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## Graphs and Combinatorics

 Short Title: Graphs Comb. Publisher: Springer Japan, Tokyo ISSN: 0911-0119; 1435-5914/e Online: http://link.springer.com/journal/volumesAndIssues/373 Comments: Indexed cover-to-cover
 Documents Indexed: 2,788 Publications (since 1985) References Indexed: 2,410 Publications with 36,633 References.
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### Latest Issues

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### Authors

 31 Henning, Michael Anthony 24 Lai, Hong-Jian 23 Alon, Noga M. 23 Tuza, Zsolt 20 Li, Xueliang 20 Urrutia Galicia, Jorge L. 20 West, Douglas Brent 18 Schiermeyer, Ingo 17 Furedi, Zoltan 17 Galeana-Sánchez, Hortensia 17 Gould, Ronald J. 16 Broersma, Hajo J. 16 Magnant, Colton 16 Nakamoto, Atsuhiro 15 Chen, Guantao 15 Maffray, Frédéric 15 Rodl, Vojtech 15 Xiong, Liming 14 Faudree, Ralph Jasper jun. 14 Frankl, Péter 14 Kano, Mikio 14 Li, Xiangwen 14 Montellano-Ballesteros, Juan José 14 Ozeki, Kenta 14 Rautenbach, Dieter 13 Egawa, Yoshimi 13 Li, Hao 13 Yeo, Anders 12 Hurtado, Ferran 12 Jendrol’, Stanislav 12 Kang, Liying 12 Kostochka, Aleksandr Vasil’evich 12 Yamashita, Tomoki 11 Demaine, Erik D. 11 Jacobson, Michael Scott 11 Klavžar, Sandi 11 Maehara, Hiroshi 11 Pach, János 11 Ren, Han 11 Woźniak, Mariusz 11 Zhang, Shenggui 10 Akbari, Saieed 10 Bryant, Darryn E. 10 Chandran, L. Sunil 10 Chen, Yaojun 10 Furuya, Michitaka 10 Jafari Rad, Nader 10 Koh, Khee Meng 10 Koolen, Jack H. 10 Langerman, Stefan 10 Li, Yusheng 10 Rodger, Christopher Andrew 10 Shan, Erfang 10 Volkmann, Lutz 10 Wang, Guanghui 10 Wang, Hong 9 Aichholzer, Oswin 9 Hiraki, Akira 9 Hoàng, Chính T. 9 Li, Binlong 9 Li, Mingchu 9 Matsumoto, Naoki 9 Meng, Jixiang 9 Rall, Douglas F. 9 Sheikholeslami, Seyed Mahmoud 9 Yuster, Raphael 9 Zhu, Xuding 8 Bannai, Eiichi 8 Bujtás, Csilla 8 Chartrand, Gary Theodore 8 Colbourn, Charles J. 8 De Bruyn, Bart 8 Griggs, Terry S. 8 Hou, Xinmin 8 Karami, Hossein 8 Kriesell, Matthias 8 Lehel, Jeno 8 Liu, Guizhen 8 Ota, Katsuhiro 8 Peng, Yuejian 8 Rivera-Campo, Eduardo 8 Saito, Akira 8 Suzuki, Yusuke 8 Wang, Wei-Fan 8 Wei, Bing 8 Wu, Jian-Liang 7 Akiyama, Jin 7 Ando, Kiyoshi 7 Billington, Elizabeth J. 7 Brown, Jason Ira 7 Chang, Gerard Jennhwa 7 Chen, Zhihong 7 Chiba, Shuya 7 Favaron, Odile 7 Fu, Hunglin 7 Horňák, Mirko 7 Ito, Hiro 7 Körner, János 7 Liu, Yanpei 7 Lu, Hongliang ...and 3,272 more Authors
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### Fields

 2,690 Combinatorics (05-XX) 118 Convex and discrete geometry (52-XX) 110 Computer science (68-XX) 73 Group theory and generalizations (20-XX) 68 Geometry (51-XX) 60 Number theory (11-XX) 56 Information and communication theory, circuits (94-XX) 43 Linear and multilinear algebra; matrix theory (15-XX) 40 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 36 Order, lattices, ordered algebraic structures (06-XX) 33 Operations research, mathematical programming (90-XX) 20 Manifolds and cell complexes (57-XX) 17 Probability theory and stochastic processes (60-XX) 14 Commutative algebra (13-XX) 10 General and overarching topics; collections (00-XX) 8 Mathematical logic and foundations (03-XX) 8 Special functions (33-XX) 8 Statistics (62-XX) 7 Biology and other natural sciences (92-XX) 5 History and biography (01-XX) 5 Algebraic geometry (14-XX) 4 Differential geometry (53-XX) 3 Functional analysis (46-XX) 3 General topology (54-XX) 3 Algebraic topology (55-XX) 3 Numerical analysis (65-XX) 3 Quantum theory (81-XX) 2 General algebraic systems (08-XX) 2 Nonassociative rings and algebras (17-XX) 2 Category theory; homological algebra (18-XX) 2 Real functions (26-XX) 2 Harmonic analysis on Euclidean spaces (42-XX) 2 Calculus of variations and optimal control; optimization (49-XX) 2 Global analysis, analysis on manifolds (58-XX) 2 Systems theory; control (93-XX) 1 Field theory and polynomials (12-XX) 1 Associative rings and algebras (16-XX) 1 Topological groups, Lie groups (22-XX) 1 Functions of a complex variable (30-XX) 1 Several complex variables and analytic spaces (32-XX) 1 Partial differential equations (35-XX) 1 Sequences, series, summability (40-XX) 1 Operator theory (47-XX) 1 Statistical mechanics, structure of matter (82-XX)

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1999
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2004
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2005
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Cutler, Jonathan; Nir, JD; Radcliffe, A. J.
2022
Predominating a vertex in the connected domination game. Zbl 1486.05200
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2022
Admissible property of graphs in terms of radius. Zbl 1479.05290
Yu, Huijuan; Wu, Baoyindureng
2022
Some constructions of quasi-strongly regular digraphs. Zbl 1479.05365
Guo, Zhengyu; Jia, Dongdong; Zhang, Gengsheng
2022
New constructions of divisible design Cayley graphs. Zbl 07451451
Crnković, Dean; Švob, Andrea
2022
Point partition numbers: perfect graphs. Zbl 1484.05080
von Postel, Justus; Schweser, Thomas; Stiebitz, Michael
2022
Thick weakly distance-regular digraphs. Zbl 1484.05209
Yang, Yuefeng; Wang, Kaishun
2022
On the existence of regular sparse anti-magic squares of odd order. Zbl 1484.05029
Chen, Guangzhou; Li, Wen; Zhong, Ming; Xin, Bangying
2022
Homogeneous coherent configurations from spherical buildings and other edge-coloured graphs. Zbl 1485.05185
Guillot, Pierre
2022
Three-weight codes over rings and strongly walk regular graphs. Zbl 07493897
Shi, Minjia; Kiermaier, Michael; Kurz, Sascha; Solé, Patrick
2022
On the connectivity of enhanced power graphs of finite groups. Zbl 07324340
Bera, Sudip; Dey, Hiranya Kishore; Mukherjee, Sajal Kumar
2021
Sharp upper bounds on the $$k$$-independence number in graphs with given minimum and maximum degree. Zbl 1459.05244
O, Suil; Shi, Yongtang; Taoqiu, Zhenyu
2021
Domination for Latin square graphs. Zbl 1470.05129
Pahlavsay, Behnaz; Palezzato, Elisa; Torielli, Michele
2021
Strongly spanning trailable graphs with small circumference and Hamilton-connected claw-free graphs. Zbl 1459.05151
Liu, Xia; Xiong, Liming; Lai, Hong-Jian
2021
Spiders and their kin: an investigation of Stanley’s chromatic symmetric function for spiders and related graphs. Zbl 1459.05337
Foley, Angèle M.; Kazdan, Joshua; Kröll, Larissa; Martínez Alberga, Sofía; Melnyk, Oleksii; Tenenbaum, Alexander
2021
Upper bounds on the $$k$$-tuple (Roman) domination number of a graph. Zbl 1459.05239
2021
Ramsey and Gallai-Ramsey numbers for two classes of unicyclic graphs. Zbl 1459.05205
Wang, Zhao; Mao, Yaping; Magnant, Colton; Zou, Jinyu
2021
On $$\ell$$-distance-balanced product graphs. Zbl 1459.05063
Jerebic, Janja; Klavžar, Sandi; Rus, Gregor
2021
Anti-Ramsey problems in complete bipartite graphs for $$t$$ edge-disjoint rainbow spanning trees. Zbl 1459.05200
Jia, Yuxing; Lu, Mei; Zhang, Yi
2021
Acyclic coloring of graphs with maximum degree 7. Zbl 1470.05063
Wang, Juan; Miao, Lianying; Song, Wenyao; Liu, Yunlong
2021
Fractional matchings, component-factors and edge-chromatic critical graphs. Zbl 1459.05263
Klopp, Antje; Steffen, Eckhard
2021
Admissible pinnacle orderings. Zbl 1469.05006
Rusu, Irena; Tenner, Bridget Eileen
2021
Egalitarian edge orderings of complete graphs. Zbl 1469.05148
Colbourn, Charles J.
2021
Strong subgraph connectivity of digraphs. Zbl 1470.05068
Sun, Yuefang; Gutin, Gregory
2021
Anti-Ramsey number of triangles in complete multipartite graphs. Zbl 1470.05113
Jin, Zemin; Zhong, Kangyun; Sun, Yuefang
2021
Sequential partitions of nonpure simplicial complexes. Zbl 1484.05212
Hachimori, Masahiro
2021
Hamiltonicity of a coprime graph. Zbl 1479.05312
Bani Mostafa A., M. H.; Ghorbani, Ebrahim
2021
Upper bounds on inclusive distance vertex irregularity strength. Zbl 1479.05079
Cichacz, Sylwia; Görlich, Agnieszka; Semaničová-Feňovčíková, Andrea
2021
Discrete geometry on colored point sets in the plane – a survey. Zbl 1459.05032
Kano, Mikio; Urrutia, Jorge
2021
Supereulerian graphs with constraints on the matching number and minimum degree. Zbl 1459.05147
Algefari, Mansour J.; Lai, Hong-Jian
2021
Graphs with diameter 2 and large total domination number. Zbl 1459.05236
Dubickas, Artūras
2021
Strict neighbor-distinguishing index of subcubic graphs. Zbl 1459.05078
Gu, Jing; Wang, Weifan; Wang, Yiqiao; Wang, Ying
2021
A size condition for diameter two orientable graphs. Zbl 1459.05061
Cochran, Garner; Czabarka, Éva; Dankelmann, Peter; Székely, László
2021
Wiener indices of maximal $$k$$-degenerate graphs. Zbl 1459.05048
Bickle, Allan; Che, Zhongyuan
2021
On Grundy and b-chromatic number of some families of graphs: a comparative study. Zbl 1459.05087
Masih, Zoya; Zaker, Manouchehr
2021
Rainbow triangles in arc-colored tournaments. Zbl 1469.05065
Li, Wei; Zhang, Shenggui; Bai, Yandong; Li, Ruonan
2021
Partitioning a graph into complementary subgraphs. Zbl 1469.05144
Nascimento, Julliano R.; Souza, Uéverton S.; Szwarcfiter, Jayme L.
2021
Number of dominating sets in cylindric square grid graphs. Zbl 1469.05079
Oh, Seungsang
2021
Extremal digraphs avoiding an orientation of the diamond. Zbl 1469.05089
Lyu, Zhenhua
2021
Exponential lower bound for Berge-Ramsey problems. Zbl 1469.05121
Pálvölgyi, Dömötör
2021
Generalized Turán number of even linear forests. Zbl 1469.05090
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# You are an accountant for the Virden Company, which has two items of long-term convertible debt...

You are an accountant for the Virden Company, which has two  items of long-term   convertible

debt on its balance sheet. The president of the company calls you into his office and says, “We are too leveraged. So, you remember that convertible debt we issued at the beginning of the year? Let’s figure out the value of the conversion feature and assign that to equity so that we can reduce the amount we report as debt. And I have also been thinking about that convertible debt we issued at par five years ago. If you remem- ber, each $1,000 bond is convertible into 25 shares. Now that our shares are trading at$70, obviously that is no longer debt. So let’s classify that debt as equity now.”

#### Required

From financial reporting and ethical perspectives, discuss the issues raised by this  situation.

format: plain


text:
# Vector (mathematics and physics) explained

In mathematics and physics, a vector is an element of a vector space.

For many specific vector spaces, the vectors have received specific names, which are listed below.

Historically, vectors were introduced in geometry and physics (typically in mechanics) before the formalization of the concept of vector space. Therefore, one often talks about vectors without specifying the vector space to which they belong. Specifically, in a Euclidean space, one considers spatial vectors, also called Euclidean vectors which are used to represent quantities that have both magnitude and direction, and may be added, subtracted and scaled (i.e. multiplied by a real number) for forming a vector space.[1]

## Vectors in Euclidean geometry

See main article: Euclidean vector.

In classical Euclidean geometry (i.e., synthetic geometry), vectors were introduced (during the 19th century) as equivalence classes under equipollence, of ordered pairs of points; two pairs and being equipollent if the points, in this order, form a parallelogram. Such an equivalence class is called a vector, more precisely, a Euclidean vector.[2] The equivalence class of is often denoted

\overrightarrow{AB}.

A Euclidean vector is thus an equivalence class of directed segments with the same magnitude (e.g., the length of the line segment) and same direction (e.g., the direction from to).[3] In physics, Euclidean vectors are used to represent physical quantities that have both magnitude and direction, but are not located at a specific place, in contrast to scalars, which have no direction.[4] For example, velocity, forces and acceleration are represented by vectors.

In modern geometry, Euclidean spaces are often defined from linear algebra. More precisely, a Euclidean space is defined as a set to which is associated an inner product space of finite dimension over the reals

\overrightarrow{E},

and a group action of the additive group of

\overrightarrow{E},

which is free and transitive (See Affine space for details of this construction). The elements of

\overrightarrow{E}

are called translations.

It has been proven that the two definitions of Euclidean spaces are equivalent, and that the equivalence classes under equipollence may be identified with translations.

Sometimes, Euclidean vectors are considered without reference to a Euclidean space. In this case, a Euclidean vector is an element of a normed vector space of finite dimension over the reals, or, typically, an element of

Rn

equipped with the dot product. This makes sense, as the addition in such a vector space acts freely and transitively on the vector space itself. That is,

Rn

is a Euclidean space, with itself as an associated vector space, and the dot product as an inner product.

The Euclidean space

Rn

is often presented as the Euclidean space of dimension . This is motivated by the fact that every Euclidean space of dimension is isomorphic to the Euclidean space

Rn.

More precisely, given such a Euclidean space, one may choose any point as an origin. By Gram–Schmidt process, one may also find an orthonormal basis of the associated vector space (a basis such that the inner product of two basis vectors is 0 if they are different and 1 if they are equal). This defines Cartesian coordinates of any point of the space, as the coordinates on this basis of the vector

\overrightarrow{OP}.

These choices define an isomorphism of the given Euclidean space onto

Rn,

by mapping any point to the -tuple of its Cartesian coordinates, and every vector to its coordinate vector.

## Specific vectors in a vector space

• Zero vector (sometimes also called null vector and denoted by

0

[5]), the additive identity in a vector space. In a normed vector space, it is the unique vector of norm zero. In a Euclidean vector space, it is the unique vector of length zero.[6]

## Vectors in specific vector spaces

• Column vector, a matrix with only one column. The column vectors with a fixed number of rows form a vector space.
• Row vector, a matrix with only one row. The row vectors with a fixed number of columns form a vector space.
• Coordinate vector, the -tuple of the coordinates of a vector on a basis of elements. For a vector space over a field, these -tuples form the vector space

Fn

(where the operation are pointwise addition and scalar multiplication).
• Displacement vector, a vector that specifies the change in position of a point relative to a previous position. Displacement vectors belong to the vector space of translations.
• Position vector of a point, the displacement vector from a reference point (called the origin) to the point. A position vector represents the position of a point in a Euclidean space or an affine space.
• Velocity vector, the derivative, with respect to time, of the position vector. It does not depend of the choice of the origin, and, thus belongs to the vector space of translations.
• Pseudovector, also called axial vector
• Covector, an element of the dual of a vector space. In an inner product space, the inner product defines an isomorphism between the space and its dual, which may make difficult to distinguish a covector from a vector. The distinction becomes apparent when one changes coordinates (non-orthogonally).
• Tangent vector, an element of the tangent space of a curve, a surface or, more generally, a differential manifold at a given point (these tangent spaces are naturally endowed with a structure of vector space)
• Normal vector or simply normal, in a Euclidean space or, more generally, in an inner product space, a vector that is perpendicular to a tangent space at a point.
• Gradient, the coordinates vector of the partial derivatives of a function of several real variables. In a Euclidean space the gradient gives the magnitude and direction of maximum increase of a scalar field. The gradient is a covector that is normal to a level curve.
• Four-vector, in the theory of relativity, a vector in a four-dimensional real vector space called Minkowski space

## Tuples that are not really vectors

The set

Rn

of tuples of real numbers has a natural structure of vector space defined by component-wise addition and scalar multiplication. When such tuples are used for representing some data, it is common to call them vectors, even if the vector addition does not mean anything for these data, which may make the terminology confusing. Similarly, some physical phenomena involve a direction and a magnitude. They are often represented by vectors, even if operations of vector spaces do not apply to them.

## Vectors in algebras

Every algebra over a field is a vector space, but elements of an algebra are generally not called vectors. However, in some cases, they are called vectors, mainly due to historical reasons.

• Vector quaternion, a quaternion with a zero real part
• Multivector or -vector, an element of the exterior algebra of a vector space.
• Spinors, also called spin vectors, have been introduced for extending the notion of rotation vector. In fact, rotation vectors represent well rotations locally, but not globally, because a closed loop in the space of rotation vectors may induce a curve in the space of rotations that is not a loop. Also, the manifold of rotation vectors is orientable, while the manifold of rotations is not. Spinors are elements of a vector subspace of some Clifford algebra.
• Witt vector, an infinite sequence of elements of a commutative ring, which belongs to an algebra over this ring, and has been introduced for handling carry propagation in the operations on p-adic numbers.

### Vector fields

A vector field is a vector-valued function that, generally, has a domain of the same dimension (as a manifold) as its codomain,

### Miscellaneous

• Ricci calculus
• Vector Analysis, a textbook on vector calculus by Wilson, first published in 1901, which did much to standardize the notation and vocabulary of three-dimensional linear algebra and vector calculus
• Vector bundle, a topological construction that makes precise the idea of a family of vector spaces parameterized by another space
• Vector calculus, a branch of mathematics concerned with differentiation and integration of vector fields
• Vector differential, or del, a vector differential operator represented by the nabla symbol

\nabla

• Vector Laplacian, the vector Laplace operator, denoted by

\nabla2

, is a differential operator defined over a vector field
• Vector notation, common notation used when working with vectors
• Vector operator, a type of differential operator used in vector calculus
• Vector product, or cross product, an operation on two vectors in a three-dimensional Euclidean space, producing a third three-dimensional Euclidean vector
• Vector projection, also known as vector resolute or vector component, a linear mapping producing a vector parallel to a second vector
• Vector-valued function, a function that has a vector space as a codomain
• Vectorization (mathematics), a linear transformation that converts a matrix into a column vector
• Vector autoregression, an econometric model used to capture the evolution and the interdependencies between multiple time series
• Vector boson, a boson with the spin quantum number equal to 1
• Vector measure, a function defined on a family of sets and taking vector values satisfying certain properties
• Vector meson, a meson with total spin 1 and odd parity
• Vector quantization, a quantization technique used in signal processing
• Vector soliton, a solitary wave with multiple components coupled together that maintains its shape during propagation
• Vector synthesis, a type of audio synthesis

## Notes and References

1. Web site: vector Definition & Facts. 2020-08-19. Encyclopedia Britannica. en.
2. In some old texts, the pair is called a bound vector, and its equivalence class is called a free vector.
3. Web site: 2013-11-07. 1.1: Vectors. 2020-08-19. Mathematics LibreTexts. en.
4. Web site: Vectors. 2020-08-19. www.mathsisfun.com.
5. Web site: 2020-03-01. Compendium of Mathematical Symbols. 2020-08-19. Math Vault. en-US.
6. Web site: Weisstein. Eric W.. Vector. 2020-08-19. mathworld.wolfram.com. en.


text:
# Energy stored in an electric field

I know the mathematical proof that $$U=\frac{\epsilon_0}{2}\int\vec{E}^2dv$$ is the energy stored in a particular volume in space due to an electric field, but I don't get what it actually means. I lack the physical intuition to this result. For example, if I want to calculate what work needs to be done in order to assemble a charged sphere with radius $$R$$, why is it required to integrate over the entire space from radius $$r=\infty$$ to radius $$r=R$$? Every insight will help, it is just a pure mathematical result for me now and I'm not even sure how to properly use it.

One needs to perform work in order to construct a charge distribution. Doing work is an energy transfer and this energy is stored in the form of the electric field.

An easy example is a parallel plate capacitor. Its capacitance is $$C=\frac{\epsilon_0A}{d},$$ whereas the energy built by moving the charge from one plate to the other is $$U=\frac{Q^2}{2C}=\frac{CV^2}{2}= \frac{\epsilon_0A}{d}\frac{E^2d^2}{2}=\frac{\epsilon_0E^2}{2}Ad,$$ i.e. exactly what we call the energy of the field in volume $$Ad$$.

• yeah but it’s not a good example since one must neglect fringing in obtaining this expression, else there are charges at $\infty$ which ruin everything. Apr 10, 2020 at 13:44
• @ZeroTheHero It is a pedagogical example: it illustrates the phenomenon/concept using familiar concepts and minimal math complexity. Apr 10, 2020 at 15:07

Suppose you have a collection of positive charges. Now, you can intuitively feel that the closer they are, more is the electrostatic potential energy stored because they would want to fly apart. Suppose this collection is in the form of uniformly charged sphere, containing very tiny positive charges spread uniformly throughout the volume of the sphere with charge density $$ρ$$.

Now, the electrostatic potential energy is somewhat related to the electric field generated by this sphere in the space around and within it. In case of a spring which is compressed, we say that the potential energy is stored inside the spring as elastic energy. But, in case of electrostatic energy, we see that the effect is not local anymore and we may say that energy is stored inside the $$field$$ which is present everywhere.

To support this viewpoint, when you perform the integral to calculate electrostatic energy for the charged sphere ( integrating from $$r=0$$ to $$r=∞$$ ), you get the result as $$U=\frac35kQ^2/r$$ As the radius of sphere decreases, potential energy increases and so does the electric field created by the configuration increases, and hence the relation involves electric field.

I hope it makes things a bit more intuitive.


text:
# Math Help - Union of subspaces is a subspace iff..

1. ## Union of subspaces is a subspace iff..

Hi everyone,

I made this post on another site thinking my proof was correct:

However, I have been told that it isn't correct. I got some help, but I still don't understand why it isn't. Can someone please help me understand why the above proof isn't valid?

Thank-you.

P.S. How can I include LaTeX code in my post?

2. ## Re: Union of subspaces is a subspace iff..

Originally Posted by BryanUrizar
P.S. How can I include LaTeX code in my post?

[TEX]\exists t\in W_1\setminus W_2~\&~ \exists s\in W_2\setminus W_1[/TEX] gives $\exists t\in W_1\setminus W_2~\&~ \exists s\in W_2\setminus W_1$
If you click on the “go advanced tab” you should see $\boxed{\Sigma}$ on the tool-bar. That gives the [TEX]..[/TEX] wrap. Your LaTeX code goes between them.

For proof by contradiction, make the above assumption.

If $W_1\cup W_2$is a subspace, then $s+t\in W_1\cup W_2~.$

Now what?

3. ## Re: Union of subspaces is a subspace iff..

Originally Posted by Plato

[TEX]\exists t\in W_1\setminnus W_2~\&~ \exists s\in W_2\setminnus W_1[/TEX] gives $\exists t\in W_1\setminnus W_2~\&~ \exists s\in W_2\setminnus W_1$
If you click on the “go advanced tab” you should see $\boxed{\Sigma}$ on the tool-bar. That gives the [TEX]..[/TEX] wrap. Your LaTeX code goes between them.

For proof by contradiction, make the above assumption.

If $W_1\cup W_2$is a subspace, then $s+t\in W_1\cup W_2~.$

Now what?
I know how to prove it by contradiction, but wanted to know specifically what was wrong with my proof. I finally understand why it isn't correct. The problem is that I haven't shown one definite conclusion for any $x$. As it stands, I have only shown that for any $x$ and any $y \in W_1, W_2$ respectively, that either $x \in W_2$ or $y \in W_1$ which isn't enough to conclude $W_1 \subset W_2$ or $W_2 \subset W_1$. For a direct proof I need to assume $W_2 \not \subset W_1$ and choose $y \in W_2-W_1$ and proceed in a similar fashion.


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## red oxide beneficiation process machinery manufacturers in ...

red iron oxide process + hematite from magnetite process + equipment. A lot of people are talking about red iron oxide process + hematite ... » Magnetite ore beneficiation process; ... Originally an original equipment manufacturer ...

## The transformation of magnetite to hematite and its ...

Iron oxide-quartz rocks from paleoproterozoic Iron Formations in southern Brazil preserve microstructures that allow a good insight into the evolution of the deformation mechanisms and fabrics during the transition from a load-bearing framework (magnetite) to an interconnected weak phase (hematite).

## Iron Oxide Pigments - 911 Metallurgist

Jan 04, 2021 · Black oxides can also be produced from red and vice versa. Hydrogen reduction of red hematite can be used to make black magnetite; conversely, calcination of magnetite will yield red iron oxide. Although brown oxide can be made directly, the more usual procedure is to blend a mixture of red, yellow, and black.

## Natural Red Iron Oxide - Applications & Uses | African ...

A logical extension from ceramic pigments to glazes is natural with hematite/red iron oxide. When combined with silica, 10% by weight of red iron oxide is more than sufficient to produce a lustrous red glaze after firing in excess of 1000 °C - with the hematite-silica combination glaze pigment possessing enhanced levels of stability over pure hematite or silica.

## Quantitative analysis of iron oxides using Fourier ...

Sep 01, 2008 · The intensity of the iron oxide peak (magnetite 570 cm −1, hematite-540 cm −1 and maghemite 630 cm −1) is represented by I and I o represents, the intensity of the 3640 cm −1 peak of CaO. This calibration curve was used to quantify the amount of iron oxides present in the field samples collected from the secondary side of CPSES.

## Beneficiation Process Of Red Oxide

red iron oxide process hematite from magnetite process ... red iron oxide process hematite from magnetite process equipment Description : Hematite ... Iron ore processing equipment and beneficiation process – Topix. Read more

## Recycling of iron from red mud by magnetic separation ...

A recycling process of iron from red mud by magnetic separation is proposed, in which iron oxide (Fe 2 O 3) in red mud transforms to magnetite (Fe 3 O 4) by anaerobic co-roasting with pyrite (FeS 2).TG-DTA and XRD results showed that hematite was reduced to magnetite by the thermo-decomposed compounds (iron monosulfide, pyrrhotite and element sulfur) of pyrite.

## Iron Oxide Red - All You Need to Know | African Pegmatite

Mar 10, 2021 · Hematite is a naturally occurring ore of iron, the most common ore of iron and the most used source of iron. It is often sold as red iron oxide. Red iron oxide has many uses from pigments and dyes through to ceramics. Its resistance to heat and antipruritic properties mean that it is often used for casings and coverings.

## Surface Treatment of Industrial-Grade Magnetite Particles ...

Magnetite (Fe3O4), or black iron oxide, is a standard ferrite with a cubic inverse spinel structure [1]. It has unique magnetic properties due to the transfer of electrons between Fe2+ and Fe3+ ions in the octahedral sites. Among the three most essential iron oxides used as pigments (red, yellow, and black), red iron oxide Fe2O3 (hematite ...

## EP1951623A2 - High purity magnetite formation process and ...

A process of producing magnetite with a high purity of greater than 90% magnetite, more typically greater than 98% magnetite, by reducing powdered hematite into magnetite under maximum temperatures of about 700 to 1300°C against a counter-current of or concurrent with methane or natural gas in a heating device. The amount of methane used to reduce the hematite may be about 0.18 and

## Hoover Color Frequently Asked Questions - Cathay ...

Goethite (Yellow Iron Oxide) Hematite (Red Iron Oxide) Magnetite (Black Iron Oxide) These minerals can be produced naturally by geologic activities or can be synthetically produced in chemical reactions. The color shade of the iron oxide is determined by the size of the individual mineral crystal.

## Evaluation of Maghemite-Rich Iron Oxide Composite Prepared ...

May 07, 2019 · The capacity and effectiveness of a composite magnetic iron oxide, consisting mainly of maghemite $$(\\gamma {\\text{Fe}}_{2} {\\text{O}}_{3} )$$ (γFe2O3), prepared by oxidation of synthetic magnetite at 350°C have been evaluated. X-ray diffraction analysis of the synthesized product indicated an iron oxide with 44%, 33%, and 23% of maghemite, hematite (Fe2O3), and magnetite (Fe3O4 ...

## beneficiation of iron ore process why with magnetite ...

Oct 26, 2019 Iron ore mining often focuses on looking for hematite deposits known as red iron ore, and magnetite, as they have naturally weak magnetic fields that aid in their purification. Hematite, however, responds better to the flotation process in iron ore beneficiation than magnetite, so it

## What is the process for polishing hematite? How does it go ...

“Almost any metal or hard plastic item can be buffed to a high polish or a satin sheen with the correct buffing wheel and the proper polishing compound. This includes copper pots, doorknobs, golf clubs and switch plates, to name a few common items...

## Difference Between Magnetite and Hematite | Compare the ...

Jan 28, 2012 · Magnetite is an iron oxide with the chemical formula Fe 3 O 4 whereas hematite is an iron oxide with the chemical formula Fe 2 O 3. Magnetite iron is in +2 and +3 oxidation states whereas, in hematite, it is only in +3 oxidation state. This is the key difference between magnetite and hematite. Moreover, magnetite has higher iron content than ...

## Hematite, the Shape-shifting Mineral Used From the Stone ...

Nov 12, 2020 · As the foremost source of iron on the planet, hematite has shaped civilizations for thousands of years. The mineral was first used as the main pigment in cave paintings often having a reddish hue, a signature trait of hematite. Later, it was – and still is – the primary source of iron, shifting the world out of the Stone Age and into the Iron Age.

## Hematite | Fe2O3 - PubChem

Iron ore is a mineral substance which, when heated in the presence of a reductant, will yield metallic iron (Fe). It almost always consists of iron oxides, the primary forms of which are magnetite (Fe3O4) and hematite (Fe2O3). Iron ore is the source of primary iron for the world's iron and steel industries.

## Infinite Flame Stone and Crystal Properties - Iron stones ...

Hematite FE2 O3 Iron with impurities Hardness: 5-6 Streak: red to red-brown Color: black, metallic grey, can oxidize to red AKA: iron oxide, haematite, Hematite is a form of iron, as anyone can tell who has lifted a piece of it. It rarely occures as pure iron, which is why it is not magnetized.

## Iron Oxide Colloidal Suspension Stabilization by Polymeric ...

(from the feed stream and/or equipment corrosion) which contributes to scale formation and deposition.1 There are 16 known iron oxides (oxides, hydroxides or oxide hydroxides); predominantly iron is present in the +3 oxidation state.2 Several iron oxides including hematite, magnetite and

## beneficiation of iron ore process why with magnetite ...

Oct 26, 2019 Iron ore mining often focuses on looking for hematite deposits known as red iron ore, and magnetite, as they have naturally weak magnetic fields that aid in their purification. Hematite, however, responds better to the flotation process in iron ore beneficiation than magnetite, so it

## mineral processing of iron ore hematite

The haematite's another name is red iron ore,and its chemical structural formula is Fe203. it's a weak magnetic iron ore and the floatability is better . ... Effectively process magnetite & hematite from iron ore. Our magnetite processing equipment will get the job done. ... Most iron ores mined today comprise the iron oxide minerals hematite ...

## Hematite | Fe2O3 - PubChem

Iron ore is a mineral substance which, when heated in the presence of a reductant, will yield metallic iron (Fe). It almost always consists of iron oxides, the primary forms of which are magnetite (Fe3O4) and hematite (Fe2O3). Iron ore is the source of primary iron for the world's iron and steel industries.

## Magnetic analysis of commercial hematite, magnetite, and ...

Dec 19, 2017 · Among the iron oxide minerals, hematite (α-Fe 2 O 3) and magnetite (Fe 3 O 4) are the most common. In this work, different commercial hematite powders, normally used as Fe precursor in laboratory synthesis of Fe-containing oxides, were characterized using X-ray diffractometry (XRD), scanning electron microscopy (SEM), and vibrating sample ...

## Photoacoustic, EPR and electrical conductivity ...

May 01, 2002 · In this paper, we disclose the preparation of three iron oxide pigments in our laboratory, and they are red-brown hematite (α-Fe 2 O 3), yellow goethite (α-FeOOH) and black magnetite (Fe 3 O 4) obtained from the oxidation process of iron(II) sulphate in aqueous solution with air oxygen. These materials have been characterized by X-ray ...

## US6302952B1 - Process for the production of black iron ...

A process for the production of a black iron oxide pigment from Fe 2 O 3 hematite iron oxide which may be an impure oxide produced by the “Ruthner” process or by other means, or which may be a mineral hematite. The process may be used to upgrade poor quality synthetic or natural hematite, or to improve their performance as red pigments, by conversion to black pigment with subsequent ...

## hematite iron oxide red grinding equipment manufacturer ...

hematite iron oxide red grinding equipment manufacturer. sovema s series lead oxide mill. sovema s series lead oxide mill Grinding Mill China. Series Lead Oxide Mill from Sovema Manufacturer of Milling,The S series represents the evolution of the well-known sovema oxide mill (more than 300 units installed to date) the new series has been developed primarily to meet the battery industry's ...

## Production of Red Iron Oxide (Mining of Mineral Ore with ...

Crude Iron is produced from iron ore. Iron ore is extracted from rock and minerals. These ores are usually wealthy in iron oxide (Fe2O3) and differ in color from grey, deep yellow, rusty red to deep purple. Iron is found within the variety of goethite (FeO (OH)), siderite (FeCO3), limonite (FeO (OH).n (H2O)), hematite (Fe2O3) and magnetite (Fe3O4).

## Foundries, Castings & Molds - QMAG

Color varies in ores from a dark grey to black, or from a pale red to a darker, rusty red. Though the process for extracting the metal from the ore varies slightly, both red iron oxide – hematite, and black iron oxide – magnetite, are great sources for natural ore and play a vital role in supplying the world with steel-making iron.

## HOW STEAM SYSTEM CHEMISTRY AND CONTROL AFFECT

oxide "hematite" Ferric oxide, hematite, "red rust" , it has a lot of names but there is no argument that it is undesirable. It is a loosely adhering oxide layer which has poor heat transfer capabilities. It also tends to deposit elsewhere in the system. It can deposit in the coil, in steam traps

## iron ore beneficiation process of hematite - Products

Hematite Iron Ore Beneficiation Process. Hematite Iron Ore Beneficiation Process Wet High Intensity Magnetic Separation. 4 Jun 2017 . The magnetite has been oxidized to hematite, and the iron silicates and . Wet high-intensity magnetic (WHIM) separation, either alone or in . these beneficiation processes, all three are being evaluated on

## Hematite: A primary ore of iron and a pigment mineral

Uses of Hematite (Iron Ore) Hematite is the world’s most important ore of iron. Although magnetite contains a higher percentage of iron and is easier to process, hematite is the leading ore because it is more abundant and present in deposits in many parts of the world. Hematite is mined in some of the largest mines in the world.

## iron processing - Ores | Britannica

The most widely distributed iron-bearing minerals are oxides, and iron ores consist mainly of hematite (Fe 2 O 3), which is red; magnetite (Fe 3 O 4), which is black; limonite or bog-iron ore (2Fe 2 O 3 ·3H 2 O), which is brown; and siderite (FeCO 3), which is pale brown. Hematite and magnetite are by far the most common types of ore.

## Opportunities in Mining of Mineral Ore with Processing and ...

Opportunities in Mining of Mineral Ore with Processing and Beneficiation for Manufacturing of Red Iron Oxide. Iron ore is a very important industrial source, is an iron oxide ore, a mineral aggregate containing iron elements or iron compounds which will be economically utilized, and there are many varieties of ore. Among them, the iron smelting products mainly include Magnetite, siderite, and ...


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# Force needed to pull a block up an incline.

by garfiegrl
Tags: block, force, incline, pull
 P: 14 1. The problem statement, all variables and given/known data Calculate the force needed to pull a mass of 20 kg at a uniform slow speed up a plane inclined at an angle of 30 with the horizontal if the coefficient of kinetic friction is 0.20. 2. Relevant equations WN= w cos $$\vartheta$$ WT= w sin $$\vartheta$$ $$\mu$$s= tan$$\vartheta$$ 3. The attempt at a solution I don't even know how to get started.


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Question

# Let U,V be subspaces of Rn. Suppose that U⊥V. Prove that {u,v} is linearly independent for any nonzero vectors u∈U,v∈V.

Vectors and spaces

Let U,V be subspaces of Rn. Suppose that $$U\bot V.$$ Prove that $$\{u,v\}$$ is linearly independent for any nonzero vectors $$u\in U,\ v\in V.$$

2021-03-03

Given that $$U\bot V$$ that is $$(u, v) = 0$$ for any $$u\in U$$ and $$v\in V$$. Let
$$au+bv=0$$
$$\Rightarrow(au+bv,u)=0 \Rightarrow a(u,v)+b(u,v)=0 \Rightarrow a(u,u)=0$$ (since u is a non zero vector) $$\Rightarrow a=0$$
Again
$$au+bv=0 \Rightarrow (au+bv,v)=0 \Rightarrow a(u,v)+b(u,v)=0 \Rightarrow b(v,v)=0$$ (since v is a non zero vector) $$\Rightarrow b=0$$
Therefore $$\displaystyle{a}{u}+{b}{v}={0}\to{a}={b}={0}$$. Hence $$\{u,v\}$$ is linearly independent.


text: References of "Journal of Physics: Condensed Matter"      in Complete repository Arts & humanities   Archaeology   Art & art history   Classical & oriental studies   History   Languages & linguistics   Literature   Performing arts   Philosophy & ethics   Religion & theology   Multidisciplinary, general & others Business & economic sciences   Accounting & auditing   Production, distribution & supply chain management   Finance   General management & organizational theory   Human resources management   Management information systems   Marketing   Strategy & innovation   Quantitative methods in economics & management   General economics & history of economic thought   International economics   Macroeconomics & monetary economics   Microeconomics   Economic systems & public economics   Social economics   Special economic topics (health, labor, transportation…)   Multidisciplinary, general & others Engineering, computing & technology   Aerospace & aeronautics engineering   Architecture   Chemical engineering   Civil engineering   Computer science   Electrical & electronics engineering   Energy   Geological, petroleum & mining engineering   Materials science & engineering   Mechanical engineering   Multidisciplinary, general & others Human health sciences   Alternative medicine   Anesthesia & intensive care   Cardiovascular & respiratory systems   Dentistry & oral medicine   Dermatology   Endocrinology, metabolism & nutrition   Forensic medicine   Gastroenterology & hepatology   General & internal medicine   Geriatrics   Hematology   Immunology & infectious disease   Laboratory medicine & medical technology   Neurology   Oncology   Ophthalmology   Orthopedics, rehabilitation & sports medicine   Otolaryngology   Pediatrics   Pharmacy, pharmacology & toxicology   Psychiatry   Public health, health care sciences & services   Radiology, nuclear medicine & imaging   Reproductive medicine (gynecology, andrology, obstetrics)   Rheumatology   Surgery   Urology & nephrology   Multidisciplinary, general & others Law, criminology & political science   Civil law   Criminal law & procedure   Criminology   Economic & commercial law   European & international law   Judicial law   Metalaw, Roman law, history of law & comparative law   Political science, public administration & international relations   Public law   Social law   Tax law   Multidisciplinary, general & others Life sciences   Agriculture & agronomy   Anatomy (cytology, histology, embryology...) & physiology   Animal production & animal husbandry   Aquatic sciences & oceanology   Biochemistry, biophysics & molecular biology   Biotechnology   Entomology & pest control   Environmental sciences & ecology   Food science   Genetics & genetic processes   Microbiology   Phytobiology (plant sciences, forestry, mycology...)   Veterinary medicine & animal health   Zoology   Multidisciplinary, general & others Physical, chemical, mathematical & earth Sciences   Chemistry   Earth sciences & physical geography   Mathematics   Physics   Space science, astronomy & astrophysics   Multidisciplinary, general & others Social & behavioral sciences, psychology   Animal psychology, ethology & psychobiology   Anthropology   Communication & mass media   Education & instruction   Human geography & demography   Library & information sciences   Neurosciences & behavior   Regional & inter-regional studies   Social work & social policy   Sociology & social sciences   Social, industrial & organizational psychology   Theoretical & cognitive psychology   Treatment & clinical psychology   Multidisciplinary, general & others     Showing results 1 to 12 of 12 1 Optical conductivity of triple point fermionsHabibi, Alireza ; Farajollahpour, Tohid; Jafari, S Ain Journal of Physics: Condensed Matter (2021), 33As a low-energy effective theory on non-symmorphic lattices, we consider a generic triple point fermion Hamiltonian, which is parameterized by an angular parameter λ. We find strong λ dependence in both ... [more ▼]As a low-energy effective theory on non-symmorphic lattices, we consider a generic triple point fermion Hamiltonian, which is parameterized by an angular parameter λ. We find strong λ dependence in both Drude and interband optical absorption of these systems. The deviation of the T2 coefficient of the Drude weight from Dirac/Weyl fermions can be used as a quick way to optically distinguish the triple point degeneracies from the Dirac/Weyl degeneracies. At the particular λ = π/6 point, we find that the 'helicity' reversal optical transition matrix element is identically zero. Nevertheless, deviating from this point, the helicity reversal emerges as an absorption channel. [less ▲]Detailed reference viewed: 10 (0 UL) Ultrafast scanning calorimetry of newly developed Au-Ga bulk metallic glassesNeuber, Nico; Frey, Maximilian; Gross, Oliver et alin Journal of Physics: Condensed Matter (2020), 32(32), 324001The isothermal crystallization times and critical cooling rates of the liquid phase are determined for the two bulk metallic glass forming alloys Au49Ag5.5Pd2.3Cu26.9Si16.3 and Au51.6Ag5.8Pd2.4Cu20.2Ga6 ... [more ▼]The isothermal crystallization times and critical cooling rates of the liquid phase are determined for the two bulk metallic glass forming alloys Au49Ag5.5Pd2.3Cu26.9Si16.3 and Au51.6Ag5.8Pd2.4Cu20.2Ga6.7Si13.3 by using fast differential scanning calorimetry, covering the whole timescale of the crystallization event of the metallic melt. In the case of Au49Ag5.5Pd2.3Cu26.9Si16.3, a typical crystallization nose was observed, whereas for the Au51.6Ag5.8Pd2.4Cu20.2Ga6.7Si13.3, a more complex crystallization behavior with two distinct crystallization noses was found. Even for the complex crystallization behavior of the Au51.6Ag5.8Pd2.4Cu20.2Ga6.7Si13.3 alloy it is shown that the minimal isothermal nose time does allow for a quantification of the macroscopic critical thickness. It is discussed in contrast to the critical cooling rate, which is found to allow less exact calculations of the critical thickness and thus does not correlate well with the critical cooling rate from macroscopic experiments. Additionally the crystallization data of Au49Ag5.5Pd2.3Cu26.9Si16.3 was modeled using classical nucleation theory with the Johnson–Mehl–Avrami–Kolmogorov (JMAK) equation, enabling a determination of the interfacial energy. [less ▲]Detailed reference viewed: 49 (0 UL) Ultrafast scanning calorimetry of newly developed Au-Ga bulk metallic glassesBaller, Jörg ; Neuber, Nico; Frey, Maximilian et alin Journal of Physics: Condensed Matter (2020), 32(32), 324001The isothermal crystallization times and critical cooling rates of the liquid phase are determined for the two bulk metallic glass forming alloys Au49Ag5.5Pd2.3Cu26.9Si16.3 and Au51.6Ag5.8Pd2.4Cu20.2Ga6 ... [more ▼]The isothermal crystallization times and critical cooling rates of the liquid phase are determined for the two bulk metallic glass forming alloys Au49Ag5.5Pd2.3Cu26.9Si16.3 and Au51.6Ag5.8Pd2.4Cu20.2Ga6.7Si13.3 by using fast differential scanning calorimetry, covering the whole timescale of the crystallization event of the metallic melt. In the case of Au49Ag5.5Pd2.3Cu26.9Si16.3, a typical crystallization nose was observed, whereas for the Au51.6Ag5.8Pd2.4Cu20.2Ga6.7Si13.3, a more complex crystallization behavior with two distinct crystallization noses was found. Even for the complex crystallization behavior of the Au51.6Ag5.8Pd2.4Cu20.2Ga6.7Si13.3 alloy it is shown that the minimal isothermal nose time ${{\tau}}_{x}^{{\ast}}$ does allow for a quantification of the macroscopic critical thickness. It is discussed in contrast to the critical cooling rate, which is found to allow less exact calculations of the critical thickness and thus does not correlate well with the critical cooling rate from macroscopic experiments. Additionally the crystallization data of Au49Ag5.5Pd2.3Cu26.9Si16.3 was modeled using classical nucleation theory with the Johnson–Mehl–Avrami–Kolmogorov (JMAK) equation, enabling a determination of the interfacial energy. [less ▲]Detailed reference viewed: 100 (11 UL) Defect-induced Dzyaloshinskii–Moriya interaction in a nanocrystalline two-phase alloyQuan, Yifan; Kohlbrecher, Joachim; Hautle, Patrick et alin Journal of Physics: Condensed Matter (2020), 32Detailed reference viewed: 92 (1 UL) Optical studies of ferroelectric and ferroelastic domain wallsNataf, G. F.; Guennou, Mael in Journal of Physics: Condensed Matter (2020), 32(18), 183001Recent studies carried out with atomic force microscopy or high-resolution transmission electron microscopy reveal that ferroic domain walls can exhibit different physical properties than the bulk of the ... [more ▼]Recent studies carried out with atomic force microscopy or high-resolution transmission electron microscopy reveal that ferroic domain walls can exhibit different physical properties than the bulk of the domains, such as enhanced conductivity in insulators, or polar properties in non-polar materials. In this review we show that optical techniques, in spite of the diffraction limit, also provide key insights into the structure and physical properties of ferroelectric and ferroelastic domain walls. We give an overview of the uses, specificities and limits of these techniques, and emphasize the properties of the domain walls that they can probe. We then highlight some open questions of the physics of domain walls that could benefit from their use. [less ▲]Detailed reference viewed: 74 (1 UL) Magnetic ordering of the martensite phase in Ni-Co-Mn-Sn-based ferromagnetic shape memory alloysSarkar, Sudip Kumar; Ahlawat, Sarita; Kaushik, S. D. et alin Journal of Physics: Condensed Matter (2019)The magnetic state of low temperature martensite phase in Co-substituted Ni-Mn-Sn-based ferromagnetic shape memory alloys (FSMAs) has been investigated, in view of numerous conflicting reports of ... [more ▼]The magnetic state of low temperature martensite phase in Co-substituted Ni-Mn-Sn-based ferromagnetic shape memory alloys (FSMAs) has been investigated, in view of numerous conflicting reports of occurrences of spin glass (SG), superparamagnetism (SPM) or long range anti- ferromagnetic (AF) ordering. Combination of dc magnetization, ac susceptibility and small angle neutron scattering (SANS) studies provide a clear evidence for AF order in martensitic phase of Ni45Co5Mn38Sn12 alloy and rule out SPM and SG orders. Identical studies on another alloy of close composition of Ni44Co6Mn40Sn10 point to presence of SG order in martensitic phase and absence of SPM behavior, contrary to earlier report. SANS results do show presence of nanometre-sized clusters but they are found to grow in size from 3 nm at 30 K to 11 nm at 300 K, and do not correlate with magnetism in these alloys.


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Date Bugs

1000ms    10000K
32    13
Beginner

Description

There are rumors that there are a lot of computers having a problem with the year 2000. As they use only two digits to represent the year, the date will suddenly turn from 1999 to 1900. In fact, there are also many other, similar problems. On some systems, a 32-bit integer is used to store the number of seconds that have elapsed since a certain fixed date. In this way, when 232 seconds (about 136 Years) have elapsed, the date will jump back to whatever the fixed date is. Now, what can you do about all that mess? Imagine you have two computers C1 and C2 with two different bugs: One with the ordinary Y2K-Bug (i.e. switching to a1:=1900 instead of b1:=2000) and one switching to a2:=1904 instead of b2:=2040. Imagine that the C1 displays the year y1:=1941 and C2 the year y2:=2005. Then you know the following (assuming that there are no other bugs): the real year can't be 1941, since, then, both computers would show the (same) right date. If the year would be 2005, y1 would be 1905, so this is impossible, too. Looking only at C1, we know that the real year is one of the following: 1941, 2041, 2141, etc. We now can calculate what C2 would display in these years: 1941, 1905, 2005, etc. So in fact, it is possible that the actual year is 2141. To calculate all this manually is a lot of work. (And you don't really want to do it each time you forgot the actual year.) So, your task is to write a program which does the calculation for you: find the first possible real year, knowing what some other computers say (yi) and knowing their bugs (switching to ai instead of bi). Note that the year ai is definitely not after the year the computer was built. Since the actual year can't be before the year the computers were built, the year your program is looking for can't be before any ai.

Input

The input file contains several test cases, in which the actual year has to be calculated. The description of each case starts with a line containing an integer n (1≤n≤20), the number of computers. Then, there is one line containing three integers yi, ai, bi for each computer (0≤ai≤yi≤bi≤10000). yi is the year the computer displays, bi is the year in which the bug happens (i.e. the first year which can't be displayed by this computer) and ai is the year that the computer displays instead of bi. The input is terminated by a test case with n=0. It should not be processed.

Output

For each test case, output the line "Case #k:'', where k is the number of the situation. Then, output the line "The actual year is z.", where z is the smallest possible year (satisfying all computers and being greater or equal to u=maxi=1nai). If there is no such year less than 10000, output "Unknown bugs detected.". Output a blank line after each case.

Sample Input

2
1941 1900 2000
2005 1904 2040
2
1998 1900 2000
1999 1900 2000
0

Sample Output

Case #1:
The actual year is 2141.

Case #2:
Unknown bugs detected.

Source

Editor

on 2006-10-02 12:13:45


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# A classical mechanics problem by Shaim Khanusiya

Classical Mechanics Level pending

What will be the effect on length(24 hours) of a day, if snow on the poles of Earth melts and water comes at the equator ?

×


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# Two brothers fishing problem

by Anthony Lahmann   Last Updated October 10, 2019 05:20 AM

Two brothers A and B are fishing. Person A says he got 10 fish. Person B says he got 12 fish. However, person A says: You need to multiply any number B says by $$\frac{3}{4}$$ to get the real answer! Person B says: You need to add $$1$$ to any number A says to get the real answer! How many fish did A catch? How many fish did B catch?

Tags :

Just follow the instructions. $$12*\frac{3}{4}=9$$, so B caught 9 fish. $$10+1=11$$, so A caught 11 fish.

H Huang
April 15, 2019 02:34 AM

B said he got 12 fish, but according to As statement he got 9 fish.

A said he got 10 fish but according to Bs statement he got 11 fish.

A:11& B:9

Samar Imam Zaidi
April 15, 2019 02:39 AM

B says: $$A_{true}(x) = x + B_{true}(1)$$

A says: $$B_{true}(x) = x \cdot A_{true}\left(\frac{3}{4}\right)$$

$$\implies B_{true}(x) = x \cdot \left(\frac{3}{4} +B_{true}(1)\right)$$

Edit: As Anthony Lahmann mentioned in the comment, the first equation itself implies

$$B_{true}(1) = \frac{3}{4} + B_{true}(1)$$

user1952500
April 15, 2019 02:56 AM

## Related Questions

Updated July 28, 2017 01:20 AM

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Updated May 24, 2017 21:20 PM


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# Scripting FontForge¶

FontForge includes two interpreters so you can write scripts to modify fonts. One of these interpreters is python, one is a legacy language I came up with. FontForge may be configured with either or both of these. If configured with both then fontforge will make a guess at which to use based on the script file’s extension (“py” means use the python interpreter, “ff” or “pe” means use the old interpreter)

## Invoking scripts¶

If you start fontforge with a script on the command line it will not put up any windows and it will exit when the script is done. The script can be in a file, or just a string presented as an argument. You may need to specify which interpreter to use with the -lang argument.

$fontforge -script scriptfile.pe {arguments}$ fontforge -c "script-string" {arguments}
$fontforge -lang={ff|py} -c "script-string"  FontForge can also be used as an interpreter to which the shell will automatically pass scripts. If you a mark your script files as executable $ chmod +x scriptfile.pe

and begin each one with the line

#!/usr/local/bin/fontforge

(or wherever fontforge happens to reside on your system) then you can invoke the script just by typing

$scriptfile.pe {fontnames}  If you wish FontForge to read a script from stdin then you can use “-” as a “filename” for stdin. (If you build FontForge without X11 then fontforge will attempt to read a script file from stdin if none is given on the command line.) You can also start a script from within FontForge with File ‣ Execute Script, and you can use the Preference Dlg to define a set of frequently used scripts which can be invoked directly by menu. The scripting language provides access to much of the functionality found in the font view’s menus. It does not currently (and probably never will) provide access to everything. (If you find a lack let me know, I may put it in for you). It does not provide commands for building up a glyph out of splines, instead it allows you to do high level modifications to glyphs. If you set the environment variable FONTFORGE_VERBOSE (it doesn’t need a value, just needs to be set) then FontForge will print scripts to stdout as it executes them. You may set the environment variable FONTFORGE_LANGUAGE to either “py” (for python) or “ff” or “pe” (for native scripting) as another way to determne what interpreter to use. ## Scripting Language¶ Note This section covers FontForge’s native scripting functionality. For the Python scripting functionality, see here. The syntax is rather like a mixture of C and shell commands. Every file corresponds to a procedure. As in a shell script arguments passed to the file are identified as$1, $2, …$n. $0 is the file name itself.$argc gives the number of arguments. $argv[<expr>] provides array access to the arguments. Terms can be • A variable name (like “$1” or “i” or “@fontvar” or “_global”)

The scope of the variable depends on the initial character of its name.

• A ‘$’ signifies that it is a built-in variable. The user cannot create any new variables beginning with ‘$’. Some, but not all, of these may be assigned to.

• A ‘_’ signifies that the variable is global, it is always available. You can use these to store context across different script files (or to access data within nested script files).

• A ‘@’ signifies that the variable is associated with the font. Any two scripts looking at the same font will have access to the same variables.

• A variable which begins with a letter is a local variable. It is only meaningful within the current script file. Nested script files may have different variables with the same names.

• an integer expressed in decimal, hex or octal

• a unicode code point (which has a prefix of “0u” or “0U” and is followed by a string of hex digits. This is only used by the select command.

• a real number (in “C” locale format – “.” as the decimal point character)

• A string which may be enclosed in either double or single quotes. String tokens are limited to 256 bytes. “n” can be used to represent a newline character. (If you need longer strings use the concatenation operator).

• a procedure to call or file to invoke.

• an expression within parentheses

• a series of expressions (separated by commas) within brackets, as [1, 2, 3, 5, 8], used to create an array.

There are three different comments supported:

• Starting with a “#” character and proceeding to end of line

• Starting with “//” and proceeding to end of line

• Starting with “/” and proceeding to “/”

### Expressions¶

Expressions are similar to those in C, a few operators have been omitted, a few added from shell scripts. Operator precedence has been simplified slightly. So operators (and their precedences) are:

• unary operators (+, -, !, ~, ++ (prefix and postfix), –(prefix and postfix), () (procedure call), [] (array index), :h, :t, :r, :e

Most of these are as expected in C, the last four are borrowed from shell scripts and are applied to strings

• :h gives the head (directory) of a pathspec

• :t gives the tail (filename) of a pathspec

• :r gives the pathspec without the extension (if any)

• :e gives the extension

• *, /, % (binary multiplicative operators)

• +, - (binary arithmetic operators)

If the first operand of + is a string then + will be treated as concatenation rather than addition. If the second operand is a number it will be converted to a string (decimal representation) and then concatenated. If the first operand of + is an array then + will do array concatenation – if the second argment is also an array the two will be concatenated ([1,2] + [3,4] yields [1,2,3,4], while [1,2] + 3 yields [1,2,3]). Otherwise these are the normal arithmetric operations.

• ==, !=, >, <, >=, <= (comparison operators, may be applied to either two integers or two strings)

• &&, & (logical and, bitwise and. (logical and will do short circuit evaluation))

• ||, |, ^ (logical or, bitwise or, bitwise exclusive or (logical or will do short circuit evaluation))

• =, +=, -=, *=, /=, %= (assignment operators as in C. The += will act as concatenation if the first operand is a string.)

Note there is no comma operator, and no “?:” operator. The precedence of “and” and “or” has been simplified, as has that of the assignment operators.

Procedure calls may be applied either to a name token, or to a string. If the name or string is recognized as one of FontForge’s internal procedures it will be executed, otherwise it will be assumed to be a filename containing another fontforge script file, this file will be invoked (since filenames can contain characters not legal in name tokens it is important to allow general strings to specify filenames). If the procedure name does not contain a directory then it is assumed to be in the same directory as the current script file. As a convenience, it is not necessary to include the script’s extension if it is either “.ff” or “.pe”. At most 25 arguments can be passed to a procedure.

Arrays are passed by reference, strings and integers are passed by value.

Variables may be created by assigning a value to them (only with the “=”), so:

i=3

could be used to define “i” as a variable. Variables are limited in scope to the current file, they will not be inherited by called procedures.

A statement may be

• an expression

• if ( expression )
statements
{elseif ( expression )
statements}
[else
statements]
endif

• while ( expression )
statements
endloop

• foreach
statements
endloop

• break

• return [ expression ]

• shift

As with C, non-zero expressions are defined to be true.

A return statement may be followed by a return value (the expression) or a procedure may return nothing (void).

The shift statement is stolen from shell scripts and shifts all arguments down by one. (argument 0, the name of the script file, remains unchanged.

The foreach statement requires that there be a current font. It executes the statements once for each glyph in the selection. Within the statements only one glyph at a time will be selected. After execution the selection will be restored to what it was initially. (Caveat: Don’t reencode the font within a foreach statement).

Statements are terminated either by a new line (you can break up long lines with backslash newline) or a semicolon.

Trivial example:

i=0;   #semicolon is not needed here, but it's ok
while ( i<3 )
if ( i==1 /* pointless comment */ )
Print( "Got to one" )   // Another comment
endif
++i
endloop

FontForge maintains the concept of a “current font”– almost all commands refer only to the current font (and require that there be a font). If you start a script with File ‣ Execute Script, the font you were editing will be current, otherwise there will be no initial current font. The Open(), New() and Close() commands all change the current font. FontForge also maintains a list of all fonts that are currently open. This list is in no particular order. The list starts with $firstfont. Similarly when working with cid keyed fonts, FontForge works in the “current sub font”, and most commands refer to this font. The CIDChangeSubFont() command can alter that. ### Variables¶ All builtin variables begin with “$”, you may not create any variables that start with “$” yourself (though you may assign to (some) already existing ones)$0

the current script filename

$1$2
$n the first, second (…) argument to the script file$argc

the number of arguments passed to the script file (this will always be at least 1 as $0 is always present)$argv

allows you to access the array of all the arguments

$curfont the name of the filename in which the current font resides$firstfont

the name of the filename of the font which is first on the font list (Can be used by Open()), if there are no fonts loaded this returns an empty string. This can be used to determine if any font at all is loaded into fontforge.

$nextfont the name of the filename of the font which follows the current font on the list (or the empty string if the current font is the last one on the list)$fontchanged

returns 1 if the current font has changed, 0 if it has not changed since it was read in (or saved).

$fontname the name contained in the postscript FontName field$familyname

the name contained in the postscript FamilyName field

$fullname the name contained in the postscript FullName field$fondname

if set this name indicates what FOND the current font should be put in under Generate Mac Family.

$weight the name contained in the postscript Weight field the name contained in the postscript Notice field$filename

the name of the file containing the font.

$fontversion the string containing the font’s version$iscid

1 if the current font is a cid keyed font, 0 if not

$cidfontname returns the fontname of the top-level cid-keyed font (or the empty string if there is none) Can be used to detect if this is a cid keyed font. $cidfamilyname, $cidfullname,$cidweight, $cidcopyright similar to cidfontname$mmcount

returns 0 for non multiple master fonts, returns the number of instances in a multiple master font.

$italicangle the value of the postscript italic angle field$loadState

$privateState a bitmask of some errors in the PostScript Private dictionary (see fontforge.font.privateState for more info).$curcid

returns the fontname of the current font

$firstcid returns the fontname of the first font within this cid font$nextcid

returns the fontname of the next font within this cid font (or the empty string if the current sub-font is the last)

$macstyle returns the value of the macstyle field (a set of bits indicating whether the font is bold, italic, condensed, etc.)$bitmaps

returns an array containing all bitmap pixelsizes generated for this font. (If the font database contains greymaps then they will be indicated in the array as (<BitmapDepth><<16)|<PixelSize>)

$order returns an integer containing either 2 (for truetype fonts) or 3 (for postscript fonts). This indicates whether the font uses quadratic or cubic splines.$em

returns the number of em-units used by the font.

$ascent returns the ascent of the font$descent

returns the descent of the font.

$selection returns an array containing one entry for each glyph in the current font indicating whether that glyph is selected or not (0=>not, 1=>selected)$panose

returns an array containing the 10 panose values for the font.

$trace if this is set to one then FontForge will trace each procedure call.$version
returns a string containing the current version of fontforge. This

should look something like “20050817”.

$haspython returns 1 if python scripting is available, 0 if it is not. ### Preferences¶ • $<Preference Item> (for example $AutoHint) allows you to examine the value of that preference item (to set it use SetPref()) The following example will perform an action on all loaded fonts: file =$firstfont
while ( file != "" )
Open(file)
/* Do Stuff */
file = $nextfont endloop  The built in procedures are very similar to the menu items with the same names. Often the description here is sketchy, look at the menu item for more information. ### Built-in procedures in alphabetic order¶ ### Built-in procedures that act like the File Menu¶ ### File Manipulation¶ ### Built-in procedures that act like the Encoding Menu¶ ### Built-in procedures that act like the Metrics Menu¶ ### Multiple master routines¶ ### User Interaction¶ ### Unicode¶ ### Arrays¶ ### Miscellaneous¶ ## Examples¶ ### Example 1:¶ #Set the color of all selected glyphs to be yellow #designed to be run within an interactive fontforge session. foreach SetCharColor(0xffff00) endloop  ### Example 2:¶ #!/usr/local/bin/fontforge #This is the sfddiff script which compares two fonts if ( Strtol($version) < 20060330 )
endif

flags=0x789
outfile=""

while ( $argc > 1 && Strsub($1,0,1)=="-" )
temp = $1 if ( Strsub(temp,1,2)=='-' ) temp = Strsub(temp,1) endif if ( temp=="-ignorehints" ) flags = flags & ~0x8 elseif ( temp=="-ignorenames" ) flags = flags & ~0x100 elseif ( temp=="-ignoregpos" ) flags = flags & ~0x200 elseif ( temp=="-ignoregsub" ) flags = flags & ~0x400 elseif ( temp=="-ignorebitmaps" ) flags = flags & ~0x80 elseif ( temp=="-exact" ) flags = flags | 0x2 elseif ( temp=="-warn" ) flags = flags | 0x44 elseif ( temp=="-merge" ) flags = flags | 0x1800 shift outfile =$1
elseif ( temp=="-help" )
Print( "sfddiff: [--version] [--help] [--usage] [--ignorehints] [--ignorenames] [--ignoregpos] [--ignoregsup] [--ignorebitmaps] [--warn] [--exact] fontfile1 fontfile2" )
Print( " Compares two fontfiles" )
Print( " --ignorehints: Do not compare postscript hints or truetype instructions" )
Print( " --ignorenames: Do not compare font names" )
Print( " --ignoregpos:  Do not compare kerning, etc." )
Print( " --ignoregsub:  Do not compare ligatures, etc." )
Print( " --ignorebitmaps: Do not compare bitmap strikes" )
Print( " --exact:       Normally sfddiff will match contours which are not exact" )
Print( "                but where the differences are slight (so you could compare" )
Print( "                truetype and postscript and get reasonable answers). Also" )
Print( "                normally sfddiff will unlink references before it compares" )
Print( "                (so you can compare a postscript font (with no references)" )
Print( "                to the original source (which does have references)). Setting")
Print( "                this flag means glyphs must match exactly.")
Print( " --warn:        Provides a warning when an exact match is not found" )
Print( " --merge outfile: Put any outline differences in the backgrounds of" )
Print( "                appropriate glyphs" )
return(0)
elseif ( temp=="-version" )
Print( "Version 1.0" )
return(0)
else
break
endif
shift
endloop

if ( $argc!=3 ||$1=="--usage" || $1=="-usage" ) Print( "sfddiff: [--version] [--help] [--usage] [--ignorehints] [--ignorenames] [--ignoregpos] [--ignoregsup] [--ignorebitmaps] [--warn] [--exact] [--merge outfile] fontfile1 fontfile2" ) return(0) endif Open($2)
Open($1) CompareFonts($2,"-",flags)
if ( outfile!="" )
Save(outfile)
endif

### Example 3:¶

#!/usr/local/bin/fontforge
#Take a Latin font and apply some simple transformations to it
#can be run in a non-interactive fontforge session.
Open($1); Reencode("KOI8-R"); Select(0xa0,0xff); //Copy those things which look just like latin BuildComposit(); BuildAccented(); //Handle Ya which looks like a backwards "R" Select("R"); Copy(); Select("afii10049"); Paste(); HFlip(); CorrectDirection(); Copy(); Select(0u044f); Paste(); CopyFgToBg(); Clear(); //Gamma looks like an upside-down L Select("L"); Copy(); Select(0u0413); Paste(); VFlip(); CorrectDirection(); Copy(); Select(0u0433); Paste(); CopyFgToBg(); Clear(); //Prepare for editing small caps K, etc. Select("K"); Copy(); Select(0u043a); Paste(); CopyFgToBg(); Clear(); Select("H"); Copy(); Select(0u043d); Paste(); CopyFgToBg(); Clear(); Select("T"); Copy(); Select(0u0442); Paste(); CopyFgToBg(); Clear(); Select("B"); Copy(); Select(0u0432); Paste(); CopyFgToBg(); Clear(); Select("M"); Copy(); Select(0u043C); Paste(); CopyFgToBg(); Clear(); Save($1:r+"-koi8-r.sfd");
Quit(0);

## The Execute Script dialog¶

This dialog allows you to type a script directly in to FontForge and then run it. Of course the most common case is that you’ll have a script file somewhere that you want to execute, so there’s a button [Call] down at the bottom of the dlg. Pressing [Call] will bring up a file picker dlg looking for files with the extension *.pe (you can change that by typing a wildcard sequence and pressing the [Filter] button). After you have selected your script the appropriate text to text to invoke it will be placed in the text area.

The current font of the script will be set to whatever font you invoked it from (in python this is fontforge.activeFont()).

Note that if you try to print from a script the output will go to stdout. If you have invoked fontforge from a window manager’s menu stdout will often be bound to /dev/null and not useful. Try starting fontforge from the command line instead.