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Topological quantum field theory

A topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants.

Although TQFTs were invented by physicists, they are also of mathematical interest, being related to, among other things, knot theory and the theory of four-manifolds in algebraic topology, and to the theory of moduli spaces in algebraic geometry. Donaldson, Jones, Witten, and Kontsevich have all won Fields Medals for work related to topological field theory.

In condensed matter physics, topological quantum field theories are the low energy effective theories of topologically ordered states, such as fractional quantum Hall states, string-net condensed states, and other strongly correlated quantum liquid states.

Overview

In a topological field theory, the correlation functions do not depend on the metric of spacetime. This means that the theory is not sensitive to changes in the shape of spacetime; if the spacetime warps or contracts, the correlation functions do not change. Consequently, they are topological invariants.

Topological field theories are not very interesting on the flat Minkowski spacetime used in particle physics. Minkowski space can be contracted to a point, so a TQFT on Minkowski space computes only trivial topological invariants. Consequently, TQFTs are usually studied on curved spacetimes, such as, for example, Riemann surfaces. Most of the known topological field theories are defined on spacetimes of dimension less than five. It seems that a few higher-dimensional theories exist, but they are not very well understood.

Quantum gravity is believed to be background-independent (in some suitable sense), and TQFTs provide examples of background independent quantum field theories. This has prompted ongoing theoretical investigation of this class of models.

(Caveat: It is often said that TQFTs have only finitely many degrees of freedom. This is not a fundamental property. It happens to be true in most of the examples that physicists and mathematicians study, but it is not necessary. A topological sigma model with target infinite-dimensional projective space, if such a thing could be defined, would have countably infinitely many degrees of freedom.)

Specific models

The known topological field theories fall into two general classes: Schwarz-type TQFTs and Witten-type TQFTs. Witten TQFTs are also sometimes referred to as cohomological field theories.

Schwarz-type TQFTs

In Schwarz-type TQFTs, the correlation functions or partition functions of the system are computed by the path integral of metric independent action functionals. For instance, in the BF model, the spacetime is a two-dimensional manifold M, the observables are constructed from a two-form F, an auxiliary scalar B, and their derivatives. The action (which determines the path integral) is

$S=\int_M B F\,$

The spacetime metric does not appear anywhere in the theory, so the theory is explicitly topologically invariant. The first example appeared in 1977 and is due to A. Schwarz, its action functional is:

$\int_M A\wedge dA.$

Another more famous example is Chern–Simons theory, which can be used to compute knot invariants. In general partition functions depend on metric but the above examples are shown to be independent on metric.

Witten-type TQFTs

The first example of the topological field theories of Witten-type appeared in Witten's paper in 1988 (Witten 1988a), i.e. topological Yang–Mills theory in four dimensions. Though its action functional contains metric gαβ, after topological twist it turns out to be metric independent. The independence of Stress-energy tensor Tαβ of the system on metric depends on whether BRST-operator is closed. Following Witten's example a lot of examples are found in string theory.

Mathematical formulations

The original Atiyah-Segal axioms

Atiyah suggested a set of axioms for topological quantum field theory (Atiyah 1988a) which was inspired by Segal's proposed axioms for conformal field theory, and Witten's idea of the geometric meaning of supersymmetry, (Witten 1982). Atiyah's axioms are constructed on gluing the boundary with differentiable (topological or continuous) transformation, while Segal's are with conformal transformation. These axioms have been relatively useful for mathematical treatments of Schwarz-type QFTs, although it isn't clear that they capture the whole structure of Witten-type QFTs. The basic idea is that a TQFT is a functor from a certain category of cobordisms to the category of vector spaces.

There are in fact two different sets of axioms which could reasonably be called the Atiyah axioms. These axioms differ basically in whether or not they study a TQFT defined on a single fixed n-dimensional Riemannian / Lorentzian spacetime M or a TQFT defined on all n-dimensional spacetimes at once.

Let Λ be a commutative ring with 1 (for almost all real-world purposes we will have Λ = Z, R or C). Atiyah originally proposed the axioms of a topological quantum field theory (TQFT) in dimension d defined over a ground ring Λ as following:

• A finitely generated Λ-module Z(Σ) associated to each oriented closed smooth d-dimensional manifold Σ (corresponding to the homotopy axiom),
• An element Z(M) ∈ Z(∂M) associated to each oriented smooth (d+1)-dimensional manifold (with boundary) M (corresponding to an additive axiom).

These data are subject to the following axioms (4 and 5 were added by Atiyah):

1. Z is functorial with respect to orientation preserving diffeomorphisms of Σ and M,
2. Z is involutory, i.e. Z(Σ*) = Z(Σ)* where Σ* is Σ with opposite orientation and Z(Σ)* denotes the dual module,
3. Z is multiplicative.
4. Z(φ) = Λ for the d-dimensional empty manifold and Z(φ) = 1 for the (d+1)-dimensional empty manifold.
5. Z(M*) = Z(M) (the hermitian axiom). Equivalently, Z(M*) is the disjoint of Z(M)

Remark. If for a closed manifold M we view Z(M) as a numerical invariant, then for a manifold with boundary we should think of Z(M) ∈ Z(∂M) as a "relative" invariant. Let f : Σ × I → Σ × I be an orientation preserving diffeomorphism, and identify opposite ends of Σ × I by f. This gives a manifold Σf and our axioms imply

$Z(\Sigma_f)=\text{Trace}\ \Sigma(f)$

where Σ(f) is the induced automorphism of Z(Σ).

Remark. For a manifold M with boundary Σ we can always form the double $M\cup_\Sigma M^*$ which is a closed manifold. The fifth shows that

$Z(M\cup_\Sigma M^*)=|Z(M)|^2$

where on the right we compute the norm in the hermitian (possibly indefinite) metric.

The relation to physics

Physically (2)+(4) is related to relativistic invariance while (3)+(5) is indicative of the quantum nature of the theory.

Σ is meant to indicate the physical space (usually, d = 3 for standard physics) and the extra dimension in Σ × I is "imaginary" time. The space Z(M) is the Hilbert space of the quantum theory and a physical theory, with a Hamiltonian H, will have an time evolution operator eitH or an "imaginary time" operator e−tH. The main feature of topological QFTs is that H = 0, which implies that there is no real dynamics or propagation, along the cylinder Σ × I. However, there can be non-trivial "propagation" (or tunneling amplitudes) from Σ0 to Σ1 through an intervening manifold M with $\partial M=\Sigma^*_0\cup\Sigma_1$; this reflects the topology of M.

If ∂M = Σ, then the distinguished vector Z(M) in the Hilbert space Z(Σ) is thought of as the vacuum state defined by M. For a closed manifold M the number Z(M) is the vacuum expectation value. In analogy with statistical mechanics it is also called the partition function.

The reason why a theory with zero Hamiltonian can be sensibly formulated in the Feynman path integral approach to QFT. This incorporates relativistic invariance (which caters for general (d+1)-dimensional "spacetimes") and the theory is formally defined by writing down a suitable Lagrangian - a functional of the classical fields of the theory. A Lagrangian which involves only first derivatives in time formally leads to a zero Hamiltonian, but the Lagrangian itself may have non-trivial features which relate it to the topology of M.

Atiyah's examples

In 1988, M. Atiyah published a paper in which he described many new examples of topological quantum field theory that were considered at that time. (Atiyah 1988) It contains some new topological invariants and the new ideas, which are Casson invariant, Donaldson invariant, Gromov's theory, Floer homology and Jones-Witten's theory.

d = 0

In this case Σ consists of finitely many points. To single point we associate a vector space V = Z(point) and to n-points the n-fold tensor product: Vn = V ⊗ ... ⊗ V. The symmetric group Sn acts on Vn. A standard way to get the quantum Hilbert space is to give a classical symplectic manifold (or phase space) and then quantize it. Let us extend Sn to compact Lie group G and consider "integrable" orbits for which the symplectic structure comes from a line bundle then quantization leads to the irreducible representations V of G. This is the physical interpretation of the Borel-Weil theorem or the Borel-Weil-Bott theorem. The Lagrangian of these theories is the classical action (holonomy of the line bundle). Thus topological QFT's with d = 0 relate naturally to the classical representation theory of Lie groups and symmetric groups.

d = 1

We should consider periodic boundary conditions given by closed loops in a compact symplectic manifold X. Along to (Witten 1982) holonomy round such loops used in the case of d = 0 as a Lagrangian is used to modify the Hamiltonian. For a closed surface M the invariant Z(M) of the theory is the number of pseudo holomorphic maps f : MX in the sense of Gromov (they are ordinary holomorphic maps if X is a Kaehler manifold). If this number becomes to infinite i.e. if there are "moduli", then we must fix further data on M. This can be done by picking some points Pi and then looking at holomorphic maps f : MX with f(Pi) constrained to lie on a fixed hyperplane. (Witten 1988b) has written down the relevant Lagrangian for this theory. Floer has given a rigorous treatment, i.e. Floer homology, based on (Witten 1982)'s Morse theory ideas, for the case when the boundary conditions are the interval instead of periodic, the initial and end-points of paths lie on two fixed Lagrangian submanifolds. This theory has been developed as Gromov–Witten invariant theory.

Another example is Holomorphic Conformal Field Theory. This might not be strictly topological quantum field theory at that time because Hilbert spaces are infinite dimensional. The conformal field theories are also related to compact Lie group G in which the classical phase consists of a central extension of the loop group LG. Quantizing these produces the Hilbert spaces of the theory of irreducible (projective) representations of LG. The group Diff+(S1) now substitutes for the symmetric group and play an important role. The partition function in such theories depends on complex structure: it is not purely topological.

d = 2

Jones-Witten theory is the most important theory in this case. Here the classical phase space, associated to a closed surface Σ is the moduli space of flat G-bundle over Σ. The Lagrangian is an integer multiple of the Chern–Simons function of a G-connection on a 3-manifold (which has to be "framed"). The integer multiple k, called the level, is a parameter of the theory and k → ∞ gives the classical limit. This theory can be naturally coupled with the d = 0 theory to produce a "relative" theory. The details have been described by Witten who shows that partition function for a (framed) link in the 3-sphere is just the value of the Jones polynomial for a suitable root of unity. The theory can be defined over the relevant cyclotomic field. By considering Riemann surface with boundary, we can couple it to the d = 1 conformal theory instead of coupling d = 2 theory to d = 0. This theory has been developed as the Jones-Witten theory and turned out to be the trigger binding the knot theory and the quantum theory.

d = 3

Donaldson has defined integer invariant of smooth 4-manifolds by using moduli spaces of SU(2)-instantons. These invariants are polynomials on the second homology. Thus 4-manifolds should have extra data consisting of the symmetric algebra of H2. (Witten 1988a) has produced a super-symmetric Lagrangian which formally reproduces the Donaldson theory. Witten's formula might be understood as an infinite-dimensional analogue of the Gauss–Bonnet theorem. At a later date, this theory is further developed and become the Seiberg–Witten gauge theory which reduces SU(2) to U(1) in N = 2, d = 4 gauge theory. The Hamiltonian version of the theory has been developed by Floer in terms of the space of connections on a 3-manifold. Floer uses the Chern–Simons function, which is the Lagrangian of the Jones-Witten theory to modify the Hamiltonian. In detail see (Atiyah 1988). (Witten 1988a) has also shown how one can couple the d = 3 and d = 1 theories together: this is quite analogous to the coupling between d = 2 and d = 0 in the Jones-Witten theory.

Now, it isn't considered on a fixed dimension but on all the dimensions at the same time, namely, topological field theory is viewed as a functor.

The case of a fixed spacetime

Let BordM be the category whose morphisms are n-dimensional submanifolds of M and whose objects are connected components of the boundaries of such submanifolds. Regard two morphisms as equivalent if they are homotopic via submanifolds of M, and so form the quotient category hBordM: The objects in hBordM are the objects of BordM, and the morphisms of hBordM are homotopy equivalence classes of morphisms in BordM. A TQFT on M is a symmetric monoidal functor from hBordM to the category of vector spaces.

Note that cobordisms can, if their boundaries match up, be sewn together to form a new bordism. This is the composition law for morphisms in the cobordism category. Since functors are required to preserve composition, this says that the linear map corresponding to a sewn together morphism is just the composition of the linear map for each piece.

There is an equivalence of categories between the category of 2-dimensional topological quantum field theories and the category of commutative Frobenius algebras.

All n-dimensional spacetimes at once

The pair of pants is a (1+1)-dimensional bordism, which corresponds to a product or coproduct in a 2-dimensional TQFT.

To consider all spacetimes at once, it is necessary to replace hBordM by a larger category. So let Bordn be the category of bordisms, i.e. the category whose morphisms are n-dimensional manifolds with boundary, and whose objects are the connected components of the boundaries of n-dimensional manifolds. (Note that any (n−1)-dimensional manifold may appear as an object in Bordn.) As above, regard two morphisms in Bordn as equivalent if they are homotopic, and form the quotient category hBordn. Bordn is a monoidal category under the operation which takes two bordisms to the bordism made from their disjoint union. A TQFT on n-dimensional manifolds is then a functor from hBordn to the category of vector spaces, which takes disjoint unions of bordisms to the tensor product of them.

For example, for (1+1)-dimensional bordisms (2-dimensional bordisms between 1-dimensional manifolds), the map associated with a pair of pants gives a product or coproduct, depending on how the boundary components are grouped – which is commutative or cocommutative, while the map associated with a disk gives a counit (trace) or unit (scalars), depending on grouping of boundary, and thus (1+1)-dimension TQFTs correspond to Frobenius algebras.

Furthermore, we consider simultaneously 4-dimensional, 3-dimensional and 2-dimensional manifolds that are related by the above bordisms, then obtain ample and important examples.

Development at a later time

Looking at the development of topological quantum field theory we should consider that it has many applications to Seiberg–Witten gauge theory, topological string theory, the relationship between knot theory and quantum theory, and quantum knot invariants. Furthermore it has provided objects of great interest to both mathematics and physics. Also of important recent interest is non-local operators in TQFT.(Gukov & Kapustin (2013)) If string theory is viewed as the fundamental, then non-local TQFTs can be viewed as non-physical models that provide a computationally efficient approximation to local string theory.


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# In the Following Figures, O is the Centre of the Circle. Find the Values of A, B and C. - Mathematics

Course

#### Question

In the following figures, O is the centre of the circle. Find the values of a, b and c.

#### Solution

Here, b=1/2xx 130°

(Angle ate he centre is double the angle at the circumference subtended by the same chord)

⇒ b = 65°

Now, a + b 180°

(Opposite angles of a cyclic quadrilateral are supplementary)

⇒ a = 180°- 65°= 115°

Is there an error in this question or solution?

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2D basic map system

i'm currently coding a 2D game in Java, and I would like to have some clues on how-to build this system :

the screen is moving on a grander map, for instance, the screen represent 800*600 units on a 100K*100K map.

When you command your unit to go to another position, the screen move on this map AND when you move your mouse on a side or another of the screen, you move the screen on the map.

Not sure that i'm clear, but we can retrieve this system in most RTS games (warcraft/starcraft for example).

I'm currently using Slick 2D.

Any idea ?

Thanks.

• What's the question? There's various ways to do a map system like this, and there's tutorials out there for most of the common methods. – thedaian Jun 2 '11 at 17:16
• Are you asking on how to move the camera? – Tetrad Jun 2 '11 at 18:57
• Yes i'm asking how to move the camera, @thedaian you're talking about tutorials, can you give me some link ? I don't how to google it... – Cyril Jun 3 '11 at 9:23
• Search for something like 2d game tile map tutorial java, then it's a matter of looking through the 68,000 results. – thedaian Jun 3 '11 at 14:34

As was said in the comments, you need to handle a camera, which in 2d is just an offset for your render method.

For exemple if you had a call like
MyObject.draw(x, y);
You replace it with
MyObject.draw(x + offsetx, y + offsety);

Then all you need to do is to modify the offset when the user uses the mouse or when a unit is moved.

You need some sort of a viewport into the larger area. Then, when the viewable area is changed you will need to draw it based on what should be displayed there (you can obtain this information from a database, for example -- in particular, PostgreSQL has some specialized indexing features available that are optimized for the needed types of queries).

PostgreSQL (free and open source)
http://www.postgresql.org/

You'll also want to make sure the areas that are no longer needed will not remain in memory for too long, otherwise a player who likes to explore with the "run 20 times faster" boots will fill up their computer's memory quickly.

One problem you might experience will be that as you load the tiles that are in view, the player will see blanks at first that get filled in shortly after. One strategy you can use to resolve this problem is to also load tiles that are slightly out of view; that way, when the player moves, those next tiles will already be pre-loaded while your code starts loading other slightly-out-of-view tiles in preparation for the next moves (no matter what direction the player goes since you can't accurately predict this anyway, especially if the player suddenly sees a dangerous giant spider that likes to eat speed-boosting-boots).

The beauty of using a database to load up on the nearby areas is that you have the option of increasing the size of your world far beyond 100k-by-100k.

For your database, you'll need to have two columns that represent the X and Y positions. If your player can move their character up (e.g., onto a rooftop) or down (e.g., into a dungeon), then you'll need to have a third column that represents the Z position as well.

If you're new to game development, I would recommend that you take a look at Tiled. Tiled is a free tile based level editor, that supports orthogonal and isometric map formats.

Tiled Page

Slick has a class, TiledMap, that can load the tmx format generated by Tiled. You can draw the background of the map simply with tiledmap and then render your units, building etc. on top of it.

You can find the examples in the Slick2d wiki: Tiled WIKI Page

• Ah sorry, didn't see the comment about the camera... You shoul'd update your question to reflect that information in the question itself, otherwise it is a bit misleading. – tom van green Sep 5 '12 at 14:32


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# There are 6 marbles in a box with numbers from 1 to 6 marked on each of them. (i) What is the probability of drawing a marble with number 2? (ii) What is the probability of drawing a marble with number 5?

2. There are 6 marbles in a box with numbers from 1 to 6 marked on each of them.
(i) What is the probability of drawing a marble with number 2?
(ii) What is the probability of drawing a marble with number 5?

Views

There are 6 marbles in the box numbered from 1 to 6

(i) Number of marble with number 2 on it = 1

$\therefore$ Probability of getting a marble with number 2

$=\frac{number\ of\ favourable\ outcomes}{number\ of\ possible\ outcomes}$

$= \frac{1}{6}$

(ii) Number of marble with number 5 on it = 1

$\therefore$ Probability of getting a marble with number 5

$=\frac{number\ of\ favourable\ outcomes}{number\ of\ possible\ outcomes}$

$= \frac{1}{6}$

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# Beautiful Array CodeChef Solution

Table of Contents

### We Are Discuss About CODECHEF SOLUTION

Beautiful Array CodeChef Solution

## Problem

You’re given an array A of N integers. You need to find the minimum cost of creating another array B of N integers with the following properties

• B_i \ge 0 for each 1 \leq i \leq N
• The GCD of adjacent elements of B is equal to 1, i.e, \gcd(B_i, B_{i+1}) = 1 for each 1 \leq i \lt N

Next Week Assignment Answers

The cost of creating B is defined as follows:

\sum_{i=1}^{N} 2^{|A_i – B_i |}

Find the minimum possible cost to create the array B. Since the answer can be huge print it modulo 10^9+7

Note: You need to minimize the value of total cost of creating the array B, and then print this minimum value modulo 10^9 + 7. For example, suppose there is a way to create the required B with a cost of 500, and another way to do it with a cost of 10^9 + 7 (which is 0 \pmod {10^9 + 7}). The output in this case would be 500 and not 0.

### Input Format

• The first line of input will contain a single integer T, denoting the number of test cases.
• The first line of each test case contains an integer N – the length of the array A
• The second line of each test case contains N space-separated integers A_1,A_2,\ldots,A_N
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### Output Format

For each test case, output on a new line the minimum cost of creating the array B, modulo 10^9 + 7.

### Constraints

• 1 \leq T \leq 150
• 1 \leq N \leq 10^4
• 1 \leq A_i \leq 10^6
• The sum of N over all test cases won’t exceed 10^4.

### Sample 1:

Input

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Output

3
3
15 16 19
2
5 10
7
9 15 7 19 10 7 1

3
3
8

### Explanation:

Test case 1: The given input already satisfies all the conditions, hence we can take B = A with a cost of 2^0 + 2^0 + 2^0 = 3.

Test case 2: One optimal way is to choose B as [ 5, 11 ], with a cost of 2^0 + 2^1 = 3.

## SOLUTION

Yhaa You have done it but next? if YOU Want to Get Others Please Visit Here ScishowEngineer   Then Follow US HERE and Join Telegram.

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The bottom line is the concept of an all-white American Flag has been a reoccurring art theme without having developed an universal meaning. The triple bar character in Unicode is code point U+2261 ≡ IDENTICAL TO (HTML ≡ · ≡). The various formatting marks are shown below. vertical synonyms, vertical pronunciation, vertical translation, English dictionary definition of vertical. The beaming across the bar line confirms this interpretation. Scan through and easily download the one you need. The joint is the basis of reference for welding symbols. You will find a wall plaque that has four lines, one vertical and three horizontal (-|--). The ISA S5. In the first example, the first violin is counting 4/8, 5/8, 3/8, 4/8; the 2nd violin is counting 4/8, 5/8, 7/8; and the viola and 'cello are counting 6/8, then two bars of 5/8. Eric Weisstein's World of Mathematics provides an extensive listing to discussions about mathematical notation. barline: a vertical line (or lines) drawn across a staff (or if there are many lines, across a number of staves) to mark off measures (or bars) of a particular length, i. It can handle horizontal and vertical tangent lines as well. Think about some of the details in design projects such as lines and curves. Lap top flickering black, purple and white vertical lines. The following examples compare and contrast the use of parentheses. If you are into textual intercourse or social media you will need a comprehensive text dictionary. Electronic Clock Switch. The displaymath environment is for formulas that appear on their own line. Line - element of art. Limits of Integration. In the schematic diagram shown in Figure 9-12, dashed lines indicate several different conditions. If the bar is grade 60 or 75,. There are many different ways to display these lines, whether you want a wrap around arm band, just two lines that make an equal sign, or you can even put a third line on the arm band to. The most common tomoe emblem has three flames (triple, or 'mitsu' tomoe), but one, two, or four are not uncommon. The main characteristic of such maps is the representation of the three-dimensional terrain or relief of the surface of the earth by the use of contour lines. The use of symbols provides a kind of lens that the work is seen through. To get the letter, character, sign or symbol "|": ( vertical-bar, vbar, vertical line or vertical slash ) on computers with GNU/Linux like Ubuntu (only with GNOME desktop): 1) Press the key combination "CTRL + SHIFT + u" on your keyboard, and not let go. 1) draw a circle 2) put on line from bottom of circle 3) put horizontal line through line as long as the other one 3) from bottom of vertical line put two lines 90 Asked in Math and Arithmetic. Tail: This is used to provide additional but important data as wells as information needed to complete the weld. The points A and B on the line are at (-15,3) and (-15,20). bols for instrument line connections is given in Figure 17. ” A circle usually means “yarnover,” “yarn over needle,” or yarn forward. The “tic” mark or “arrow” perpendicular to the strike line indicates the dip quadrant direction. Most of you might realize it looks like the commonly known yin/yang symbol and that's because. For example, to put a Σ (the Greek capital letter sigma) use the code Σ (i. Vertical angles are called that because they share a common vertex. To see the full set of symbols: 2. Every once in a while my laptop will flicker these colors of purple, black, and white. Essentially there are three meanings. But from one civilization to another, from one era to another, the symbol takes on a different meaning in each case. This list is provided by The International Association of Cemetery Preservationists, Inc. Single pole and double pole switches are shown, but a switch may have as many poles as it requires to perform its function. But there are so many texting symbols that its difficult to remember all of these. If your hand has many smaller lines. WindowsBBS. A floor plan is a picture of a level of a home sliced horizontally about 4ft from the ground and looking down from above. Question: what does the symbol with the lightning bolt in the middle of the two lines mean? Answer: That is the indicator that the electronic throttle system has detected a malfunction and set a code. Read, understand the situation, identify and pull out important information. 🔃 Clockwise Vertical Arrows 🔄 Counterclockwise Arrows Button 🔙 Back Arrow 🔚 End Arrow 🔛 On! Arrow 🔜 Soon Arrow 🔝 Top Arrow 🛐 Place of Worship ⚛ Atom Symbol 🕉 Om Star of David ☸ Wheel of Dharma ☯ Yin Yang Latin Cross ☦ Orthodox Cross ☪ Star and Crescent ☮ Peace Symbol 🕎 Menorah 🔯 Dotted Six-Pointed. 1 Openingsymbols,\mathopen 3 2 Closingsymbols,\mathclose 5 3 Fencesymbols,\mathfence 7 4 Punctuationsymbols,\mathpunct 7 5 'Over'symbols,\mathover 7 6 'Under'symbols,\mathunder 7 7 Accents,\mathaccent 9 8 Bottomaccents,\mathbotaccent 10 9 Bigoperators,\mathop 11 10 Binaryrelations,\mathbin 15. History of Text Messaging Symbols Unbelievably, the use of the emoticons can be traced back to the year 1857. Learn vocabulary, terms, and more with flashcards, games, and other study tools. Because--at least within cultures--symbols are recognized and immediatly understood, they provide a context for the work. However, I also refer to the app's Main Menu within the same section of the document. Originally Posted by dtang4. These three lines have gained a personality, and a name - the hamburger button, because stripped back to its constituent parts, it looks slightly like a hamburger inside two halves of a bun. Surface Finish Symbols, Callouts, and Standards. Video Examples: Example of Tangent Line. 2 Hydraulic 3. Solving Two-Step Linear Equations with Rational Numbers. The devotees of Shiva usually use sacred ashes (Bhasma) for the tilak. Hint: combine a symbol table and a randomized queue. For additional information contact: Investor line 1-844-464-2820 or Lynn Ball, at 1-403-264-2820. Topographic maps provide a detailed represention of the natural and man-made features of the earth's surface. One number is either greater than, less than or equal to the other number. what do's a sign on a grave stone with 3 vertical line instead of two mean. The term "pole," as used to describe the switches in Figure 3, refers to the number of points at which current can enter a switch. Each descendant has his own vertical line. The lines are made with the sacred ash… Everton F. Wáng, by the way, has three lines which represent heaven, earth, and humanity joined together by the vertical line representing the emperor, who apparently didn't lack for ego. Heraldry is the practice of devising, granting, displaying, describing, and recording coats of arms and heraldic badges. I quite like the three-line symbol as a symbol to represent menus. All emoji pictures here has a text label that explains it's exact meaning to avoid ambiguity and possible confusion when typing and reading messages with emoji symbols and smileys on Facebook, Twitter and messaging applications. 5, may be added with option. How to Make Keyboard Symbols with the "Alt" Key and Numeric Keypad: Another way is to use the "Alt" key and numbers - making sure to use your numeric keypad, not the numbers on the top of the keyboard. And you wouldn’t be alone as research has revealed that hordes of people draw a blank at what the symbols actually mean, reports The Sun. The points A and B on the line are at (-15,3) and (-15,20). We have the meanings of the most popular thin lines below. Musical dynamics control the volume of a song, and may be signified by words, symbols, or both. Problem: The equation of the vertical line that passes through the point (4, -3). Subtended angle. Four elevations are customarily drawn, one for each side of the house. These useful Mathematical terms with video lessons help English students and ESL learners improve their English vocabulary. Limit from Above. The beginning of symbols appearing on Chinese coins. Click to listen to Pronunciation >>> The meaning of this symbol is characterized slightly differently in “The Adinkra Dictionary” and “The Values of Adinkra Symbols”; the former emphasizes more abstract qualities of feminine goodness, love and care, while the latter has a more literal interpretation, looking one. under strip and pad foundations, the induced vertical total stresses will decrease both with depth and horizontal distance from the load. Lines often define the edges of a form. This is the generic symbol to describe a married couple no longer living together. Vertical pipe. Plan symbols. ) in Greece. Vertical lines do not cross each other. The equals sign is placed between the things stated to be exactly equal or the same. One horizontal line means dry flat. In other cases, the three lines can be symbolic for three significant people, periods of time, events, or any other place or idea. Here are all emoji meanings. The Flame This shape is formed by three lines encircling an implied sphere. If the square has three vertical lines in it, this means drip dry your clothes on a rack indoors If the square has one horizontal line, this means dry the garment flat to keep its shape. When Mind and Body are combined, they form the cross of matter. Indeed, symbols themselves serve as messages from our higher selves and the Universe. tured by natural forces--by contour lines. 1,058 3 3 gold badges 11 11 silver badges 22 22 bronze badges Yup, the update (2. You'll need to get familiar with floor plan symbols if you're looking at floor plans. In the schematic diagram shown in Figure 9-12, dashed lines indicate several different conditions. Discussion in ' PC Hardware ' started by screamer, 2004/06/04. The evidence was scant. These useful Mathematical terms with video lessons help English students and ESL learners improve their English vocabulary. A copy and paste line symbols collection for easy access. Figure 3-45 shows how a weld symbol is applied to the reference line. An arrow facing to the left is meant to ward off evil. Partition lines are those which divide the field of the escutcheon into any numbers of parts. 8 m (6 ft) gaps. In other cases, the three lines can be symbolic for three significant people, periods of time, events, or any other place or idea. Inserts a blank space that will stretch accordingly to fill the vertical space available. some symbols and lettering are hand- drawn. bols for instrument line connections is given in Figure 17. In Word, you can insert mathematical symbols into equations or text by using the equation tools. Decorative lines – This type of line is generally used to add something to an element: texture, depth, shading, function or any other decorative embellishment. Write a program that takes a command line input N and prints N random phone numbers of the form (xxx) xxx-xxxx. Characters used in compounds. Supplementary angles. 5 is also de standard for the number and the diameter of the bolt holes in a flange. This is how an axis system is located from the intersecting planes. This diagram represents a C Chord. Electrical symbols are always shown in their off or deenergized position. A right-angle arrow pointing to the left represents a line break, inserted with Shift+Enter. The serpent in this. The length is chosen, according to some scale, to represent the magnitude of the vector, and the direction of the directed line segment represents the direction of the vector. /ˈɛvərtən/ are a football club in Liverpool, England, which plays in the Premier League. Travel & Places. To the extent permitted by law, Moody’s and its directors, officers, employees, representatives, licensors and suppliers disclaim liability for: (i) any indirect, special, consequential, or incidental losses or damages whatsoever arising from or in connection with use of the Information; and (ii) any direct or compensatory damages caused to any person or entity, including but not limited. The horizontal line indicates eternity past and future. Each Unicode character has its own number and HTML-code. The lines represent the three angels of Revelation. There are multiple meanings for the Awen Celtic symbol. You can use a line break to start a new line without starting a new paragraph. (Contributed by Richard Murray, 28 August 2010) In the text at the end of Section 4. The word NOT and its equivalent symbols are also prefix. In the first page of Disquisitiones Arithmeticae Gauss says that he has adopted this symbol from Legendre but Legendre used this symbol to show both equality and congruence while Gauss uses this symbol only for congruence to distinguish it from equality. Hey all just saw on my notification bar i have 3 vertical wavy lines (like 3 sideways "m's") to the left of my 3G symbol. Primarily used to show secondary options. U+2589 LEFT SEVEN EIGHTHS BLOCK. I'm reading up on Wilson's Theorem, and see a symbol I don't know what does an equal sign with three lines mean? I'm looking at the example table and I still can't infer what they are trying to say about that relationship between equations. A CIRCLE AND THREE LINES is my first film. Meaning: Something between playful and desperate, depending on usage. Lines often define the edges of a form. However, if a hotkey needs to execute only a single line, that line can be listed to the right of the double-colon. Long Division - One-Digit Divisor and a Three-Digit Quotient with No Remainder (1005 views this week) Long Division - One-Digit Divisor and a Two-Digit Quotient with No Remainder (949 views this week) Division Facts to 100 No Zeros (592 views this week) Division Facts with Divisors and Quotients from 1 to 12 with Long Division Symbol (586 views this week) 4-Digit by 2-Digit Long Division with. Vertical line = represents man's presence. Cross on the Jupiter Mount generally indicates happy and successful marriage. Even in curved lines the repetition creates order, however one more dynamic than straight lines. Accurate vertical centering needs character metric information on individual characters which is only available on some devices. Absolute Value (| |): This symbol means to find the absolute value of the number between the two vertical lines, which is the number itself or its distance from zero on the number line. The number 13 was drawn by depicting two parallel horizontal lines with three dots above them. To understand why, please read the following step by step solution. Among Vaishnavites (followers of Vishnu), the many tilak variations usually include two or more vertical. The lambda symbol seems to be one of the most controversial symbols concerning its meaning. I’ll use the baseline grid for alignment (length & vertical position) of the line, lock the position of the text box, and use distribute spacing to ensure my lines are always the same distance from the side of the text. This symbol will appear if you’re using a Sony smartphone, and you have the vibrate function turned on. Letters with Accents - (e. Use a symbol table to avoid choosing the same number more than once. AA A-6 Building section A-A can be seen on drawing No. That's why the line "Text at the bottom of the page. Picture worksheets contain skills involving reading, drawing, grouping and more. Understanding the letters provides essential insight into the deeper meanings of the Torah or the Bible. Note: These symbols use combining characters and may not appear correctly in some older browsers. It almost always means "and," both in and outside of mathematics. 2 Examples of typical door and window symbols shown in both plan (as it may appear on a blueprint) and in pictorial form. A large and a small line is suppose to represent one battery cell so that the image below would suggest a two-cell battery of 3 V. The pressure rose and then fell over the past three hours, a total. It is one-dimensional and can vary in width, direction, and length. This symbol means, "the set with nothing in it". Only one line of text can be printed on 1/4” labels. Sureños and Norteños sometimes use the ancient Aztec language. THE CREATION. A square with a single horizontal line indicates to lay the garment flat for drying. History of Text Messaging Symbols Unbelievably, the use of the emoticons can be traced back to the year 1857. Then finger through it while saying the letters. When two points have the same y-value, it means they lie on a horizontal line. Line Segment. Excepting the symbol of „Without removal of material‟, both the other symbols are defined with value of Ra required for a particular manufacturing process. Contours make it possible to measure the height of mountains, depths of the ocean bottom, and steep-ness of slopes. If we have to pick one, I'm all for this one. It indicates or signifies as representing an idea, object, or relationship. Start studying Blueprint Reading Lessons 1-3, 5-7. The combination symbol means the button does not truly disconnect from power. geodetic leveling) to best fit the surface of the earth. List of ESL Math vocabulary words and Math terms in English with pictures. Introduction to the Elements of Design: Point The elements are components or parts which can be isolated and defined in any visual design or work of art. The vertical axis equates directly to the human spine and to the tree of life, as well as to the axis mundi, the great pole around which the constellations of the zodiac revolve. You would strum the bottom 5 strings of the guitar in one motion if you were to read this tab properly: And in this case you would strum the ‘C Chord’ three times:. We explain all the symbols, clues, subtle references, hidden meanings, and cult-like ceremonies in the new horror movie 'Midsommar. (Contributed by Richard Murray, 28 August 2010) In the text at the end of Section 4. Valve Symbols Author: Department of Veterans Affairs, Office of Acquisition Logistics and Construction, Office of Construction and Facilities Management, Office of Facilities Planning, Facilities Standards Service Subject: Standard Detail Created Date: 10/18/2017 5:25:14 PM. Index – Symbols(exclamation) in a command interpreter; in curses module; in formatted string literal. This is the chief Adinkra symbol. Blank - Marked in 1s - with large spacing (with option of 1 to 8 lines/ page) Large Scale Number Line: 0 to 100 (13 Pages) Number Line Generator. It indicates or signifies as representing an idea, object, or relationship. The symbols on the left side are potential negative outcomes, while the symbols on the right are potential positive outcomes. Essentially there are three meanings. The Z is has no lines of symmetry. This is the generic symbol to describe a married couple no longer living together. geodetic leveling) to best fit the surface of the earth. Spiritual Meanings of the Hebrew Alphabet Letters. such as gauge lines shall be drawn the same as the line to which it connects. Butterfly - The soul, Although quite rare, it is occasionally seen on graves (most often of children). A vertical line symbolizes the spiritual world, and a horizontal line the temporal. Interestingly, this same symbol was worn by the (in)famous Freemason Albert Pike, founder of the 33rd degree of the Masons. Arbitrary quantities (areas, volumes, angles) were represented by the lengths of lines and the product of two such quantities was represented by a rectangle with sides representing the respective factors. Symbol of Beauty and Cleanliness; Symbols of desirable feminine qualities. U+2593 DARK SHADE. The tomoe is related to the yin yang symbol, and has a similar meaning, representing the play of forces in the cosmos. The Statue of Liberty stands in Upper New York Bay, a universal symbol of freedom. Diagonal lines are slanted - they look like they are leaning over. Check the depth of the line, since deeper lines can. I would consider it odd if local Folk/Blood members were just tagging up territory with the pitchfork alone, but it’s probably not. I mean practically all the lingual, technical, economical text characters and signs and the easiness of pressing three buttons. Tripundra, believed to be a spiritual symbol in the tradition of Lord Shiva, consists of three horizontal lines and a red mark (usually dot) on the center line. The mountain-shaped logo conveys the idea of overcoming challenges and pursuing your goals no matter what. Choose More Symbols… from the bottom of the list - the Symbol dialog box appears: The characters that show initially will depend on the last symbol that was inserted using this method. Meaning: Something between playful and desperate, depending on usage. #N#Thank you for making this an award winning site. 16 Emoji Changelog 📙 Emojipedia Lookups At All Time High 🗓 What the 2021 Unicode Delay Means for Emoji Updates 🦠 Spread of the Coronavirus Emoji 📋 What's New in Unicode 13. Source (s): https://shrink. I'm not a senior; I'm taking the ACT in 7th grade to send to colleges in advance, and since I never took any classes discussing this symbol, I was curious. Line Symbols take the form of words. The first term is point. Om/Aum ## What does Om mean? In Hinduism, Om (also spelled Aum) is a Hindu sacred sound that is considered the greatest of all mantras. IHS can also stand for the Latin phrase Iesus Hominum Salvator , or Jesus saviour of mankind. Not open for further replies. Dots inside the bucket represent the temperature of water to use -- more dots mean more heat. & This symbol is called an ampersand. While these two words sound the same, they imply very different things. This will help you enjoy texting and decipher the various symbols send to you through messages. For more information on the SYMBOL statement, see About SYMBOL Definitions. Eric Weisstein's World of Mathematics provides an extensive listing to discussions about mathematical notation. The perpendicular axis intersect at a point called. - Nick T Aug 9 '11 at 15:59. The Wikipedia article Triple bar describes other less common meanings of this symbol. Finding the domain and range in these situations is simple, as long as we remember what the terms mean. Lemniscate. The beaming across the bar line confirms this interpretation. , symbols, emoji’s and random letters) Reviews on the product and not the customer’s Sales or Service experience close Search anytime by typing. robertskmiles on June 15, 2012 My Algorithms lecturer called it the "equivils" symbol, as in "like equals but meaning exactly equivalent to". The characters that appear in the Unicode Character column of the following table are generated from Unicode numeric character references, and so they should appear correctly in any Web browser that supports Unicode and that has suitable fonts available, regardless of the operating system. The bearer will get understandable,loving and caring life partner. 1, ISO 10628, and BS 5070 cover the standardization of P&ID symbols and guide process engineers in their plant design. U+258A LEFT THREE QUARTERS BLOCK. 15 lines in Word 2007 and Word 2010 and to 1. It forms the basis of Adinkra printings. This kind of people always pulls through the illness and lives to a ripe old age. The following values vertically align the element relative to the entire line: Aligns the top of the element and its descendants with the. , means is defined to be equal to) and congruence (e. Symbolism in artwork, whether paintings or on gravestones, reflects the culture which created the work and can have multiple. Catholic Charities of Santa Rosa (CCSR), in addition to its many social services, operates three homeless shelters, each serving specific populations: Samuel L. The four most commonly used feature classes in the geodatabase are points, lines, polygons, and annotation (the geodatabase name for map text). I was looking through a MENSA book and came across it. But before we discuss the Psychology of Shapes in Logo Design & Marketing, let us first look deep into the different kinds of shapes. These pages are reproduced from Apples iPhone Users Guide. Tab consists of 6 horizontal lines that represent the. However, I also refer to the app's Main Menu within the same section of the document. The hexagram is associated with the Biblical Solomon, best known as the Star of David in the Jewish religion. Meaning behind the Adidas logo. To add a vertical axis title, execute the following steps. The process of making it has been transformational for me both personally and professionally. But in algebra we can. Realistically, anything you draw on paper is a line segment because it cannot go on forever!. The joint is the basis of reference for welding symbols. THE WORLD OF THE SYMBOLS MEANINGS - BUT THE ACTUAL TRUE MEANING IS CLEAR WHEN YOU STUDY OUT THE PAGAN SYMBOLS. When two straight lines intersect (i. This is how. You can use it on Facebook, Twitter, Line, Whatsapp, Blog, WeChat, Tumblr, Blog, those social profile, status, comment and reply. Use this list of area codes to avoid printing out bogus area codes. 5 Flow Direction of 3. It signifies ‘greatness’ and ‘royalty’. It is symbolic of the resurrection of Christ. The second term is plane. Reply Delete. The line is a symbol of boundary and division. Perpendicular line. The Five Dots Tattoo. A cross shape can appear in the form of a “t” or “x” or combination of intersecting lines and spaces. Understanding the letters provides essential insight into the deeper meanings of the Torah or the Bible. The three vertical lines in Urdhva Pundra reminds Lord Brahma, Lord Vishnu, and Lord Shiva. It seems to have gotten a bit worse lately. 🔃 Clockwise Vertical Arrows 🔄 Counterclockwise Arrows Button 🔙 Back Arrow 🔚 End Arrow 🔛 On! Arrow 🔜 Soon Arrow 🔝 Top Arrow 🛐 Place of Worship ⚛ Atom Symbol 🕉 Om Star of David ☸ Wheel of Dharma ☯ Yin Yang Latin Cross ☦ Orthodox Cross ☪ Star and Crescent ☮ Peace Symbol 🕎 Menorah 🔯 Dotted Six-Pointed. The second pair of numbers is 3 and 4. On the contrary, a sesquisharp is known to raise the pitch of a natural note by three quarters of a tone. A vertical line symbolizes the spiritual world, and a horizontal line the temporal. Two lines (or dashes) underneath this symbol indicate even less tolerance to wash action and mean you should wash on a delicates or other gentle wash cycle. 03 Edge Line Pavement Markings for Roundabout Circulatory Roadways Section 3C. Arrow Symbols with Unicode Values. If your hand has many smaller lines. Inserts a blank space that will stretch accordingly to fill the vertical space available. We chose about half of them, for you not to get lost in them. It is found on cars, Bible covers, tattoos, and more. A perpendicular symbol is simply an upside down capital letter T. • Almost always parabolic to provide constant rate of change of graderate of change of grade. Horizontal lines do not cross each other. A copy and paste line symbols collection for easy access. And you wouldn’t be alone as research has revealed that hordes of people draw a blank at what the symbols actually mean, reports The Sun. In order to understand tab, you need to visualize a guitar neck as if you are playing the guitar and looking down on the neck. 1,058 3 3 gold badges 11 11 silver badges 22 22 bronze badges Yup, the update (2. com is completely free, paid for by advertisers and donations. epsilon (looks like c with horizontal line in middle) means 'an element of' or 'in' of 'belongs to' 4. Sureño Tattoos and Symbols Gangs • March 3, 2010 • by Richard Valdemar In Los Angeles California alone, more than 50,000 Hispanic gang members are members over 500 separate street gangs, and although they are usually rivals on the streets, they are united in their alliance and support of the Mexican Mafia prison gang. To do this the part is rotated or tilted, so that the viewer can see three sides. There is no special symbol to distinguish a civil marriage and/or a religious marriage. Realistically, anything you draw on paper is a line segment because it cannot go on forever!. Arch = the world. So, and represent lines. In other cases, the three lines can be symbolic for three significant people, periods of time, events, or any other place or idea. ⊃ may mean the same as ⇒ (the symbol may also mean superset). A right-angle arrow between two vertical lines represents a text-wrapping break. Square with two vertical lines either side. Tripundra, believed to be a spiritual symbol in the tradition of Lord Shiva, consists of three horizontal lines and a red mark (usually dot) on the center line. asked May 17 '17 at 16:54. Three Line Equal Sign. It forms the basis of Adinkra printings. If the line from one process to another crosses over an already existing flow line, draw the secondary line as shown below: Secondary Proess Existing Process Number the symbols. It is a vertical line. Quality German engineering made the heater silent so there was no way to tell by listening. Three Lines Tattoo Meaning Three bold, thick, black lines that wrap around the arm or leg, or any other body part, is often an expression of symmetry and the uniformity that is found in nature. The bar size, which is a numeral. The symbol ">" is used to mean greater than. One symbol that may cloud your judgement (no pun intended) may be the symbol that shows a cloud with two lines. 3rd Grade, Line - YouTube. The link between Christ and a simple line. That's why we are going to show you some of the most really useful texting symbols out there that are commonly used when sending and receiving text messages. " I just wondered why/how they decided to simplify it and if there was a less-broad name than "epsilon/ampersand" for it. Picture graph worksheets contain cut-paste activity and reading graph. Not all of these elementary geometric facts are true in the 3-D context. This is how an axis system is located from the intersecting planes. We're going to look at the "how" to create this symbol in a bunch of different ways. Matrix Algebra Symbols. 1 Openingsymbols,\mathopen 3 2 Closingsymbols,\mathclose 5 3 Fencesymbols,\mathfence 7 4 Punctuationsymbols,\mathpunct 7 5 'Over'symbols,\mathover 7 6 'Under'symbols,\mathunder 7 7 Accents,\mathaccent 9 8 Bottomaccents,\mathbotaccent 10 9 Bigoperators,\mathop 11 10 Binaryrelations,\mathbin 15. For vertical attitudes the dip direction indicator points in. geodetic leveling) to best fit the surface of the earth. Three line tattoos as armbands. 1 male child. BRAIN BUSTERS by Ed Pegg Jr BRAIN BUSTERS by Ed Pegg Jr 1 3 4 5 2 A D G C F E B H 6 21 26 22 27 23 28 19 24 29 20 25 30 1 3 4 5 6 2 The water hyacinth is a fast. Any welded joint indicated by a symbol will always have an arrow side and an other side. /ˈɛvərtən/ are a football club in Liverpool, England, which plays in the Premier League. Brackets are the supportive curves which connect the serif to the stroke. To display these. If we have two lines (they don't have to be parallel) and have a third line that crosses them as in the figure below - the crossing line is called a transversal: In the following figure: If we draw to parallel lines and then draw a line transversal through them we will get eight different angles. I only want the pointer not the vertical lines. One interpretation is main outside lines are symbolic of both man and women while the inside line represents balance. To make special characters and accented letters show up on your pages, use a special set of codes called character entities, which you insert into your HTML code and which your. A third kind of Chinese symbols for words provide hints about their pronunciation. Write a formula for his present age in years? Source(s): horizontal lines equals sign 2 lines mean: https://shortly. The symbols are shorthand and are much more efficient than writing out the words or meanings. Note: These symbols use combining characters and may not appear correctly in some older browsers. 1 Openingsymbols,\mathopen 3 2 Closingsymbols,\mathclose 5 3 Fencesymbols,\mathfence 7 4 Punctuationsymbols,\mathpunct 7 5 ‘Over’symbols,\mathover 7 6 ‘Under’symbols,\mathunder 7 7 Accents,\mathaccent 9 8 Bottomaccents,\mathbotaccent 10 9 Bigoperators,\mathop 11 10 Binaryrelations,\mathbin 15. Each line represents five and each dot is one. three sisters) Pregnancy loss (i. 3) for the Droid X made the icon even more abstract and didn't include a phone thing. For vertical attitudes the dip direction indicator points in. The symbol together with signifying infinity also signals to perfection, as symbol 8-s in double infinity can mean infinity of time, and infinity of space separately, giving more possibilities to achieve perfection together. Proofreaders' marks or, as they are often called "proofreading symbols", are still used, but with much less frequency than in the past. A line does not have any thickness. An r-squared of 1. Two vertical lines on either side of a mathematical expression (not equation) means “the absolute value of that expression”, i. Have you ever wandered through a cemetery and wondered about the meanings of the designs carved on old gravestones? Thousands of different religious and secular symbols and emblems have adorned tombstones through the ages, indicating attitudes towards death and the hereafter, membership in a fraternal or social organization, or an individual's trade, occupation or even ethnic identity. You can also look for squares with one. QNT561 QNT 561 FINAL EXAM 1) A difference between calculating the sample mean and the population mean is A) Only in the symbols, we use instead of μ and n instead of N B) We divide the sum of the observations by n - 1 instead of n. TeX - LaTeX Stack Exchange is a question and answer site for users of TeX, LaTeX, ConTeXt, and related typesetting systems. A symbol for 2 pills would have 2 vertical lines crossed at the top with a horizonal line (like a double TT) and have 2 dots on top, 3 pills would be TTT with 3 dots on top. Line and Page Break. They can indicate warnings of difficulties, separations, traumatic events or health problems. 大 丈 尢 犬 太 尤 六 穴. Remember, a line itself is infinite, and a line segment is a finite portion of that. ₡ Colón: Two vertical or diagonal bars through a lowercase letter "c". In some cases, however, three vertical lines, particularly if they are incused, are interpreted as meaning "river" (chuan 川) and are found on the wu zhu coins of the Kingdom of Shu (蜀) (221-263 AD). For vertical attitudes the dip direction indicator points in. What is the official name of these buttons/icons? It would be nice if developers could consistently use a particular icon for menus. There are several differing opinions as to why the lambda was chosen as a gay symbol and what it means. The combination symbol means the button does not truly disconnect from power. Answer plus sign. An arrow at the end of the leader line - In essence, the leader line is the shaft of the arrow, while the arrow refers to the actual point itself. It is symbolic of the resurrection of Christ. Arch = the world. How to write tabs? Read this first!. 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# Smooth structures on closed $3$-manifolds are unique up to diffeomorphism?

Hi!

I'm using the theorem stated in the question, but so far, I haven't found a source that does explicitely state and prove it or at least give a proper citation. It must be me though, since the theorem seems rather well-established.

So far, I have found dmoskovich's blog and a few papers by googling that are using the theorem as well.

Any pointers to a source that explicitely states and proves the above theorem are appreciated. I need the citation for my thesis, so I don't think I'm allowed to take any hints on how to actually prove the theorem :)

Thanks!

-


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# Free answers to math word problem

Recent questions in Math Word Problem
Wierzycaz 2020-11-12

### In DISCRETE MATHEMATICS & APPLICATIONS Write a note about application of mathematical induction in daily life. Give two examples (150 words).

Wribreeminsl 2020-11-07

### A farmer has a vegetable garden which is 35 feet wide by 80 feet long. The farmer wishes to divide the garden equally between tomatoes, peppers, and cabbage. However, the farmer would like to keep a one-foot border around the garden free for a fence. How many square feet can be used for the tomato planting?

Wierzycaz 2020-11-01

### Math word problem that states t-shirts are $10 each and sweatshirts are$15 each, if 100 items were sold and they raised \$1120, how many sweatshirts were sold, how would i set up this problem

BolkowN 2020-10-28

### a.)Explain in your own words what this problem is asking. b.)Explain the meaning of any notation used in the problem and in your solution. c.)Describe the mathematical concept(s) that appear to be foundational to this problem. d.)Justified solution to or proof of the problem. Find the greatest common divisior of a,b, and c and write it in the form ax+by+cz for integers x,y, and z. a=26,b=52,c=60

The Math word problem, in simple terms, is a problem that is presented in words, meaning that it can relate to integrals, statistics, analysis, basically anything that deals with calculations. It also helps to look at various questions and answers to help yourself see the logic. If you are challenged with math word problem help, do not forget to explore several answers below. Likewise, looking for a Math word problem solver, take your time to study your instructions first and see how certain solutions may be similar to yours. It can be achieved by looking into math word problem answers.


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# A non-trivial absolute value on an algebraic number field $K$ restricts to $Q$ non-trivially.

Let $|\cdot|$ be a non-trivial absolute value over an algebraic number field over $K$. Then restriction of $|\cdot|$ to $Q$ is non-trivial.(i.e. $i:Z\to K$ embedding allows $Q\to K$ embedding. $|\cdot|$ is restricted to the image of $Q\to K$ embedding.)

$\textbf{Q:}$ Since non-trivial absolute values over $Q$ are either equivalent to the absolute value defined by $p-$valuation($p$ is a prime in $Z$) or equivalent absolute value in $R$, can I say this restricted absolute value must be the absolute value coming from $R$(real number)?(In particular, do I know this absolute value ultrametric or not?) I do not see any reason this has to be the case.

This is related to Taylor-Frohlich Pg 70. (2.21) statement.

• The absolute values on $K$ are either Archimedean or $\mathfrak p$-adic, where $\mathfrak p$ is a prime of $\mathcal O_K$ above $p$. In the latter case, the restriction to $\mathbb Q$ will be $p$-adic. By the way, blackboard bold letters can be produced by writing $\mathbb Q$ for example. – Mathmo123 Apr 14 '18 at 23:24
• What is your intuition for saying $| \cdot |$ would induce the archimedean absolute value on $\mathbb Q$? Why do you think it could not induce a $p$-adic absolute value? – D_S Apr 16 '18 at 1:29
• @D_S I do not have any goo reason to say that. However, on $R$, I have a very good "absolute value" which is standard absolute value. There is no reason to guess that I will get $p-$adic absolute value here. So I cannot see any good reason to induce $p-$adic value from $R$. Of course, irrational may have $p-$adic expansion but there is no guarantee that they will converge in $p-$adic sense. At this time point, hensel lifting has not been covered by the book. I do not know why it should be obvious. – user45765 Apr 16 '18 at 13:56

Here is a counterexample where an absolute value on $K$ restricts to a nonarchimedean absolute. Up to equivalence of absolute value, it is the only counterexample.

Let $p$ be a prime number, and let $\mathfrak p$ be a prime of $A = \mathcal O_K$ lying over $p\mathbb Z$. Since $A$ is a Dedekind domain, the localization $A_{\mathfrak p}$ is a discrete valuation ring with unique maximal ideal $\mathfrak p A_{\mathfrak p}$. Let $\varpi$ be a generator of the unique maximal ideal of $A_{\mathfrak p}$. Then

$$pA_{\mathfrak p} = \mathfrak p^e A_{\mathfrak p} = \varpi^e A_{\mathfrak p}$$

where $e \geq 1$ is the ramification index of $\mathfrak p$ over $p$. Every nonzero element of $K$ can be written uniquely as $u \omega^n$, for $u \in A_{\mathfrak p}^{\ast}$ and $n \in \mathbb{Z}$. Define an absolute value on $K$ by

$$|u \omega^n| = \rho^{n}$$

where $\rho = p^{\frac{-1}{e}} \in (0,1)$. This absolute value is nonarchimedean, since it is coming from a DVR. We can write $p = \varpi^e u_1$, for $u_1 \in A_{\mathfrak p}^{\ast}$.

If we restrict $| \cdot |$ to $\mathbb{Q}$, we get the $p$-adic absolute value: every nonzero rational number can be uniquely written as $ap^n$, for $a$ a rational number relatively prime to $p$. Then

$$|ap^n| = |a \varpi^{en} u_1^n| = \rho^{en} = p^{-n}$$

which is the $p$-adic absolute value.

Using Ostrowski's theorem, the absolute values on $K$ can be completely classified by how they restrict to $\mathbb Q$; if the restriction is archimedean, the absolute value comes from an embedding of $K$ into $\mathbb C$. If it is archimedean, the absolute value comes from a unique prime $\mathfrak p$ of $\mathcal O_K$ as above.


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# Question about the Structure group of a circle bundle over a Riemann surface

The last two pages of Appendix C in Milnor's Characteristic Classes gives an example of a flat bundle with nonzero Euler class. I have a question about the structure group of this bundle.

The example starts with a Riemann surface of genus larger than 2. Its fundamental group, $G$, is represented as a subgroup of $\text{PSL}(2,\Bbb R)$ and acts on the upper half-plane $H$ as the group of covering transformations of the surface.

This action preserves the extended real axis and gives a circle bundle over the surface, $E = (H \times \Bbb{RP}^1)/G \to H/G = S$.

Proof is given that it is isomorphic to the bundle of tangent directions to the surface.

This bundle has a 2-fold cover corresponding to the cohomology class in $H^1(E;\Bbb Z_2)$ that maps into the Thom class of the associated 2-plane bundle under the connecting homomorphism. This class exists because the second Stiefel-Whitney class of the bundle is zero (the surface has even Euler characteristic).

So the 2-fold cover is another circle bundle and each fiber circle in the original is covered by a fiber circle of the two fold cover.

Now it is claimed that the structure group of this two fold cover comes from a representation of $G$ in $\text{SL}(2,\Bbb R)$. I think this means that it is the quotient $(H\times \Bbb{RP}^1)/G$ where $G$ now acts on the projective line as a subgroup of $\text{SL}(2,\Bbb R)$ rather than $\text{PSL}(2,\Bbb R)$. This I do not understand.

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## 1 Answer

This all comes down to the lift of the action of $PSL(2,R)$ on $RP^1$ to the 2-fold cover:

The unique connected 2-fold cover of $PSL(2,R)$ is $SL(2,R)$, it acts naturally on the unit circle $S^1$, which is understood as the set of oriented lines in $R^2$. The circle $S^1$ is the unique 2-fold cover of $RP^1$ (defined by forgetting the orientation of the lines). Thus, the group $PSL(2,R)$ lifts to $SL(2,R)$ under the covering map $S^1\to RP^1$. The kernel consists of all $SL(2,R)$ matrices which send each line to itself, i.e. equals $\{\pm I\}$. In particular, the diagram (where the horizontal arrow are group actions and vertical arrow consist of the covering maps and their products) $$\begin{array}{ccc} SL(2,R)\times S^1 & \longrightarrow & S^1\\ \downarrow &~& \downarrow\\ PSL(2,R)\times RP^1 & \longrightarrow &RP^1 \end{array}$$ is commutative.

The rest is done by applying this consideration to each fiber. Hence, the bundle which is the 2-fold cover Milnor is talking about, is $$(H\times S^1)/G$$ where $G$ acts on $S^1$ via its isomorphic lift from $PSL(2,R)$ to $SL(2,R)$.

The confusion stems from the fact that $S^1$ is homeomorphic to $RP^1$, but in this situation they should be regarded as different entities.

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# How can I get a similar chapter style in the report class?

I'm relatively new to LaTeX, and I'm trying to find the right style for my PhD thesis. I want something simple and elegant, inspired by the layout of some books published by Oxford University Press.

Specifically, my goal is to get a chapter style that looks like this (including the chapter title style, the quote and if possible the indented section title style):

I know that some of this can be done with the titlesec package in principle. However, I'm at a loss when it comes to actually writing to the corresponding code. I've tried to provide a MWE, but in that case I'm really not sure where to begin.

Any help would be greatly appreciated!

Here is a way to go:

\documentclass[12pt, a4paper]{report}%
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[lf]{ebgaramond}
\usepackage{geometry}
\usepackage[newlinetospace]{titlesec}%

\titleformat{\chapter}[display]{\fontsize{30pt}{36pt}\selectfont\filright}{\fontsize{36pt}{48pt}\bfseries\thechapter}{2ex}{}

\begin{document}
\setcounter{chapter}{1}

\chapter{The History of the Seven Families \\ of the Lake Pipple-Popple}

In former days -- that is to say, once upon a time, there lived in the Land of Gramblamble, Seven Families. They lived by the side of the great Lake Pipple-popple (one of the Seven Families, indeed, lived in the Lake), and on the outskirts of the City of Tosh, which, excepting when it was quite dark, they could see plainly. The names of all these places you have probably heard of, and you have only not to look in your Geography books to find out all about them.

Now the Seven Families who lived on the borders of the great Lake Pipple-popple, were as follows in the next Chapter.

\end{document}

Added: if you want for the title a wider right margin (than that of the text), you can add a \parbox{some width} s the very last argument of \titleformat, like this:

 \titleformat{\chapter}[display]{\fontsize{30pt}{36pt}\selectfont\filright}{\fontsize{36pt}{48pt}\bfseries\thechapter}{2ex}{\parbox{0.75\textwidth}}

• That's amazing, thank you so much! One small thing: would you know how to reduce the margins of the title (and the epigraph), as in the example I provided? And speaking of the epigraph, do you have a suggestion about the snippet of code I could use for that? Many thanks in advance! – random_grad_student May 24 '19 at 20:36
• Do you mean have shorter margins for the title as compared to the margins of the document? As to the epigraph, you have Robert Wilson's excellent epigraph package (maybe you already used it?). – Bernard May 24 '19 at 21:13
• Yes, that's exactly what I meant. I've found plenty of information to change the spacing before/after the title, but not the left/right spacing. Thanks, I'll check out the epigraph package! – random_grad_student May 25 '19 at 11:17
• For the title margin problem, please see my update, – Bernard May 25 '19 at 11:55
• Thank you Bernard! Is it possible at all to do the same with the left margin of the title, as in the example I provided? – random_grad_student May 25 '19 at 12:34

The memoir class, which encompasses the book, report and article classes and extends them provides some 29 different chapter styles; unfortunately your desired style is not among them but perhaps you would like one of those provided. If not then in the manual (> texdoc memoir) code is given for many of these and you could make your own.

The class also provides for epigraphs which you want to put after the chapter title.

For indenting the section title the class provides the macro \setsecindent{<length>} for setting the indent before section titles.


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# Trigonometric Ratios CBSE NCERT Solutions Chapter 8 Exercise 8.2 Question 4

Trigonometric Ratios CBSE NCERT Solutions Chapter 8 Exercise 8.2 Question 4

4.   State whether the following are true or false. Justify your answer.

(i)  sin (A+B) = sin A + sin B

It is False.

Let A = 60$^\circ$ and B =30$^\circ$

L.H.S = sin (A+B) = sin(60$^\circ$+30$^\circ$) = sin(90$^\circ$) = 1

R.H.S = sin A + sin B = sin 60$^\circ$ + sin 30$^\circ$ = $\frac{\sqrt{3}}{2} + \frac{1}{2}=\frac{\sqrt{3}+1}{2}$

L.H.S $\ne$ R.H.S

Therefore, the given statement is False.

(ii)  The value of sin $\theta$ increases as $\theta$ increases.

True.

You can check here values of six trigonometric ratios (sin, cos, tan, sec, cot and cosec) for 0, 30, 45, 60 and 90 degrees. You can also learn from this article about how to memorize all the values in an easy way.

(iii)  The value of cos $\theta$ increases as $\theta$ increases.

False.

You can check here values of six trigonometric ratios (sin, cos, tan, sec, cot and cosec) for 0, 30, 45, 60 and 90 degrees. You can also learn from this article about how to memorize all the values in an easy way.

(iv) sin $\theta$ = cos $\theta$ for all values of $\theta$.

False

Example, sin 30$^\circ=\frac{1}{2}$ but cos 30$^\circ=\frac{\sqrt{3}}{2}$

(v) cot A is not defined for A = 0$^\circ$

True

cot 0$^\circ= \frac{cos 0^\circ}{sin 0^\circ}$           (1)

But, sin 0$^\circ=0$     (2)

Putting (2) in (1), we get

cot 0$^\circ= \frac{cos 0^\circ}{0}$ which is not defined.

Posted in : NCERT Solutions, Trigonometry

Tags : One Comment


text:
# Web3

For web3 to be successful (or avoid just being Web 2.0 all over again) it needs to solve one huge, currently unsolved problem.

That is the problem of aggregator-wins.

Let’s look at a few popular examples:

• Amazon aggregating online shopping. Amazon are the most popular online retailer. They aggregate other retailers and bring customers together.
• Linux Distros aggregating open-source software. Although the distributions are open-source, and consist almost entirely of open-source software, it’s the distros that make the money (through support / licensing agreements etc). Whereas the guy who maintains grep, -a vital part of that distro- is probably doing it for the love of it.
• Social networks aggregate audience. This is a well-understood problem: the biggest social networks are the most profitable.
• Google + Facebook aggregate online advertising. They have built out a common aggregation platform so they don’t need to compete with each other.
• AWS “owning” ElasticSearch (et al.) Although ElasticSearch is a viable product, they’re seeing their profits bleed away as Amazon are able to offer ElasticSearch as a SaaS option within AWS. The aggregator wins again.

Arguably, this is exactly the problem Web3 is trying to solve. So, why doesn’t it solve it?

To understand this, we need to understand why it happens in the first place. There are two forces at play: The Virtuous Circle of Evolution and Coordination Trumping Competition.

## The Virtuous Circle of Evolution

As shown in the above diagram, successful platforms beget success. This is how the big get bigger. This is why we see a power-law distribution of wealth in the world, and also why platforms like Wordpress are orders of magnitude more successful than its rivals like Drupal. As we discussed in Boundary Risk:

“Did WordPress gain this march because it was always better than Drupal? That’s arguable. Certainly, they’re not different enough that WordPress is 16x better. That it’s this way round could be entirely accidental, and a result of the Network Effect.” - Boundary Risk, Risk First

## Coordination Trumps Competition

Secondly, Coordination beats Competition. Why is this? Because of the law of diminishing returns: sharing out a bag of candy means a higher total utility than one person scoffing the lot and feeling ill. The first candy is really nice, the second is OK but eventually you’re worse off from eating them.

We can see this in the chart above. Let’s say A and B compete over a resource, of which there are 5 units available. For every extra A takes, B loses one. The X axis shows A’s consumption of the resource. While the biggest benefit to A is in taking all of the resources, the greatest increase in benefit comes from the consumption of the first unit.

As you can see, by sharing, it’s possible that the total benefit is greater than it can be for either individual. But sharing requires coordination. Further, the more competitors involved, the worse a winner-take-all outcome is for total benefit.

## Blockchains

But, you might say, “this is exactly what decentralised block-chains and so on are designed to prevent!” But currently, Cryptocurrency doesn’t solve aggregator-wins. Here’s why:

Anyone can mine crypto. However, the problem is, as a crypto-miner, you compete with everyone else. This is a problem, because it means that multiple parties are racing with you to mine the same blocks. This is pure competition.

However, competition is not efficient. Efficiency relies on Coordination. By coordinating the use of scarce resources, you can achieve much better average outcomes than by competing over them, which ends up with a winner-takes-all outcome.

So it goes with crypto: this is why we have mining pools, where everyone coordinates rather than competes. The problem is that coordination is (or should be) capped at 50%, because if a mining consortium owns a majority of the compute power, they can literally say anything happened on the network.

The effect is that you will end up with a duopoly of mining pools - the exact opposite of what was intended.

## Where We Go Next

The Internet has enabled people and organisations to coordinate like never before, so it’s not surprising that we would see today’s mega-corporations of people coordinating together and avoiding the costs of competition. Also, we can’t expect to replace these highly coordinated entities with less-efficient competition: that virtuous circle is bringing innovation (even if it is in walled gardens).

Perhaps we need to look more closely at why we think decentralisation / competition is a desirable characteristic, or whether there are better models we can aspire to.

(More to come on this.)