text:
## Introduction

Quantum coherence has been the focus of investigation in numerous fields such as quantum optics where the fundamental nature of coherence has been investigated using phase-space distributions and higher order correlation functions1,2,3. It was quantified recently in a quantum-information theoretic way4 and the modern view is that it is the broadest quantum properties and is at the root of various quantum quantifiers such as discord, entanglement, EPR steering, and Bell correlations5,6. A set of axioms were formally introduced which need to be satisfied by a coherence quantifier4. This gave rise to the field of resource theories of quantum coherence7,8,9,10, along with an explosion of interest in the measurement of coherence11,12,13,14,15 and its applications16,17,18,19.

Quantum coherence has some unique features not present in other quantifiers such as entanglement and quantum discord (for a review see ref. 9). One feature is that coherence is a basis dependent property and hence the amount of coherence depends upon the chosen measurement basis. Additionally, coherence can localize in a unipartite system as quantum superposition, or be present as correlations between different qubits12. For example, in a bipartite entangled state $$(\left|00\right\rangle +\left|11\right\rangle )/\sqrt{2}$$, the coherence is delocalized and cannot be attributed to any particular qubit. On the other hand, in a separable state $$\left|++\right\rangle =(\left|0\right\rangle +\left|1\right\rangle )(\left|0\right\rangle +\left|1\right\rangle )/2$$, the coherence is localized within the qubits. In fact, a maximally entangled state has only global coherence and no local coherence; meanwhile product states are the opposite. This example illustrates the presence of a trade-off between the local and the global coherence in a quantum system. This trade-off is the simplest case and highlights the different complementary distributions of coherence in two qubits. In a multipartite system, there are more possible distributions of coherence and hence other types of trade-off relations.

In this work, we use an adiabatically evolved spin system to experimentally measure coherence and investigate the trade-off relations in tripartite systems. The tripartite trade-off relations are an application of the theory presented in ref. 12, where it was found that the total coherence could be decomposed into local and global coherence. We theoretically define three new inequality trade-off relations for the tripartite system which further characterize the quantum state. In the experiment, we consider two different classes of quantum systems with two and three-body interactions. The quantum coherence is measured at different stages of an adiabatic evolution and various trade-off relations are verified. An example of this process is shown in Fig. 1a, where initially the coherence is completely localized within the qubits. When the system is adiabatically evolved, it has both local and global coherence. At the end of the adiabatic evolution, the system has only global coherence. In addition to verifying the trade-off relations and coherence distributions, we analyze the monogamy of coherence. Monogamy, first introduced in the context of entanglement20,21, implies that, when Alice and Bob are maximally entangled, they are impossible to be simultaneously entangled with a third party Charlie. This concept was later extended to quantum correlation22,23 and quantum coherence12. We show that using the various coherence quantifiers and monogamy, one can reveal that despite the apparent differences between the two Hamiltonians, the decompositions of the coherence are in fact similar. This illustrates the utility of our approach where coherence can be used to characterize a state to reveal hidden similarities between different systems.

We note that several other works have examined quantum coherence experimentally recently24,25,26,27. A coherence witness was introduced in ref. 24 to detect the total coherence through a violation of Leggett–Garg type inequality28. Meanwhile the amount of coherence in a single photonic qubit was measured experimentally25 using the robustness of coherence13. These works studied coherence in unipartite systems and did not analyze the coherence decompositions, their distribution and the consequent trade-off relations in multipartite systems. Such experimental demonstrations, and our work in this paper, show that quantum coherence can be an effective characterization tool for quantum states. In particular, using the coherence decomposition methods as introduced in this work and ref. 12, one can obtain insights into the nature of the quantum state that may not be obvious from the examination of the density matrix.

## Results

### Models

In this work, we study two different tripartite quantum systems. The first system is an Ising model described by the Hamiltonian

$${H}_{{{{{zz}}}}}(t)={\omega }_{z}\mathop{\sum}\limits_{i=1,2,3}{S}_{i}^{z}+{\omega }_{x}\mathop{\sum}\limits_{i=1,2,3}{S}_{i}^{x}+2{J}_{2}(t)\mathop{\sum}\limits_{1\le i < j\le 3}{S}_{i}^{z}{S}_{j}^{z},$$
(1)

where $${S}_{i}^{z/x}$$ is the nuclear spin in the z/x-direction and J2 represents the two-body interaction strength with ωz = −2 being the magnetic field in the longitudinal direction. A small transverse field ωx = 0.1 is provided to lift the degeneracy of the ground state so that the adiabatic evolution is possible. Initially when we set J2 = 0 and ωxωz, the ground state is nearly a separable state $$\left|000\right\rangle$$. The state is adiabatically evolved by increasing J2 from 0 to ωz in order to obtain a state close to $$\left|W\right\rangle =(\left|001\right\rangle +\left|010\right\rangle +\left|100\right\rangle )/\sqrt{3}$$ at the end of the evolution29. The fidelity between the final ground state and $$\left|W\right\rangle$$ is 0.9978 while our experimental final state has a fidelity to $$\left|W\right\rangle$$ as high as 0.9578.

The second quantum system we consider has the following form:

$${H}_{{{{{zzz}}}}}(t)={\omega }_{x}\mathop{\sum}\limits_{i=1,2,3}{S}_{i}^{x}+4{J}_{3}(t){S}_{1}^{z}{S}_{2}^{z}{S}_{3}^{z},$$
(2)

where J3 is the three-body interaction strength which varies from 0 to 5 during the adiabatic evolution. The corresponding initial and final ground states are $$\left|---\right\rangle$$ and, in the sense of zero-order perturbation, $$\left|G\right\rangle =(\left|001\right\rangle +\left|010\right\rangle +\left|100\right\rangle +\left|111\right\rangle )/2$$, respectively. The fidelity between the final ground state and $$\left|G\right\rangle$$ is 0.9996 while our experimental final state has a fidelity to $$\left|G\right\rangle$$ as high as 0.9661. The final state has both bipartite and tripartite coherences. For both the quantum systems, the coherence is measured at each stage of the evolution using quantum tomography methods.

### Quantifying coherence

To measure coherence we use the square root of quantum version of the Jensen–Shannon divergence (QJSD)30,31,32,33

$${{{\mathcal{D}}}}(\rho ,\sigma )=\sqrt{\frac{1}{2}\left[{{{{\mathcal{S}}}}}_{{{{\rm{r}}}}}(\rho \parallel (\rho +\sigma )/2)+{{{{\mathcal{S}}}}}_{{{{\rm{r}}}}}(\sigma \parallel (\rho +\sigma )/2)\right]}.$$
(3)

Here ρ and σ are two density matrices of the same dimensionality and $${{{{\mathcal{S}}}}}_{{{{\rm{r}}}}}({\rho }_{1}\parallel {\rho }_{2})={{{\rm{tr}}}}{\rho }_{1}{{\mathrm{log}}}\,({\rho }_{1}/{\rho }_{2})$$ is the quantum relative entropy. Using this measure, the total coherence in the system is

$${{{{\mathcal{C}}}}}_{{{{\rm{T}}}}}(\rho )\equiv {{{\mathcal{D}}}}(\rho ,{\rho }_{{{{\rm{d}}}}}),$$
(4)

where ρ is the density matrix and $${\rho }_{{{{\rm{d}}}}}={\sum }_{k}\left\langle k\right|\rho \left|k\right\rangle \left|k\right\rangle \left\langle k\right|$$ is the diagonal density matrix with $$\left|k\right\rangle$$ representing the eigenstates of $${S}_{j}^{z}$$. The global and local coherence are defined, respectively, as12

$${{{{\mathcal{C}}}}}_{{{{\rm{G}}}}}(\rho )\equiv {{{\mathcal{D}}}}(\rho ,\pi (\rho ));\qquad {{{{\mathcal{C}}}}}_{{{{\rm{L}}}}}\equiv {{{\mathcal{D}}}}(\pi (\rho ),{[\pi (\rho )]}_{{{{\rm{d}}}}}).$$
(5)

Here π(ρ) ≡ iρi, where $${\rho }_{i}={{{{\rm{tr}}}}}_{\forall j\ne i}\,\rho$$ and footnotes i, j are indices of subsystems. Subscript d indicates the diagonal part of the density matrix in the Sz-basis. In terms of the coherence trade-off, the more relevant quantity is the absolute coherence defined as

$${{{{\mathcal{C}}}}}_{{{{\rm{A}}}}}(\rho )\equiv {{{\mathcal{D}}}}(\rho ,{[\pi (\rho )]}_{{{{\rm{d}}}}}),$$
(6)

which is different from the total coherence $${{{{\mathcal{C}}}}}_{{{{\rm{T}}}}}$$ and is the total amount of coherence in the product basis. The reference state [π(ρ)]d for absolute coherence contains neither coherence or correlations between the subsystems, while the reference state for total coherence ρd can potentially contain classical correlations. Since our measure $${{{\mathcal{D}}}}(\rho ,\sigma )$$ satisfies the triangle inequality for a multipartite system up to five qubits according to numerical studies34, we have the trade-off relation (see Fig. 1b).

$${{{{\mathcal{C}}}}}_{{{{\rm{A}}}}}\le {{{{\mathcal{C}}}}}_{{{{\rm{L}}}}}+{{{{\mathcal{C}}}}}_{{{{\rm{G}}}}}.$$
(7)

The total coherence $${{{{\mathcal{C}}}}}_{{{{\rm{T}}}}}$$ does not satisfy this trade-off relation since the reference state π(ρ) is used.

One of the interesting aspects of tripartite systems is that coherence can be distributed in different ways. It is well-known that for entanglement, GHZ and W states are two different classes of tripartite entangled states35. The entanglement in a GHZ state is genuinely tripartite, whereas in a W state, the entanglement is bipartite in nature. In this context, it is interesting to examine the coherence additionally in a bipartite fashion. To this end, we introduce a geometrical picture as shown in Fig. 1b, by considering ρ1ρ23. We evaluate the coherence between qubit 1 and the bipartite block 23 according to

$${{{{\mathcal{C}}}}}_{1:23}\equiv {{{\mathcal{D}}}}(\rho ,{\rho }_{1}\otimes {\rho }_{23}),\quad {{{{\mathcal{C}}}}}_{2:3}\equiv {{{\mathcal{D}}}}({\rho }_{23},{\rho }_{2}\otimes {\rho }_{3}).$$
(8)

Here $${{{{\mathcal{C}}}}}_{1:23}$$ measures the coherence between qubit 1 and the bipartite block 23 and $${{{{\mathcal{C}}}}}_{2:3}$$ is the coherence in the bipartite block 23. We note that including ρ1 does not make any difference to the global coherence, and $${{{{\mathcal{C}}}}}_{2:3}={{{\mathcal{D}}}}({\rho }_{1}\otimes {\rho }_{23},\pi (\rho ))$$; the proof is given in the Supplementary Note 1. Complementary to the coherence $${{{{\mathcal{C}}}}}_{1:23}$$ we have the contribution

$${{{{\mathcal{C}}}}}_{{{{\rm{A}}}}}^{1:23}\equiv {{{\mathcal{D}}}}({\rho }_{1}\otimes {\rho }_{23},{[\pi (\rho )]}_{{{{\rm{d}}}}}).$$
(9)

Based on these coherence distributions we have the following trade-off relations

$${{{{\mathcal{C}}}}}_{{{{\rm{A}}}}}\le {{{{\mathcal{C}}}}}_{1:23}+{{{{\mathcal{C}}}}}_{{{{\rm{A}}}}}^{1:23};\qquad {{{{\mathcal{C}}}}}_{{{{\rm{A}}}}}^{1:23}\le {{{{\mathcal{C}}}}}_{2:3}+{{{{\mathcal{C}}}}}_{{{{\rm{L}}}}}.$$
(10)

Since $${{{{\mathcal{C}}}}}_{1:23}$$ and $${{{{\mathcal{C}}}}}_{2:3}$$ are global coherences, they give another trade-off relation

$${{{{\mathcal{C}}}}}_{{{{\rm{G}}}}}\le {{{{\mathcal{C}}}}}_{1:23}+{{{{\mathcal{C}}}}}_{2:3}.$$
(11)

The four equations in (7), (10) and (11) correspond to the four triangles that are present in Fig. 1b. They are also four trade-off relations that can be verified for the generated tripartite states.

The variation of the coherence contributions during the evolution is shown in Fig. 2. There are mainly four types of errors leading to the deviation between experimental and theoretical results: initial state imperfection, decoherence, pulse error and measurement error. Please refer to Supplementary Note 4 for a detailed analysis. For Hamiltonian Hzz, at J2 = 0, the interactions are turned off and the ground state is a product state. It is locally rotated from the state $$\left|000\right\rangle$$ ($${{{{\mathcal{C}}}}}_{{{{\rm{L}}}}}={{{{\mathcal{C}}}}}_{{{{\rm{G}}}}}=0$$), due to the transverse field ωx which induces a small local coherence. We observe that there are two regions, J2 [0, 1) with $${{{{\mathcal{C}}}}}_{{{{\rm{L}}}}}$$ and $${{{{\mathcal{C}}}}}_{{{{\rm{G}}}}}$$ increasing and J2 [1, 2] with $${{{{\mathcal{C}}}}}_{{{{\rm{L}}}}}$$ decreasing and $${{{{\mathcal{C}}}}}_{{{{\rm{G}}}}}$$ increasing. The crossover at J2 = 1 corresponds to a quantum phase transition in a spin system with two-body interactions29. For the Hamiltonian Hzzz, at J3 = 0, the ground state is $$\left|---\right\rangle$$, a coherent product state and hence $${{{{\mathcal{C}}}}}_{{{{\rm{G}}}}}=0$$ and $${{{{\mathcal{C}}}}}_{{{{\rm{L}}}}}$$ is maximal. At J3 = 5, the ground state is nearly $$\left|G\right\rangle$$ for which $${{{{\mathcal{C}}}}}_{{{{\rm{L}}}}}=0$$ and $${{{{\mathcal{C}}}}}_{{{{\rm{G}}}}}$$ is maximal. The two distinct regions of Hzzz are J3 [0, 0.25) ($${{{{\mathcal{C}}}}}_{{{{\rm{L}}}}}\, > \,{{{{\mathcal{C}}}}}_{{{{\rm{G}}}}}$$) and J3 [0.25, 5] ($${{{{\mathcal{C}}}}}_{{{{\rm{L}}}}}\, < \,{{{{\mathcal{C}}}}}_{{{{\rm{G}}}}}$$) with the crossover at J3 = 0.25. They are related to a critical point at J3 = ωx for a spin system with three-body interaction in the thermal dynamic limit29,36,37,38,39,40,41. We note that there are regions where $${{{{\mathcal{C}}}}}_{{{{\rm{G}}}}}\, > \,{{{{\mathcal{C}}}}}_{{{{\rm{T}}}}}$$ for both Hzz and Hzzz. This is due to our definition of global coherence, where all correlations between the qubits are broken by forming a product state, whereas in the definition of total coherence, classical correlations can be present in the decohered state. This verifies that $${{{{\mathcal{C}}}}}_{{{{\rm{A}}}}}$$ is the more appropriate quantity in the context of trade-off relations.

To visualize the expected trade-off relations according to (4), (7) and (11), we look at the ratios $${{{{\mathcal{C}}}}}_{{{{\rm{G}}}}}/{{{{\mathcal{C}}}}}_{{{{\rm{L}}}}}$$, $${{{{\mathcal{C}}}}}_{2:3}/{{{{\mathcal{C}}}}}_{{{{\rm{L}}}}}$$, $${{{{\mathcal{C}}}}}_{1:23}/{{{{\mathcal{C}}}}}_{{{{\rm{A}}}}}^{1:23}$$ and $${{{{\mathcal{C}}}}}_{2:3}/{{{{\mathcal{C}}}}}_{1:23}$$ using both experimental data and the corresponding theoretical results as shown in Fig. 3. We observe three types of trade-off behavior corresponding to complete, partial and no trade-off. For ratios of $${{{{\mathcal{C}}}}}_{{{{\rm{G}}}}}$$ and $${{{{\mathcal{C}}}}}_{2:3}$$ to $${{{{\mathcal{C}}}}}_{{{{\rm{L}}}}}$$, there is a complete trade-off between these quantities, since there is a complete exchange from locally to collectively distributed coherence. Meanwhile the comparison of $${{{{\mathcal{C}}}}}_{2:3}$$ and $${{{{\mathcal{C}}}}}_{{{{\rm{A}}}}}^{1:23}$$ to $${{{{\mathcal{C}}}}}_{1:23}$$ only results in a partial trade-off, where the ratios saturate to a finite value. In these cases, since both quantities in the ratio are types of global coherence, the ratios saturate to these particular values decided by large J2, J3. For Hamiltonian Hzz, the ratio $${{{{\mathcal{C}}}}}_{2:3}/{{{{\mathcal{C}}}}}_{1:23}$$ remains constant throughout. We attribute this to the fact that for this Hamiltonian, there is a complete qubit symmetry, such that the entangled component of the state is always of the form of a W state. Hence when comparing two types of correlation-type coherences, although the amount of coherence both become small as J2 → 0, their ratio remains the same.

The different kinds of coherences can also be visualized geometrically as shown in Fig. 3c, d. Here, we plot the various coherences by assigning them Euclidean distances in three dimensional space. From the results, we find that while Hzz and Hzzz have different kinds of interactions, their coherence distributions evolve similarly. The general behavior is that the states π(ρ) and ρ1ρ12 start in the vicinity of ρ, then eventually move to a location near [π(ρ)]d, along different trajectories. The primary difference between the two Hamiltonians is that Hzzz always has a constant $${{{{\mathcal{C}}}}}_{{{{\rm{A}}}}}$$, hence the size of the tetrahedron is of the same order, whereas for Hzz the tetrahedron starts from a point. However, apart from the overall magnitude of the coherence, the distribution of coherences are remarkably similar for both cases.

### Monogamy of coherence

The monogamy of coherence describes the trade-off between the bipartite and tripartite global coherences of a three-body quantum system12. We can quantify the monogamy according to $$M={{{{\mathcal{C}}}}}_{1:2}+{{{{\mathcal{C}}}}}_{1:3}-{{{{\mathcal{C}}}}}_{1:23}$$, where M > 0 corresponds to a polygamous system and M ≤ 0 to a monogamous system. The monogamy of coherence for the two Hamiltonians is shown in Fig. 3a, b. We find that the quantum systems are polygamous for every value of the interaction parameter except for the initial value J2, J3 = 0. This points to the fact that for both the quantum systems, the most dominant form of global coherence is the bipartite global coherence. Since the coherences $${{{{\mathcal{C}}}}}_{1:23}$$ and $${{{{\mathcal{C}}}}}_{2:3}$$ are global coherences, it is only natural that they are related to $${{{{\mathcal{C}}}}}_{{{{\rm{G}}}}}$$, the total global coherence as explained in (11). This confirms the picture provided by Fig. 3c, d, that the coherence generated in the two Hamiltonians is of the same type. This arises fundamentally because of the similar nature of the $$\left|W\right\rangle$$ and $$\left|G\right\rangle$$ state, which both have an bipartite-like entanglement structure.

## Discussion

We extended the notion of coherence trade-offs introduced in ref. 12 and experimentally studied all the trade-offs that are possible with the four-point decompositions as shown in Fig. 1b. Each point in the diagram corresponds to removing a coherence contribution. For example, the state π(ρ) removes all the inter-qubit coherence and the state [π(ρ)]d removes all the coherence including that lying within the qubits. Since we are dealing with a tripartite system, we further performed a bipartite decomposition where the coherence between site 1 and bipartite block 23 is removed. Our results point to the fact that the trade-off relations are generic behavior and are always obeyed as we move from a separable state to an entangled state. The trade-off behavior is also consistent with approaches where coherence is considered a resource, and coherence is converted into different forms42, which may have different sensitivities to decoherence19,43. We also examined the distribution of global coherence using the property of monogamy of coherence and it was found that both the states were polygamous except when the interactions were turned off. The characterization of a quantum system through the coherence distribution diagrams in Fig. 3c, d was shown to be an effective tool to visualize the quantum state.

It is interesting to note that while Hzz and Hzzz have different types of interactions and different initial states, the final quantum states have similar quantum properties. In fact, they are both highly entangled states and symmetric under spin permutations, which is not necessarily obvious by simply examining their wavefunctions. For instance, the final state of Hzzz is $$\left|G\right\rangle =(\left|001\right\rangle +\left|010\right\rangle +\left|100\right\rangle +\left|111\right\rangle )/2$$, which is not a well-known entangled state in comparison to the W state, the final state of Hzz. As a matter of fact, $$\left|G\right\rangle$$ is a result of a local Hadamard operation of a GHZ state, which is more entangled than a W state. By comparing the two final states’ coherence contributions quantitatively, it is revealed that the global coherence of Hzz is lower than that of Hzzz. Hence, using the coherence decompositions and the trade-off relations, one can gain insights into the essential character of a given state, which may not be obvious simply by examining the wavefunction.

## Methods

### Setup

We experimentally simulate Hzz and Hzzz with nuclear spin qubits in this work. Diethyl fluoromalonate molecules are used to perform the target adiabatic evolution via a Trotter decomposition on a 400 MHz (9.4 T) NMR spectrometer at 303K. The molecular structure of diethyl fluoromalonate is shown in Fig. 4a. The three nuclear spins 13C, 1H and 19F in the molecule acts as the qubits. The natural Hamiltonian of the system is

$${H}_{{{{\rm{spin}}}}}=\mathop{\sum}\limits_{i=1,2,3}2\pi {\delta }_{i}{S}_{i}^{z}+\mathop{\sum}\limits_{1\le i\le j\le 3}2\pi {J}_{ij}{S}_{i}^{z}{S}_{j}^{z}$$
(12)

where δi is the chemical shift of the nuclear spin and Jij is the coupling between the ith and the jth nucleus as given in Fig. 4b.

In the first step of the experiment, a pseudopure state (PPS) of the form $$\rho =(1-\mu )I/8+\mu \left|\psi \right\rangle \left\langle \psi \right|$$ is prepared from thermal equilibrium state using a line-selective method44, where $$\left|\psi \right\rangle$$ is an arbitrary pure state. Here the mixing parameter μ ≈ 10−5 and I denotes the 8 × 8 identity matrix. The adiabatic pathway is numerically optimized to generate the desired ground state. The schematic diagram of the sequence to fulfill Hzz and Hzzz are shown in Fig. 4c, d. At each stage, the corresponding density matrices are reconstructed using tomographic techniques.

### Sequence design

The adiabatic evolution is performed in discrete steps in the experiment29,45,46. We label Hamiltonians Hzz and Hzzz as Hk, where index k {zz, zzz}. Notice that Hk can be decomposed into different basis, $${H}_{k}={H}_{k}^{x}+{H}_{k}^{y}$$, in which $${H}_{k}^{x}$$ contains only $${S}_{i}^{x}$$ and $${H}_{k}^{y}$$ contains only $${S}_{i}^{y}$$. The evolution of each segment $${U}_{k,\exp }({t}_{m})$$, is a Trotter expansion of the ideal one Uk,ide(tm), which can be expressed as

$$\begin{array}{lll}{U}_{k,{{{\rm{ide}}}}}({t}_{m})\,=\,{{\rm{e}}}^{-{\rm{i}}[{H}_{k}^{x}+{H}_{k}^{z}({t}_{m})]{\tau }_{k}}\\ \qquad\qquad\,\,\,\,=\,{{\rm{e}}}^{-{\rm{i}}{H}_{k}^{x}{\tau }_{k}/2}{{\rm{e}}}^{-{\rm{i}}{H}_{k}^{z}({t}_{m}){\tau }_{k}}{{\rm{e}}}^{-{\rm{i}}{H}_{k}^{x}{\tau }_{k}/2}+O({\tau }^{3})\\ \qquad\qquad\,\,\,\,=\,{U}_{k,\exp }({t}_{m})+O({\tau }^{3})\end{array}$$
(13)

where τk is the interval of each step and m [0, Mk] is the index of each step. We use a refocusing scheme to achieve each step in our work. In this method, tuned pulses are applied during each Trotter slice, and the Hamiltonian in each short time period is accurately controlled.

For Hamiltonian (1), the quantum system is adiabatically evolved by tuning the two qubit interaction strength adiabatically over the range [0, 2]. Experimentally, the adiabatic state transfer (ASP) is performed in discrete steps, such that J2(t) assumes discrete value J2(tm) with m = 0, . . . , Mzz. At each time step, the evolution is generated using multipulse sequence $${U}_{{{{{zz,{\rm{exp}}}}}}}({t}_{m})$$ using Trotter expansion formula as described in Eq. (13). The resulting Hamiltonian is

$$\begin{array}{lll}{H}_{{{{{zz}}}}}^{x}\,=\,{\omega }_{x}\mathop{\sum}\limits_{i=1,2,3}{S}_{i}^{x}, {H}_{{{{{zz}}}}}^{z}({J}_{2}({t}_{m}))\\ \qquad=\,{\omega }_{z}\mathop{\sum}\limits_{i=1,2,3}{S}_{i}^{z}+2{J}_{2}({t}_{m})\mathop{\sum}\limits_{1\le i\le j\le 3}{S}_{i}^{z}{S}_{j}^{z}\end{array}$$
(14)

A schematic description of the refocusing scheme is shown in Fig. 4c where the narrow unfilled rectangles denote π/2 pulses, and the wide ones show π pulses. By defining dij = 1/(2Jij), the width of filled pulse in the figure are all ωxτzz/2 and the radio-frequency offsets for three channels are set as FQ1m = ωz/(4J2(tm)d12), FQ2m = ωz/(4J2(tm)(d12 + d13 + d23)) and FQ3m = ωz/(4J2(tm)d23), the delays are $${\tau }_{m}^{1}=\frac{{J}_{2}({t}_{m}){\tau }_{{{{{zz}}}}}}{\pi }\times ({d}_{12}+{d}_{23}),\,{\tau }_{m}^{2}=\frac{{J}_{2}({t}_{m}){\tau }_{{{{{zz}}}}}}{\pi }\times ({d}_{12}+{d}_{13})$$, and $${\tau }_{m}^{3}=\frac{{J}_{2}({t}_{m}){\tau }_{{{{{zz}}}}}}{\pi }\times ({d}_{13}+{d}_{23})$$.

Next we consider the Hamiltonian of a tripartite quantum system with J3 being the three-body interaction strength as shown in Eq. (2). The interaction parameter J3 is tuned adiabatically in the range [0, 5]. Again, we use a discrete refocusing scheme in which J3(t) is discretized into tm, m = 0, . . . , Mzzz. The schematic diagram is shown in Fig. 4d in which the width of the filled pulse are all ωxτzzz/2 and the delay $${d}_{m}=\frac{{J}_{3}({t}_{m}){\tau }_{{{{{zzz}}}}}}{\pi }\times {d}_{12}$$.

From above, one can see that the unit of studied quantities like J2, ωz, τk always cancel out when they come into the parameters of the experiment. This means that the units of them do not matter in the experiment, only the relative relations between them matter. So, they are in arbitrary units and we need not mention the unit during discussion. We use 0.7 and 0.4 as the value of τzz and τzzz when we design the experimental sequences. Please refer to Supplementary Note 3 for the optimization details of the parameters.


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Global dynamics beyond uniform hyperbolicity

The conference will focus on the global qualitative study (topological and ergodic) of differentiable dynamical systems, especially diffeomorphisms and vector fields, that are not uniformly hyperbolic. Among the main topics to be covered are:

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Answer to the main question: It is a well regarded fact that the terminology unified electroweak interaction is a bit of an abuse of terminology. What the term means is that both Quantum Field Theories, the Hypercharge ($U(1)_Y$) and Weak ($SU(2)_L$), are unified in a common framework, which predicts the low energy electromagnetism ($U(1)_{em}$) through the ...

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For anyone interested, I found a good document with all Feynman rules and different notations: http://arxiv.org/abs/1209.6213

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First of all, the oblique parameters S,T and U are defined to be zero within the Standard Model (SM). This means, that the SM is a reference and therefore, these parameters are indications for physics beyond the Standard Model (BSM). They account for corrections in the vacuum polarizations of the EW gauge bosons and are chosen in a way, that different BSM ...

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Research Papers

# Analytic and Geometric Technique for the Singularity Analysis of Multi-Degree-of-Freedom Spherical Mechanisms

[+] Author and Article Information
Raffaele Di Gregorio

Department of Engineering,
University of Ferrara,
Via Saragat, 1,
FERRARA 44122, Italy
e-mail: rdigregorio@ing.unife.it

Following [3,4], the connectivity between two links of a mechanism is the dof number of the relative motion between the two links. Hereafter, the phrase “local connectivity” will denote the dof number of the instantaneous relative motion at a given mechanism configuration.

It is worth noting that, since the output link belongs to the mechanism, its local connectivity, gq, must respect the inequality: gq ≤ min(f, nq), where f is the dof number of the unconstrained rigid body in the space the mechanism's links move in.

Superposition principle states that “In a linear and homogeneous system, say

$Ax=By,$
IF (x1,y1) and (x2,y2) are both solutions, THEN (k1x1+k2x2, k1y1+k2y2) where k1 and k2 are any two constant coefficients is solution, too.” [9].

It is worth noting that relationship (1) defines ujk and ωjk but a sign (i.e., {ujk, ωjk} and {-ujk, -ωjk} yield the same angular velocity ωjk)

It is worth stressing that Eq. (2) yields a type-I singularity iff d is equal to zero, a type-II singularity iff c is equal to zero and, accordingly, a type-III singularity iff c and d are both equal to zero.

Hereafter, mechanism's geometric variables that are not input variables will be called “secondary variables” (e.g., nonactuated joints variables).

Hereafter, the notation (P, a) denotes an oriented line passing through point P and with the direction of the unit vector a.

In fact, in this case, since link 4 is at a stationary configuration, if link 5 is locked link 6 cannot move (i.e., this is not a type-II singularity); vice versa, if link 6 is locked link 5 cannot move (i.e., this is not a type-I singularity).

In fact, in this case, since link 4 is at a stationary configuration, if link 3+5 is locked link 6 cannot move (i.e., this is not a type-II singularity); vice versa, if link 6 is locked link 3+5 cannot move (i.e., this is not a type-I singularity).

1The right subscript m × n denotes the size of the matrix. In this case, the number of rows, m, is equal to the number of velocity constraints, and the number of column, n, is equal to the total number of joint variables. In a mechanism, n is always greater than m since a number of independent constraints equal to n would lead to zero mobility (i.e., to a structure).

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANISMS AND ROBOTICS. Manuscript received September 5, 2013; final manuscript received August 30, 2014; published online December 4, 2014. Assoc. Editor: Andrew P. Murray.

J. Mechanisms Robotics 7(3), 031008 (Aug 01, 2015) (9 pages) Paper No: JMR-13-1175; doi: 10.1115/1.4028625 History: Received September 05, 2013; Revised August 30, 2014; Online December 04, 2014

## Abstract

Mechanisms' instantaneous kinematics is modeled by linear and homogeneous mappings whose coefficient matrices are also meaningful to understand their static behavior through the virtual work principle. The analysis of these models is a mandatory step during design. The superposition principle can be used for building and studying linear and homogeneous models. Here, multi-degree-of-freedom (multi-DOF) spherical mechanisms are considered. Their instantaneous-kinematics model is written by exploiting instantaneous-pole-axes' (IPA) properties and the superposition principle. Then, this general model is analyzed and an exhaustive analytic and geometric technique to identify all their singular configurations is deduced. Eventually, the effectiveness of the deduced technique is shown with two relevant case studies.

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## Figures

Fig. 1

General scheme: O is the center of the spherical motion; with reference to the angular velocity of the output link, u and ω are its unit vector and signed magnitude, respectively, in the actual case, whereas ui and ωi, for i = 1,…,g, are its unit vector and signed magnitude, respectively, in the ith single-DOF mechanism

Fig. 2

Spherical 6R single-loop linkage: notations

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text:
OpenStudy (anonymous):

How to graph r=6sin2θ?

OpenStudy (anonymous):

you have to first find the amplitude and period.

OpenStudy (anonymous):

Can you help me with that process?

OpenStudy (anonymous):

yeah sure, usually amplitude is in front of sin or cos, and it is never negative. so do you know the amplitude now?

OpenStudy (anonymous):

6

OpenStudy (anonymous):

yeah, okay now for period. period is $2\pi/w$

OpenStudy (anonymous):

I have no clue. :(


text:
# Problems with Excel

23 thoughts
last posted Aug. 26, 2016, 2:18 p.m.

16 earlier thoughts

0

Functions should not ‘live’ in cells. Function definitions should simply be part of the spreadsheet’s code .

The text-representation of the sheet would include not only a series of cell definitions (coordinates of cells and their contents) but proper functions as well.

6 later thoughts


text:
In an experiment with sonometer a tuning fork of frequency 256 Hz resonates with a length of 25 cm and another tuning fork resonates with a length of 16 cm. Tension of the string remaining constant the frequency of the second tuning fork is

(1) 163.84 Hz

(2) 400 Hz

(3) 320 Hz

(4) 204.8 Hz

(2) In case of sonometer frequency is given by

$n=\frac{p}{2l}\sqrt{\frac{T}{m}}$$\frac{{n}_{2}}{{n}_{1}}=\frac{{l}_{1}}{{l}_{2}}⇒{n}_{2}=\frac{25}{16}×256=400\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}Hz$

Difficulty Level:

• 0%
• 70%
• 26%
• 6%


text:
# Outreachy Final Blog : An awesome Journey

by lekhidugtal | | 110 views | 6 comments | 31 Aug 12:32

lekhidugtal was awarded the Basic Barnstar by gauravano for their work in this research note.

Here in this post I would be summing up my contribution for this summer. We do have this one single link to checkout the whole contribution in project -

https://github.com/publiclab/plots2/projects/10

### What was the aim ?

My project was about revamping the design of various sections pages, sidebar etc. Me and gautamig did it together. We divided an issue of one section into small checklists and get ourselves assigned with those small parts and then starting working on it. We have covered Tags, People,Profile,Individual tags, Questions pages along with sidebar. Along with that, it included various bug-fixing.

### What I did and What I learnt ?

The first section we covered in the summer of code was people's section as we had already begin working on the major parts. I started off with my parts which were addition of cards for the list of users which included info and other stuff. #5740. It was great experience with review process and working on responsiveness (@~_@~). After that I moved forward with small fixes of responsiveness. #5754. Then till now, we had a card with details like user info but not tags the user is associated with. So I added in the most recent tags by user on its card #5765.

Gautami completed her portions and then we went to next section.

Yay !! Done with one section. (✖╭╮✖)

The next Section was Questions section , I created a new shadow page for questions page and added first section which included few info #5787. The idea of shadow thing still looks so COOL !!! (⊙⊙)(☉_☉)(⊙⊙)

After that I added featured Question part #5861 for which I took bit of time as it included changes in lots of cells (^_^;). I need to check the whole path of calling and adding a shadow function in back-end for popular questions only. We needed to show two features questions and this was third section for questions part.

Yay !! Done with with another one (✖╭╮✖)

Then we went to Individual Section page. I started off with #5903 absolute fixing of wikis page if included for that tag in background and if not then add default grey with the addition of followers to that particular tag. I guess this PR of mine did create lots of bugs and broke many major things. (. _ . )( . _ .)(⌣̩̩́_⌣̩̩̀)

Then we went onto to next, I moved forward to as per new design draft changing navs, tabs to dropdowns and vice versa. One of that included distincting between notes questions [#5925] (https://github.com/publiclab/plots2/pull/5925). I wrote my very first extra-small screenshot test here after checking out the code written by jywarren. (✖╭╮✖) It was of getting redirected to tag page, clicking the button and getting a screenshot of dropdown.

This is another design change of side-vertical search results 5851

And then then then, I moved to next part which was actually something major #5948. Its not one or two anymore but changing the design of all the cards with all the info for that particular page. I did this kind of work into parts before. But here i went for all in one go. (︶︿︶) And it also included changing the UI test as per new design. Woah !! Done with another one (✖╭╮✖)

And Finally we came to profile section. I tried something completely new here. _へ__(‾◡◝ )> I tried building whole profile page in one single go #5990. (︶︿︶) No taking small cases anymore. It was long review process with a lot of requested changes. As it was building up whole section in one PR , so lots of bugs and problems were there (~ _ ~;). And finally after 26 days, It was merged :'). Hurray !! ♪

(✖╭╮✖)

Me and Gautami tried to do various portions altogether. As the portions were dependent on each other and we had to wait for some parts. till that time, I went for sidebar. It was addition of toggle icons 😄 #6011, We can't leave our sidebar out when all his siblings sections are upgrading with new designs ¯(°_o)/¯

Ohk !! trying to build profile page in one go left some features (⌣̩̩́_⌣̩̩̀) and bugs. Then addition of some buttons and info required as per new design #6079. And after that addition of popular tags by that user to be shown on its profile section was added #6120.

I did few small bug-fixes of bootstrap upgrades : 5755,5756, #5767, 5772 (˘⌣˘ )

Also, few of the bug-fixing because of the new designs - 5766

And more than that, I somehow created lots of bugs ¯(° _ o)/¯, And thus created many first-timer and other bug issues for others to get them fixed. (҂⌣̀ _ ⌣́)

First-timers are: First-timer only issues link

Otherwise, we have lot more issues - Click to see issues

And some of the recent PRs which are still under revewing process (・_・;) - #6103 One of them is fixing the mismatch working of subscribe button as earlier we were subscribing to related tags but not that particular tags. And having dropdown for related tag. And if a user is subscribed, then we should have post option.

#6217 And another is addition of all the extra info of questions,notes,wiki to sidebar into ...

### What is left

So Right now, we have one main section left which is Dashbaord. I'll keep on contributing. So My next target is dashboard now. (@~_@~).

I had an awesome collaboration with Gautami. Special shoutout to @warren for being such a patient mentor.(ø_ø) Thanks to mentors for solving and answering the silly questions I asked.

And Thanks to PublicLab for being such a wonderful community.

I guess I did miss various openhour calls sessions and was not much of active speaker if attended. (˘_˘٥)

I'll keep on contributing to PublicLab in future too. Thanks !!

Thanks. Both, you and Gautami did amazing job of beautifying the website.

Is this a question? Click here to post it to the Questions page.

Thanks for the response @gauravano . Surely, I'll update the post with images.

Hi @gauravano I just updated the blog with images !! Thanks !!

@gauravano awards a barnstar to lekhidugtal for their awesome contribution!


text:
5 views

The majority carriers in an $n$-type semiconductor have an average drift velocity $\mathrm{V}$ in a direction perpendicular to a uniform magnetic field $\text{B}.$ The electric field $\mathrm{E}$ induced due to Hall effect acts in the direction

1. $\mathrm{V} \times \mathrm{B}$
2. $\mathrm{B} \times \mathrm{V}$
3. along $\text{V}$
4. opposite to $\mathrm{V}$


text:
# Chapter 1 - 1.7 - Problems - Introduction to Circuits - Page 40: P1.80

$I = 1.5A$ $V = 52.5V$

#### Work Step by Step

First, we apply Kirchhoff's junction rule to obtain: $I = .5I_x + I_x \\ I = 1.5I_x$ We now use Ohm's law to find $I_x$: $I_x = \frac{V}{R} = \frac{20V}{20\Omega} = 1A$ Thus, we find $I$: $I = 1.5 I_x = 1.5(1) = \fbox{1.5A}$ We now use Kirchhoff's Voltage Law, which states that the voltage around any closed loop in a circuit is 0, to find: $-v + (15\Omega)(1.5A)+20V + (10\Omega)(1A) = 0 \\ v = (15\Omega)(1.5A)+20V + (10\Omega)(1A)\\ v = \fbox{52.5V}$

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


text:
Mirror Formula and Magnification

Mirror Formula and Magnification

Mirror formula gives the relationship between image distance, object distance and the focal length of the mirror and this formula is valid in all situations for all spherical mirrors for all positions of the object. Magnification produced by a spherical mirror gives the relative extent to which the image of an object is magnified with respect to the object size. It is expressed as the ratio of the height of the image to the height of the object.

For a spherical mirror:

u = Distance of object from pole,

v = Distance of image from pole,

f = Focal length,

R = Radius of curvature,

O = Size of object,

I = Size of image.

1) Mirror Formula: $$\frac{1}{f}=\frac{1}{v}+\frac{1}{u}$$.

2) Lateral Magnification: When an object is placed perpendicular to the principle axis, then linear magnification is called lateral or transverse magnification.

$$m=\frac{I}{O}=-\frac{v}{u}=\frac{f}{f-u}=\frac{f-v}{f}$$.

3) Axis Magnification: When object lies along the principle axis then its axial magnification is:

$$m=\frac{I}{O}=\frac{-\left( {{v}_{2}}-{{v}_{1}} \right)}{\left( {{u}_{2}}-{{u}_{1}} \right)}$$; If object is small, $$m=-\frac{dv}{du}={{\left( \frac{v}{u} \right)}^{2}}={{\left( \frac{f}{f-u} \right)}^{2}}={{\left( \frac{f-v}{f} \right)}^{2}}$$.

4) Areal Magnification: If 2D – object is placed with its plane perpendicular to principle axis and then its areal magnification is:

$${{m}_{s}}=\frac{Area\,of\,image\left( {{A}_{i}} \right)}{Area\,of\,object\left( {{A}_{o}} \right)}\Rightarrow {{m}_{s}}={{m}^{2}}=\frac{{{A}_{i}}}{{{A}_{o}}}$$.


text:
# Prove $(r\sin A \cos A)^2+(r\sin A \sin A)^2+(r\cos A)^2=r^2$

How would I verify the following trig identity?

$$(r\sin A \cos A)^2+(r\sin A \sin A)^2+(r\cos A)^2=r^2$$

My work thus far is

$$(r^2\cos^2A\sin^2A)+(r^2\sin^2A\sin^2A)+(r^2\cos^2A)$$

But how would I continue? My math skills fail me.

-
Sorry for not being able to type my post in mathjax I just lack the technical skill. –  jose Jul 18 '12 at 22:28
Look for common factors in each term of the expression. What do you get when you factor these out? –  Code-Guru Jul 18 '12 at 22:30
I think you lost a plus sign in there. –  Mike Jul 18 '12 at 22:34
You can't solve for $r$, as $r^2$ is a common factor of your equation. If you assume $r \ne 0$ you can divide out the $r^2$. You can prove that it is an identity. –  Ross Millikan Jul 18 '12 at 22:41

Just use the distributive property and $\sin^2(x)+\cos^2(x)=1$: \begin{align} &(r\sin(A)\cos(A))^2+(r\sin(A)\sin(A))^2+(r\cos(A))^2\\ &=r^2\sin^2(A)(\cos^2(A)+\sin^2(A))+r^2\cos^2(A)\\ &=r^2\sin^2(A)+r^2\cos^2(A)\\ &=r^2(\sin^2(A)+\cos^2(A))\\ &=r^2\tag{1} \end{align} This can be generalized to \begin{align} &(r\sin(A)\cos(B))^2+(r\sin(A)\sin(B))^2+(r\cos(A))^2\\ &=r^2\sin^2(A)(\cos^2(B)+\sin^2(B))+r^2\cos^2(A)\\ &=r^2\sin^2(A)+r^2\cos^2(A)\\ &=r^2(\sin^2(A)+\cos^2(A))\\ &=r^2\tag{2} \end{align} $(2)$ verifies that spherical coordinates have the specified distance from the origin.
To continue on what you have: $$(\color{red}{r^2}\cos^2A\sin^2A)+(\color{red}{r^2}\sin^2A\sin^2A)+(\color{red}{r^2}\cos^2A) \\ = \color{red}{r^2}( \cos^2A\color{blue}{\sin^2A}+\sin^2A\color{blue}{\sin^2A}+\cos^2A) \\ = r^2( (\color{red}{\cos^2A+\sin^2A})\color{blue}{\sin^2A}+\cos^2A) \\ = r^2( (\color{red}{1})\sin^2A+\cos^2A) \\ = r^2( \color{red}{\cos^2A+\sin^2A}) \\ = r^2( \color{red}{1} ) \\ = r^2$$


text:
# minipage layout not correct

I want to generate a document with graph at left and tikz code at the right. I try to use two minipage but it seems it always layout it as up and down. The minipage is 0.45\textwidth, why latex still can not put it side by side?

The example code as below:

\documentclass[12pt]{article}
\usepackage[margin=0.5in]{geometry}
\usepackage{listings}
\usepackage{color}
\usepackage{tikz}

\lstset{
backgroundcolor=\color{white},
breaklines=true,
language=TeX,
frame=single}

\begin{document}

\begin{minipage}{0.45\textwidth}
\begin{tikzpicture}
\draw (0,0) circle (5cm);
\end{tikzpicture}
\end{minipage}

\begin{minipage}{0.45\textwidth}
\begin{lstlisting}
\begin{tikzpicture}
\draw (0,0) circle (5cm);
\end{tikzpicture}
\end{lstlisting}
\end{minipage}

\end{document}

Current output by pdflatex as bleow:

You should consider using the showexpl package and use the LTXexample environment.

If you wish to have the code in an external file use the \LTXinputExample environment instead as the second example does in the MWE.

## Code:

\documentclass[12pt]{article}

%\usepackage{filecontents}% <-- Commented to prevent overwrite of test.tex
\begin{filecontents*}{test.tex}
\begin{tikzpicture}
\draw [ultra thick, blue] (0,0) circle (2.5cm);
\end{tikzpicture}
\end{filecontents*}

\usepackage[margin=0.5in]{geometry}
\usepackage{tikz}
\usepackage{showexpl}

\lstset{
backgroundcolor=\color{white},
breaklines=true,
language=TeX,
frame=single,
}

\begin{document}
\begin{LTXexample}[
width=0.6\linewidth,
backgroundcolor=\color{yellow!40},
basicstyle=\small\ttfamily,
numbers=none
]
\begin{tikzpicture}
\draw [ultra thick, blue] (0,0) circle (2.5cm);
\end{tikzpicture}
\end{LTXexample}

\LTXinputExample[
width=0.6\linewidth,
backgroundcolor=\color{green!40},
basicstyle=\small\ttfamily,
numbers=none,
]{test}
\end{document}

• Good suggestion, but if I use \input{test.tikz} replace the tikzpicture, I will get "\input{test.tikz}" at the right, I wish the code at the right but not the command. any idea? Dec 30 '14 at 23:51
• @Beatlej: Have updated the example to show how to do that. Dec 31 '14 at 0:46
• Thanks, it's really good now. BTW, just now I see pgfplot gallery at pgfplots.sourceforge.net/gallery.html. it really looks nice. do you know which package can get the same behavior? Jan 4 '15 at 2:33
• @Beatlej: I am not sure I fully understand your question, but below each image is .tex file link that gives the entire .tex file and all the required packages. If that wasn't what you were asking I'd recommend you post a new question. Jan 5 '15 at 5:42


text:
# American Institute of Mathematical Sciences

ISSN:
1547-5816

eISSN:
1553-166X

All Issues

## Journal of Industrial & Management Optimization

2007 , Volume 3 , Issue 3

Select all articles

Export/Reference:

2007, 3(3): 415-427 doi: 10.3934/jimo.2007.3.415 +[Abstract](301) +[PDF](178.1KB)
Abstract:
Equivalence between a constrained scalar optimization problem and its three conjugate dual models is established for the class of generalized C-subconvex functions. Applying these equivalent relations, optimality conditions in terms of conjugate functions are obtained for the constrained multiobjective optimization problem.
2007, 3(3): 429-444 doi: 10.3934/jimo.2007.3.429 +[Abstract](408) +[PDF](204.8KB)
Abstract:
We propose a heuristic and two stochastic approximation based learning algorithms for finite horizon, finite state-action constrained Markov decision models. We include models and numerical examples arising from risk management in fund allocation, retailer-depot product availability in a supply chain and admission control in a simple queue, that have to satisfy performance based constraints.
2007, 3(3): 445-456 doi: 10.3934/jimo.2007.3.445 +[Abstract](260) +[PDF](148.0KB)
Abstract:
This paper considers the problem of scheduling a single job family, where each job has non-identical job-sizes, non-identical job-dimensions and a furnace (a batch processing machine) that can process up to $B (B < n)$ jobs as a batch simultaneously. The motivation for this problem is the heat-treatment operation in the post casting stage of steel casting manufacturing. We propose (0-1) integer non-linear programming for minimizing the completion time of the last job, makespan. We also propose heuristic and a simple computational analysis which indicates heuristic algorithm has very good solution quality.
2007, 3(3): 457-479 doi: 10.3934/jimo.2007.3.457 +[Abstract](345) +[PDF](305.6KB)
Abstract:
Today's industries are highly complex in nature offering multiple customized quality products with shorter product life-cycles, volatile demand and tighter due-dates etc. to the customers. Manufacturers are focussing on Available-To-Promise (ATP) to their customers as a retention strategy. In other words manufacturers are forced to commit in advance to the customers the amount they can deliver by the specified due-date. In the current work we address a single manufacturer and multi-customer supply chain setting wherein there are multiple products, stochastic demands, varying profit rates, different learning rates etc. We restrict our focus to the multi-product ATP (MATP) strategies that maximize net profit of the manufacturer. We present optimization models in which there is a possibility of cancelling prior committed orders. We also model the dynamic pricing decision integrated with revenue management in MATP setting. We present the results of weak concavity of the MATP models and related structural insights. We support our thesis with rigorous numerical experimental results.
2007, 3(3): 481-487 doi: 10.3934/jimo.2007.3.481 +[Abstract](297) +[PDF](112.5KB)
Abstract:
In this article we propose a finite algorithm for minimizing the product of two linear functions over a polyhedron. Preliminary computational results are reported.
2007, 3(3): 489-501 doi: 10.3934/jimo.2007.3.489 +[Abstract](287) +[PDF](176.2KB)
Abstract:
We describe the second step in a two-step approach for the development of new and improved alloys. The first step, proposed by Golodnikov et al [3], entails using experimental data to statistically model tensile yield strength and the 20th percentile of the impact toughness, as a function of alloy composition and processing variables. We demonstrate how the models can be used in the second step to search for combinations of the variables in small neighborhoods of the data space, that result in alloys having optimal levels of the properties modeled. The optimization is performed via the efficient frontier methodology. Such an approach, based on validated statistical models, can lead to a substantial reduction in the experimental effort and cost associated with alloy development. The procedure can also be used at various stages of the experimental program, to indicate what changes should be made in the composition and processing variables in order to shift the alloy development process toward the efficient frontier. Data from these more refined experiments can then be used to adjust the model and improve the second step, in an iterative search for superior alloys.
2007, 3(3): 503-517 doi: 10.3934/jimo.2007.3.503 +[Abstract](248) +[PDF](197.8KB)
Abstract:
In this paper, based on semilocal convexity, invexity and prequasi-invexity, a new kind of generalized convexity called (strictly/ semistrictly) semilocally prequasi-invex functions is presented, and some of their basic characterizations are discussed. Then, several necessary and sufficient conditions for the proposed generalized convexity are established. Finally, some important properties of the optimal solutions to the associated generalized convex optimization problems are discussed.
2007, 3(3): 519-528 doi: 10.3934/jimo.2007.3.519 +[Abstract](498) +[PDF](152.8KB)
Abstract:
This paper investigates the semicontinuity of the solution set map for a parametric set-valued weak vector variational inequality in Banach spaces. The upper semicontinuity and closedness of the solution set map are obtained. A parametric gap function is proposed by using a nonlinear scalarization function. By virtue of the parametric gap function and a key assumption, the lower semicontinuity of the solution set map is established.
2007, 3(3): 529-542 doi: 10.3934/jimo.2007.3.529 +[Abstract](466) +[PDF](180.8KB)
Abstract:
Support vector machine (SVM) is a very popular method for binary data classification in data mining (machine learning). Since the objective function of the unconstrained SVM model is a non-smooth function, a lot of good optimal algorithms can't be used to find the solution. In order to overcome this model's non-smooth property, Lee and Mangasarian proposed smooth support vector machine (SSVM) in 2001. Later, Yuan et al. proposed the polynomial smooth support vector machine (PSSVM) in 2005. In this paper, a three-order spline function is used to smooth the objective function and a three-order spline smooth support vector machine model (TSSVM) is obtained. By analyzing the performance of the smooth function, the smooth precision has been improved obviously. Moreover, BFGS and Newton-Armijo algorithms are used to solve the TSSVM model. Our experimental results prove that the TSSVM model has better classification performance than other competitive baselines.
2007, 3(3): 543-552 doi: 10.3934/jimo.2007.3.543 +[Abstract](256) +[PDF](136.3KB)
Abstract:
This paper presents a mathematical model for Layout optimization of structure with discrete variables. The optimization procedure is composed of two kinds of sub-procedures of optimization: the topological optimization and the shape optimization. In each one, a comprehensive algorithm is used to treat the problem. The two kinds of optimization procedures are used in turn until convergence appears. After the dimension of the structure is reduced, the delimiting combinatorial algorithm is used to search for the better objective value. A couple of classical examples are presented to show the efficiency of the method. Numerical results indicate that the method is efficient and the optimal results are satisfactory.
2007, 3(3): 553-567 doi: 10.3934/jimo.2007.3.553 +[Abstract](371) +[PDF](317.3KB)
Abstract:
In this paper, we present some new optimization approaches to solve optimal power flow (OPF) problems. By using a so-called Nonlinear Complementarity Problem (NCP) function, the optimality condition (KKT system) of the original optimization problem is reformulated into a set of nonsmooth equations. The advantage of the new reformulation lies in that the inequality constraints are transformed into equations. The semismooth Newton-type method is applied to solve the reformulated equations. Moreover, we present a decoupled semismooth Newton method according to the inherent weak-coupling characteristics of power systems. The convergence of the new methods, especially for the decoupled method, are established. Numerical examples of both OPF and available transfer capability (ATC) problems demonstrate that the new algorithms are effective.
2007, 3(3): 569-584 doi: 10.3934/jimo.2007.3.569 +[Abstract](313) +[PDF](203.3KB)
Abstract:
In this paper, a non-interior-point smoothing algorithm is applied to solve the $P_*$ nonlinear complementarity problem (NCP). The algorithm is proved to be globally convergent under an assumption that the $P_*$ NCP has a nonempty solution set. In particular, the solution obtained by the algorithm is shown to be a maximally complementary solution of the $P_*$ NCP. The results we obtained strictly generalize the relative results appeared in the literature.
2007, 3(3): 585-596 doi: 10.3934/jimo.2007.3.585 +[Abstract](321) +[PDF](356.7KB)
Abstract:
A dynamic model of an aluminium trihydroxide batch crystallization is considered in this work. The process model, which takes into account kinetics of nucleation, growth and agglomeration, is based on the mass balance of the process and the population balance of the dispersed crystals. Assuming that the temperature of the solution and the seeding policy are variable, optimal control techniques are applied to the model to optimize various performance criteria. Some interesting numerical results for the behaviour of the model are presented.
2007, 3(3): 597-618 doi: 10.3934/jimo.2007.3.597 +[Abstract](449) +[PDF](442.0KB)
Abstract:
This paper discusses the operating parameters of a two-echelon 'm' Vendors - 'n' Buyers Vendor Managed Inventory (VMI) System with outsourcing (MVMBO). The operational parameters to the above model are the selling prices at the buyer's market and the contract prices between the vendors and the buyers. Selling prices depend on sales quantities and determines the channel profit of the supply chain (SC). Contract prices depend on the understanding between partners on their revenue sharing agreement. A mathematical model of the MV MBO model is formulated to find optimal sales quantities (summation of optimal transaction quantities) for maximum channel profit. Optimal outsourcing quantities, selling prices and acceptable contract prices are derived from the obtained optimal transaction quantities. The mathematical model formulation of MV MBO involves mixed integer variable, a non-linear objective function and linear constraints which fall under the category of Mixed Integer Non-linear Programming (MINP) optimization problem. Simulated Annealing Algorithm (SAA) based heuristic is proposed to find the optimal operational parameters of the MVMBO problem. The proposed methodology is evaluated for its solution quality.

2017  Impact Factor: 0.994


text:
## k-path

### Set

 context $V,E$ … set context $k\in\mathbb N$
 definiendum $\langle V,E,\psi\rangle \in \mathrm{it}$
 inclusion $\langle V,E,\psi\rangle$ … path
 postulate $|\ E\ |=k$


text: - Completely positive semidefinite rank Anupam Prakash(aprakashntu.edu.sg) Jamie Sikora(cqtjwjsnus.edu.sg) Antonios Varvitsiotis(avarvitsgmail.com) Zhaohui Wei(weizhaohuigmail.com.) Abstract: An $n\times n$ matrix $X$ is called completely positive semidefinite (cpsd) if there exist $d\times d$ Hermitian positive semidefinite {matrices} $\{P_i\}_{i=1}^n$ (for some $d\ge 1$) such that $X_{ij}= {\rm Tr}(P_iP_j),$ for all $i,j \in \{ 1, \ldots, n \}$. The cpsd-rank of a cpsd matrix is the smallest $d\ge 1$ for which such a representation is possible. In this work we initiate the study of the cpsd-rank which we motivate twofold. First, the cpsd-rank is a natural non-commutative analogue of the completely positive rank of a completely positive matrix. Second, we show that the cpsd-rank is physically motivated as it can be used to upper and lower bound the size of a quantum system needed to generate a quantum behavior. In this work we present several properties of the cpsd-rank. Unlike the completely positive rank which is at most quadratic in the size of the matrix, no general upper bound is known on the cpsd-rank of a cpsd matrix. In fact, we show that the cpsd-rank can be exponential in terms of the size. Specifically, for any $n\ge1,$ we construct a cpsd matrix of size $2n$ whose cpsd-rank is $2^{\Omega(\sqrt{n})}$. Our construction is based on Gram matrices of Lorentz cone vectors, which we show are cpsd. The proof relies crucially on the connection between the cpsd-rank and quantum behaviors. In particular, we use a known lower bound on the size of matrix representations of extremal quantum correlations which we apply to high-rank extreme points of the $n$-dimensional elliptope. Lastly, we study cpsd-graphs, i.e., graphs $G$ with the property that every doubly nonnegative matrix whose support is given by $G$ is cpsd. We show that a graph is cpsd if and only if it has no odd cycle of length at least $5$ as a subgraph. This coincides with the characterization of cp-graphs. Keywords: completely positive semidefinite cone, cpsd-rank, Lorentz cone, elliptope, Bell scenario, quantum behaviors, quantum correlations, cpsd-graphs Category 1: Linear, Cone and Semidefinite Programming (Other ) Citation: Download: [PDF]Entry Submitted: 04/25/2016Entry Accepted: 04/25/2016Entry Last Modified: 04/25/2016Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Optmization Society.


text:
# Statistics/Testing Data/z-tests2

## Indications

The Null Hypothesis should be an assumption about the difference in the population means for two populations (note that the same quantitative variable must have been measured in each population). The data should consist of two samples of quantitative data (one from each population). The samples must be obtained independently from each other.

## Requirements

The samples must be drawn from populations which have known Standard Deviations (or Variances). Also, the measured variable in each population (generically denoted x1 and x2) should have a Normal Distribution.

Note that if the distributions of the variables in the populations are non-normal (or unknown), the two-sample z-test can still be used for approximate results, provided the combined sample size (sum of sample sizes) is sufficiently large. Historically, a combined sample size of at least 30 has been considered sufficiently large; reality is (of course) much more complicated, but this rule of thumb is still in use in many textbooks.

## Procedure

• The Null Hypothesis:
H0 : μ1 - μ2 = δ

in which δ is the supposed difference in the expected values under the null hypothesis.

• The Alternate Hypothesis:
H0 : μ1 - μ2 < δ
H0 : μ1 - μ2 > δ
H0 : μ1 - μ2 ≠ δ

For more information about the Null and Alternate Hypotheses, see the page on the z test for a single mean.

• The Test Statistic:
${\displaystyle z={\frac {({\bar {x}}_{1}-{\bar {x}}_{2})-\delta }{\sqrt {{\frac {\sigma _{1}^{2}}{n_{1}}}+{\frac {\sigma _{2}^{2}}{n_{2}}}}}}}$

Usually, the null hypothesis is that the population means are equal; in this case, the formula reduces to

${\displaystyle z={\frac {{\bar {x}}_{1}-{\bar {x}}_{2}}{\sqrt {{\frac {\sigma _{1}^{2}}{n_{1}}}+{\frac {\sigma _{2}^{2}}{n_{2}}}}}}}$

In the past, the calculations were simpler if the Variances (and thus the Standard Deviations) of the two populations could be assumed equal. This process is called Pooling, and many textbooks still use it, though it is falling out of practice (since computers and calculators have all but removed any computational problems).

${\displaystyle {\frac {{\bar {x}}_{1}-{\bar {x}}_{2}}{\sigma {\sqrt {{\frac {1}{n_{1}}}+{\frac {1}{n_{2}}}}}}}}$
• The Significance (p-value)

Calculate the probability of observing a value of z (from a Standard Normal Distribution) using the Alternate Hypothesis to indicate the direction in which the area under the Probability Density Function is to be calculated. This is the Attained Significance, or p-value.

Note that some (older) methods first chose a Level Of Significance, which was then translated into a value of z. This made more sense (and was easier!) in the days before computers and graphics calculators.

• Decision

The Attained Significance represents the probability of obtaining a test statistic as extreme, or more extreme, than ours—if the null hypothesis is true.

If the Attained Significance (p-value) is sufficiently low, then this indicates that our test statistic is unusual (rare)—we usually take this as evidence that the null hypothesis is in error. In this case, we reject the null hypothesis.

If the p-value is large, then this indicates that the test statistic is usual (common)—we take this as a lack of evidence against the null hypothesis. In this case, we fail to reject the null hypothesis.

It is common to use 5% as the dividing line between the common and the unusual; again, reality is more complicated.

## Worked Examples

### Do Professors Make More Money at Larger Universities?

Universities and colleges in the United States of America are categorized by the highest degree offered. Type IIA institutions offer a Master's Degree, and type IIB institutions offer a Baccalaureate degree. A professor, looking for a new position, wonders if the salary difference between type IIA and IIB institutions is really significant.

He finds that a random sample of 200 IIA institutions has a mean salary (for full professors) of $54,218.00, with standard deviation$8,450. A random sample of 200 IIB institutions has a mean salary (for full professors) of $46,550.00, with standard deviation$9,500 (assume that the sample standard deviations are in fact the population standard deviations).

Do these data indicate a significantly higher salary at IIA institutions?

The null hypothesis is that there is no difference; thus

H0 : μA = μB

(where μA is the true mean full professor salary at IIA institutions, and μB is the mean at IIB institutions)

He is looking for evidence that IIA institutions have a higher mean salary; thus the alternate hypothesis is

H1 : μA > μB

Since the hypotheses concern means from independent samples (we'll assume that these are independent samples), a two sample test is indicated. The samples are large, and the standard deviations are known (assumed?), so a two sample z-test is appropriate.

${\displaystyle z={\frac {\mu _{A}-\mu _{B}}{\sqrt {{\frac {\sigma _{A}^{2}}{n_{A}}}+{\frac {\sigma _{B}^{2}}{n_{B}}}}}}={\frac {54218-46550}{\sqrt {{\frac {8450^{2}}{200}}+{\frac {9500^{2}}{200}}}}}=8.5292}$

Now we find the area to the right of z = 8.5292 in the Standard Normal Distribution. This can be done with a table of values or software—I get 0.

If the null hypothesis is true, and there is no difference in the salaries between the two types of institutions, then the probability of obtaining samples where the mean for IIA institutions is at least \$7,668 higher than the mean for IIB institutions is essentially zero. This occurs far too rarely to attribute to chance variation; it seems quite unusual. I reject the null hypothesis (at any reasonable level of significance!).

It appears that IIA schools have a significantly higher salary than IIB schools.


text:
# The IntMath Newsletter - 22 Apr 2008 (Earth Day)

By Murray Bourne, 22 Apr 2008

1. Earth Day Math
2. Math tips - (a) I dunno how to do; (b) Use the resources
3. From the math blog

## 1. Earth Day Math

Today is Earth Day. Let's think about our most basic needs.

Air: Humans are currently pumping around 27 gigatonnes of carbon into the air every year. (Source.) A gigatonne is 109 tonnes or 1012 kg. (In the old imperial system, a tonne is around 2205 lb.)

Petroleum is the biggest contributor to carbon pollution, followed by coal, natural gas and cement production. The USA is responsible for 20% of this pollution, followed closely by China at 18%. (Source)

Without air, we can live for about 3 minutes only.

Water: Over 1 billion people lack access to clean water and 2.6 billion have poor or non-existent sanitation (Source).

Without water, we can live for about 3 days only.

Food: Have you heard the term "agflation"? This describes what is happening in the world right now. Food costs are rapidly increasing, partly due to the use of grain to make biofuels. Urbanisation, droughts and floods also add to the problem. Already agflation is causing social unrest in many countries. As global warming gets worse, the amount of available food will continue to drop and prices will rise.

Without food, we can live for about 3 weeks only.

Energy: Many people believe that unlimited water, electricity and car use are essential. That's because our lifestyles have developed around these luxuries.

But do you really need that heater up at 28°C (82°F) in winter and that air-con down to 18°C (64°F) in summer? You can live for a long time at 23°C (73°F).

And, do you really need that gas-guzzling SUV (sports utility vehicle)? Around half of all vehicles sold in the USA are SUVs and because they are regarded as light trucks, they do not have to conform to the same pollution standards as cars.

Without an SUV, you can live a very long time. And the rest of us will breathe a lot easier too.

Happy Earth Day, 2008.

The decisions you make now will affect the world that you - and your children - live in.

## 2. Math tips

a. I dunno how to do: A lot of students take one look at their math homework and freeze up because they have no idea where to start.

Most of the time the answer is right there in your notes or in your textbook. You just need to go back a few steps and dive into the stuff you can do. Build up your skills and confidence first, and then attack the current homework.

b. Use the resources: Many students never go near their school library to check out the math books there. Sure, it sounds a nerdy thing to do, but do you want go well in math or what?

Most school libraries have a range of textbooks similar to the one you are using. It is amazing how helpful it is to read another version of the math you are studying. The new examples might be just a bit easier and will give you a better idea of what your own textbook (and teacher) are on about.

Try also the local library or the library of your local college. There's a good chance they have some useful math books too.

Also, be on the look out for 2nd-hand textbook sales. They are a good source for math books.

## 3. From the math blog

1) Some cool number patterns - and how to give 100%
Here's some interesting number patterns - they could be good for a party trick!

2) Where did matrices and determinants come from?
Many of our mathematical discoveries are named after European mathematicians, even though they originated in China, India or the Middle East. Gaussian Elimination is one such example.

3) Friday Math Movie - Uncounted
Bush only just scraped through the 2000 election. What really happened? And what happened to keep the Bush team in power through the 2004 election? This disturbing documentary is about the fundamental math of democracy - correct, honest and incorruptible counting.

See the 1 Comment below.

### One Comment on “The IntMath Newsletter - 22 Apr 2008 (Earth Day)”

1. Miss Loi says:

Regarding the SUV situation, most tai-tais believe that they won't last 3 minutes on Singapore's road without a big SUV chassis to protect them 😛

### Comment Preview

HTML: You can use simple tags like <b>, <a href="...">, etc.

To enter math, you can can either:

1. Use simple calculator-like input in the following format (surround your math in backticks, or qq on tablet or phone):
a^2 = sqrt(b^2 + c^2)
(See more on ASCIIMath syntax); or
2. Use simple LaTeX in the following format. Surround your math with $$ and $$.
$$\int g dx = \sqrt{\frac{a}{b}}$$
(This is standard simple LaTeX.)

NOTE: You can't mix both types of math entry in your comment.


text:
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# If the length of an edge of cube X is twice the length of an edge of

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If the length of an edge of cube X is twice the length of an edge of [#permalink]

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If the length of an edge of cube X is twice the length of an edge of cube Y, what is the ratio of the volume of cube Y to the volume of cube X?

(A) 1/2
(B) 1/4
(C) 1/6
(D) 1/8
(E) 1/27
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Re: If the length of an edge of cube X is twice the length of an edge of [#permalink]

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06 Dec 2017, 08:04
length of edge of cube x = 2a
length of edge of cube y = a

Ratio of volume y / volume x = {(length of edge of cube y)/(length of edge of cube x)}^3 = 1/8 . Hence answer is D.
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Re: If the length of an edge of cube X is twice the length of an edge of [#permalink]

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06 Dec 2017, 08:50
Bunuel wrote:
If the length of an edge of cube X is twice the length of an edge of cube Y, what is the ratio of the volume of cube Y to the volume of cube X?

(A) 1/2
(B) 1/4
(C) 1/6
(D) 1/8
(E) 1/27

Let Length of edge of cube Y be 1, so edge of cube X is 2
Volume of Cube Y = 1 & Volume of Cube X = 8

So,ratio of the volume of cube Y to the volume of cube X is $$\frac{1}{8}$$, thus answer will be (D)

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Re: If the length of an edge of cube X is twice the length of an edge of [#permalink]

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06 Dec 2017, 08:54
Bunuel wrote:
If the length of an edge of cube X is twice the length of an edge of cube Y, what is the ratio of the volume of cube Y to the volume of cube X?

(A) 1/2
(B) 1/4
(C) 1/6
(D) 1/8
(E) 1/27

Let Length of edge of cube Y be 1, so edge of cube X is 2
Volume of Cube Y = 1 & Volume of Cube X = 8

So,ratio of the volume of cube Y to the volume of cube X is $$\frac{1}{8}$$, thus answer will be (D)

_________________

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Abhishek....

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Re: If the length of an edge of cube X is twice the length of an edge of [#permalink]

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06 Dec 2017, 21:58
Let the edge of Cube Y = p
so as per the question the edge of Cube X = 2p

as we know the edge of cube Y is p so its volume is = p^3
similarly as we know the edge of cube X is 2p so its volume is = (2p)^3 = 8p^3

so what is asked in the question is the ratio of Volume of cube Y to Volume of cube X that is = (p^3/8p^3) ---> 1/8

So answer is D.

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Re: If the length of an edge of cube X is twice the length of an edge of   [#permalink] 06 Dec 2017, 21:58
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# Interpolating rotated objects

I've finally got interpolation working to 'smooth' out movement, however it doesn't seem to work for objects that are rotating (around their own center) - when I try to do this, the objects in question 'wobble'.

Note: I'm not talking about interpolating the actual rotation but rather the movement of a rotated object

So, let's say I have an object moving from left to right. Using the code below (with interpolation), it moves nice and smoothly. However, after rotating the object (let's say by 45 degrees, but this could be any angle), it's no longer moving smoothly, it still moves along it's path (left to right in this case) but it's no longer smooth.

Main Game Loop

@Override
public void onDrawFrame(GL10 gl) {

//Grab time
newTime = System.currentTimeMillis()*0.001;
frameTime = newTime - currentTime;
if ( frameTime > (dt*25))
frameTime = (dt*25);

currentTime = newTime;

accumulator += frameTime;

//Save game-state and update logic
while (accumulator >= dt){
saveGameState();  //Back up positions before updating them
updateLogic(); //Update logic
t += dt;
accumulator -= dt;
}

//Get the amount to interpolate by
interpolation = (float) (accumulator / dt);

//Interpolate positions and render
interpolate();
render(interpolation);
}

Rotation Code

public void rotate(float x, float y, float angle){

Matrix.setIdentityM(mRotationMatrix, 0);
Matrix.translateM(mRotationMatrix, 0, centreX, centreY, 0f);
Matrix.rotateM(mRotationMatrix, 0, angle, 0, 0, 0.1f);
Matrix.translateM(mRotationMatrix, 0, -centreX, -centreY, 0f);
}

Edit

OK so it appears this is the problem:

• During logic update, sprite is moved and rotates it around it's center (using it's new/current X and Y to calculate it's center).
• Get's to render()
• Render method interpolates position creating new 'throwaway' coordinates at which object should be drawn
• Render method draws sprite at it's 'render' x and y coordinates
• Logic update moves and rotate around it actual x and y - not it's 'rendering x and y
• and so on

So as you can see, it's being rotated, not around the coordinates that it's been drawn at but at it's actual coordinates. I can't rotate it around it's render coordinates, because the rotation is done within the logic update, not during the render call, therefore, it would be rotating it around it 'old' render position, not the next one and I won't know the new render coordinates until the next render call which means.........

That the only way I can do this, is to actually translate / rotate the sprite within the render routine and not the logic routine.

I've tried this and it works, but not sure I'm happy about piling more 'logic' into my render routine.

Is this an 'acceptable' model? How else can I achieve rotation while keeping the object nicely interpolated?!

• working through your code in my head, im curious why in rotate you are translating around a negative origin? Matrix.translateM(mRotationMatrix, 0, -centreX, -centreY, 0f);
– Joe
Sep 8, 2014 at 18:33
• Hi @Joe, yeah, you're right. I will post another question regarding that :-) that's been a long standing problem, I must have something wrong in my shader because if I specify 45 as rotation amount, it rotates in the opposite direction, so I simply negated this to make my code more readable. I'll correct this for this question to take away any confusion, but I will definitely post another question regarding this because I've never been able to work out why this is..... cheers! Sep 8, 2014 at 18:39
• id venture a guess your issues are happening here by first translating, rotating then translating back along x & y
– Joe
Sep 8, 2014 at 18:40
• I should point out @Jo that if I remove the interpolation completely, then the rotation doesn't cause any 'wobbling' (although then of course, I would get choppiness which the interpolation sorts out), the rotation is only causing an issue when used in conjunction with interpolation. Hope this makes sense. Would appreciate if you could elaborate. Thanks Sep 8, 2014 at 18:52

from looking at the docs I have found your issue

public static void rotateM (float[] m, int mOffset, float a, float x, float y, float z) Added in API level 1

Rotates matrix m in place by angle a (in degrees) around the axis (x, y, z).

Parameters:

• m source matrix
• mOffset index into m where the matrix starts a angle to rotate in degrees
• x X axis component
• y Y axis component
• z Z axis component

Matrix.rotateM(mRotationMatrix, 0, angle, 0, 0, 0.1f);

looks like this function is meant to rotate around an axis and your passing the axis of ( 0,0,0.1f ) when you probably mean to be rotating this object around its local origin

local origin rotation: (this is what you want to be doing)

rotation around world point: (this is what you are currently doing)

this would very probably change position, thus ruining your interpolation

you can change your rotateM call to look like this

Matrix.rotateM(mRotationMatrix, 0, angle, ORIGIN_X, ORIGIN_Y, ORIGIN_Z);

• Joe is on the right track, I think. Most matrix rotation functions that work in 3D expect the axis to be passed in a normalized form. Try normalizing your axis before passing it in i.e. Matrix.rotateM(mRotationMatrix, 0, angle, 0, 0, 1.0f); Sep 8, 2014 at 19:20
• Thanks @Joe, I attempted to use 1.0f instead of 0.1f (as user47946 suggested), but this didn't make any difference. Is this what you meant? Thanks for your help. Sep 8, 2014 at 20:34
• no take a second look ive added an edit... you are attempting to rotate an object around the 3d point (0,0,0.1) which is causing position distortion... you want to rotate it around its own center point aka local origin (originX, originY, originZ)
– Joe
Sep 8, 2014 at 20:36
• OK @Joe thanks, I'll have to have a go at this, I don't quite understand as I tought that Matrix.translateM(mRotationMatrix, 0, centreX, centreY, 0f), centered the object in the center if the GL Viewport then you rotated it around the center, then translated it back into position (hence the use of negative values for the 2nd translateM line). I may have misunderstood. Also, rotation (as I'm doing it) works fine if I don't interpolate (or if the object isn't moving) - it does rotate around it's own center, I'll re-read the docs and see if I've got this all wrong! Cheers Sep 8, 2014 at 21:26
• not a problem imagine the difference between the earth revolving around the sun and the earth rotating around its axis... very loose analogy but i hope it helps
– Joe
Sep 8, 2014 at 21:29


text:
Produce a digest (MD5) for the irreversible and invariant passwords encryption. It is represented by a invariant of 32 bytes hexa. It is internally used in the passwords of users / groups.

# 1 crypt.digest

<crypt.digest>    <var /> !</crypt.digest>

#### Exceptions

requires 1 argument

The entry parameter has been specified.

Example

Encrypt the text 'hello', passing the argument as String and as byte[] (file.read).

Copy
<xsql-script name='string_crypt_digest_sample1'>
<body>
<set name='a'>abc</set>
<println>
<crypt.digest><a /></crypt.digest>
</println>
</body>
</xsql-script>


text:
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text:
×

# 0.9999999.....

Does 0.999999... has a fractional form ??? If it has , will you show please.. :(( Coz I always get 1 in solving it .. But it isnt equal to 1 right ??

Note by Noel Quirol
4 years, 1 month ago

Sort by:

It is equal to 1. Just like $$0.00000... = 0$$. :) · 4 years, 1 month ago

i just dont get it.. · 4 years, 1 month ago

$$100N = 99.9999.... \implies 99N = 99.999999... - 0.999999.....\implies 99N = 99 \implies N=1$$ · 4 years ago

https://brilliant.org/discussions/thread/explain-why-0999-1/ · 4 years, 1 month ago

Find any number between $$0.99999....$$ and $$1$$. You can't! So they have to be equal! · 4 years, 1 month ago

ohh , so thats it.. hehe :D Thanks.. · 4 years, 1 month ago

yes, the fractional form is 1/1..... · 4 years, 1 month ago

Thanks... :D · 4 years, 1 month ago


text:
# Configuration file structure and format (YAML)¶

Grond is configured with plain text files in YAML format. The YAML format has been chosen because it can represent arbitrarily nested data structures built from mappings, lists, and scalar values. It also provides an excellent balance between human and machine readability. When working with such files, it is good to know that the indentation is part of the syntax and that comments can be introduced with the # symbol.

The configuration file can be made to work with multiple events. A basic configuration of some example types can be generated using the subcommand (see section Initializing an empty project)

grond init <example>

The following commented configuration presents the overall structure of a Grond configuration file. It has a top level container (mapping), introduced with the line --- !grond.Config with several child elements: path_prefix, rundir_template, dataset_config, etc. Some of these entries may again contain their own child elements (indented blocks of lines) or lists (lines introduced with dashes). The type markers, like e.g. !grond.DatasetConfig, select the Grond object type of the following mapping and their documentation can likely be found in the Library reference.

Example Grond YAML config
 %YAML 1.1
--- !grond.Config

# All file paths referenced below are treated relative to the location of this
# configuration file, here we may give a common prefix. E.g. setting it to '..'
# if the configuration file is in the sub-directory '${project_root}/config' # allows us to give the paths below relative to '${project_root}'.

path_prefix: '..'

# Path, where to store output (run directories). The placeholder
# '${problem_name}' will be expanded to a name configured below in # problem_config.name_template and will typically include a config identifier # and the event name. rundir_template: 'runs/${problem_name}.run'

# Configuration section for dataset (input data)

dataset_config: !grond.DatasetConfig
...

# Configuration section for the forward modelling engine (configures where
# to look for GF stores)

engine_config: !grond.EngineConfig
...

# Configuration section selecting data to be included in the data optimisation.
# Amongst other parameters, the objective function for the optimisation is
# defined for each target group. The targets can be composed of one or more
# contributions, each represented by a !grond.TargetConfig section.

target_groups:

- !grond.WaveformTargetGroup       # Setup for seismic waveforms
...

- !grond.SatelliteTargetGroup      # Setup for InSAR
...

- !grond.GNSSCampaignTargetGroup   # Setup for coseismic GNSS displacements
...

# Definition of the problem to be solved - source model, parameter space, and
# global misfit configuration settings. Only one problem can be defined per
# configuration file.

#problem_config: !grond.RectangularProblemConfig  # setup for an extended source
#problem_config: !grond.DoubleDCProblemConfig     # setup for combination of two double-couples
problem_config: !grond.CMTProblemConfig           # setup for a general moment tensor

# Name used to identify the output the placeholder '${event_name}' will # be replaced with the current event name name_template: 'cmt_${event_name}'
...

# Configuration of pre-optimisation analysis phase; e.g. balancing weights are
# determined during this phase. Analysers can be combined.

analyser_configs:

- !grond.TargetBalancingAnalyserConfig  # balancing weights
...

- !grond.NoiseAnalyserConfig            # pre-event noise based weights
...

# Configuration of the optimisation procedure.

optimiser_config: !grond.HighScoreOptimiserConfig
...