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Gini coefficient refers to the different in poverty and wealth in any given country. On a scale of 0 to 1, the lower the Gini coefficient, the more evenly distributed the wealth.

Gini coefficient or Gini index is an economic measure of inequality in income distribution. Gini coefficient is specified as a ratio between 0 and 1. A society that tallies 0.0 on the Gini scale is said to possess perfect equality in income distribution. A society with a score of 1 suggests total inequality. Gini coefficient expects that no person in a society has negative wealth or net income. The coefficient is named after its inventor, the Italian statistician Corrado Gini (23rd May, 1884-13th March, 1965).

The Gini coefficient is usually defined mathematically based on the Lorenz curve, which plots the proportion of the total income of the population (y axis) that is cumulatively earned by the bottom x% of the population (see diagram). The line at 45 degrees thus represents perfect equality of incomes. The Gini coefficient can then be thought of as the ratio of the area that lies between the line of equality and the Lorenz curve (marked 'A' in the diagram) over the total area under the line of equality (marked 'A' and 'B' in the diagram); i.e. G=A/(A+B).

The Gini coefficient can range from 0 to 1; it is sometimes multiplied by 100 to range between 0 and 100. A low Gini coefficient indicates a more equal distribution, with 0 corresponding to complete equality, while higher Gini coefficients indicate more unequal distribution, with 1 corresponding to complete inequality. To be validly computed, no negative goods can be distributed. Thus, if the Gini coefficient is being used to describe household income inequality, then no household can have a negative income. When used as a measure of income inequality, the most unequal society will be one in which a single person.

Antimicrobial peptides are usually classified according to structure (Table 1 ), since no taxonomic relationships appear to exist between them. 5,6 For instance, peptides from entirely different sources may have very similar structures and overlapping target specificities, whereas peptides from similar origins often have completely different structures and specificities. A good example of the latter are mammalian cathelicidins, which are found in human epithelial surfaces and are released by neutrophiles in response to an infection. 7 All cathelicidins are produced as large, two-domain peptides that are divided into a conserved, N-terminal domain and a highly variable, C-terminal domain. which has antimicrobial activity. The common N-terminal domain shows sequence similarity to a cysteine protease inhibitor, cathepsin-L inhibitor, and is referred to as cathelin-like. The cathelin-like domain, from which the cathelicidins derive their name, is cleaved by tissue-specific proteases, leading to the liberation of the antimicrobial peptide. In contrast to the conserved N-terminal domain, the antimicrobial domains of different cathelicidins bear no sequence or structural relationship to each other and can fall in any of the structural classes described below.

Linear antimicrobial peptides represent one of the largest structural groups (Figure 1 Figure 1 Structures of antimicrobial peptides representative of the most important classes. Blue segments are hydrophilic; orange are hydrophobic. Disulfide bridges are shown in gold. (a) Magainin 2; (b) gaegurin; (c) indolicidin; (d) tachyplesin; (e) α-defensin (hNP-1); (f) β-defensin (hBD-2); (g) θ-defensin; (h) leucocin (bacteriocin); (i) cyclic, cystine-knot plant peptide Kalata B2. Graphics produced with Pymol. DeLano, W.L. The PyMOL Molecular Graphics System. DeLano Scientific LLC, Palo Alto, CA, USA. http://www.pymol.org.. Table 1 ). Peptides in this class, of which cecropins, magainin. melittin. dermaseptins, and LL-37 (a human cathelicidin ) are some well-known members, are about 20–50 amino acids long, and the vast majority are cationic at neutral pH. Some representative exceptions to this general rule are maximin H5, an anionic peptide identified in the toad *Bombina maxima*. and dermcidin. a peptide present in human sweat, with a net negative charge of −2.5 at physiological pH. 8–10 Antimicrobial peptides typically form amphipathic structures when bound at the membrane-water interface. Very often, the structure adopted by linear peptides is that of an amphipathic *α* -helix, in which polar and non-polar amino acids are segregated along the helical axis. 6 This is true for dermaseptins, cecropins, magainin. and related peptides (Figure 1(a) ). In fact, type II brevinins, esculentins, and gaegurins produced by frogs of *Rana* species are also mainly helical. The N-terminal region of these peptides is an amphipathic *α* -helix, whereas the C-terminus contains a disulfide bridge, which is otherwise a rare structural element in predominantly *α* -helical peptides (Figure 1(b) ). A number of linear peptides adopt less well-defined structures when bound to membrane-water interfaces, often due to the presence of proline and glycine residues. For instance, the mammalian cathelicidins histatin and indolicidin, as well as the synthetic, cathelicidin -based tritrpticin, form wedge-type structures in which the amphipathic character is generated by clustering the hydrophobic amino acids (Figure 1(c) ).

Among the simplest versions of disulfide-bridged antimicrobial peptides are small *β* hairpins containing two or four cysteine residues. Examples of these are thanatin, produced by the spined soldier bug (“stink bug”), and the well-studied peptides tachyplesin and protegrin 11–13 (Table 1 ). Protegrin. a cathelicidin isolated from pig, and tachyplesin, from the horseshoe crab (Figure 1(d) ), have been proposed to exert their antimicrobial activity by forming oligomeric pores upon binding to the cytoplasmic membrane . 14 The vertebrate *α* - and *β* -defensins are 30 to 40 residue peptides that contain a central *β* -sheet hairpin core. 15,16 In the *α* -defensins, the N-terminal stretch adopts an additional *β* -sheet conformation, leading to a compact, triple-stranded structure (Figure 1(e) ), whereas the corresponding stretch forms an *α* -helix in the human *β* -defensin-2, hBD-2 (Figure 1(f) ). The *θ* -defensins of *Rhesus* monkeys consist of two truncated *α* -defensins ligated head to tail (Figure 1(g) ). The defensins are not restricted to vertebrates; a series of related peptides have been identified in insects and even plants.

Bacterocins are a heterogeneous group of antimicrobial proteins and peptides of bacterial origin. 17 The colicins, large multi-subunit proteins, produced by *Escherichia coli*. that bind to cell surface receptors in the target cell, are among the oldest known antibacterial proteins. However, some of the bacteriocins produced by gram-positive bacteria, in particular the class I and II bacteriocins of lactic acid bacteria, bear a strong resemblance to the antimicrobial peptides of eukaryotic origin discussed earlier. Class I bacteriocins are also known as lantibiotics due to multiple post translationally introduced thioether bridges, known as lanthionine rings. Lanthionine rings are generated from serine and threonine residues that are dehydrated to yield 2-aminoprop-2-enoic acid (dehydroalanine ) and 2-aminobut-2-enoic acid (dehydrobutyrine). Intramolecular conjugate addition of cysteine residues to these unsaturated amino acid derivatives results in the formation of lanthionine and methyllanthionine rings. Nisin. easily the most studied lantibiotic, is a 37-amino-acid peptide that inhibits bacterial replication by binding to the pyrophosphate moiety of lipid II, an important precursor in bacterial cell wall synthesis. 18 Leucocin (Figure 1(h) ) is a typical representative of the class IIa bacteriocins, 19 a family of small, heat-stable peptides. usually <10kDa. The structure of leucocin resembles the *Rana* peptides described earlier, with an amphipathic, C-terminal *α* -helix and a short, disulfide-bridged N-terminal domain.

Cyclization among naturally occurring antimicrobial peptides is not uncommon and may be a strategy to evade the action of proteolytic enzymes. However, cyclization may have other functional consequences. Artificial cyclization of linear peptides often leads to reduced hemolytic activity combined with increased activity under high-salt conditions where many linear peptides lose activity. 20,21 Among the oldest members of this class, in evolutionary terms, are daptomycin and gramicidin S. Daptomycin is a nonribosomally synthesized, cyclic lipopetide secreted by the soil bacterium *Streptomyces roseosporus* and is currently in use as an antibiotic of last resort against methicillin- and vancomycin-resistant *Staphylococcus aureus.* 22 Gramicidin S is also a non ribosomally synthesized circular decapeptide produced by *Bacillus brevis* that is active against gram-positive and gram-negative bacteria but exhibits considerable toxicity toward eukaryotic cells. 23 Ribosomally synthesized cyclized peptides include the *θ* -defensins, already described, ranacyclins, produced by frogs of *Rana* species, 24 and the complex, cyclic, cystine-knot peptides found in plants (Figure 1(i) ). 25

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The β-sheet structure is another common secondary structure in peptides /proteins. The major structural element of many native proteins is β-sheet. Different from α-helical structures, the SFG spectra of antiparallel β-sheet structure are centered at 1625, 1690, and 1730cm −1. which correspond to B, B, and B modes, respectively. Besides this difference, orientation information of antiparallel β-sheet structure includes both the tilt (*θ* ) and the twist (*ψ* ) angles. Therefore, only one SFG intensity ratio obtained from the ppp and ssp spectra is not enough to determine the orientation. In addition, due to the *D* symmetry, β-sheet structures exhibit apparent chiral SFG signal with the polarization combination of spp, psp, and pps. Although there have been many researches on helical structures using SFG, only several SFG applications on antiparallel β-sheet structure were reported since Chen group first detected amide I signal from tachyplesin I (which forms antiparallel β-sheet structure) on polystyrene (PS) surface. Previous studies mainly focus on characterizing the formation of β-sheet structure in terms of the SFG special spectral features (Chen, Wang, Sniadecki, Even, & Chen, 2005; Wang, Chen, et al. 2005 ). For example, Castner group probed the orientation of electrostatically immobilized protein G B onto both amine (NH + ) and carboxyl (COO − ) functionalized gold (Baio et al. 2012 ). Yan group investigated proton exchange in antiparallel β-sheets at interfaces (Fu, Xiao, Wang, Batista, & Yan, 2013 ). As mentioned above, Chen group developed a methodology to determine the orientation of antiparallel β-sheet structure using SFG amide I spectra collected with different polarization combinations by treating antiparallel β-sheet structure as having *D* symmetry (Nguyen, King, et al. 2010 ). They estimated orientations of antiparallel β-sheet structures (tachyplesin I) in DPPG/d-DPPG lipid bilayers by using the relative SFG signal intensities of the B and B modes and found that the tilt angle (*θ* ) has a range of 75–90° and the twist angle (*ψ* ) has a range of 75–90°. It is evident to show that tachyplesin I adopts quite a different twist angle on the DPPG/d-DPPG lipid bilayer than it does on PS or deuterated-PS polymer surface (Nguyen, King, et al. 2010 ).

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*Correlation Coefficient, r :*

The quantity *r*. called the *linear correlation coefficient*. measures the strength and

the direction of a linear relationship between two variables. The linear correlation

coefficient is sometimes referred to as the *Pearson product moment correlation coefficient* in

honor of its developer Karl Pearson.

The mathematical formula for computing *r* is:

where *n* is the number of pairs of data.

(Aren't you glad you have a graphing calculator that computes this formula?)

The value of *r* is such that -1 __<__*r*__<__ +1. The + and � signs are used for positive

linear correlations and negative linear correlations, respectively. *Positive correlation:* If *x* and *y* have a strong positive linear correlation, *r* is close

to +1. An *r* value of exactly +1 indicates a perfect positive fit. Positive values

indicate a relationship between *x* and *y* variables such that as values for *x* increases,

values for *y* also increase. *Negative correlation:* If *x* and *y* have a strong negative linear correlation, *r* is close

to -1. An *r* value of exactly -1 indicates a perfect negative fit. Negative values

indicate a relationship between *x* and *y* such that as values for *x* increase, values

for *y* decrease. *No correlation:* If there is no linear correlation or a weak linear correlation, *r* is

close to 0. A value near zero means that there is a random, nonlinear relationship

between the two variables

Note that *r* is a dimensionless quantity; that is, it does not depend on the units

employed.

A *perfect* correlation of � 1 occurs only when the data points all lie exactly on a

straight line. If *r* = +1, the slope of this line is positive. If *r* = -1, the slope of this

line is negative.

A correlation greater than 0.8 is generally described as *strong*. whereas a correlation

less than 0.5 is generally described as *weak*. These values can vary based upon the

"type" of data being examined. A study utilizing scientific data may require a stronger

correlation than a study using social science data.

*Coefficient of Determination, r 2 or R 2 :*

The *coefficient of determination,**r* 2. is useful because it gives the proportion of

the variance (fluctuation) of one variable that is predictable from the other variable .

It is a measure that allows us to determine how certain one can be in making

predictions from a certain model/graph.

The *coefficient of determination* is the ratio of the explained variation to the total

variation.

The *coefficient of determination* is such that 0 __<__*r* 2 __<__ 1, and denotes the strength

of the linear association between *x* and *y*.

The *coefficient of determination* represents the percent of the data that is the closest

to the line of best fit. For example, if *r* = 0.922, then *r* 2 = 0.850, which means that

85% of the total variation in *y* can be explained by the linear relationship between *x* and

in

The

represents the data. If the regression line passes exactly through every point on the

scatter plot, it would be able to explain all of the variation. The further the line is

away from the points, the less it is able to explain.

Coefficient of Friction

By Omar Ramadan

Partners: Samuel Saarinen

Brian Urbancic

Feb 23, 2012

Abstract:

The coefficient of friction is a number that determines how much force is required to move an object that is held back by friction.

The goal of our experiment was to measure the static and kinetic sliding coefficient of friction between two surfaces by using a ramp and measuring its inclination. The premise is that when a solid object is placed on a ramp and the ramp is tilted upward, there is a point that the object starts to slide. That is the angle where the force of gravity is strong enough to overcome the kinetic and static friction. Once the angle, or the inclination, is known, we can then calculate the sliding coefficient of friction between the two surfaces.

In this experiment, to determine the coefficients of friction, both static and kinetic, our cart with an aluminum bottom was allowed to slide down an inclined plane. Several choices of covering for the inclined plane were given including aluminum, polyethylene, or left as bare wood. We raised the plane to an angle such that the cart would slightly start moving, then we measured the angle of the plane by using protractor and the sonic ranger to determine the acceleration of the cart (the calibration of sonic ranger – 0.012 percent error).This procedure was performed three times, each with a different inclined plane.

Data and Analysis:

The coefficient of static friction was derived in the following manner:

Coefficient of static friction = (force of friction) / (normal force)

Force of friction = horizontal component of gravity = m * g * sin α

Force = vertical component of gravity = m * g * cos α

Coefficient of static friction = m*g*sin α/ (m * g * cos α) = tan α

Coefficient for kinetic friction = [g*m*sin α-(a/m)]/ [g*m* cos α]

Where α is the angle of the incline surface.

We calculated the coefficients for each combination, using the.

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Fortran Exercise #2: Coefficient Of Restitution

In this lab, we shall be studying the coefficient of restitution of a superball. The coefficient of restitution, COR, is the ratio of the bounce-back velocity to the original velocity of an object undergoing impact (such as a ball impacting the ground after being dropped from an initial height). Using the principle of the conservation of energy, the COR can be used to relate the bounce-back height to the original height of a ball dropped from rest. In fact, this ratio is equal to the square of the COR. Mathematically, if a ball is dropped from a height h(0), and bounces back to a height h(1), then:

As the object hits the ground, it is compressed (like a spring) such that the energy of the ball is transferred into the compression of the material. Some of this energy is converted to heat, and lost. The remainder is available to launch the object back up into space after the bounce. The fact that the ball bounces back to less than its original height indicates that the maximum value of COR is 1, and values progressively less than this indicate that less and less of the original energy is available after the bounce.

The COR is a mechanical property of a specific impact, and involves the material and geometries of both bodies. Consider a superball being bounced upon a concrete floor. A superball (made of superball plastic) bouncing upon the concrete floor has a different coefficient of restitution than a sphere of the same size made of rubber or of another plastic bouncing upon the same concrete floor. Conversely, a superball would bounce back a different amount from a concrete floor than if it bounced upon a flat surface of mud (as an extreme example). Also, the coefficient of restitution of a rubber sphere is quite likely different from that of a rubber cube dropped upon the same surface. So, the COR is dependent upon the geometries of the two objects undergoing impact and their compositions (materials).

In this exercise, we shall begin by doing an experiment. We shall drop a superball upon the floor from an initial height, h(0), of one meter. After the first bounce, we shall call the height above the floor h(1); after the second bounce, h(2); after the ith bounce, h(i). Thus, the COR for each bounce may be defined as:

Notice that COR is a ratio of heights and as such is unitless.

This will allow us to fill out the following table: