Angles Of Triangle Research Articles

Modeling of velocity distribution among microchannels with triangle manifolds

AbstractA model of velocity distribution among microchannels with triangle manifolds is proposed. According to the flow behaviors analyzed by Fluent, the manifolds are divided into several approximate rectangular channels, and then an equivalent simplified resistance network model is developed to establish the relationships between the velocities and pressure drops in microchannels and approximate rectangular channels. The velocity distributions are calculated under two situations, respectively, considering and ignoring singular losses. The outcomes of the present study are compared with Fluent's simulated results to analyze the effects of singular losses on the velocity distributions. It indicates that the proposed model is suitable for the calculation of velocity distribution among microchannels with obtuse angled or right triangle manifolds under low Reynolds numbers. The premise of ignoring singular losses is that the frictional pressure drops are three times larger than the singular pressure drops in each flow loop. The manifold optimization results indicate that the velocity distribution among microchannels with right triangle manifolds is much more uniform than that of the corresponding one with obtuse angled manifolds. © 2009 American Institute of Chemical Engineers AIChE J, 55: 1969–1982, 2009

Flavour tagging performance in LHCb

To do precise CP violation measurements, the best possible determination of the flavour of the B-meson is necessary. This report summarizes the flavour tagging performances for the LHCb experiment. The flavour tagging is obtained through a combination of several methods, based on different signatures. The use of control channels, which are decays to flavour-specific final states, will allow to determine the wrong tag fraction ω (the probability of a tag to be wrong), which can be used as an input for the determination of CKM unitarity triangle angles.

Triangulation refinement by using edge subdivision

It is proved that any triangulation of a flat polygonal region can be refined by using repeated subdivisions of an edge so that: (1) the maximum diameter of the triangles would be less than any pre-assigned positive number, and (2) the minimum interior angle of the triangles of the triangulation obtained would be not less than the minimum interior angle of the triangles of the original triangulation divided by 9. The required triangulation refinement is constructed in two steps: first, the triangulation is refined so that the triangles of the triangulation obtained can be combined into pairs, and only boundary triangles may be left unpaired; at this step each triangle is split into at most 4 parts. Then the triangulation obtained is refined once again in order that the diameter of each triangle be less then a prescribed ɛ. At each of the steps, the minimum interior angle of triangles is reduced by at most 3 times. This is guaranteed by the lemma saying that the interior angles of the triangles into which the original triangle is divided by a median are at least as great as one-third of the minimum interior angle of the original triangle.

On the measurement of the unitarity triangle angleγfromB0→DK*0decays

The decay ${B}^{0}\ensuremath{\rightarrow}D{K}^{*0}$ is well known to provide excellent potential for a precise measurement of the unitarity triangle angle $\ensuremath{\gamma}$ in future experiments. It is noted that the sensitivity can be significantly enhanced by studying the amplitudes relative to those of the flavor-specific decay ${B}^{0}\ensuremath{\rightarrow}{D}_{2}^{*\ensuremath{-}}{K}^{+}$, which can be achieved by analyzing the ${B}^{0}\ensuremath{\rightarrow}D{\ensuremath{\pi}}^{\ensuremath{-}}{K}^{+}$ Dalitz plot.

A method of measuring cos(2β) using a time-dependent dalitz plot analysis of B0 → DCPπ+π−

We present a feasibility study of a previously outlined method for measuring both the sine and the cosine of twice the Cabibbo–Kobayashi—Maskawa (CKM) Unitarity Triangle angle β using a time-dependent Dalitz plot analysis of B0 → Dπ+π− decays, where the neutral D meson is reconstructed in decays to CP eigenstates. We show that this method can be used at the B factories to make a measurement of cos(2β) that is competitive with, or more precise than, other techniques using different quark-level transitions, while sin(2β) can be measured to a better precision than any existing measurement using transitions. Furthermore, this technique has great potential to be employed at LHCb.

Measurement of ratios of branching fractions andCP-violating asymmetries ofB±→D*K±decays

We report a study of B -> D*K decays with D* decaying to D pi0 or D gamma, using 383 * 10**6 B Bbar pairs collected at the Upsilon(4S) resonance with the BaBar detector at the SLAC PEP-II B-Factory. The D meson decays under study include a non-CP mode (Kpi), CP-even modes (K+K-, pi^+pi^-) and CP-odd modes (KS pi0, KS phi, KS omega). We measure ratios (R*_{CP+-}) of branching fractions of decays to CP eigenmode states and to flavor-specific states as well as CP asymmetries (A*_{CP+-}). These measurements are sensitive to the unitarity triangle angle gamma. We obtain A*_{CP+} = -0.11 pm 0.09 pm 0.01, R*_{CP+} = ~1.31 pm 0.13 pm 0.04, and A*_{CP-} = 0.06 pm 0.10 pm 0.02, R*_{CP-} = ~1.10 pm 0.12 pm 0.04, where the first error is statistical and the second error is systematic. Translating our results into an alternative parametrization, widely used for related measurements, we obtain x*_{+}=0.11 pm 0.06 pm 0.02 and x*_{-} = 0.00 pm 0.06 pm 0.02. No significant CP-violating charge asymmetry is found in either the flavor-specific mode D -> K pi or in B -> D* pi decays.

Determination of the angles of the unitarity triangle at BABAR⋆

We report recent measurements of the three angles of the unitarity triangle using the data collected with the BABAR detector at the PEP-II asymmetric energy e + e - collider.

Measurement of the branching fraction, polarization, andCPasymmetries inB0→ρ0ρ0decay, and implications for the CKM angleα

We study B0->rho0rho0 decays in a sample of 465*10^6 Upsilon (4S)->BBbar events collected with the BABAR detector at the PEP-II asymmetric-energy e+e- collider located at the Stanford Linear Accelerator Center (SLAC). We measure the branching fraction B = (0.92 +/- 0.32 +/- 0.14)*10^{-6} and longitudinal polarization fraction f_L = 0.75^{+0.11}_{-0.14} +/- 0.04, where the first uncertainty is statistical and the second is systematic. The evidence for the B0->rho0rho0 signal has a significance of 3.1 standard deviations, including systematic uncertainties. We investigate the proper-time dependence of the longitudinal component in the decay and measure the CP-violating coefficients S^{00}_L = (0.3 +/- 0.7 +/- 0.2) and C^{00}_L = (0.2 +/- 0.8 +/- 0.3). We study the implication of these results for the unitarity triangle angle alpha.

Flavour physics at CLEO-c

Three CLEO-c results related to flavour-physics are presented: the determination of the strong-phase difference between D0->K+pi- and D0->K-pi+, delta, the measurement of the coherence parameter and average strong-phase difference between D0->K+pi-pi-pi+ and D0 -> K-pi+pi+pi- and measurements of the variation of strong-phase difference between D0 and D0bar decays to K0Spi+pi- over phase space. All measurements are important for determining the unitarity triangle angle gamma from B->D(*)K(*) decays. Furthermore, the measurement of delta is important for interpreting D0-D0bar mixing.

Combined mandibular angle resection with lateral cortex splitting ostectomy around mandibular angle by intraoral approach for correction of square face

Objective To search for a procedure for correction of square face. Methods 68 patients with square face (aged from 21～40 years, with 65 females and 3 males) underwent osteetomy combined mandibular angle triangle resection with lateral cortex around the mandibular angle-splitting by intraoral approach. The mandible outer cortex was splited with bone chisel and mandible angle was resected by the high-speed pneumatic drill. The bulky masseter muscle and the enbonpoint buecal fat-pad were scissored. Results The application of high-speed pneumatic drill and bone chisel facilitated resection of the mandibular angle with lateral cortex splitting osteetomy. The operative effects were satisfactory in 56 patients during 3-24 months of follow-up. The bigonial distance was effectively re-duced, and facial shape became ellipse. The gross asymmetry and mandibular fracture were not conse-quences. The lateral contour of the mandibular angle was natural. Conclusion Ostectomy of mandible angle and lateral cortex of the mandible allows the reduction of the mandibular contour of the frontal view and the lateral contour of mandibular angle becomes soft, giving a more natural appearance. This new method is worthy to be applied widely because of the good result with effective reduction of the mandibular width. Key words: Square face; Mandibuloplasty; Ostectomy

Permutation asymmetry of the relativistic velocity addition lawand non-Euclidean geometry

The asymmetry of the relativistic addition law for noncollinear velocities under the velocity permutation leads to two modified triangles on a Euclidean plane depicting the addition of unpermuted and permuted velocities and the appearance of a nonzero angle ω between two resulting velocities. A particle spin rotates through the same angle ω under a Lorentz boost with a velocity noncollinear to the particle velocity. Three mutually connected three-parameter representations of the angle ω, obtained by the author earlier, express the three-parameter symmetry of the sides and angles of two Euclidean triangles identical to the sine and cosine theorems for the sides and angles of a single geodesic triangle on the surface of a pseudosphere. Namely, all three representations of the angle ω, after a transformation of one of them, coincide with the representations of the area of a pseudospherical triangle expressed in terms of any two of its sides and the angle between them. The angle ω is also symmetrically expressed in terms of three angles or three sides of a geodesic triangle, and therefore it is an invariant of the group of triangle motions over the pseudo-sphere surface, the group that includes the Lorentz group. Although the pseudospheres in Euclidean and pseudo-Euclidean spaces are locally isometric, only the latter is isometric to the entire Lobachevsky plane and forms a homogeneous isotropic curved 4-velocity space in the flat Minkowski space. In this connection, relativistic physical processes that may be related to the pseudosphere in Euclidean space are especially interesting.

Dihedral F-Tilings of the Sphere by Equilateral and Scalene Triangles - II

The study of dihedral f-tilings of the Euclidean sphere $S^2$ by triangles and $r$-sided regular polygons was initiated in 2004 where the case $r=4$ was considered [5]. In a subsequent paper [1], the study of all spherical f-tilings by triangles and $r$-sided regular polygons, for any $r\ge 5$, was described. Later on, in [3], the classification of all f-tilings of $S^2$ whose prototiles are an equilateral triangle and an isosceles triangle is obtained. The algebraic and combinatorial description of spherical f-tilings by equilateral triangles and scalene triangles of angles $\beta$, $\gamma$ and $\delta$ $(\beta>\gamma>\delta)$ whose edge adjacency is performed by the side opposite to $\beta$ was done in [4]. In this paper we extend these results considering the edge adjacency performed by the side opposite to $\delta$.

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.

Global shape versus local feature: An angle illusion

We have shown previously that the precision of angle judgments depends strongly on the global stimulus configuration: discrimination thresholds for angles that form part of isosceles triangles are up to 3 times lower than for those that form part of scalene triangles [Kennedy, G. J., Orbach, H. S., & Loffler, G. (2006). Effects of global shape on angle discrimination. Vision Research, 46(8–9), 1530–1539]. Here, we investigated whether or not the perceived size of an angle (accuracy) is also affected by the overall shape of which it forms a part. Observers compared the relative sizes of angles contained in isosceles triangles with those of angles in scalene triangles and points of subjective equality were determined. For a reference angle of 60°, angles embedded in isosceles triangles were judged to be on average 14° larger than angles embedded in scalene triangles. This result is largely independent of the reference angle, triangle orientation and triangle size. Moreover, the effect is present whether or not triangles of different shapes enclose the same area, whether or not the side of the triangle opposite the angle is present and whether the triangle is outlined or defined by dots at its vertexes. In sum, our results provide evidence for a novel illusion where an angle embedded in an isosceles triangle is judged substantially larger than the same angle embedded in a scalene triangle. This finding demonstrates that mechanisms for computing angles are sensitive to the context within which angles are presented.

Variation of the obturator foramen and pubic arch of the female bony pelvis

The purpose of this study was to determine dimensions of the obturator foramen and pubic arch of the female pelvis and to assess for variability. Ninety-six female pelvises were selected from the Cleveland Natural History Museum. The obturator foramen area, pubic arch angle, pubic ramus, pubic symphysis, and anterior urogenital triangle area were measured. Linear regression was used to evaluate independent associations. There was considerable variability of the bony architecture. The mean obturator foramen area was 12.2 +/- 2.1 cm(2) (range, 7.38-18.22 cm(2)). After controlling for height, the pubic ramus width, pubic symphysis length, and interobturator foramina distance were significantly greater in European American women compared with African American women. Obturator foramen area increased with increasing height (P = .0008) but was not associated with race. There is considerable variability in the bony architecture of the obturator foramen and pubic arch of the female pelvis. Race and height may account for some of these variabilities.

The Erdős-Mordell inequality is equivalent to non-positive curvature

We prove that, in plane absolute geometry, the Erdős-Mordell inequality is equivalent to the statement that the sum of the angles of a triangle is less than or equal to two right angles.

LATOR: ITS SCIENCE PRODUCT AND ORBITAL CONSIDERATIONS

In a LATOR mission to measure the non-Euclidean relationship between three sides and one angle of a light triangle near the Sun, the primary science parameter, to be measured to part-in-109 precision, is shown to include not only the key parametrized post-Newtonian (PPN) γ, but also the Sun's additional mass parameter, MΓ, which appears in the spatial metric field potential. MΓ may deviate from the Sun's well-measured gravitational mass due to post-Newtonian features of gravitational theory not previously measured in relativistic gravity observations. Under plausible assumptions, MΓ is a linear combination of the Sun's gravitational and inertial masses. If LATOR's two spacecraft lines of sight are kept close to equal and opposite relative to the Sun during the mission's key measurements of the light triangle, it is found that the navigational requirements for the spacecraft positions are greatly relaxed, eliminating the need for on-board drag-free systems. Spacecraft orbits from the Earth to achieve the equal and opposite passages by the Sun's, line of sight are illustrated.

Entropy production due to free convection in partially heated isosceles triangular enclosures

A numerical analysis of the entropy production has been performed due to natural convection heat transfer and fluid flow in isosceles triangular enclosures with partially heated from below and symmetrically cooled from sloping walls. Governing equations are solved by finite difference method. Governing parameters on flow and temperature fields are Rayleigh number (10 3 ⩽ Ra ⩽ 8.8 × 10 5), dimensionless length of heater (0.25 ⩽ (ℓ′ = ℓ/ L) ⩽ 1.0), dimensionless location of heater (0.25 ⩽ ( c′ = c/ L) ⩽ 0.75) and inclination angle of slopping walls (30° ⩽ β ⩽ 60°). Heat transfer results are presented in terms of local and mean Nusselt numbers ( Nu) while entropy production results are shown with entropy production number ( N s) and Bejan number ( Be). Isotherms, streamlines, contours of entropy production due to heat transfer and fluid friction irreversibility are plotted. It is observed that entropy production number increases but Bejan number decreases with increasing of Rayleigh number. However, both entropy production due to heat transfer and fluid friction irreversibility are affected by higher inclination angle of triangle and length of heater.

Measurements of γ (Dalitz Analysis) and Future Projections

We summarize the status of the measurement of the unitarity triangle angle γ performed by the BaBar and Belle collaborations using Dalitz plot analyses of the three-body decay of the D 0 meson from the B ± → D ( ∗ ) K ( ∗ ) ± process. Future prospects at the B factories are given.

Formula omitted] Prospects at LHCb

The LHCb experiment will collect very large samples of B -mesons decaying to charmless two-body final states. The combined measurements of the B 0 → π + π − and B s 0 → K + K − time-dependent CP asymmetries allow a determination of the unitarity triangle angle γ , under the assumption of U-spin flavour symmetry. We present the status of the LHCb simulated analysis of the charged two-body charmless modes.