text:
Article | Open | Published:

# Improved phase field model of dislocation intersections

## Abstract

Revealing the long-range elastic interaction and short-range core reaction between intersecting dislocations is crucial to the understanding of dislocation-based strain hardening mechanisms in crystalline solids. Phase field model has shown great potential in modeling dislocation dynamics by both employing the continuum microelasticity theory to describe the elastic interactions and incorporating the γ-surface into the crystalline energy to enable the core reactions. Since the crystalline energy is approximately formulated by linear superposition of interplanar potential of each slip plane in the previous phase field model, it does not fully account for the reactions between dislocations gliding in intersecting slip planes. In this study, an improved phase field model of dislocation intersections is proposed through updating the crystalline energy by coupling the potential of two intersecting planes, and then applied to study the collinear interaction followed by comparison with the previous simulation result using discrete dislocation dynamics. Collinear annihilation captured only in the improved phase field model is found to strongly affect the junction formation and plastic flow in multislip systems. The results indicate that the improvement is essential for phase field model of dislocation intersections.

## Introduction

Interactions between dislocations gliding in intersecting slip planes play fundamental roles in the strain hardening during plastic deformation of crystalline materials.1,2,3,4 Dislocation intersections lead to the formation of dislocation junctions in minimum energy configurations. In face-centered cubic (FCC) crystals, the sessile junction known as Lomer–Cottrell lock5,6 is assumed to be the most stable barrier to further dislocation motion in the traditional view.1,3 However, Madec and co-workers7 reported that the interaction between two intersecting dislocations with collinear Burgers vectors is the strongest of all dislocation reactions. Previous modeling and simulations7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24 on investigating the dislocation intersections emphasized the importance of the interaction details between individual dislocations, including not only the long-range elastic interactions but also the short-range core reactions. Most of these researches suggested that the dislocation interactions can be studied by multiscale approaches, where the discrete dislocation dynamics (DDD) based on continuum elasticity theory deals with the long-range elastic attraction or repulsion, and atomistic simulations the short-range reaction rules.9 Different from the DDD techniques that require priori rules as inputs,10 another simulation method also based on continuum elastic theory called phase field model (PFM)25,26,27,28,29,30 can straightforwardly account for the effects of short-range core reactions by incorporating the generalized stacking fault energy (γ-surface)11 from atomistic or first principle calculations, such as dislocation dissociation.31,32,33,34,35 With this advantage, the PFM model of dislocation shows great potential in modeling dislocation intersections.

The evolution of dislocations is determined by minimizing the total system energy in the PFM. The total energy terms include the elastic energy, the crystalline energy and gradient energy. The elastic energy accounts for the long-range elastic interactions among dislocations coupled with applied stress, and it is given by the Khachaturyan–Shatalov linear elasticity theory36,37,38 combined with the exact 3-D Green function. The crystalline energy and gradient energy induced by the local disregistry of atomic positions describe the short-range core–core reactions. In the original PFM,25 the simplest approximation of the crystalline energy can describe the states of dislocation gliding individually in the different slip planes, but it does not account for dislocation reaction within the same slip plane. Shen and Wang have extended the approximation of this energy by coupling different slip modes within one slip plane to describe coplanar dislocation reactions and dislocation network formation.39 However, for different slip planes, previous PFM of dislocation formulated the crystalline energy by simple linear superposition of the interplanar potential of each individual slip plane. Thus, the core interactions between dislocations gliding in intersecting slip planes are not completely considered in this particular form of the crystalline energy although the long-range elastic interactions among multiple slip modes on different slip planes are already taken into account. For example, Ruffini et al.40 adopted this crystalline energy formulation to simulate the collinear annihilation of two dislocation loops with opposite Burgers vector gliding in the intersecting slip planes, but the results showed that the two loops cannot be annihilated physically at their intersection.

In this paper, we refine the formulation of the crystalline energy in the previous PFM by coupling the potential of two intersecting planes to update PFM for the dislocation intersections, and then as a proof of concept, we simultaneously utilize the previous and the improved PFMs to study the collinear reaction of intersecting perfect dislocations in FCC or simple cubic (SC) crystals. This paper is organized as follows: in “Results and discussion” section, the previous PFM is briefly reviewed and the updated PFM with the improvement of the crystalline energy formulation is introduced. In “Methods” section, both the PFMs with or without updating the crystalline energy are applied to simulate interaction between the two intersecting dislocation loops with opposite Burgers vector in FCC crystals. The calculated result for the collinear interaction in FCC crystals is also compared with the previous simulation result using discrete dislocation dynamics to show the distinct difference between the two PFMs. And then the differences of the dislocation structures and mechanical responses between the two PFMs are discussed during the collinear junction formations and the glide dislocations interacting with forest dislocations in a SC crystal for simplicity, one by one. On the basis of these results, the main conclusion is summarized.

## Results and discussion

### Model

In PFM of dislocation, a set of non-conserved phase field variables η(α, m α , r) is introduced to represent the amount of relative slips in the slip plane α and the slip direction m α in units of the Burgers vector b(α, m α ). The regions with η(α, m α , r) ≠ 0 and η(α, m α , r) = 0 describe the slipped and unslipped parts in a crystal, respectively. The dislocation lines are viewed as the boundaries between the slipped and unslipped regions. The eigenstrain $$\varepsilon _{ij}^0\left( {\bf{r}} \right)$$ induced by the inelastic slips can be expressed as25

$$\varepsilon _{ij}^0\left( {\bf{r}} \right) = \mathop {\sum}\limits_{\alpha ,m_\alpha } {\frac{{b_i\left( {\alpha ,m_\alpha } \right)n_j\left( \alpha \right) + b_j\left( {\alpha ,m_\alpha } \right)n_i\left( \alpha \right)}}{{2d}}\eta \left( {\alpha ,m_\alpha ,r} \right)},$$
(1)

where n j (α) is the component of the unit normal vector of the slip plane α and d is the inter-planar distance between slip planes.

The evolution of η(α, m α , r) characterizing the dislocation assembly is governed by minimizing the total system free energy. In the original formulations, the total system free energy includes the elastic energy, the crystalline energy and gradient energy. According to the careful discussions on the similarities and differences between the PFM and Peierls-Nabarro (PN) model41,42 in ref. 29, the PFM is allowed to fully converge to the PN model through some amendments. Therefore, in this study, we remove the gradient term and strictly confine Burgers vector distribution within the slip planes by using $$\eta _m^\alpha$$ (α, m α , r) = $$\eta _m^\alpha$$ (α, m α , r α )δ(r − r α ) in the PFM for keeping in line with the PN model, where r α is the position vector on the slip plane α and δ defined as a Dirac function is equal to one on the slip plane α. In this situation, the equilibrium dislocation core width is determined by balancing the elastic energy and the crystalline energy. Thus the total energy is expressed as

$$E^{\mathrm{tot}} = E_{\mathrm{ela}} + E_{\mathrm{cry}}.$$
(2)

The Khachaturyan-Shatalov theory provides the exact solution of the elastic energy Eela as a function of the eigenstrain and the external applied stress $$\sigma _{ij}^{{\mathrm{ext}}}$$:

$$\begin{array}{*{20}{l}} {E_{ela}} \hfill & = \hfill & {\frac{1}{2}\mathop {\int}\limits_{\left| {\bf{\xi }} \right| \ne 0} {\left[ {C_{ijkl}\tilde \varepsilon _{ij}^0\left( {\bf{\xi }} \right)\tilde \varepsilon _{kl}^0\left( {\bf{\xi }} \right)^ \ast } \right.} } \hfill \\ {} \hfill & {} \hfill & {\left. { - e_i\tilde \sigma _{ij}^0\left( {\bf{\xi }} \right)\Omega _{jk}\left( {\bf{e}} \right)\tilde \sigma _{kl}^0\left( {\bf{\xi }} \right)^ \ast e_l} \right]\frac{{{\mathrm{d}}^3\xi }}{{\left( {2\pi } \right)^3}}} \hfill \\ {} \hfill & \hfill & - {{\int} {\sigma _{ij}^{{\mathrm{ext}}}} \varepsilon _{ij}^0\left( {\bf{r}} \right){\mathrm{d}}^3r - \frac{V}{2}C_{ijkl}^{ - 1}\sigma _{ij}^{{\mathrm{ext}}}\sigma _{kl}^{{\mathrm{ext}}},} \hfill \end{array}$$
(3)

where C ijkl is the elastic moduli tensor, ξ is the vector in the Fourier space, and $${\bf{e}} = {\bf{\xi }}{\mathrm{/}}\left| {\bf{\xi }} \right|$$ is the unit vector along ξ. $$\tilde \varepsilon _{ij}^0\left( {\bf{\xi }} \right)$$ is the Fourier transformation of $$\varepsilon _{ij}^0\left( {\bf{r}} \right)$$ given by the equation $$\tilde \varepsilon _{ij}^0\left( {\bf{\xi }} \right) = {\int} {\varepsilon _{ij}^0\left( {{\bf{r}},t} \right)e^{-i{\bf{\xi }} \cdot {\bf{r}}}d^3r}$$. $$\tilde \sigma _{ij}^0\left( {\bf{\xi }} \right) = C_{ijkl}\tilde \varepsilon _{kl}^0\left( {\bf{\xi }} \right)$$. $${\mathrm{\Omega }}\left( e \right)_{jk} = \left( {C_{jlpk}e_le_p} \right)^{ - 1}$$ is the elastic Green function, and the symbol * denotes the complex conjugation. The integral $$\smallint _{\left| {\bf{\xi }} \right| \ne 0}$$ is in the Fourier space excluding the points at $$\left| {\bf{\xi }} \right| = 0$$ and V is the total volume of the system. The average strain is derived as $$\bar \varepsilon _{ij} = C_{ijkl}^{ - 1}\sigma _{kl}^{\mathrm{ext}} + \bar \varepsilon _{ij}^0$$ for stress-controlled condition and $$\bar \varepsilon _{ij} = \varepsilon _{ij}^{\mathrm{ext}}$$ for strain-controlled condition. After substitution of Eq. (1) into Eq. (3), the elastic energy is an explicit functional of the phase field variables in the Fourier space.

The crystalline energy is given by the integral of the periodical potential f cry describing the atomic misfit energy in the dislocation core:

$$E_{\mathrm{cry}} = {\int} {f_{\mathrm{cry}}} d^3r.$$
(4)

The general expression of the periodical function fcry can be proposed as a Fourier expansion series:43

$$f_{\mathrm{cry}} = \mathop {\sum}\limits_{\bf{H}} {A\left( {\bf{H}} \right)} \left\{ {1 - {\mathrm{exp}}\left[ {i2\pi {\bf{H}} \cdot {\bf{b}}\left( {\bf{r}} \right)} \right]} \right\},$$
(5)

where b(r) = $$\mathop {\sum}\limits_{\alpha ,m_\alpha } {{\bf{b}}\left( {\alpha ,m_\alpha } \right)}$$ $$\eta \left( {\alpha ,m_\alpha ,{\bf{r}}} \right)$$ is the total Burgers vector, H are the reciprocal lattice vectors of the host lattice and A(H) are positive Fourier coefficients reflecting the symmetry of the crystal lattice. In the original PFM,25 the simplified approximation of fcry is formulated by imitating the periodic Peierls potential:

$$f_{\mathrm{cry}} = \mathop {\sum}\limits_{\alpha ,m_\alpha } {A{\kern 1pt} {\mathrm{sin}}^2\pi } \eta \left( {\alpha ,m_\alpha ,{\bf{r}}} \right),$$
(6)

where A = μb2/(2π2d2), μ is the shear modulus and b is the magnitude of the Burgers vector for the corresponding slip system. In this expression of fcry given by Eq. (6), there is no coupling among the different phase field variables, therefore the core reactions among dislocations are not considered. Shen and Wang have demonstrated the limitation of the crystalline energy derived from Eq. (6) in details and introduced the new expressions of this energy to account for the dislocation core-core reactions in ref. 39.

FCC {111} slip planes are chosen for discussion by Shen and Wang.39 As shown in Fig. 1, the total Burgers vector caused by glide of the three perfect dislocations on the single (111) α plane is given:

$${\bf{b}}^\alpha \left( {\bf{r}} \right) = \eta _1^\alpha \left( {\bf{r}} \right){\bf{b}}_1^\alpha + \eta _2^\alpha \left( {\bf{r}} \right){\bf{b}}_2^\alpha + \eta _3^\alpha \left( {\bf{r}} \right){\bf{b}}_3^\alpha ,$$
(7)

where $${\bf{b}}_{\mathrm{1}}^\alpha$$ = a/2(i − k), $${\bf{b}}_{\mathrm{2}}^\alpha$$ = a/2(j − i), $${\bf{b}}_{\mathrm{3}}^\alpha$$ = a/2(k − j), a is the lattice parameter, i, j, k are unit vectors of the coordinate system. The total Burgers vector on the (111) plane bα(r) is expressed by the field variables:

$${\bf{b}}^\alpha \left( {\bf{r}} \right) = \frac{a}{2}\left[ {\left( {\eta _{\mathrm{1}}^\alpha - \eta _{\mathrm{2}}^\alpha } \right){\bf{i}} + \left( {\eta _{\mathrm{2}}^\alpha - \eta _{\mathrm{3}}^\alpha } \right){\bf{j}} + \left( {\eta _{\mathrm{3}}^\alpha - \eta _{\mathrm{1}}^\alpha } \right){\bf{k}}} \right].$$
(8)

A periodical potential that reflects the symmetry in FCC crystal is employed in ref. 39, and it is dependent on the Burgers vector $${\bf{b}} = \left( {b_i,b_j,b_k} \right)$$:

$$f\left( {\bf{b}} \right) = A\left( {1 - {\mathrm{cos}}{\kern 1pt} 2\pi \frac{{b_i}}{a}{\mathrm{cos}}{\kern 1pt} 2\pi \frac{{b_j}}{a}{\mathrm{cos}}{\kern 1pt} 2\pi \frac{{b_k}}{a}} \right).$$
(9)

Substituting Eq. (8) into Eq. (9), the periodical potential is

$$f_{\mathrm{cry}}^\alpha = A\left[ {1 - {\mathrm{cos}}{\kern 1pt} \pi \left( {\eta _2^\alpha - \eta _3^\alpha } \right){\mathrm{cos}}{\kern 1pt} \pi \left( {\eta _3^\alpha - \eta _1^\alpha } \right){\mathrm{cos}}{\kern 1pt} \pi \left( {\eta _1^\alpha - \eta _2^\alpha } \right)} \right].$$
(10)

Therefore, the crystalline energy derived from Eq. (10) is easily proved to be able to describe the core reactions among dislocations gliding in single (111) α plane such as $${\bf{b}}_1^\alpha + {\bf{b}}_2^\alpha = - {\bf{b}}_3^\alpha$$. To derive a more general expression of the crystalline energy including different slip planes, however, Shen and Wang adopt the simple linear superposition of the potential given by Eq. (10). According to this simplest approximation, the expression of the integrand of the crystal energy for two intersecting slip planes (111) α and $$\left( {\bar 1\bar 11} \right)$$ β in FCC crystal as shown in Fig. 1 is given here for simplicity:

$$\begin{array}{c}f_{\mathrm{cry}} = A\left[ {2 - {\mathrm{cos}}{\kern 1pt} \pi \left( {\eta _2^\alpha - \eta _3^\alpha } \right){\mathrm{cos}}{\kern 1pt} \pi \left( {\eta _3^\alpha - \eta _1^\alpha } \right){\mathrm{cos}}{\kern 1pt} \pi \left( {\eta _1^\alpha - \eta _2^\alpha } \right)} \right.\\ \left. { - {\mathrm{cos}}{\kern 1pt} \pi \left( {\eta _2^\beta - \eta _3^\beta } \right){\mathrm{cos}}{\kern 1pt} \pi \left( {\eta _3^\beta - \eta _1^\beta } \right){\mathrm{cos}}{\kern 1pt} \pi \left( {\eta _1^\beta - \eta _2^\beta } \right)} \right].\end{array}$$
(11)

In this formulation, there is coupling among the phase field variables in single slip planes but no coupling among the phase field variables in different slip planes. Thus the crystalline energy given by Eq. (11) does not take into account the core reactions between dislocations gliding in intersecting slip planes in ref. 39. For examples, we consider the collinear reaction between two dislocations with Burgers vector $${\bf{b}}_2^\alpha = a\left[ {\bar 110} \right]{\mathrm{/}}2$$ gliding on the (111) plane and $${\bf{b}}_2^\beta = a\left[ {1\bar 10} \right]{\mathrm{/}}2$$ gliding on the $$\left( {\bar 1\bar 11} \right)$$ plane as shown in Fig. 1, respectively. When the two dislocations collide, they can annihilate completely, leaving no product. However, according to Eq. (11), the integrand of the crystalline energy at the intersection of these two planes is fcry = A$$\left( {{\mathrm{sin}}^2{\kern 1pt} \pi \eta _2^\alpha + {\mathrm{sin}}^2{\kern 1pt} \pi \eta _2^\beta } \right)$$, which is twice of that of either unreacted dislocation as shown in Eq. (6) rather than zero.

To describe the core reactions among the dislocations gliding in the intersecting slip planes, the expressions of the crystalline energy need to be improved. To proceed, we consider two intersecting slip planes (111) and $$\left( {\bar 1\bar 11} \right)$$ in FCC crystal as shown in Fig. 1. Since the dislocation reactions at the intersection may be influenced by the dislocations on both slip planes, the total Burgers vector is expressed by the combination of all phase field variables on the two planes:

$$\begin{array}{*{20}{l}} {{\bf{b}}\left( {\bf{r}} \right)} \hfill & = \hfill & {\eta _1^\alpha \left( {\bf{r}} \right){\bf{b}}_1^\alpha + \eta _2^\alpha \left( {\bf{r}} \right){\bf{b}}_2^\alpha } \hfill \\ {} \hfill & {} \hfill & { \hskip-9pt + \eta _3^\alpha \left( {\bf{r}} \right){\bf{b}}_3^\alpha + \eta _1^\beta \left( {\bf{r}} \right){\bf{b}}_1^\beta } \hfill \\ {} \hfill & {} \hfill & { \hskip-9pt + \eta _2^\beta \left( {\bf{r}} \right){\bf{b}}_2^\beta + \eta _3^\beta \left( {\bf{r}} \right){\bf{b}}_3^\beta } \hfill \\ {} \hfill & {\mathrm{ = }} \hfill & {b_i{\bf{i}} + b_j{\bf{j}} + b_k{\bf{k}},} \hfill \end{array}$$
(12)

where b i  = $$\frac{a}{2}\left( {\eta _1^\alpha - \eta _2^\alpha - \eta _1^\beta + \eta _2^\beta } \right)$$, b j  = $$\frac{a}{2}\left( {\eta _2^\alpha - \eta _3^\alpha - \eta _2^\beta + \eta _3^\beta } \right)$$, b k  = $$\frac{a}{2}\left( {\eta _3^\alpha - \eta _1^\alpha + \eta _3^\beta - \eta _1^\beta } \right)$$. Substituting Eq. (12) into Eq. (9), the integrand of the crystalline energy is

$$\begin{array}{*{20}{l}} {f_{\mathrm{cry}}} \hfill & = \hfill & {A\left[ {1 - {\mathrm{cos}}{\kern 1pt} \pi \left( {\eta _2^\alpha - \eta _3^\alpha - \eta _2^\beta + \eta _3^\beta } \right)} \right.} \hfill \\ {} \hfill & {} \hfill & {{\mathrm{cos}}{\kern 1pt} \pi \left( {\eta _3^\alpha - \eta _1^\alpha + \eta _3^\beta - \eta _1^\beta } \right)} \hfill \\ {} \hfill & {} \hfill & {\left. {{\mathrm{cos}}{\kern 1pt} \pi \left( {\eta _1^\alpha - \eta _2^\alpha - \eta _1^\beta + \eta _2^\beta } \right)} \right].} \hfill \end{array}$$
(13)

We now use this improved expression of the integrand of the crystalline energy to account for the collinear interaction. Equation 13 is a generalization of Eq. (11). On the individual slip plane, Eq. (13) is the same as Eq. (11), while at the intersection on both slip planes, Eq. (13) has the coupling term with respect to the Burgers vectors on both slip planes and Eq. (11) does not. For example, according to Eqs. (11), (12), and (13), when two dislocations with Burgers vector $${\bf{b}}_2^\alpha$$ gliding on the (111) plane and $${\bf{b}}_2^\beta$$ gliding on the $$\left( {\bar 1\bar 11} \right)$$ plane meet each other, the integrand of the crystalline energy at the intersection is $$f_{\mathrm{cry}} = A{\kern 1pt} {\mathrm{sin}}^2\pi$$ $$\left( {\eta _2^\alpha - \eta _2^\beta } \right)$$, which is exactly equal to zero if $$\eta _2^\alpha = \eta _2^\beta$$. Therefore the core annihilation in the collinear interaction is correctly represented by the improved form of the crystalline energy.

For the description of the core reactions in dislocation intersections, we have briefly discussed the limitations of the crystalline energy given by Eq. (11) in the previous PFM and introduced the new expressions of this energy term given by Eq. (13) in the improved model. It should be noted that both Eq. (11) in the previous PFM and Eq. (13) in the improved model are special forms of the general expressions of Eq. (5), however Eq. (13) appears as a better approximation describing collinear interactions among dislocations from intersecting glide planes. Since the integrands of the crystalline energy expressed by Eqs. (5) and (13) are solely dependent on the total Burgers vector, they do not provide any barriers in the crystalline energy for cross-slip of a perfect dislocation according to the discussions in ref.39.

Besides the collinear interaction, other binary junctions formed by the perfect dislocation intersections in FCC crystals are the Hirth lock, the glissile junction and the Lomer lock. We carefully check the changes in the PFM with the crystalline energy of Eq. (11) and the PFM with Eq. (13) in describing the core energy for all the four kinds of junctions in FCC crystals, and the evaluation results are shown in Table 1. In this evaluation, we assume that the two dislocations before reaction and the product junction after the reaction have equivalent slip characterized by the field variable η. From Table 1, it is easily found that the crystalline energy after the reaction derived from Eq. (11) in the PFM model is twice as that of either unreacted dislocation for all the four junction types because of the simple linear superposition. On the other hand, the crystalline energy derived by Eq. (13) in the improved PFM model correctly describes the core energy of the product junction for all the four types of reactions, since Eq. (13) is a function of solely the total Burgers vector. Comparing the two energy terms derived from Eqs. (11) and (13), only the Hirth lock can be described by the PFM with the crystalline energy of Eq. (11).

It is worth noting that the GSFE similar to γ surface is not incorporated into the crystalline energy yet. Describing the intersections of partial dislocations requires more accurate 3D periodical potential associated the γ surface as the integrand of the crystalline energy, and this will be addressed in the future study.

For the slip systems with two slip planes as shown in Fig. 1, the evolution processes of the six phase field variables are governed by the time-dependent Ginzburg-Landau kinetic equations:25

$$\frac{{\partial \eta _n^m\left( {{\bf{r}},t} \right)}}{{\partial t}} = - L\frac{{\delta E^{tot}}}{{\delta \eta _n^m\left( {{\bf{r}},t} \right)}},$$
(14)

where m = α, β, n = 1, 2, 3, t is the time, L is the positive kinetic coefficients related to dislocation mobility. The equilibrium configuration corresponds to $$\delta E^{tot}{\mathrm{/}}\delta \eta _n^m = 0$$.

The collinear interaction is the strongest of all the reactions and can provide considerable contribution to strain hardening.7 Thus we focus our discussions on the collinear interactions between perfect dislocations lying in the intersecting slip planes.

### Intersecting of two dislocation loops with the opposite Burgers vector

The interaction between two dislocation loops with collinear Burgers vector gliding in two intersecting slip planes in an FCC crystal is considered here. A dislocation loop with Burgers vector $${\bf{b}}_2^\alpha$$ on the (111) plane and another dislocation loop with Burgers vector $${\bf{b}}_2^\beta$$ on the $$\left( {\bar 1\bar 11} \right)$$ plane are initially located in their own planes, as shown in Fig. 2a. The computational cell is 128a × 128a × 128a and remote periodic boundary conditions are used. The two dislocation loops can expand under an applied stress σ11 = −0.05μ, and interact with each other when they reach the intersecting line of the two slip planes. Physically, the two loops will be annihilated at their intersection because of $${\bf{b}}_2^\alpha + {\bf{b}}_2^\beta = 0$$. The simulation results of the dislocation intersection by the PFMs with the crystalline energy of Eqs. (11) or (13) are shown in Fig. 2b, c respectively, and the dislocation line is represented by the non-zero crystalline energy distribution fcry. In Fig. 2b, as the same as the simulation results of Ruffini et al.40 a dislocation segment with non-zero crystalline energy along the intersection line still forms during the reactions of the two loops. It is not surprising because the previous PFM employs the crystalline energy of Eq. (11) formulated by a linear superposition which is not equal to zero in this situation. Contrary to the result of the PFM with the crystalline energy of Eq. (11), simulation by the improved PFM employing the crystalline energy formulation of Eq. (13) displays that the two loops are annihilated when they meet each other as shown in Fig. 2c, because the Eq. (13) is formulated by a coupling form and the integrand is always zero while two dislocation segments are fully encountered. These results indicate that the PFM with the crystalline energy of Eq. (13) can correctly capture the collinear annihilation between the intersecting dislocations with opposite Burgers vector.

### Formations of the collinear junctions

A multiscale approach was employed to investigate the dependence of the collinear annihilation on the orientation of the dislocation lines in an FCC crystal by Madec et al. in ref.7. In this subsection, we also study the same problem to check the capability of the previous and improved PFMs. Figure 3a shows the initial configuration of two intersecting dislocation in a FCC crystal. Dislocation segment with Burgers vector $${\bf{b}}_2^\alpha = a\left[ {\bar 110} \right]{\mathrm{/}}2$$ on the (111) plane and another dislocation segment with Burgers vector $${\bf{b}}_2^\beta = a\left[ {1\bar 10} \right]{\mathrm{/}}2$$ on the $$\left( {\bar 1\bar 11} \right)$$ plane are intersected at their midpoint. Initially, the two segments are straight and make angles of ϕα and ϕβ with the intersection $$\left[ {1\bar 10} \right]$$ of the two slip planes respectively. In the absence of applied stress, when the two dislocation segments are pinned at their ends, they are allowed to relax driven by their interaction forces until they reach a stable configuration. Three states appear in the final configuration:7 The two intersecting dislocations can zip a junction when collinear annihilation is energetically favorable, or mutually pin each other at their intersection point at a crossed state, or move apart from each other under their repulsive forces. In the PFMs with the crystalline energy of Eqs. (11) or (13), such structures and formation processes of the dislocation configurations are captured by numerically solving Eq. (14) in the periodic computational domain (128a)3. Figure 3b,c show the simulated final equilibrium structures of the two intersection dislocations under ϕα = ϕβ = 45° without applied stress by using the PFMs with the crystalline energy of Eqs. (11) or (13), respectively. The simulated results in Fig. 3b show that the two dislocations move apart from each other. However, in Fig. 3c, partial annihilation of the two dislocation lines produces a junction with zero Burgers vector, which is the same as the simulation results by DDD for ϕα = ϕβ = 45° in ref. 7. As shown in Fig. 2b, c, the junction formed by collinear annihilation in the PFM with the crystalline energy of Eq. (13) leads to a perfect crystal without generating any kind of energy, but residual crystalline energy still exists for the collinear junction in the PFM model with the crystalline energy of Eq. (11). Therefore, for the two intersecting dislocations shown in Fig. 3a, junction formation is energetically favorable in the improved PFM. Obviously, the two dislocations repel each other in the initial state as shown in Fig. 3a, but they still zip a junction simulated by the improved PFM in Fig. 3c. Madec et al. in ref. 7 have also given the reason: the dislocation lines with flexibility can annihilate each other through bending and twisting, although they are elastically repulsive.

We continue to investigate the equilibrium configurations of the collinear junctions with varying ϕα and ϕβ ranging from −180° to 180° by the PFMs with the crystalline energy of Eqs. (11) or (13). By using squares to indicate junction formation, circles to indicate repulsive interaction and crosses to indicate crossed state, the final state of the two intersecting dislocation simulated by the PFMs with the crystalline energy of Eq. (11) or (13) are mapped with respect to ϕα and ϕβ in Fig. 3d, e, respectively. For direct comparisons with the DDD simulation results, Fig. 3f shows the mapping of the collinear interaction obtained from ref. 7. The results simulated by the improved PFM in Fig. 3e are in good agreement with the DDD’s results in Fig. 3f. In the improved PFM and the DDD model, the collinear annihilations appear in a vast majority in the relaxed configurations, although half of the initial configurations correspond to segments that are elastically repulsive. On the other hand, in the PFM with the crystalline energy of Eq. (11), the probability of occurrence of collinear junctions is relatively low, which is against the theory that the high strength of the collinear reaction is due to intrinsic stability and high probability of occurrence.7

### Glide dislocations interacting with forest dislocations

In this subsection, the effects of the collinear annihilation emerging only in the improved PFM on the stress-strain curves of the crystalline solids are investigated. For simplicity, we consider a simple cubic (SC) crystal with the size of (128a)3 and only two slip systems are activated here. The glide dislocations are located in the parallel (001) planes and the forest dislocations are lying in the (010) planes. External pure shear strain ε13 is applied to drive the glide dislocations to move and interact with the forest dislocations, and we simultaneously calculate the corresponding shear stress σ13 to obtain the stress-strain curve. The initial dislocation density of this system is ρ = 5 × 1016/m2.

Figure 4a shows the strain-controlled stress-strain curves of glide dislocations interacting with forest dislocations obtained from the simulation results of the PFMs with the crystalline energy of Eqs. (11) or (13) under the quasi-static conditions. The results obviously show that both the yield stress and the flow stress in the improved PFM with the crystalline energy of Eq. (13) are higher than the ones in the PFM with the crystalline energy of Eq. (11) because of low energy state of large dislocation junctions formed by collinear annihilations. It is important to note that the yield stress and elastic strain limit predicted by the PFMs here and elsewhere33 are much higher than the ones obtained by the DDD models,7 since limited activated slip systems are considered in single crystals within small computational domain. In this limited computational system, the strain-controlled stress–strain curves here exhibiting features of work softening is very different from the stress-controlled curves exhibiting features of work hardening. The unzipping of the junction and further slip of the glide dislocation leads to an abrupt stress drop after yielding in the strain-controlled case but causes a strain burst in the stress-controlled case,33 and the collecting behaviors of many unzipping events result in the whole stress-strain curves. In addition, dislocation annihilations reduce the dislocation density, and dislocation multiplication such as Frank-Read source is limited in the small computational domain. This is possibly why no work hardening is observed.

For direct view of the dislocation evolution, we capture the dislocation structures of the crystal at ε13 = 8% in the PFMs with the crystalline energy of Eqs. (11) or (13) as shown in Fig. 4b, c, respectively. It is found that the dislocation density in the improved PFM is much lower, because the dislocation annihilations occur only in this model. Thus, the inability of simulating the collinear annihilation in the PFM with the crystalline energy of Eq. (11) can significantly affect the plastic flow in multislip systems. We should note the reason for choosing SC crystals here. In the SC crystals the activated planes are parallel to the coordinate planes, thus there are relatively low “image” stresses, which have little effects on the forest dislocations under the applied pure shear strain. However for the activated intersecting {111} planes in FCC crystals, the relatively large “image” forces are generated when the periodic boundary conditions are used. The forest dislocations in FCC crystals in the small computational domain are unstable and always slip away before reacting with the glide dislocations. One has to increase the size of the computational domain to capture glide dislocations interacting with forest dislocations in FCC crystals by using the improved PFM.

To further explain the effects of the dislocation annihilations on the mechanical response in details, we study the interaction between a single glide dislocation and a single forest dislocation by the PFMs with the crystalline energy of Eqs. (11) or (13) in a SC crystal with the size of (128a)3. A screw dislocation segment with Burgers vector $${\bf{b}}_0^\alpha = a\left[ {\bar 100} \right]$$ on the (001) plane tends to transmit across an intersecting forest edge dislocation with Burgers vector $${\bf{b}}_0^\beta = a\left[ {100} \right]$$ on the (010) plane under an external applied stress τ13 as shown in Fig. 5a. Initially, the two dislocation segments are pinned at their ends. The transmission processes under τ13 = 0.05μ simulated by the PFMs with the crystalline energy of Eqs. (11) or (13) are shown in Fig. 5b, c, respectively. In the PFM with the crystalline energy of Eq. (11) as shown in Fig. 5b, the screw dislocation easily transmits across the edge dislocation and continues to glide in the slip plane, indicating that 0.05μ is sufficient to overcome the barrier of the edge dislocation. However, in the PFM with the crystalline energy of Eq. (13) as shown in Fig. 5c, the glide dislocation is blocked by the forest dislocation through collinear annihilation, higher applied stress (nearly 0.09μ) is required to unzip the collinear junction and drive the glide dislocation to move. Full annihilation of the interacting dislocation segments occurring only in the improved PFM make the collinear junction much stronger due to its zero energy state, thus the glide dislocation needs higher applied stress to unzip the junction and transmit across the forest dislocation as compared with the PFM with the crystalline energy of Eq. (11). This required higher unzipping and transmission stress also can give the reason of the higher yield stress and the higher flow stress in the stress-strain curve simulated by the improved PFM as shown in Fig. 4a.

In summary, we have improved expressions of the crystalline energy in previous PF models to describe the core interactions between the perfect dislocations gliding on the intersecting planes in FCC or SC crystalline materials. The improved formulation of this energy is fully dependent on the total Burgers vector of all the slip planes at the intersection line of these slip planes and automatically accounts for the reactions during dislocation intersections such as collinear annihilation, Lomer lock and so on. The improved and previous PFMs have been applied to study the collinear interactions, and the collinear annihilation appears only in the results simulated by the improved PFM. The collinear annihilation absence in the previous PFM strongly affects the formation of collinear junction and stress-strain response of the crystals. These applications of the improved and previous PFMs indicate that the improvement is absolutely necessary for PFM of dislocation intersections. Additionally, the improved model may be extended with the help of 3D potentials associated γ surface to quantify the cross-slip and core dissociation processes within multiple slip planes in future.

## Methods

By substituting Eqs. (1), (3), (4), and (11) into Eq. (2) in the previous PFM or by substituting Eqs. (1), (3), (4), and (13) into Eq. (2) in the improved PFM, the total system energy Etot becomes functional of the six phase field variables. By using a finite difference method combined with FFT algorithm, Eq. (14) can be numerically solved in its reduced form with all physical lengths measured in unit a, all the stresses in unit μ (the shear modulus) and the time in unit t(t = 1/). The dimensionless coefficient in the crystalline energy is A* = 1/2π2. All the simulations are performed in the computational cell 128 × 128 × 128 with periodic boundary conditions.

### Data availability

The data and codes supporting the findings of this study are available from the corresponding author on reasonable request.

Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## References

1. 1.

<NAME>. Sur le durcissement dû à la recombinaison des dislocations. Acta Metall. 8, 841–847 (1960).

2. 2.

<NAME>. On dislocation interactions in the fcc lattice. J. Appl. Phys. 32, 700–706 (1961).

3. 3.

<NAME>., <NAME>. & <NAME>. Latent hardening in copper and aluminium single crystals. Acta Metall. 28, 273–283 (1980).

4. 4.

<NAME>. & <NAME>. A constitutive description of the deformation of copper based on the use of the mechanical threshold stress as an internal state variable. Acta Metall. 36, 81–93 (1988).

5. 5.

<NAME>. A dislocation reaction in the face-centred cubic lattice. Philos. Mag. 42, 1327–1331 (1951).

6. 6.

<NAME>. The formation of immobile dislocations during slip. Philos. Mag. 43, 645–647 (1952).

7. 7.

<NAME>., <NAME>., <NAME>., <NAME>. & <NAME>. The role of collinear interaction in dislocation-induced hardening. Science 301, 1879–1882 (2003).

8. 8.

<NAME>., <NAME>. & <NAME>. Core effects in dislocation intersection. Scr. Mater. 36, 707–712 (1997).

9. 9.

<NAME>., <NAME>., <NAME>., <NAME>. & <NAME>. Connecting atomistic and mesoscale simulations of crystal plasticity. Nature 391, 669–672 (1998).

10. 10.

<NAME>., <NAME>., <NAME>., <NAME>. & <NAME>. Atomistically informed dislocation dynamics in fcc crystals. J. Mech. Phys. Solids 56, 869–895 (2008).

11. 11.

<NAME>. Intrinsic stacking faults in body-centred cubic crystals. Philos. Mag. 18, 773–786 (1968).

12. 12.

<NAME>., <NAME>., <NAME>. & <NAME>. Large-scale molecular dynamics simulations of dislocation intersection in copper. Science 279, 1525–1527 (1998).

13. 13.

<NAME>., <NAME>., <NAME>. & <NAME>. Short-range dislocation interactions using molecular dynamics: annihilation of screw dislocations. J. Mater. Res. 13, 3478–3484 (1998).

14. 14.

<NAME>. & <NAME>. Structure and strength of dislocation junctions: an atomic level analysis. Phys. Rev. Lett. 82, 1704–1707 (1999).

15. 15.

<NAME>. & <NAME>. Short-range dislocation interactions. Phys. D 133, 498–504 (1999).

16. 16.

<NAME>., <NAME>. & <NAME>. Mesoscopic analysis of structure and strength of dislocation junctions in fcc metals. Phys. Rev. Lett. 84, 1491–1494 (2000).

17. 17.

<NAME>. et al. Formation and strength of dislocation junctions in fcc metals: a study by dislocation dynamics and atomistic simulations. J. Phys. IV 11, 19–26 (2001).

18. 18.

<NAME>., <NAME>. & <NAME>. From dislocation junctions to forest hardening. Phys. Rev. Lett. 89, 255508 (2002).

19. 19.

<NAME>. & <NAME>. A study of dislocation junctions in FCC metals by an orientation dependent line tension model. Acta Mater. 50, 4873–4885 (2002).

20. 20.

<NAME>., <NAME>., <NAME>., <NAME>. & <NAME>. Molecular dynamics simulation of dislocation intersections in aluminum. Mater. Sci. Eng. A 363, 234–241 (2003).

21. 21.

Bulatov, V. V. et al. Dislocation multi-junctions and strain hardening. Nature 440, 1174–1178 (2006).

22. 22.

<NAME>. & <NAME>. Solute friction and forest interaction. Philos. Mag. 86, 1555–1565 (2006).

23. 23.

<NAME>. & <NAME>. Asymmetric dislocation junctions exhibit a broad range of strengths. Scr. Mater. 62, 508–511 (2010).

24. 24.

<NAME>., <NAME>., <NAME>. & <NAME>. Determination of dislocation interaction strengths using discrete dislocation dynamics of curved dislocations. J. Eng. Mater. Technol. 134, 021018 (2012).

25. 25.

<NAME>., <NAME>., <NAME>. & <NAME>. Nanoscale phase field microelasticity theory of dislocations: model and 3D simulations. Acta Mater. 49, 1847–1857 (2001).

26. 26.

<NAME>. Phase-field models for microstructure evolution. Annu. Rev. Mater. Res. 32, 113–140 (2002).

27. 27.

<NAME>., <NAME>. & Ortiz, M. A phase-field theory of dislocation dynamics, strain hardening and hysteresis in ductile single crystals. J. Mech. Phys. Solids 50, 2597–2635 (2002).

28. 28.

<NAME>., <NAME>. & <NAME>. Phase field methods and dislocations. Acta Mater. 51, 17–30 (2003).

29. 29.

<NAME>. & <NAME>. Phase field modeling of defects and deformation. Acta Mater. 58, 1212–1235 (2010).

30. 30.

<NAME>. & <NAME>. Understanding dislocation mechanics at the mesoscale using phase field dislocation dynamics. Philos. Trans. R. Soc. A 374, 20150166 (2016).

31. 31.

<NAME>. & <NAME>. Incorporation of γ-surface to phase field model of dislocations: simulating dislocation dissociation in fcc crystals. Acta Mater. 52, 683–691 (2004).

32. 32.

<NAME>., <NAME>., <NAME>. & <NAME>. Influence of the stacking fault energy surface on partial dislocations in fcc metals with a three-dimensional phase field dislocations dynamics model. Phys. Rev. B 84, 144108 (2011).

33. 33.

<NAME>., <NAME>., <NAME>. & <NAME>. The role of partial mediated slip during quasi-static deformation of 3d nanocrystalline metals. J. Mech. Phys. Solids 78, 415–426 (2015).

34. 34.

<NAME>., <NAME>. & <NAME>. Phase field modeling of a glide dislocation transmission across a coherent sliding interface. Model. Simul. Mater. Sci. Eng. 23, 035002 (2015).

35. 35.

<NAME>., <NAME>. & <NAME>. Alternative transmission mode and long stacking fault formation during a dissociated screw dislocation across a coherent sliding interface. J. Phys. D 48, 395301 (2015).

36. 36.

<NAME>. Some questions concerning the theory of phase transformations in solids. Sov. Phys. Solid State 8, 2163–2168 (1967).

37. 37.

<NAME>. & <NAME>. Elastic interaction potential of defects in a crystal. Sov. Phys. Solid State 11, 118–123 (1969).

38. 38.

<NAME>. Theory of Structural Transformations in Solids (Wiley, Hoboken, 1983).

39. 39.

<NAME>. & <NAME>. Phase field model of dislocation networks. Acta Mater. 51, 2595–2610 (2003).

40. 40.

<NAME>., <NAME>. & <NAME>. Three-dimensional phase-field model of dislocations for a heterogeneous face-centered cubic crystal. J. Mech. Phys. Solids 105, 95–115 (2017).

41. 41.

<NAME>. The size of a dislocation. Proc. Phys. Soc. 52, 34–37 (1940).

42. 42.

<NAME>. N. Dislocations in a simple cubic lattice. Proc. Phys. Soc. 59, 256–272 (1947).

43. 43.

Khachaturyan, <NAME>. in The Science of Alloys for the 21st Century: A Hume–Rothery Symposium Celebration (eds Turchi, <NAME>., Shull, <NAME>. & Gonis, A.) 293–308 (St Louis, 2000).

## Acknowledgements

This work was supported by the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB22040502), the National Natural Science Foundation of China (Grant Nos. 11672285 and 11402243), the Science Challenge Program (Grant Nos. TZ2016001 and TZ2018001), the Collaborative Innovation Center of Suzhou Nano Science and Technology, and the Fundamental Research Funds for the Central Universities.

## Author information

### Affiliations

1. #### CAS Key Laboratory of Mechanical Behavior and Design of Materials, and Department of Modern Mechanics, University of Science and Technology of China, 230026, Hefei, Anhui, China

• <NAME>
• , <NAME>
• , <NAME>
•  & <NAME>
2. #### National Key Laboratory of Shockwave Physics and Detonation Physics, Institute of Fluid Physics, China Academy of Engineering Physics, 621900, Mianyang, Sichuan, China

• <NAME>

### Contributions

Y.N. initiated and supervised the study; S.Z. developed the model and carried out most numerical simulations. D.Z. carried out some numerical simulations. All authors analyzed the data, wrote the manuscript and gave final approval for publication.

### Competing interests

The authors declare no competing interests.

### Corresponding author

Correspondence to <NAME>.

### DOI

https://doi.org/10.1038/s41524-018-0075-x


text:
### R-15.1, r. 6 - Regulation respecting supplemental pension plans

5.1. (Replaced).
O.C. 1073-2009, s. 2; O.C. 1183-2017, s. 4.
5.1. Where the provision for adverse deviation is calculated on the basis of estimates authorized by section 60.5, the report must contain the following information:
(1)  the amount;
(2)  a certification of the actuary certifying that a complete actuarial valuation of the plan carried out at the valuation date would have established an amount for the provision for adverse deviation equal to or less than the amount indicated in paragraph 1;
(3)  the maximum amount of surplus assets that may be appropriated to the payment of employer contributions;
(4)  the maximum amount of the reduction to which the pension committee may agree under section 15.0.0.5;
(5)  the maximum amount of the reduction to which the pension committee may agree under the first paragraph of section 15.0.0.6, specifying that the amount is established on the assumption that the surplus assets of the plan will be in no way appropriated to the payment of employer contributions;
(6)  a certification of the actuary certifying that, should complete actuarial valuation be carried out, the resulting amounts would be at least equal to those indicated in paragraphs 3 to 5.
O.C. 1073-2009, s. 2.

format: plain


text:
# Can a “proportionality” metric designed for compositional data analysis be used for non-compositional representations of the data?

I have been diving deep into compositional data analysis. Here is a great thread containing a conversation with <NAME>, who is pushing this type of research in microbiology, about some intricacies of compositional data analysis . In that thread proportionality is defined but essentially it can be used as a stand-in replacement for correlation that is compositionally valid.

My question is out of scope for that thread so I thought of asking here. I have a compositional dataset X that is made up of genes as components and samples as compositions. I've made a transformation (that might not be the correct term) where I sum up the counts for different groups of genes. However, a single component sometimes belong to multiple groups so it's not compositional anymore. Furthermore, I normalize by the length of the genes in each group so the data is certainly not compositional.

My question is whether or not I can use the proportionality rho metric in this case?

Certainly I can use it and get an answer out with propr but would this be valid after I've applied these types of data manipulation?


text:
# Tutorial: Using styles¶

Styles a way of specifying how plots created by pyCreeper.crGraphs should look like. Styles are encapsulated in a pyCreeper.crGraphStyle.crGraphStyle object and the object’s properties may be set to change different settings.

There are several advantages of using pyCreeper’s crGraphStyle compared to the standard matplotlib way:

1. crGraphStyle disallows setting of properties that are invalid. This means that python will let you know when you try to set a style that does not exist.
2. When you set crGraphStyle properties, a type check is performed in the background. Python will let you know if you are trying to set a property value that does not make sense
3. Enumerations instead of strings are used where appropriate, for example, to set a grid style. This meas that you can easily select from a set of valid values.
4. You can easily revert to the default look by calling the style reset function

When no styles are specified, pyCreeper.crGraphs uses a default style. For example, a default line plot may look like this:

To change the way the plot looks, for example, to change the colours, markers, font sizes, etc., properties of a pyCreeper.crGraphStyle.crGraphStyle need to be set. There are two ways of doing this.

One way is to create a brand new pyCreeper.crGraphStyle.crGraphStyle object, set its properties and then assign it to pyCreeper.crGraphs:

from python.pyCreeper import crGraphStyle;

#-- create new style object
myStyle = crGraphStyle.crGraphStyle();
style.markers = ['o','o','s'];
style.colors = ['g','c','r'];
style.titleFontSize = 40;
style.numOfLegendColumns = 1;
style.gridType = crGraphStyle.GRID_TYPE.MAJOR_HORIZONTAL;

#-- assign it to crGraphs
crGraphs.setStyle(style);

Another way is to directly set properties of the pyCreeper.crGraphStyle.crGraphStyle instance that is already stored in pyCreeper.crGraphs:

crGraphs.getStyle().markers = ['o','o','s'];
crGraphs.getStyle().colors = ['g','c','r'];
crGraphs.getStyle().titleFontSize = 40;
crGraphs.getStyle().numOfLegendColumns = 1;
crGraphs.getStyle().crGraphStyle.GRID_TYPE.MAJOR_HORIZONTAL;

Whichever way you use, pyCreeper.crGraphs now contains the new style:

Have a look at the pyCreeper.crGraphStyle.crGraphStyle documentation to see what style properties you can set.

The style will be applied to all pyCreeper.crGraphs function calls in the same python document. To reset the styles back to defaults, use

myStyle = crGraphStyle.crGraphStyle();
crGraphs.setStyle(style);

or simply

crGraphs.getStyle().reset();


text:
# Solve equation involving logarithms

Printable View

• Apr 6th 2011, 12:01 AM
mathguy80
Solve equation involving logarithms
Hi All,

I was studying logarithms today, and came across this problem. Need a hint on how to proceed.

$2^{4x} - 3^{3x} + 4^{2x} = 0$

I simplified it down to,

$\frac{2^{4x}}{3^{3x}} = \frac{1}{2}$

How do I proceed further? Thanks!
• Apr 6th 2011, 12:08 AM
earboth
Quote:

Originally Posted by mathguy80
Hi All,

I was studying logarithms today, and came across this problem. Need a hint on how to proceed.

$2^{4x} - 3^{3x} + 4^{2x} = 0$

I simplified it down to,

$\frac{2^{4x}}{3^{3x}} = \frac{1}{2}$

How do I proceed further? Thanks!

I take your last line:

$\dfrac{2^{4x}}{3^{3x}} = \dfrac{16^x}{27^x} = \left(\dfrac{16}{27} \right)^x = \dfrac{1}{2}$

Now use logarithms to the base $\frac{16}{27}$
• Apr 6th 2011, 01:00 AM
mathguy80
Thanks @earboth. Looks simple now!


text:
$\require{\cancel}$ $\require{\stix[upint]}$

### HENRYTAIGO

#### Cambridge International AS and A Level

 Name of student HENRYTAIGO Date Adm. number Year/grade HenryTaigo Stream HenryTaigo Subject Pure Mathematics 3 (P3) Variant(s) P31, P32, P33 Start time Duration Stop time

Qtn No. 1 2 3 4 5 6 Total
Marks 9 8 8 9 9 8 51
Score

Get Mathematics 9709 Topical Questions (2010-2021) for $14.5 per Subject. Attempt all the 6 questions Question 1 Code: 9709/31/M/J/10/7, Topic: Complex numbers The complex number$2+2 \mathrm{i}$is denoted by$u$.$\text{(i)}$Find the modulus and argument of$u$.$[3]\text{(ii)}$Sketch an Argand diagram showing the points representing the complex numbers 1 , i and$u$. Shade the region whose points represent the complex numbers$z$which satisfy both the inequalities$|z-1| \leqslant|z-\mathrm{i}|$and$|z-u| \leqslant 1[4]\text{(iii)}$Using your diagram, calculate the value of$|z|$for the point in this region for which$\arg z$is least.$[3]$Question 2 Code: 9709/32/M/J/10/7, Topic: Differentiation The variables$x$and$t$are related by the differential equation $$\mathrm{e}^{2 t} \displaystyle\frac{\mathrm{d} x}{\mathrm{~d} t}=\cos ^{2} x$$ where$t \geqslant 0$. When$t=0, x=0$.$\text{(i)}$Solve the differential equation, obtaining an expression for$x$in terms of$t$.$[6]\text{(ii)}$State what happens to the value of$x$when$t$becomes very large.$[1]\text{(iii)}$Explain why$x$increases as$t$increases.$[1]$Question 3 Code: 9709/33/M/J/10/7, Topic: Trigonometry, Integration$\text{(i)}$Prove the identity$\cos 3 \theta \equiv 4 \cos ^{3} \theta-3 \cos \theta$.$[4]\text{(ii)}$Using this result, find the exact value of$[4]$$$\displaystyle\int_{\frac{1}{3} \pi}^{\frac{1}{2} \pi} \cos ^{3} \theta \mathrm{d} \theta$$ Question 4 Code: 9709/31/O/N/10/7, Topic: Vectors With respect to the origin$O$, the points$A$and$B$have position vectors given by$\overrightarrow{O A}=\mathbf{i}+2 \mathbf{j}+2 \mathbf{k}$and$\overrightarrow{O B}=3 \mathbf{i}+4 \mathbf{j}.$The point$P$lies on the line$A B$and$O P$is perpendicular to$A B$.$\text{(i)}$Find a vector equation for the line$A B$.$[1]\text{(ii)}$Find the position vector of$P$.$[4]\text{(iii)}$Find the equation of the plane which contains$A B$and which is perpendicular to the plane$O A B$, giving your answer in the form$a x+b y+c z=d$.$[4]$Question 5 Code: 9709/32/O/N/10/7, Topic: Vectors Question 6 Code: 9709/33/O/N/10/7, Topic: Integration, Numerical solutions of equations$\text{(i)}$Given that$\displaystyle\int_{1}^{a} \frac{\ln x}{x^{2}} \mathrm{~d} x=\frac{2}{5}$, show that$a=\frac{5}{3}(1+\ln a)$.$[5]\text{(ii)}$Use an iteration formula based on the equation$a=\frac{5}{3}(1+\ln a)$to find the value of$a$correct to 2 decimal places. Use an initial value of 4 and give the result of each iteration to 4 decimal places.$[3]\$

Worked solutions: P1, P3 & P6 (S1)

If you need worked solutions for P1, P3 & P6 (S1), contact us @ [email protected] | +254 721 301 418.

1. Send us the link to these questions ( https://stemcie.com/view/152 ).
2. We will solve the questions and provide you with the step by step worked solutions.
3. We will then schedule a one to one online session to take you through the solutions (optional).


text:
# echarlaix /bert-base-uncased-qqp-f87.8-d36-hybrid

## bert-base-uncased model fine-tuned on QQP

This model was created using the nn_pruning python library: the linear layers contains 36% of the original weights.

The model contains 50% of the original weights overall (the embeddings account for a significant part of the model, and they are not pruned by this method).

## Fine-Pruning details

This model was fine-tuned from the HuggingFace model checkpoint on task, and distilled from the model textattack/bert-base-uncased-QQP. This model is case-insensitive: it does not make a difference between english and English.

A side-effect of block pruning is that some of the attention heads are completely removed: 54 heads were removed on a total of 144 (37.5%).

## Details of the QQP dataset

Dataset Split # samples
QQP train 364K
QQP eval 40K

### Results

Pytorch model file size: 377MB (original BERT: 420MB)

Metric # Value
F1 87.87
Mask token: [MASK]


text:
# Writing a filter¶

PDAL can be extended through the development of filter functions.

For more on filters and their role in PDAL, please refer to PDAL Architecture Overview.

Every filter stage in PDAL is implemented as a plugin (sometimes referred to as a “driver”). Filters native to PDAL, such as filters.ferry, are implemented as _static_ filters and are statically linked into the PDAL library. Filters that require extra/optional dependencies, or are external to the core PDAL codebase altogether, such as filters.pmf, are implemented as _shared_ filters, and are built as individual shared libraries, discoverable by PDAL at runtime.

In this tutorial, we will give a brief example of a filter, with notes on how to make it static or shared.

First, we provide a full listing of the filter header.

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 // MyFilter.hpp #pragma once #include #include #include namespace pdal { class Options; class PointLayout; class PointView; class PDAL_DLL MyFilter : public Filter { public: MyFilter() : Filter() {} std::string getName() const; private: double m_value; Dimension::Id m_myDimension; virtual void addDimensions(PointLayoutPtr layout); virtual void addArgs(ProgramArgs& args); virtual PointViewSet run(PointViewPtr view); MyFilter& operator=(const MyFilter&); // not implemented MyFilter(const MyFilter&); // not implemented }; } // namespace pdal

This header should be relatively straightforward, but we will point out three methods that must be declared for the plugin interface to be satisfied.

private:
double m_value;

In many instances, you should be able to copy this header template verbatim, changing only the filter class name, includes, and member functions/variables as required by your implementation.

## The source¶

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 // MyFilter.cpp #include "MyFilter.hpp" #include #include #include #include namespace pdal { static PluginInfo const s_info = PluginInfo("filters.name", "My awesome filter", "http://link/to/documentation"); CREATE_SHARED_PLUGIN(1, 0, MyFilter, Filter, s_info) std::string MyFilter::getName() const { return s_info.name; } void MyFilter::addArgs(ProgramArgs& args) { args.add("param", "Some parameter", m_value, 1.0); } void MyFilter::addDimensions(PointLayoutPtr layout) { layout->registerDim(Dimension::Id::Intensity); m_myDimension = layout->registerOrAssignDim("MyDimension", Dimension::Type::Unsigned8); } PointViewSet MyFilter::run(PointViewPtr input) { PointViewSet viewSet; viewSet.insert(input); return viewSet; } } // namespace pdal

For your filter to be available to PDAL at runtime, it must adhere to the PDAL plugin interface. As a convenience, we provide the macros in pdal_macros.hpp to do just this.

We begin by creating a PluginInfo struct containing three identifying elements - the filter name, description, and a link to documentation.

 1 2 3  "http://link/to/documentation"); CREATE_SHARED_PLUGIN(1, 0, MyFilter, Filter, s_info)

PDAL requires that filter names always begin with filters., and end with a string that uniquely identifies the filter. The description will be displayed to users of the PDAL CLI (pdal --drivers).

Next, we pass the following to the CREATE_STATIC_STAGE macro, passin in the name of the stage and the PluginInfo struct.

std::string MyFilter::getName() const

To create a shared stage, we simply change CREATE_STATIC_STAGE to CREATE_SHARED_STAGE.

Finally, we implement a method to get the plugin name, which is primarily used by the PDAL CLI when using the --drivers or --options arguments.

 1 2 3 4  return s_info.name; } void MyFilter::addArgs(ProgramArgs& args)

Now that the filter has implemented the proper plugin interface, we will begin to implement some methods that actually implement the filter. First, getDefaultOptions() is used to advertise those options that the filter provides. Within PDAL, this is primarily used as a means of displaying options via the PDAL CLI with the --options argument. It provides the user with the option names, descriptions, and default values.

 1 2 3 4  args.add("param", "Some parameter", m_value, 1.0); } void MyFilter::addDimensions(PointLayoutPtr layout)

The addArgs() method is used to register and bind any provided options to the stage. Here, we get the value of param, if provided, else we populate m_value with the default value of 1.0.

 1 2 3 4 5 6  layout->registerDim(Dimension::Id::Intensity); m_myDimension = layout->registerOrAssignDim("MyDimension", Dimension::Type::Unsigned8); } PointViewSet MyFilter::run(PointViewPtr input)

In addDimensions() we make sure that the known Intensity dimension is registered. We can also add a custom dimension, MyDimension, which will be populated within run().

 1 2 3 4 5 6  PointViewSet viewSet; viewSet.insert(input); return viewSet; } } // namespace pdal

Finally, we define run(), which takes as input a PointViewPtr and returns a PointViewSet. It is here that we can transform existing dimensions, add data to new dimensions, or selectively add/remove individual points.

We suggest you take a closer look at our existing filters to get an idea of the power of the Filter stage and inspiration for your own filters!

## Compilation¶

Set up a CMakeLists.txt file to compile your filter against PDAL:

 1 2 3 4 5 6 7 8 9 cmake_minimum_required(VERSION 2.8.12) project(FilterTutorial) find_package(PDAL 1.6.0 REQUIRED CONFIG) add_library(pdal_plugin_filter_myfilter SHARED MyFilter.cpp) target_link_libraries(pdal_plugin_filter_myfilter PRIVATE ${PDAL_LIBRARIES}) target_include_directories(pdal_plugin_filter_myfilter PRIVATE${PDAL_INCLUDE_DIRS})


text:
Below I have implemented a class that will validate and upload files based on the specified rules.

class Uploader
{
protected $finfo; // extensions corresponding mime types protected$mimes = [
'jpg' => 'image/jpeg',
'pdf' => 'application/pdf',
'png' => 'image/png',
'gif' => 'image/gif'
];

// default settings
protected $defaults = [ 'auto_rename' => true, // rename the uploaded file to a random ID 'new_name' => null, // rename the uploaded file to this name 'ext_whitelist' => null, // mime types corresponding to the types of files allowed to be uploaded //'max_height' => null, // maximum height (in pixels) that the file can be; null for no limit 'max_size' => null, // maximum size (in megabytes) that the file can be; null for no limit //'max_width' => null, // maximum width (in pixels) that the file can be; null for no limit 'path' => null, // path to the folder where the upload should be placed 'required' => false // require that file needs to be uploaded ]; // settings for fields to be uploaded protected$rules = array();

// files that were processed properly
protected $processed = array(); // errors found during processing or uploading protected$errors = array();

/**
*/
public function __construct()
{
$this->finfo = new finfo(); } /** * Set default settings. * @param array$params
*/
public function setConfig(array $params) { foreach ($this->defaults as $name =>$value) {
isset($params[$name]) && $this->defaults[$name] = $params[$name];
}
}

/**
* Set the settings for the fields to be uploaded.
* @param   array   $params */ public function setRules(array$params)
{
foreach ($params as$name => $value) { // merge the default settings with field's settings if (is_array($value)) {
$this->rules[$name] = $value +$this->defaults;
} else {
$this->rules[$value] = $this->defaults; } } } /** * Process files uploaded based on the settings given. * @param array$params
* @return  true on success or false on failure
*/
public function process()
{
foreach ($this->rules as$name => $rules) { // check file was given if (empty($_FILES[$name]) || !$this->isUploadedFile($_FILES[$name])) {
if ($rules['required']) {$this->errors[$name] = 'No file was provided.'; } continue; }$file = $_FILES[$name];
// check file error
if ($file['error'] !== UPLOAD_ERR_OK) {$this->errors[$name] = 'An error occurred while trying to upload file.'; continue; }$file['ext'] = $this->getExtension($file);
$file['size'] =$this->getSize($file); // create directory if it does not exist if (!$rules['path'] || !$this->mkdir($rules['path'])) {
$this->errors[$name] = 'An error occurred while trying to upload file.';
error_log('Unable to write to directory, ' . $rules['path'], 3, 'log.txt'); continue; } // check file extension if ($rules['ext_whitelist'] && strpos($rules['ext_whitelist'],$file['ext']) === false) {
$this->errors[$name] = 'This file is not a valid format.' ;
continue;
}
// check file size
if ($rules['max_size'] &&$file['size'] > $rules['max_size']) {$this->errors[$name] = 'This file is too large to upload.'; continue; } // rename filename using given new name if ($rules['new_name']) {
$file['name'] =$rules['new_name'];
// rename filename using random number
} elseif ($rules['auto_rename']) {$file['name'] = uniqid() . '.' . $file['ext']; } // mark as processed$this->processed[$name] =$file;
}
return !$this->errors; } /** * Upload processed files. * @return true on success or false on failure */ public function run() { foreach ($this->processed as $name =>$file) {
$rules =$this->rules[$name];$destination = $rules['path'] . '/' .$file['name'];
// save file to destination
if (!$this->move($file, $destination)) {$this->errors[$name] = 'An unexpected error has occurred m.'; error_log('Unable to upload file, ' .$destination, 3, 'log.txt');
}
}
return !$this->errors; } /** * Get processed file. * @return array */ public function getFile($name)
{
return isset($this->processed[$name]) && !isset($this->errors[$name]) ? $this->processed[$name] : false;
}

/**
* @return  array
*/
public function getErrors()
{
return $this->errors; } /** * Gets file extension. * @param array$file
* @return  extension of the file if it is found
*/
protected function getExtension(array $file) { return array_search($this->finfo->file($file['tmp_name'], FILEINFO_MIME_TYPE),$this->mimes, true);
}

/**
* Gets file size in MB.
* @param   array   $file * @return size of the file */ protected function getSize(array$file)
{
return round((filesize($file['tmp_name']) / 1048576), 2); } /** * Tells whether the file was uploaded via HTTP POST. * @param array$file
* @return  true on success, false on failure
*/
protected function isUploadedFile(array $file) { return file_exists($file['tmp_name']) && is_uploaded_file($file['tmp_name']); } /** * Make directory if it does not already exist. * @param array$file
* @return  true on success, false on failure
*/
protected function mkdir($path) { return is_dir($path) && is_writable($path) || mkdir($path, 0755, true);
}

/**
* Delete file if it does exist.
* @param   array   $file * @return true on success, false on failure */ protected function unlink($path)
{
return !is_file($path) || unlink($path);
}

/**
* Moves an uploaded file to a new location.
* @param   array   $file * @return true on success, false on failure */ protected function move(array$file, $path) { // delete old file if one already exist and upload new file return !$this->unlink($path) ?: move_uploaded_file($file['tmp_name'], $path); } }  This is how it would be used when a form is submitted: require 'Uploader.php'; if (isset($_POST['submit'])) {
$uploader = new Uploader(); // set global settings$uploader->setConfig(array(
));
// set settings for individual fields
$uploader->setRules(array( 'file1' => array( 'auto_rename' => false, // do not rename file 'ext_whitelist' => 'gif|jpg|png', // accept only GIF, JPEG, and PNG 'max_size' => 1 // accept only sizes up to 1 MB ), 'file2' => array( 'required' => true, // require file 'ext_whitelist' => 'pdf', // accept only PDF 'max_size' => 20 // accept only sizes up to 20 MB ) )); if ($uploader->process() && $uploader->run()) { if ($file1 = $uploader->getFile('file1')) { echo$file1['name'] . ' was uploaded<br>';
}
if ($file2 =$uploader->getFile('file2')) {
echo $file2['name'] . ' was uploaded'; } } else { var_dump($uploader->getErrors());
}
}

I know there are many existing uploading classes, but I have yet to find one that is exactly to my liking. So I wanted to try and write my own. My goal with this class is to be able to validate files (for now, images and PDFs) and upload files. Though this class is probably not as strong as existing ones, it works as intended.

So, any suggestions on how my class can be improved, especially when it comes to error handling?

I like your code. The public interface is mostly well thought out, it seems easy to use, the code is mostly well structured and readable, well commented, well formatted, etc.

A couple of issues do exist though:

It is possible to bypass finfo check (depending on the system), and thus to upload files with a dangerous type, such as PHP files.

You should get the extension from the file name, not from a mimetype check.

You could add an additional mimetype check, but the extension check needs to be independent of it.

Security: Weak Extension Check

Now, even if your extension check would work, it is rather weak as it doesn't check for equality. Instead, it checks if the extension exists in a given string, which is a weaker check.

For example, a user may want to allow phtml (which doesn't lead to code execution with most current server configurations), but not pht (which does lead to code execution with most current server configurations). The user will set 'ext_whitelist' => 'foobar|phtml', and they will expect that phtml files can be uploaded, and that pht files cannot. But that isn't what happens.

Misc

• process and run are a bit generic. run could probably be upload, and process may be checkValidity or something.
• $defaults should be renamed, to either $defaultConfig or $defaultRules (whichever one is actually correct). • Do you really need separate process and run methods? It seems to just complicate the interface. Or is there ever a situation in which a user does want to call one, but not the other? If not, just make both private, and add a new upload method which calls both. • The difference between config and rules isn't quite clear to me. Config seems to be the default rules + the upload path? Or just the default rules? If that is the case, setConfig should really be named setDefaultRules (no reason to confuse a user with config, if there isn't anything to configure except rules). If that's wrong, the interface and documentation should make the difference clearer. • $rules doesn't just contain rules, but actually contains the file reference + its associated rules. The naming here is a bit confusing. Reading foreach ($this->rules as$name => \$rules), it looks as if you were iterating over rules, while you are actually iterating over files.
• magic numbers: they make code hard to understand. What's 1048576? I would save it in a properly named variable, or at a minimum add a comment (or even a function) which makes it clear what is happening here.
• I must have been living in a bubble as I never heard of phtml or pht. Great suggestions and critiques! I stupidly spent a longtime trying to figure out appropriate names for the variables and functions. Much appreciated. – Mikey Apr 13 '16 at 23:51


text:
Question

# Two right circular cylinders of equal volumes have their heights in the ratio 1 : 2. What is the ratio of their radii?

Solution

## So, the ratio of their radii is $\sqrt{2}$ : 1.MathematicsRS Aggarwal (2018)Standard X

Suggest Corrections

0

Similar questions
View More

Same exercise questions
View More


text:
# colorbox and text placement

I was playing up with \colorbox for my resume and really loved the way I can create styling for the document.

Anyway I was trying to center align the text but failed to achieve it using the \centering command and if I use \begin{center} to allign the text, \colorbox width increase two fold which I don't want.

\newcommand \reshead [1]{\par \noindent \colorbox{black}{\begin{minipage}
{\textwidth}{\color{white} \centering \bfseries {#1}} \hfill \end{minipage}}}

Apart from this issue, I wanted to my resume to look like this, the top line should be lesser in length compared to \colorbox

However with my code both are equal, is there a way I can stretch the \colorbox a bit more.? -

%% (c) <NAME>
%% Resume - C.V of <NAME>
%% September 17th 2012

\documentclass[letterpaper,12pt]{article}
%\usepackage[charter]{mathdesign}
% \usepackage{charter}
% \usepackage{mathpazo}
%\usepackage{kpfonts}
%\usepackage{geometry}
\usepackage{palatino}
\usepackage{url}
\usepackage[empty]{fullpage}
\usepackage{color}
%\definecolor{mygrey}{gray}{0.95}
%\usepackage{xcolor}

\newcommand \reshead [1]{\par \noindent \colorbox{black}{\begin{minipage}{\textwidth}{  \color{white} \centering \bfseries {#1}} \hfill \end{minipage}}}

\begin{document}

\begin{center}

\bfseries {\huge {Shashwat Pant}} \\
\bfseries E-Mail - <EMAIL> ~ Voice -
\hrule \hrule
\end{center}

\noindent Seeking a challenging career to gain experience in rich evolving computing in the areas of :
\begin{itemize}
\item
\item Parallel and Vector computing (Open CL)

\end{itemize}

\end{document}

With my code the output look like this

Is there a way to stretch the colorbox and underlying text a bit towards left so that are not in same symmetry?

-
As an aside if you really want a space before the :, you should use areas of~: instead of areas of : so that the colon does not end up on its own line. –  <NAME> Sep 18 '12 at 16:04
Thanks :) will keep that in mind :) –  Shashwat Sep 18 '12 at 16:07

Here is an option that you can use:

\documentclass[letterpaper,12pt]{article}
\usepackage[empty]{fullpage}% http://ctan.org/pkg/fullpage
\usepackage{xcolor}% http://ctan.org/pkg/xcolor
\usepackage{changepage}% http://ctan.org/pkg/changepage

\par\noindent%
\colorbox{black}{\makebox[\dimexpr\linewidth-2\fboxrule-2\fboxsep][c]{\textcolor{white}{\bfseries #1}}}}

\begin{document}

\begin{center}
\bfseries
{\huge <NAME>} \\
E-Mail - <EMAIL> ~ Voice - \\
\end{center}

\noindent\hspace*{-1em}\rule{\dimexpr\linewidth+2em}{2pt}

\noindent Seeking a challenging career to gain experience in rich evolving computing in the areas of :
\begin{itemize}
\item
\item Parallel and Vector computing (Open CL)
\end{itemize}

\bigskip

\end{document}

Some things I've changed:

• Instead of \hrule\hrule, use a specified rule width: \rule{\dimexpr\linewidth-2em}{2pt}. With the horizontal movement, this rule overhangs on both sides by 1em;
• Added chngpage which provides the adjustwidth environment for modifying the margins locally; and
• Used \makebox instead of minipage to specify the contents of your \reshead. Also updated the width to fix within the margins exactly (\linewidth-2\fboxrule-2\fboxsep).
-

I wouldn't stretch the \colorbox, but reduce the top rule:

%% (c) <NAME>
%% Resume - C.V of <NAME>
%% September 17th 2012

\documentclass[letterpaper,12pt]{article}
%\usepackage[charter]{mathdesign}
% \usepackage{charter}
% \usepackage{mathpazo}
%\usepackage{kpfonts}
%\usepackage{geometry}
\usepackage{palatino}
\usepackage{url}
\usepackage[empty]{fullpage}
\usepackage{color}
%\definecolor{mygrey}{gray}{0.95}
%\usepackage{xcolor}

\par\noindent
\colorbox{black}{%
\begin{minipage}{\dimexpr\textwidth-2\fboxsep\relax}
\color{white}\centering\bfseries#1
\end{minipage}%
}%
}

\begin{document}

\begin{center}
\bfseries
{\huge <NAME>} \\[1ex]
E-Mail - <EMAIL> ~ Voice -

% Here I reduce the rule by 1in on both sides (adjust to suit)
% The rule thickness is 1pt; it's 2pt from the bottom of the text above it
\nointerlineskip
\moveright 1in \vbox{\hsize=\dimexpr\hsize - 2in \relax
\vskip 2pt % separation from the text above
\hrule width \hsize height 1pt % rule thickness
}
\end{center}

\noindent Seeking a challenging career to gain experience
in rich evolving computing in the areas of:
\begin{itemize}
\item \LaTeX
\item Parallel and Vector computing (Open CL)

\end{itemize}

\end{document}

You have to reduce a bit the width of the minipage, otherwise the \colorbox would be too wide and overflow on the right: this is the purpose of

\dimexpr\textwidth-2\fboxsep\relax

The \moveright trick is a low level one, but once in a while it comes handy. :)

-


text:
Gadgets Cleaning Devices BetterAir: Reviewing Biotica800 Probiotic Air Purifier Research

# BetterAir: Reviewing Biotica800 Probiotic Air Purifier Research

BetterAir Biotica800 probiotic air purifier uses an organic, filterless, and ecological balancing system to naturally eliminate allergens, pathogens, pet dander and mold spores on objects and surfaces.

BetterAir is billed as the world's first probiotic air purifier. Priced at $300, the only air, object and surface purifier promises to eliminate 99.99% of certain viruses and pathogens within hours. The Biotica800 ecological balancing system has no filter and is 100% organic and works by releasing a probiotic mist into the room that took 9 years of research and development before being able to be available for effective at-home indoor pollution technology. You’ve heard of probiotics, and you've likely heard of the air purifiers exploding in popularity amid the COVID-19 pandemic and subsequent scramble for cleaning devices that are shown to provide extra protection and added defense against harmful particulate matter (PM0.3, PM2.5, PM10). Probiotic purifiers uniquely combine the two popular alternative health trends into one product type. When such statements as “according to the EPA our indoor air is 5X more polluted than the air outside” or “who could have imagined our homes and offices would become the most dangerous places for us to be” are made, it raises a cause for concern as to just how subpar indoor breathing environments may be on a day to day basis – especially if not “aired out” frequently. Now, you can run the air purifier to eliminate 99.99% of particles in your home within 6 hours. BetterAir purportedly uses an innovative natural and organic purification system (probiotics) to get rid of harmful pathogens, including bacteria, allergens, pet dander, dust mites, mold spores and other wellness-threatening contaminants. Much like the rise in popularity for ionizers and UV sanitizers, air purifying systems are trending in demand because of their modern high-tech effectiveness at eliminating invisible airborne pathogens and allergens within hours. When most of us hear “probiotics,” we think of gut bacteria and the leaky gut supplements that work to promote bacterial balance in the gut. However, probiotics is a catch-all term for beneficial bacteria. Probiotics are helpful bacteria. BetterAir’s research has shown that by spraying probiotics into the air, these good bacteria particles can eliminate bad bacteria in multiple ways. BetterAir is priced at$300 and is said to be 100% organic, natural and safe for infants, humans and pets. You can buy it either through Amazon or from the official product website.

Is BetterAir worth the price? How does the BetterAir Probiotic Air Purifier compare with the Molekule Air or Dyson Pure Cool air purifier? Can you really use probiotics to cleanse the air of 99.99% of particles? Our BetterAir Probiotic Purifier review will shed light on many of the most important components of this new ‘restoring ecological balance' product that rebalances ‘indoor ecology by mimicking nature's healthy outdoor ecology'.

## What is the BetterAir Probiotic Purifier?

The BetterAir Probiotic Purifier is an air purification system designed for use in your home. Each air purifier covers 800 square feet of space, according to the product's manufacturers.

The device is marketed under several names, including: Biotica800, BetterAir, and Betterair. On Amazon, the device is marketed as BetterAir. On other websites, it’s marketed as the Biotica800 by BetterAir.

The manufacturer claims the purifier is “proven to eliminate the threat of pathogens on indoor surfaces and objects within 6 hours.” Just plug in the purifier, let it run, and it will eliminate 99.99% of certain particles within 6 hours.

“BetterAir has taken the science behind probiotics and transposed it to indoor environments,” explains the official website. “Now, you can rebalance your home and rid it of harmful pathogens and allergens naturally.” As the world's first environmental probiotic, BetterAir says the organic air purification product can actually boost your immune system too because it not only targets pathogens in the air, but on objects and surfaces as well. On the BetterAir website, it shows a statement directly from the CDC where 4 in every 5 infectious diseases are transmitted by touching a contaminated surface.

This is important because of the nature in how germs only breed on surfaces, not in the air. The Biotica800 ecological balancing system attacks these harmful pollutants at their source and in the same process reduces the likelihood of these allergens, viruses and pathogens to return.

Probiotics, for those out of the loop, are helpful bacteria that help balance your digestive tract. Some people take a probiotic supplement. Others eat probiotic yogurt.

BetterAir cites a study from Dr. <NAME>, PhD, Head of Antimicrobial Atmospheres at AirRx Anti Microbial Sciences, which proved that BetterAir “reduces FCV by more than 99.99% within 6 hours.”

“FCV” is not a common acronym in the air purification industry, and oddly, the official BetterAir website doesn’t clarify what it means. However, FCV stands for feline calicivirus, a contagious upper respiratory infection. It’s the flu for cats.

Each BetterAir unit comes with one cartridge. You plug the cartridge in, and it lasts approximately three months (90 days). The air purifier periodically dispenses liquid probiotic particles from the cartridge, and each cartridge contains billions of probiotic bacteria. You can buy replacement cartridges for $50 apiece. ## How Does BetterAir Work? BetterAir claims to work by spraying probiotic bacteria into the air. These probiotic bacteria start to work within 15 minutes. After six hours of use, the device has purportedly killed 99.99% of certain pathogens in the surrounding environment. BetterAir is smaller than most air purifiers. It looks like an extra long toaster. You can use the included wall mount. Or, you can stand the device on its stabilizer feet (both the wall mount and stabilizer feet are included with your purchase). A single BetterAir unit claims to cover 800 square feet. Here is a video to visualize how the probiotic air purifier by BetterAir works: ## How Do Probiotics Clean the Air? Here’s how the company explained the system in a 2019 interview with Forbes: “BetterAir is the only air purification system that treats the root cause of indoor air pollution. Allergens, pathogens, germs etc. do not grow in the air, but are born and propagate on surfaces and objects. Since they are microscopic they are then propelled into the air by minute movements of air such as the wave of a hand across a tabletop or fall off a foot upon stepping onto carpet.” The proprietary formula consumes organic allergens like pollen, mold spores, pet dander, and dust mite fecal matter. The probiotics also consume the food sources for these organic matter, which also causes them to die. BetterAir calls its probiotic cartridges “Enviro-Biotics”. Here’s how the official website explains how they work: “Betterair®'s Enviro-Biotics® repopulate the environment with probiotics consuming the food sources that all bacteria require resetting the indoor ecology to be healthier for our bodies.” In other words, BetterAir sprays liquid probiotic into the air, filling your air with bacteria that consume food sources, which means the pathogens in your air have nothing to consume, so they die. BetterAir claims to be superior to an ordinary air purifier, which cleanses the air and removes probiotics (beneficial bacteria) from the air, creating an overly sterile environment. BetterAir adds bacteria to the air, helping protect your home long-term. ## Scientific Evidence for BetterAir Biotica800 Probiotic Mist BetterAir makes big claims about its effectiveness, claiming to eliminate all pathogens and allergens and 99.99% of particles in the air, which is a claim only usually accessed by the most high-quality air purifiers. Is there any scientific evidence that BetterAir works as advertised? Or is BetterAir exaggerating the effectiveness of their product? First, BetterAir cites a study performed by Dr. <NAME>, PhD. That study purportedly found that BetterAir resulted in a “complete kill (>99.99%) of the pathogens” in the air. It also found that BetterAir reduced FCV by more than 99.99%. Dr. <NAME> is the Head of Antimicrobial Atmospheres at AirRx Antimicrobial Sciences, a Baltimore, Maryland-based company dedicated to developing aerosol and vapor disinfectants. You can view her LinkedIn page here. In 2017, that company received patents for an aerosol-based insecticide system. The company created a system that sprayed essential oils into the air to kill pests naturally. As far as we can tell, this document from the official BetterAir website is the only source for Dr. <NAME>’s study. It’s more of an email than a formal study, and the study was not published in any peer-reviewed journal. This raises some red flags, as it makes it impossible to verify the legitimacy of the study or the likelihood of its claims. In that study, Dr. Neff exposed glass slides to probiotic bacteria, then sealed the container and observed the results. Dr. Neff found that the probiotics killed the bacteria more than a placebo. The probiotics were particularly effectively against feline calicivirus (FCV). Probiotics caused a complete kill of that virus (more than 99.99% of the virus was removed) after 15 minutes of exposure and a 6 hour hold time. The study showed that probiotic bacteria, when applied to a slide, could kill certain viruses. It did not test the effects of airborne probiotic bacteria in an 800 square foot room – the area purportedly covered by BetterAir. Other studies, however, have shown that probiotics can cleanse the air effectively. In a 2019 article on BetterAir, Forbes described how BetterAir’s probiotic formula was based on eight years of research, claiming that the environmental probiotics are cultured in an ISO 9001 certified laboratory, then mixed with water as a suspension agent. You can view a summary of BetterAir’s research here. In one case study performed by the University of Wisconsin, researchers tested BetterAir at an assisted living facility, distributing the company’s probiotics through the facility’s HVAC system. After four months, researchers noticed a significant reduction in harmful bacteria. Other studies have been performed by researchers at the University of Oregon, researchers in Australia, a team in South Korea, and on El Al Airways international flights. Vice also summarized scientific evidence for BetterAir in a December 2016 article, concluding that: “At almost$300 a pop, these purifiers purportedly “clean” your home by releasing probiotics into the air. To be sure, the word purifier is a bit misleading: These devices don't actually purify the air with a filter that conventional air purifiers use. Instead, they pump “good” bacteria into the air to rebalance the ecosystem of bacteria in your house and make the air cleaner.”

Overall, there’s genuine scientific evidence that BetterAir’s probiotics work as advertised to re-balance the air and eliminate pathogens/bacterial particles.

## BetterAir Features & Benefits

BetterAir advertises all of the following features and benefits:

• 99.99% Effective. BetterAir claims to be 99.99% effective at removing certain particles from the air within 6 hours.
• Get Rid of Harmful Pathogens and Allergens. BetterAir claims to remove pathogens and allergens from your home, reducing your risk of disease and illness.
• Control Odor. In addition to attacking pathogens, BetterAir attacks odor-causing microbes like pet dander, mold spores, and more.
• Probiotic Cartridges.Your BetterAir purchase comes with one probiotic cartridge. You plug in the cartridge, and it dispenses liquid probiotic bacteria into the air, which purportedly removes 99.99% of airborne particles. Replacement cartridges cost $50 apiece. • Reduce Asthma Symptoms. BetterAir claims to reduce symptoms of asthma by removing certain particles from the air. • 800 Square Feet of Coverage. BetterAir claims to cover 800 square feet with each unit. • Works Immediately. BetterAir claims you’ll notice the difference immediately after plugging the unit in. ## BetterAir Creators BetterAir was founded in Israel in 2010. It launched in the United States in 2017. The company is led by President <NAME>. There seems to be very little information available online about the nature and history of the company behind this product. Consumers should be aware of this, particularly, if they are interested in a manufacturer with elevated levels of transparency. You can contact the company toll-free at (833) 692-3883. ## BetterAir Customer Reviews BetterAir is an unusual product. But what do customers have to say about it? Do customers claim the air purifier works as advertised? BetterAir has an average rating of 3.5 stars out of 5 on Amazon with 30+ reviews. 46% of reviewers gave it 5 stars, while 23% gave it 1 star. On the positive side of things, reviewers described BetterAir as “absolutely amazing”. One customer claimed that she and her family had been suffering from allergies for years without relief. After two weeks with BetterAir, they noticed significant improvement. BetterAir also seems popular among pet owners, especially pet owners with allergies to the hair of their furry friends. Other reviewers claim to have experienced improvements in sleeping, skin issues, and breathing. On the negative side of things, reviewers complain about poor craftsmanship. Multiple reviewers claim the air purifier broke after six weeks, for example, which is just beyond the return period. Other customers criticize the cartridge system, claiming that paying$50-$60 for a replacement cartridge every three months is too expensive. Overall, customers seem to either love or hate BetterAir, with very little reviews in-between. ## Frequently Asked Questions About BetterAir This section will answer a few of the most commonly asked questions many consumers have about BetterAir, as well as the company behind it. ### Q: How does BetterAir work? A: Like most air purifiers, this device is plugged into a wall and works to promote air quality by eliminating pathogens and other negative particles. The system relies on cartridges, which are a consistent expenditure but may help to improve the purification process. ### Q: Is BetterAir scientifically supported? A: This is a difficult question. Some scientific evidence may support the efficiency of probiotic air purifiers, but the evidence cited by this particular organization is difficult to verify. Users should conduct their own research before making a purchase. ### Q: How do you use BetterAir? A: Simply plug the device into a wall of the room you want to purify. Each device should cover around 800 square feet of space, so more than one device may be required to purify the air of an entire home. ## BetterAir Pricing The BetterAir Biotica800 is priced at$299.95.

Included in the box are the BetterAir base unit, a single Enviro-Biotics cartridge, a wall mount, stabilizer feet, power cord, and instruction manual.

Your purchase comes with one probiotic cartridge. Each cartridge lasts approximately 90 days (three months) of normal use. Each refill cartridge is \$50.

### BetterAir Refund Policy

BetterAir has a 60 day refund policy. You can request a refund within 60 days and receive 100% of your money back.

## Final Thoughts

Betterair is an air purification system that claims to remove 99.99% of certain particles from the environment using probiotics. The device specifically targets pathogens, allergens, germs, and more.

The manufacturer has spent eight years developing the probiotic formula, and there’s scientific evidence that the device works as advertised. Much like plug-in ionizers and the ultraviolet UV-C light sanitizers, home indoor air purification systems are a near-vital addition to consider adding should anyone be spending ample amounts of time in the same room, home and also consider the family's lung health well-being as well.


text:
Carbon Stuff

Help Support The Rocketry Forum:

REK

Alright I got three items for you. Two are 38mm diameter carbon tubes. One is all UNI and the other is all twill fabric. The twill is 31.5" long and the UNI is 20.875" long. Both of these still have the shrink wrap spirals on them. I sanded the twill in attempt to get rid of the spirals, but I stopped in worry of ruining the beauty of the twill. The UNI tube is still all glossy.

Finally I got some UNI fabric with a veil layer that I have no use for. The weight is 9oz. Per square yard. The veil layer is fiberglass. The fabric is 24" wide by 16" long.

I'm asking $60 plus$10 shipping. Paypal is prefered.

Pm if interested.

<NAME> - TRA Level 1 - Mariah 54 - CTI-I100 Red Lightning Longburn - 6,345 Feet


text:
# How to add colors and legends to plots with "Show" [duplicate]

I have lets say three plots A, B, C. I know that in order to plot them on top of each other I can use Show function and it will show them all together, Is there a way to change the color (and/or legend) of the plots within the Show command? without going back to the original Plot for each A, B and C and changing the color and legend.

I tried something like this

Show[{A, PlotStyle -> Black}, {B, PlotStyle -> Red}, {C, PlotRange -> Pink}]

But i got an error (Show::gcomb: Could not combine the graphics objects in ).

• Put PlotStyle -> correspondingstyle as option in each plot, and then use Show[A,B,C] Jan 30, 2020 at 18:56
• @Fraccalo while this is what I would have suggested as well, I think that Erf does not want to go back to the individual plots and if I understand the solution has to be at the level of Show
– kcr
Jan 30, 2020 at 19:00
• That is true @Darth_Bane , I don't want to go back to the each plot,
– Erf
Jan 30, 2020 at 19:02
• sorry didn't read properly, see answer below. Jan 30, 2020 at 19:12

a = Plot[x, {x, 0, 2}];
b = Plot[x^2, {x, 0, 2}];
c = Plot[x^3, {x, 0, 2}];

defaultMMAcolor=RGBColor[0.368417, 0.506779, 0.709798]

Show[a /. {defaultMMAcolor -> Black},
b /. {defaultMMAcolor -> Red},
c /. {defaultMMAcolor -> Pink}]

Is this something that you had in mind?

Generate a random plot

A = Plot[x^3, {x, -3, 3}]

And then

Show[A /. _RGBColor -> Red]

The above is the simplest resolution I know off and I have used in the past. I found this answer on this site. I am giving the link, where you can also find more sophisticated approaches you can check

Regarding the question at the comment: is the following acceptable?

Legended[Show[A /. _RGBColor -> Red], Placed[SwatchLegend[{Red, Green, Blue},
{"something"}], {1, 1}]]

• the problem with this approach is that it changes all the colors: say for example your plot is: a = Plot[x, {x, 0, 2}, AxesStyle -> Green]; with your code the axes color would be overwritten to Red Jan 30, 2020 at 19:16
• @Fraccalo that is a valid point. I did not know that because I don't change the colours of the axes. Thanks for pointing it out anyway
– kcr
Jan 30, 2020 at 19:19
• @Darth_Bane thank you, this actually works. But do you know how to add legends in similar way ?
– Erf
Jan 30, 2020 at 19:21
• @Erf I just want to point out that the other answer is better and more precise and I have voted for it. Just to be clear
– kcr
Jan 30, 2020 at 19:23
• @Erf while I still suggest that you try the other answer as it is more complete, I think I have answered the PlotLegends comment. please have a look
– kcr
Jan 30, 2020 at 19:31


text:
@RPi؁jeXgI2018.3.1
 悢RB܂͊eXgłB͂VȂ̃eXgłB̂UZ͊eNXŃeXgɌŌ̊wK̎ԂƂȂ܂Bwψ쐬̗\zuƏovɃ`WāAŏI`FbN܂BeXg׋ɂ̂ނ݂Ȃ̎p܂APڂ̋Ȃŗ͔͂łł傤HQڂɌĂƌߌ̎Ԃɏ߂Ă܂傤I

@QQU(j⋳2018.2.26
 یɎ⋳eȂŊJÂ܂BȒSɉāAñ{eBA̕ɂĂȂAeXgɌwK߂܂B

@QQR(jiHwKifI2018.2.23
 PɎg񂾐iHwKɂuEƃ|[gvKix̊wNfɌfĂ܂BeNX炳܂܂ȐEƂ̍i𐔓_IяoĂ܂BPNłȂÅwN̐kĂp܂B摜NbNƊg\܂B

@QQPijwψƎsψ̎g2018.2.21
 یɊwψƍZOwKsψ̃o[ŁAZOwK̐UԂ̃|X^[Â߂Ă܂BwψɂwψBeʐ^NX̒ԂzȂǂ͑ɂ܂Ƃ߂Ƃ߂Ă܂B܂sψɂȂǂ߂wN[V̂悤܂Ƃ߂Ƃ߂Ă܂Bꂼ̃NX炵̂f炵i̊y݂łB

@QQP(jeψ̕2018.2.21
 یɂQ̊eψJÂ܂B̒̉ɂāAψS犈e̕񍐂܂BAP[g{ψ܂BR̈ψŌψ͍ŏIłB

 P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 P15 P16 P17 P18 P19 P20 P21 P22 P23 P24 P25 P26 P27 P28 P29 P30 P31 P32 P33 P34 P35 P36 P37 P38 P39 P40 P41 P42 P43 P44 P45 P46 P47 P48 P49 P50 P51 P52 P53 P54 P55 P56 P57 P58 P59 P60 P61 P62 P63 P64 P65 P66 P67 P68 P69 P70 P71 P72 P73 P74 P75 P76 P77 P78 P79 P80 P81 P82 P83 P84 P85 P86 P87 P88 P89 P90 P91 P92 P93 P94 P95 P96 P97 P98 P99 P100 P101 P102 P103 P104 P105 P106 P107 P108 P109 P110 P111 P112 P113 P114 P115 P116 P117 P118 P119 P120 P121 P122 P123 P124
[ҏW]
CGI-design


text:
## Equality Set Projection: A new algorithm for the projection of polytopes in halfspace representation

In this paper we introduce a new algorithm called Equality Set Projection (ESP) for computing the orthogonal projection of bounded, convex polytopes. Our solution addresses the case where the input polytope is represented as the intersection of a finite number of halfplanes and its projection is given in an irredundant halfspace form. Unlike many existing approaches, the key advantage offered by ESP is its output sensitivity, i.e., its complexity is a function of the number of facets in the projection of the polytope. This feature makes it particularly suited for many problems of theoretical and practical importance in which the number of vertices far exceeds the number of facets. Further, it is shown that for non-degenerate polytopes of fixed size (dimension and number of facets) the complexity is linear in the number of facets in the projection. Numerical results are presented that demonstrate that high dimensional polytopes can be projected efficiently.

Year:
2004
Publisher:
Cambridge, Cambridge University Engineering Dept
Laboratories:


text:
# Simplify rational expressions: common monomial factors

Factor the numerator and the denominator of a rational expression and cancel out common monomial factors.

### Problem

Ivan and Javier manipulated the rational expression start fraction, 6, x, plus, 4, divided by, 12, x, plus, 4, end fraction. Their responses are shown below.
IvanJavier
start fraction, 3, x, plus, 2, divided by, 6, x, plus, 2, end fractionstart fraction, 6, x, divided by, 12, x, end fraction
Which student wrote an expression that is equivalent to the original expression?
Please choose from one of the following options.


text:
the cedar ledge

# Call Center Stress Recognition with Person-Specific Models

Date: March 27 2020

Summary: Assessment of stress in call center employees using skin conductance sensors to measure SCR/EDA.

Keywords: ##bibliography #eda #mit #affectivecomputing #medialab #stress #archive

# Bibliography

<NAME>, <NAME>, and <NAME>, "Call Center Stress Recognition with Person-Specific Models," in Affective Computing and Intelligent Interaction, vol. 6974, <NAME>, <NAME>, <NAME>, and <NAME>, <NAME>, Heidelberg: Springer Berlin Heidelberg, 2011, pp. 125–134.

They had a small cohort of nine individuals working at a call center using skin conductance sensors to measure EDA. They used exponential smoothing and chose a smoothing factor, $\alpha$, of $0.8$ – it was unclear why this was chosen. They also elected to normalize the skin conductance signals on each individual between $0$ and $1$ to reduce variability between the cohort members.