text:
# Kernel (matrix)

In linear algebra, the kernel or null space (also nullspace) of a matrix A is the set of all vectors x for which Ax = 0. The kernel of a matrix with n columns is a linear subspace of n-dimensional Euclidean space.[1] The dimension of the null space of A is called the nullity of A.

If viewed as a linear transformation, the null space of a matrix is precisely the kernel of the mapping (i.e. the set of vectors that map to zero). For this reason, the kernel of a linear transformation between abstract vector spaces is sometimes referred to as the null space of the transformation.

## Definition

The kernel of an m × n matrix A is the set

$\text{Ker}(\mathbf{A}) = \left\{ \textbf{x}\in\textbf{R}^n : \mathbf{A}\textbf{x} = \textbf{0} \right\}\text{,}$[2]

where 0 denotes the zero vector with m components. The matrix equation Ax = 0 is equivalent to a homogeneous system of linear equations:

\mathbf{A}\textbf{x}=\textbf{0} \;\;\Leftrightarrow\;\; \begin{alignat}{6} a_{11} x_1 &&\; + \;&& a_{12} x_2 &&\; + \cdots + \;&& a_{1n} x_n &&\; = 0& \\ a_{21} x_1 &&\; + \;&& a_{22} x_2 &&\; + \cdots + \;&& a_{2n} x_n &&\; = 0& \\ \vdots\;\;\; && && \vdots\;\;\; && && \vdots\;\;\; && \vdots\,& \\ a_{m1} x_1 &&\; + \;&& a_{m2} x_2 &&\; + \cdots + \;&& a_{mn} x_n &&\; = 0. & \end{alignat}

From this viewpoint, the null space of A is the same as the solution set to the homogeneous system.

## Example

Consider the matrix

$\mathbf{A} = \begin{bmatrix}\,\,\,2 & 3 & 5 \\ -4 & 2 & 3\end{bmatrix}.$

The null space of this matrix consists of all vectors (xyz) ∈ R3 for which

$\begin{bmatrix}\,\,\,2 & 3 & 5 \\ -4 & 2 & 3\end{bmatrix}\begin{bmatrix} x \\ y \\ z\end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}\text{.}$

This can be written as a homogeneous system of linear equations involving x, y, and z:

\begin{alignat}{7} 2x &&\; + \;&& 3y &&\; + \;&& 5z &&\; = \;&& 0, \\ -4x &&\; + \;&& 2y &&\; + \;&& 3z &&\; = \;&& 0.\\ \end{alignat}

This can be written in matrix form as:

$\left[\begin{array}{ccc|c} 2 & 3 & 5 & 0 \\ -4 & 2 & 3 & 0 \end{array}\right].$

Using Gauss-Jordan reduction, this reduces to:

$\left[\begin{array}{ccc|c} 1 & 0 & .0625 & 0 \\ 0 & 1 & 1.625 & 0 \end{array}\right].$

Rewriting yields:

\begin{alignat}{7} x = \;&& -.0625z \\ y = \;&& -1.625z. \end{alignat}

Now we can write the null space (solution to Ax = 0) in terms of z (which is our free variable), where z is scalar:

$\begin{bmatrix} x \\ y \\ z\end{bmatrix} = z \begin{bmatrix}\,\,\,-.0625 \\ -1.625 \\ 1\end{bmatrix}.$

The null space of A is precisely the set of solutions to these equations (in this case, a line through the origin in R3).

## Subspace properties

The null space of an m × n matrix is a subspace of Rn. That is, the set Null(A) has the following three properties:

1. Null(A) always contains the zero vector.
2. If x ∈ Null(A) and y ∈ Null(A), then x + y ∈ Null(A).
3. If x ∈ Null(A) and c is a scalar, then c x ∈ Null(A).

Here are the proofs:

1. A0 = 0.
2. If Ax = 0 and Ay = 0, then A(x + y) = Ax + Ay0 + 0 = 0.
3. If Ax = 0 and c is a scalar, then A(cx) = cAxc0 = 0.

## Basis

The null space of a matrix is not affected by elementary row operations. This makes it possible to use row reduction to find a basis for the null space:

Input An m × n matrix A.
Output A basis for the null space of A
1. Use elementary row operations to put A in reduced row echelon form.
2. Interpreting the reduced row echelon form as a homogeneous linear system, determine which of the variables x1x2, ..., xn are free. Write equations for the dependent variables in terms of the free variables.
3. For each free variable xi, choose the vector in the null space for which xi = 1 and the remaining free variables are zero. The resulting collection of vectors is a basis for the null space of A.

For example, suppose that the reduced row echelon form of A is

\left[ \begin{alignat}{6} 1 && 0 && -3 && 0 && 2 && -8 \\ 0 && 1 && 5 && 0 && -1 && 4 \\ 0 && 0 && 0 && 1 && 7 && -9 \\ 0 && \;\;\;\;\;0 && \;\;\;\;\;0 && \;\;\;\;\;0 && \;\;\;\;\;0 && \;\;\;\;\;0 \end{alignat} \,\right]\text{.}

Then the solutions to the homogeneous system given in parametric form with x3, x5, and x6 as free variables are

\begin{alignat}{7} x_1 &&\; = \;&& 3x_3 &&\; - \;&& 2x_5 &&\; + \;&& 8x_6 & \\ x_2 &&\; = \;&& -5x_3 &&\; + \;&& x_5 &&\; - \;&& 4x_6 & \\ x_4 &&\; = \;&& &&\; - \;&& 7x_5 &&\; + \;&& 9x_6 &. \end{alignat}

Which can be rewritten as

$\begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \\ x_5 \\ x_6 \end{bmatrix} = x_3 \begin{bmatrix} 3 \\ -5 \\ 1 \\ 0 \\ 0 \\ 0 \end{bmatrix} + x_5 \begin{bmatrix} -2 \\ 1 \\ 0 \\ -7 \\ 1 \\ 0 \end{bmatrix} + x_6 \begin{bmatrix} 8 \\ -4 \\ 0 \\ 9 \\ 0 \\ 1 \end{bmatrix} \text{.}$

Therefore, the three vectors

$\left[\!\! \begin{array}{r} 3 \\ -5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array} \right],\; \left[\!\! \begin{array}{r} -2 \\ 1 \\ \mathbf{0} \\ -7 \\ \mathbf{1} \\ \mathbf{0} \end{array} \right],\; \left[\!\! \begin{array}{r} 8 \\ -4 \\ \mathbf{0} \\ 9 \\ \mathbf{0} \\ \mathbf{1} \end{array} \right]$

are a basis for the null space of A.

## Relation to the row space

Let A be an m by n matrix (i.e., A has m rows and n columns). The product of A and the n-dimensional vector x can be written in terms of the dot product of vectors as follows:

$\mathbf{A}\textbf{x} = \begin{bmatrix} \textbf{a}_1 \cdot \textbf{x} \\ \textbf{a}_2 \cdot \textbf{x} \\ \vdots \\ \textbf{a}_m \cdot \textbf{x} \end{bmatrix}\text{.}$

Here a1, ..., am denote the rows of the matrix A. It follows that x is in the null space of A if and only if x is orthogonal (or perpendicular) to each of the row vectors of A (because if the dot product of two vectors is equal to zero they are by definition orthogonal).

The row space of a matrix A is the span of the row vectors of A. By the above reasoning, the null space of A is the orthogonal complement to the row space. That is, a vector x lies in the null space of A if and only if it is perpendicular to every vector in the row space of A.

The dimension of the row space of A is called the rank of A, and the dimension of the null space of A is called the nullity of A. These quantities are related by the equation

$\text{rank}(\mathbf{A}) + \text{nullity}(\mathbf{A}) = n\text{.}\,$

The equation above is known as the rank-nullity theorem.

## Nonhomogeneous equations

The null space also plays a role in the solution to a nonhomogeneous system of linear equations:

\mathbf{A}\textbf{x}=\textbf{b}\;\;\;\;\;\;\text{or}\;\;\;\;\;\;\begin{alignat}{7} a_{11} x_1 &&\; + \;&& a_{12} x_2 &&\; + \cdots + \;&& a_{1n} x_n &&\; = \;&&& b_1 \\ a_{21} x_1 &&\; + \;&& a_{22} x_2 &&\; + \cdots + \;&& a_{2n} x_n &&\; = \;&&& b_2 \\ \vdots\;\;\; && && \vdots\;\;\; && && \vdots\;\;\; && &&& \;\vdots \\ a_{m1} x_1 &&\; + \;&& a_{m2} x_2 &&\; + \cdots + \;&& a_{mn} x_n &&\; = \;&&& b_m \\ \end{alignat}

If u and v are two possible solutions to the above equation, then

$\mathbf{A}(\textbf{u}-\textbf{v}) = \mathbf{A}\textbf{u} - \mathbf{A}\textbf{v} = \textbf{b} - \textbf{b} = \textbf{0}\,$

Thus, the difference of any two solutions to the equation Ax = b lies in the null space of A.

It follows that any solution to the equation Ax = b can be expressed as the sum of a fixed solution v and an arbitrary element of the null space. That is, the solution set to the equation Ax = b is

$\left\{ \textbf{v}+\textbf{x} \,:\, \textbf{x}\in\text{Null}(\mathbf{A})\, \right\}\text{,}$

where v is any fixed vector satisfying Av = b. Geometrically, this says that the solution set to Ax = b is the translation of the null space of A by the vector v. See also Fredholm alternative.

## Left null space

The left null space of a matrix A consists of all vectors x such that xTA = 0T, where T denotes the transpose of a column vector. The left null space of A is the same as the null space of AT. The left null space of A is the orthogonal complement to the column space of A, and is the cokernel of the associated linear transformation. The null space, the row space, the column space, and the left null space of A are the four fundamental subspaces associated to the matrix A.

## Null space of a transformation

If V and W are vector spaces, the null space (or kernel) of a linear transformation TV → W is the set of all vectors in V that map to zero:

$\ker(T) = \left\{\textbf{v}\in V : T(\textbf{v}) = \textbf{0} \right\}\text{.}$

If we represent the linear transformation by a matrix, then the kernel of the transformation is precisely the null space of the matrix.

## Numerical computation of null space

Algorithms based on row or column reduction, that is, Gaussian elimination, presented in introductory linear algebra textbooks and in the preceding sections of this article are not suitable for a practical computation of the null space because of numerical accuracy problems in the presence of rounding errors. Namely, the computation may greatly amplify the rounding errors, which are inevitable in all but textbook examples on integers, and so give completely wrong results. For this reason, methods based on introductory linear algebra texts are generally not suitable for implementation in software; rather, one should consult contemporary numerical analysis sources for an algorithm like the one below, which does not amplify rounding errors unnecessarily.

A state-of-the-art approach is based on singular value decomposition (SVD). This approach can be also easily programmed using standard libraries, such as LAPACK. SVD of matrix A computes unitary matrices U and V and a rectangular diagonal matrix S of the same size as A with nonnegative diagonal entries, such that

$\mathbf{U}\mathbf{S}\mathbf{V}^T = \mathbf{A}.$

Denote the columns of V by

$v_1,\dots,v_n,$

the diagonal entries of S by

$s_1,\dots,s_{\min\{m,n\}},$

and put

$s_{\min\{m,n\}+1}=\cdots=s_{\max\{m,n\}}=0.$

(The numbers si are called the singular values of A.) Then the columns vi of V such that the corresponding $s_i\,=\,0$ form an orthonormal basis of the nullspace of A. This can be seen as follows: First note that if we have one solution y of the equation USy = 0, then also US(y + ei) = 0 for unit vectors ei with si = 0. Now if we solve (y + ei) = VTz for z, then z = V(y + ei) because of VTV = Id, which means that the i'th column of V spans one direction of the null space.

In a numerical computation, the singular values si are taken to be zero when they are less than some small tolerance. For example, the tolerance can be taken to be

max{m,n}max{si}ε,

where ε is the machine epsilon of the computer, that is, the smallest number such that in the floating point arithmetics of the computer, $1\,+\,\varepsilon>1$. For the IEEE 64 bit floating point format, $\varepsilon\,\approx\, 2.2\cdot 10^{-16}$.

Computation of the SVD of a matrix generally costs about the same as several matrix-matrix multiplications with matrices of the same size when state-of-the art implementation (accurate up to rounding precision) is used, such as in LAPACK. This is true even if, in theory, the SVD cannot be computed by a finite number of operations, so an iterative method with stopping tolerances based on rounding precision must be employed. The cost of the SVD approach is several times higher than computing the null space by reduction, but it should be acceptable whenever reliability is important. It is also possible to compute the null space by the QR decomposition, with the numerical stability and the cost both being between those of the SVD and the reduction approaches. The computation of a null space basis using the QR decomposition is explained in more detail below.

Let A be a mxn matrix with m < n. Using the QR factorization of $\mathbf{A}^T$, we can find a matrix such that

$\mathbf{A}^T \, \mathbf{P} = \mathbf{Q} \, \mathbf{R} = [\mathbf{Q}_1 \; \mathbf{Q}_2] \, \mathbf{R}$,

where P is a permutation matrix, Q is nxn and R is nxm. Matrix $\mathbf{Q}_1$ is nxm and consists of the first m columns of Q. Matrix $\mathbf{Q}_2$ is nx(n-m) and is made up of Q 's last n-m columns. Since $\mathbf{A} \, \mathbf{Q}_2 = 0$, the columns of $\mathbf{Q}_2$ span the null space of A.

## Notes

1. ^ Linear algebra, as discussed in this article, is a very well-established mathematical discipline for which there are many sources. Almost all of the material in this article can be found in Lay 2005, Meyer 2001, and Strang 2005.
2. ^ This equation uses set-builder notation.

## References

### Textbooks

• Axler, Sheldon Jay (1997), Linear Algebra Done Right (2nd ed.), Springer-Verlag, ISBN 0387982590
• Lay, David C. (August 22, 2005), Linear Algebra and Its Applications (3rd ed.), Addison Wesley, ISBN 978-0321287137
• Meyer, Carl D. (February 15, 2001), Matrix Analysis and Applied Linear Algebra, Society for Industrial and Applied Mathematics (SIAM), ISBN 978-0898714548
• Poole, David (2006), Linear Algebra: A Modern Introduction (2nd ed.), Brooks/Cole, ISBN 0-534-99845-3
• Anton, Howard (2005), Elementary Linear Algebra (Applications Version) (9th ed.), Wiley International
• Leon, Steven J. (2006), Linear Algebra With Applications (7th ed.), Pearson Prentice Hall

### Numerical analysis textbooks

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Kernel — may refer to:Computing* Kernel (computer science), the central component of most operating systems ** Linux kernel * Kernel (programming language), a Scheme like language * kernel trick, in machine learningLiterature* Kernel ( Lilo Stitch ),… …   Wikipedia

• Matrix decoder — is an audio technology where a finite number of discrete audio channels (e.g., 2) are decoded into a larger number of channels on play back (e.g., 5). The channels are generally, but not always, arranged for transmission or recording by an… …   Wikipedia

• Kernel smoother — A kernel smoother is a statistical technique for estimating a real valued function f(X),,left( Xin mathbb{R}^{p} ight) by using its noisy observations, when no parametric model for this function is known. The estimated function is smooth, and the …   Wikipedia

• Kernel (algebra) — In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. An important special case is the kernel of a matrix, also… …   Wikipedia

• Kernel (mathematics) — In mathematics, the word kernel has several meanings. Kernel may mean a subset associated with a mapping:* The kernel of a mapping is the set of elements that map to the zero element (such as zero or zero vector), as in kernel of a linear… …   Wikipedia

• Kernel (linear operator) — Main article: Kernel (mathematics) In linear algebra and functional analysis, the kernel of a linear operator L is the set of all operands v for which L(v) = 0. That is, if L: V → W, then where 0 denotes the null vector… …   Wikipedia

• Matrix (mathematics) — Specific elements of a matrix are often denoted by a variable with two subscripts. For instance, a2,1 represents the element at the second row and first column of a matrix A. In mathematics, a matrix (plural matrices, or less commonly matrixes)… …   Wikipedia

• Matrix decomposition — In the mathematical discipline of linear algebra, a matrix decomposition is a factorization of a matrix into some canonical form. There are many different matrix decompositions; each finds use among a particular class of problems. Contents 1… …   Wikipedia

• Multivariate kernel density estimation — Kernel density estimation is a nonparametric technique for density estimation i.e., estimation of probability density functions, which is one of the fundamental questions in statistics. It can be viewed as a generalisation of histogram density… …   Wikipedia

• Positive definite kernel — In operator theory, a positive definite kernel is a generalization of a positive matrix. Definition Let :{ H n } {n in {mathbb Z be a sequence of (complex) Hilbert spaces and :mathcal{L}(H i, H j)be the bounded operators from Hi to Hj . A map A… …   Wikipedia

### Share the article and excerpts

Do a right-click on the link above


text:
# Browse College of Engineering by Author "Xia, Xiaoyu"

• (2001)
It is believed that all classical fluids could form glasses if cooled sufficiently fast so as to avoid crystallization. Various phenomena including violation of the usual Arrhenius law, stretched relaxations, deviations ...

application/pdf

PDF (3MB)
• (2001)
It is believed that all classical fluids could form glasses if cooled sufficiently fast so as to avoid crystallization. Various phenomena including violation of the usual Arrhenius law, stretched relaxations, deviations ...

application/pdf

PDF (2MB)

format: plain


text:
CryptoDB

Paper: Sum-of-Squares Meets Program Obfuscation, Revisited

Authors: Boaz Barak Samuel B. Hopkins Aayush Jain Pravesh Kothari Amit Sahai DOI: 10.1007/978-3-030-17653-2_8 (login may be required) Search ePrint Search Google We develop attacks on the security of variants of pseudo-random generators computed by quadratic polynomials. In particular we give a general condition for breaking the one-way property of mappings where every output is a quadratic polynomial (over the reals) of the input. As a corollary, we break the degree-2 candidates for security assumptions recently proposed for constructing indistinguishability obfuscation by Ananth, Jain and Sahai (ePrint 2018) and Agrawal (ePrint 2018). We present conjectures that would imply our attacks extend to a wider variety of instances, and in particular offer experimental evidence that they break assumption of Lin-Matt (ePrint 2018).Our algorithms use semidefinite programming, and in particular, results on low-rank recovery (Recht, Fazel, Parrilo 2007) and matrix completion (Gross 2009).
BibTeX
@article{eurocrypt-2019-29336,
title={Sum-of-Squares Meets Program Obfuscation, Revisited},
booktitle={Advances in Cryptology – EUROCRYPT 2019},
series={Advances in Cryptology – EUROCRYPT 2019},
publisher={Springer},
volume={11476},
pages={226-250},
doi={10.1007/978-3-030-17653-2_8},
author={Boaz Barak and Samuel B. Hopkins and Aayush Jain and Pravesh Kothari and Amit Sahai},
year=2019
}


text:
Interlayer twist between van der Waals graphene crystals led to the discovery of superconducting and insulating states near the magic angle. In this work, we exploit this mechanical degree of freedom by twisting the graphene middle layer in a trilayer graphene spacer between two metallic lead (Magnetic and nonmagnetic). A large difference in conductance is found depending on the angle of twist between the middle layer graphene and the ones at the interface this difference, called twisting resistance, reach more than 1000% in the non-magnetic Cu case. For the magnetic Ni case, the magneto-resistance decreases and the difference in conductance between twisted and not twisted depends strongly on the relative magnetization configuration. For the parallel configuration, the twisting resistance is about -40%, while for the anti-parallel configuration it can reach up to 130%. Furthermore, we show that the twisting resistance can be enhanced by inserting a thin Cu layer at the interface of Ni/graphene where it reaches a value of 200% and 1600% for parallel and antiparallel configurations, respectively. These finding could pave the way toward the integration of 2D materials on novel spinmechatronics based devices.

Published : "arXiv Mesoscale and Nanoscale Physics".


text:
# Genetic Diversity

Genetic diversity is a measure of local variation, either at the allele or the genotype level. In this topic, we explore some of the more common measures of gneetic diversity.

## Lecture Content

For this topic, I will be spending a bit of time covering the foundations of this topic as well as giving some concrete examples. To do so, you have access to the following slide set1.

Here is a recording of the Diversity talk I gave on Zoom.

### Supplemental Materials

In addition to this lecture content, you may find the following information helpful as you go though these exercises.

## Activity

Here is the activity associated with this particular topic.

Here is some code that I used to solve the questions in the activity.

1. This is a presentation writing in Xaringan, an RMarkdown presentation type. It is an active embed in this page (and hosted elsewhere). It can be extracted my clicking on the slide and hitting the c key on your keyboard—to clone it into its own window (more commands available using ? for viewing).↩

Previous


text:
TheInfoList

Voltage, electric potential difference, electromotive force emf, electric pressure or electric tension is the difference in electric potential between two points, which (in a static electric field) is defined as the work needed per unit of charge to move a test charge between the two points. In the International System of Units, the derived unit for voltage (potential difference) is named ''volt''. In SI units, work per unit charge is expressed as joules per coulomb, where 1 volt = 1 joule (of work) per 1 coulomb (of charge). The old SI definition for ''volt'' used power and current; starting in 1990, the quantum Hall and Josephson effect were used, and recently (2019) fundamental physical constants have been introduced for the definition of all SI units and derived units. Voltage or electric potential difference is denoted symbolically by , simplified ''V'', or ''U'', for instance in the context of Ohm's or Kirchhoff's circuit laws. Electric potential differences between points can be caused physically by electric charge build up or imbalance (eg. well known "static" and electronic capacitor) also by electric current through a magnetic field, and by time-varying magnetic fields (eg. dynamo or generator), or some combination of these three. Additionally on a macroscopic scale potential difference can be caused by electrochemical processes (cells and batteries) and pressure induced piezoelectric effect and heat induced emf across metal junctions. These latter processes at microscopic level have the physical origins previously mentioned. A voltmeter can be used to measure the voltage (or potential difference) between two points in a system; often a common reference potential such as the ground of the system is used as one of the points. A voltage may represent either a source of energy (electromotive force) or lost, used, or stored energy (potential drop).

Definition

There are multiple useful ways to define voltage, including the standard definition mentioned at the start of this page. There are also other useful definitions of work per charge (see this section). Voltage is defined so that negatively charged objects are pulled towards higher voltages, while positively charged objects are pulled towards lower voltages. Therefore, the conventional current in a wire or resistor always flows from higher voltage to lower voltage. Historically, voltage has been referred to using terms like "tension" and "pressure". Even today, the term "tension" is still used, for example within the phrase "high tension" (HT) which is commonly used in thermionic valve (vacuum tube) based electronics.

Definition as potential of electric field

The voltage increase from some point $x_A$ to some point $x_B$ is given by :$\begin \Delta V_ &= V\left(x_B\right) - V\left(x_A\right) \\ &= - \int_^ \vec \cdot d\vec - \left\left( - \int_^ \vec \cdot d\vec \right\right)\\ &= -\int_^ \vec \cdot d\vec \end$ In this case, the voltage increase from point A to point B is equal to the work done per unit charge, against the electric field, to move the charge from A to B without causing any acceleration. Mathematically, this is expressed as the line integral of the electric field along that path. Under this definition, the voltage difference between two points is not uniquely defined when there are time-varying magnetic fields since the electric force is not a conservative force in such cases. If this definition of voltage is used, any circuit where there are time-varying magnetic fields,If there are time-varying electric fields or accelerating charges, then there will be time-varying magnetic fields. This means in AC circuits, there are always some non-confined magnetic fields. However, except at higher frequencies, these are neglected. such as circuits containing inductors, will not have a well-defined voltage between nodes in the circuit. However, if magnetic fields are suitably contained to each component, then the electric field is conservative in the region exteriorThis relies on the fact that each component has a finite volume. If a component had an infinite extent, the region exterior to the components would not be simply connected, and thus integrals through it would still depend on the path taken. to the components, and voltages are well-defined in that region. In this case, the voltage across an inductor, viewed externally, turns out to be $U = \Delta V = -L\frac$ despite the fact that, internally, the electric field in the coil is zero (assuming it is a perfect conductor).

Definition via decomposition of electric field

Using the above definition, the electric potential is not defined whenever magnetic fields change with time. In physics, it is sometimes useful to generalize the electric potential by only considering the conservative part of the electric field. This is done by the following decomposition used in electrodynamics: $\vec=-\nabla V-\frac$ where $\vec$ is the magnetic vector potential. The above decomposition is justified by Helmholtz's theorem. In this case, the voltage increase from $x_A$ to $x_B$ is given by $\begin \Delta V_&=-\int_^\vec_\mathrm\cdot d\vec\\ &=-\int_^\left\left(\vec+\frac\right\right)\cdot d\vec\\ &=-\int_^\left(\vec-\vec_\mathrm\right)\cdot d\vec \end$ where $\vec_\mathrm$ is the rotational electric field due to time-varying magnetic fields. In this case, the voltage between points is always uniquely defined.

Treatment in circuit theory

In circuit analysis and electrical engineering, the voltage across an inductor is not considered to be zero or undefined, as the standard definition would suggest. This is because electrical engineers use a lumped element model to represent and analyze circuits. When using a lumped element model, it is assumed that there are no magnetic fields in the region surrounding the circuit and that the effects of these are contained in 'lumped elements', which are idealized and self-contained circuit elements used to model physical components. If the assumption of negligible leaked fields is too inaccurate, their effects can be modelled by parasitic components. In the case of a physical inductor though, the ideal lumped representation is often accurate. This is because the leaked fields of the inductor are generally negligible, especially if the inductor is a toroid. If leaked fields are negligible, we find that $\int_\mathrm\vec\cdot d\vec=-L\frac$ is path-independent, and there is a well-defined voltage across the inductor's terminals. This is the reason that measurements with a voltmeter across an inductor are often reasonably independent of the placement of the test leads.

Volt

The volt (symbol: ) is the derived unit for electric potential, electric potential difference, and electromotive force. The volt is named in honour of the Italian physicist Alessandro Volta (1745–1827), who invented the voltaic pile, possibly the first chemical battery.

Hydraulic analogy

A simple analogy for an electric circuit is water flowing in a closed circuit of pipework, driven by a mechanical pump. This can be called a "water circuit". Potential difference between two points corresponds to the pressure difference between two points. If the pump creates a pressure difference between two points, then water flowing from one point to the other will be able to do work, such as driving a turbine. Similarly, work can be done by an electric current driven by the potential difference provided by a battery. For example, the voltage provided by a sufficiently-charged automobile battery can "push" a large current through the windings of an automobile's starter motor. If the pump isn't working, it produces no pressure difference, and the turbine will not rotate. Likewise, if the automobile's battery is very weak or "dead" (or "flat"), then it will not turn the starter motor. The hydraulic analogy is a useful way of understanding many electrical concepts. In such a system, the work done to move water is equal to the "pressure drop" (compare p.d.) multiplied by the volume of water moved. Similarly, in an electrical circuit, the work done to move electrons or other charge-carriers is equal to "electrical pressure difference" multiplied by the quantity of electrical charges moved. In relation to "flow", the larger the "pressure difference" between two points (potential difference or water pressure difference), the greater the flow between them (electric current or water flow). (See "electric power".)

Applications

Specifying a voltage measurement requires explicit or implicit specification of the points across which the voltage is measured. When using a voltmeter to measure potential difference, one electrical lead of the voltmeter must be connected to the first point, one to the second point. A common use of the term "voltage" is in describing the voltage dropped across an electrical device (such as a resistor). The voltage drop across the device can be understood as the difference between measurements at each terminal of the device with respect to a common reference point (or ground). The voltage drop is the difference between the two readings. Two points in an electric circuit that are connected by an ideal conductor without resistance and not within a changing magnetic field have a voltage of zero. Any two points with the same potential may be connected by a conductor and no current will flow between them.

The voltage between ''A'' and ''C'' is the sum of the voltage between ''A'' and ''B'' and the voltage between ''B'' and ''C''. The various voltages in a circuit can be computed using Kirchhoff's circuit laws. When talking about alternating current (AC) there is a difference between instantaneous voltage and average voltage. Instantaneous voltages can be added for direct current (DC) and AC, but average voltages can be meaningfully added only when they apply to signals that all have the same frequency and phase.

Measuring instruments

Instruments for measuring voltages include the voltmeter, the potentiometer, and the oscilloscope. Analog voltmeters, such as moving-coil instruments, work by measuring the current through a fixed resistor, which, according to Ohm's Law, is proportional to the voltage across the resistor. The potentiometer works by balancing the unknown voltage against a known voltage in a bridge circuit. The cathode-ray oscilloscope works by amplifying the voltage and using it to deflect an electron beam from a straight path, so that the deflection of the beam is proportional to the voltage.

Typical voltages

A common voltage for flashlight batteries is 1.5 volts (DC). A common voltage for automobile batteries is 12 volts (DC). Common voltages supplied by power companies to consumers are 110 to 120 volts (AC) and 220 to 240 volts (AC). The voltage in electric power transmission lines used to distribute electricity from power stations can be several hundred times greater than consumer voltages, typically 110 to 1200 kV (AC). The voltage used in overhead lines to power railway locomotives is between 12 kV and 50 kV (AC) or between 0.75 kV and 3 kV (DC).

Galvani potential vs. electrochemical potential

Inside a conductive material, the energy of an electron is affected not only by the average electric potential, but also by the specific thermal and atomic environment that it is in. When a voltmeter is connected between two different types of metal, it measures not the electrostatic potential difference, but instead something else that is affected by thermodynamics. The quantity measured by a voltmeter is the negative of the difference of the electrochemical potential of electrons (Fermi level) divided by the electron charge and commonly referred to as the voltage difference, while the pure unadjusted electrostatic potential (not measurable with a voltmeter) is sometimes called Galvani potential. The terms "voltage" and "electric potential" are ambiguous in that, in practice, they can refer to ''either'' of these in different contexts.

History

The term ''electromotive force'' was first used by Volta in a letter to Giovanni Aldini in 1798, and first appeared in a published paper in 1801 in ''Annales de chimie et de physique''. Volta meant by this a force that was not an electrostatic force, specifically, an electrochemical force.Robert N. Varney, Leon H. Fisher
"Electromotive force: Volta's forgotten concept"
''American Journal of Physics'', vol. 48, iss. 5, pp. 405–408, May 1980.
The term was taken up by Michael Faraday in connection with electromagnetic induction in the 1820s. However, a clear definition of voltage and method of measuring it had not been developed at this time.C. J. Brockman
"The origin of voltaic electricity: The contact vs. chemical theory before the concept of E. M. F. was developed"
''Journal of Chemical Education'', vol. 5, no. 5, pp. 549–555, May 1928
Volta distinguished electromotive force (emf) from ''tension'' (potential difference): the observed potential difference at the terminals of an electrochemical cell when it was open circuit must exactly balance the emf of the cell so that no current flowed.

* Alternating current (AC) * Direct current (DC) * Electric potential * Electric shock * Electrical measurements * Electrochemical potential * Fermi level * High voltage * Mains electricity (an article about domestic power supply voltages) * Mains electricity by country (list of countries with mains voltage and frequency) * Ohm's law * Ohm * Open-circuit voltage * Phantom voltage

References

Footnotes


text:
## CryptoDB

### Paper: Streamlined blockchains: A simple and elegant approach (tutorial)

Authors: Elaine Shi , Cornell University Search ePrint Search Google Invited talk A blockchain protocol (also called state machine replication) allows a set of nodes to agree on an ever-growing, linearly ordered log of transactions. In this tutorial, we present a new paradigm called “streamlined blockchains”. This paradigm enables a new family of protocols that are extremely simple and natural: every epoch, a proposer proposes a block extending from a notarized parent chain, and nodes vote if the proposal’s parent chain is not too old. Whenever a block gains enough votes, it becomes notarized. Whenever a node observes a notarized chain with several blocks of consecutive epochs at the end, then the entire chain chopping off a few blocks at the end is final. By varying the parameters highlighted in blue, we illustrate two variants for the partially synchronous and synchronous settings respectively. We present very simple proofs of consistency and liveness. We hope that this tutorial provides a compelling argument why this new family of protocols should be used in lieu of classical candidates (e.g., PBFT, Paxos, and their variants), both in practical implementation and for pedagogical purposes.
##### BibTeX
@misc{asiacrypt-2019-30081,
title={Streamlined blockchains: A simple and elegant approach (tutorial)},
note={Invited talk},
author={Elaine Shi},
year=2019
}


text:
# What do the Contour Levels Produced by Matlab Function Contourc Represent

I have been trying to understand what the contour levels are.

As far as I understand a contour of the function $$Z(x,y)$$ is a set of points in the $$(x,y)$$ plane, such that $$Z(x,y)$$ is fixed at some constant value, that constant value is the contour level.

Matlab docs state that the contourc function takes an argument that determines the levels upon which the contours are computed.

I wanted to try that out, so I constructed a picture with $$10$$ rectangles, each of which has a distinct depth starting from $$0.1$$ ending up at $$1.0$$, and I ran the contour function to see the highlighted contours according to the specified levels in the input, but the results weren't expected.

For instance, I ran the code on $$0.1$$ contour level and the contour function produces all the contours in the picture, not only the ones with $$0.1$$ level.

Here is the Matlab code used to construct the picture and find its contours:

pic = zeros(500, 500);
val = 0.1;
shift = 40;

for j = 1:10
for i= 50:450
pic(shift*j,i) = val;
end
for i= 50:450
pic(shift*j+25,i) = val;
end
for i = shift*j: shift*j+25
pic(i, 50) = val;
pic(i,450) = val;
end
val = val + 0.1;

end

imshow(pic);
figure(2);
contour(pic, [0.1 0.1]);

I'd be so grateful if someone could clue me in.

• Try contour without the second argument – Irreducible Oct 22 '18 at 14:31


text:
# Generalized Distributive Law

This is problem 3.11, p.29, Introduction to Set Theory, Hrbacek and Jech.

$$\big( \bigcap_{a \in A} F_a \big) \cup \big( \bigcap_{b \in B} G_b \big) = \bigcap_{(a,b) \in A \times B} (F_a \cup G_b).$$

It is my attempt.

$$$$x \in (LHS) \ \Leftrightarrow x \in \bigcap_{a \in A} F_a ~~ or ~~ x \in \bigcap_{b \in B} G_b \ \Leftrightarrow (x \in F_a ~~for ~~all~~a\in A)~~or~~ (x\in G_b~~ for~~all~~b\in B).$$$$

And,

$$$$x\in(RHS) \ \Leftrightarrow x \in F_a \cup G_b~~for ~~all~~(a,b) \in A \times B \ \Leftrightarrow (x\in F_a~~or ~~ x \in G_b) ~~for ~~all~~(a,b) \in A \times B.$$$$

I want to change the position of phrases in each last sentences, but I’m not sure doing it preserves if and only if condition.

Please help my problem.


text:
Sections
Chapter Review

# Chapter Review

### Concept Items

#### 10.1Postulates of Special Relativity

1.
Why was it once believed that light must travel through a medium and could not propagate across empty space?
1. The longitudinal nature of light waves implies this.
2. Light shows the phenomenon of diffraction.
3. The speed of light is the maximum possible speed.
4. All other wave energy needs a medium to travel.
2.

Describe the relative motion of Earth and the sun:

1. if Earth is taken as the inertial frame of reference and
2. if the sun is taken as the inertial frame of reference.
1. Earth is at rest and the sun orbits Earth.
2. The sun is at rest and Earth orbits the sun.
1. The sun is at rest and Earth orbits the sun.
2. Earth is at rest and the sun orbits Earth.
1. The sun is at rest and Earth orbits the sun.
2. The sun is at rest and Earth orbits the sun.
1. Earth is at rest and the sun orbits Earth.
2. Earth is at rest and the sun orbits Earth.

#### 10.2Consequences of Special Relativity

3.
A $β$ particle (a free electron) is speeding around the track in a cyclotron, rapidly gaining speed. How will the particle’s momentum change as its speed approaches the speed of light? Explain.
1. The particle’s momentum will rapidly decrease.
2. The particle’s momentum will rapidly increase.
3. The particle’s momentum will remain constant.
4. The particle’s momentum will approach zero.
4.
An astronaut goes on a long space voyage at near the speed of light. When she returns home, how will her age compare to the age of her twin who stayed on Earth?
1. Both of them will be the same age.
2. This is a paradox and hence the ages cannot be compared.
3. The age of the twin who traveled will be less than the age of her twin.
4. The age of the twin who traveled will be greater than the age of her twin.
5.

A comet reaches its greatest speed as it travels near the sun. True or false— Relativistic effects make the comet’s tail look longer to an observer on Earth.

1. True
2. False

### Critical Thinking Items

#### 10.1Postulates of Special Relativity

6.

Explain how the two postulates of Einstein’s theory of special relativity, when taken together, could lead to a situation that seems to contradict the mechanics and laws of motion as described by Newton.

1. In Newtonian mechanics, velocities are multiplicative but the speed of a moving light source cannot be multiplied to the speed of light because, according to special relativity, the speed of light is the maximum speed possible.
2. In Newtonian mechanics, velocities are additive but the speed of a moving light source cannot be added to the speed of light because the speed of light is the maximum speed possible.
3. An object that is at rest in one frame of reference may appear to be in motion in another frame of reference, while in Newtonian mechanics such a situation is not possible.
4. The postulates of Einstein’s theory of special relativity do not contradict any situation that Newtonian mechanics explains.
7.
It takes light $6.0minutes$ to travel from the sun to the planet Venus. How far is Venus from the sun?
1. $18×106km$
2. $18×108km$
3. $1.08×1011km$
4. $1.08×108km$
8.

In 2003, Earth and Mars were the closest they had been in 50,000 years. The two planets were aligned so that Earth was between Mars and the sun. At that time it took light from the sun 500 s to reach Earth and 687 s to get to Mars. What was the distance from Mars to Earth?

1. 5.6×107 km
2. 5.6×1010 km
3. 6.2×106 km
4. 6.2×1012 km
9.
Describe two ways in which light differs from all other forms of wave energy.
1. Light travels as a longitudinal wave.
2. Light travels through a medium that fills up the empty space in the universe.
1. Light travels as a transverse wave.
2. Light travels through a medium that fills up the empty space in the universe.
1. Light travels at the maximum possible speed in the universe.
2. Light travels through a medium that fills up the empty space in the universe.
1. Light travels at the maximum possible speed in the universe.
2. Light does not require any material medium to travel.
10.

Use the postulates of the special relativity theory to explain why the speed of light emitted from a fast-moving light source cannot exceed 3.00×108 m/s.

1. The speed of light is maximum in the frame of reference of the moving object.
2. The speed of light is minimum in the frame of reference of the moving object.
3. The speed of light is the same in all frames of reference, including in the rest frame of its source.
4. Light always travels in a vacuum with a speed less than 3.00×108 m/s, regardless of the speed of the source.

#### 10.2Consequences of Special Relativity

11.

Halley’s Comet comes near Earth every 75 years as it travels around its 22 billion km orbit at a speed of up to 700, 000 m/s. If it were possible to put a clock on the comet and read it each time the comet passed, which part of special relativity theory could be tested? What would be the expected result? Explain.

1. It would test time dilation. The clock would appear to be slightly slower.
2. It would test time dilation. The clock would appear to be slightly faster.
3. It would test length contraction. The length of the orbit would appear to be shortened from Earth’s frame of reference.
4. It would test length contraction. The length of the orbit would appear to be shortened from the comet’s frame of reference.
12.

The nucleus of the isotope fluorine-18 (18 F) has mass defect of 2.44×10-28 kg. What is the binding energy of 18F?

1. 2.2×10-11 J
2. 7.3×10-20 J
3. 2.2×10-20 J
4. 2.4×10-28 J

### Problems

#### 10.2Consequences of Special Relativity

13.

Deuterium (2 H) is an isotope of hydrogen that has one proton and one neutron in its nucleus. The binding energy of deuterium is 3.56×10-13 J. What is the mass defect of deuterium?

1. 3.20×10-4 kg
2. 1.68×10-6 kg
3. 1.19×10-21 kg
4. 3.96×10-30 kg
14.

The sun orbits the center of the galaxy at a speed of 2.3×105 m/s. The diameter of the sun is 1.391684×109 m. An observer is in a frame of reference that is stationary with respect to the center of the galaxy. True or false—The sun is moving fast enough for the observer to notice length contraction of the sun’s diameter.

1. True
2. False
15.

Consider the nuclear fission reaction $n+U92235→B56144a+K3689r+3n+En+U92235→B56144a+K3689r+3n+E$. If a neutron has a rest mass of 1.009u, $U92235U92235$ has a rest mass of 235.044u, $B56144aB56144a$ has rest mass of 143.923u, and $K3689rK3689r$ has a rest mass of 88.918u, what is the value of E in joules?

1. $1.8×10−111.8×10−11$J
2. $2.8×10−112.8×10−11$J
3. $1.8×10−101.8×10−10$J
4. $3.3×10−103.3×10−10$J
16.

Consider the nuclear fusion reaction $H12+H13→H24e+n+EH12+H13→H24e+n+E$. If $H12H12$ has a rest mass of 2.014u, $H13H13$ has a rest mass of 3.016u, $H24eH24e$ has a rest mass of 4.003u, and a neutron has a rest mass of 1.009u, what is the value of E in joules?

1. $2.7×10−142.7×10−14$J
2. $2.7×10−132.7×10−13$J
3. $2.7×10−122.7×10−12$J
4. $2.7×10−112.7×10−11$J


text:
### 2000 IMO SL #N4

Find all triplets of positive integers $(a,m,n)$ such that $a^m + 1 \mid (a + 1)^n$.

The answer is $\boxed{(a,1,n),(1,m,n)},$ and $\boxed{(2,3,n)\text{ where }n>1}$.

We have 3 cases.

For our first case, we have $m=1$. Then we have $a+1 \mid (a+1)^n$ which is true for all positive integers $a, n$. So, $(a,m,n)=(a,1,n)$.

For our second case, we have $a=1$. This implies $2 \mid 2^n$ which is true for all positive integers $m,n$. So, $(a,m,n)=(1,m,n)$.

For our final case, we consider $a,m>1$. Because $\gcd(a,1)=1$, we use Zsigmondy's theorem assuming that $(a,m) \ne (2,3)$ so let $p$ such that $p \mid a^m+1$ and $p \not \; \mid a+1$. However, $p \mid a^m+1 \mid (a+1)^n$, so we get a contradiction. Therefore, $a=2$ and $m=3$ and because $9 \mid 3^n$ is true for all $n>1$ the only possibility for this case is $(a,m,n)=(2,3,n)$ where $n>1$.

We are done. $\square$


text:
# StringCases is Confused with Patterns in Version 10.0.1

Bug introduced in 10.0.1 and fixed in 10.0.2

I observed the following with StringCases when trying to use rules/patterns. Here is a screenshot of the behavior:

Not only is the pattern highlighted in red, hovering the mouse over it claims it's an unknown option as shown above. Can anyone confirm this behavior? I'm on Windows 8.1. Regardless of this behavior, it does appear to work as expected.

• Red color also under Linux. I don't have the hint. Maybe because I disabled some annoying hinting features in options. – ybeltukov Sep 19 '14 at 20:04
• @ybeltukov. Thanks for confirming. – RunnyKine Sep 19 '14 at 20:08
• It's even red in the help. So maybe it is on purpose? If you do not want to see red things, just do StringCases["item13", {"item"~~x:DigitCharacter.. :> x}] – Rolf Mertig Sep 19 '14 at 20:19
• @RolfMertig, why would it be on purpose, when it has worked that way forever and there's no documentation of a change in behavior? – RunnyKine Sep 19 '14 at 20:23
• I can also confirm (Win 8.1). On top of that, if I try an example (pick one at random, showing red) from the Documentation for StringCases in a clean NB, then hover my mouse over some the other red hints on the Documentation page, version 10.0.1 crashes. Seems related to hovering over more than 1 or 2 of the red hints. Very weird. Anyone else see this happen? – david Sep 19 '14 at 20:29

• @Mr.Wizard A belated response, but it first appeared in 10.1.0 (together with the operator form of StringCases). – ilian Aug 6 '15 at 21:09


text:
# Thread: help me to solve the equations

1. ## help me to solve the equations

i want to find A,B & C from the equation below,

100 = (4A+2B)*EXP^-(2c)
120 = (16A+4B)*EXP^-(4c)
130 = (36A+6B)*EXP^-(6c)

help me to find these coefficients (A,B,C)..

2. If its negative exponent we have x^-1 = 1/x

So you could rewrite the problems with denominators. Then you would have something like

$100 = 2C/(4A + 2B)$

Then multiply the whole equation by the denominator to get rid of the denominators. Then solve for one of the variables and substitute into another equation and keep going until you find all the variables.

3. thanks caity for the reply

but i didnt know how you got the above equation. please explain me.

(Then multiply the whole equation by the denominator to get rid of the denominators)

i didnt understand how to do this for this equation too.

please clear me guys....

4. I used the formula x^-1 = 1/x

so I let the -2C be the -1 and 4A + 2B be the x but I tried to work this problem for you and ended up with 0's each time. I think that step is right but think need to solve it differently from there and I'm not sure how. I'll play with it some more or maybe someone else can help us out here. You could also try guess and check when its in the fraction form because it would be easier to work with.

5. thansk caity

i think the equations are not clear it is
100 = (4A+2B) * e^(-2C) = (4A+2B) / e^(2C)
120 = (16A+4B) * e^(-4C) = (16A+4B) / e^(4C)
130 = (36A+6B) * e^(-6C) = (36A+6B) / e^(66C)

ya i also tried the ways by keeping them to denominator, but it will make A in terms of B and either ways tats the problem. took logarthimic methods also that was also not coming good, and expansion also same, i dont know how to play with this equation.

when we divide these equations it leads to a fraction in LHS.

6. Ok I see now... I suggest try sending one of the big guys a pm and see if they can help... sorry I didnt see the e in the beginning I though you just meant exponent.

7. ## Business Math, need help!

I am stuck on the following problem.

The list price for a radio is 22% higher than its net price. If the net price is $29.00, what is the list price? What is the amount of the trade discount? HELP!!! 8. Originally Posted by wgazell I am stuck on the following problem. The list price for a radio is 22% higher than its net price. If the net price is$29.00, what is the list price? What is the amount of the trade discount? HELP!!!
Hey wgazell! Don't hijack someone else's thread. Make one of your own.

9. Hello, sanmo4!

I had a good start on it . . . then got bogged down.

Find $a, b\text{ and }c\!:\;\;\begin{array}{cccc} 100 &=&(4a + 2b) e^{-2c} \\ 120 &=& (16a + 4b)e^{-4c} \\130 &=& (36a + 6b)e^{-6c} \end{array}$
The equations can be simplified . . .

. . $\begin{array}{cccccccc}
100 &=& \dfrac{2(2a+b)}{e^{2c}} & \Longrightarrow & e^{2c} &=& \dfrac{2a+b}{50} & {\color{blue}[1]} \\ \\[-3mm]

120 & = & \dfrac{4(4a+b)}{e^{4c}} & \Longrightarrow & e^{4c} &=& \dfrac{4a+b}{30} & {\color{blue}[2]} \\ \\[-3mm]

130 &=& \dfrac{6(6a+b)}{e^{6c}} & \Longrightarrow & e^{6c} &=& \dfrac{3(6a+b)}{65} & {\color{blue}[3]} \end{array}$

$\begin{array}{cccc}\text{Square [1]:} & e^{4c} &=& \dfrac{(2a+b)^2}{50^2} \\ \\[-3mm] \text{Equate to [2]:} & e^{4c} &=& \dfrac{4a+b}{30} \end{array}\qquad\Rightarrow\qquad \frac{(2a+b)^2}{50^2} \:=\:\frac{4a+b}{30}\;\;{\color{blue}[4]}$

$\begin{array}{cccc} \text{Multiply [1] and [2]:} & e^{6c} &=&\dfrac{(2a+b)(4a+b)}{1500} \\ \\[-3mm] \text{Equate to [3]:} & e^{6c} &=& \dfrac{3(6a+b)}{65}\end{array}$ . $\Rightarrow\qquad \frac{(2a+b)(4a+b)}{1500} \:=\:\frac{3(6a+b)}{65}\;\;{\color{blue}[5]}$

I thought that, between [4] and [5], I could solve for $a\text{ and }b$, but . . .

10. thanks sorobon and caity...

in the previous post, if we equate eqns 4 & 5 we get a in terms of b and viceversa.

i have a approacing in another way but i am not sure whether it is correct or not, please clarify me.
my approach is below..

2a + b = 50e^2c ....... 1
4a + b = 30e^4c ....... 2
18a + 3b = 65e^6c ....... 3

from 1, b = 50e^2c - 2a

substitute for b in eqn 2, we get.

4a + 50e^2c - 2a = 30e^4c

2a = 30e^4c - 50e^2c
a = ( 30e^4c - 50e^2c )/2

substitute for b, we get..

b = 50e^2c - 30e^4c - 50e^2c
b = 100e^2c - 30e^4c

substitute for a & b in eqn 3.. we get,

18 * ( 30e^4c - 50e^2c )/2 + 3* 100e^2c - 30e^4c = 65e^6c
270e^4c - 450e^2c + 300e^2c - 90e^4c = 65e^6c
180e^4c - 150e^2c = 65e^6c

divide by e^2c, we get

65e^4c -180e^2c + 150 = 0

assume e^2c = x, therefore eqn bcomes

65x^2 -180x +150 =0

by solving the quadratic equation we can get c, a & b can be found subsequently.

but i tried two problems in this form but in both it comes solution doesnt exixt because it leads to sqrt( negative no)...

please clarify me whether the approach is correct...

11. hi guys,

i thank everyone for your replies.... i back substitued the answer in the formula, the formula and solution is correct.. thankyou for your ideas....


text:
# ATTiny Default AREF

I am trying to get the Adafruit Electret Microphone Amplifier to work with an ATTiny85 but am having some trouble with the reference voltage. I am using a 5V circuit because I am also powering a strip of LEDs and I am trying to avoid making the circuit too complicated with a bunch of extra regulators. Since the chip should use the supply voltage as reference by default I figured I would be fine in using 5V across the board. But I get basically no response out of the mic when doing this. However, if I hook up 5V to AREF and then call analogReference(EXTERNAL), everything works fine.

Any thoughts as to why this would be the case? This completely goes against my understanding (and experience) of the default AREF on the ATMega chips.

It would be nice to be able to use that AREF pin for something else :)

Side Note: I am going to be powering this with batteries so maybe I just need it anyways? But again, I thought the default mode was to use VCC as the reference...

• You mention three different platforms: ATtiny85, ATmega and Arduino. They are closely related, but which one are you using? Please include a link to the appropriate datasheet and the appropriate parts of the code. – jippie Mar 8 '13 at 19:31
• If you suspect you're assumption about default reference voltage is wrong, why not explicitly force it in your code, it only takes couple of bytes. – jippie Mar 8 '13 at 19:32
• I am using an ATTiny85 but on the Arduino code platform. I know it only takes up a couple of bytes in code but it takes up a whole I/O pin that I would rather be able to use... – Adam Haile Mar 8 '13 at 20:11
• You can force it to the (assumed) default Vcc in a few bytes and without sacrificing the pin. – jippie Mar 8 '13 at 20:12
• Ok... could you tell me how to do that? All I know is to call analogReference() and none of it's options seem to do what you are suggesting. – Adam Haile Mar 8 '13 at 20:20

## 1 Answer

Perhaps you are changing the A/D reference mode somewhere else in your code? You can double check this by adding a call to analogReference(DEFAULT) just before your call to analogRead(). This should select Vcc as the input to the A/D Mux on the ATTINY85 just as you expected.

Here is the ADMUX register from the ATTINY85 datasheet...

Here is the relevant code excerpted from the Arduino library....

#if defined(__AVR_ATtiny24__) || defined(__AVR_ATtiny44__) || defined(__AVR_ATtiny84__) || defined(__AVR_ATtiny25__) || defined(__AVR_ATtiny45__) || defined(__AVR_ATtiny85__)
65  #define DEFAULT 0
66  #define EXTERNAL 1
67  #define INTERNAL 2

uint8_t analog_reference = DEFAULT;

void analogReference(uint8_t mode)
{
// can't actually set the register here because the default setting
// will connect AVCC and the AREF pin, which would cause a short if
// there's something connected to AREF.
analog_reference = mode;
}

int analogRead(uint8_t pin)
{
uint8_t low, high;

if (pin >= 14) pin -= 14; // allow for channel or pin numbers

// set the analog reference (high two bits of ADMUX) and select the
// channel (low 4 bits).  this also sets ADLAR (left-adjust result)
// to 0 (the default).
ADMUX = (analog_reference << 6) | (pin & 0x07);
...


text:
PreprintPDF Available

# Compatible meshfree discretization of surface PDEs

Authors:

## Abstract and Figures

Meshfree discretization of surface partial differential equations are appealing, due to their ability to naturally adapt to deforming motion of the underlying manifold. In this work, we consider an existing scheme proposed by Liang et al. reinterpreted in the context of generalized moving least squares (GMLS), showing that existing numerical analysis from the GMLS literature applies to their scheme. With this interpretation, their approach may then be unified with recent work developing compatible meshfree discretizations for the div-grad problem in R^d. In this way, we obtain a compatible meshfree discretization of elliptic problems on manifolds which is naturally stable for problems with material interfaces, without the need to introduce numerical dissipation or local enrichment near the interface. We provide convergence studies illustrating the high-order convergence and stability of the approach for manufactured solutions and for an adaptation of the classical five-strip benchmark to a cylindrical manifold.
Content may be subject to copyright.
Noname manuscript No.
(will be inserted by the editor)
Compatible meshfree discretization of surface PDEs
Abstract Meshfree discretization of surface partial diﬀerential equations are
appealing, due to their ability to naturally adapt to deforming motion of the
underlying manifold. In this work, we consider an existing scheme proposed
by Liang et al. reinterpreted in the context of generalized moving least squares
(GMLS), showing that existing numerical analysis from the GMLS literature
applies to their scheme. With this interpretation, their approach may then be
uniﬁed with recent work developing compatible meshfree discretizations for
the div-grad problem in Rd. In this way, we obtain a compatible meshfree
discretization of elliptic problems on manifolds which is naturally stable for
problems with material interfaces, without the need to introduce numerical
dissipation or local enrichment near the interface. We provide convergence
studies illustrating the high-order convergence and stability of the approach
for manufactured solutions and for an adaptation of the classical ﬁve-strip
benchmark to a cylindrical manifold.
Keywords Generalized moving least squares ·Compatible discretization ·
Surface PDE ·Meshfree
Center for Computing Research, Sandia National Laboratories
Mailstop 1320, P.O. Box 5800, Albuquerque, NM 87125-1320
P. Kuberry
Center for Computing Research, Sandia National Laboratories
Mailstop 1320, P.O. Box 5800, Albuquerque, NM 87125-1320
Sandia National Laboratories is a multimission laboratory managed and operated by Na-
tional Technology and Engineering Solutions of Sandia, LLC.,a wholly owned subsidiary of
Honeywell International, Inc., for the U.S. Department of Energys National Nuclear Security
Administration under contract DE-NA-0003525. This paper describes objective technical re-
sults and analysis. Any subjective views or opinions that might be expressed in the paper do
not necessarily represent the views of the U.S. Department of Energy or the United States
Government.
1 Background and model problem
A wide class of mechanics problems are modelled by surface partial diﬀerential
equations (PDEs), whereby transport and material response are restricted to
a manifold M ⊂ Rd. For example, mechanical response of thin structures are
often modeled eﬃciently by plate or shell formulations of elasticity[26,12,1].
Diﬀusive transport is often modelled via the Laplace-Beltrami problem[31];
for example in the context of surfactant transport on interfaces in multiphase
ﬂows[27,8]. In biomechanics and material science, the mechanics of lipid bilay-
ers are governed by the Stokes equations, wherein the ﬂow is restricted to lie
within a manifold whose conﬁguration may evolve in response to ﬂuid ﬂow[25,
20]. In ocean/atmosphere dynamics, meshfree techniques are used to solve the
shallow-water equations in the context of semi-Lagrangian schemes[22,15,7,
23].
To solve surface PDEs, one typically requires a means to compute metric
information characterizing the local curvature and conﬁguration of the man-
ifold. For many of these applications, the manifold evolves in a Lagrangian
fashion, and it is desirable to obtain a scheme wherein this metric information
is easily computed as the conﬁguration evolves. Recent interest has formed
around the cut-FEM method, in which the manifold is evolved as a level set
which may then be treated as an immersed problem in an ambient Cartesian
ﬁnite element simulation[9,5].
Meshfree methods on the other hand oﬀer an attractive alternative in which
a point cloud sampling of the manifold is used to characterize both the mani-
fold itself and also the solution of the PDE. The points may thus evolve under
Lagrangian motion, and meshfree techniques may be used to recalibrate metric
information “on-the-ﬂy”. Several meshfree techniques have emerged relatively
recently for this class of problems, see for example [23,24, 2, 20, 19, 28,16,14].
Oftentimes however, these techniques rely on approaches such as hyperviscos-
ity and under-integration to handle the lack of stability theory that trouble
some meshfree discretizations. These methods may be characterized as treat-
ing the manifold intrinsically, working in coordinates internal to the manifold,
or extrinsically, working in an ambient space and projecting back to the man-
ifold. The approach considered in this work is an intrinsic method.
In recent work we have shown that generalized moving least squares (GMLS)
may be used together with a primal/dual strategy to obtain accurate dis-
cretization of H(div)-type problems[30, 29]. We informally categorize this a
spatially compatible meshfree discretization because its computational proper-
ties parallel those of div-compatible mesh-based schemes [3]. In the current
work, we re-interpret a meshfree scheme for surface PDE used by Liang et al.
[16] as a GMLS discretization. We then introduce an extension of our compat-
ible meshfree approach into this surface PDE setting, and provide numerical
results highlighting the high-order accuracy and stability of the approach.
For the sake of brevity, we will consider as a simpliﬁed model H(div)-
problem the Laplace-Beltrami problem without boundary conditions, written
Compatible meshfree discretization of surface PDEs 3
in mixed form:
s·u=f(1)
uκsφ= 0,(2)
where sdenotes the surface gradient, and κdenotes a diﬀusion parameter
that is potentially discontinuous.
2 Generalized moving least squares
We ﬁrst brieﬂy summarize the GMLS framework, referring to previous works
for technical details [16,30, 18, 32].
Consider φof function class V. Consider a collection of samples Λ=
{λi(φ)}N
i=1 corresponding to a quasi-uniform[32] collection of data sites Xh=
{xi} ⊂ Rdcharacterized by ﬁll distance h. To approximate a given linear target
functional τ˜xassociated with a target site ˜x, we seek a reconstruction pVh,
where VhVis a ﬁnite dimensional space chosen to provide good approxi-
mation properties, with basis P={P}dim(Vh)
i=1 . We perform this reconstruction
in the following weighted 2sense:
p= argmin
qVh
N
X
i=1
(λi(φ)λi(q))2ω(λi, τ˜x) (3)
where ωis a locally supported positive function. In this paper we restrict
attention to ω=Φ(|˜xxi|), where |·| denotes the Euclidean norm and
Φ(r) = (1 r/)4
+, where is a parameter controlling the support of ω, and
f+denotes the positive ﬂoor function.
With the optimal reconstruction pin hand, the target functional is ap-
proximated via τ˜x(φ)τh
˜x(φ) := τ˜x(p). As an unconstrained 2-optimization
problem, this process admits the explicit form
τh
˜x(φ) = τ˜x(P)|(Λ(P)|WΛ(P))1Λ(P)|WΛ(φ),(4)
where we denote:
τ˜x(P)Rdim(Vh)is a vector with components consisting of the target
functional applied to each basis function.
W RN×Nis a diagonal matrix with diagonal entries consisting of {ω(λi, τ˜x)}i=1,...,N .
Λ(P)RN×dim(Vh)is a rectangular matrix whose (i, j) entry corresponds
to the application of the ith sampling functional applied to the jth basis
function.
Λ(φ)RNis a vector consisting of the Nsamples of the function φ.
We note that by taking the contraction of the tensors appearing in Equation 4
and exploiting the compact support of ω, we may interpret the output of the
GMLS process as a ﬁnite diﬀerence-like stencil of the form
τh
˜x(φ) = X
xiBx)
αiλi(φ),(5)
where Bx) denotes the -ball neighborhood of the target site ˜x. Therefore,
GMLS admits an interpretation as an automated process for generating gen-
eralized ﬁnite diﬀerence methods on unstructured point clouds. We note that
the computational cost of solving the GMLS problem amounts to inverting
a small linear system which may be assembled using only information from
neighbors within the support of ω, and construction of such stencils across the
entire domain is embarrassingly parallel.
3 Liang’s scheme recast as GMLS problem
We next show that the scheme used by Liang et al. may be recast as a GMLS
problem. This means that accuracy results from GMLS literature may imme-
diately be applied to their scheme. For the sake of brevity, we only draw a
high-level comparison and refer the interested reader to Liang’s original work
for details[16]. In their work they consider the setting κ= 1.
We assume for the remainder of this work that Xhsamples a continuous
two-dimensional manifold M ⊂ R3. For a given point xiXh, Liang seeks
an approximation to the Laplace-Beltrami operator, deﬁned in contravariant
form as
LB φ=1
p|g|ip|g|gij jφ,
where gij denotes the inverse metric tensor and |g|denotes its determinant.
First, they solve an eigenvalue problem to obtain an approximation to
the tangent plane TxiM. The speciﬁc calculation of this approximate tangent
plane is unimportant for the current work; we assume we are given two linearly
independent vectors ξ1, ξ2which are suﬃciently “close” to the actual tangent
plane, whose cross product deﬁnes a vector normal to the manifold, ˆn. Their
scheme then consists of solving two GMLS problems. First, an approximation
ghto the metric tensor is obtained to approximate Mlocally in a neighborhood
of xi. This approximate metric tensor is then used to deﬁne a second GMLS
problem calculating the action of the Laplace-Beltrami operator.
First, we adapt as coordinate system the triple (ξ1, ξ2,ˆn). We seek a pa-
rameterization of the manifold from local coordinates Γ(ξ1, ξ2) : TxiM→M,
and assume that this mapping takes the form
Γ(ξ1, ξ2) = hξ1, ξ2, q(ξ1, ξ2)i(6)
where qis an unknown smooth function which we would like to infer. If qis
known, then the metric tensor gij is deﬁned componentwise as
gij =Γξi·Γξj,
where Γξi=hxiξ1, ∂xiξ2, ∂xiq(ξ1, ξ2)idenotes the gradient of the parameter-
ization with respect to the local coordinate ξi.
We deﬁne the ﬁrst GMLS problem as follows:
Compatible meshfree discretization of surface PDEs 5
As samples Λ, select the point evaluation of Γat all points in the -ball
neighborhood of xi.
As target functional, select the point evaluation of the metric tensor at xi.
As reconstruction space, select the collection of mth
1-order polynomials,
where m1is an integer parameter.
Upon recovering the approximate metric tensor gh, we may compute and store
its inverse.
The second GMLS problem may then be deﬁned:
As samples Λ, select the point evaluation of φin the neighborhood of of xi
As target functional, select
τxi(φ) = 1
p|gh|ip|gh|gij
hjφ
As reconstruction space, select the collection of mth
2-order polynomials,
where m2is another integer parameter.
In this manner, the GMLS process provides a local, high-order stencil for
the Laplace-Beltrami problem of the form
h
LB φ(xi) = X
jB(xi)
αij φj,(7)
which may be assembled into a global stiﬀness matrix as in standard mesh-
based schemes. The key feature of this approach is that the coeﬃcients in
the stencil encode a high-order reconstruction of the manifold, providing a
high-order scheme that may be eﬃciently generated “on the ﬂy” from a qua-
siuniform point cloud sampling of the manifold.
4 Compatible generalization of Liang’s scheme
In [30] we presented a compatible meshfree discretization of the div-grad prob-
lem in Rd. The key premise of our approach is the treatment of the mixed
problem (Equation 1) on a virtual primal/dual grid. In the manifold setting,
we solve the ﬁrst GMLS problem in Liang’s scheme without modiﬁcation, so
that we again have access to ghat the point xi. We consider the -ball graph of
neighbors in the vicinity of xias a surrogate for a mesh, informally identifying
nodes as virtual dual cells and virtual dual faces. We interpret the nodes and
edges of this graph as a physical mesh constituted of only zero- and one-chains.
To discretize the surface gradient over the edge eij connecting xiand xjon
the primal grid, we deﬁne the following mimetic gradient operator
κsφ·dl(8)
We make the added assumption that κmay be approximated as piecewise
constant over the edge, yielding the following discrete gradient from the fun-
damental theorem of calculus.
ij (φ) = κxi+xj
2(φjφi) (9)
ij (φ) exactly for
any integrable φ.
To obtain a surface divergence operator on the virtual dual grid, we deﬁne
the following GMLS problem:
As target functional, select the surface divergence,
τ(u) = s·u(xi) = 1
p|g|ip|g|ui
As reconstruction space, select the gradient of mth
2-order scalar polynomi-
als,
Vh=nu=q, where q(Pm2)do
As sampling functionals, select λj(u) = Reij u·dl.
Solving the GMLS problem, we obtain again a stencil for the surface di-
vergence of the form
s·u=X
j
αij λj(u)
To discretize the full div-grad problem, we chain the surface divergence
s·u=X
j
ij (φ).(10)
5 Numerical approach
In the results that follow, we will consider both spherical and cylindrical man-
ifolds. To document convergence, we seek a hierarchy of quasi-uniform point
clouds parameterized by a ﬁll distance h. To do this in the spherical case we
take a quasi-uniform MPAS[21] grid and generate points at cell centers. In the
cylindrical case we take a Cartesian lattice with grid spacing hand apply the
mapping (x, y)(rcos θ, z ), where ris the cylinder radius.
We select for parameters m1=m2=m, so that the same polynomial
order is used for both the manifold and function reconstructions, and mmay
be varied to benchmark the high-order convergence in a preﬁnement setting.
It is well-understood in GMLS that the existence of a solution to Equation
3 stems from unisolvency of the samples over the reconstruction space[32];
practically this means that the support must be chosen suﬃciently large,
Compatible meshfree discretization of surface PDEs 7
to obtain suﬃcient samples, to uniquely obtain a polynomial ﬁt. We select
the support adaptively following a process detailed in previous work[29]. In
short, for each particle we ﬁnd the distance to the dim(Vh)-st nearest neighbor
rneigh, and then set the support as 1.5rneigh . In this manner, we are guaranteed
to obtain suﬃcient neighbors for unisolvency, provided the underlying point
cloud is quasi-uniform.
At each point of the point cloud, the GMLS approach is used to generate
a local stencil, which is summed into a row of a sparse global stiﬀness matrix.
We use the Compadre toolkit [13] and multiple packages from Trilinos[10]
(MueLu[11], Belos, Tpetra, Teuchos, and Kokkos[6]) to exploit latent paral-
lelism in this construction: each small optimization problem may be solved on
GPUs, and the global matrix is distributed across multiple processors. At this
point, the matrix may be solved using standard algebraic multigrid precondi-
tioning techniques.
To document convergence, we deﬁne the following 2norm
||φ||2=sPN
i=1 φ2
i
N.(11)
Fig. 1 Solution to manufactured smooth solution on spherical manifold for scalar variable
φ.
6 Results: Manufactured solution on a sphere
We consider ﬁrst the case on the unit sphere in which κ= 1. Using the
method of manufactured solutions, we denote the true solution φ=YM
l(θ, r),
f=λl,M φ, where YM
ldenotes a spherical harmonic with associated eigenvalue
λl,M . We select as parameters l= 5 and M= 4. A representative point cloud
is given in Figure 1, corresponding to a resolution of h0.035. In Table 6,
we summarize the convergence of the solution in the 2-norm with respect to
h, and observe algebraic convergence of the form ||uuexact||2hm.
m = 2 m = 4 m = 6
h2-error Rate 2-error Rate 2-error Rate
0.07 1.3199e-03 - 4.0960e-04 - 6.9184e-07 -
0.035 3.2997e-04 2.00 4.5474e-05 3.17 1.0538e-08 6.04
0.0175 8.2581e-05 2.00 4.9164e-06 3.21 1.6135e-10 6.03
0.00875 2.1249e-05 1.96 2.9869e-07 4.04 3.6536e-12 5.46
Table 1 Convergence in 2norm with respect to particle spacing h, illustrating (m)th-order
convergence with respect to the order of reconstruction min the GMLS problem.
7 Results: Five-strip problem on a cylinder
We consider now a case in which κis piecewise constant. For such problems
involving material interfaces, there is an implicit interface condition that the
jump in ﬂux is zero ([κφ·ˆ
n] = 0). In Rd, the ﬁve-strip test is a standard
benchmark to test H(div)-conformity, and has been used in the past to high-
light areas where e.g. Lagrangian ﬁnite elements will qualitatively fail but
compatible spatial discretizations will reproduce the solution exactly[17, 4]. In
the Euclidean setting, the problem is set up by partitioning the unit square
into ﬁve strips ({i}i∈{1,...,5}) of equal height ∆y = 0.2, each with corre-
sponding diﬀusion coeﬃcient µi∈ {16,6,1,10,2}. A ﬂux boundary condition
is imposed consistent with an analytic solution φ= 1 x. It is easily seen
that φis orthogonal to the normal associated with the material interfaces,
and thus the interface condition is trivially satisﬁed. Thus, the problem stands
as a form of patch test for interfacial problems. We have shown in previous
work that in Rd, the staggered compatible meshfree approach pursued in this
work provides predictive results for this problem reminiscent of mesh-based
spatially compatible approaches[30].
We generalize the classical benchmark by constructing a similar solution
on the cylinder with the property that the interface condition is similarly
upheld. We consider a cylinder with height 2 and unit diameter, and partition
the cylinder into a stack of ﬁve cylinders, each with height ∆z = 0.4 and
diﬀusion coeﬃcient µi∈ {16,6,1,10,2}. We construct the problem to provide
Compatible meshfree discretization of surface PDEs 9
Fig. 2 Plot of analytic ﬂux u=κsφof ﬁve-strip problem on the cylinder, corresponding
to a resolution of 8 points per strip.
the analytic solution φ=, so that sφ= ˆeθ, and ﬂux is again parallel to
the interface. For all results we select m= 2.
The cylinder diﬀers from the Euclidean case in that there are no boundaries
from which the ﬂow may be driven by a Neumann condition, and additionally
the exact solution φ=is discontinuous at θ= 0. To address this, we impose
that the analytic solution hold exactly as a Dirichlet condition over the sector
θ∈ {−π
2,π
2}, so that there is no ambiguity in handling the discontinuity at
θ= 0. To enforce this Dirichlet condition, particles lying in the sector are
identiﬁed, and their rows in the global stiﬀness matrix replaced with zeros
and a one on the diagonal, and the corresponding entry on the right-hand-side
replaced with the analytic solution.
A characteristic point cloud, with corresponding ﬂuxes, is plotted in Figure
2. We plot the proﬁle of ﬂuxes along the θ=πline, using nearest point
interpolation to post-process probe data from the point cloud. We include
both coarse resolution results corresponding to two points per strip, and ﬁne
resolution results corresponding to 32 points per strip. From this proﬁle, we
see that the gradient is non-oscillatory in the vicinity of the jump, similar to
the results one would obtain with a spatially compatible mesh-based approach.
To postprocess the ﬂuxes, we must ﬁrst recover the surface gradient of the
solution. The scheme only provides as a solution point-values of φ; the ﬂuxes
uare only available in terms of their integrals over edges λij(u) = Rκu·dl.
Fig. 3 Convergence of post-processed ﬂux to analytic solution along probe at θ=π. Coarse
and ﬁne grids correspond to two and thirty points per strip, respectively. Numerical ﬂux is
non-oscillatory near material interfaces for both coarse and ﬁne resolution.
Fill distance hFlux 2-error Rate
0.2 0.05621 -
0.1 0.01472 1.93
0.05 0.00369 2.00
0.025 0.00092 2.01
0.0125 0.00023 2.00
Table 2 Convergence with respect to hof 2-norm of ﬂux magnitude normalized by maxi-
mum ﬂux, illustrating second-order convergence for the choice m= 2.
To obtain instead the point evaluation of the ﬂux at each particle, we compute
the following GMLS problem in post-processing:
As target functional, select the point evaluation,
τ(u) = κ(xi)sφ(xi)
As reconstruction space, select the space of mth
2-order scalar polynomials.
As sampling functionals, select the point evaluation λj(u) = φ(xj).
In this manner, we obtain a point evaluation of the ﬂux at each point.
We quantify convergence of ﬂuxes in Table 7 for the choice m= 2, where
we observe algebraic convergence of the ﬂux magnitude in the `2norm with
respect to h.
Compatible meshfree discretization of surface PDEs 11
8 Conclusion
In this work, we have used the GMLS framework to unify both Liang’s ap-
proach for solving surface PDEs and our previous compatible meshfree scheme
for Euclidean space, to obtain a new compatible meshfree scheme applicable
to general manifolds. The scheme has been shown to directly generalize re-
sults from the Euclidean setting. In other works, we have used the Euclidean
compatible meshfree scheme as a foundation for developing schemes for the
stationary Stokes problem[29]. We intend to similarly extend the current work
to obtain a meshfree capability for stationary Stokes on manifolds. Such a
framework is foundational to developing mechanics solvers capable of operat-
ing in near incompressible regimes.
Acknowledgements The authors acknowledge support under the Sandia National Lab-
oratories LDRD program, and thank Dr. Pavel Bochev for reviewing an early draft of the
work. On behalf of all authors, the corresponding author states that there is no conﬂict of
interest. The views expressed in the article do not necessarily represent the views of the U.S.
Department of Energy or the United States Government.
References
1. Amiri, F., Mill´an, D., Shen, Y., Rabczuk, T., Arroyo, M.: Phase-ﬁeld modeling of frac-
ture in linear thin shells. Theoretical and Applied Fracture Mechanics 69, 102–109
(2014)
2. Amiri, F., Ziaei-Rad, S., Valizadeh, N., Rabczuk, T.: On the use of local maximum en-
tropy approximants for cahn–hilliard phase-ﬁeld models in 2d domains and on surfaces.
Computer Methods in Applied Mechanics and Engineering 346, 1–24 (2019)
3. Arnold, D.N., Bochev, P.B., Lehoucq, R.B., Nicolaides, R.A., Shashkov, M.: Compatible
spatial discretizations, vol. 142. Springer Science & Business Media (2007)
4. Bochev, P.B., Gunzburger, M.D.: Least-squares ﬁnite element methods, vol. 166.
Springer Science & Business Media (2009)
5. Burman, E., Hansbo, P., Larson, M.G.: A stabilized cut ﬁnite element method for partial
diﬀerential equations on surfaces: The laplace–beltrami operator. Computer Methods
in Applied Mechanics and Engineering 285, 188–207 (2015)
6. Edwards, H.C., Trott, C.R., Sunderland, D.: Kokkos: Enabling many-
core performance portability through polymorphic memory access pat-
terns. Journal of Parallel and Distributed Computing 74(12), 3202
– 3216 (2014). DOI https://doi.org/10.1016/j.jpdc.2014.07.003. URL
http://www.sciencedirect.com/science/article/pii/S0743731514001257. Domain-
Speciﬁc Languages and High-Level Frameworks for High-Performance Computing
7. Flyer, N., Lehto, E., Blaise, S., Wright, G.B., St-Cyr, A.: A guide to rbf-generated ﬁnite
diﬀerences for nonlinear transport: Shallow water simulations on a sphere. Journal of
Computational Physics 231(11), 4078–4095 (2012)
8. Ganesan, S., Tobiska, L.: Arbitrary lagrangian–eulerian ﬁnite-element method for com-
putation of two-phase ﬂows with soluble surfactants. Journal of Computational Physics
231(9), 3685–3702 (2012)
9. Hansbo, P., Larson, M.G., Zahedi, S.: A cut ﬁnite element method for a stokes interface
problem. Applied Numerical Mathematics 85, 90–114 (2014)
10. Heroux, M., Bartlett, R., Hoekstra, V.H.R., Hu, J., Kolda, T., Lehoucq, R., Long, K.,
Pawlowski, R., Phipps, E., Salinger, A., Thornquist, H., Tuminaro, R., Willenbring, J.,
Williams, A.: An Overview of Trilinos. Tech. Rep. SAND2003-2927, Sandia National
Laboratories (2003)
11. Hu, J.J., Prokopenko, A., Siefert, C.M., Tuminaro, R.S., Wiesner, T.A.: MueLu multi-
grid framework. http://trilinos.org/packages/muelu (2014)
12. Krysl, P., Belytschko, T.: Analysis of thin shells by the element-free galerkin method.
International Journal of Solids and Structures 33(20-22), 3057–3080 (1996)
13. Kuberry, P., Bosler, P., Trask, N.: Compadre toolkit version 1.0.1 (2019). DOI
10.5281/zenodo.2560287. URL https://doi.org/10.5281/zenodo.2560287
14. Lai, R., Liang, J., Zhao, H.: A local mesh method for solving pdes on point clouds.
Inverse Problems & Imaging 7(3) (2013)
15. Lauritzen, P.H., Jablonowski, C., Taylor, M.A., Nair, R.D.: Numerical techniques for
global atmospheric models, vol. 80. Springer Science & Business Media (2011)
16. Liang, J., Zhao, H.: Solving partial diﬀerential equations on point clouds. SIAM Journal
on Scientiﬁc Computing 35(3), A1461–A1486 (2013)
17. Masud, A., Hughes, T.J.: A stabilized mixed ﬁnite element method for darcy ﬂow.
Computer methods in applied mechanics and engineering 191(39-40), 4341–4370 (2002)
18. Mirzaei, D., Schaback, R., Dehghan, M.: On generalized moving least squares and diﬀuse
derivatives. IMA Journal of Numerical Analysis 32(3), 983–1000 (2012)
19. Petras, A., Ling, L., Ruuth, S.J.: An rbf-fd closest point method for solving pdes on
surfaces. Journal of Computational Physics 370, 43–57 (2018)
20. Rahimi, M., DeSimone, A., Arroyo, M.: Curved ﬂuid membranes behave laterally as
eﬀective viscoelastic media. Soft Matter 9(46), 11033–11045 (2013)
21. Ringler, T., Petersen, M., Higdon, R.L., Jacobsen, D., Jones, P.W., Maltrud, M.: A
multi-resolution approach to global ocean modeling. Ocean Modelling 69, 211–232
(2013)
22. Salmon, R.: Lectures on geophysical ﬂuid dynamics. Oxford University Press (1998)
23. Shankar, V., Wright, G.B.: Mesh-free semi-lagrangian methods for transport on a sphere
using radial basis functions. Journal of Computational Physics 366, 170–190 (2018)
24. Shankar, V., Wright, G.B., Kirby, R.M., Fogelson, A.L.: A radial basis function (rbf)-
ﬁnite diﬀerence (fd) method for diﬀusion and reaction–diﬀusion equations on surfaces.
Journal of scientiﬁc computing 63(3), 745–768 (2015)
25. Sigurdsson, J.K., Atzberger, P.J.: Hydrodynamic coupling of particle inclusions embed-
ded in curved lipid bilayer membranes. Soft matter 12(32), 6685–6707 (2016)
26. Stolarski, H., Belytschko, T., Lee, S.H.: A review of shell ﬁnite elements and corotational
theories. Computational mechanics advances 2(2), 125–212 (1995)
27. Stone, H.: A simple derivation of the time-dependent convective-diﬀusion equation for
surfactant transport along a deforming interface. Physics of Fluids A: Fluid Dynamics
2(1), 111–112 (1990)
28. Suchde, P., Kuhnert, J.: A meshfree generalized ﬁnite diﬀerence method for surface
pdes. arXiv preprint arXiv:1806.07193 (2018)
29. Trask, N., Maxey, M., Hu, X.: A compatible high-order meshless method for the stokes
equations with applications to suspension ﬂows. Journal of Computational Physics 355,
310–326 (2018)
30. Trask, N., Perego, M., Bochev, P.: A high-order staggered meshless method for elliptic
problems. SIAM Journal on Scientiﬁc Computing 39(2), A479–A502 (2017)
31. Walker, S.W.: The shape of things: a practical guide to diﬀerential geometry and the
shape derivative, vol. 28. SIAM (2015)
32. Wendland, H.: Scattered data approximation, vol. 17. Cambridge university press (2004)
ResearchGate has not been able to resolve any citations for this publication.
Article
Full-text available
We apply a local maximum entropy (LME) approximation scheme to the fourth order phase-field model for the traditional Cahn-Hilliard theory. The discretization of the Cahn-Hilliard equation by Galerkin method requires at least $C^1$-continuous basis functions. This requirement can be fulfilled by using LME shape functions which are $C^\infty$-continuous. In this case, the primal variational formulations of the fourth-order partial differential equation is well defined and integrable. Hence, there is no need to split the fourth-order partial differential equation into two second-order partial differential equations; this splitting scheme is a common practice in mixed finite element formulations with $C^0$-continuous Lagrange shape functions. Furthermore, we use a general and simple numerical method such as statistical manifold learning techniques that allows dealing with general point set surfaces avoiding a global parametrization, which can be applied to tackle surfaces of complex geometry and topology, and to solve Cahn-Hilliard equation on general surfaces. Finally, the flexibility and robustness of the presented methodology is demonstrated for several representative numerical examples.
Article
Full-text available
Partial differential equations (PDEs) on surfaces appear in many applications throughout the natural and applied sciences. The classical closest point method (Ruuth and Merriman, J. Comput. Phys. 227(3):1943-1961, [2008]) is an embedding method for solving PDEs on surfaces using standard finite difference schemes. In this paper, we formulate an explicit closest point method using finite difference schemes derived from radial basis functions (RBF- FD). Unlike the orthogonal gradients method (Piret, J. Comput. Phys. 231(14):4662-4675, [2012]), our proposed method uses RBF centers on regular grid nodes. This formulation not only reduces the computational cost but also avoids the ill-conditioning from point clustering on the surface and is more natural to couple with a grid based manifold evolution algorithm (Leung and Zhao, J. Comput. Phys. 228(8):2993-3024, [2009]). When compared to the standard finite difference discretization of the closest point method, the proposed method requires a smaller computational domain surrounding the surface, resulting in a decrease in the number of sampling points on the surface. In addition, higher-order schemes can easily be constructed by increasing the number of points in the RBF-FD stencil. Applications to a variety of examples are provided to illustrate the numerical convergence of the method.
Article
Full-text available
We present three new semi-Lagrangian methods based on radial basis function (RBF) interpolation for numerically simulating transport on a sphere. The methods are mesh-free and are formulated entirely in Cartesian coordinates, thus avoiding any irregular clustering of nodes at artificial boundaries on the sphere and naturally bypassing any apparent artificial singularities associated with surface-based coordinate systems. For problems involving tracer transport in a given velocity field, the semi-Lagrangian framework allows these new methods to avoid the use of any stabilization terms (such as hyperviscosity) during time-integration, thus reducing the number of parameters that have to be tuned. The three new methods are based on interpolation using 1) global RBFs, 2) local RBF stencils, and 3) RBF partition of unity. For the latter two of these methods, we find that it is crucial to include some low degree spherical harmonics in the interpolants. Standard test cases consisting of solid body rotation and deformational flow are used to compare and contrast the methods in terms of their accuracy, efficiency, conservation properties, and dissipation/dispersion errors. For global RBFs, spectral spatial convergence is observed for smooth solutions on quasi-uniform nodes, while high-order accuracy is observed for the local RBF stencil and partition of unity approaches.
Article
Full-text available
We present a new meshless method for scalar diffusion equations which is motivated by their compatible discretizations on primal-dual grids. Unlike the latter though, our approach is truly meshless because it only requires the graph of nearby neighbor connectivity of the discretization points $\bm{x}_i$. This graph defines a local primal-dual grid complex with a \emph{virtual} dual grid, in the sense that specification of the dual metric attributes is implicit in the method's construction. Our method combines a topological gradient operator on the local primal grid with a Generalized Moving Least Squares approximation of the divergence on the local dual grid. We show that the resulting approximation of the div-grad operator maintains polynomial reproduction to arbitrary orders and yields a meshless method, which attains $O(h^{m})$ convergence in both $L^2$ and $H^1$ norms, similar to mixed finite element methods. We demonstrate this convergence on curvilinear domains using manufactured solutions. Application of the new method to problems with discontinuous coefficients reveals solutions that are qualitatively similar to those of compatible mesh-based discretizations.
Article
Full-text available
We develop theory and computational methods to investigate particle inclusions embedded within curved lipid bilayer membranes. We consider the case of spherical lipid vesicles where inclusion particles are coupled through (i) intramembrane hydrodynamics, (ii) traction stresses with the external and trapped solvent fluid, and (iii) intermonolayer slip between the two leaflets of the bilayer. We investigate relative to flat membranes how the membrane curvature and topology augment hydrodynamic responses. We show how both the translational and rotational mobility of protein inclusions are effected by the membrane curvature, ratio of intramembrane viscosity to solvent viscosity, and inter-monolayer slip. For general investigations of many-particle dynamics, we also discuss how our approaches can be used to treat the collective diffusion and hydrodynamic coupling within spherical bilayers.
Article
Full-text available
The lateral mobility of membrane inclusions is essential in biological processes involving membrane-bound macromolecules, which often take place in highly curved geometries such as membrane tubes or small organelles. Probe mobility is assisted by the lateral fluidity, which is thought to be purely viscous for lipid bilayers and synthetic systems such as polymersomes. In previous theoretical studies, the hydrodynamical mobility is estimated assuming a fixed membrane geometry. However, fluid membranes are very flexible out-of-plane. By accounting for the deformability of the membrane and in the presence of curvature, we show that the lateral motion of an inclusion produces a normal force, which results in a nonuniform membrane deformation. Such a deformation mobilizes the bending elasticity, produces extra lateral viscous and elastic forces, and results in an effective lateral viscoelastic behavior. The coupling between lateral and out-of-plane mechanics is mediated by the interfacial hydrodynamics and curvature. We analyze the frequency and curvature dependent rheology of flexible fluid membranes, and interpret it with a simple four-element model, which provides a background for microrheological experiments. Two key technical aspects of the present work are a new formulation for the interfacial hydrodynamics, and the linearization of the governing equations around a cylindrical geometry.
Article
A stable numerical solution of the steady Stokes problem requires compatibility between the choice of velocity and pressure approximation that has traditionally proven problematic for meshless methods. In this work, we present a discretization that couples a staggered scheme for pressure approximation with a divergence-free velocity reconstruction to obtain an adaptive, high-order, finite difference-like discretization that can be efficiently solved with conventional algebraic multigrid techniques. We use analytic benchmarks to demonstrate equal-order convergence for both velocity and pressure when solving problems with curvilinear geometries. In order to study problems in dense suspensions, we couple the solution for the flow to the equations of motion for freely suspended particles in an implicit monolithic scheme. The combination of high-order accuracy with fully-implicit schemes allows the accurate resolution of stiff lubrication forces directly from the solution of the Stokes problem without the need to introduce sub-grid lubrication models.
Article
The development of shell finite elements is reviewed. As a background for these discussions, some of the controversies and difficulties in classical shell theories are reviewed and the features of shell bending behavior which were elucidated analytically are recalled. The description of shell elements focuses on three types of methods: mixed methods, assumed strain methods and discrete Kirchhoff methods. In Section 4, a corotational theory and its application to shells are presented; as part of this development, the kinematics of degenerated continuum formulations are compared to the Koiter-Sanders theory.
Article
We present a general framework for solving partial differential equations on manifolds represented by meshless points, i.e. point clouds, without parameterization or connection information. Our method is based on a local approximation of the manifold as well as functions defined on the manifold, such as using least squares, simultaneously in a local intrinsic coordinate system constructed by local principal component analysis using K nearest neighbors. Once the local reconstruction is available, differential operators on the manifold can be approximated discretely. The framework extends to manifolds of any dimension. The complexity of our method scales well with the total number of points and the true dimension of the manifold (not the embedded dimension). The numerical algorithms, error analysis, and test examples are presented.


text:
## Trimetallaborides as starting points for the syntheses of large metal-rich molecular borides and clusters

Please always quote using this URN: urn:nbn:de:bvb:20-opus-191511
• Treatment of an anionic dimanganaborylene complex ([{Cp(CO)$$_2$$Mn}$$_2$$B]$$^-$$) with coinage metal cations stabilized by a very weakly coordinating Lewis base (SMe$$_2$$) led to the coordination of the incoming metal and subsequent displacement of dimethylsulfide in the formation of hexametalladiborides featuring planar four-membered M$$_2$$B$$_2$$ cores (M = Cu, Au) comparable to transition metal clusters constructed around four-membered rings composed solely of coinage metals. The analogies between compounds consisting of B$$_2$$M$$_2$$Treatment of an anionic dimanganaborylene complex ([{Cp(CO)$$_2$$Mn}$$_2$$B]$$^-$$) with coinage metal cations stabilized by a very weakly coordinating Lewis base (SMe$$_2$$) led to the coordination of the incoming metal and subsequent displacement of dimethylsulfide in the formation of hexametalladiborides featuring planar four-membered M$$_2$$B$$_2$$ cores (M = Cu, Au) comparable to transition metal clusters constructed around four-membered rings composed solely of coinage metals. The analogies between compounds consisting of B$$_2$$M$$_2$$ units and M$$_4$$ (M = Cu, Au) units speak to the often overlooked metalloid nature of boron. Treatment of one of these compounds (M = Cu) with a Lewis-basic metal fragment (Pt(PCy$$_3$$)$$_2$$) led to the formation of a tetrametallaboride featuring two manganese, one copper and one platinum atom, all bound to boron in a geometry not yet seen for this kind of compound. Computational examination suggests that this geometry is the result of d$$^{10}$$-d$$^{10}$$ dispersion interactions between the copper and platinum fragments.


text:
# 7: Elliptic Equations of Second Order

Here we consider linear elliptic equations of second order, mainly the Laplace equation

$$\triangle u=0.$$

Solutions of the Laplace  equation are called potential functions or harmonic functions. The Laplace equation is called also potential equation. The general elliptic equation for a scalar function $$u(x)$$, $$x\in\Omega\subset\mathbb{R}^n$$, is

$$Lu:=\sum_{i,j=1}^na^{ij}(x)u_{x_ix_j}+\sum_{j=1}^n b^j(x)u_{x_j}+c(x)u=f(x),$$

where the matrix $$A=(a^{ij})$$ is real, symmetric and positive definite. If $$A$$ is a constant matrix, then a transform to principal axis and stretching of axis leads to

$$\sum_{i,j=1}^na^{ij}u_{x_ix_j}=\triangle v,$$

where $$v(y):=u(Ty)$$, $$T$$ stands for the above composition of mappings.

### Contributors

• Integrated by Justin Marshall.