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# Scientists develop new process for more eco-friendly liquid crystals — ScienceDaily

Liquid crystals might quickly be produced extra effectively and in a extra environmentally pleasant means. A brand new course of has been developed by researchers at Martin Luther College Halle-Wittenberg (MLU) in Germany, Bangalore College in India and Cairo College in Egypt. In comparison with typical strategies, it’s sooner, extra energy-efficient and guarantees a excessive yield because the crew stories within the Journal of Molecular Liquids. Liquid crystals are utilized in most smartphone, pill and pc shows.

The manufacturing of liquid crystals is a posh course of with many intermediate steps. “Usually it requires numerous solvents and costly catalysts,” says Dr Mohamed Alaasar, a chemist at MLU. The crew from Germany, India and Egypt was searching for a strategy to simplify the method and make it extra environmentally pleasant. The thought: as an alternative of the chemical reactions happening one after the opposite, sure steps might be mixed in a so-called multicomponent response wherein a number of substances react instantly with each other.

The crew developed an strategy for producing liquid crystals which doesn’t require environmentally dangerous solvents and depends on cheaper catalysts. “We had been in a position to obtain a yield of about 90 per cent. Because of this a lot of the chemical substances are used within the course of and comparatively few residues are produced,” explains Alaasar. This protects vitality and in the end additionally cash. At room temperature the newly created liquid crystals are in a nematic part — a particular association of molecules utilized in most liquid crystal shows or LCDs.

Thus far, the researchers have solely examined their new course of within the laboratory. Nonetheless, Alaasar is assured that it is also applied on an industrial scale. “Nonetheless, producers must rebuild elements of their manufacturing. This has not occurred up to now with different promising supplies,” says the scientist. Nonetheless, customers began valuing sustainability and extra environmentally pleasant merchandise of the final years. That might be an extra argument in favour for the brand new strategy.

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# Innovative crystals for future computer electronics — ScienceDaily

Whereas trendy computer systems are already very quick, in addition they eat huge quantities of electrical energy. For some years now a brand new know-how has been a lot talked about, which though it’s nonetheless in its infancy may sooner or later revolutionise laptop know-how — spintronics. The phrase is a portmanteau which means “spin” and “electronics,” as a result of with these parts electrons now not movement by laptop chips, however the spin of the electrons serves as the knowledge provider. A crew of researchers with employees from Goethe College Frankfurt has now recognized supplies which have surprisingly quick properties for spintronics. The outcomes have been printed within the specialist journal “Nature Supplies.”

“You need to think about the electron spins as in the event that they had been tiny magnetic needles that are hooked up to the atoms of a crystal lattice and which talk with each other,” says Cornelius Krellner, Professor for Experimental Physics at Goethe College Frankfurt. How these magnetic needles react with each other essentially depends upon the properties of the fabric. Up to now ferromagnetic supplies have been examined in spintronics above all; with these supplies — equally to iron magnets — the magnetic needles want to level in a single route. Lately, nonetheless, the main target has been positioned on so-called antiferromagnets to a larger diploma, as a result of these supplies are mentioned to permit for even sooner and extra environment friendly switchability than different spintronic supplies.

With antiferromagnets the neighbouring magnetic needles all the time level in reverse instructions. If an atomic magnetic needle is pushed in a single route, the neighbouring needle turns to face in the wrong way. This in flip causes the following however one neighbour to level in the identical route as the primary needle once more. “As this interaction takes place in a short time and with just about no friction loss, it provides appreciable potential for completely new types of digital componentry,” explains Krellner.

Above all crystals with atoms from the group of uncommon earths are considered attention-grabbing candidates for spintronics as these comparatively heavy atoms have robust magnetic moments — chemists name the corresponding states of the electrons 4f orbitals. Among the many rare-earth metals — a few of that are neither uncommon nor costly — are components akin to praseodymium and neodymium, that are additionally utilized in magnet know-how. The analysis crew has now studied seven supplies with differing rare-earth atoms in whole, from praseodymium to holmium.

The issue within the improvement of spintronic supplies is that completely designed crystals are required for such parts because the smallest discrepancies instantly have a unfavorable influence on the general magnetic order within the materials. That is the place the experience in Frankfurt got here into play. “The uncommon earths soften at about 1000 levels Celsius, however the rhodium that can be wanted for the crystal doesn’t soften till about 2000 levels Celsius,” says Krellner. “That is why customary crystallisation strategies don’t operate right here.”

As an alternative the scientists used sizzling indium as a solvent. The uncommon earths, in addition to the rhodium and silicon which might be required, dissolve on this at about 1500 levels Celsius. The graphite crucible was saved at this temperature for a couple of week after which gently cooled. In consequence the specified crystals grew within the type of skinny disks with an edge size of two to a few millimetres. These had been then studied by the crew with the help of X-rays produced on the Berlin synchrotron BESSY II and on the Swiss Mild Supply of the Paul Scherrer Institute in Switzerland.

“An important discovering is that within the crystals which we’ve grown the rare-earth atoms react magnetically with each other in a short time and that the power of those reactions may be particularly adjusted by the selection of atoms,” says Krellner. This opens up the trail for additional optimisation — finally spintronics remains to be purely elementary analysis and years away from the manufacturing of business parts.

There are nonetheless a fantastic many issues to be solved on the trail to market maturity, nonetheless. Thus, the crystals — that are produced in blazing warmth — solely ship convincing magnetic properties at temperatures of lower than minus 170 levels Celsius. “We suspect that the working temperatures may be raised considerably by including iron atoms or comparable components,” says Krellner. “Nevertheless it stays to be seen whether or not the magnetic properties are then simply as constructive.” Because of the brand new outcomes the researchers now have a greater thought of the place it is smart to vary parameters, nonetheless.

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# Fragile nanosheets stripped from crystals

Nature, Printed on-line: 23 February 2022; doi:10.1038/d41586-022-00466-z

Two-dimensional supplies have been restricted to programs wherein sturdy chemical bonds maintain atoms collectively in sheets. Now, 2D supplies consisting of molecules linked by weak non-covalent bonds have been peeled from crystals.

# Free-standing homochiral 2D monolayers by exfoliation of molecular crystals

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• # Hyperbolic shear polaritons in low-symmetry crystals

Crystal symmetry performs a important function in dictating the optical, digital, mechanical and thermal properties of a cloth. Lowered symmetry is on the coronary heart of quite a few emergent phenomena, together with structural part transitions11, charge-density waves13 and topological physics14. The interplay of sunshine with low-symmetry supplies is especially vital, because it permits superb management over the part, propagation path and polarization1,2,3,4. This management may be particularly pronounced for sub-diffractional floor waves, for example, floor phonon polaritons (SPhPs)15 and floor plasmon polaritons (SPPs), supported on the floor of polar crystals and conductors, respectively. Each SPhPs and SPPs are quasiparticles comprising photons and coherently oscillating costs, that’s, polar lattice vibrations or free-carrier plasmas, respectively, and they’re strongly influenced by crystal symmetry. As a related instance, low-symmetry polaritonic supplies can help hyperbolic gentle propagation16, constituting an unique class of sunshine waves which can be extremely directional with very giant momenta. Hyperbolic polaritons happen in supplies during which the actual a part of the permittivity alongside not less than one crystal path is destructive and constructive alongside not less than one different. This excessive anisotropy is related to free carriers and optic phonons in anisotropic lattices. In flip, hyperbolic polaritons allow deeply sub-wavelength gentle confinement over broad bandwidths8,9. In polar crystals with symmetries that help a single optical axis (uniaxial), corresponding to hexagonal boron nitride (hBN), hyperbolic polaritons (HPs) of kind I or kind II can come up5,8,9, for which the hyperbolic isofrequency surfaces do or don’t intersect the optical axis, respectively. Supplies or metamaterials exhibiting decrease symmetry, during which all three main polarizability axes are totally different (biaxial) however orthogonal, corresponding to alpha-phase molybdenum trioxide (α-MoO3)1,3, Li-intercalated vanadium pentoxide (V2O5)4 or nanostructured metasurfaces17, exhibit a number of distinct spectral regimes of hyperbolic modes propagating alongside totally different crystal axes. Notably, in-plane hyperbolicity inside α-MoO3 movies has been proven to be low loss1,3, with reconfigurable options2 and able to supporting topological transitions2. Extra unique polaritonic responses could also be anticipated in crystals with additional decreased symmetry, corresponding to monoclinic and triclinic lattices.

Monoclinic crystals make up the most important crystal system, with round one-third of the minerals of Earth belonging to one in every of its three lessons18. These low-symmetry Bravais lattices exhibit non-orthogonal principal crystal axes (Fig. 1a), in distinction to orthorhombic (for instance, biaxial α-MoO3 (ref. 1)), tetragonal, hexagonal, trigonal (for instance, uniaxial α-quartz, aQ, Fig. 1b) or cubic crystal methods. As a consequence, their dielectric permittivity tensor has main polarizability instructions that strongly depend upon the frequency, with off-diagonal phrases that can’t be utterly eliminated via coordinate rotation6,7, and reveals shear phrases analogous to viscous circulation19. These options come up as a result of non-trivial relative orientation (neither parallel nor orthogonal) of a number of optical transitions that, at a given frequency, contribute to a web polarization that can’t be aligned with the crystal axes. In flip, this property ends in unique gentle propagation not supported by higher-symmetry crystals6,7,20. Right here we present exemplary penalties of those materials options for nanophotonics, specifically, the emergence of a brand new type of waves — hyperbolic shear polaritons (HShPs) — which haven’t been beforehand noticed.

On this work, we theoretically and experimentally exhibit the emergence of HShPs in monoclinic crystals. As an exemplary materials to exhibit this phenomenon, we examine beta-phase Ga2O3 (bGO), which has gained a considerable amount of analysis and industrial consideration for its excessive breakdown discipline21 and functions in photovoltaics22, optical shows23 and gasoline sensors24. Within the low-energy vary, bGO options a number of robust infrared-active, non-orthogonal phonon resonances6, making the permittivity tensor of bGO naturally non-diagonalizable. Its low symmetry has two penalties on the polariton propagation when put next with extra standard hyperbolic supplies with a diagonal permittivity tensor, corresponding to hBN, aQ and α-MoO3. First, each the bGO polariton wavelength and the propagation path strongly disperse with frequency. Second, we exhibit that the uneven nature of optical loss in such crystals offers rise to shear, leading to polariton propagation with tilted wavefronts. Such tilted wavefronts are a direct consequence of the low symmetry of the fabric and are one of the crucial notable and distinctive options of HShPs. New alternatives for polaritonics come up for HShPs stemming instantly from their non-Hermitian and topological nature. But, surprisingly, they are often noticed in low-loss, naturally occurring supplies, with out the necessity for synthetic structuring of a cloth floor17.

To focus on the function of the asymmetry of monoclinic crystals of their polariton response, we examine HShPs with HPs supported by higher-symmetry anisotropic crystals, corresponding to aQ25. On this vein, we examine the crystal construction of monoclinic bGO in Fig 1a with the trigonal crystal of aQ in Fig. 1b, illustrating the low crystal symmetry current in bGO. Basically, the outline of the dielectric response of monoclinic crystals requires inclusion of similar off-diagonal components within the monoclinic airplane throughout the frequency-dependent, complex-valued dielectric tensor.

$$overline{overline{varepsilon (omega )}}=[begin{array}{ccc}{varepsilon }_{xx}(omega ) & {varepsilon }_{xy}(omega ) & 0 {varepsilon }_{xy}(omega ) & {varepsilon }_{yy}(omega ) & 0 0 & 0 & {varepsilon }_{zz}(omega )end{array}]$$

(1)

The coordinate methods used to outline the response of bGO and aQ are sketched in Fig. 1a, b, respectively. To analyse the properties of HShPs in monoclinic supplies, we first rigorously remedy Maxwell’s equations (see Strategies) to calculate the dispersion relation of the polaritonic modes supported by bGO and — for comparability — aQ, every at two distinct frequencies. Initially, we contemplate the lossless case, during which the imaginary a part of every time period within the dielectric tensor is uncared for for each bGO and aQ. The options for the polariton wavevectors in each supplies at two totally different frequencies are offered in Fig. 1c, d. For aQ, we observe two open hyperboloid surfaces — as anticipated for uniaxial hyperbolic supplies — during which a change in frequency ends in a corresponding change in wavevector, whereas preserving the hyperboloid orientation, that’s, the path of polariton propagation (Fig. 1d). In contrast, as we alter the frequency, not solely does the bGO polariton wavevector magnitude change however the path of the hyperboloid additionally rotates throughout the monoclinic airplane, as may be appreciated by analyzing the okayz = 0 projections (Fig. 1c). It is a direct consequence of the non-trivial relative orientation of the phonon resonances supporting the hyperbolic response6, which ends up in polariton bands that disperse in azimuth angle as a perform of frequency. This function represents a signature of the decreased symmetry related to HShPs supported in monoclinic crystals (and can be anticipated in triclinic crystals), in distinction to HPs noticed in higher-symmetry lattices.

Once we additionally account for pure materials losses ensuing from inherent phonon-scattering processes, the polariton propagation in bGO reveals a decreased symmetry compared with hyperbolic polaritons in aQ, even at particular person frequencies, as illustrated in Fig. 1e, f. In these panels, we present the outcomes of full-wave calculations of dipole-launched floor polaritons propagating throughout the floor of a semi-infinite slab of bGO and y-cut aQ, during which — in each circumstances — pure materials losses had been explicitly taken into consideration. For in-plane hyperbolic supplies, these floor waves present a hyperbolic dispersion throughout the floor airplane and are known as hyperbolic floor or hyperbolic Dyakonov polaritons26,27, constituting a subset of HPs supported in these supplies much like volume-confined HPs in skinny movies. For aQ, HPs unfold out alongside one crystal axis of the floor and are symmetric with respect to the crystal axes, as may be confirmed by a Fourier rework of the real-space electrical discipline profile (Fig. 1h). Nevertheless, for bGO (Fig. 1e), we observe that HPs are rotated with respect to the coordinate system of the monoclinic airplane, as anticipated by the isofrequency contours (Fig. 1c). As well as, the wavefronts are tilted with respect to the main propagation path, with no obvious mirror symmetry. This function can be clearly seen by analyzing the Fourier rework of the real-space profile (Fig. 1g), exhibiting a stronger depth alongside one facet of the hyperbola. These observations represent the invention of HShPs in low-symmetry crystals.

To experimentally exhibit the consequences of decreased symmetry in polariton propagation in bGO in distinction to higher-symmetry supplies, we examine the azimuthal dispersion of HShPs in bGO to the one in every of HPs in aQ utilizing an Otto-type prism-coupling geometry28,29 (sketched in Fig. 2a; for particulars, see Strategies). The experimental azimuthal dispersion of HPs on the floor of aQ is proven in Fig. 2b (see additionally Prolonged Knowledge Fig. 3), in glorious settlement with the corresponding simulations (Fig. 2c). The dips within the reflectance spectra present the polariton resonances, that are solely observable alongside particular azimuth angles and are symmetric in regards to the crystal axes, ϕ = 0° (180°) and 90°. In contrast, the experimental azimuthal dispersion of HShPs on monoclinic bGO (Fig. 2nd) reveals no mirror symmetry, once more in glorious settlement with the simulated dispersion (Fig. 2e).

To experimentally entry the in-plane hyperbolic dispersion of the HShP in bGO noticed in Fig. 1e, we mapped out the frequency–momentum dispersion in shut spectral proximity of that mode (680–720 cm−1) at many azimuth angles (see Prolonged Knowledge Fig. 4). The ensuing map of polariton resonance frequencies is proven in Fig. 2f, in glorious settlement with the simulated resonance frequencies proven in Fig. 2g. These knowledge enable extraction of single-frequency in-plane dispersion curves proven in Fig. 2h, i from experiment and simulations, respectively, for a number of chosen frequencies, clearly demonstrating a hyperbolic dispersion, in glorious settlement with Fig. 1e. Notably, the bottom of the hyperbola shifts repeatedly with frequency, as marked by the symmetry axes for every curve in Fig. 2i, which instantly results in an uneven distribution of the group velocity alongside the hyperbolic dispersion curve, as proven in Fig. 2j (see additionally Prolonged Knowledge Fig. 7).

The decreased symmetry noticed within the polaritonic dispersion of bGO (Fig. 2nd) is a direct consequence of the dearth of symmetry in its vibrational construction6. Subsequently, the HShPs aren’t propagating alongside fastened axes however present a steady rotation of the HShP propagation path because the frequency is diverse. To explain the character of this rotation, we diagonalize the actual a part of the permittivity tensor of bGO ({rm{Re}}[overline{overline{varepsilon (omega )}}]) individually at every frequency, by rotating the monoclinic airplane by the frequency-dependent angle.

$$gamma (omega )=frac{1}{2}arctan left(frac{2{rm{Re}}({varepsilon }_{xy})(omega )}{,{rm{Re}}({varepsilon }_{xx})(omega )-{rm{Re}}({varepsilon }_{yy})(omega )}proper)$$

(2)

The dispersion of γ(ω) is proven in Fig. 2e (white strains), illustrating that the main polarizability instructions throughout the monoclinic airplane, denoted as m and n, range broadly throughout the vary. This frequency-dependent coordinate system permits a neater understanding and classification of the polaritonic response (see Strategies and Prolonged Knowledge Fig. 1 for particulars). The rotated coordinate axes are proven in Fig. 1e, g (see additionally Prolonged Knowledge Fig. 2 for additional modes), illustrating their alignment with the hyperbolic dispersion.

Though equation (2) describes the frequency variation of the polariton propagation path, it doesn’t seize the tilted wavefronts noticed in Fig. 1e. It is because, as we select the rotated coordinate system [mnz], we nonetheless retain a purely imaginary off-diagonal permittivity part (see Prolonged Knowledge Fig. 1). These phrases are related to the non-orthogonal relative orientation of the fabric resonances, coupling the 2 crystal axes within the monoclinic airplane. Consequently, even within the rotated coordinate system [mnz], the dielectric tensor has off-diagonal phrases related to shear phenomena.

To selectively analyse the function of those shear phrases, we simulate the polariton propagation within the rotated coordinate system [mnz] at 718 cm−1. Specifically, we embody a scaling issue for the magnitude of the off-diagonal imaginary part, indicated as i × fIm(εmn), with f = 0, 0.5 and 1 (proven in Fig. 3a–c), whereas retaining the diagonal loss phrases. Once we take away the off-diagonal part (f = 0), bGO basically turns into a shear-free biaxial materials, akin to α-MoO3 and much like uniaxial aQ, with polaritons propagating alongside the optical axes (Fig. 3a). Subsequently, whereas polariton propagation in such a fictional type of bGO is anisotropic in particular spectral ranges (much like polaritons in MoO3 (refs. 1,3)), mode propagation with out shear phenomena is symmetric in regards to the (frequency-dependent) main polarizability axes (Fig. 3a). As we steadily enhance the magnitude of the off-diagonal imaginary phrases again to its pure worth (f = 1), the wavefronts change into more and more skewed from the main polarizability axis (Fig. 3b, c). The respective reciprocal house maps in Fig. 3d–f present a robust symmetry breaking within the depth distribution throughout the hyperbolic isofrequency curves. This statement gives additional proof that the propagation of polaritons is non-trivial inside low-symmetry monoclinic — and, by extension, triclinic — methods, and it can’t be anticipated in higher-symmetry supplies during which polariton propagation patterns are symmetric in regards to the principal crystal axes1,3,4.

To attach the decreased symmetry of the floor subset of HShPs noticed right here experimentally (Fig. 2) and thru our simulations (Fig. 3a–c) to the extra basic HShPs within the bulk, we now calculate isofrequency surfaces for polariton modes in bGO explicitly together with loss, to account for the impact of shear phenomena. To this finish, we remedy Maxwell’s equations for actual momentum values, yielding advanced frequency eigenvalues, whose imaginary half accounts for the finite lifetime of the supported modes (see Strategies for particulars). The outcomes of those calculations are proven in Fig. 3g–i. Right here the actual a part of the eigenfrequency is fastened and we discover its imaginary half ωi (Fig. 3h), which is proportional to the modal lifetime, and the corresponding worth of okayz (Fig. 3g) for every pair of okaym and okayn. The calculations are carried out within the rotated coordinate system for each f = 0 and f = 1, exhibiting that each the form of the isofrequency surfaces in addition to their lifetimes change drastically with the inclusion of the off-diagonal imaginary parts. Notably, these calculations show that, at particular person frequencies and within the main polarizability body, mirror symmetry of polariton propagation is misplaced in monoclinic supplies as a direct consequence of shear.

To narrate the isofrequency contours of HShPs to the floor mode dispersions in Fig. 3d–f, we plot the okayz = 0 resolution in Fig. 3i, with the color scale indicating the relative loss ωi of the mode. Two vital observations may be made: first, the mirror symmetry of the isofrequency curves is damaged for f > 0 and it requires higher-order phrases to account for the uneven form; second, the mode losses are redistributed asymmetrically, with losses reducing in a single arm of the hyperbolae however rising on the opposite arm. We notice that, additionally within the experimental knowledge (Prolonged Knowledge Fig. 4), we observe a sign of uneven distribution of polariton high quality components alongside the hyperbolic dispersion curves (see Prolonged Knowledge Fig. 5).

These observations naturally hyperlink HShPs in monoclinic crystals to the wealthy, rising space of non-Hermitian and topological photonics. Though loss in orthogonal methods alone can have already got fascinating penalties for polariton propagation30, the off-diagonal shear phrases highlighted right here can present new alternatives for non-Hermitian photonics and for manipulation of topological polaritons in low-symmetry supplies. As an example, we foresee uneven topological transitions skilled by HShPs, generalizing earlier ends in orthorhombic methods2 by exploiting the distinctive non-Hermitian options rising in low-symmetry supplies. As well as, current research recommend the connection between Dyakonov floor waves and floor states rising from one-dimensional band degeneracy (nodal strains) of topological nature of high-symmetry metacrystals31. We anticipate that HShPs might generalize these alternatives to uneven topological bands during which non-Hermiticity within the pure supplies performs a dominant function.

Right here now we have demonstrated that low-symmetry crystals can help a brand new class of hyperbolic polariton modes with damaged symmetry attributable to shear phenomena, which we confer with as HShPs. We introduce bGO as an exemplary materials to allow the statement of those phenomena and experimentally exhibit the symmetry-broken dispersion of the supported floor waves. The non-diagonalizable dielectric permittivity performs a key function within the distinctive properties of low-symmetry crystals, together with monoclinic and triclinic lattices. Our findings are generalizable to engineered photonic methods with not less than two non-orthogonal oscillators, together with new metasurface designs capturing these physics. Past the outcomes offered right here for intrinsic, compensation-doped bGO, the presence of free cost carriers in bGO32 might enable for strategies for direct steering of the HShP propagation path (see Prolonged Knowledge Fig. 6). Lastly, exfoliation of skinny flakes of single-crystal bGO has additionally been not too long ago reported33, which is able to enable to utilize volume-confined HShPs in such bGO skinny movies or — doubtlessly — even in monolayers34. We anticipate that HShPs might have vital implications within the manipulation of part and directional power switch, together with radiative warmth transport35, ultra-fast uneven thermal dissipation within the close to discipline35 and gate-tunability for on-chip all-optical circuitry36. Past advances in nanophotonics, infrared polariton propagation has been demonstrated as a method for quantifying crystal pressure37, polytypes38, variations in free-carrier density, in addition to phononic and digital properties round defects39, thereby additionally promising a brand new metrology instrument for characterizing low-symmetry ultra-wide-bandgap semiconductors. We spotlight that our outcomes are relevant to any materials with non-orthogonal optically energetic transitions and should subsequently be prolonged to different optical phenomena, corresponding to excitons in triclinic ReSe2 (ref. 40).


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# Trying to prove the earth is flat

Geometry Level 1

In 1838, Samuel Birley Rowbotham conducted observations leading him to conclude the earth was flat. His idea was the following: If light goes straight and I'm standing at $$A$$ with my friend, there must be a point $$B$$ where I can't see him.

Assuming light does not bend because of atmospheric refraction, which of these is closest to the minimum arc length of $$\stackrel\frown{AB}$$ that would prevent Samuel from seeing his friend? Assume they are both $$1.8$$ meters tall and the radius of the earth is $$6,371,009\text{ m}.$$

Note: Though the earth is (of course) not flat, Samuel was able to see his friend at a farther distance than he was supposed to. This is due to atmospheric refraction, i.e. the non-homogeneous density of air creating a slight deviation of light. Without knowing this effect, a newspaper editor performed in 1896 the same experiment and concluded the earth was concavely curved.

×

format: plain


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# 应用经济|ECON10162T Applied Economics代写 manchester代写

For completeness, we define an approximate integral that is also a linear spline, with a breakpoint sequence $z_{i+1}=\left(\nu_{i}+\nu_{i+1}\right) / 2$ for $i=2, \ldots, n$ and with additional break points defined by extrapolating beyond the original sequence: $z_{1}=\left(3 \nu_{1}-\nu_{2}\right) / 2$ and $z_{n+1}=\left(3 \nu_{n}-\nu_{n-1}\right) / 2$. The approximation to the integral,
$$F(x)=\int_{\nu_{1}}^{x} f(x) d x$$
at the new breakpoints is
$$F\left(z_{i}\right)=F\left(z_{i-1}\right)+\left(z_{i}-z_{i-1}\right) f\left(\nu_{i-1}\right),$$
where
$$F\left(z_{1}\right)=\frac{1}{2}\left(\nu_{1}-\nu_{2}\right) f\left(\nu_{1}\right)$$

(this ensures the normalization that $\left.F\left(\nu_{1}\right)=0\right) .{ }^{3}$ This definition produces an approximation to the integral at the original breakpoints that is equal to the approximation obt ained by applying the trapezoid rule :
$$\int_{\nu_{i}}^{\nu_{i+1}} f(x) d x \approx \frac{1}{2}\left(\nu_{i+1}-\nu_{i}\right)\left(f\left(\nu_{i+1}\right)+f\left(\nu_{i}\right)\right) .$$
(we leave the verification of this assertion as exercise for the reader).

myassignments-help数学代考价格说明

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Math作业代写、数学代写常见问题

myassignments-help擅长领域包含但不是全部:


text:
Question
In Example 25.7, the magnification of a book held 7.50 cm from a 10.0 cm focal length lens was found to be 3.00. (a) Find the magnification for the book when it is held 8.50 cm from the magnifier. (b) Do the same for when it is held 9.50 cm from the magnifier. (c) Comment on the trend in m as the object distance increases as in these two calculations.
1. $6.67$
2. $20.0$
3. The magnification increases as the object distance gets closer to the focal length.
Solution Video


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# Non-parametric models as solutions to Partial Differential Equations

In the realm of scientific computing, there are a plethora of techniques developed to solve Partial Differential Equations (PDEs). Many of the popular methods are variants of common techniques such as Finite Differences, Finite Volume, Finite Elements and the theory behind where they work or not is pretty extensive. Each method has pretty clear pros and cons which might influence its use over another for any given research problem.

I was just now contemplating the idea of non-parametrically modeling a solution for some system of PDEs, say some hyperbolic system of PDEs to keep things interesting. The idea could be to essentially having some cloud of spacetime points with some solution coefficients that represent the approximate solution to the PDEs. Then a separate cloud of spacetime points could be used as collocation points used to "train" the model. Whenever we wish to evaluate our model and compute gradients at some spacetime location, we could perhaps do some least square fit based on data that is "close" to that spacetime location. The problem of actually solving the PDE might then be viewed as an optimization of the solution coefficients at the model's spacetime points such that the residual of the PDEs and boundary conditions are minimized.

In the context of hyperbolic PDEs, do you see such an approach as described working out? Do you see problems arising due to characteristics of the system of hyperbolic PDEs that might influence how one should evaluate the non-parametric model in spacetime? Are there any papers you know of covering similar concepts?

This idea came about thinking about some paper I read a while back about someone using a similar collocation approach but with a neural network as the approximate PDE solution instead. It seems to me a non-parametric model would be easier to optimize and would still be capable of capturing potential nonlinearities, at the expense of having to keep track of solution data at a likely good number of spacetime locations.

I haven't thought too hard about this so perhaps there is something obvious I am missing but any thoughts, comments, or references about similar material would be appreciated!


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FreeSASA  2.0.3 Open source SASA calculations View on GitHub
Selection syntax

FreeSASA uses a subset of the Pymol select commands to give users an easy way of summing up the SASA of groups of atoms. This is done by the function freesasa_selection_new() in the C API, freesasa.selectArea() in the Python interface and the option --select for the command line tool. All commands are case insensitive. A basic selection has a selection name, a property selector and a list of arguments

<selection-name>, <selector> <list>

For example

aromatic, resn phe+tyr+trp+his+pro

Several selectors can be joined using boolean logic and parentheses,

<selection-name>, (<s1> <l1>) and not (<s2> <l2> or <s3> <l3>)

where s1, s2 and s3 are selectors and l1, l2 and l3 are lists. The operator and has precedence over or, so the second parentheses is necessary but not the first, in the example above. The selection name can include letters, numbers and underscores. The name can't be longer than FREESASA_MAX_SELECTION_NAME characters.

The following property selectors are supported

• resn Residue names like "ala", "arg", "du", etc
• resi Residue index (positive or negative integers)
• chain Chain labels (single characters)
• name Atom names, such as "ca", "c", "oxt", etc
• symbol Element symbols, such as "C", "O", "Se", "Fe", etc.

A list of residues can be selected using

resn ala+val+leu+ile+met

and similarly for the other four selectors. In addition resi and chain support ranges

resi 1-10             (residues 1 to 10)
resi -10              (residues indices < 10)
resi 10-              (residues indices > 10)
resi 1-10+20-30+35-   (residues 1 to 10, 20 to 30 and above 35)
resi \-20-\-15+\-10-5 (residues -20 to -15 and -10 to 5)
chain A+C-E           (chains A and C to E, no open intervals allowed here)

Combining ranges with plus signs, as in the three last lines, is not allowed in Pymol but supported by FreeSASA.

If a selection list contains elements not found in the molecule that is analyzed, a warning is printed and that part of the list does not contribute to the selection. Not finding a list element can be because it specifies a residue that does not exist in the particular molecule, or because of typos. The selector does not keep a list of valid elements, residue names, etc.


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# Homework Help: Numerical analysis confusion

1. Nov 9, 2008

### JamesF

I'm having trouble with a sample exam question. I don't really understand the question, don't know what section of the book it relates to, and don't have any idea on how to solve it. I might be in trouble :)

Can anyone provide any suggestions or guidance on how I might go about solving this problem? Again, I'm not even really sure what's being asked.

1. The problem statement, all variables and given/known data
Suppose that $$L = \lim_{h \rightarrow 0} f(h)$$
and $$L -f(h) = c_6 h^6 + c_9 h^9 + \cdots$$

Find a combination of f(h) and f(h/2) that is a much better estimate of L

2. Nov 9, 2008

### gabbagabbahey

Well, if $L -f(h) = c_6 h^6 + c_9 h^9 + \ldots$ , then surely you can say that $f(h) =L-( c_6 h^6 + c_9 h^9 + \ldots)$ right?....What does that make $f(h/2)$?....Basically you want to use this to find some linear combination of f(h) and f(h/2) that is closer to L than f(h) is.

3. Nov 9, 2008

### JamesF

if $$f(h) = L - c_6 h^6 - c_9 h^9 - \cdots$$
and $$f(\frac{h}{2}) = L - \frac{c_6 h^6}{64} - \frac{c_9 h^9}{512}$$

what I'm trying to find is $$a_0, a_1$$ such that $$a_0 f(h) + a_1 f(\frac{h}{2}) \approx L$$. Is that right?

I'm still somewhat confused as to what this problem relates to, ie what methods have we studied would this problem apply to

4. Nov 9, 2008

### gabbagabbahey

You're not even looking for something this restrictive, you just want $a_0 f(h) + a_1 f(\frac{h}{2})$
to be closer to L than f(h) was....so as long as $a_0 f(h) + a_1 f(\frac{h}{2})-L< c_6 h^6 +c_9 h^9 + \cdots$, then it is mission accomplished. ...what happens if you take $a_0=1$ and $a_1=-2^6$?

I'm not sure what methods you've studied :tongue:, but I think this is basically the beginning of an algorithm to numerically approximate the limit L.


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# 1 R Basics

“8/27/2020 | Last Compiled: 2022-04-26”

## 1.1 Reading and walkthrough video

Vokey & Allen,11 Chapter 1, available online here: http://people.uleth.ca/~vokey/pdf/thinking.pdf.

## 1.2 Overview

The labs for this course are designed to give students exposure to the free and open-source statistical programming language R. The assumption is that students may have zero prior experience with scripting, coding, or computer programming. Over the semester we will use R as a tool for data-analysis, and as a tool to sharpen understanding of statistical concepts.

Before starting this lab follow the getting started instructions to install R, R-studio, create a Github.com account, download github Desktop; and, make sure that you test the github pipeline. Throughout the semester you will be posting your assignments on Github.com, and submitting links to your repositories on blackboard.

## 1.3 Problem 1: Summing 1 to 100

Vokey & Allen12 tell the story of a teacher giving challenging students to add up all of the numbers from 1 to 100. This was supposed to stump one of the students, a young Gauss. Apparently, it did not, and Gauss very quickly wrote down the sum 5050.

Gauss didn’t have R, but we do, so we can solve the problem very quickly too:

1. Use R to find the sum of the sequence of numbers from 1 to 100:
sum(1:100)
#> [1] 5050

Using sum(), it is simple and fast to solve Gauss’s problem in R. You could easily find other sums by changing the 1 or the 100

sum(5:10)
#> [1] 45
sum(100:200)
#> [1] 15150

There are many details going on behind the scenes in R that allow the sum() function do its work. One detail is how to get R to create a sequence of numbers, another is how to take an action like adding up all the numbers in the sequence. Two major concepts here are variables for storing information (like sequences of numbers), and functions that take actions to transform an input (a number sequence) to a desired output (a sum of the number sequence).

## 1.4 R Basics Background

### 1.4.1 Creating sequences of numbers in R

There are multiple ways to create number sequences in R. A sequence of integers can be generated with x:y, where x is the starting value, and y is the ending value.

1:5
#> [1] 1 2 3 4 5
1:10
#>  [1]  1  2  3  4  5  6  7  8  9 10
5:-5
#>  [1]  5  4  3  2  1  0 -1 -2 -3 -4 -5

### 1.4.2 seq()

Sequences incremented by any constant value can be created using the seq() function. Look up “help” for any R function by typing ?name_of_function into the console

?seq

R comes pre-packaged with many functions like seq(), and you can write your own functions, and download libraries of functions that other people have written to extend the base functionality of R. We will look more closely at functions throughout the semester.

An R function usually has three components. It will 1) receive some kind of input, 2) “do something”, and 3) return some kind of output. In R, you use functions by writing the name of the function and parentheses name(). If the function takes inputs, then you define the inputs inside the parentheses name(x=1). Functions can have multiple inputs, that are separated by commas.

Let’s take a look at using the seq() function to generate sequences of numbers.

#lines beginning with # are comments and not run

#seq(from, to)
seq(from = 1, to = 5)
#> [1] 1 2 3 4 5
seq(1, 5)
#> [1] 1 2 3 4 5

#seq(from, to, by= )
seq(from = 1, to = 5, by = 2)
#> [1] 1 3 5
seq(1, 5, .5)
#> [1] 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
seq(1, 10, 2)
#> [1] 1 3 5 7 9

#seq(from, to, length.out= )
seq(from = 1, to = 2, length.out =5)
#> [1] 1.00 1.25 1.50 1.75 2.00

seq(5)
#> [1] 1 2 3 4 5

## 1.5 Problem 2: Summing any constant series

Now that we have seen the sum() and seq() functions, you should be able to use them to find the sum of any constant series.

For example, find the sum of the series 100 to 200, going up by five.

sum( seq(100,200,5) )
#> [1] 3150

It is also possible to write the analytic formula in R, and compare results, remember that:

$$X_1 + X_2 + \ldots + X_n = (\frac{X_n-X_1}{c}+1)(\frac{X_1+X_n}{2})$$

Where $$X_1$$ is the starting value, $$X_n$$ is the ending value and $$c$$ is the constant step value.

Here is an example writing this formula in R. We create variables with the names X1, Xn, and step, and assign (<-) them any value we want. Then we compute the formula. This should give the same value as the previous example, because it is the same sequence.

X1 <- 100
Xn <- 200
step <- 5

(((Xn - X1)/step) + 1) * ((X1 + Xn)/2)
#> [1] 3150

( ( (Xn-X1)/step ) + 1 ) * ( (X1+Xn)/2 )
#> [1] 3150

## 1.6 Vectors

### 1.6.1 The Gaussian trick

Remember Gauss added up the numbers from 1 to 100 by imagining two number lines:

1:100
#>   [1]   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18
#>  [19]  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36
#>  [37]  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54
#>  [55]  55  56  57  58  59  60  61  62  63  64  65  66  67  68  69  70  71  72
#>  [73]  73  74  75  76  77  78  79  80  81  82  83  84  85  86  87  88  89  90
#>  [91]  91  92  93  94  95  96  97  98  99 100
100:1
#>   [1] 100  99  98  97  96  95  94  93  92  91  90  89  88  87  86  85  84  83
#>  [19]  82  81  80  79  78  77  76  75  74  73  72  71  70  69  68  67  66  65
#>  [37]  64  63  62  61  60  59  58  57  56  55  54  53  52  51  50  49  48  47
#>  [55]  46  45  44  43  42  41  40  39  38  37  36  35  34  33  32  31  30  29
#>  [73]  28  27  26  25  24  23  22  21  20  19  18  17  16  15  14  13  12  11
#>  [91]  10   9   8   7   6   5   4   3   2   1

He noted that the sum of the columns always added up to 101 (e.g., 1+100 = 101, 2+99 = 101, etc.). It is possible to demonstrate this in R because we can directly add both number lines:

1:100 + 100:1
#>   [1] 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101
#>  [19] 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101
#>  [37] 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101
#>  [55] 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101
#>  [73] 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101
#>  [91] 101 101 101 101 101 101 101 101 101 101

Each time we created a sequence of numbers (e.g., from 1 to 100) we were creating an object called a numeric vector in R. Vectors can store multiple numbers.

Vectors are a kind of variable in R. Vectors can have names, they can be saved, and they can be manipulated. Let’s take a quick look at vectors:

a is the name of the new vector. <- is called the assignment operator. 1:5 creates a vector of length 5, containing the sequence of numbers 1 to 5. In plain language, we “assign” the object on the right (the 1:5), to the “name” a written on the left side of the <-. In other words, we put the numbers 1 to 5 into something called a.

#creates a variable a
a <- 1:5

When you create a variable like the a above, its name will appear in the global environment (top right environment tab). When the variable is created for the first time (by executing the code in the console), it becomes registered or saved in your computer’s memory for the current R session.

If you want to double-check that a variable exists in memory, enter its name into the console, and press enter:

a
#> [1] 1 2 3 4 5

You can clear (remove) a variable using rm(). And, you can clear the entire global environment using Session > Clear Workspace... from the RStudio menu.

rm(a)

You can also check the class of a variable in R using the class() function. Let’s create two vectors and check their classes. The a vector contains an integer class, and b vector contains a numeric class because there are decimal values.

a <- 1:10
class(a)
#> [1] "integer"

b <- seq(1,2,.25)
class(b)
#> [1] "numeric"

### 1.6.2 c()

It is also possible to create vectors that contain characters, rather than numbers. To illustrate an example of a character vector, we introduce one more basic R function c(), short for “combine”.

?c

Vectors are like trains, they have slots (train cars) that can contain things, like passengers, or oil or coal. Trains can have any number of train cars, just like a vector can have any length (number of slots).

Importantly, train cars need to be connected together to form a whole train. Similarly, in R, to make a vector you need to concatenate or connect individual slots of a vector together. This is what the c() function is all about. It combines individual units together to form a vector.

To use c(), insert individual items separated by commas between the parentheses. To insert characters, wrap each character (or string of characters) between quotations.

letters <- c("a","b","c")
numbers <- c(1,2,3)
numbers_as_chars <- c("1","2","3")
words <- c("this","is","a","vector","of","strings")

The elements between the commas are like individual train cars, and the c() function connects the elements together into a single entity comprised of multiple units, like a train, called a vector.

### 1.6.3 length()

Just like a train has a specific number of cars, a vector has a specific number of slots. This is called the length of the vector. R has a length() function that reports the length of a vector.

length(letters)
#> [1] 3
length(words)
#> [1] 6

### 1.6.4 More on combining

The c() function is flexible because it can combine all sorts of elements, including vectors or variables.

some_numbers <- c(1, 2, 3, 1:5)

some_characters <- c(letters, words)

Remember that vectors have different classes depending on what kind of elements are inside the vector. This is important, because R requires that all of the elements in a vector have the same class.

class(c(1,2,3))
#> [1] "numeric"
class(c("A","B","C"))
#> [1] "character"
class(c(TRUE,FALSE,TRUE))
#> [1] "logical"

It is possible to combine vectors that start with different classes, but R may give an error, or it will convert one class into another. To go back to the train car analogy, R doesn’t like trains that have different kinds of cars…it wants the whole train to be passenger cars, or the whole train to be oil tankers.

# the numbers are converted to characters
c(1,2,3,"a","b","c")
#> [1] "1" "2" "3" "a" "b" "c"

### 1.6.5 Indexing a vector

Vector indexing iallows elements to be inspected and changed. This is like a train, where you could inspect the contents of cars 3 to 5, or unload car 7 and put something else in it.

Square bracket [] notation indexes a vector, with variable_name[x]; where, x is another vector specifying the indexed slots.

a <- c(1,6,3,2,8,9) # make a vector

The following examples use [] to index specific elements of the vector a. The outcome is that the elements specified inside the square brackets are printed to the console.

a[1] # first element
#> [1] 1
a[2] # second element
#> [1] 6
a[1:3] # 1st to 3rd elements
#> [1] 1 6 3
a[c(1,5)] # elements 1 and 5
#> [1] 1 8

It is also possible to assign new values to specific elements of a vector:

# assign 100 to the first slot of a
a[1] <- 100
a
#> [1] 100   6   3   2   8   9

# assign the value 1 to slots 5 to 6 of a
a[5:6] <- 1
a
#> [1] 100   6   3   2   1   1

### 1.6.6 Growing a vector

In upcoming labs we will use vectors to store information. Sometimes you know in advance how many slots you need for your vector, and other times you might not know, and instead you could decide to build your vector one slot at a time.

Below I begin with an empty (NULL) vector. I use the c() command, but I don’t combine anything together. This like starting a train with no cars at all.

a <- c()
a
#> NULL

We can add a slot to this vector by combining a new element to the existing variable. Here we combine a with 1, and then assign the result back into a, replacing it’s original NULL value.

a <- c(a,1)
a
#> [1] 1

If we keep doing this, we will keep adding 1s, to the end of a.

a <- c(a,1)
a
#> [1] 1 1
a <- c(a,1) # c(1,1,1)
a
#> [1] 1 1 1
a <- c(a,1)
a
#> [1] 1 1 1 1
a <- c(a,1)
a
#> [1] 1 1 1 1 1

Consider this alternative method for growing a vector:

a <- c()
a
#> NULL

a[1] <- 1
a
#> [1] 1

a[2] <- 1
a
#> [1] 1 1

a[3] <- 1
a
#> [1] 1 1 1

a[10] <- 1
a
#>  [1]  1  1  1 NA NA NA NA NA NA  1

## 1.7 Problem 3: Writing a sum function in R

We have already used R to solve the problems in Chapter in 1 of Vokey & Allen (2018). We can create sequences of numbers, we can create custom vectors of numbers, and we can use the sum() function to find the sum. However, we haven’t discussed how the sum() function actually works. How does R know how to find the sum?

This is an example of writing your own sum function in R. This example involves understanding for loops and writing custom functions, both of which are explained in the next sections.

my_sum <- function(x) {
sum <- 0
for(i in x) sum <- sum + i
return(sum)
}

my_sum(1:100)
#> [1] 5050

## 1.8 Algorithms

To understand how functions like sum() work in R, we need to understand the more general concept of an algorithm. I’ll define algorithm as a recipe, or series of steps/actions that result in a particular outcome. With a scripting language like R, it is possible to define algorithms that are infallible. That is, given an input, they always apply all of the steps and arrive at the same answer demanded by the algorithm.

When you sum up a series of numbers in your head, say the numbers 1 to 5, you are probably applying a simple algorithm that could be describe like this:

1. Take the first number and add it to the second (1+2 = 3)
2. Take the sum (3) and add it to the next number (3+3 = 6)
3. Repeat step 2 until there are no more numbers in the series
• 6+4 = 10
• 10+5 = 15
1. report the final sum (15)
sum(1:5)
#> [1] 15

Consider how this could look in R. This is one example of producing an algorithm in R. Everytime we run the below script, we always end up with a sum of 15, because the answer is demanded by the series of steps that I wrote down.

a <- 1:5
a
#> [1] 1 2 3 4 5

the_sum <- a[1]+a[2]
the_sum
#> [1] 3
the_sum <- the_sum + a[3]
the_sum
#> [1] 6
the_sum <- the_sum + a[4]
the_sum
#> [1] 10
the_sum <- the_sum + a[5]
the_sum
#> [1] 15

## 1.9 Loops

The above example shows an algorithm written by hand in R. It would be tiresome to write out a sum for a sequence with many more numbers. Fortunately, there are ways to automate repetitive processes in R. A common method for repeating commands in R is to use a for loop.

Check R help for on Control Flow ?Control.

for(){} for(loop control){do something each iteration}

The basic syntax for loops is as follows:

for(iterator in vector){
#do something
}

Loop control is defined in between the parentheses. The name of the iterator is placed on the left of in(can be assigned any name you want, does not need to be declared in advance). During the execution of the loop, the iterator takes on the values inside the vector which is placed on the right side of in. Specifically, the following is happening.

Loop steps: 1. iterator <- vector[1] 2. iterator <- vector[2] 3. iterator <- vector[3] 4. etc.

The loop will automatically stop once it reaches the last item in the vector. The loop can be stopped before that using the break command.

# Make a loop do something 5 times
# i is the iterator
# 1:5 creates a vector with 5 numbers in it, 1, 2, 3, 4, 5
# the loop will run 5 times, because there are five things to assign to i
for(i in 1:5){
print("hello")
}
#> [1] "hello"
#> [1] "hello"
#> [1] "hello"
#> [1] "hello"
#> [1] "hello"
# show the value of i each step of the loop
for(i in 1:5){
print(i)
}
#> [1] 1
#> [1] 2
#> [1] 3
#> [1] 4
#> [1] 5
# define the vector to loop over in advance
my_sequence <- 1:5
for(i in my_sequence){
print(i)
}
#> [1] 1
#> [1] 2
#> [1] 3
#> [1] 4
#> [1] 5
# Reminder that i becomes the next value in the vector
# your vector can have any order
my_sequence <- c(1,5,2,3,4)
for(i in my_sequence){
print(i)
}
#> [1] 1
#> [1] 5
#> [1] 2
#> [1] 3
#> [1] 4
# index vector does not need to be numbers
my_things <- c("A","B","C","D")
for(i in my_things){
print(i)
}
#> [1] "A"
#> [1] "B"
#> [1] "C"
#> [1] "D"

### 1.9.1 Breaking a loop

break stops a loop. Used with logical statements to define the conditions necessary to cause the break.

for(i in 1:10){
if(i <= 5){
print(i)
} else {
break
}
}
#> [1] 1
#> [1] 2
#> [1] 3
#> [1] 4
#> [1] 5

### 1.9.2 While loops

While loops run until a logical condition is met. Here there is no iterator, just a logic statement that needs to be met.

This one prints i while i is less than 6. As soon as i becomes “not less than 6”, then the loop stops. Critically, inside the loop, the value of i increases each iteration.

i <- 1 # create an variable
while (i < 6) {
print(i)
i = i+1 #add one each step of the loop
}
#> [1] 1
#> [1] 2
#> [1] 3
#> [1] 4
#> [1] 5

### 1.9.3 Repeat loops

Similar to while, but let’s do things until a condition is met.

i<-0
repeat{
i<-i+1
print(i)
if(i==5){
break
}
}
#> [1] 1
#> [1] 2
#> [1] 3
#> [1] 4
#> [1] 5

### 1.9.4 Examples

Braces are not needed on one line

for(i in 1:5) print(i)
#> [1] 1
#> [1] 2
#> [1] 3
#> [1] 4
#> [1] 5

Using the value of the iterator to assign in values systematically to another variable.

# put 1 into the first five positions of x
x <- c() # create empty vector
for(i in 1:5){
x[i] <- 1  # assign 1 to the ith slot in x
}
x
#> [1] 1 1 1 1 1

# put the numbers 1-5 in the first 5 positions of x
x <-c()
for(i in 1:5){
x[i] <- i
}
x
#> [1] 1 2 3 4 5

Using a loop to add up numbers in a vector.

a <- c(10,20,30,40,50) #some numbers

the_sum <- 0 # initialize a variable that will keep track of the sum

for (i in a) {
the_sum <- the_sum + i
}

the_sum
#> [1] 150
sum(a) #check against the sum function
#> [1] 150

The above example shows the use of a for loop to compute the sum of the numbers in the vector a. This is an example of using a loop in an algorithm. At the end we checked out custom script against the sum() function, and we see that we arrived at the same answer.

A final task for this lab is take a look at R functions, and learn how to write our own basic functions like sum().

## 1.10 Functions

Functions are re-useable algorithms. For example, rather re-writing all of the code necessary to compute a sum everytime we want to find a sum, we instead store the necessary code inside a named variable called sum(), then we “call” the function by writing its name and providing inputs.

It is fairly straightforward to write your own custom functions in R, and learning how to write your own functions is an excellent method to improve your understanding of R fundamentals.

### 1.10.1 function syntax

This is the general syntax for writing functions:

function_name <- function(input1,input2){
#code here
return(something)
}

### 1.10.2 Example functions

This function has no input between the (). Whenever you run this function, it will simply return whatever is placed inside the return statement.

# define the function
print_hello_world <- function(){
return(print("hello world"))
}

# use the function
print_hello_world()
#> [1] "hello world"

This function simply takes an input, and then returns the input without modifying it.

return_input <- function(input){
return(input)
}

# the variable input is assigned a 1
# then we return(input), which will result in a 1
# because the function internally assigns 1 to the input
return_input(1)
#> [1] 1
a <- "something"
return_input(a)
#> [1] "something"

This function takes an input, then creates an internal variable called temp and assigns input+1. Then the contents of temp is returned. Note there, is no checking of the input, so it will return an erro if you input a character (can’t add one to a character in R)

add_one <- function(input){
temp <- input+1
return(temp)
}
#> [1] 2
add_one("a") #this will cause an error
#> Error in input + 1: non-numeric argument to binary operator

This function adds some input checking. We only add one if the input is a numeric type. Otherwise, we use stop() to return an error message to the console

add_one <- function(input){
if(class(input) == "numeric"){
temp <- input+1
return(temp)
} else {
return(stop("input must be numeric"))
}
}
#> [1] 2
#> Error in add_one("a"): input must be numeric

A function with three inputs

add_multiply <- function(input, x_plus, x_times){
temp <- (input+x_plus)*x_times
return(temp)
}

# input is 1
# x_plus <- 2
# x_times <- 3
# will return (1+2)*3 = 9
#> [1] 9

## 1.11 Lab 1 Generalization Assignment

Follow the instructions below to complete the assignment for lab 1 and hand it in by the due date on blackboard. For the first lab I have taken the extra step of pretending I was student in this course, and completed the first lab myself. This next video shows how I would complete the lab if I was a student. It is important that you try to solve the problems on your own, but please use this video as a resource to help you if you get stuck.

### 1.11.1 Instructions

In general, labs will present a discussion of problems and issues with example code like above, and then students will be tasked with completing generalization assignments, showing that they can work with the concepts and tools independently.

Your assignment instructions are the following:

1. Make a new R project (initialized as a git repository) called “StatsLab1’.
2. Create a new R Markdown document called “Lab1.Rmd”
4. Use Lab1.Rmd to show your work attempting to solve the following generalization problems. Commit your work regularly so that it appears on your Github repository.
6. There are six problems to solve, each worth 1 point.

Refer to the getting started videos for examples of creating a new R project and uploading it to Github. If you have problems with these steps and they have not been resolved in our first class, then please email me about them, or create an issue on the course github page https://github.com/CrumpLab/psyc7709Lab/issues

### 1.11.2 Problems

1. Compute the sum of the sequence 100 to 1000, going up by a constant value of 100 (100,200,300,400,500,600,700,800,900,1000).
2. Compute the sum of these numbers (1,3,2,4,3,5,4,3,4,5,6,5,6,7,6,5,6,5,4,3,4,5)
3. Write a custom sequence generator function using a for loop that generates a sequence from a starting integer value to an ending integer value in steps of 1. Demonstrate that it can produce the sequence 1 to 10.
4. Write a custom function to implement the following general equation to find the sum of any constant series:

$$X_1 + X_2 + \ldots + X_n = (\frac{X_n-X_1}{c}+1)(\frac{X_1+X_n}{2})$$

Demonstrate that your function correctly produces the sum for the series below:

seq(10,100,10)
#>  [1]  10  20  30  40  50  60  70  80  90 100
1. Write a custom function that generates a constant series between any start and end values, with any constant, and finds the sum. Have your function output both the sequence and the sum. For this problem, feel free to use the existing seq() and sum() functions in your custom function. Demonstrate the function correctly prints out the above sequence (10 to 100 in steps of 10), and its sum.
2. Use the sum() and the length() functions to calculate the mean (average) of the numbers x = c(1,2,3,4,5).


text:
# Revision history [back]

### Some strings don't appear in rosetta

The jinja2_makemessages command isn't picking up some strings -- they don't even appear in the .po file.

For example "Sort by »" on the main /questions/ page is picked up correctly, but all the "by activity", "by date" texts next to it are not.

I looked in the source at skins/default/templates/main_page/tab_bar.html:

    {{macros.reversible_sort_button(
button_sort_criterium = 'activity',
label = gettext('by activity'),
asc_tooltip = gettext('click to see the least recently updated questions'),
desc_tooltip = gettext('click to see the most recently updated questions'),
current_sort_method = sort
)
}}

Why is the gettext directive ignored? And how to fix this?


text:
# GE Electric Motor - Power hookup for use in DIY projects

Thread Starter

#### Aues

Joined Jun 2, 2021
14
I have a GE electric motor that came out of a washing machine that I would like to use to power DIY tools, like a sander. It is a 3 phase motor, but runs from a standard plug, not a 240v. The label shows "120v 60Hz 6.1 LRA 3 Phase". I am a total rookie when it comes to this kind of electronics, so if my questions are stupid, please bear with me!
Is there a way to connect this to a power cord with an on/off switch so the motor will run at full speed all the time, or maybe with the speed controlled by a switch of some kind? From what I have been reading from Google searches, generally 3 phase motors need a capacitor to start them, but this one has a PCB board attached to it with 2 large capacitors on it. (See the Pictures). So I am going to assume that is already taken care of.
I have labeled the photos as best as I could with what connections I could figure out. There are two connections that go from the board on the motor to the windings, and there doesn't appear to be brushes on this motor.

Thanks in advance for any help you can give me with this

#### LowQCab

Joined Nov 6, 2012
2,499
"" Is there a way to connect this to a power cord with an on/off switch so the motor will run at full speed all the time, or maybe with the speed controlled by a switch of some kind? ""
No,
and 3-Phase Motors don't need Capacitors, but a
"Variable-Frequency-Drive" Inverter/Controller does.

Trying to make a 3-Phase-Motor, run on Single-Phase Power,
by using various Capacitor Schemes is always very tricky, and disappointing,
and Single-Phase to 3-Phase VFD-Controllers are quite expensive.
.
.
.

#### MaxHeadRoom

Joined Jul 18, 2013
25,973
If you can decifer the present hook up of the VFD to motor, it may be possible to use both if they have no issues/faults etc.
The means of control signal would be needed, is there a print of the machine inside the cover etc?
Or if you know the model it came from you may be lucky in finding a print via Google.

#### shortbus

Joined Sep 30, 2009
9,631
Is what the TS has a real 3PH motor or is it a BLDC motor with 3 phases? Many of the newer appliances are now using that type of motor. I can't believe this is really a 3 PH motor running on 120VAC.

#### MaxHeadRoom

Joined Jul 18, 2013
25,973
There where some 3ph induction motors that were made expressly for washing machine use.
If the motor leads are shorted and the rotor spun, if physical resistance is felt, it will show right away if PM, which I doubt.
.

Thread Starter

#### Aues

Joined Jun 2, 2021
14
"" Is there a way to connect this to a power cord with an on/off switch so the motor will run at full speed all the time, or maybe with the speed controlled by a switch of some kind? ""
No,
and 3-Phase Motors don't need Capacitors, but a
"Variable-Frequency-Drive" Inverter/Controller does.

Trying to make a 3-Phase-Motor, run on Single-Phase Power,
by using various Capacitor Schemes is always very tricky, and disappointing,
and Single-Phase to 3-Phase VFD-Controllers are quite expensive.
.
.
.
That is interesting, as this machine does indeed run when plugged into a standard electrical socket, 3 prong, 110v US standard plug in. Does this mean that there is an expensive VFD Controller built into the control boards of the washing machine then? Or is this motor perhaps a different type than what you think it is?

Thread Starter

#### Aues

Joined Jun 2, 2021
14
If you can decifer the present hook up of the VFD to motor, it may be possible to use both if they have no issues/faults etc.
The means of control signal would be needed, is there a print of the machine inside the cover etc?
Or if you know the model it came from you may be lucky in finding a print via Google.
Not sure what you mean when you talk about the present hook up of the VFD to motor...do you mean there is already a VFD built into the washing machine somewhere? There are 3 pcb boards on the machine, which were the ones that were attached to the plastic control panel. (where the knobs you use to set the cycles/spin/etc... and the on button are). I doesn't look like there is anything special about these boards, no large components or anything, but they do control the entire process of the machine....and yes, there was a print of a sort-of schematic on the inside of the control panel. It is labeled Control Circuit Board, and shows a very limited schematic of the panel. I will post a picture of that in a few minutes.

Thread Starter

#### Aues

Joined Jun 2, 2021
14
Is what the TS has a real 3PH motor or is it a BLDC motor with 3 phases? Many of the newer appliances are now using that type of motor. I can't believe this is really a 3 PH motor running on 120VAC.
If by BLDC you mean a brushless DC motor, then no I don't think it is that. It does run on AC, as it is plugged into a standard 3 prong 110v US electric socket. I suppose there could be an AC-DC adapter built into the control boards somewhere though. You can see on the label that it states "120V 60Hz 6.1LBA 3 Phase" and does not have anything like Induction or Inverter on the label anywhere. Nor does it have VFD noted anywhere.

Thread Starter

#### Aues

Joined Jun 2, 2021
14
Here is a copy of the Control Circuit Board printout that was included in the control panel of the washing machine.
The box labeled C4 Inverter is showing the connectors on the motor. Could the PCB on the motor itself be the VFD controller?

Thread Starter

#### Aues

Joined Jun 2, 2021
14
I suppose I could simply wire the three wires that are noted as power and ground to a plug with a switch on it and try this, but I really don't want to destroy the motor if I am wrong...Not that it's an expensive motor or anything...I think you can buy this particular one NEW on Amazon from $55 to$100, Depending on who you buy it from. And I actually got it free from tearing down the washing machine I got from a curbside pickup! Still hate to waste things for no reason though...

#### MaxHeadRoom

Joined Jul 18, 2013
25,973
The VFD is shown pictured in the OP is it not ?
Presumably all you may need is the 120v power in, the motor hooked up and find out the means of control signal input.

Thread Starter

#### Aues

Joined Jun 2, 2021
14
The VFD is shown pictured in the OP is it not ?
Presumably all you may need is the 120v power in, the motor hooked up and find out the means of control signal input.
I think I understand that you are saying the VFD is the board that is connected directly to the motor. I was thinking this was the case myself, but wasn't too sure. The three other wires that connect to the motor must be the control signal input, so I am under the impression that power to one or more of these would determine motor speed. They aren't connected to any power line though, just directly to the control circuit. The connections labeled Rx Yx and Nx on the control circuit board printout. I have no way of tracing the connections except by visually tracing the lines on the PCB, which with my eyesight might prove to be out of my capability! Thanks for your input!!

#### MaxHeadRoom

Joined Jul 18, 2013
25,973
You have the three 3ph power leads from motor to board and then the AC power supply input, then there should be some kind of input from what never programs or controls the washer cycles etc.

Thread Starter

#### Aues

Joined Jun 2, 2021
14
You have the three 3ph power leads from motor to board and then the AC power supply input, then there should be some kind of input from what never programs or controls the washer cycles etc.
Ok. I think I'm going to try hooking up the power connections, and then either the common or power connections one at a time to the power leads...I'm thinking that there are 3 speeds, one for gentle cycle, one for normal, and one for spin cycle. Slowest to fastest. Thanks again for your replies...I do appreciate it greatly!

#### MaxHeadRoom

Joined Jul 18, 2013
25,973
Did you try and look for machine diagram/print?
Often they are out there if you know the model of the washer.

Thread Starter

#### Aues

Joined Jun 2, 2021
14
Did you try and look for machine diagram/print?
Often they are out there if you know the model of the washer.
I posted a copy of what was in the machine a couple posts ago...but that's all I have. I did try hooking this motor up to power and different combinations with the 3 wires from the circuit board, but no luck. I also hooked up the entire board like it was in the machine, but obviously without all the sensors detecting what they should that didn't work either...I did get a single beep from it though LOL!
I think since this is such a specific motor I am going to move on and see if I can't find an older one or one that is more like the ones on the youtube videos...hopefully I can find one here in the US, as all the vids are apparently from either the UK or Germany.

Thanks to everyone that replied to this...You guys are great!!

#### MisterBill2

Joined Jan 23, 2018
13,088
Is what the TS has a real 3PH motor or is it a BLDC motor with 3 phases? Many of the newer appliances are now using that type of motor. I can't believe this is really a 3 PH motor running on 120VAC.
It certainly looks like a brushess DC motor and so it need to have not only that circuit board that we see, but also whatever additional control circuits are part of the system. There is probably a service document some place inside that washer that will include all of the needed connection information, and so it will be very worth the effort to locate and rescue that document.

Thread Starter

#### Aues

Joined Jun 2, 2021
14
It certainly looks like a brushess DC motor and so it need to have not only that circuit board that we see, but also whatever additional control circuits are part of the system. There is probably a service document some place inside that washer that will include all of the needed connection information, and so it will be very worth the effort to locate and rescue that document.
I already posted the only documentation that came with it, it was inside the control panel, so I'm pretty certain it is the service document, or at least what they consider to be that. Very few control boards on this machine, but I am also certain that you are correct about them being needed for this motor to run. It is way past the point of being a simple to start motor though, so I think I will continue to look for ones similar to the ones on the YouTube posts...apparently they will be older models than what I got. Thanks for your input!

#### MisterBill2

Joined Jan 23, 2018
13,088
OK, I look at the post #1 and the images there show a board that wms to be part of the motor assembly. That might be the inverter board shown on the paper that you have. It appears that here are two motor speeds , and so it may be useful to examine whatever markings are on that motor driver board as they may give a clue about the power connections.

#### MaxHeadRoom

Joined Jul 18, 2013
25,973
As per post #5, the shorted motor lead test will soon indicate whether it is BLDC or not!


text:
# Introduction

Conic programs are a class of convex nonlinear optimization problems which use cones to represent the nonlinearities. They have the form:

\begin{align} & \min_{x \in \mathbb{R}^n} & f_0(x) \\ & \;\;\text{s.t.} & f_j(x) \in \mathcal{S}_j & \;\; j = 1 \ldots m \end{align}

Mixed-integer conic programs (MICPs) are extensions of conic programs in which some (or all) of the decision variables take discrete values.

## How to choose a solver

JuMP supports a range of conic solvers, although support differs on what types of cones each solver supports. In the list of Supported solvers, "SOCP" denotes solvers supporting second-order cones and "SDP" denotes solvers supporting semidefinite cones. In addition, solvers such as SCS and Mosek have support for the exponential cone. Moreover, due to the bridging system in MathOptInterface, many of these solvers support a much wider range of exotic cones than they natively support. Solvers supporting discrete variables start with "(MI)" in the list of Supported solvers.

## How these tutorials are structured

Having a high-level overview of how this part of the documentation is structured will help you know where to look for certain things.

• The following tutorials are worked examples that present a problem in words, then formulate it in mathematics, and then solve it in JuMP. This usually involves some sort of visualization of the solution. Start here if you are new to JuMP.
• The Tips and tricks tutorial contains a number of helpful reformulations and tricks you can use when modeling conic programs. Look here if you are stuck trying to formulate a problem as a conic program.
• The remaining tutorials are less verbose and styled in the form of short code examples. These tutorials have less explanation, but may contain useful code snippets, particularly if they are similar to a problem you are trying to solve.


text:
# Area of Triangles in Trapezoid

• Jun 18th 2011, 06:42 PM
Area of Triangles in Trapezoid
Hi, quick question. For any trapezoid ABCD. with diagonals BD and CA and midpoint O , it creates 4 traingles AOB, COD, BOC, AOD, With Area of AOD=a & Area of BOC=b. With this Information how do i find the area of the whole trapezoid?

Now i know that Area of Triangle BOA=COD. But how would i find the areas of those triangles with the given information as well?

Here's a picAttachment 21694
• Jun 18th 2011, 07:10 PM
skeeter
Re: Area of Triangles in Trapezoid
Quote:

Hi, quick question. For any trapezoid ABCD. with diagonals BD and CA and midpoint O , it creates 4 traingles AOB, COD, BOC, AOD, With Area of AOD=a & Area of BOC=b. With this Information how do i find the area of the whole trapezoid?

Now i know that Area of Triangle BOA=COD. But how would i find the areas of those triangles with the given information as well?

Here's a picAttachment 21694

let $h_a$ = the perpendicular from O to AD

$h_b$ = the perpendicular from O to BC

$a = \frac{1}{2}h_a \cdot |AD|$

$b = \frac{1}{2}h_b \cdot |BC|$

area of the trapezoid ...

$A = \frac{1}{2}(h_a+h_b)(|BC| + |AD|)$

... finish it.

http://www.mathhelpforum.com/math-he...oid-trap8a.jpg
• Jun 18th 2011, 07:42 PM
Re: Area of Triangles in Trapezoid
Isn't that the regular formula for the area of a trapezoid though? im confused
• Jun 18th 2011, 11:25 PM
earboth
Re: Area of Triangles in Trapezoid
Quote:

Hi, quick question. For any trapezoid ABCD. with diagonals BD and CA and midpoint O <-- is this correct? , it creates 4 traingles AOB, COD, BOC, AOD, With Area of AOD=a & Area of BOC=b. With this Information how do i find the area of the whole trapezoid?

Now i know that Area of Triangle BOA=COD. But how would i find the areas of those triangles with the given information as well?

If and only if the text of the question is correct (see the highlighted part in red) then the trapezium is a rectangle and

$a = b$

and consequently the area of the quadrilateral is

$A = 4\cdot a = 4 \cdot b$
• Jun 19th 2011, 04:12 AM
Zellator
Re: Area of Triangles in Trapezoid
Quote:

Hi, quick question. For any trapezoid ABCD. with diagonals BD and CA and midpoint O , it creates 4 traingles AOB, COD, BOC, AOD, With Area of AOD=a & Area of BOC=b. With this Information how do i find the area of the whole trapezoid?

Now i know that Area of Triangle BOA=COD. But how would i find the areas of those triangles with the given information as well?

You probably meant centroid here, right?

The way skeeter did it,
I can only consider that he used a as triangle ACD and b as triangle BAC, not the way the graph indicates.
Moreover, that BC is equal to the height of ABC
and AD is equal to the height of ACD. Am I right?

I don't understand how can you use the diagonals to calculate the height and make it a regular trapezoid formula.
$h_a+h_b=H$?

I am confused also :)
• Jun 19th 2011, 07:33 AM
skeeter
Re: Area of Triangles in Trapezoid
Quote:

Originally Posted by Zellator
I don't understand how can you use the diagonals to calculate the height and make it a regular trapezoid formula.
$h_a+h_b=H$?

$h_a$ is the shortest distance from O to base AD

$h_b$ is the shortest distance from O to base BC

let $H$ = shortest distance from AD to BC through point O

$H = h_a+h_b$

I was under the impression that the bases were given values along with the given areas a and b ... solve for h_a and h_b in terms of the given areas and the bases and you can derive an expression for the area of the trapezoid.

$A = \left(\frac{a}{|AD|} + \frac{b}{|BC|} \right) \cdot (|AD|+|BC|)$

also, as pointed out by earboth ...

Quote:

For any trapezoid ABCD. with diagonals BD and CA and midpoint O ,
... no way O can be the midpoint of the respective diagonals. That would make ABCD a rectangle or a rhombus.
• Jun 19th 2011, 09:12 AM


text:
# How fast would a polynomial time factoring algorithm compute?

I know factoring is the chief means of breaking RSA keys. I know an algorithm that runs in polynomial time would be able to break an RSA key pair "quickly". But how quickly is "quickly"? Note, I'm not talking about any quantum computing at all here.

• Big-O Notation (poly time is usually the union $\cup_{x \in \mathbb{N}} O(n^x)$) does not indicate actual computation time. It is a statement about the asymptotic computation time for $n \rightarrow \infty$ and ignores constant factors. – tylo Feb 5 '15 at 12:25

I know an algorithm that runs in polynomial time would be able to break an RSA key pair "quickly". But how quickly is "quickly"?

No way to say, it might be microseconds, and it might be large multiplies of the age of the universe.

When we say that an algorithm runs in polynomial time, we're not saying anything about how fast the algorithm runs given any particular input. Instead, what we're saying that, as we give the algorithm increasingly large inputs, the time it takes doesn't increase that quickly.

How polynomial time is generally expressed is that there are values $c, e$ such that, given a problem of size $N$ (and in the factorization case, $N$ would be the number of bits in the RSA key), the algorithm takes time less than $cN^e$. Now, there are no limits on how big $c$ and $e$ might be, and so this doesn't give any limits on how much time a specific instance might take.

On the other hand, for all known factorization algorithms, this is not true -- no matter how large values we select for $c$ and $e$, we can find problem sizes $N$ large enough that the algorithm takes more than $cN^e$ time; hence saying that an algorithm is "polytime" does say something -- it just doesn't say what you're hoping it did.

• Suggestions: "what we're saying is that, as we give the algorithm increasingly large inputs, the time it takes doesn't increase more quickly than some limit. $\;$ Perhaps also, change $N$ to $n$ and $e$ to $k$ as this is more consistent with standard notation in an RSA context. – fgrieu Feb 4 '15 at 17:52

The rationale behind polynomial vs. exponential is in tweaking the size of the keys. We need to achieve mainly two goals:

• Encryption and decryption by legitimate users is reasonable fast.
• Decryption by adversary without private key knowledge is prohibitively slow. (One way for decrypting by adversary might be computing private key from public key and subsequent decryption with the private key.)

When the size of the key grows, time of all these operations also grows bot both legitimate users and adversaries. We want to hugely (prohibitively) increase the computation time for adversaries while maintaining only small increase of time for legitimate users (i.e. for encryption with public key and decryption with private key). The encryption schemes are usually designed to increase the time polynomially for legitimate users and exponentially for adversaries. Note that if legitimate users can do the tasks in polynomial time, we can't make it any harder than exponential for adversary, because the best attack can't perform worse than brute-force.

This is the way we make the gap between legitimate users and adversaries huge. When a polynomial algorithm for factorisation is found, the gap will at least get lower. It might be a practical algorithm (and thus effectively a RSA break), but not necessarily:

• It might be an algorithm with a high degree of polynomial (e.g. $n^{10000}$), which will perform better than today's algorithms only on very very large inputs and which is clearly impractical.
• It might be an algorithm with a promising asymptotic complexity (e.g. $n^2$), but it would have a significant multiplicative or additive constant (e.g. $2^{128}$ seconds).

Nevertheless, when a polynomial-time integer factorisation algorithm is found, we have to at least reconsider usage of RSA. Such algorithm would be a warning sign that RSA is weaker than we thought.

• Maybe that's nitpicking but the distinction is polynomial vs. non-polynomial rather than polynomial vs exponential. We know subexponetial algorithms to factor large numbers as well as finding discrete logarithms in finite fiels. – Alexandre Yamajako Feb 4 '15 at 22:45
• @AlexandreYamajako I agree there are some super-polynomial and sub-exponential algorithms. They are usually $2^{\frac{n}{k}}$ for some constant $k$. How should I name them? Almost-exponential? (By the way, I am not sure if any algorithm harder than polynomial and easier than $2^{\frac{n}{k}}$.) – v6ak Feb 5 '15 at 6:33