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International audience; We consider the problem of controlling the diffusion coefficient of a diffusion with constant negative drift rate such that the probability of hitting a given lower barrier up to some finite time horizon is minimized. We assume that the diffusion rate can be chosen in a progressively measurable way with values in the interval [0, 1]. We prove that the value function is regular, concave in the space variable, and that it solves the associated HJB equation. To do so, we show that the heat equation on a right triangle, with a boundary condition that is discontinuous in the corner, possesses a smooth solution.

This paper examines the maximum cogging torque (MCT) distribution characteristic of surface-mounted permanent-magnet (SMPM) machines. The MCT distributions of SMPM machines with different pole-arc coefficients and different ratios of slot opening to slot pitch are studied using: 1) the Fourier series analysis together with the Maxwell stress tensor and 2) the finite-element method. Right triangle distribution regions of the optimal MCT are determined by comparing the MCT distribution contour maps obtained by the afore-mentioned two methods. Finally, a combined MCT optimization method for an SMPM with substantial savings in the optimization time, when compared with conventional methods, is proposed.

The article initially reviews various works describing the physical model (PM) of Michelson’s interferometric experiment (ME), represented by the race between two swimmers Sw1, Sw2 (or boats, or planes, or sound signals, etc.). The two swimmers must each swim the same distance, but Sw1 will swim along the river flow, and Sw2 will swim perpendicularly to this direction. In all such works, it is considered that Sw2’s path will require less time and that it will reach the start point first. However, in this work, it has been determined that in order to make this possible, Sw2 must not observe the orthogonality rule of his start direction. This action would be deceitful to the arbiters and thus considered as non-fair-play towards Sw1. The article proves by swimming times calculus, that if the fair-play rules are observed, then the correct crosswise path (in water reference frame) is a right triangle instead of the isosceles triangle considered by Michelson. Consequently, the two times shall be perfectly equal and the race ends in a tie, and the myth of Sw2 as the race winner shall be debunked. Note that the same result shall also be applicable to Michelson’s interferometric experiment (ME) as well as to any similar experiment. Therefore, utilising the isosceles triangle as the transversal path in PM and also in ME is an erroneous act.

Using the bond-propagation algorithm, we study the critical Ising model on the square lattice with the shape of right triangle and free boundaries. The length of short leg is N and that of long leg is M . The aspect ratio between the two legs ρ = M ∕ N is fixed. For six ratios: ρ = 1 , 2 , 3 , 4 , 8 , 16 , the critical free energy density, internal energy density and specific heat on the lattices with 30 ≤ N ≤ 600 are calculated. Based on these accurate data, we determine exact expansions of the critical free energy, internal energy, and specific heat. With these expansions, we extract the bulk, surface, and corner parts of the free energy, internal energy, and specific heat. The corner term in free energy density proportional to ln N is consistent to the conformal field theory in the accuracy of 1 0 − 10 at the least. It is found that the edge terms in the internal energy proportional to ln N related to the hypotenuse are the same for ρ = 1 , 2 , 3 , 4 , 8 , 16 , i.e. this term is geometry independent. However the edge terms in the specific heat proportional to ln N are geometry dependent, i.e. they are different for different ρ . The corner terms in the internal energy and specific heat are also obtained.

This chapter continues with the basic concepts involved with analytic geometry. The topics covered include reducing dimensionality, dimensional projection, trigonometric functions of a right triangle, and dimensional rotation. Mathematical examples and illustrations are given for each topic.

A set X in a real Hilbert space H is called an a t-set if every three-element subset of X forms either an acute-angled triangle or a right-angled triangle. The maximal cardinality of an a t-set in an infinite-dimensional H is found. Furthermore, the number of right angles in the unit cube [ 0 , 1 ] n ${[0,1]^n}$ is calculated. As an application, a simple solution of a well-known problem is given, concerning the maximal cardinality of a strong a t-set in the Euclidean space ℝ n .

SummaryBy fixing either one leg or the hypotenuse of a right triangle and letting the other sides vary, we obtain three new proofs of the Pythagorean theorem.

A reference triangular quadratic Lagrange finite element consists of a right triangle \(\hat K\) with unit legs S1, S2, a local space \(\hat L\) of quadratic polynomials on \(\hat K\) and of parameters relating the values in the vertices and midpoints of sides of \(\hat K\) to every function from \(\hat L\). Any isoparametric triangular quadratic Lagrange finite element is determined by an invertible isoparametric mapping \({F_h} = ({F_1},{F_2}) \in \hat L \times \hat L\). We explicitly describe such invertible isoparametric mappings Fh for which the images Fh(S1), Fh(S2) of the segments S1, S2 are segments, too. In this way we extend the well-known result going back to W.B. Jordan, 1970, characterizing those invertible isoparametric mappings whose restrictions to the segments S1 and S2 are linear.

Two exact solutions of the problem on harmonic vibrations of a thermoelastic rod with a cross section representing a right triangle have been obtained with the use of multiaction logic operations. The influence of the connectedness of the problem as well as the temperature and elastic properties of the indicated rod on the wave process of its deformation has been investigated. Expressions for the velocities of the temperature, longitudinal, and shear waves were obtained. A criterion M0 for the expediency of taking into account the connectedness in the formulation of the problem was determined.

The Gromov-hyperbolic δ or “fatness” of a hyperbolic geodesic triangle, defined to be the infimum of the perimeters of all inscribed triangles, is given an explicit analytical expression in term of the angle data of the triangle. By a hyperbolic extension of Fermat’s principle, the optimum inscribed triangle is easily constructed as the orthic triangle, that is, the triangle with its vertices at the feet of the altitudes of the original triangle. From the analytical expression of the optimum perimeter δ , a Tarski–Seidenberg computer algebra argument demonstrates that the δ , scaled by the diameter of the triangle, never exceeds 3/2 in a Riemannian manifold of constant nonpositive curvature. As probably the most important corollary, a finite metric geodesic space in which the ratio δ /diam is (strictly) bounded from above by 3/2 for all geodesic triangles exhibits the same metric properties as a negatively curved Riemannian manifold. The specific applications targeted here are those involving such very large but finite graphs as the Internet and the Protein Interaction Network. It is indeed argued that negative curvature is the precise mathematical formulation of their visually intuitive core concentric property.

The Pythagorean Theorem is one of the oldest, best known, and most useful theorems in all of mathematics, and it has also surely been proved in more different ways than any other. Euclid gave two proofs of it in the Elements, as Proposition I,47, and also as Proposition VI,31, a more general but less well-known formulation concerning arbitrary ‘figures’ described on the sides of a right triangle. The first of those demonstrations is based on a comparison of areas and the second on similarity theory, a basic distinction that can be used as a first step in classifying many other proofs of the theorem as well.

When N-beads slide along a frictionless hoop, their collision sequence gives rise to a dynamical system that can be studied via matrix products. It is of general interest to understand the distribution of velocities and the corresponding eigenvalue spectrum that a given collision sequence can produce. We formulate the problem for general N and state some basic theorems regarding the eigenvalues of the collision matrices and their products. The case of three beads of masses m1, m2, m3 is studied in detail. We exploit the fact that each collision sequence can be viewed as a billiard trajectory in a right triangle with non-standard reflection rules. Existence of families of periodic orbits are proven, and orbits that densely fill the triangle are computed. Eigenvalue distributions and position and velocity histograms are computed as a function of the restitution coefficient, both periodic and dense collision sequences are discussed, and a series of conjectures based on computational evidence are formulated. Comparisons are made between the eigenvalue distributions and autocorrelation matrices associated with dense trajectories generated from a chaotic collision sequence and spectra from matrix sequences generated from random orderings, and we describe how the three-bead system could be used as the basis for a random number generating algorithm that is computationally efficient.

Two-dimensional (2-D) TEM horns are modeled using the finite-difference time-domain (FDTD) method. The boundary walls are perfect electric conductors and one wall, which does not align with the Cartesian grid, is approximated using a staircased representation. By carefully comparing the FDTD results to those of the analytic solution, one can make conclusions about the coarseness with which a boundary can be represented. It is found that staircasing errors are small when the staircase diagonal (the hypotenuse of the right triangle created by the stairstep) is smaller than half a wavelength at the highest significant frequency in the excitation. This rule-of-thumb is put forward as a necessary condition for the discretization of general problems. Results are also provided for some simple FDTD schemes that are designed to reduce staircasing errors. By using large aspect-ratio cells, a grid can be constructed that satisfies the rule-of-thumb given above. While this approach eliminates general staircasing errors, some errors persist owing to the presence of step discontinuities immediately adjacent to the horn feed. These errors can be further reduced by using a cell-splitting approach. It is shown that the contour path FDTD technique can be used to eliminate nearly all staircasing errors, while some additional improvement is shown to be provided by using a stabilized contour path FDTD approach. Finally, a recently proposed conformal technique that permits simple implementation is shown to provide results comparable with those of the stabilized contour path approach.

Billiard problems are simple examples of Hamiltonian dynamical systems. These problems have been used as model systems to study the link betwen classical and quantum chaos. The heart of this linkage is provided by the periodic orbits in the classical system. In this article we will show that for an arbitrary right triangle, almost all trajectories that begin perpendicular to a side are periodic, that is, the set of points on the sides of a right triangle from which nonperiodic (perpendicular) trajectories begin is a set of measure zero. Our proof incorporates the previous result for rational right triangles (where the angles are rational multiples of \ensuremath{\pi}), while extending the result to nonrational right triangles.

A frictionless cart in the shape of a right triangle (with the vertical side forward) is elastically impacted by vertically falling raindrops. The speed of the cart as a function of time can be analytically deduced as an exercise in the use of trigonometric and hyperbolic functions. A characteristic time defines the approach to a terminal speed which is a sizeable fraction of the speed of the rain. The treatment is accessible to a student in a calculus-based mechanics course.

We prove convergence of difference schemes to generalized solutions of the Helmholtz operator with a coefficient from Lp(Ω), where Ω is a right triangle.

An O(h) accurate difference scheme is constructed for the eigenvalue problem for the Helmholtz operator in a right triangle. The convergence of the difference scheme is analyzed under conditions ensuring that the eigenfunctions of the differential problem are in the space W21 (Ω).

A well-known theorem of Hardy and Littlewood gives a three-term asymptotic formula, counting the lattice points inside an expanding, right triangle. In this paper a generalisation of their theorem is presented. Also an analytic method is developed which enables one to interpret the coefficients in the formula. These methods are combined to give a generalisation of a “heightcounting” formula of Györy and Pethö which itself was a generalisation of a theorem of Lang.

We present an efficient method to solve Schr\"odinger's equation for perturbations of low rank. In particular, the method allows to calculate the level counting function with very little numerical effort. To illustrate the power of the method, we calculate the number variance for two pseudointegrable quantum billiards: the barrier billiard and the right triangle billiard (smallest angle $\pi/5$). In this way, we obtain precise estimates for the level compressibility in the semiclassical (high energy) limit. In both cases, our results confirm recent theoretical predictions, based on periodic orbit summation., Comment: 4 pages

Equivalent fields are often used in radiation oncology for calculation of dose. This avoids the need to make a scatter integration but has limited applicability and some inaccuracy. We have evaluated the alternative of explicit integration of phantom-scatter dose using a functional representation, sigma, of the ratio between the scatter dose and the primary dose. The independent parameters are depth and the side of the square field. The function chosen has the advantage that the integral for a right triangle is available in closed form, which simplifies the determination of the dose from phantom-scattered photons for irregular fields by summation over such triangles. This approach accounts for the influence of depth and beam quality, which the commonly used equivalent-field tables and the area-over-perimeter relation ignore. The accuracy of this procedure is determined by the accuracy of the function sigma. This has about a 1% error of total dose for high-energy x-rays. We conclude that the tables and rules can be replaced by a computer-implemented integration of the phantom-scatter dose represented by this function sigma and using sectors or right triangles. Summing the closed-form contributions from component right triangles reduces the calculation time, which is particularly desirable when many fields are employed, as for intensity-modulated techniques and inverse planning. Measurements performed on irregular MLC-shaped fields compared well with the result from calculations.

An integral equation technique is employed to obtain eigenvalues and eigenmodes of the homogeneous Holmholtz equation for a two‐ or three‐dimensional closed region of arbitrary shape with arbitrary first‐order homogeneous boundary conditions. The method is described for general (i.e., nonseparable) geometries, with a discussion of the simplification introduced by having a separable geometry given in an Appendix. A numerical example is given for a nonseparable geometry, i.e., a two‐dimensional right triangle of arbitrary enclosed angle with Neumann boundary conditions. Results for the special case of an isosceles right triangle agree very well with a known analytical solution.

The flow of a classical particle bouncing elastically inside an arbitrary polygon is investigated. If every interior angle is a rational multiple of π, there exists precisely one isolating integral in addition to the energy; this integral is described in detail; any possible third integral is nonisolating. If one or more interior angles is an irrational multiple of π, the second integral becomes everywhere nonisolating and non‐Lebesgue‐measurable, i.e., the second integral disappears. The flow of two hard points bouncing elastically in a finite one‐dimensional box is equivalent to the flow of a point particle moving elastically inside a right triangle having interior angle tan−1 (m2/m1)1/2, so the preceding remarks apply to this model. Nonrigorous arguments are given in support of the notion that the polygon model is ergodic and mixing, but is not a C‐system.

This paper proves the existence of non-periodic and not everywhere dense billiard trajectories in convex polygons and polyhedrons. For anyn≧3 there exists a corresponding convexn-agon (forn=3 this will be a right triangle with a small acute angle), while in three-dimensional space it will be a prism, then-agon serving as the base. The results are applied for investigating a mechanical system of two absolutely elastic balls on a segment, and also for proving the existence of an infinite number of periodic trajectories in the given polygons. The exchange transformation of two intervals is used for proving the theorems. An arbitrary exchange transformation of any number of intervals can also be modeled by a billiard trajectory in some convex polygon with many sides.

Publisher Summary To deduce the relations between the sides and angles of a triangle in the h-plane the use of the so-called hyperbolic functions should be made, which are related in the manner of the trigonometric functions. This chapter presents a derivation of the relations between the sides and angles in a right triangle. In the process of derivation, the principle of special position is used that consists in using reflection, etc., to obtain a triangle congruent to the initial triangle but located more conveniently than the initial triangle. This procedure is justified by the fact that reflections, parallel translations, and similitudes do not change angles and lengths. The derivation of the trignometric formulas of hyperbolic geometry in the chapter involves the use of the Poincare model. However, these relations can be derived directly from the axionms of the hyperbolic geometry.

A simple analog computer, which solves the oblique and the right spherical triangle equation, is described based on the law of cosines. With three parameters of the oblique triangle as inputs, the fourth is computed with an accuracy better than ± 10 minutes of arc for most of the practical problems. The computer can be used to solve navigational problems such as the hour angle, course angle and great circle distance. Solutions for the right spherical triangle are obtained when one parameter is set to 90°.

describes a method for the evaluation of integrals over a triangle, based on a formula developed in [1]. The mid-points of the sides of the triangle are used as the points of evaluation in this method. Other possibilities may be used; for example, the trisection points of the medians that are not the centroid could be used as the points of evaluation. These points are symmetrically spaced and the weights, wi, again are equal. The centroid could also be used as a single point of evaluation with wi equal to the area of the triangle. As another modification to this method, one could calculate the appropriate mid-points of the reference triangle for a reasonable number of subdivisions, then retain them as constants rather than calculate each point as it is used. Recently, (1958) Mr. Walter Leffin carried out computations by the use of these methods.