text:
## FANDOM

18 Pages

Programming & Mathematics

This section contains an overview of the math in this system, and how we built it to be versatile, but at the same time fast and efficient. The method we use may seem a bit weird at first, but it is very fast and works well.

NOTE: This document is still a draft. Still working on correcting typos etc.

## Matrix system basicsEdit

### IntroductionEdit

The basic single-qubit matrices we use are these.

The Pauli matrices:

$I = \left[\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}\right]$

$X = \left[\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}\right]$

$Y = \left[\begin{matrix} 0 & -i \\ i & 0 \end{matrix}\right]$

$I = \left[\begin{matrix} 1 & 0 \\ 0 & -1 \end{matrix}\right]$

$H = \frac{1}{\sqrt{2}}\left[\begin{matrix} 1 & 1 \\ 1 & -1 \end{matrix}\right]$

$S = \left[\begin{matrix} 1 & 0 \\ 0 & i \end{matrix}\right]$

$T = \left[\begin{matrix} 1 & 0 \\ 0 & e^{\frac{i\pi}{4}} \end{matrix}\right]$

We also have a programmable phase gate, where you set the angle yourself.

$P(\theta) = \left[\begin{matrix} 1 & 0 \\ 0 & e^{i \theta} \end{matrix}\right]$

All of these gates, except for the Hadamard, are general permutation matrices. A general permutation matrix (in our system) is a matrix on the form G = DP, where P is a permutation matrix, and D is a diagonal matrix with complex entries (that are allowed to be 0).

$D = \left[\begin{matrix} z_0 & 0 \\ 0 & z_1 \end{matrix}\right]$

This prompted us to add a data structure for general permutation matrices. But before getting into that, we will look at vectors.

### VectorsEdit

The vectors are qubit states, which means their dimension is always a power of two. Vectors with (at most) one non-zero element occurs frequently in this application, for example - any vector that represent a (computational) basis state, regardless of dimension, is a vector of that kind. Also - any such vector that passes through a single- or multiqubit gate composed only out of general permutation matrices, would also be in that form. In our system, that would be every gate where Hadamard gates are not involved. This holds regardless of whether or not the gates are controlled.

We call such a vector ComplexVectorPerm, as they can be used to represent a row in a general permutation matrix. Let 'z' be a complex value and |b> an N-dimensional computational basis state. All ComplexVectorPerm instances is then on the form $z|b\rangle$.

The data type used to represent a ComplexVectorPerm looks like this:

ComplexVectorPerm implements ComplexVector {

Variables:

byte type

int dimension

int position

Complex value

Methods:

}

We have an interface for vectors called ComplexVector, which all vectors must implement.

'type' is just a flag. There are three types of vectors at this point - GPWZ, genPerm, and sparse. GPWZ means "general permutation with zeroes", and is used both for all-zero vectors, and general permutation matrices that has at least one all-zero row. ComplexVectorPerm can be of type GPWZ or genPerm.

'position' is the position of the non-zero element, and is an int.

'value' is the value of that element.

'dimension' is the dimension of the vector An example vector would be:

ComplexVectorPerm cvp

cvp.type = genPerm

cvp.dimension = 4

cvp.position = 0

cvp.value = i

This would be the vector (i,0,0,0) or i|00>

If the vector only contains zero elements, the type is "GPWZ", the position -1, and the value 'null'. Here is an example:

ComplexVectorPerm cvp

cvp.type = GPWZ

cvp.dimension = 8

cvp.position = -1

cvp.value = null

This would be the vector (0,0,0,0,0,0,0,0). It's not a valid qubit state, of course, but its valid as a row in some matrices.

ComplexVectorSparse is the other vector type. It essentially just holds a (pos,value)-map and implements the vector interface. We use it even for fully dense vectors at this point, and its just a regular hashmap, which brings great shame to me and my family. It will be replaced with a faster collection later (when external dependencies works better), and the dense vector will be re-introduced and used whenever the number of elements exceed 0.75 * dimension or so.

### MatricesEdit

There are only two matrix types at this time (2014-02-03). The first one is ComplexMatrixPerm.

ComplexMatrixPerm is a general permutation matrix, and is essentially just an array of ComplexVectorPerm.

MatrixGenPerm implements ComplexMatrix {

byte type

int dimension

ComplexVectorPerm[] rows

int[] rowByPos

Methods:

}

'type' is the same as with vectors. There are three matrix types as well, GPWZ, genPerm and partition.

'dimension' is the dimension. All gate matrices are square, so only one value is needed.

'rows' is the row vectors, and the size of the array is 'dimension'.

'rowByPos' is just an array of ints storing row by position, so that you can fetch the row based on the position of the element, and not just the position by fetching the row. Its not essential, but allows for O(1) matrix-vector multiplication and O(n) matrix-matrix between genPerms.

General permutation matrices has got (at most) one non-zero element per row and column, which makes them more akin to sparse vectors then matrices, only you have to associate each value with a row and a position, instead of just a position. They are super light, and super fast to work with, and are used a lot of the time. This in itself is not that useful, however. The performance would be pretty much the same if using a generic sparse/dense system. The usefulness becomes more clear when working with larger and more complex matrices.

Anyways, the second matrix type is ComplexMatrixPartition, which contains an array of genPerms. The genPerms are treated as a sum when performing operations. For instance, if a ComplexMatrixPartition has the array of genPerms {A,B}, a matrix-vector addition would be computed (A + B)v = Av + Bv = u

ComplexMatrixPartition implements ComplexMatrix {

ComplexMatrixPerm[] genPerms;

Methods:

}

The point of the partition is to support matrices like the Hadamard. We have $H = \frac{X}{\sqrt 2} + \frac{Z}{\sqrt 2}$, which means it is the sum of two genPerms, and therefore a ComplexMatrixPartition. We can also use partition matrices to represent any dense 2x2 unitaries, not just the Hadamard.

This is also useful when it comes to representing large tensor products containing 1 Hadamard and the rest genPerms, such as (H x Y x Y x Y). We often encounter these type of matrices since they represent non-controlled N qubit gates (a 4 qubit gate in this case). The tensor product is distributive over addition. If we omit the square-root term for brevity, this can be rewritten:

$H \otimes Y \otimes Y \otimes Y = (X + Z) \otimes Y \otimes Y \otimes Y = (X \otimes Y \otimes Y \otimes Y) + (Z \otimes Y \otimes Y \otimes Y)$

A tensor product of any number of genPerms is itself a genPerm, so what we got here is nothing other then the sum of two genPerms, and thus a ComplexMatrixPartition.

### Dense MatricesEdit

The result above makes the genPerm matrix practical, but its real strength shows when we must compute and apply very dense matrices.

Take the matrix M = (H x H x H x H), a fully dense 16x16 matrix. We could partition this matrix like before, but it would result in 16 genPerms. However, as you will see now, we never have to consider terms with more then one "non-genPerm" in them. We start by using a property of the tensor product to write this as a matrix product of four terms:

$H \otimes H \otimes H \otimes H = (H \otimes I \otimes I \otimes I)(I \otimes H \otimes I \otimes I)(I \otimes I \otimes H \otimes I)(I \otimes I \otimes I \otimes H)$

Each term is a 2 term partition matrix, that contains 32 elements (2 * 16). This means we have to generate 128 elements in total. The 16x16 fully dense matrix requires 256. Actually, as the matrix dimension grows, the number of elements we have to calculate when partitioning shrinks exponentially. The number of factors is linear in the number of gates (N), and each factor has only got 2 elements per row, which means we have to compute 2N elements in total per row. The number of elements per row in the dense version is $2^N$. Furthermore, we never have to multiply these matrices together to create the dense matrix, because gates always acts on qubit states, which means that we can simply multiply the state vectors with one factor at a time in succession. For example, we have:

$M|\psi\rangle = |\phi\rangle$

Without loss of generality we assume the state vector is fully dense. M has 16 elements per row, so we must do 16 multiplications and 15 additions per matrix row to get the product. This means 16*16 = 256 multiplications in all (+ the addition).

Now let $M_k$ be the matrix with H in place k, and I otherwise:

$M|\psi\rangle = (H \otimes H \otimes H \otimes H)|\psi\rangle = M_0M_1M_2M_3|\psi\rangle = M_0M_1M_2|\psi_1\rangle =$ $M_0M_1|\psi_2\rangle = M_0|\psi_3\rangle = |\phi\rangle$

The Factors has only 2 elements per row, so each mat-vec multiplication requires 2*16 = 32 element multiplications, and 16 additions. There are four factors in total, so 128 multiplications in all. We have exponential gain here as well.

This adds to the speed, but it also reduces memory usage. Technically we only compute one of these "factors" at a time, so we only have to allocate memory for one. In terms of matrix dimension, this reduces the memory needed from d*d to 2*d. Also, this works for controlled gates as well, but to verify that we must look at the gate generation algorithm.

### Gate generationEdit

At the core of the gate matrix generation system is an algorithm DennisMcKinnon wrote to generate matrices for any possible combination of gates and controls. The input to the generation function 'G' is an array of gate indices (java unsigned bytes), where each index represents either a control or a gate. The function returns the gate matrix:

M = G(arg0,arg1,...)

If 'C' is used for controls, a cNOT gate would be specified like this:

M = G(C,X)

The Toffoli gate and its permutations would be specified like this:

G(C,C,X), G(C,X,C), G(X,C,C).

This algorithm lets the user create multigates using any combination of single qubit gates.

When it comes to combining gates - those that has no controls in them are considered un-linked, and each single qubit gate is applied locally. Let's say two qubits enter into two Z gates. If they do not share their states with any other qubits,this is done simply by multiplying the state of each qubit with Z. If they have a shared state, the gate applied to qubit 0 would be Z ⊗ I, and the other I ⊗ Z. It works this way because tensoring all gates together would require us to tensor all qubit states together, and that should really not be done unless absolutely necessary. If all qubits are already in a combined state, however, the matrices are all tensored into one.

There are cases when the qubits passing through a multigate are linked with other qubits, but they aren't affected by the gate. In cases like that, the gate is padded with identity matrices. Let's say qubits 1, 2 and 3 share a state, and they pass through a cNOT that acts only on qubits 1 and 2, the gate G(C,X,I) would be applied to the entire combined state, instead of G(C,X).

Finally, two qubits sharing a state (qubit 1 and 2) might enter a multi-gate in such a way that qubit 2 would have position 1 in the gate, and vice versa.

Qubit 1 |----X---->

Qubit 2 |----C---->

There are ways to prevent ordering issues like that. One way would be to apply a swap gate on the qubits first, then run the cNOT, then do another swap. That is what we did at first. The method we use now is to apply an algorithm that switches and organize the gate indices instead, before applying the actual gate creation algorithm. This is more efficient, as it runs in O(N), where N is the number of gates (or qubits), while computing and applying a swap matrix is done in O(2N).

As an example of this, lets say two qubits are passed through a Toffoli gate, G(C,C,X). Qubits 1 and 3 have a shared state, and qubit 2 is free. In this case, the combined state would be formed by tensoring the combined state of 1 and 3 with the state of 2 (and the state would get the internal qubit ordering 1, 3, 2). The gate is then switched to G(C,X,C). This is faster then splitting the state, or performing swaps, and the effect is the same. The ordering of the qubits in the 3-qubit combined state would of course be (1,3,2), but after the state has been combined, the indices are changed to (1,2,3), and all qubits that are affected by this change is updated.

If qubit 2 above was in a combined state as well (lets say in position 1 in that state), the other qubit would have been unaffected by the gate, but we would have had to include it in the calculation. We would have created a 4-qubit combined state, and instead of the ordering (1,3,2) we would have had (1, 3, 2-1, 2-2), and the gate would have been G(C,X,C,I).

### Controlled gate generationEdit

The way we generate control matrices as fast as possible is to always sort the controls first, then the unitaries. For example, lets say the gate sorting stuff gave us the gate G(C,X,X,Z,C,Y). The first thing we do is to move the gates so that all controls end up first. In this case it would mean swapping gates 1 and 4 (counting from 0). The result would be: G(C,C,X,X,Z,X,Y).

During the first step, all the new mappings are stored, so that we know which gates were swapped. This 'pattern' is then used to do an optimized swap on the vector (more on this later), then apply the gate, then swap again. With highly optimized I mean the algorithm runs in O(ab) time, where a is the number of controls, and b the number of non-zero elements in the vector. It is capped at O(n log n). Also, the swap algorithm is essentially just a series of bit operations.

NOTE: The optimized swap approach to controlled gate generation is new /2014, february), so I will probably be able to remove the old swap-avoiding gate-sorting stuff, as a swap will have to be done anyways. Or two swaps, rather. The reason we avoided swaps altogether at first, was because of how controlled gates were generated (through a recursion algorithm), but that has since been replaced.

Factoring controlled gates

In terms of the gate generation function G, the factoring rule of non-controlled gates could be written:

$G(U_1,U_2,\ldots,U_N) = G(U_1,I,\ldots,I)G(I,U_2,\ldots,I) \ldots G(I,\ldots,I,U_N)$

Here, the U's are unitary 2x2 Matrices. Technically it would work for larger unitary matrices as well, to a lesser degree, but we'll not get into that now.

To show that this works for controlled gates as well, We will start with controlled gates on this form:

$G(C_1,C_2,...,C_M,U_1,U_2,...,U_N)$

The C's are control gates, and the U's are 2x2 unitary matrices. Technically the C gates does not need to be indexed, as they all do the same thing, but its useful when discussing swaps as we will do later.

What we want to show is that factoring is possible:

$G(C_1,C_2,\ldots,C_M,U_1,U_2,\ldots,U_N) =$

$G(C_1,C_2,\ldots,C_M,U_1,I,\ldots,I) \ldots G(C_1,C_2,\ldots,C_M,I,\ldots,I,U_N)$

First of all - the dimension of this matrix is $2^{M + N}$, where M is the number of controls, and N is the number of 2x2 unitary gates. If M = 0, this would be just a regular non-controlled gate, for which we know that factoring works. If N = 0, this matrix would simply be I x I x ... x I (M terms), and do nothing to the qubit state its applied to. This state has no unitaries, and therefore does not even need to be factored. To show the general case (when neither M nor N is 0) we will need the following symbols:

$I_N =\left[\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}\right]^{\otimes N}$

$0_N =\left[\begin{matrix}0 & 0 \\ 0 & 0 \end{matrix}\right]^{\otimes N}$

$U = U_1 \otimes U_2 \otimes \ldots \otimes U_N$.

Using these symbols, The matrix for G can be expressed as a diagonal block matrix:

$\left[\begin{matrix} I_N & 0_N & \cdots & 0_N \\ 0_N & I_N & \cdots & 0_N \\ \vdots & \vdots & \ddots & \vdots \\ 0_N & 0_N & \cdots & U \end{matrix}\right]$

There would be 2^M matrices in each row and column. This is the type of matrices we use to represent control gates.

Now, let T be the tensor product factorization:

$T_j = I \otimes I \otimes \ldots \otimes U_j \otimes I \otimes \ldots \otimes I$

We get:

$\left[\begin{matrix} I_N & 0_N & \cdots & 0_N \\ 0_N & I_N & \cdots & 0_N \\ \vdots & \vdots & \ddots & \vdots \\ 0_N & 0_N & \cdots & T_0 \end{matrix}\right] \left[\begin{matrix} I_N & 0_N & \cdots & 0_N \\ 0_N & I_N & \cdots & 0_N \\ \vdots & \vdots & \ddots & \vdots \\ 0_N & 0_N & \cdots & T1 \end{matrix}\right] \ldots \left[\begin{matrix} I_N & 0_N & \cdots & 0_N \\ 0_N & I_N & \cdots & 0_N \\ \vdots & \vdots & \ddots & \vdots \\ 0_N & 0_N & \cdots & T_N \end{matrix}\right] =$

$\left[\begin{matrix} I_N & 0_N & \cdots & 0_N \\ 0_N & I_N & \cdots & 0_N \\ \vdots & \vdots & \ddots & \vdots \\ 0_N & 0_N & \cdots & T_0 T_1 \ldots T_N \end{matrix}\right] = \left[\begin{matrix} I_N & 0_N & \cdots & 0_N \\ 0_N & I_N & \cdots & 0_N \\ \vdots & \vdots & \ddots & \vdots \\ 0_N & 0_N & \cdots & U \end{matrix}\right]$

We can therefore factor G in this case as well.

The next step is to show that this applies to all controlled gates, regardless of how controls and unitaries are placed. We will use swap gates for that.

If we want to change the gate G(X,C) into G(C,X), for example, we can do that with swap gates. For example, if $S_{(0,1)}$ is the matrix that swaps qubits 0 and 1, we have the have the following relation:

$G(C,X) = S_{(0,1)} G(X,C) S_{(0,1)}$

An individual qubit swap is self-inverse, that is $S_{(a,b)}S_{(a,b)} = I$. We also have $S_{(a,b)} = S_{(b,a)}$

We can create any permutation of qubits by applying a series of individual qubit swap gates in succession. The inverse of the operation would be to apply the individual qubit swap gates in reversed order. If indexed S stands for arbitrary single qubit swaps, we have:

$S = S_0S_1 \ldots S_n$

$S^{-1} = S_nS_{n - 1} \ldots S_0$

If we refer to gates that has all its controls before the unitaries as being "split", this statement holds:

For any gate matrix $H = G(A_0,A_1,...,A_{M + N})$, where M is the number of controls, N is the number of 2x2 unitaries, and $A_k$ represents either a 2x2 unitary or a control, there exists a matrix S that splits H. That is:

$SHS^{-1} = G(C_0,C_1,...,C_M,U_1,U_2,...,U_N)$

Furthermore, the matrix S is a swap matrix.

Using this result, the proof is trivial. Let $J = SHS^{-1}$. Also, let $H_k$ be the version of H where each unitary except the k'th one is replaced with an I. S would split that as well, and we choose S so that we get $SH_kS^{-1} = J_k$for all k.

J is split, so it can be factored. We therefore have $H = S^{-1}JS = S^{-1}(J_0J_1J_2 \ldots J_N)S$. Since $SS^{-1} = S^{-1}S = I$, we can also write this:

$H = S^{-1}JS = S^{-1}J_0(SS^{-1})J_1(SS^{-1})J_2S \ldots S^{-1}J_NS =$

$(S^{-1}J_0S)(S^{-1}J_1S)(S^{-1}J_2S) \ldots (S^{-1}J_NS) = H_0H_1 \ldots H_N$.

### Optimized swapEdit

We do not apply swap gate matrices to the qubit states. Our swap instead works like this:

Imagine you have a vector that represents the combined state of three arbitrary qubits:

$q_0 = \left[\begin{matrix} \alpha_0 \\ \alpha_1 \end{matrix}\right], q_1 = \left[\begin{matrix} \beta_0 \\ \beta_1 \end{matrix}\right], q_2 = \left[\begin{matrix} \gamma_0 \\ \gamma_1 \end{matrix}\right]$

The combined state would be:

$|\psi\rangle = \left[\begin{matrix} \alpha_0 \beta_0 \gamma_0 \\ \alpha_0 \beta_0 \gamma_1 \\ \alpha_0 \beta_1 \gamma_0 \\ \alpha_0 \beta_1 \gamma_1 \\ \alpha_1 \beta_0 \gamma_0 \\ \alpha_1 \beta_0 \gamma_1 \\ \alpha_1 \beta_1 \gamma_0 \\ \alpha_1 \beta_1 \gamma_1 \end{matrix}\right]$

Now let us ignore the alphas and betas and whatnot, and just use the indices. Keep in mind - this is still a vector. There is 1 element per row. The n'th number is just the index to use for the n'th qubit, for example: 0 0 1 means the element is $\alpha_0 \beta_0 \gamma_1$.

$|\psi\rangle = \left[\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 0 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \\ 1 & 1 & 1 \end{matrix}\right]$

Notice these numbers form the binary value representing the row number, except that qubit 0 is farthest to the left instead of the right, but that is simple to reverse. Also, this can of course be extended to any number of qubits.

If we want to swap qubits 0 and 1, we would have to ensure the indices of column 0 and 1 are switched, whilst the index of column 2 remains the same. This could be done by swapping rows 2->4 and 3->5. It would take two swaps (counting rows from 0).

In the general case, if we want to swap qubits 'i' and 'j', and the total number of qubits is N, this is the algorithm:

1) Invert i and j using the formula bit_k = N - 1 - k.

1) Find the states where bit_i is 0, and the bit_j is 1. There are $2^{N - 2}$ such states. This could be done fast and easy by holding bit_i = 0 and bit_j = 1 constant, and create all possible combinations of the other bits..

2) For each such state, get the decimal value from the bit string. Call it 's'. Then get the decimal value after switching bit i to 1, and j to 0. Call it 't'. Then swap row s with row t.

Note that this holds when we're dealing with an entangled state as well, for the same reason a swap gate works regardless of the type of state.

Example

Lets use the 3 qubit case and swap qubits 1 and 2 (still counting from 0). We have i = 1, j = 2 and N = 3. This means we have 2^(3-1) = 2 rows we need to find:.

001 = 1, 101 = 5

Note that with the bit reversal, bit 2 (which would represent qubit 0 when reading right-to-left) is the variable one. Bits 0 and 1 are fixed (representing qubits 2 and 1, respectively).

The "opposites" would be:

010 = 2, 110 = 6

This means we need to swap row 1 with row 2, and row 5 with row 6.

Let us confirm by calculating the swap matrix 'M' that swaps qubits 1 and 2. Let S be the typical 4x4 swap matrix. We get:

$M = I \otimes S = \left[\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}\right] \otimes \left[\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}\right] = \left[\begin{matrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix}\right]$

As we can see, it prompts us to do the exact same switching of rows.

Example

Lets say we have a 4 qubit state, and we want to swap qubits 1 and 2 (counting from 0). We need to find the states where column 1 = 0 and column 2 = 1. There are 4 such states. It is easy to find out which they are by inspection:

0010 = 2, 0011 = 3, 1010 = 10, 1011 = 11

Now we need to find the rows with column 1 = 1 and column 2 = 0.

0100 = 4, 0101 = 5, 1100 = 12, 1101 = 13

If we swap these rows, so that the two swapped rows has the same values for bit 0 and 3, we will have swapped qubits 1 and 2. It would mean swapping row 2 with 4, 3 with 5, 10 with 12, and 11 with 13.

Example

Lets test this for the 2 qubit case, just to show. There is only one swap possible, which is qubit 0 to qubit 1.

The four possible states are these:

$|\psi\rangle = \left[\begin{matrix} 0 & 0 \\ 0 & 1 \\ 1 & 0 \\ 1 & 1 \end{matrix}\right]$

Now, since N = 2, and 2^(N - 2) = 2^(2 - 2) = 1, this means we have 1 state with column 0 = 0 and column 1 = 1. That happens to be 01 = 1.

We now find the index for 10, which is 2. So we swap rows 1 and 2 (again - counting from 0), giving us the proper swap.

Lets look at the swap matrix:

$\left[\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}\right]$

This matrix of course tells us to change rows 1 and 2 (still counting from 0).


text:
# $a+b\sqrt{-3}$ and $a-b\sqrt{-3}$ are coprime in $\mathbb{Z}+ \omega \mathbb{Z}$

I want to prove that $a+b\sqrt{-3}$ and $a-b\sqrt{-3}$ are coprime in $\mathbb{Z}+ \omega \mathbb{Z}$ with $\omega$ a primitive third root of unity and $\gcd(a,b)=1$.

Approach: Assume they are not, let $s$ be an irreducible factor of $a+b\sqrt{-3}$ and $a-b\sqrt{-3}$, then $s|2b\sqrt{-3}$. Then $N(s)|6b^2$ and $N(s)|(a^2+3b^2)$….

Can someone help me how to approach a contradiction from this? Thanks in advance!!

#### Solutions Collecting From Web of "$a+b\sqrt{-3}$ and $a-b\sqrt{-3}$ are coprime in $\mathbb{Z}+ \omega \mathbb{Z}$"

Even if $a,b$ are relatively prime, the claim is false.

For example, let

$$y = 1 + \sqrt{-3}$$

$$z = 1 – \sqrt{-3}$$

Then $y,z$ are non-units, but each divides the other, hence they have a common non-unit factor.

So many things that need clarification here. What are $a$ and $b$? Are they integers, that is $a, b \in \mathbb Z$? And are you sure you’re only looking at numbers of the form $a + b \sqrt{-3}$, and not also numbers of the form $a + b \omega$?

It needs to be said explicitly that $$\omega = -\frac{1}{2} + \frac{\sqrt{-3}}{2}.$$ With that in mind, we can rewrite $1 – \sqrt{-3}$ and $1 + \sqrt{-3}$ as $-2 \omega$ and $2 + 2 \omega$ respectively. We have changed $a$ and $b$, of course, but it should still be clear that $2 \omega$ and $-2 \omega$ have at least one prime factor in common, as do $2 – 2 \omega$ and $2 + 2 \omega$.

Also note that $N(a + b \omega) = a^2 – ab + b^2$.

format: plain


text:
# Geometry question with pentagon and circle

Inside a very large field there is a shed in the shape of a regular pentagon of side $12$ m. A goat is tied at one vertex of the pentagon by a rope of length $16$ m. The goat cannot access the area inside the shed. Find the area that can be accessed by the goat inside the field.

No figure was provided in the original question. Hence note that the figure attached is my own interpretation. Please correct the figure if it seems wrong.
Figure:

As I have interpreted we need to find the area of the shaded region of the circle with radius $16$ m.
The problem I am facing is that I am not allowed to use the values of any trigonometric ratios besides the standard $45^{\circ},30^{\circ},60^{\circ},90^{\circ},0^{\circ}$. I could have solved it had a calulator been allowed but no value except these has to be used.

-
There is some discussion of a similar problem at sosmath.com/CBB/viewtopic.php?t=28503 –  Gerry Myerson Oct 12 '13 at 11:23
Your main mistake is that you assume that the outer boundary is a piece of circle. If you think a bit about what happens when the goat goes around the corner, you'll realize that the problem requires no trigonometry at all, just the general observation that the area of a circular sector is proportional to its aperture. –  fedja Oct 12 '13 at 11:49
@fedja i can't follow you Could you please be a little more descriptive. –  Suy Oct 12 '13 at 11:52
Take a box and a piece of rope (or thread) longer than the side of the box. Attach one end of the rope to one corner of the box (barn) and attach the other end to your finger (goat). Start walking the goat around the box and watch what happens with the rope. It is better to see it once than to hear about it 100 times. :-) –  fedja Oct 12 '13 at 12:11
@fedja thanks I got it –  Suy Oct 12 '13 at 14:16

As fedja said you assumption that the area that the goat can reach is a piece of circle is wrong, because the rope can't go through the walls of the shed, that means that when the rope will touch the wall and we want go into wall's direction, then we would have another anchor point at the another vertex. Because the side is of length $12$ and the rope of length $16$. We have rope of length $4$ left, i.e. we can make a circle of radius $4$ from the new anchor point. I hope it's clearer to you with picture:
You can calculate the area of the sector $GDF$ using the formula:
$$P = \frac{r\pi\theta}{180}$$
We know that the interior angle of pentagon is $108^{\circ}$ so the angle $\theta = 360^{\circ} - 108^{\circ} = 252^{\circ}$
And for the sectors $FEH$ and $GCI$ use the same formula, but this time use the exterior angle of pentagon, which we know that is $72^{\circ}$. And to get the final result just add those 3 areas together.


text:
# Nonhomogeneous wave equation with vanishing initial conditions

1. Jul 9, 2010

### Mmmm

1. The problem statement, all variables and given/known data

Let u(x,t) be the solution of the following initial value problem for the nonhomogeneous wave equation,

$$u_{x_1x_1}+u_{x_2x_2}+u_{x_3x_3}-u_{tt}=f(x_1,x_2,x_3,t)$$

$$u(x,0)=0$$ and $$u_t(x,0)=0$$

$$x\in\Re^3 , t>0$$

Use Duhamel's Principle and Kirchoff's formula to show that

$$u(x,t)=-\frac{1}{4\pi}\int_{\overline{B}(x,t)}\frac{f(x',t-r)}{r}dx'_1dx'_2dx'_3$$

where

$$r=\left|x-x'\right|=[(x_1-x'_1)^2+(x_2-x'_2)^2+(x_3-x'_3)^2]^\frac{1}{2}$$

and $$\overline{B}(x,t)[/itex] is the ball in $\Re^3$ with center at x and radius t. 2. Relevant equations Duhamel's Principle Let $v(x,t;\tau)$ be the solution of the associated (to the above initial value problem) "pulse problem" [tex]v_{x_1x_1}+v_{x_2x_2}+v_{x_3x_3}-v_{tt}=0$$

$$v(x,\tau;\tau)=0$$ and $$v_t(x,\tau;\tau)=-f(x,\tau)$$
$$x\in\Re^3 , t>\tau$$

then
$$u(x,t)=\int^t_0v(x,t;\tau)d\tau$$

Kirchoff's Formula

Suppose $p\in C^k(\Re^3)$ where k is any integer $\geq$2 Then the solution of

$$u_{x_1x_1}+u_{x_2x_2}+u_{x_3x_3}-u_{tt}=0$$

$$u(x,0)=0$$ and $$u_t(x,0)=p(x)$$

$$x\in\Re^3 , t>0$$

is given by

$$\frac{1}{4\pi t}\int_{S(x,t)}p(x')d\sigma_t$$

where S(x,t) is the surface of the sphere with radius t and centre at the point x. $d\sigma_t$ is the element of surface on S and x' is the variable point of integration.

3. The attempt at a solution

I should split this into two parts, one for each formula.
first I need to use Kirchoff's formula to find v and then Duhamel's principle to find u.

The problem with using Kirchoff is that the initial conditions are given at t=0 whereas our initial conditions for v are at $t=\tau$

So a transformation into kirchoff's formula to give the required initial conditions, $t'=t-\tau$

so

$$v(x,\tau;\tau)=0$$ and $$v_t(x,\tau;\tau)=-f(x,\tau)$$

becomes

$$v(x,0;\tau)=?$$ and $$v_t(x,0;\tau)=-f(x,?)$$

which doesn't give me anything because of that parameter in v. The t value and the parameter must be equal to give a known value.

Can anyone give me a clue as to how to get Kirchoff's formula to work with this? Or am I going in completely the wrong direction?
I'm sure that this is only the first of many sticking points in this question but I thought I'd ask one question at a time...:)


text:
Lower and Upper Bounds for Deniable Public-Key Encryption

Rikke Bendlin, Jesper Buus Nielsen, Peter Sebastian Nordholt, and Claudio Orlandi

Abstract

A deniable cryptosystem allows a sender and a receiver to communicate over an insecure channel in such a way that the communication is still secure even if the adversary can threaten the parties into revealing their internal states after the execution of the protocol. This is done by allowing the parties to change their internal state to make it look like a given ciphertext decrypts to a message different from what it really decrypts to. Deniable encryption was in this way introduced to allow to deny a message exchange and hence combat coercion. Depending on which parties can be coerced, the security level, the flavor and the number of rounds of the cryptosystem, it is possible to define a number of notions of deniable encryption. In this paper we prove that there does not exist any non-interactive receiver-deniable cryptosystem with better than polynomial security. This also shows that it is impossible to construct a non-interactive bi-deniable public-key encryption scheme with better than polynomial security. Specifically, we give an explicit bound relating the security of the scheme to how efficient the scheme is in terms of key size. Our impossibility result establishes a lower bound on the security. As a final contribution we give constructions of deniable public-key encryption schemes which establishes upper bounds on the security in terms of key length. There is a gap between our lower and upper bounds, which leaves the interesting open problem of finding the tight bounds.

Metadata
Available format(s)
Category
Public-key cryptography
Publication info
Published elsewhere. Full version of an ASIACRYPT 2011 paper.
Keywords
deniable encryption
Contact author(s)
jbn @ cs au dk
History
2011-09-13: revised
2011-01-25: received
See all versions
Short URL
https://ia.cr/2011/046
License

CC BY

BibTeX

@misc{cryptoeprint:2011/046,
author = {Rikke Bendlin and Jesper Buus Nielsen and Peter Sebastian Nordholt and Claudio Orlandi},
title = {Lower and Upper Bounds for Deniable Public-Key Encryption},
howpublished = {Cryptology ePrint Archive, Paper 2011/046},
year = {2011},
note = {\url{https://eprint.iacr.org/2011/046}},
url = {https://eprint.iacr.org/2011/046}
}

Note: In order to protect the privacy of readers, eprint.iacr.org does not use cookies or embedded third party content.


text:
# Very-long-baseline interferometry

(Redirected from Very Long Baseline Interferometry)
Some of the Atacama Large Millimeter Array radio telescopes.
The eight radio telescopes of the Smithsonian Submillimeter Array, located at the Mauna Kea Observatory in Hawai'i.

Very-long-baseline interferometry (VLBI) is a type of astronomical interferometry used in radio astronomy. In VLBI a signal from an astronomical radio source, such as a quasar, is collected at multiple radio telescopes on Earth. The distance between the radio telescopes is then calculated using the time difference between the arrivals of the radio signal at different telescopes. This allows observations of an object that are made simultaneously by many radio telescopes to be combined, emulating a telescope with a size equal to the maximum separation between the telescopes.

Data received at each antenna in the array include arrival times from a local atomic clock, such as a hydrogen maser. At a later time, the data are correlated with data from other antennas that recorded the same radio signal, to produce the resulting image. The resolution achievable using interferometry is proportional to the observing frequency. The VLBI technique enables the distance between telescopes to be much greater than that possible with conventional interferometry, which requires antennas to be physically connected by coaxial cable, waveguide, optical fiber, or other type of transmission line. The greater telescope separations are possible in VLBI due to the development of the closure phase imaging technique by Roger Jennison in the 1950s, allowing VLBI to produce images with superior resolution.

VLBI is most well known for imaging distant cosmic radio sources, spacecraft tracking, and for applications in astrometry. However, since the VLBI technique measures the time differences between the arrival of radio waves at separate antennas, it can also be used "in reverse" to perform earth rotation studies, map movements of tectonic plates very precisely (within millimetres), and perform other types of geodesy. Using VLBI in this manner requires large numbers of time difference measurements from distant sources (such as quasars) observed with a global network of antennas over a period of time.

## Scientific results

Geodesist Chopo Ma explains some of the geodetic uses of VLBI.

Some of the scientific results derived from VLBI include:

• Imaging the surfaces of nearby stars at radio wavelengths (see also interferometry) – similar techniques have also been used to make infrared and optical images of stellar surfaces
• Definition of the celestial reference frame
• Motion of the Earth's tectonic plates
• Regional deformation and local uplift or subsidence.
• Variations in the Earth's orientation and length of day.
• Maintenance of the terrestrial reference frame
• Measurement of gravitational forces of the Sun and Moon on the Earth and the deep structure of the Earth
• Improvement of atmospheric models
• Measurement of the fundamental speed of gravity
• The tracking of the Huygens probe as it passed through Titan's atmosphere, allowing wind velocity measurements

## VLBI arrays

There are several VLBI arrays located in Europe, Canada, the United States, Russia, Korea, Japan, Mexico and Australia. The most sensitive VLBI array in the world is the European VLBI Network (EVN). This is a part-time array which brings together the largest European radiotelescopes for typically week-long sessions, with the data being processed at the Joint Institute for VLBI in Europe (JIVE). The Very Long Baseline Array (VLBA) uses ten dedicated, 25-meter telescopes spanning 5351 miles across the United States, and is the largest VLBI array that operates all year round as both an astronomical and geodesy instrument.[1] The combination of the EVN and VLBA is known as Global VLBI. When one or both of these arrays are combined with one or more space-based VLBI antennas such as HALCA (previously) and now with RadioAstron (Spektr-R), the resolution obtained is higher than any other astronomical instrument, capable of imaging the sky with a level of detail measured in microarcseconds. VLBI generally benefits from the longer baselines afforded by international collaboration, with a notable early example in 1976, when radio telescopes in the United States, USSR and Australia were linked to observe hydroxyl-maser sources.[2] This technique is currently being used by the Event Horizon Telescope, whose goal is to observe the supermassive black holes at the centers of the Milky Way Galaxy and Messier 87.[3]

## e-VLBI

Image of the source IRC+10420. The lower resolution image on the left image was taken with the UK's MERLIN array and shows the shell of maser emission produced by an expanding shell of gas with a diameter about 200 times that of the Solar System. The shell of gas was ejected from a supergiant star (10 times the mass of our sun) at the centre of the emission about 900 years ago. The corresponding EVN e-VLBI image (right) shows the much finer structure of the masers because of the higher resolution of the VLBI array.

VLBI has traditionally operated by recording the signal at each telescope on magnetic tapes or disks, and shipping those to the correlation center for replay. Recently, it has become possible to connect VLBI radio telescopes in close to real-time, while still employing the local time references of the VLBI technique, in a technique known as e-VLBI. In Europe, six radio telescopes of the European VLBI Network (EVN) are now connected with Gigabit per second links via their National Research Networks and the Pan-European research network GEANT2, and the first astronomical experiments using this new technique were successfully conducted in 2011.[4]

The image to the right shows the first science produced by the European VLBI Network using e-VLBI. The data from 6 telescopes were processed in real time at the European Data Processing centre at JIVE. The Netherlands Academic Research Network SURFnet provides 6 x 1 Gbit/s connectivity between JIVE and the GEANT2 network.

## Space VLBI

In the quest for even greater angular resolution, dedicated VLBI satellites have been placed in Earth orbit to provide greatly extended baselines. Experiments incorporating such space-borne array elements are termed Space Very Long Baseline Interferometry (SVLBI).

The first space VLBI experiment was on Salyut-6 orbital station with KRT-10 (ru:КРТ-10), a 10-meter radio telescope, which was launched in July 1978.

The first such dedicated VLBI mission was HALCA, an 8-meter radio telescope, which was launched in February 1997 and made observations until October 2003, but due to the small size of the dish only very strong radio sources could be observed with SVLBI arrays incorporating it.

Another space VLBI mission, Spektr-R (or RadioAstron), was launched in July 2011. The Radioastron project is a radiointerferometer composed of the Spectrum-R space telescope and terrestrial radio telescopes. The orbit height of the Spectrum-R is 390 thousand km, which makes it the longest radiointerferometer to date. The resolution of the system will reach 8 microseconds of the arc.

## How VLBI works

Recording data at each of the telescopes in a VLBI array. Extremely accurate high-frequency clocks are recorded alongside the astronomical data in order to help get the synchronization correct

In VLBI interferometry, the digitized antenna data are usually recorded at each of the telescopes (in the past this was done on large magnetic tapes, but nowadays it is usually done on large arrays of computer disk drives). The antenna signal is sampled with an extremely precise and stable atomic clock (usually a hydrogen maser) that is additionally locked onto a GPS time standard. Alongside the astronomical data samples, the output of this clock is recorded on the tape/disk media. The recorded media are then transported to a central location. More recent experiments have been conducted with "electronic" VLBI (e-VLBI) where the data are sent by fibre-optics (e.g., 10 Gbit/s fiber-optic paths in the European GEANT2 research network) and not recorded at the telescopes, speeding up and simplifying the observing process significantly. Even though the data rates are very high, the data can be sent over normal Internet connections taking advantage of the fact that many of the international high speed networks have significant spare capacity at present.

At the location of the correlator the data are played back. The timing of the playback is adjusted according to the atomic clock signals on the (tapes/disk drives/fibre optic signal), and the estimated times of arrival of the radio signal at each of the telescopes. A range of playback timings over a range of nanoseconds are usually tested until the correct timing is found.

Playing back the data from each of the telescopes in a VLBI array. Great care must be taken to synchronize the play back of the data from different telescopes. Atomic clock signals recorded with the data help in getting the timing correct.

Each antenna will be a different distance from the radio source, and as with the short baseline radio interferometer the delays incurred by the extra distance to one antenna must be added artificially to the signals received at each of the other antennas. The approximate delay required can be calculated from the geometry of the problem. The tape playback is synchronized using the recorded signals from the atomic clocks as time references, as shown in the drawing on the right. If the position of the antennas is not known to sufficient accuracy or atmospheric effects are significant, fine adjustments to the delays must be made until interference fringes are detected. If the signal from antenna A is taken as the reference, inaccuracies in the delay will lead to errors ${\displaystyle \epsilon _{B}}$ and ${\displaystyle \epsilon _{C}}$ in the phases of the signals from tapes B and C respectively (see drawing on right). As a result of these errors the phase of the complex visibility cannot be measured with a very-long-baseline interferometer.

The phase of the complex visibility depends on the symmetry of the source brightness distribution. Any brightness distribution can be written as the sum of a symmetric component and an anti-symmetric component. The symmetric component of the brightness distribution only contributes to the real part of the complex visibility, while the anti-symmetric component only contributes to the imaginary part. As the phase of each complex visibility measurement cannot be determined with a very-long-baseline interferometer the symmetry of the corresponding contribution to the source brightness distributions is not known.

R. C. Jennison developed a novel technique for obtaining information about visibility phases when delay errors are present, using an observable called the closure phase. Although his initial laboratory measurements of closure phase had been done at optical wavelengths, he foresaw greater potential for his technique in radio interferometry. In 1958 he demonstrated its effectiveness with a radio interferometer, but it only became widely used for long-baseline radio interferometry in 1974. At least three antennas are required. This method was used for the first VLBI measurements, and a modified form of this approach ("Self-Calibration") is still used today.

## References

1. ^ "Archived copy". Archived from the original on June 11, 2012. Retrieved May 30, 2012.
2. ^ First Global Radio Telescope, Sov. Astron., Oct 1976
3. ^ Webb, Jonathan (8 January 2016). "Event horizon snapshot due in 2017". BBC News. Retrieved 2017-10-22.
4. ^ "Astronomers Demonstrate a Global Internet Telescope". Retrieved 2011-05-06.


text:
# Change Engineering

In software design, we’re often obsessed with perfecting the qualities software has right now, or the qualities of how the software should be under ideal conditions in the future. We don’t often focus on designing changes to software – we mostly focus on the start and end points.

Imagine different components of a software system dancing together – each step of the dance is a change to the component. In an ideal situation, when software development is complete the dance stops and players remain perfectly still in their final state of interlock. No change is required, because the software is “finished”.

But no software is like this. When a software project first commences the dance floor is empty. Gradually new players/dancers are introduced to the dance floor. Dancer’s positions and poses are change to accommodate each other. Bug fixes and refactorings change the position or pose of dancers that needed fixing.

But by today’s software design practices this dance is nothing but a crowd of players, each one rigid in his/her shape. We go to great lengths to fix solid the interface between the players (components) – their interlocking posture – and then we hope never to change it again. Instead of a dance, this is like a weird game of twister. Players have to hold their position as steadily as possible. It’s considered a design flaw if changes to one player cause changes to another player – this is software coupling.

Let me emphasize this point: todays design practices emphasize immobility and decoupling between components of the system. This is very valid way of dealing with the dancing problem: limit the movement of the dancers so that they don’t run into each other. Give them each enough space so that when they do need to move a little they won’t influence others around them.

But we know all this comes at enormous cost. The interfaces have to be clearly defined and agreed upon – never to be changed – so they better be correct right at the beginning. We have to use complicated interfacing patterns – facades, indirection, visitors, injection. No longer can we just call function foo. We now have to add foo to the interface, and inject the concrete object through the construct, saving it until we need to call it, and then call it through an adapter because it didn’t quite have the right structure1.

So these principles solve half the problem: slow down the game so that it’s easier to keep everyone separate. But there’s another solution which also deserves attention: formalize methods of change. That is, formalize the dance moves.

For example, here’s a situation I’ve encountered. Let’s say a program X calls a web service Y. So X and Y hold a single dance position with each other. Now lets say we need to add a new parameter to the service. What moves can we make?

• We can simply change X and Y, and deploy them at the same time.
• We can change Y first to accept the new parameter, but not require the new parameter, and deploy Y. Then we can change X to provide the new argument, and deploy it
• We can change X first to provide the new argument, and deploy X. Then we can change Y to use the argument, and deploy Y.
• We can create a new service Z which supports the new parameter, and then update X to use Z.

The order of footwork is important: we can’t change X to require that Y have the parameter if it doesn’t yet. We can’t change Y to require the new argument, if X isn’t supplying it yet. We can’t deploy X to require the new service Z, if the new service hasn’t been implemented yet. There are a number of steps in this part of the dance, and the order is important.

The above example is simple, and could just be solved by reasoning it out, but you can see it could get complicated quite quickly. If some data had to be passed back from the service as well, then the moves become more intricate.

Do any of these dance moves have a name? Not that I know of (but correct me if I’m wrong). We don’t have names for them, and we also don’t really have a standard notation to describe the moves. And what about tools, and what about static analysis? What if the development tools could give a compilation error if we make an illegal dance move?2

This doesn’t just apply to distributed systems. It applies equally to self contained programs, or within modules and components themselves. For one thing, things don’t get developed atomically. A developer can only write one line of code at a time, and each new line of code is a small dance move. If the move is illegal, you might get a compilation error or perhaps your unit tests fail. It’s much better to be able to develop in small increments, rather than having to go a week with failing tests. I’ve certainly had times like that: where I started a “small” refactoring but it was weeks before the code even compiled again. Instead of engineering a sequence of small legal moves, I made one multi-week leap – flying dangerously through the air. This is the same thing that I’ve seen happen with producing new major versions of software: it takes months or years to develop the new major version from scratch, and only then can it be introduced to the dance floor with the customers. It would be better to engineer a sequence of small changes to get from the old version to the new one, even if the new one has a completely different architecture.

This is also important for collaboration, since software teams don’t make changes atomically either. There are dependencies between different people’s development, and order is often important. Every time you commit code to the common repository you’re making a dance move with your team mates, and you don’t want to tread on their feet. A formal discipline of change engineering would make it easier to coordinate this dance without having to resort to tons of scaffolding or indirection code.

## Coupling can be good

I’d like to make one last point and say that coupling can be a good thing. The reason is perhaps obvious: the more connected two pieces of code are, the easier it is to directly access information between them. All the rules we have for decoupling generally add layers of indirection, which in turn makes it harder to see what’s really going on (the runtime structure looks less and less like the code structure), and harder to add new functionality because you have to weave new pieces of data through all the decoupling layers and interfaces. The “god-class” anti-pattern is, practically speaking, a great pattern to use if you have a program that’s only a few hundred lines of code.

The apparent problem with coupling is that solidifies the program. You can’t change one thing because it requires a change to something else. Each coupling link is an iron beam that connects the two pieces: when one moves, so does the other. Enough iron beans and the whole system becomes completely rigid. You don’t want to change anything because you would risk breaking everything else.

Although decoupling addresses this issue, so does dancing. That is, if we can define formal rules for changing components that are coupled together, we are expanding the number of coupled pieces that we can handle in the system. We’re replacing the iron beams with hinges, and allowing the dancers to interact directly with each other without being rigid.

## Conclusion

In software engineering and design, there is perhaps too much emphasis on the fixed state of code. You generally design software with a single endpoint in mind, rather than the sequence of atomic changes required to get there without breaking the system at each checkpoint. These are almost independent of each other, and there should be a new discipline for change engineering, so that we can build more robust systems, minimize risk, and reduce code. For large systems there should perhaps even be dedicated engineers focusing on the problem of change design, formally engineering the sequences needed to move the software towards long-term goals of the company, rather than letting the dancers to wonder undirected into a dead-end of rotted software before starting again on a new version.

1. Don’t think to hard about this example. The point is that it’s normally a long process to do something that should be simple

2. Don’t underestimate the power of this possibility


text:
# Homework HELP!

1. Jan 4, 2005

### Stargate

The force of gravity is twice as great on a 2-kg rock on a 1-kg rock. Why does the 2-kg rock not fall with twice the acceleration?

2. Jan 4, 2005

### dextercioby

$$F=ma$$
If the force is the product between the mass and the acceleration,figure out yourself...

Daniel.

3. Jan 4, 2005

### Stargate

They both have the same mass and acceleration! Right?

Last edited: Jan 4, 2005
4. Jan 4, 2005

### dextercioby

Okay,for the first body write
$$G_{1}=m_{1}a_{1}$$(1)
And for the second
$$G_{2}=m_{2}a_{2}$$(2)

Divide (1) through (2)
$$\frac{G_{1}}{G_{2}}=\frac{G_{1}a_{1}}{G_{2}a_{2}}$$(3)
Use the fact that $G_{1}$ is two times $G_{2}$ and that $m_{1}$ is 2Kg,which is two times the value of $m_{2}$ which is 1Kg.Then
$$2=\frac{2a_{1}}{a_{2}} \Rightarrow a_{1}=a_{2}$$

Daniel.


text:
# 05 Jun 2018: UPSC Exam Comprehensive News Analysis

A. GS1 Related
B. GS2 Related
GOVERNANCE
1. Ban proposed on obscene depiction of women on Net
2. Be agents of change, President tells Governors
INTERNATIONAL RELATIONS
1. India seeks new security forum
2. War games to hone anti-submarine skills
3. In S-400 deal, U.S. is the elephant in the room
4. China hails Modi’s Singapore speech
C. GS3 Related
ECONOMY
1. Centre to start measuring ‘green GDP’ of States
2. NPAs increasing, IBA tells panel
3. TRAI moots network to help in disaster relief
4. SEBI slashes expenses charged by mutual funds
5. Oil price falls as U.S. supply grows
6. Fitch downgrades PNB on credit profile
7. IBC: delisting norms eased for firms
8. India flouting laws by taxing global air tickets: IATA chief
D. GS4 Related
E. Editorials
AGRICULTURE
1. Farm friction: on the malaise in agriculture
POLITY AND GOVERNANCE
1. Governance and the Governor
F. Prelims Fact
G. UPSC Prelims Practice Questions
H. UPSC Mains Practice Questions

A. GS1 Related

Nothing here for today!

B. GS2 Related

1. Ban proposed on obscene depiction of women on Net

• The Ministry of Women and Child Development has proposed to ban obscene depiction of women on the Internet and through SMS/MMS by amending the Indecent Representation of Women Act, 1986.
• The Ministry has also suggested that stricter punishments be awarded for such crimes on par with those recommended under the IT Act, 2008.
• The Ministry has proposed amendment in definition of the term advertisement to include digital form or electronic form or hoardings, or through SMS, MMS, etc.
• It has also proposed setting up a Central authority under the National Commission of Women, which will include representatives from the Advertising Standards Council of India, the Press Council of India, the Ministry of Information and Broadcasting and one member with experience of working on women’s issues.

Indecent Representation of Women Act, 1986

• It was enacted to prohibit indecent representation of women through advertisement or in publications, writings, paintings, figures or in any other manner.
• The IRW Act provides for punishment of up to two years in jail for an offence committed for the first time and imprisonment of six months to five years for a second conviction.
• Sections 67 and 67A of the IT Act lay down a punishment of three to five years for circulating obscene material and five to seven years for circulating sexually explicit material respectively.

2. Be agents of change, President tells Governors

• President Ram Nath Kovind said that the people of a State look at the offices of the Governor as the fount of ideals and values.
• He asked these constitutional functionaries to be the agents of change.
• Against the backdrop of the government formation efforts recently in Karnataka, when the Congress and the Janata Dal(S) had accused Governor Vajubhai Bala as partisan, President Kovind’s remarks are significant.
• The President said a Governor was not only an important link in the federal structure of the country but also a mentor and a guide to the State governments in preparing a road map for development of 100 million tribal people and other marginalised sections.
• As Governors, they can help in the shaping of a road map for the betterment of lives of our fellow citizens who have not benefited as much as expected from our development journey.
• The President also urged Governors to serve as guardians to India’s youth in their role of Chancellors of universities.
• He said that nearly 94% of the students enrolled in higher education study in universities under the purview of the State government.
• It is for the Governors to inspire State universities to maintain discipline and integrity.
• President Kovind also sought suggestions from Governors on the celebrations for the 150th birth anniversary of Mahatma Gandhi. Beginning October 2, this year, the Centre has decided to celebrate it over a 24-month period.

Prime Minister

• Prime Minister Narendra Modi, who spoke at the inaugural session, asked Governors to use their public experience to ensure that people benefited from the flagships schemes of the Central government, and asked them to visit some of the villages that had been electrified for the first time.
• Mr Modi said that the institution of Governor has a pivotal role to play within the federal structure and constitutional framework of our country.
• Prime Minister Modi suggested that universities be focal point for celebrations like the 150th birth anniversary of Mahatma Gandhi.
• The Prime Minister also urged Governors of States with a substantial tribal population to ensure that they benefited from government programmes on education, sports and financial inclusion.
• Modi also mentioned some key themes of development such as the National Nutrition Mission, rural electrification and development parameters in aspirational districts and the move towards Gram Swaraj Abhiyan from April 14, right now being implemented across 16,000 village, focussing on seven key schemes.

1. India seeks new security forum

• India urged for support from Russia, China, South Africa and Brazil to create a new security forum to counter terrorism and radicalisation.
• Speaking at the BRICS Foreign Ministers meeting, External Affairs Minister Sushma Swaraj said that the forum could be realised through an understanding among the National Security Advisors (NSAs) of the member countries of the grouping.
• Swaraj also emphasised the contribution of the BRICS countries in ensuring stability in a world faced with growing insecurity.
• The National Security Advisers of the BRICS member- states have a dialogue mechanism to counter radicalisation, terrorism, money laundering and other international crimes.
• During Prime Minister Narendra Modi’s recent meetings with President Vladimir Putin and President Xi Jinping, India had expressed willingness to intensify work with the BRICS member- states to address pressing issues of collective concern.

2. War games to hone anti-submarine skills

• The Navies of India, Japan and the U.S. will enhance their anti-submarine warfare skills in this year’s Malabar naval war games to be held off the coast of Guam from June 7 to 16.
• For the first time in a Malabar exercise, all three Navies are deploying their maritime reconnaissance (MR) aircraft to sharpen those skills. MR  is a fixed-wing aircraft designed to operate for long durations over water in maritime patrol roles.
• In addition, Japan and the U.S. have anti-submarine warfare helicopters on board their helicopter carrier JS Ise and aircraft carrier USS Ronald Reagan, respectively.

India

• However, the Indian Navy does not have any anti-submarine warfare helicopter in the exercise.
• Indian effort would be limited to P-8I and the sonars on ships.
• The Navy’s anti-submarine warfare helicopter fleet is dependent on the ageing Sea Kings and is in urgent need of new helicopters.
• The exercise contributes towards increasing the level of mutual understanding, interoperability and sharing of the best practices between the three navies.

3. In S-400 deal, U.S. is the elephant in the room

• India’s ambitions to procure the Russian-developed S-400 Triumf long-range air-defence system has landed New Delhi right in the middle of global strategic complexities.

The S-400

• The S-400 is a complex military system comprising several radars, command post, different types of missiles and launchers that can track several dozen incoming objects simultaneously from hundreds of kilometres away, launch counter-missiles within seconds and shoot them down with great efficiency.
• The S-400 (SA-21 Growler), when properly operated, is a potent medium-to-long-range surface-to-air missile system.
• To be most effective, however, it needs to be integrated with other air defence systems and components — such as radars — operated by the purchasing country.
• This however, presents problems if some of these have been bought from the U.S. or potentially other Western states, where the required levels of integration will not be possible because of security concerns.
• Given its advanced capabilities, the weapon system has added a new strategic angle to ongoing stand-offs around the world — in Syria, between Saudi Arabia and Qatar, and in China and its neighbourhood.
• As the Defence Ministry prepares to present the procurement proposal for the S-400 systems before the Cabinet Committee on Security (CCS), officials are divided on whether New Delhi will push the deal through in the face of possible U.S. sanctions and warnings that it will adversely affect transfer of U.S. military technology.
• But Washington will likely soon withdraw this threat and quietly acquiesce to the purchase once Indian diplomats have the opportunity to explain to their U.S. counterparts how the S-400 acquisition supports the U.S.-India shared goal of enhancing regional capabilities to deter Chinese aggression.

4. China hails Modi’s Singapore speech

• China welcomed keynote remarks by Prime Minister Narendra Modi at the Shangrila dialogue in Singapore, linking them with the spirit of the Wuhan summit in April with Chinese President Xi Jinping.
• They had in-depth exchange of views in international landscape and bilateral relations and reached consensus on many issues during the informal meeting of the Indian PM and Chinese President at Wuhan.

Stable relations

• In his remarks at the Asia Security Summit in Singapore, Mr. Modi had referred to the Wuhan summit.
• He said that a two-day informal Summit with President Xi in Wuhan had helped Indian and China cement their understanding that strong and stable relations between the two nations are an important factor for global peace and progress.
• He also said that he firmly believes that, Asia and the world will have a better future when India and China work together in trust and confidence, sensitive to each other’s interests.
• Modi made it plain that India rejected the doctrine of containment, opposed protectionism, and backed globalisation as a template of progress.

C. GS3 Related

1. Centre to start measuring ‘green GDP’ of States

• Starting this year, the government will begin a five-year exercise to compute district-level data of the country’s environmental wealth.
• The numbers will eventually be used to calculate every State’s ‘green’ Gross Domestic Product (GDP).
• The metric will help with a range of policy decisions, such as compensation to be paid during land acquisition, calculation of funds required for climate mitigation, and so on.
• This is the first time such a national environment survey is being undertaken. India’s environmental diversity and riches are universally recognised but have never been quantified.

Pilot project

• A pilot project is set to begin this September in 54 districts.
• Land will be demarcated into grids with about 15-20 grids per district.
• These will capture the diversity in the State’s geography, farmland, wildlife, and emissions pattern, and will be used to compute a value.
• The funds for the pilot project are already available. But the budget has not been specified.
• Much of the data required for the inventory would be sourced from datasets that already exist with other government ministries.

Green skilling programme

• The government has also launched a ‘green skilling’ programme under which youth, particularly school dropouts, would be trained in a range of ‘green jobs’.
• Some of the labour required for the survey would also be sourced from the green-skilled workforce.

2. NPAs increasing, IBA tells panel

• NPA is one of the major concerns for the banking system around the globe and the Indian banking system is not an exception to this universal phenomenon.
• With non-performing assets (NPA) increasing and banks’ capital positions not improving despite the additional capital infused by the government in public sector banks, the Indian Banks’ Association (IBA) told the Standing Committee on Finance.
• The data provided by the IBA showed that the ratio of gross NPAs to gross advances — basically a metric of bad loans — had grown from 2.36% by March 2011 to 4.11% in March 2014, and grew further to 10.41% by December 2017.
• However, the IBA said this growth was also due to the Asset Quality Review conducted by the Reserve Bank of India, which revealed a lot of loans as NPAs, which were earlier classified as standard assets.
• Another major issue it raised was the strain put on banks’ resources by the fast pace of technological advancement.

Way forward (containing NPAs)

• The issue of shortage of capital with banks.
• Extending the time frame for full implementation of Basel III norms (international standards that lay strict requirements on banks’ equity and capital ratios) from the current deadline of March 2019. The RBI has been implementing the norms in a phased manner from April 2013.
• A realistic assessment of capital requirement of PSBs and need for further infusion of capital by promoters/owners.
• Periodic visits to the borrowers’ business units for the verification of stocks, and the plant and machinery along with the proper scrutiny of the quarterly financial statements and projections received from the borrower.

3. TRAI moots network to help in disaster relief

• The Telecom Regulatory Authority of India recommended setting up of a pan-India broadband public protection disaster relief (PPDR) communications network based on 3GPP PS-LTE technology for an advanced, reliable, robust and responsive communication networks to be used during disasters.
• Existing PPDR networks in the country are analog and digital systems supporting narrowband voice and data communications.
• Introduction of advanced broadband PPDR communication networks can be a great enabler in decision-making and handling of PPDR operations for personnel and organisations involved.
• PPDR communication supports a range of services such as maintenance of law and order, protection of life and property, disaster relief and emergency responses.
• TRAI recommended that a hybrid model of broadband PPDR network in India should be set up.
• The regulator recommended creating a Special Purpose Vehicle under the Ministry of Home Affairs to plan, coordinate and steer the nationwide network besides implementation and subsequent operation.
• TRAI proposed that operators should be mandated to provide mobile BTS and backpack devices in case of disasters when the terrestrial network gets damaged or becomes dysfunctional.

Read more on Telecom Regulatory Authority Of India

4. SEBI slashes expenses charged by mutual funds

• Markets regulator SEBI has slashed the additional expense charged by mutual funds to just 5 basis points to help increase the penetration of such products among investors.
• The move will help cut the cost of investing in mutual funds and industry players believe that it may result in lower commissions for distributors.
• The extra expense of 20 basis points has been reduced to 5 basis points across all schemes.

5. Oil price falls as U.S. supply grows

• Oil prices fell as U.S. production hit a record high and OPEC members considered boosting supply.
• The benchmark Brent crude oil lost \$1.26 a barrel.
• A sea of red is washing over the energy complex as rising U.S. production coupled with a looming relaxation in OPEC-led cuts sends bulls scurrying for the exits.

OPEC

• Arab oil ministers agreed on the need for continued cooperation between members of the Organization of the Petroleum Exporting Countries (OPEC) and other big producers to balance global supply.
• OPEC will meet formally on June 22 to set oil policy.
• It is expected to agree to raise output to cool the market amid worries over Iranian and Venezuelan supply and after Washington raised concerns that the oil rally was going too far.
• Saudi Arabia, the effective OPEC leader, and Russia have discussed boosting output to compensate for supply losses from Venezuela and to address concerns about the impact of U.S. sanctions on Iranian output.

6. Fitch downgrades PNB on credit profile

• Fitch Ratings has downgraded the Viability Rating (VR) of Punjab National Bank to ‘b’ from ‘bb-’ and maintained negative ratings.
• The two-notch downgrade is a reflection of the significant deterioration in its standalone credit profile, mainly due to a drop in its core capital ratio that was bigger than Fitch’s expectation.
• The deterioration in its core capitalisation was caused by a sharp increase in its non-performing loans (NPLs) and the related increase in credit costs, which resulted in large losses in the financial year ended March 2018 (FY18).
• The decline also highlights the management’s weaker execution and previous underwriting and oversight gaps.

7. IBC: delisting norms eased for firms

• Market regulator SEBI has relaxed requirements to comply with delisting norms for companies facing insolvency proceedings provided the resolution plan lays down the procedure for delisting that particular entity from the exchanges.
• The move is expected to provide succour to various listed companies that have been referred for resolution under the Insolvency and Bankruptcy Code (IBC).
• More than 750 entities, including many listed companies, are facing insolvency proceedings under the Code.

Watch BYJU’S video on Insolvency & Bankruptcy Code and Ordinance

8. India flouting laws by taxing global air tickets: IATA chief

• The International Air Transport Association (IATA) castigated India for taxing international tickets, even as it asked governments to facilitate the growth of worldwide connectivity by avoiding creeping re-regulation, maintaining the integrity of global standards and addressing a capacity crisis.

Background

• India helped develop ICAO resolutions prohibiting tax on international tickets.
• But India was taxing international tickets in contravention of the resolutions of the UN body International Civil Aviation Organisation (ICAO).
• Imposition of Goods and Services Tax (GST) has led to the enhancement of rates on international air tickets, especially business class.
• The tax covers airline products and services including tickets, ancillary, change, refund and other products and fees.

Capacity crisis

• IATA also warned that governments need to urgently tackle a capacity crisis facing airports as demand for international travel grows.
• But they should be cautious about private sector involvement.
• With passenger levels projected to nearly double to 7.8 billion by 2036, infrastructure such as airports and air traffic control systems are not keeping pace.

IATA (The International Air Transport Association)

• It is a trade association of the world’s airlines.
• It consists of 278 airlines, primarily major carriers, representing 117 countries.
• It supports airline activity and helps formulate industry policy and standards.
• IATA is headquartered in Montreal, Quebec, Canada with Executive Offices in Geneva, Switzerland.

D. GS4 Related

Nothing here for today!!!

E. Editorials

1. Farm friction: on the malaise in agriculture

Present Issue

• Since June 1, many farmers are on an unusual 10-day ‘strike’ to draw the government’s attention to distress in the fields.
• A federation of 130 farmer bodies has decided to stop supplies of vegetables and dairy produce to major cities and hold a dharna on 30 national highways, without blocking vehicular passage.
• Prices of vegetables and fruits are inching up in urban centres given the supply shock created by this ‘Gaon Bandh’; in cities like Mumbai, fishermen have joined the cause.

What are the Farmers’ Demands?

• Enhancement of the minimum support price regime for crops in line with the M.S. Swaminathan Commission’s recommendations, higher prices for milk procurement and loan waivers to offset low or negative returns on investment.
• This stir, which ends with a Bharat Bandh call on June 10, is not about the immediate crisis faced by specific sections of farmers (such as the cash-strapped sugarcane-growers for whom a cess is being considered under the GST regime). It is a culmination of multiple attempts made over a year to red-flag the systemic malaise in agriculture.
• The current stir also derives from lack of tangible action on assurances made earlier and imperceptible movement on the Centre’s grand promises such as doubling farm incomes and raising MSPs.

Triggering Point

• On June 6, 2017, some farmers were killed in police firing in Mandsaur, Madhya Pradesh, during an agitation for better crop prices.
• There have been agitations across the country since then, including in Tamil Nadu, Punjab, Haryana, Maharashtra (where thousands of farmers walked nearly 200 km to the State capital in March).

Way Forward

• The government must move purposefully to address the systemic malaise in agriculture
• The minister in charge had promised this over a year ago, but nothing happened. Blaming the agitators is easy; policy responses are where the heavy lifting is needed.

To Think/Know

• Doubling the farmers’ income by 2022 – Can this be a reality?
• M.S. Swaminathan Formula to solve the farmers’ problems.

1. Governance and the Governor

Governance and the Governor

• The recent developments in Karnataka makes us to think “Do we need the office of the Governor?”

Role of Governor in Governance

• Under the constitutional scheme, the Governor’s mandate is substantial. From being tasked with overseeing government formation, to reporting on the breakdown of constitutional machinery in a State, to maintaining the chain of command between the Centre and the State, he can also reserve his assent to Bills passed by the State Legislature and promulgate ordinances if the need arises.
• Further, under Article 355, the Governor, being the Central authority in a State, acts as an overseer in this regard.
• Undoubtedly, the most crucial issue relates to the exercise of gubernatorial (relating to or connected with the post of governor) discretion. The Governor has the task of inviting the leader of the largest party/alliance, post-election, to form the government; overseeing the dismissal of the government in case of a breakdown of the Constitution in the State; and, through his report, recommending the imposition of President’s rule.
• In India, the balance in power is tilted towards the Union. The importance of the Governor’s position arises not from the exceptional circumstances that necessitate the use of his discretion, but as a crucial link within this federal structure in maintaining effective communication between the Centre and a State.
• As a figurehead who ensures the continuance of governance in the State, even in times of constitutional crises, his role is often that of a neutral arbiter in disputes settled informally within the various strata of government, and as the conscience keeper of the community.

Why the abuse of power?

• There are numerous examples of the Governor’s position being abused, usually at the behest of the ruling party at the Centre.
• The root lies in the process of appointment itself. The post has been reduced to becoming a retirement package for politicians for being politically faithful to the government of the day.
• Consequently, a candidate wedded to a political ideology could find it difficult to adjust to the requirements of a constitutionally mandated neutral seat. This could result in bias, as appears to have happened in Karnataka.

Nonpartisan Politicians

• On the other hand, there are instances of politicians who have risen above partisan politics, and performed their role with dignity and without fear or favour.
• In this one can think of former President Pranab Mukherjee, former Lok Sabha Speaker Somnath Chatterjee, and former West Bengal Governor Gopalkrishna Gandhi.

Way forward

• A possible solution would be not to nominate career politicians and choose “eminent persons” from other walks of life. Both the Sarkaria and M.M. Punchhi Commissions seem to hint at this. But this could also lead to the creation of sycophants within the intelligentsia, an equally worrisome prospect.
• One has to consider the verdict of the Supreme Court in B.P. Singhal v. Union of India, on interpreting Article 156 of the Constitution and the arbitrary removal of Governors before the expiration of their tenure. This judgment is crucial since a fixed tenure for Governors could go quite far in encouraging neutrality and fairness in the discharge of their duties, unmindful of the dispensation at the Centre.
• Since the Bommai verdict allows the Supreme Court to investigate claims of mala fide in the Governor’s report, a similar extension to cover mala fide in the invitation process could be a potential solution.

Conclusion

• Misuse of a position of power should not serve as a justification for removing the office altogether, unless such a position has totally lost its relevance.
• Rather, these debates on limitations on the power of constitutional functionaries should be allowed unimpeded to ensure the organic development of our polity.

F. Prelims Fact

Nothing here for today!!!

G. Practice Questions for UPSC Prelims Exam

Question 1. Consider the following statements:
1. 5th June is celebrated as the World Environment Day.
2. The theme for this year’s World Environment Day is ‘beat air pollution’.
3. India is the global host nation for the 43rd edition of the event.

Which of the above statement/s is/are correct?

1. 1 and 2 only
2. 1 and 3 only
3. 2 and 3 only
4. All of the above

See

Question 2. Consider the following statements with respect to Central Pollution Control Board
(CPCB):
1. It is a statutory organisation.
2. It was constituted under the Air (Prevention and Control of Pollution) Act, 1981.
3. It was entrusted with the powers and functions under the Water (Prevention and Control of Pollution) Act, 1974.
4. It has to advise the Central Government on any matter concerning prevention and control of water and air pollution and improvement of the quality of air.

Which of the above statement/s is/are incorrect?

1. 1 and 2 only
2. 2 and 3 only
3. 1 and 4 only
4. None of the above

See

Question 3. Which of the following statements is incorrect?
1. The Department of Telecommunications is a department of the Ministry of Communications.
2. The Department of Telecom has been formulating developmental policies.
3. The Department is also responsible for grant of licenses for various telecom services.
4. None of the above

See

Question 4. Which of the following statements is correct with respect to The International
Civil Aviation Organization?
1. It is a specialized agency of the United Nations.
2. Its headquarters are located in the Quartier International of Montreal, Quebec, Canada.
3. It codifies the principles and techniques of international air navigation.
4. All of the above

See

Also, check previous Daily News Analysis


text:
# How do I find the kernel and range of a linear transformation?

For example: $T: P_2 \to P_5$ given by $T(p(x)) = x^3(p(x))$

How would I find the kernel and range for this?

All I know is that $\dim(\ker T)+ \dim(\operatorname{image}T)= \dim(p_2)$.

Therefore, $\dim(\ker T)+ \dim(\operatorname{image}T)=3$?

Also, how do I represent T by a matrix with respect to bases for the domain and codomain?

Any help at all is appreciated, as I really have absolutely no idea what I am doing.

-

For your specific problem, you can find by inspection that the kernel is just 0 (the only polynomial that gives 0 when multiplied by $x^3$ is just 0) and the image is all the 5th-degree polynomials with no quadratic, linear, or constant terms (as everything has just been multiplied by $x^3$ and there can be no terms with degree less than 2 because of that).

However, in general, we can proceed this way: your linear transformation can be represented as a matrix:

$\left[ \begin{matrix} 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} \right]$

To find the kernel, row reduce this matrix as augmented with the identity, and look for all the zero rows: $\left[ \begin{array}{cccccc|ccc} 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 \end{array} \right]$

Conveniently, it is already row-reduced. So the kernel has no basis and therefore is trivial. If it did have a basis, the basis vectors would be the entries on the right matrix that corresponded to zero rows on the left matrix (remember that the kernel is expressed in terms of the domain space, not the range space).

I believe that to find a basis of the image, you look for all the nonzero rows after row reduction, but I forgot that part. (It seems to work in this case, however, as $\{x^3, x^4, x^5\}$ is indeed a basis for the image.)

-
 Ok, I used another problem because I didn't want to cheat myself out of finding out how to do it. . . but the problem in my HW is T:P2- - >P4 given by T(p(x)) = x^2(P(x)) so I guess kerT is still 0 then. . . but I am still confused about how to find the range. . . is the basis of the image analogous to the range? – So Confused Dec 17 '12 at 4:12 Oh, by "image" I mean "range". They're used interchangeably. – Joe Z. Dec 17 '12 at 4:13 Oh okay, thank you! – So Confused Dec 17 '12 at 4:14 You are correct in saying that the kernel of T is still 0 in your case. The range is slightly different - just shift the exponents in the answer above down by 1. – Joe Z. Dec 17 '12 at 5:43