronomical measurements and is frequently modelled with a parametric model using polynomials in the cosine of the angle between the line of sight and the emergent intensity. Two-parameter laws are particularly popular for cases where one wishes to fit freely for the limb darkening coefficients (i.e. an uninformative prior) due to the compact prior volume and the fact that more complex models rarely obtain unique solutions with present data. In such cases, we show that the two limb darkening coefficients are constrained by three physical boundary conditions, describing a triangular region in the two-dimensional parameter space. We show that uniformly distributed samples may be drawn from this region with optimal efficiency by a technique developed by computer graphical programming: triangular sampling. Alternatively, one can make draws using a uniform, bivariate Dirichlet distribution. We provide simple expressions for these parametrizations for both techniques applied to the case of quadratic, square-root and logarithmic limb darkening laws. For example, in the case of the popular quadratic law, we advocate fitting for q_1 = (u_1+u_2)^2 and q_2 = 0.5u_1(u_1+u_2)^{-1} with uniform priors in the interval [0,1] to implement triangular sampling easily. Employing these parametrizations allows one to derive model parameters which fully account for our ignorance about the intensity profile, yet never explore unphysical solutions, yielding robust and realistic uncertainty estimates. Furthermore, in the case of triangular sampling with the quadratic law, our parametrization leads to significantly reduced mutual correlations and provides an alternative geometric explanation as to why naively fitting the quadratic limb darkening coefficients precipitates strong correlations in the first place.

Geometric principles in the assembly of α-helical bundles

α-Helical coiled coils are usually stabilized by hydrophobic interfaces between the two constituent α-helices, in the form of 'knobs-into-holes' packing of non-polar residues arranged in repeating heptad patterns. Here we examine the corresponding 'hydrophobic cores' that stabilize bundles of four α-helices. In particular, we study three different kinds of bundle, involving four α-helices of identical sequence: two pack in a parallel and one in an anti-parallel orientation. We point out that the simplest way of understanding the packing of these 4-helix bundles is to use Crick's original idea that the helices are held together by 'hydrophobic stripes', which are readily visualized on the cylindrical surface lattice of the α-helices; and that the 'helix-crossing angle'--which determines, in particular, whether supercoiling is left- or right-handed--is fixed by the slope of the lattice lines that contain the hydrophobic residues. In our three examples the constituent α-helices have hydrophobic repeat patterns of 7, 11 and 4 residues, respectively; and we associate the different overall conformations with 'knobs-into-holes' packing along the 7-, 11- and 4-start lines, respectively, of the cylindrical surface lattices of the constituent α-helices. For the first two examples, all four interfaces between adjacent helices are geometrically equivalent; but in the third, one of the four interfaces differs significantly from the others. We provide a geometrical explanation for this non-equivalence in terms of two different but equivalent ways of assembling this bundle, which may possibly constitute a bistable molecular 'switch' with a coaxial throw of about 12 Å. The geometrical ideas that we deploy in this paper provide the simplest and clearest description of the structure of helical bundles. In an appendix, we describe briefly a computer program that we have devised in order to search for 'knobs-into-holes' packing between α-helices in proteins.

Geometric Satake, Springer correspondence and small representations

For a simply-connected simple algebraic group $$G$$ over $$\mathbb C $$ , we exhibit a subvariety of its affine Grassmannian that is closely related to the nilpotent cone of $$G$$ , generalizing a well-known fact about $$GL_n$$ . Using this variety, we construct a sheaf-theoretic functor that, when combined with the geometric Satake equivalence and the Springer correspondence, leads to a geometric explanation for a number of known facts (mostly due to Broer and Reeder) about small representations of the dual group.

Earthquake Surface Slip-Length Data is Fit by Constant Stress Drop and is Useful for Seismic Hazard Analysis

Abstract We present a new method to use directly observable surface‐slip measurements in seismic hazard estimates. We present measures of scaling‐relation fits to slip‐length data. These fits show sublinear scaling, a slowing in the rate of slip increase for the longest ruptures, so that L scaling—scaling with the length of the rupture—does not hold out to very large aspect ratio events. We find the best fitting for a constant stress‐drop model, followed next by a square root of length model. The constant stress‐drop model, newly introduced here, provides a geometrical explanation for a long‐standing puzzle of why slip only begins to saturate at large aspect ratios. The good fit of the constant stress‐drop model to the slip‐length data lends further support to the observations of constant stress‐drop scaling across the whole range of magnitudes of earthquakes, from small to great earthquakes. The good fit of the constant stress‐drop model is also reflected by the low variability about the mean, with an average of less than a factor‐of‐2 variability in stress drop about the mean observed. Converting magnitude‐area scaling into implied slip‐length scaling, we determine qualitative consistency in the functional forms, but a quantitative difference of, on average, ∼30% more slip estimated from magnitude area compared with slip length. Online Material: Tables of magnitude‐length‐width, magnitude‐area, and surface slip‐length relations.

Application of unified array calculus to connect 4-D spacetime sensing with string theory and relativity

Array algebra of photogrammetry and geodesy unified multi-linear matrix and tensor operators in an expansion of Gaussian adjustment calculus to general matrix inverses and solutions of inverse problems to find all, or some optimal, parametric solutions that satisfy the available observables. By-products in expanding array and tensor calculus to handle redundant observables resulted in general theories of estimation in mathematical statistics and fast transform technology of signal processing. Their applications in gravity modeling and system automation of multi-ray digital image and terrain matching evolved into fast multi-nonlinear differential and integral array calculus. Work since 1980’s also uncovered closed-form inverse Taylor and least squares Newton-Raphson-Gauss perturbation solutions of nonlinear systems of equations. Fast nonlinear integral matching of array wavelets enabled an expansion of the bundle adjustment to 4-D stereo imaging and range sensing where real-time stereo sequence and waveform phase matching enabled data-to-info conversion and compression on-board advanced sensors. The resulting unified array calculus of spacetime sensing is applicable in virtually any math and engineering science, including recent work in spacetime physics. The paper focuses on geometric spacetime reconstruction from its image projections inspired by unified relativity and string theories. The collinear imaging equations of active object space shutter of special relativity are expanded to 4-D Lorentz transform. However, regular passive imaging and shutter inside the sensor expands the law of special relativity by a quantum geometric explanation of 4-D photogrammetry. The collinear imaging equations provide common sense explanations to the 10 (and 26) dimensional hyperspace concepts of a purely geometric string theory. The 11-D geometric M-theory is interpreted as a bundle adjustment of spacetime images using 2-D or 5-D membrane observables of image, string and waveform matching in the unified array calculus of applied mathematics.

Gauge and moduli hierarchy in a multiply warped braneworld scenario

Discovery of Higgs-like boson near the mass scale ∼126 Gev generates renewed interest to the gauge hierarchy problem in the standard model related to the stabilisation of the Higgs mass within Tev scale without any unnatural fine tuning. One of the successful attempts to resolve this problem has been the Randall–Sundrum warped geometry model. Subsequently this 5-dimensional model was extended to a doubly warped 6-dimensional (or higher) model which can offer a geometric explanation of the fermion mass hierarchy in the standard model of elementary particles (D. Choudhury and S. SenGupta, 2007 [1]). In an attempt to address the dark energy issue, we in this work extend such 6-dimensional warped braneworld model to include non-flat 3-branes at the orbifold fixed points such that a small but non-vanishing brane cosmological constant is induced in our observable brane. We show that the requirements of a Planck to Tev scale warping along with a vanishingly small but non-zero cosmological constant on the visible brane with non-hierarchical moduli, each with scale close to Planck length, lead to a scenario where the 3-branes can have energy scales either close to Tev or close to Planck scale. Such a scenario can address both the gauge hierarchy as well as fermion mass hierarchy problem in standard model without introducing hierarchical scales between the two moduli. Thus simultaneous resolutions to the gauge hierarchy problem, fermion mass hierarchy problem and non-hierarchical moduli problem are closely linked with the near flatness condition of our universe.

Horseshoes of periodically kicked van der Pol oscillators

This paper contains a numerical study of the periodically forced van der Pol system. Our aim is to determine the extent to which chaotic behavior occurs in this system as well as the nature of the chaos. Unlike previous studies, which used continuous forcing, we work with instantaneous kicks, for which the geometry is simpler. Our study covers a range of parameters describing nonlinearity, kick sizes, and kick periods. We show that horseshoes are abundant whenever the limit cycle is kicked to a specific region of the phase space and offer a geometric explanation for the stretch-and-fold behavior which ensues.

Free-Form Conjugation Modeling and Gear Tooth Profile Design

The paper presents the first theory and practice of free-form conjugation modeling. It introduces the concept of master-slave that broadens the traditional computer aided single geometry modeling to dual geometry modeling. The concept is then applied to gear geometry to establish the first free-form conjugation modeling technique, in which a fictitious free-form rack-cutter or free-form contact path is proposed as the master geometry. Simple geometric relationship suitable for conjugation modeling and undercutting analysis is found between the master and the conjugate profiles. With a free-form master geometry, free-form conjugation modeling has desirable properties of guaranteed continuity, flexibility, and controllability. It offers unlimited representation capability crucial for conjugation modification and optimization. Undercutting is examined rigorously and extensively under the free-form technique via differential geometry. For general planar conjugation, including noncircular gearing, necessary and sufficient undercutting conditions in terms of the master geometry are obtained. Simple geometric explanation of undercutting is demonstrated in terms of the distance between a contact path and two centrode centers. The technique is demonstrated with B-spline as the master geometry.

Constraints on the relative sizes of intervening Mg II-absorbing clouds and quasar emitting regions

Context: A significantly higher incidence of strong (rest equivalent width W_r > 1 {\AA}) intervening Mg II absorption is observed along gamma-ray burst (GRB) sight-lines relative to those of quasar sight-lines. A geometrical explanation for this discrepancy has been suggested: the ratio of the beam size of the source to the characteristic size of a Mg II absorption system can influence the observed Mg II equivalent width, if these two sizes are comparable. Aims: We investigate whether the differing beam sizes of the continuum source and broad-line region of Sloan Digital Sky Survey (SDSS) quasars produce a discrepancy between the incidence of strong Mg II absorbers illuminated by the quasar continuum region and those of absorbers illuminated by both continuum and broad-line region light. Methods: We perform a semi-automated search for strong Mg II absorbers in the SDSS Data Release 7 quasar sample. The resulting strong Mg II absorber catalog is available online. We measure the sight-line number density of strong Mg II absorbers superimposed on and off the quasar C IV 1550 {\AA} and C III] 1909 {\AA} emission lines. Results: We see no difference in the sight-line number density of strong Mg II absorbers superimposed on quasar broad emission lines compared to those superimposed on continuum-dominated spectral regions. This suggests that the Mg II-absorbing clouds typically observed as intervening absorbers in quasar spectra are larger than the beam sizes of both the continuum-emitting regions and broad line-emitting regions in the centers of quasars, corresponding to a lower limit of the order of 10^17} cm for the characteristic size of a Mg II absorbing cloud.

Robust control oriented identification of errors-in-variables models based on normalised coprime factors

A robust control oriented identification approach is proposed to deal with the identification of errors-in-variables models (EIVMs), which are corrupted with input and output noises. Based on normalised coprime factor model (NCFM) representations, a frequency-domain perturbed NCFM for an EIVM is derived according to a geometrical explanation for the v-gap metric. As a result, identification of the EIVM is converted into that of the NCFM. Besides an identified nominal NCFM, its worst case error has to be quantified. Unlike other traditional control-oriented identification methods, the v-gap metric is employed to measure the uncertainties including a priori information on the disturbing noises and the worst case error for the resulting nominal NCFM. Since this metric is also used as an optimisation criterion, the associate parameter estimation problem can be effectively solved by linear matrix inequalities. Finally, a numerical simulation shows the effectiveness of the proposed method.

单目视频运动目标轨迹三维重建的平滑约束法

In this paper, generic smoothness constrained 3D trajectory reconstruction of moving object from monocular video is proposed. By introducing the generic smoothness constraint on the 3D trajectory, uncon- strained optimization model for 3D trajectory reconstruction is achieved and a closed-form solution is derived. Compared with the prede ned basis methods such as Discrete Cosine Transform based method and polynomial basis based method, the proposed method is more generic and can be applied to incomplete measurement case, thus having broad applicability. Geometric explanation and uniqueness of 3D trajectory reconstruction are given. Experimental results on both synthetic data and real monocular video sequences have shown the validity and advancement of the proposed method.

One-port (uniportal) video-assisted thoracic surgical resections—A clear advance

One-port (uniportal) video-assisted thoracic surgery (VATS) consists of approaching an intrathoracic target lesion through a sagittal, craniocaudal plane through 1 single-port incision. The use of articulating instruments inserted parallel to the videothoracoscope enables the surgeon to mimic inside the chest the maneuvers that are usually performed during open surgery. Through this VATS approach, several thoracic conditions can be addressed, including lung cancer in selected patients. Unlike conventional, 3-port VATS, the uniportal VATS technique enables the surgeon to bring the operative fulcrum inside the chest when the target lunge lesion is approached through a sagittal plan, thanks to articulating instruments. Uniportal wedge VATS resections of peripheral nodules can help in solving diagnostic dilemmas, be of therapeutic benefit, and provide tissue for biomolecular studies.

Analysis and acceleration strategy of endmember extraction algorithms based on convex geometry

Under a linear mixture model with non-negativity and sum-to-one constraints, the spectral unmixing problem can be seen as a convex geometry problem. This article first analyses three commonly used endmember extraction criteria including the extreme projection criterion, the maximum simplex volume criterion and the minimum volume enclosing simplex criterion, which are derived from the geometrical explanation of the linear mixture model. And then an acceleration strategy is introduced to shorten the computing time of endmember extraction algorithms. The acceleration strategy exploits two facts: (1) the endmembers corresponding to the vertices of a simplex composed of the mixed pixels can be determined only by the boundary points, with little or no affect by the interior points; (2) the boundary points can be found in a series of two-dimensional subspace. Experiments using simulated data on eight popular endmember extraction algorithms show that the proposed acceleration strategy can reduce the computing time and then improve the speed of endmember extraction, while maintaining the same results or little sacrifice of computing precision.