Simple Geometric Criteria Research Articles

Quantitative comparison of the eye formation in cheese using radiography and computed tomography data

Eye formation is an important quality parameter in the dairy industry for (semi-)hard cheeses in Switzerland. To monitor the formation of eyes in cheese, radiography and, more recently, computed tomography (CT) technology, are employed. In the present study, two quality indicators (eye number and total eye volume) are calculated from radiographs by two recent procedures and compared with the results from CT data. The radiograph analysis procedures underestimate both quantifiers systematically and the deviation increases with the number of eyes. In contrast to radiographic data, the distribution of the eye volumes and positions as well as their geometric shape can be calculated from the CT data. These data allowed the observation that the eyes in the considered cheeses accumulate in the centre of the cheese, are predominately of spherical shape and non-spherical eyes can be extracted by a simple geometrical criterion. These findings illustrate the advantages of the CT technology.

Optimal Definition of Inter-Residual Contact in Globular Proteins Based on Pairwise Interaction Energy Calculations, Its Robustness, and Applications

Although a contact is an essential measurement for the topology as well as strength of non-covalent interactions in biomolecules and their complexes, there is no general agreement in the definition of this feature. Most of the definitions work with simple geometric criteria which do not fully reflect the energy content or ability of the biomolecular building blocks to arrange their environment. We offer a reasonable solution to this problem by distinguishing between "productive" and "non-productive" contacts based on their interaction energy strength and properties. We have proposed a method which converts the protein topology into a contact map that represents interactions with statistically significant high interaction energies. We do not prove that these contacts are exclusively stabilizing, but they represent a gateway to thermodynamically important rather than geometry-based contacts. The process is based on protein fragmentation and calculation of interaction energies using the OPLS force field and relies on pairwise additivity of amino acid interactions. Our approach integrates the treatment of different types of interactions, avoiding the problems resulting from different contributions to the overall stability and the different effect of the environment. The first applications on a set of homologous proteins have shown the usefulness of this classification for a sound estimate of protein stability.

Competitive modes as reliable predictors of chaos versus hyperchaos and as geometric mappings accurately delimiting attractors

We consider real quadratic dynamics in the context of competitive modes, which allows us to view chaotic systems as ensembles of competing nonlinear oscillators. We find that the standard competitive mode conditions may in fact be interpreted and employed slightly more generally than has usually been done in recent investigations, with negative values of the squared mode frequencies in fact being admissible in chaotic regimes, provided that the competition among them persists. This is somewhat reminiscent of, but of course not directly correlated to, “stretching (along unstable manifolds) and folding (due to local volume dissipation)” on chaotic attractors. This new feature allows for the system variables to grow exponentially during time intervals when mode frequencies are imaginary and comparable, while oscillating at instants when the frequencies are real and locked in or entrained. In addition to an application of the method to chaotic attractors, we consider systems exhibiting hyperchaos and conclude that the latter exhibit three competitive modes rather than two for the former. Finally, in a novel twist, we reinterpret the components of the Competitive Modes analysis as simple geometric criteria to map out the spatial location and extent, as well as the rough general shape, of the system attractor for any parameter sets corresponding to chaos. The accuracy of this mapping adds further evidence to the growing body of recent work on the correctness and usefulness of competitive modes. In fact, it may be considered a strong “a posteriori” validation of the Competitive Modes conjectures and analysis.

ON THE ORIGIN OF THE ANGULAR MOMENTUM PROPERTIES OF GAS AND DARK MATTER IN GALACTIC HALOS AND ITS IMPLICATIONS

We perform a set of non-radiative hydrodynamical simulations of merging spherical halos in order to understand the angular momentum (AM) properties of the galactic halos seen in cosmological simulations. The universal shape of AM distributions seen in simulations is found to be generically produced as a result of mergers. The universal shape is such that it has an excess of low AM material and hence cannot explain the exponential structure of disk galaxies. A resolution to this is suggested by the spatial distribution of low AM material which is found to be in the centre and a conical region close to the axis of rotation. A mechanism that preferentially discards the material in the centre and prevents the material along the poles from falling onto the disc is proposed as a solution. We implement a simple geometric criteria for selective removal of low AM material and show that in order for 90% of halos to host exponential discs one has to reject at least 40% of material. Next, we explore the physical mechanisms responsible for distributing the AM within the halo during a merger. For dark matter there is an inside-out transfer of AM, whereas for gas there is an outside-in transfer, which is due to differences between collisionless and gas dynamics. We also explain the apparent high spin of dark matter halos undergoing mergers and show that a criteria stricter than what is currently used, would be required to detect such unrelaxed halos. Finally, we demonstrate that the misalignment of AM between gas and dark matter only occurs when the intrinsic spins of the merging halos are not aligned with the orbital AM of the system. The self-misalignment (orientation of AM when measured in radial shells not being constant), which could be the cause of warps and anomalous rotation in disks galaxies, also occurs under similar conditions.

An optimized video system for augmented reality in endodontics: a feasibility study

We propose an augmented reality system for the reliable detection of root canals in video sequences based on a k-nearest neighbor color classification and introduce a simple geometric criterion for teeth. The new software was implemented using C++, Qt, and the image processing library OpenCV. Teeth are detected in video images to restrict the segmentation of the root canal orifices by using a k-nearest neighbor algorithm. The location of the root canal orifices were determined using Euclidean distance-based image segmentation. A set of 126 human teeth with known and verified locations of the root canal orifices was used for evaluation. The software detects root canals orifices for automatic classification of the teeth in video images and stores location and size of the found structures. Overall 287 of 305 root canals were correctly detected. The overall sensitivity was about 94 %. Classification accuracy for molars ranged from 65.0 to 81.2 % and from 85.7 to 96.7 % for premolars. The realized software shows that observations made in anatomical studies can be exploited to automate real-time detection of root canal orifices and tooth classification with a software system. Automatic storage of location, size, and orientation of the found structures with this software can be used for future anatomical studies. Thus, statistical tables with canal locations will be derived, which can improve anatomical knowledge of the teeth to alleviate root canal detection in the future. For this purpose the software is freely available at: http://www.dental-imaging.zahnmedizin.uni-mainz.de/.

Paperfolding morphisms, planefilling curves, and fractal tiles

An interesting class of automatic sequences emerges from iterated paperfolding. The sequences generate curves in the plane with an almost periodic structure. We generalize the results obtained by Davis and Knuth on the self-avoiding and planefilling properties of these curves, giving simple geometric criteria for a complete classification. Finally, we show how the automatic structure of the sequences leads to self-similarity of the curves, which turns the planefilling curves in a scaling limit into fractal tiles. For some of these tiles we give a particularly simple formula for the Hausdorff dimension of their boundary.

Singularities of Type-Q ABS Equations

The type-Q equations lie on the top level of the hierarchy introduced by Adler, Bobenko and Suris (ABS) in their classification of discrete counterparts of KdV-type integrable partial differential equations. We ask what singularities are possible in the solutions of these equations, and examine the relationship between the singularities and the principal integrability feature of multidimensional consistency. These questions are considered in the global setting and therefore extend previous considerations of singularities which have been local. What emerges are some simple geometric criteria that determine the allowed singularities, and the interesting discovery that generically the presence of singularities leads to a breakdown in the global consistency of such systems despite their local consistency property. This failure to be globally consistent is quantified by introducing a natural notion of monodromy for isolated singularities.

High-dimensional inference with the generalized Hopfield model: Principal component analysis and corrections

We consider the problem of inferring the interactions between a set of N binary variables from the knowledge of their frequencies and pairwise correlations. The inference framework is based on the Hopfield model, a special case of the Ising model where the interaction matrix is defined through a set of patterns in the variable space, and is of rank much smaller than N. We show that maximum likelihood inference is deeply related to principal component analysis when the amplitude of the pattern components ξ is negligible compared to √N. Using techniques from statistical mechanics, we calculate the corrections to the patterns to the first order in ξ/√N. We stress the need to generalize the Hopfield model and include both attractive and repulsive patterns in order to correctly infer networks with sparse and strong interactions. We present a simple geometrical criterion to decide how many attractive and repulsive patterns should be considered as a function of the sampling noise. We moreover discuss how many sampled configurations are required for a good inference, as a function of the system size N and of the amplitude ξ. The inference approach is illustrated on synthetic and biological data.

Quantum Experimental Data in Psychology and Economics

We prove a theorem which shows that a collection of experimental data of probabilistic weights related to decisions with respect to situations and their disjunction cannot be modeled within a classical probabilistic weight structure in case the experimental data contain the effect referred to as the ‘disjunction effect’ in psychology. We identify different experimental situations in psychology, more specifically in concept theory and in decision theory, and in economics (namely situations where Savage’s Sure-Thing Principle is violated) where the disjunction effect appears and we point out the common nature of the effect. We analyze how our theorem constitutes a no-go theorem for classical probabilistic weight structures for common experimental data when the disjunction effect is affecting the values of these data. We put forward a simple geometric criterion that reveals the non classicality of the considered probabilistic weights and we illustrate our geometrical criterion by means of experimentally measured membership weights of items with respect to pairs of concepts and their disjunctions. The violation of the classical probabilistic weight structure is very analogous to the violation of the well-known Bell inequalities studied in quantum mechanics. The no-go theorem we prove in the present article with respect to the collection of experimental data we consider has a status analogous to the well known no-go theorems for hidden variable theories in quantum mechanics with respect to experimental data obtained in quantum laboratories. Our analysis puts forward a strong argument in favor of the validity of using the quantum formalism for modeling the considered psychological experimental data as considered in this paper.

Comparison of biomechanical failure criteria for abdominal aortic aneurysm

Medical doctors consider a surgery option for the expanding abdominal aortic aneurysm (AAA) when its maximum diameter reaches 5.5 cm. This simple geometrical criterion may possibly underestimate the risks of rupture of small aneurysms as well as overestimate the risks of rupture of large aneurysms. Biomechanical criteria of the AAA failure are desired. Various local criteria of the AAA failure are used in the literature though their experimental validation is needed. In the present work, we use the experimentally calibrated AAA model, which includes a failure description, to examine various popular criteria of the local failure. Particularly, we analyze various states of the biaxial tension of the AAA material and evaluate the following criteria of the local failure: (1) the maximum principal stretch; (2) the maximum principal stress; (3) the maximum shear stress; (4) von Mises stress; and (5) the strain energy. The results show that the strain energy is almost constant for the failure states induced by the loads varying from the uniaxial to the equal biaxial tension. The von Mises stress exhibits a wider range of scattering as compared to the strain energy. The maximum stresses and stretches vary significantly with the variation of loads from the uniaxial to the equal biaxial tension.

Rapid chain tracing of polypeptide backbones in electron-density maps

A method for the rapid tracing of polypeptide backbones has been developed. The method creates an approximate chain tracing that is useful for visual evaluation of whether a structure has been solved and for use in scoring the quality of electron-density maps. The essence of the method is to (i) sample candidate C(alpha) positions at spacings of approximately 0.6 A along ridgelines of high electron density, (ii) list all possible nonapeptides that satisfy simple geometric and density criteria using these candidate C(alpha) positions, (iii) score the nonapeptides and choose the highest scoring ones, and (iv) find the longest chains that can be made by connecting nonamers. An indexing and storage scheme that allows a single calculation of most distances and density values is used to speed up the process. The method was applied to 42 density-modified electron-density maps at resolutions from 1.5 to 3.8 A. A total of 21 428 residues in these maps were traced in 24 CPU min with an overall r.m.s.d. of 1.61 A for C(alpha) atoms compared with the known refined structures. The method appears to be suitable for rapid evaluation of electron-density map quality.

Ternary fluorides BaMF4 (M = Zn, Mg and Mn) at low temperatures

Ternary fluorides BaMF4 (M = Zn, Mg, Mn) have been studied in the temperature range from 300 to 10 K using synchrotron and laboratory powder and single-crystal diffraction. The first two compounds are stable down to 10 K, while the third one undergoes a phase transition to an incommensurately modulated structure at approximately 250 K. The modulated phase is stable down to 10 K. The magnetic anomalies at 45 and 27 K observed previously in BaMnF4 are exclusively reflected in the behavior of the gamma component of the q vector, which assumes an irrational value of approximately 0.395 A(-1) at the temperature corresponding to the onset of the magnetic ordering and then stays constant down to 10 K. Mn-Mn distances do not indicate any structural response to the magnetic ordering. The formation of the modulated phase can be explained on the basis of simple geometrical criteria. The incorporation of the large Mn cation in the octahedral sheets implies an increase of the cavity in which the Ba ion is incorporated. This leads to the formation of the modulated structure to adapt the coordination sphere around Ba in such a way that the bond-valence sums can be kept close to the ideal value of two. With further lowering of the temperature, the charge balance around the Ba ion requires an increasingly anharmonic character of the modulation function of Ba, until finally at 10 K a crenel-like shape is assumed for the modulation of this atom.

Exterior Monge–Ampère solutions

We discuss the Siciak–Zaharjuta extremal function of a real convex body in C n , a solution of the homogeneous complex Monge–Ampère equation on the exterior of the convex body. We determine several conditions under which a foliation by holomorphic curves can be found in the complement of the convex body along which the extremal function is harmonic. We study a variational problem for holomorphic disks in projective space passing through prescribed points at infinity. The extremal curves are all complex quadratic curves, and the geometry of such curves allows for the determination of the leaves of the foliation by simple geometric criteria. As a byproduct we encounter a new invariant of an exterior domain, the Robin indicatrix, which is in some cases the dual of the Kobayashi indicatrix for a bounded domain. Finally, we construct extremal curves for two non-convex bodies in R 2 .

Simulation of two‐phase flow with sub‐scale droplet and bubble effects

AbstractWe present a new Eulerian‐Lagrangian method for physics‐based simulation of fluid flow, which includes automatic generation of sub‐scale spray and bubbles. The Marker Level Set method is used to provide a simple geometric criterion for free marker generation. A filtering method, inspired from Weber number thresholding, further controls the free marker generation (in a physics‐based manner). Two separate models are used, one for sub‐scale droplets, the other for sub‐scale bubbles. Droplets are evolved in a Newtonian manner, using a density‐extension drag force field, while bubbles are evolved using a model based on Stokes' Law. We show that our model for sub‐scale droplet and bubble dynamics is simple to couple with a full (macro‐scale) Navier‐Stokes two‐phase flow model and is quite powerful in its applications. Our animations include coarse grained multiphase features interacting with fine scale multiphase features.

Nucleosome Shape Dictates Chromatin Fiber Structure

In addition to being the gateway for all access to the eukaryotic genome, chromatin has in recent years been identified as carrying an epigenetic code regulating transcriptional activity. Though much is known about the biochemistry of this code, little is understood regarding the different fiber structures through which the regulation is mediated. Over the last three decades many fiber models have been suggested, but none are able to predict even the basic characteristics of the fiber. In this work, we characterize the set of all possible dense fibers, which includes, but is not limited to, all previously suggested structures. To guide future experimental efforts, we show which fiber characteristics depend on the underlying structure and, crucially, which do not. Addressing the predictive power of these models, we suggest a simple geometric criterion based on the nucleosome shape alone. This enables us to predict the observed characteristics of the condensed chromatin fiber, and how these change with varying nucleosome repeat length. Our approach sheds light on how the in vivo observed heterogeneity in linker lengths can be accommodated within the 30 nm fiber, and suggest an important role for nucleosome surface interactions in the regulation of chromatin structure and function.

Robust recognition of zinc binding sites in proteins

Metals play a variety of roles in biological processes, and hence their presence in a protein structure can yield vital functional information. Because the residues that coordinate a metal often undergo conformational changes upon binding, detection of binding sites based on simple geometric criteria in proteins without bound metal is difficult. However, aspects of the physicochemical environment around a metal binding site are often conserved even when this structural rearrangement occurs. We have developed a Bayesian classifier using known zinc binding sites as positive training examples and nonmetal binding regions that nonetheless contain residues frequently observed in zinc sites as negative training examples. In order to allow variation in the exact positions of atoms, we average a variety of biochemical and biophysical properties in six concentric spherical shells around the site of interest. At a specificity of 99.8%, this method achieves 75.5% sensitivity in unbound proteins at a positive predictive value of 73.6%. We also test its accuracy on predicted protein structures obtained by homology modeling using templates with 30%-50% sequence identity to the target sequences. At a specificity of 99.8%, we correctly identify at least one zinc binding site in 65.5% of modeled proteins. Thus, in many cases, our model is accurate enough to identify metal binding sites in proteins of unknown structure for which no high sequence identity homologs of known structure exist. Both the source code and a Web interface are available to the public at http://feature.stanford.edu/metals.

Systematic pathway generation and sorting in martensitic transformations: Titaniumαtoω

Martensitic phase transitions appear in a diverse range of natural and engineering material systems. Examination of the energetics and kinetics of the transformation requires an understanding of the atomic mechanism for the transformation. A systematic pathway generation and sorting algorithm is presented and applied to the problem of the titanium $\ensuremath{\alpha}$ to $\ensuremath{\omega}$ transformation under pressure. The transformation pathways are separated into strain and shuffle components. All pathways are constructed within energetically motivated strain and shuffle constraints, and efficiently sorted by their energy barriers. The geometry and symmetry details of the seven lowest energy barrier pathways are given. The lack of a single simple geometric criterion for determining the lowest energy pathway shows the necessity of atomistic studies for pathway determination. The general algorithm can determine the pathway for any martensitic transformation.

Winding number criterion for existence and uniqueness of equilibrium in linear rational expectations models

This paper develops a simple geometric criterion for the existence and uniqueness of equilibrium in linear rational expectations (LRE) models. The class of models considered is very general and includes models with infinitely many lags and/or leads of endogenous variables. I show that the criterion is tightly related to the familiar Blanchard and Kahn (Econometrica 48(5) (1980) 1305), Hansen and Sargent (Rational Expectations and Econometric Practice, 1981), and Whiteman (Linear Rational Expectations Models: A User's Guide, 1983) conditions for the existence and uniqueness of equilibrium in LRE models. The new criterion can be used to extend the policy rule robustness analysis of Onatski and Stock (Macroeconomic Dynamics 6(1) (2002) 85) from purely backward-looking models to forward-looking models.

Intramolecular γ‐Hydroxylations of Nonactivated CH Bonds with Copper Complexes and Molecular Oxygen

Copper(I) complexes incorporating the isomeric bidentate ligands IMPY (iminomethyl-2-pyridines) or AMPY (aminomethylene-2-pyridines) are quite unusual in their ability to bind and activate molecular oxygen. Using these complexes, hydroxylations of nonactivated CH, CH2, or CH3 groups in the gamma-position in relation to the imino-nitrogen atom, and with a specific orientation of one H atom with respect to the binuclear Cu-O species, can be achieved in synthetically useful yields. Through mechanistic studies employing conformationally well-defined molecules (for example, cyclic isoprenoids), coupled with solid-state X-ray structure analyses and force-field calculations, we postulate a seven-membered transition state for this reaction in which six atoms lie approximately in a plane. This plane is defined by the positions of the lone pairs on the nitrogen atoms, as well as the copper and the oxygen atoms. For a successful hydroxylation, one hydrogen atom should be located close to this plane. Prediction of the stereochemical course of these reactions is possible based on a simple geometrical criterion. The convenient introduction of IMPY and AMPY groups as auxiliaries into oxo and primary amino compounds and the simple hydrolysis after the hydroxylation procedure has allowed the synthesis of 3-hydroxy-1-oxo and 3-hydroxy-1-amino compounds. If desired, the 3-hydroxy-1-IMPY and -1-AMPY compounds can be reduced with NaBH4 to obtain 3-hydroxy-1-aminomethylpyridines. For a successful hydroxylation procedure, the method employed for the synthesis of the CuI complexes is very important. Starting either from CuI salts or from CuII salts with a subsequent reduction with benzoin/triethylamine may turn out to be the better way, depending on the ligand and the molecular structure.

How tightly can you fold a sphere?

Consider a compact, connected Lie group G acting isometrically on a sphere S n of radius 1. The quotient of S n by this group action, S n / G , has a natural metric on it, and so we may ask what are its diameter and q- extents. These values have been computed for cohomogeneity one actions on spheres. In this paper, we compute the diameters, extents, and several q-extents of cohomogeneity two orbit spaces resulting from such actions, and we also obtain results about the q-extents of Euclidean disks. Additionally, via a simple geometric criterion, we can identify which of these actions give rise to a decomposition of the sphere as a union of disk bundles. In addition, as a service to the reader, we give a complete breakdown of all the isotropy subgroups resulting from cohomogeneity one and two actions.