One-dimensional Sphere Research Articles

Shaping in the Third Direction; Synthesis of Patterned Colloidal Crystals by Polyester Fabric-Guided Self-Assembly.

A polyester fabric with rectangular openings was used as a sacrificial template for the guiding of a sub-micron sphere (polystyrene (PS) and silica) aqueous colloid self-assembly process during evaporation as a patterned colloidal crystal (PCC). This simple process is also a robust one, being less sensitive to external parameters (ambient pressure, temperature, humidity, vibrations). The most interesting feature of the concave-shape-pattern unit cell (350 μm × 400 μm × 3 μm) of this crystal is the presence of triangular prisms at its border, each prism having a one-dimensional sphere array at its top edge. The high-quality ordered single layer found inside of each unit cell presents the super-prism effect and left-handed behavior. Wider yet elongated deposits with ordered walls and disordered top surfaces were formed under the fabric knots. Rectangular patterning was obtained even for 20 μm PS spheres. Polyester fabrics with other opening geometries and sizes (~300–1000 μm) or with higher fiber elasticity also allowed the formation of similar PCCs, some having curved prismatic walls. A higher colloid concentration (10–20%) induces the formation of thicker walls with fiber-negative replica morphology. Additionally, thick-wall PCCs (~100 μm) with semi-cylindrical morphology were obtained using SiO2 sub-microspheres and a wavy fabric. The colloidal pattern was used as a lithographic mask for natural lithography and as a template for the synthesis of triangular-prism-shaped inverted opals.

Numerical analysis of the influence of RBCs on oxygen transport within a tissue with an embedded capillary network

In this paper, the oxygen transport in a capillary network and the surrounding tissue was simulated by considering the influence of red blood cells (RBCs). To simplify this problem, the erythrocytes as the oxygen carriers were modelled as one-dimensional (1 D) spheres, and oxygen and haemoglobin diffusion took place only in the radial direction. The direction of movement of the RBCs at the network bifurcations was determined by a probability function related to the ratio of the flow rate in the mother vessel to that in a daughter vessel. The capillaries were reduced to a skeleton structure that included vessels with different radii, lengths, and three-dimensional (3 D) coordinates. The capillary network was embedded in a 3 D tissue, which was considered as a homogeneous porous medium. The interactions between the capillary network and the surrounding tissue were solved by the immersed boundary method. The influence of RBC rheological behaviours, such as deformation capacity and flux, on oxygen transport were analysed. The oxygen dissociation capacity of the RBCs was also investigated. The results showed that when the RBCs’ deformation capacity was weakened, the probability of RBCs choosing large-flow daughter vessel at the bifurcation increased considerably, leading to a highly non-uniform distribution of RBCs in the network. The non-uniformity of the RBCs further led to partial hypoxia in regions through which no RBCs passed. In contrast, a lower haematocrit and weakening of the oxygen release capability led to the overall decline of oxygen partial pressure in the whole region.

Polinômio de Alexander via Linguagem Python

Um nó clássico é um mergulho de uma esfera unidimensional S1 em um ambiente tridimensional real, geralmente R3. Nestas condições é possível considerar o diagrama do nó, isto é, a projeção planar do mergulho. Esta assemelha-se a uma curva na qual os cruzamentos são trocados por interrupções no traço da mesma, indicando desta maneira que um arco passa por sobre o outro. Na Teoria dos Nós estudam-se invariantes algébricos extraídos do complementar do mergulho, e este complementar é visível no caso do mergulho S1 --> R3. Um dos invariantes extraídos do diagrama do nó é o polinômio de Alexander. Neste artigo mostramos como o processo de determinar o polinômio de Alexander de um nó pode ser transportado para um algoritmo implementado no Python e obtido a partir de um diagrama desenhado com o auxílio do mouse.

Algorithms for Constructing Isolating Sets of Phase Flows and Computer-Assisted Proofs with the Use of Interval Taylor Models

This work continues the research devoted to considering interval Taylor models (TM) as applied to proving the existence of periodic trajectories in systems of ordinary differential equations (ODEs). The Taylor models are used here within a topological approach to constructing an isolating set for the ODE phase flow. We prove auxiliary assertions and propose constructive algorithms for validated numerics over TMs with the aim to expand the domain of their applicability. Necessary topological assertions are proved in order to establish the properties of isolating sets constructed with the aid of TMs. As a result, constructive algorithms are formulated and the main theorem is proved. This theorem makes it possible to construct and verify the homotopy equivalence of the isolating set and the one-dimensional sphere and the homotopy of the mapping of the isolating set into itself to the identity mapping for given systems of ODEs. We also prove the computational complexity of the main algorithm and provide an example of its usage.

Research on the dissipative characteristics of unsteady heat conduction for the one-dimensional sphere

In this paper, the dissipative characteristics of unsteady heat conduction process for the one-dimensional sphere is studied. The dissipation function can be regarded as a Lyapunov function for the heat conduction system, which determines the evolution direction of the system and the stability of the steady state. By use of the vector formula, the relationship between the thermal potential and dissipation function is derived, and its similarity with the dissipation system of mechanical energy is shown. The expression of dissipation function is obtained when the boundary temperature is fixed. In addition, an example for optimization of heat conduction process is discussed based on the entransy dissipation extremum principle.

Non-robust Phase Transitions in the Generalized Clock Model on Trees

Pemantle and Steif provided a sharp threshold for the existence of a RPT (robust phase transition) for the continuous rotator model and the Potts model in terms of the branching number and the second eigenvalue of the transfer operator, where a robust phase transition is said to occur if an arbitrarily weak coupling with symmetry-breaking boundary conditions suffices to induce symmetry breaking in the bulk. They further showed that for the Potts model RPT occurs at a different threshold than PT (phase transition in the sense of multiple Gibbs measures), and conjectured that RPT and PT should occur at the same threshold in the continuous rotator model. We consider the class of 4- and 5-state rotation-invariant spin models with reflection symmetry on general trees which contains the Potts model and the clock model with scalarproduct-interaction as limiting cases. The clock model can be viewed as a particular discretization which is obtained from the classical rotator model on the continuous one-dimensional sphere. We analyze the transition between PT=RPT and PT is unequal to RPT, in terms of the eigenvalues of the transfer matrix of the model at the critical threshold value for the existence of RPT. The transition between the two regimes depends sensitively on the third largest eigenvalue.

Lipschitz Retractions in Hadamard Spaces via Gradient Flow Semigroups

AbstractLet X(n), for n ∊ ℕ, be the set of all subsets of a metric space (X, d) of cardinality at most n. The set X(n) equipped with the Hausdorff metric is called a finite subset space. In this paper we are concerned with the existence of Lipschitz retractions r: X(n)→ X(n − 1) for n ≥ 2. It is known that such retractions do not exist if X is the one-dimensional sphere. On the other hand, Kovalev has recently established their existence if X is a Hilbert space, and he also posed a question as to whether or not such Lipschitz retractions exist when X is a Hadamard space. In this paper we answer the question in the positive.

A class of nonergodic interacting particle systems with unique invariant measure

We consider a class of discrete $q$-state spin models defined in terms of a translation-invariant quasilocal specification with discrete clock-rotation invariance which have extremal Gibbs measures $\mu'_{\varphi}$ labeled by the uncountably many values of $\varphi$ in the one-dimensional sphere (introduced by van Enter, Opoku, K\"{u}lske [J. Phys. A 44 (2011) 475002, 11]). In the present paper we construct an associated Markov jump process with quasilocal rates whose semigroup $(S_t)_{t\geq0}$ acts by a continuous rotation $S_t(\mu'_{\varphi})=\mu'_{\varphi +t}$. As a consequence our construction provides examples of interacting particle systems with unique translation-invariant invariant measure, which is not long-time limit of all starting measures, answering an old question (compare Liggett [Interacting Particle Systems (1985) Springer], question four, Chapter one). The construction of this particle system is inspired by recent conjectures of Maes and Shlosman about the intermediate temperature regime of the nearest-neighbor clock model. We define our generator of the interacting particle system as a (noncommuting) sum of the rotation part and a Glauber part. Technically the paper rests on the control of the spread of weak nonlocalities and relative entropy-methods, both in equilibrium and dynamically, based on Dobrushin-uniqueness bounds for conditional measures.

Second Order Differences of Cyclic Data and Applications in Variational Denoising

In many image and signal processing applications, as interferometric synthetic aperture radar (SAR), electroencephalogram (EEG) data analysis or color image restoration in HSV or LCh spaces the data has its range on the one-dimensional sphere $\mathbb S^1$. Although the minimization of total variation (TV) regularized functionals is among the most popular methods for edge-preserving image restoration such methods were only very recently applied to cyclic structures. However, as for Euclidean data, TV regularized variational methods suffer from the so called staircasing effect. This effect can be avoided by involving higher order derivatives into the functional. This is the first paper which uses higher order differences of cyclic data in regularization terms of energy functionals for image restoration. We introduce absolute higher order differences for $\mathbb S^1$-valued data in a sound way which is independent of the chosen representation system on the circle. Our absolute cyclic first order difference is just the geodesic distance between points. Similar to the geodesic distances the absolute cyclic second order differences have only values in [0,{\pi}]. We update the cyclic variational TV approach by our new cyclic second order differences. To minimize the corresponding functional we apply a cyclic proximal point method which was recently successfully proposed for Hadamard manifolds. Choosing appropriate cycles this algorithm can be implemented in an efficient way. The main steps require the evaluation of proximal mappings of our cyclic differences for which we provide analytical expressions. Under certain conditions we prove the convergence of our algorithm. Various numerical examples with artificial as well as real-world data demonstrate the advantageous performance of our algorithm.

Spherical shields perturbed to ellipsoids in transport theory

One-dimensional spheres are perturbed to ellipsoids, and perturbation theory for inhomogeneous transport problems is applied to estimate the leakage of an uncollided decay gamma ray, a neutron thermal capture gamma ray, and a neutron inelastic scatter gamma ray. Only the shielding is perturbed, not the source. The surface transformation function for the sphere-to-ellipsoid change-of-shape perturbation is derived. Schwinger, Roussopolos, and combined perturbation estimates are applied. The perturbation estimates are defined to estimate the total (4π) flux at an external spherical surface detector, and they were accurate for point-detector fluxes when the leakage estimated from a point detector was similar to the total external surface flux. For uncollided line fluxes, the Schwinger estimate worked very well when the response of interest was the total external surface flux, but perturbation theory did not work well when the response of interest was the flux measured at a single external point (unless extra care was taken to account for geometric effects). For thermal capture line fluxes, the Roussopolos estimate was extremely accurate for one point detector location but its accuracy depended on the detector location. For inelastic scatter line fluxes, the detector fluxes were relatively insensitive to the detector location and the perturbation estimates were fairly accurate.

Surface and Volume Integrals of Uncollided Adjoint Fluxes and Forward-Adjoint Flux Products

Ray analysis techniques are standard for computing the uncollided component of a detector response to radiation. In this paper, uncollided adjoint flux integrals and forward-adjoint inner product integrals in volumes and on surfaces are derived for general geometries. In numerical test problems using a one-dimensional sphere and a two-dimensional (r-z) cylinder, deterministic and stochastic evaluations of the integrals yielded the same results. A semianalytic benchmark for the adjoint flux integral on a cylindrical surface is also used.

Quantum Fractals on n–Spheres. Clifford Algebra Approach

Using the Clifford algebra formalism we extend the quantum jumps algorithm of the Event Enhanced Quantum Theory (EEQT) to convex state figures other than those stemming from convex hulls of complex projective spaces that form the basis for the standard quantum theory. We study quantum jumps on n-dimensional spheres, jumps that are induced by symmetric configurations of non-commuting state monitoring detectors. The detectors cause quantum jumps via geometrically induced conformal maps (Mobius transformations) and realize iterated function systems (IFS) with fractal attractors located on n-dimensional spheres. We also extend the formalism to mixed states, represented by “density matrices” in the standard formalism, (the n-balls), but such an extension does not lead to new results, as there is a natural mechanism of purification of states. As a numerical illustration we study quantum fractals on the circle (one-dimensional sphere and pentagon), two–sphere (octahedron), and on three-dimensional sphere (hypercubetesseract, 24 cell, 600 cell, and 120 cell). The attractor, and the invariant measure on the attractor, are approximated by the powers of the Markov operator. In the appendices we calculate the Radon-Nikodym derivative of the SO(n + 1) invariant measure on Sn under SO(1, n + 1) transformations and discuss the Hamilton’s “icossian calculus” as well as its application to quaternionic realization of the binary icosahedral group that is at the basis of the 600 cell and its dual, the 120 cell.

Modeling of Diffusion in Ellipsoidal Solids: A Comparative Study

Most food materials are usually measured in three mutually perpendicular dimensions (length, width, and thickness) and considered approximately as general ellipsoids. In this article, we briefly reviewed the available studies on how to deal with the diffusion problems in ellipsoidal solids. A three-dimensional (3-D) model was proposed to accurately represent the kernel geometry and rigorously interpret the diffusion phenomena. By the finite difference method, the numerical solution of the diffusion equation for kernels of different shape factors was obtained and compared with those predicted by the one-dimensional (sphere) model and two-dimensional (axi-symmetrical body) model. Meanwhile, the effectiveness of the latter two models was evaluated.

Computation of Radiative Properties in One-Dimensional Sphere Packings for Sintering Applications

Rapid sintering is an important manufacturing process. Because of the rapid rise of the powder system to the sintering temperature, thermal gradients are typically induced in the system. This gradient can be important in determining the microstructure of the final product. To obtain the temperature distribution, the radiative properties such as the phase function, as well as the absorption and scattering coefficients, are required. Typically, the conjugate gradient method is utilized to acquire these properties. The method employs a minimization technique on a single-objective function composed of several sets of other parameters. Thus although this method has been shown to be useful in many situations, it may not be suitable for certain configurations. In the current study, another method is developed to compute the radiative properties. The phase function was removed from the objective function and computed separately. In addition, a dual-objective function technique was developed. The results from benchmarking showed the present technique not only to produce more accurate results than the conjugate gradient method, but also to require significantly less computational time.

Effective Action in Multidimensional (Super)Gravities and Spontaneous Compactification (Quantum Aspects of Kaluza-Klein Theories)

The review of modern status of problem of quantum effects in Kaluza-Klein theories is given. The effective action (EA) in multidimensional (super)gravities (SG's) on the compactified background is investigated. The standard gauge dependent EA in d=5 Einstein gravity and d=5 R2-gravity on the background R4 × S1, where R4 is 4-dimensional space, S1 is one-dimensional sphere is calculated. Gauge and parametrization independent Vilcovisky-De Witt EA in d=5 Einstein gravity and d=5 R2-gravity on the background R4×S1 at zero and non-zero temperature is obtained. We have found that there are no physically acceptable self-consistent solutions of the form R4×S1 at the one-loop level in d=5 Einstein gravity. We calculated also EA for arbitrary multidimensional SG on the background R4×Td-n where Td is d-dimensional torus as expansion on the curvature and its derivatives. The mechanizm of induced of four-dimensional gravity with zero Λ-term is proposed. The Vilcovisky-De Witt EA in d=5 SG's on the background R4×S1 at non-zero temperature is obtained. The three gauge parameter dependent off-shell EA in N=2, d=5 gauged SG on R40×S1 where R40 is flat four-dimensional space is calculated. The expression for vacuum energy for bosonic string with torus compactification is presented. Vacuum energy for superstrings with supersymmetry broken as the result of choice of boundary conditions on background R4×T6 is calculated.

Unique effective action in Kaluza-Klein quantum theories and spontaneous compactification

The one-loop unique effective action by Vilkovisky in the d = 5 Einstein gravity and R 2-gravity on the space M 4×S 1 is calculated, where M 4 is the Minkowski space and S 1 is a one-dimensional sphere. Also, the one-loop unique effective action for d-dimensional Einstein gravity on M 4×T d−4 is calculated. The results of the calculation of the standard effective action are compared with those in the same theories.

On the car parking problem

The radial distribution functions of randomly distributed one-dimensional spheres are studied. The differences between equilibrium fluids and sequentially constructed systems are discussed in terms of the corresponding configuration spaces and numerical simulations. It is shown that for the sequentially constructed systems, which correspond to the car parking problem, a 'memory effect' occurs, such that the sequential spheres are temporally distinguishable. Exact solutions for some simple cases and numerical simulations for large systems are shown, discussed and compared with the equilibrium fluid results.

Solution of nonstationary heat and mass conductivity problems by using the imaginary-frequency characteristics

A method for the approximate solution of the internal problem of nonstationary heat and mass conductivity is elucidated which is based on a rational fraction mode of approximating transcendental functions. Computational formulas are presented for one-dimensional plates, cylinders, and spheres.

On almost rational co-𝐻-spaces

Let X X be a 0 0 -connected co- H H -space whose homotopy groups π n ( X ) {\pi _n}(X) are Q Q vector spaces if n > 1 n > 1 and whose fundamental group π 1 ( X ) {\pi _1}(X) is arbitrary. We prove that X X is homotopy equivalent to a wedge of rational spheres of dimension at least two and of ordinary one-dimensional spheres.

An infinite-dimensional Schoenflies theorem

Although important high-dimensional cases of the polyhedral Schoenflies theorem are yet unsettled (see [4] and [11]), there has been considerable success in the past ten years in generalizing the theorem, proved by Schoenflies in 1906, to the effect that every one-dimensional sphere embedded in the plane is tame. Of course Alexander's horned sphere and others show the necessity of a further condition to obtain a generalization to embeddings of S in E +1 for n> 1. Alexander proved tameness in the combinatorial and differential categories for n=2, M. H. A. Newman extended the result to include all finite dimensions in the combinatorial category [10] and Barry Mazur accomplished the same for the differential category [7]. In the topological category, Marston Morse showed that Mazur's result implies tameness of semilocally tame (bicollared) n-spheres in En+, [9] and at the same time Morton Brown gave an elegant proof independent of Mazur's result [3]. As might be expected, in infinite-dimensional normed linear spaces even the counterpart of the Jordan curve theorem does not hold. However, it is shown in this paper that with the bicollared condition one does get Jordan separation. In fact tameness is proved by a modification of Brown's technique. Hopefully this infinite-dimensional Schoenflies theorem will be as useful in infinite-dimensional topology as has been the case in finite dimensions. To give substance to this hope a number of applications are given including an annulus theorem, a contribution to the classification of convex sets and partial answers to questions raised by R. H. Bing and R. D. Anderson [2]. 1.0 DEFINITIONS. Unless otherwise stated, X denotes a topological space and E (with or without subscripts or superscripts) an infinite-dimensional normed linear space with the norm topology. Let Bt={xeFII xI ? t} and St={xeFII xI=t} for t > 0. If A c X, Bd A denotes the point-set boundary of A and 'A its interior. If h: (B2, S2) -> (A, Bd A) is a homeomorphism, then A is called a (closed) E-cell in X, h(B1) is a collared E-cell in X, collared by h and A is a tame E-cell if X= E and h extends to a space homeomorphism. On the other hand, if h: (B3 -B1, S3 u S1) -(A, Bd A) is a homeomorphism, A is called a (closed) E-annulus in X and h(S2) a bicollared E-sphere in X, bicollared by h. The prefix Ewill be omitted if no confusion is likely.