Triangle In Terms Research Articles

Hyperbolic Lunes

Summary The formula for the area of a hyperbolic triangle in terms of its angle defect is derived using hyperbolic lunes, in analogy with the argument using (elliptic) lunes to express the area of an elliptic triangle in terms of its angle excess. Several pedagogical features of this construction are then discussed.

A sangaku problem involving two circles and an equilateral triangle

Summary We generalize a problem in Wasan geometry involving two circles and an equilateral triangle in terms of oriented circles and oriented lines, and get an invariant property of the figure consisting of an isosceles triangle of fixed height and a line passing through the apex of the triangle. Several results for the incircle and the excircles of the isosceles triangle and the right triangle are obtained as special cases of the generalization.

Conception of spherical triangle in terms of learning styles

Penelitian ini bertujuan untuk mendeskripsikan konsepsi mahasiswa pada materi segitiga bola ditinjau dari gaya belajar. Konsepsi mahasiswa dideskripsikan berdasarkan identifikasi mahasiswa terhadap segitiga bola serta penerapan segitiga bola untuk menentukan jarak pelayaran dan haluan kapal dalam pelayaran. Representasi mahasiswa terhadap segitiga bola difokuskan pada menghitung besarnya bagian-bagian segitiga bola, penggunaan aturan-aturan segitiga bola untuk menghitung jarak dan haluan kapal dalam pelayaran. Penelitian ini merupakan penelitian eksploratif dengan pendekatan kualitatif. Subyek penelitian ini adalah mahasiswa semester I Program Studi Nautika yang bergaya belajar visual, auditori, dan kinestetik. Hasil penelitian menunjukkan bahwa terdapat perbedaan konsepsi antara mahasiswa yang bergaya belajar visual, auditori, dan kinestetik. Dosen diharapkan memiliki strategi yang tepat dalam menyampaikan materi perkuliahan, sehingga materi dapat dengan mudah dipahami oleh mahasiswa yang memiliki gaya belajar yang berbeda-beda.

Interpolation with C2 quartic macro-elements based on 10-splits

In this paper interpolation methods for the construction of quartic splines on triangulations refined with 10-splits are proposed. After examining the C2 macro-structure on a single triangle in terms of the Bernstein–Bézier representation, three methods that can be applied on general triangulations are developed. The methods make use of Hermite interpolation data prescribed at the vertices and (optionally) on the edges of the triangulation. The first two approaches lead to splines that are C2 continuous on triangles and C1 continuous across the interior edges of the triangulation. The third method gives rise to splines with overall C2 continuity, which is an exceptionally high order of smoothness for splines of degree four, but comes at the cost of solving a global system of linear equations. The derived results are accompanied with a few numerical examples that show an interesting behavior of splines in dependence of interpolated data.

Counting with Borel’s triangle

Borel’s triangle is an array of integers closely related to the classical Catalan numbers. In this paper we study combinatorial statistics counted by Borel’s triangle. We present various combinatorial interpretations of Borel’s triangle in terms of lattice paths, binary trees, and pattern avoiding permutations and matchings, and derive a functional equation that is useful in analyzing the involved structures.

Triple point of a scalar field theory on a fuzzy sphere

The model of a scalar field with quartic self-interaction on the fuzzy sphere has three known phases: a uniformly ordered phase, a disordered phase and a non-uniformly ordered phase, the last of which has no classical counterpart. These three phases are expected to meet at a triple point. By studying the infinite matrix size limit, we locate the position of this triple point to within a small triangle in terms of the parameters of the model. We find the triple point is closer to the coordinate origin of the phase diagram than previous estimates but broadly consistent with recent analytic predictions.

On certain new refinements of Finsler-Hadwiger inequalities

Several refinements of the Finsler-Hadwiger inequality and its reverse in the triangle are discussed. A new one parameter family of Finsler-Hadwiger inequalities and their reverses are proved. This allows us to obtain new bounds for the sum of the squares of the side lengths of a triangle in terms of other elements in the triangle. Finally, these new bounds are compared to known ones.

(Breaking) The Iron Triangle of Evaluation

Ideas from complexity science and systems thinking are demonstrably helpful in a shift from exploring (systematic) linear net effects of an intervention towards exploring wider (systemic) effects occurring elsewhere. But where these ideas of ‘impact’ are coupled with a narrow use of the contingency approach, some less helpful ‘triangulated’ relationships might be evident. These relationships might be regarded in terms of an ‘iron triangle’, a metaphor used frequently to exemplify pernicious relations of power. The most notable expression of the iron triangle is the ‘military–industrial complex’. This article briefly outlines generic features of the iron triangle in terms of ‘systemic triangulation’ – an idea linking three core systems concepts of interrelationships, perspectives and boundaries. Drawing on a tradition of systems thinking in practice, an associated systemic triangulator is introduced as both a diagnostic and planning heuristic; a device for not only diagnosing symptoms of an evaluation–industrial complex but for prompting ideas towards a more benign evaluation–adaptive complex for impact evaluation.

AB0809 Teriparatide – Better Efficacy in Primary Osteoporosis and Lower T-Score

BackgroundTeriparatide is an anabolic agent that has been shown to increase bone mineral density (BMD) and reduce the risk of fracture. Despite its unique mechanism, there are still few reviews...

A new hypothesis about hematopoietic Pbx-interaction protein (HPIP): Can it be a key factor in neurodegeneration in the post-menopausal period?

Neuronal degeneration in the post-menopausal term leads to cognitive symptoms such as anxiety, difficulty in concentrating, overreacting to minor upsets, quickly becoming irritated and forgetfulness in approximately 70–80% of all women around the world. These symptoms, which result from microtubule damage in the axon extensions of hippocampal neurons in during menopause, greatly reduce individuals’ life quality. Thus, an investigation of the estrogen receptor-signaling pathway–microtubule dynamic triangle and the possible links between them is important when it comes to explaining the possible mechanism of neurodegeneration. Hematopoietic Pbx-interaction protein (HPIP), a microtubule-binding protein, is a novel scaffolding protein. The detection of this protein on neurons represents the most important step in our hypothesis. The importance of the hypothesis is that it might provide important clues about the possible role of HPIP and its mechanism through in vivo and in vitro studies of estrogen receptors–microtubules and the HPIP triangle in terms of neuronal degeneration in the post-menopausal period.A preliminary study was performed to test the main part of our hypothesis using real-time PCR. According to the results, the mRNA expression of HPIP was found in hippocampal neurons. After the detection of this novel protein in neurons, it was observed that there were differences in the experimental groups when compared with the control group relating to the mRNA expression of this protein.An important scientific question remains concerning the mechanisms of neurodegeneration appearing in the post-menopausal period and the receptors, proteins, and signaling pathways that play a role in this degeneration. In consideration of the data from in vivo and in vitro studies used to test our hypothesis, we will try to address the relevant questions. As this issue is resolved, new studies and treatment procedures that can help to prevent the possible difficulties in the menopausal period will be illuminated.

Against Matricide: Rethinking Subjectivity and the Maternal Body

In this article I critically re‐examine Julia Kristeva's view that becoming a speaking subject requires psychical matricide: violent separation from the maternal body. I propose an alternative, non‐matricidal conception of subjectivity, in part by drawing out anti‐matricidal strands in Kristeva's own thought, including her view that early mother–child relations are triangular. Whereas she understands this triangle in terms of a first imaginary father, I re‐interpret this triangle using Donald Winnicott's idea of potential space and Jessica Benjamin's idea of an intersubjective space of thirdness. I argue that this space provides a maternal third term: a relation of connection and difference between two, a relation that inherits the affective, mobile, generative qualities of the maternal body as the infant (according to Kristeva) imagines it. This connecting space allows both mothers and children to emerge as subjects in their own right. I then suggest that potential‐maternal space expands into language, so that language intrinsically allows the possibility of a speaking position of connection with the mother. Entrance into language need not entail separation or matricide: the problem is not language as such but the particular way that speech and logos have been defined historically.

Parametric adjustment of a junction triangle in terms of the precise construction of haulage drives in underground mines

In total costs of ore mining, haulage participates with 30-60%; the total number of workers engaged in the transportation exceeds 35%, while the participation of transportation in the total power consumption amounts to over 40%. All these values indicate that transportation is one of the most signifi cant challenges in the underground mining of ore deposits and that haulage analysis is very important both in selecting the haulage type and its design, utilization and maintenance. In addition to various technical mining parameters, together with mining and geological conditions, haulage effi ciency depends on the selection of optimal haulage drives, which can be achieved by accurately connecting the underground mine workings with mine network points on the surface of a site. The selection of the best mine haulage routes is one the most critical tasks in mine surveying. This paper shows how to adjust a junction triangle on a working level using the parametric adjustment method.

Elementary notions of lattice trigonometry

In this paper we study properties of lattice trigonometric functions of lattice angles in lattice geometry. We introduce the definition of sums of lattice angles and establish a necessary and sufficient condition for three angles to be the angles of some lattice triangle in terms of lattice tangents. This condition is a version of the Euclidean condition: three angles are the angles of some triangle iff their sum equals \pi. Further we find the necessary and sufficient condition for an ordered n-tuple of angles to be the angles of some convex lattice polygon. In conclusion we show applications to theory of complex projective toric varieties, and a list of unsolved problems and questions.

Upper and lower bounds for the first Dirichlet eigenvalue of a triangle

We prove some new upper and lower bounds for the first Dirichlet eigenvalue of a triangle in terms of the lengths of its sides.

An Intuitive Derivation of Heron's Formula

From elementary geometry we learn that two triangles are congruent if their edges have the same lengths, so it should come as no surprise that the edge lengths of a triangle determine the area of that triangle. On the other hand, the explicit formula for the area of a triangle in terms of its edge lengths, named for Heron of Alexandria (although attributed to Archimedes [4]), seems to be less commonly remembered (as compared with, say, the formulas for the volume of a sphere or the area of a rectangle). One reason why Heron's formula is so easily forgotten may be that proofs are usually presented as unwieldy verifications of an already known formula, rather than as expositions that derive a formula from scratch in a constructive and intuitive manner. Perhaps the derivation that follows, while not truly elementary, will render Heron's formula more memorable for its symmetric and intuitive factorization. The first step of this derivation is to recall that the square of the area of a triangle is a polynomial in the edge lengths. More generally, suppose that T is a simplex in RT with vertices xo, xi, ... , x,, where x0 = 0, the origin. Let A denote the n x n matrix whose columns are given by the vectors xl, ..., xn, and suppose that the xi are ordered so that A has positive determinant. The volume of T is then given by det(A) = n! V(T), implying that

Thomas rotation and the mixed state geometric phase

It is shown that Uhlmann's parallel transport of purifications along a path of mixed states represented by 2 × 2 density matrices is just the path-ordered product of Thomas rotations. These rotations are invariant under hyperbolic translations inside the Bloch sphere that can be regarded as the Poincare ball model of hyperbolic geometry. A general expression for the mixed state geometric phase for an arbitrary geodesic triangle in terms of the Bures fidelities is derived. The formula gives back the solid angle result well known from studies of the pure state geometric phase. It is also shown that this mixed state anholonomy can be reinterpreted as the pure state non-Abelian anholonomy of entangled states living in a suitable restriction of the quaternionic Hopf bundle. In this picture Uhlmann's parallel transport is just the Pancharatnam transport of quaternionic spinors.

Area of a Triangle in Terms of the Taxicab Distance

Area of a Triangle in Terms of the Taxicab Distance

Inscribed polygons and Heron polynomials

Heron's well-known formula expressing the area of a triangle in terms of the lengths of its sides is generalized in the following sense to polygons inscribed in a circle: it is proved that the area is an algebraic function of the lengths of the edges of the polygon. Similar results are proved for the diagonals and the radius of the circumscribed circle. The resulting algebraic equations are studied and elementary geometric applications of the algebraic results obtained are presented.

Breaking of the flavor permutational symmetry: Mass textures and the CKM matrix

Different Ans\"atze for the breaking of flavor permutational symmetry according to ${S}_{L}(3)\ensuremath{\bigotimes}{S}_{R}(3)\ensuremath{\supset}{S}_{L}(2)\ensuremath{\bigotimes}{S}_{R}(2)$ give different Hermitian mass matrices of the same modified Fritzsch type, which differ in the symmetry breaking pattern. In this work we obtain a clear and precise indication on the preferred symmetry breaking scheme from a fit of the predicted $|{\mathbf{V}}^{\mathrm{th}}|$ to the experimentally determined absolute values of the elements of the CKM matrix. The preferred scheme leads to simple mass textures and allows us to compute the CKM mixing matrix, the Jarlskog invariant $J$, and the three inner angles of the unitarity triangle in terms of four quark mass ratios and only one free parameter: the $\mathrm{CP}$ violating phase $\ensuremath{\Phi}$. Excellent agreement with the experimentally determined absolute values of the entries in the CKM matrix is obtained for $\ensuremath{\Phi}=90\ifmmode^\circ\else\textdegree\fi{}$. The corresponding computed values of the Jarlskog invariant and the inner angles are $J=3.00\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5},$ $\ensuremath{\alpha}=84\ifmmode^\circ\else\textdegree\fi{},$ $\ensuremath{\beta}=24\ifmmode^\circ\else\textdegree\fi{}$, and $\ensuremath{\gamma}=72\ifmmode^\circ\else\textdegree\fi{}$ in very good agreement with current data on $\mathrm{CP}$ violation in the neutral kaon-antikaon system and oscillations in the ${B}_{s}^{0}\ensuremath{-}{B}_{s}^{0}$ system.

A generalized Heron-Tartaglia formula and some of its consequences

The well-known formula for finding the area of a triangle in terms of its sides is generalized to volumes of polyhedra in the following way. It is proved that for a polyhedron (with triangular faces) with a given combinatorial structure and with a given collection of edge lengths there is a polynomial such that the volume of the polyhedron is a root of it, and the coefficients of the polynomial depend only on and and not on the concrete configuration of the polyhedron itself. A number of problems in the metric theory of polyhedra are solved as a consequence.