6.1: Clastic Sedimentary Rocks
A clast is a fragment of rock or mineral, ranging in size from less than a micron [1] (too small to see) to as big as an apartment block. Various types of clasts are shown in Figure \(\PageIndex{1}\) and in Exercise 5.3. The smaller ones tend to be composed of a single mineral crystal, and the larger ones are typically composed of pieces of rock. As we’ve seen in Chapter 5, most sand-sized clasts are made of quartz because quartz is more resistant to weathering than any other common mineral. Many of the clasts that are smaller than sand size (less than 1 / 16 th millimetre) are made of clay minerals. Most clasts larger than sand size (greater than 2 millimeters) are actual fragments of rock, and commonly these might be fine-grained rock like basalt or andesite, or if they are bigger, coarse-grained rock like granite or gneiss. Sedimentary rocks that are made up of “clasts” are called clastic sedimentary rocks. A comparable term is “detrital sedimentary rocks”.
Grain-Size Classification
Geologists that study sediments and sedimentary rocks use the Udden-Wentworth grain-size scale for describing the sizes of the grains in these materials (Table 6.1).
| [Skip Table] | |||
| Type | Description | Size range (millimeters) | Size range (microns) |
|---|---|---|---|
| Boulder | large | 1024 and up | |
| medium | 512 to 1024 | ||
| small | 256 to 512 | ||
| Cobble | large | 128 to 256 | |
| small | 64 to 128 | ||
| Pebble (Granule) | very coarse | 32 to 64 | |
| coarse | 16 to 32 | ||
| medium | 8 to 16 | ||
| fine | 4 to 8 | ||
| very fine | 2 to 4 | ||
| Sand | very coarse | 1 to 2 | 1000 to 2000 |
| coarse | 0.5 to 1 | 500 to 1000 | |
| medium | 0.25 to 0.5 ( 1 / 4 to 1 / 2 mm) | 250 to 500 | |
| fine | 0.125 to 0.25 ( 1 / 8 th to 1 / 4 mm) | 125 to 250 | |
| very fine | 0.063 to 0.125 (or 1 / 16 th to 1 / 8 th mm) | 63 to 125 | |
| Silt | very course | 32 to 63 | |
| course | 16 to 32 | ||
| medium | 8 to 16 | ||
| fine | 4 to 8 | ||
| very fine | 2 to 4 | ||
| Clay | clay | 0 to 2 |
There are six main grain-size categories; five are broken down into subcategories, with clay being the exception. The diameter limits for each successive subcategory are twice as large as the one beneath it. In general, a boulder is bigger than a toaster and difficult to lift. There is no upper limit to the size of boulder. [2] A small cobble will fit in one hand, a large one in two hands. A pebble is something that you could throw quite easily. The smaller ones—known as granules —are gravel size, but still you could throw one. You can’t really throw a single grain of sand . Sand ranges from 2 millimeters down to 0.063 millimeters, and its key characteristic is that it feels “sandy” or gritty between your fingers—even the finest sand grains feel that way. Silt is essentially too small for individual grains to be visible, and while sand feels sandy to your fingers, silt feels smooth to your fingers but gritty in your mouth. Clay is so fine that it feels smooth even in your mouth.
Providing that your landscape isn’t covered in deep snow at present, visit a beach somewhere nearby—an ocean shore, a lake shore, or a river bank. Look carefully at the size and shape of the beach sediments. Are they sand, pebbles, or cobbles? If they are not too fine, you should be able to tell if they are well rounded or more angular.
The beach in Figure \(\PageIndex{1}\) is at Sechelt, B.C. Although there is a range of clast sizes, it’s mostly made up of well-rounded cobbles interspersed with pebbles. This beach is subject to strong wave activity, especially when winds blow across the Strait of Georgia from the south. That explains why the clasts are relatively large and are well rounded.
See Appendix 3 for Exercise 6.1 answers .
If you drop a granule into a glass of water, it will sink quickly to the bottom (less than half a second). If you drop a grain of sand into the same glass, it will sink more slowly (a second or two depending on the size). A grain of silt will take several seconds to get to the bottom, and a particle of fine clay may never get there. The rate of settling is determined by the balance between gravity and friction, as shown in Figure \(\PageIndex{2}\). Large particles settle quickly because the gravitational force (which is proportional to the mass, and therefore to the volume of the particle) is much greater than the frictional resistance (which is proportional to the surface area of the particle). For smaller particles the difference between gravitational push and frictional resistance is less, so they settle slowly.
Small particles that settle slowly spend longer suspended in the water, and therefore tend to get moved farther than large particles if the water is moving.
Transportation
One of the key principles of sedimentary geology is that the ability of a moving medium (air or water) to move sedimentary particles—and keep them moving—is dependent on the velocity of flow. The faster the medium flows, the larger the particles it can move. This is illustrated in Figure \(\PageIndex{3}\). Parts of the river are moving faster than other parts, especially where the slope is greatest and the channel is narrow. Not only does the velocity of a river change from place to place, but it changes from season to season. During peak discharge [3] at the location of Figure \(\PageIndex{3}\), the water is high enough to flow over the embankment on the right, and it flows fast enough to move the boulders that cannot be moved during low flows.
Clasts within streams are moved in several different ways, as illustrated in Figure \(\PageIndex{4}\). Large bed load clasts are pushed (by traction) or bounced along the bottom (by saltation), while smaller clasts are suspended in the water and kept there by the turbulence of the flow. As the flow velocity changes, different-sized clasts may be either incorporated into the flow or deposited on the bottom. At various places along a river, there are always some clasts being deposited, some staying where they are, and some being eroded and transported. This changes over time as the discharge of the river changes in response to changing weather conditions.
Other sediment transportation media, such as waves, ocean currents, and wind, operate under similar principles, with flow velocity as the key underlying factor that controls transportation and deposition.
Clastic sediments are deposited in a wide range of environments, including glaciers, slope failures, rivers—both fast and slow—lakes, deltas, and ocean environments—both shallow and deep. If the sedimentary deposits last long enough to get covered with other sediments they may eventually form into rocks ranging from fine mudstone to coarse breccia and conglomerate.
Lithification is the term used to describe a number of different processes that take place within a deposit of sediment to turn it into solid rock (Figure \(\PageIndex{5}\)). One of these processes is burial by other sediments, which leads to compaction of the material and removal of some of the intervening water and air. After this stage, the individual clasts are touching one another. Cementation is the process of crystallization of minerals within the pores between the small clasts, and especially at the points of contact between clasts. Depending on the pressure, temperature, and chemical conditions, these crystals might include a range of minerals, the common ones being calcite, hematite, quartz and clay minerals.
The characteristics and distinguishing features of clastic sedimentary rocks are summarized in Table 6.2. Mudrock is composed of at least 75% silt- and clay-sized fragments. If it is dominated by clay, it is called claystone . If it shows evidence of bedding or fine laminations, it is shale ; otherwise, it is mudstone. Mudrocks form in very low energy environments, such as lakes, river backwaters, and the deep ocean.
| [Skip Table] | ||
| Group | Examples | Characteristics |
|---|---|---|
| Mudrock | mudstone | Greater than 75% silt and clay, not bedded |
| shale | Greater than 75% silt and clay, thinly bedded | |
| Coal | Dominated by fragments of partially decayed plant matter often enclosed between beds of sandstone or mudrock. | |
| Sandstone | quartz sandstone | Dominated by sand, greater than 90% quartz |
| arkose | Dominated by sand, greater than 10% feldspar | |
| lithic wacke | dominated by sand, greater than 10% rock fragments, greater than 15% silt and clay | |
| Conglomerate | Dominated by rounded clasts, granule size and larger | |
| Breccia | Dominated by angular clasts, granule size and larger |
Most coal forms in fluvial or delta environments where vegetation growth is vigorous and where decaying plant matter accumulates in long-lasting swamps with low oxygen levels. To avoid oxidation and breakdown, the organic matter must remain submerged for centuries or millennia, until it is covered with another layer of either muddy or sandy sediments. It is important to note that in some textbooks coal is described as an “organic sedimentary rock.” In this book, coal is included with the clastic rocks for two reasons: first, because it is made up of fragments of organic matter; and second, because coal seams (sedimentary layers) are almost always interbedded with layers of clastic rocks, such as mudrock or sandstone. In other words, coal accumulates in environments where other clastic rocks accumulate.
It’s worth taking a closer look at the different types of sandstone because sandstone is a common and important sedimentary rock. Typical sandstone compositions are shown in Figure \(\PageIndex{6}\). Sandstones are mostly made up of sand grains of course, but they also include finer material—both silt and clay. The term arenite applies to a so-called clean sandstone, meaning one with less than 15% silt and clay. Considering the sand-sized grains only (the grains larger than 1 / 16 th mm), arenites with 90% or more quartz are called quartz arenites. If they have more than 10% feldspar and more feldspar than rock fragments, they are called feldspathic arenites or arkosic arenites (or just arkose ). If they have more than 10% rock fragments, and more rock fragments than feldspar, they are lithic arenites . [4] A sandstone with more than 15% silt or clay is called a wacke (pronounced wackie ). The terms quartz wacke, lithic wacke , and feldspathic wacke are used with limits similar to those on the arenite diagram. Another name for a lithic wacke is greywacke .
Some examples of sandstones, magnified in thin section are shown in Figure \(\PageIndex{7}\). (A thin section is rock sliced thin enough so that light can shine through.)
Clastic sedimentary rocks in which a significant proportion of the clasts are larger than 2 millimeters are known as conglomerate if the clasts are well rounded, and breccia if they are angular. Conglomerates form in high-energy environments such as fast-flowing rivers, where the particles can become rounded. Breccias typically form where the particles are not transported a significant distance in water, such as alluvial fans and talus slopes. Some examples of clastic sedimentary rocks are shown on Figure \(\PageIndex{8}\).
Table 6.3 below shows magnified thin sections of three sandstones, along with descriptions of their compositions. Using Table 6.1 and Figure \(\PageIndex{6}\), find an appropriate name for each of these rocks.
| Magnified Thin Section | Description |
|---|---|
| Angular sand-sized grains are approximately 85% quartz and 15% feldspar. Silt and clay make up less than 5% of the rock. | |
| Rounded sand-sized grains are approximately 99% quartz and 1% feldspar. Silt and clay make up less than 2% of the rock. | |
| Angular sand-sized grains are approximately 70% quartz, 20% lithic, and 10% feldspar. Silt and clay make up about 20% of the rock. |
See Appendix 3 for Exercise 6.2 answers .
Image Descriptions
Figure \(\PageIndex{8}\) image description: (A) Mudrock with bivalve impressions, Cretaceous Nanaimo group, Browns River, Vancouver Island. A very fine-grained rock with shell impressions. (B) Coarse sandstone with cross-bedding, Cambrian Tapeats Formation Chino Valley, Arizona. (C) Conglomerate with imbricate (aligned, tilted down to the left) cobbles, Cretaceous Geoffrey Formation, Hornby Island, BC. (D) Sedimentary breccia, the Pre-Cambrian Toby Formation, east of Castlegar, BC. [Return to Figure \(\PageIndex{8}\)]
Media Attributions
- Figures 6.1.1, 6.1.2, 6.1.3, 6.1.4, 6.1.5, 6.1.6, 6.1.7, 6.1.8: © Steven Earle. CC BY.
- Exercise 6.2, first image: Aplite Red © Rudolf Pohl. CC BY-SA.
- A micron is a millionth of a metre. There are 1,000 microns in a millimetre. ↵
- The largest known free-standing rock (i.e., not part of bedrock) is Giant Rock in the Mojave Desert, California. It’s about as big as an apartment building—seven stories high! ↵
- Discharge of a stream is the volume of flow passing a point per unit time. It’s normally measured in cubic meters per second (m 3 /s). ↵
- “Lithic” means “rock.” Lithic clasts are rock fragments, as opposed to mineral fragments. ↵


15.5: Alternating Renewal Processes
Basic Theory
Preliminaries
An alternating renewal process models a system that, over time, alternates between two states, which we denote by 1 and 0 (so the system starts in state 1). Generically, we can imagine a device that, over time, alternates between on and off states. Specializing further, suppose that a device operates until it fails, and then is replaced with an identical device, which in turn operates until failure and is replaced, and so forth. In this setting, the times that the device is functioning correspond to the on state, while the replacement times correspond to the off state. (The device might actually be repaired rather than replaced , as long as the repair returns the device to pristine, new condition.) The basic assumption is that the pairs of random times successively spent in the two states form an independent, identically distributed sequence. Clearly the model of a system alternating between two states is basic and important, but moreover, such alternating processes are often found embedded in other stochastic processes.
Let's set up the mathematical notation. Let \( \bs{U} = (U_1, U_2, \ldots) \) denote the successive lengths of time that the system is in state 1, and let \( \bs{V} = (V_1, V_2, \ldots)\) the successive lengths of time that the system is in state 0. So to be clear, the system starts in state 1 and remains in that state for a period of time \( U_1 \), then goes to state 0 and stays in this state for a period of time \( V_1 \), then back to state 1 for a period of time \( U_2 \), and so forth. Our basic assumption is that \( \bs{W} = \left((U_1, V_1), (U_2, V_2), \ldots\right) \) is an independent, identically distributed sequence. It follows that \( \bs{U} \) and \( \bs{V} \) each are independent, identically distributed sequences, but \( \bs{U} \) and \( \bs{V} \) might well be dependent. In fact, \( V_n \) might be a function of \( U_n \) for \( n \in \N_+ \). Let \( \mu = \E(U) \) denote the mean of a generic time period \( U \) in state 1 and let \( \nu = \E(V) \) denote the mean of a generic time period \( V \) in state 0. Let \( G \) denote the distribution function of a time period \( U \) in state 1, and as usual, let \( G^c = 1 - G \) denote the right distribution function (or reliability function) of \( U \).
Clearly it's natural to consider returns to state 1 as the arrivals in a renewal process. Thus, let \( X_n = U_n + V_n \) for \( n \in \N_+ \) and consider the renewal process with interarrival times \( \bs{X} = (X_1, X_2, \ldots) \). Clearly this makes sense, since \( \bs{X} \) is an independent, identically distributed sequence of nonnegative variables. For the most part, we will use our usual notation for a renewal process, so the common distribution function of \( X_n = U_n + V_n \) is denoted by \( F \), the arrival time process is \( \bs{T} = (T_0, T_1, \ldots) \), the counting process is \( \{N_t: t \in [0, \infty)\} \), and the renewal function is \( M \). But note that the mean interarrival time is now \( \mu + \nu \).
The renwal process associated with \( \bs{W} = ((U_1, V_1), (U_2, V_2), \dots) \) as constructed above is known as an alternating renewal process .
The State Process
Our interest is the state \( I_t \) of the system at time \( t \in [0, \infty) \), so \( \bs{I} = \left\{I_t: t \in [0, \infty)\right\} \) is a stochastic process with state space \( \{0, 1\} \). Clearly the stochastic processes \( \bs{W} \) and \( \bs{I} \) are equivalent in the sense that we can recover one from the other. Let \( p(t) = \P(I_t = 1) \), the probability that the device is on at time \( t \in [0, \infty) \). Our first main result is a renewal equation for the function \( p \).
The function \( p \) satisfies the renewal equation \( p = G^c + p * F \) and hence \( p = G^c + G^c * M \).
Proof
By now, the approach should be clear—we're going to condition on the first arrival \( X_1 \): \[ \P(I_t = 1) = \P(I_t = 1, X_1 \gt t) + \P(I_t = 1, X_1 \le t) = \P(I_t = 1, X_1 \gt t) + \int_0^t \P(I_t = 1 \mid X_1 = s) \, dF(s) \] But \( \{I_t = 1, X_1 \gt t\} = \{U_1 \gt t) \) so \( \P(I_t = 1, X_1 \gt t) = \P(U_1 \gt t) = G^c(t) \). By the fundamental renewal property (the process restarts, independently of the past, at each arrival) \( \P(I_t = 1 \mid X_1 = s) = p(t - s) \) for \( s \le t \). Hence we have \[ p(t) = G^c(t) + \int_0^t p(t - s) \, dF(s), \quad t \in [0, \infty) \] or equivalently, \( p = G^c + p * F \). By the fundamental theorem on renewal equations, the solution is \( p = G^c + G^c * M \), so \( p(t) = G^c(t) + \int_0^t G^c(t - s) \, dM(s), \quad t \in [0, \infty) \)
We can now apply the key renewal theorem to get the asymptotic behavior of \( p \).
If the renewal process is non-arithmetic, then \[ p(t) \to \frac{\mu}{\mu + \nu} \text{ as } t \to \infty\]
Proof
From the result above , \( p = G^c + G^c * m \). First, \( G^c(t) \to 0 \) as \( t \to \infty \) as a basic property of the right distribution function. Next, by the key renewal theorem, \[ (G^c * M)(t) \to \frac{1}{\mu + \nu} \int_0^\infty G^c(s) \, ds \text{ as } t \to \infty \] But by another basic property of the right distribution function, \( \int_0^\infty G^c(s) \, ds = \mu \).
Thus, the limiting probability that the system is on is simply the ratio of the mean of an on period to the mean of an on-off period. It follows, of course, that \[ \P(I_t = 0) = 1 - p(t) \to \frac{\nu}{\mu + \nu} \text{ as } t \to \infty\] so in particular, the fact that the system starts in the on state makes no difference in the limit. We will return to the asymptotic behavior of the alternating renewal process in the next section on renewal reward processes.
Applications and Special Cases
With a clever definition of on and off , many stochastic processes can be turned into alternating renewal processes, leading in turn to interesting limits, via the basic limit theorem above.
Age Processes
The last remark applies in particular to the age processes of a standard renewal process. So, suppose that we have a renewal process with interarrival sequence \( \bs{X} \), arrival sequence \( \bs{T} \), and counting process \( \bs{N} \). As usual, let \( \mu \) denote the mean and \( F\) the probability distribution function of an interarrival time, and let \( F^c = 1 - F \) denote the right distribution function (or reliability function).
For \( t \in [0, \infty) \), recall that the current life , remaining life and total life at time \( t \) are \[ C_t = t - T_{N_t}, \quad R_t = T_{N_t + 1} - t, \quad L_t = C_t + R_t = T_{N_t+1} - T_{N_t} = X_{N_t + 1} \] respectively. In the usual terminology of reliability, \( C_t \) is the age of the device in service at time \( t \), \( R_t \) is the time remaining until this device fails, and \( L_t \) is total life of the device. We will use limit theorem above to derive the limiting distributions these age processes. The limiting distributions were obtained earlier, in the section on renewal limit theorems, by a direct application of the key renewal theorem. So the results are not new, but the method of proof is interesting.
If the renewal process is non-arithmetic then \[ \lim_{t \to \infty} \P(C_t \le x) = \lim_{t \to \infty} \P(R_t \le x) = \frac{1}{\mu} \int_0^x F^c(y) \, dy, \quad x \in [0, \infty) \]
Proof
Fix \( x \in [0, \infty) \). For the current life limit, define the on period corresponding to the interarrival time \( X_n \) to be \( U_n = \min\{X_n, x\} \) for \( n \in \N_+ \), so that the off period is \(V_n = X_n - \min\{X_n, x\} \). Note that the system is on at time \( t \in [0, \infty) \) if and only if \( C_t \le x \), and hence \( p(t) = \P(C_t \le x) \). It follows from the limit theorem above that \[ \P(C_t \le x) \to \frac{\E[\min\{X, x\}]}{\mu} \text{ as } t \to \infty \] where \( X \) is a generic interarrival time. But \[G^c(y) = \P[\min\{X, x\} \gt y] = \P(X \gt y) \bs{1}(x \gt y) = F^c(y) \bs{1}(x \gt y), \quad y \in [0, \infty)\] Hence \[ \E[\min(X, x)] = \int_0^\infty G^c(y) \, dy = \int_0^x F^c(y) \, dy \] For the remaining life limit we reverse the on-off periods. Thus, define the on period corresponding to the interarrival time \( X_n \) to be \( U_n = X_n - \min\{X_n, x\} \) for \( n \in \N_+ \), so that the off period is \( V_n = \min\{X_n, x\} \). Note that the system is off at time \( t \) if and only if \( R_t \le x \), and hence \( 1 - p(t) = \P(R_t \le x) \). From the limit theorem above, \[ \P(R_t \le x) \to \frac{\E[\min\{X, x\}]}{\mu} \text{ as } t \to \infty \]
As we have noted before, the fact that the limiting distributions are the same is not surprising after a little thought. After a long time, the renewal process looks the same forward and backward in time, and reversing the arrow of time reverses the roles of current and remaining time.
If the renewal process is non-arithmetic then \[ \lim_{t \to \infty} \P(L_t \le x) = \frac{1}{\mu} \int_0^x y \, dF(y), \quad x \in [0, \infty) \]
Proof
Fix \( x \in [0, \infty) \). For \( n \in \N_+ \), define the on period associated with interarrival time \( X_n \) by \( U_n = X_n \bs{1}(X_n \gt x) \). Of course, the off period corresponding to \( X_n \) is \(V_n = X_n - U_n \). Thus, each renewal period is either totally on or totally off, depending on whether or not the interarrival time is greater than \( x \). Note that the system is on at time \( t \in [0, \infty) \) if and only if \( L_t \gt x \), so from the basic limit theorem above , \[ \P(L_t \gt x) \to \frac{1}{\mu} \E[X \bs{1}(X \gt x)] \] where \( X \) is a generic interarrival time. But \[ \E[X \bs{1}(X \gt x)] = \int_x^\infty y \, dF(y)\]


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5.4: Enantioselective Sulfoxidation
Enantiopure sulfoxides serve as chiral auxiliary as well as intermediates for the synthesis of optically active compounds. Optically active sulfoxide structural unit is also present in many compounds that exhibit interesting biological properties (Scheme \(\PageIndex{1}\)). Development of methods for the asymmetric sulfoxidation has thus been active topic in asymmetric catalysis. This lecture covers the common methods that are used for the synthesis of optically active sulfoxides.
5.4.1 Enzyme-Catalyzed Reactions
Enzyme catalyzed asymmetric oxidation of sulfides provides effective methods for the synthesis of optically active sulfoxides. For example, cyclohexanone monooxygenase (CHMO), a bacterial flavoenzyme, catalyzes the oxidation of prochiral thioethers with excellent enantioselectivity (Scheme \(\PageIndex{2}\)).
5.4.2 Chiral Reagents Based Reactions
Chiral reagents have been used for the oxidation of prochiral sulfides. For example, chiral hydroperoxides, N -sulfonyl oxaziridines and chiral oxaziridines can oxidize prochiral sulfides to optically active sulfoxides with moderate to good enantioselectivity (Scheme \(\PageIndex{3}\)).
In addition, chiral sulfinates are precursors of chiral sulfoxides (Scheme \(\PageIndex{4}\)). This approach is of preparative interest to provide the sulfoxides with high enantioselectivity. The important issue is need to prepare the menthyl- p -tolylsulfinates from L-(-)-menthol and then to separate them.
Furthermore, N -tosyl-norephedrine can be reacted with thionyl chloride to afford heterocyclic compound A, which could be reacted via the sequential addition of R 1 MgX and R 2 MgX, in a one-pot procedure to give sulfoxides in >99% ee (Scheme \(\PageIndex{5}\)). The configuration depends on the order of introduction of the two Grignard reagents.
5.4.3 Metal-Catalyzed Reactions
<IP_ADDRESS> Reactions with Diethyl Tartrates
In the middle of 1980, Kagan and Modena groups independently modified the conditions that were employed by Sharpless group for the asymmetric epoxidation of allylic alcohols, and used for the oxidation of sulfides. The modified conditions involve the combination of Ti(O i Pr) 4 , ( R,R)- diethyl tartrate (DET) and t -BuOOH (TBHP) in water (Scheme \(\PageIndex{6}\)). The replacement of TBHP with cumyl hydroperoxide (CHP) led to improvement in the enantioselectivity of the sulfoxide.
<IP_ADDRESS> Reactions with Tridentate Ligands
In the middle of 1990, vanadium complexes having the tridentate Schiff base ligands L1-2 derived from optically active amino alcohols and aryl aldehydes have been studied for the oxidation of sulfides in the presence of aq. H 2 O 2 as terminal oxidant (Scheme \(\PageIndex{7}\)). The catalysts are prepared in situ and the effect of series Schiff base ligands is studied.
In case of di- tert -butyldisulfide, monoxidation occurs selectively with up to > 90% ee (Scheme \(\PageIndex{8}\)).
Subsequently, the reaction has also been found to be effective with Fe(acac) 3 in the presence of additive such as p-methoxybenzoic acid (Scheme \(\PageIndex{9}\)). For example, the oxidation of p -chlorophenyl methyl sulfide can be accomplished with 92% ee and 60% yield. In some cases, kinetic resolution is observed.
<IP_ADDRESS> Reactions with Salen Based Ligands
Chiral Ti-salen has been found to be effective catalyst for the oxidation of sulfides in the presence of urea hydrogen peroxide (UHP) or aqueous H 2 O 2 . First, Ti-salen is converted into cis - μ -dioxo Ti-dimer that reacts with H 2 O 2 to give peroxo species. The latter can oxidize the sulfide to sulfoxide (Scheme \(\PageIndex{10}\)). The oxidation of several alkyl aryl sulfides can be accomplished with 92–99% ee.
Subsequently, Fe(salan) has been found to catalyze the oxidation of sulfides in the presence of a queous H 2 O 2 in water (Scheme \(\PageIndex{11}\)). This procedure has the advantages of high catalytic turnover number (TON) of 8000 as well as the use of water as reaction medium.
Furthermore, Al(salalen), which is compatible in water, catalyzes the oxidation of sulfides with aqueous H 2 O 2 at room temperature in phosphate buffer condition (Scheme \(\PageIndex{12}\)). The reactions of a variety of sulfides have been demonstrated with high enantioselectivity.
Meanwhile, chiral Ru(NO)-salen has been found to catalyze the sulfoxidation under aerobic conditions in the presence of water under visible light irradiation at room temperature (Scheme \(\PageIndex{13}\)). Unlike biological oxygen atom transfer reactions that need a proton and electron transfer system, this aerobic oxygen atom transfer reaction requires neither such a system nor a sacrificial reductant.
Although the mechanism of this oxidation has not been completely clarified, some experimental results support the notion that an aqua ligand coordinated with the ruthenium ion serves as a proton transfer agent for the oxygen activation process, and it is recycled and used as the proton transfer mediator during the process.


43: 48 Hours as a Muslim American: A Professor Reflects (Khalil)
Mohammad Hassan Khalil
#civilrights #politics #religion #currentevents #global
"Muhammad ALI Funeral Procession on 06-10-16" by Louisville Images is licensed under CC BY-NC-ND 4.0
What a difference 48 hours can make.
Last Friday afternoon, before a global audience, former president Bill Clinton (a Christian) and comedian Billy Crystal (a Jew) eulogized “the Greatest,” the most famous Muslim American of all time, Muhammad Ali.
The televised audience also took in Islamic invocations, recitations from the Qur’an, and, if they listened carefully, gleeful shouts of “Allahu akbar” (“God is the greatest”) from many of the thousands of attendees who packed Louisville’s KFC Yum! Sports Arena for the memorial service. President Barack Obama declared that Ali “will always be America.” And prior to his passing, a moment of silence in his honor was taken before tip-off at two NBA Finals games, once at Oracle Arena in Oakland, another time at Quicken Loans Arena in Cleveland.
On Sunday, Americans awoke to the tragic news that a young Muslim American had perpetrated the deadliest mass shooting in American history when he took the lives of dozens of innocents at Orlando’s Pulse nightclub.
His proclaimed affiliation to ISIS and reports of his homophobia and anger problems were all over the news.
Presidential candidate Donald Trump tweeted that the shooter reportedly shouted “Allahu akbar” during the horrific rampage. President Obama condemned what “was an act of terror and an act of hate.” And a moment of silence in honor of the victims was taken prior to tip-off at the most recent NBA Finals game in Oakland.
To say this past week has been a turbulent one would be an understatement.
Numerous Muslim Americans expressed how pleasantly surprised they were at seeing widely broadcast positive portrayals of their coreligionists in a nonetheless sobering Ali memorial.
At the Louisville, KY Islamic funeral service, the prominent Muslim American scholar Sherman Jackson had declared, “Ali put the question as to whether you could be a Muslim and an American to rest.” What is more, “Ali made being a Muslim cool.”
But with vile criminals such as the above-mentioned shooter (whose name I shall not bother to mention) dominating the headlines, it is easy to see why many Americans see Muslims as being very uncool.
Surveys show that non-Muslims who do not regularly interact with Muslims tend to have a significantly more negative impression of them. This should not come as a surprise given recent events and the media coverage of these events.
It certainly does not help that there exist numerous misconceptions and oversimplifications about both Islam and Muslims that are widely propagated online. (Of course, one could say something similar about America itself. If Islam and Muslims have a “public relations problem” in the West, the same is true for America and Americans in many countries overseas.)
Indeed, as a professor of religious studies, I find that I spend much of my time debunking popular myths.
The demographics
The reality is that with a population of over three million, most Muslim Americans are not nearly as “great” as Ali; and they are certainly nothing like the Orlando shooter. The reality is that most Muslims are everyday people.
Muslim Americans are extremely diverse: 63 percent are immigrants hailing from 77 countries . They are, on average, relatively young. Their levels of education mirror those of the U.S. population as a whole. As a Pew survey put it several years ago, they are “decidedly American in their outlook, values and attitudes.”
What is more, they have been here for a long time.
Coming to America
Some arrived on slave ships centuries ago. Others – including famous figures such as Ali, Malcolm X, basketball legend Kareem Abdul-Jabbar, U.S. Representative Keith Ellison, author G. Willow Wilson , singer Jermaine Jackson, comedian Dave Chappelle and a 19th-century writer and U.S. Consul to the Philippines named Alexander Russell Webb (1846-1916) – chose to convert to Islam.
Most – including numerous physicians, researchers, business owners and cab drivers – immigrated to this country to seek a better life, and, in the process, like other immigrants, made America a better country.
One such immigrant was the late Fazlur Khan , an architect originally from Bangladesh who designed the iconic Sears Tower (now called Willis Tower) and the John Hancock Center in Chicago. And just this past year, another immigrant, this time from Turkey, scientist Aziz Sancar, received the Nobel Prize, making him the second Muslim American Nobel Laureate in Chemistry. (The first was Egyptian-American Ahmed Zewail in 1999.)
Finally, many Muslims – from hip-hop artist Lupe Fiasco to U.S. Army Specialist and Purple Heart recipient Kareem Khan (who was killed in combat in 2007) – were born into Muslim families right here in the United States. According to the Pentagon, there are almost 6,000 Muslims currently serving in the U.S. military.
The Islamic State does not even account for one percent of one percent of the worldwide Muslim population. And among the many devout Muslim Americans I know personally, I have yet to meet or even hear about a single ISIS sympathizer; I see only looks of disgust whenever they’re mentioned.
Such sympathizers obviously exist. But this relatively small collection of individuals represent only themselves.
A friend on Facebook shared a story about how he came to learn about the Orlando shootings. He was passing through an airport when he noticed a crowd huddled around a television screen. When it was revealed that the shooter came from a Muslim family, a man in the crowd remarked, “Those damn Muslims.”
Three million diverse, overwhelmingly peaceful and productive Muslim Americans reduced to “those damn Muslims.”
Interestingly, just hours before the attack in Orlando I was discussing the Ali memorial service with a group of Muslim friends. Though sad about the passing of “the Greatest,” they all had smiles on their faces as they recounted the speeches from the service and imagined the effects those speeches might have on the broader American public. They had never felt better represented. At that moment, at least, they felt cool.
____________________
Mohammad Hassan Khalil is an associate professor of Religious Studies, an adjunct professor of Law, and Director of the Muslim Studies Program at Michigan State University. He is the is author of Islam and the Fate of Others: The Salvation Question (Oxford University Press, 2012) and editor of Between Heaven and Hell: Islam, Salvation, and the Fate of Others (Oxford University Press, 2013). His essay was previously published in The Conversation.
48 Hours as a Muslim American: A Professor Reflects by
Mohammad Hassan Khalil
is licensed under a
Creative Commons Attribution-NoDerivatives 4.0 International License
.


3.1: System Software Learning Objectives
Learning Objectives
If you do well in this unit, you should be able to:
- Distinguish between Operating Systems software and Application Systems software.
- Describe commonly used operating systems.
- Identify the primary functions of an Operating System.
- Describe the “boot” process.
- Identify Desktop and Windows features.
- Use Utility programs.
- Discuss the pros and cons of the three major operating systems.
CC licensed content, Original
- Introduction to Computer Applications and Concepts System Software Objectives. Authored by : Jim Shannon. Provided by : Extended Learning Institute of Northern Virginia Community College. Located at : http://www.nvcc.edu/eli/index.html . License : CC BY: Attribution


3.5: Conclusions and Review Questions
Conclusions
The opportunity to apply green chemistry to reducing the toxicity in industrial processes is of paramount importance for a sustainable future. In this chapter, we investigated the concept of hazards from the perspective of their meaning, nature, implications, control, and ultimate reengineering. Green chemical principles hold the key to ensuring that society provides a viable channel for all of our physical needs and maintains the environment.
Review Questions
- Think about the additive impact of hazards from a LD50 perspective. Why would materials with different values not be additive in their cumulative effects on lethality? Can you think of two chemicals, however, whose lethality may be more than the sum of their individual values?
- Would phosgene be more or less reactive as a function of relative humidity? Why or why not? In a wartime situation, for example, would an opposing army to whom phosgene is being directed favor a low humidity?
- Can phosgene be reacted with sodium azide in a 1:2 molar ratio? If so, what would be the reaction and product? If not, why not?
- What do these GHS labels tell us about ammonia? (See Figure \(\PageIndex{1}\) )
Further Reading
- blogs.ei.columbia.edu/2014/12...try-theanswer/
- https://play.google.com/books/reader...ec=frontcover& output=reader&hl=en_US&pg=GBS.PP1(Bretherick'sHandbookofReactive Chemical Hazards: Edition 4, L. Bretherick, October 22, 2013, Elsevier)
- http://ccc.chem.pitt.edu/wipf/Web/HCH.pdf (Hazardous Chemicals Handbook)


4.1: Cyclic Groups
Recall that if \(G\) is a group and \(g\in G\) , then the cyclic subgroup generated by \(g\) is given by \[\langle g\rangle =\{g^k\mid k\in\mathbb{Z}\}.\] It is important to point out that \(\langle g\rangle\) may be finite or infinite. In the finite case, the Cayley diagram with generator \(g\) gives us a good indication of where the word “cyclic" comes from (see Problem \(\PageIndex{11}\)). If there exists \(g\in G\) such that \(G=\langle g\rangle\) , then we say that \(G\) is a cyclic group .
List all of the elements in each of the following cyclic subgroups.
- \(\langle r\rangle\) , where \(r\in D_3\)
- \(\langle r\rangle\) , where \(r\in R_4\)
- \(\langle rs\rangle\) , where \(rs\in D_4\)
- \(\langle r^2\rangle\) , where \(r^2\in R_6\)
- \(\langle i\rangle\) , where \(i\in Q_8\)
- \(\langle 6\rangle\) , where \(6\in \mathbb{Z}\) and the operation is ordinary addition
Consider the group of invertible \(2\times 2\) matrices with real number entries under the operation of matrix multiplication. This group is denoted by \(\mathrm{GL}_2(\mathbb{R})\) . List the elements in the cyclic subgroups generated by each of the following matrices.
- \(\begin{bmatrix} 0 & -1\\ -1 & 0\end{bmatrix}\)
- \(\begin{bmatrix} 0 & -1\\ 1 & 0\end{bmatrix}\)
- \(\begin{bmatrix} 2 & 0\\ 0 & 1\end{bmatrix}\)
Determine whether each of the following groups is cyclic. If the group is cyclic, find at least one generator.
- \(S_2\)
- \(R_3\)
- \(R_4\)
- \(V_4\)
- \(R_5\)
- \(R_6\)
- \(D_3\)
- \(R_7\)
- \(R_8\)
- \(\text{Spin}_{1\times 2}\)
- \(D_4\)
- \(Q_8\)
Determine whether each of the following groups is cyclic. If the group is cyclic, find at least one generator. If you believe that a group is not cyclic, try to sketch an argument.
- \((\mathbb{Z},+)\)
- \((\mathbb{R},+)\)
- \((\mathbb{R}^+,\cdot)\)
- \((\{6^n\mid n\in\mathbb{Z}\},\cdot)\)
- \(\textrm{GL}_2(\mathbb{R})\) under matrix multiplication
- \(\{(\cos(\pi/4) +i\sin(\pi/4))^n\mid n\in \mathbb{Z}\}\) under multiplication of complex numbers
If \(G\) is a cyclic group, then \(G\) is abelian.
Provide an example of a finite group that is abelian but not cyclic.
Provide an example of an infinite group that is abelian but not cyclic.
If \(G\) is a group and \(g\in G\) , then \(\langle g\rangle=\langle g^{-1}\rangle\) .
If \(G\) is a cyclic group such that \(G\) has exactly one element that generates all of \(G\) , then the order of \(G\) is at most order 2.
If \(G\) is a group such that \(G\) has no proper nontrivial subgroups, then \(G\) is cyclic.
Recall that the order of a group \(G\) , denoted \(|G|\) , is the number of elements in \(G\) . We define the order of an element \(g\) , written \(|g|\) , to be the order of \(\langle g\rangle\) . That is, \(|g|=|\langle g\rangle|\) . It is clear that \(G\) is cyclic with generator \(g\) if and only if \(|G|=|g|\) .
What is the order of the identity in any group?
Find the orders of each of the elements in each of the groups in Problem \(\PageIndex{3}\).
Consider the group \((\mathbb{Z},+)\) . What is the order of 1? Are there any elements in \(\mathbb{Z}\) with finite order?
Find the order of each of the matrices in Problem \(\PageIndex{2}\).
The next result follows immediately from Theorem \(\PageIndex{2}\).
If \(G\) is a group and \(g\in G\) , then \(|g|=|g^{-1}|\) .
The next result should look familiar and will come in handy a few times in this chapter. We’ll take the result for granted and not worry about proving it.
If \(n\) is a positive integer and \(m\) is any integer, then there exist unique integers \(q\) (called the quotient ) and \(r\) (called the remainder ) such that \(m=nq+r\) , where \(0\leq r<n\) .
Suppose \(G\) is a group and let \(g\in G\) . The subgroup \(\langle g\rangle\) is finite if and only if there exists \(n\in\mathbb{N}\) such that \(g^n=e\) .*
- *
-
For the forward implication, if \(\langle g\rangle\) is finite, then there exists distinct positive integers \(i\) and \(j\) such that \(g^i=g^j\) . Can you find a useful way to rewrite this equation? For the reverse implication, let \(m\in\mathbb{Z}\) and use the Division Algorithm with \(m\) and \(n\) .
If \(G\) is a finite group, then for all \(g\in G\) , there exists \(n\in\mathbb{N}\) such that \(g^n=e\) .
Suppose \(G\) is a group and let \(g\in G\) such that \(\langle g\rangle\) is a finite group. If \(n\) is the smallest positive integer such that \(g^n=e\) , then \(\langle g\rangle = \{e, g, g^2, \ldots, g^{n-1}\}\) and this set contains \(n\) distinct elements. *
- *
-
Note that Theorem \(\PageIndex{7}\) together with the Well-Ordering Principle guarantees the existence of a smallest positive integer \(n\) such that \(g^n=e\) . The claim that the set contains \(n\) distinct elements is not immediate. You need to argue that there are no repeats in the list. Choose distinct \(i,j\in\{0,1,\ldots,n-1\}\) such that \(i\neq j\) and then show that \(g^i\neq g^j\) . Consider a proof by contradiction and try to contradict the minimality of \(n\) .
The next result provides an extremely useful interpretation of the order of an element.
If \(G\) is a group and \(g\in G\) such that \(\langle g\rangle\) is a finite group, then the order of \(g\) is the smallest positive integer \(n\) such that \(g^n=e\) .
Suppose \(G\) is a finite cyclic group such that \(G=\langle g\rangle\) . Using the generating set \(\{g\}\) , what does the Cayley diagram for \(G\) look like?
Suppose \(G\) is a finite cyclic group of order \(n\) with generator \(g\) . If we write down the group table for \(G\) using \(e, g, g^2, \ldots, g^{n-1}\) as the labels for the rows and columns, are there any interesting patterns in the table?
Notice that in the definition for \(\langle g\rangle\) , we allow the exponents on \(g\) to be negative. Explain why we only need to use positive exponents when \(\langle g\rangle\) is a finite group
The Division Algorithm should come in handy when proving the next theorem.
Suppose \(G\) is a group and let \(g\in G\) such that \(|g|=n\) . Then \(g^i=g^j\) if and only if \(n\) divides \(i-j\) .
Suppose \(G\) is a group and let \(g\in G\) such that \(|g|=n\) . If \(g^k=e\) , then \(n\) divides \(k\) .
Recall that for \(n\geq3\) , \(R_n\) is the group of rotational symmetries of a regular \(n\) -gon, where the operation is composition of actions.
For all \(n\geq 3\) , \(R_n\) is cyclic.
Suppose \(G\) is a finite cyclic group of order \(n\) . Then \(G\) is isomorphic to \(R_n\) if \(n\geq 3\) , \(S_2\) if \(n=2\) , and the trivial group if \(n=1\) .
Most of the previous results have involved finite cyclic groups. What about infinite cyclic groups?
Suppose \(G\) is a group and let \(g\in G\) . The subgroup \(\langle g\rangle\) is infinite if and only if each \(g^k\) is distinct for all \(k\in\mathbb{Z}\) .*
- *
-
For the forward implication, try a proof by contradiction and suppose there exists integers \(i\) and \(j\) such that \(g^i=g^j\) .
If \(G\) is an infinite cyclic group, then \(G\) is isomorphic to \(\mathbb{Z}\) (under the operation of addition).
The upshot of Theorems \(\PageIndex{13}\) and \(\PageIndex{11}\) is that up to isomorphism, we know exactly what all of the cyclic groups are.
We now turn our attention to two new groups. Recall that two integers are relatively prime if the only positive integer that divides both of them is 1. That is, integers \(n\) and \(k\) are relatively prime if and only if \(\gcd(n,k)=1\) .
Let \(n\in\mathbb{N}\) and define the following sets.
- \(\mathbb{Z}_n:=\{0,1,\ldots,n-1\}\)
- \(U_n:=\{k\in\mathbb{Z}_n\mid \gcd(n,k)=1\}\)
For example, \(\mathbb{Z}_{12}=\{0,1,2,3,4,5,6,7,8,9,10,11\}\) while \(U_{12}=\{1,5,7,11\}\) since 1, 5, 7, and 11 are the only elements in \(\mathbb{Z}_{12}\) that are relatively prime to 12.
For each set in Definition: \(n\in\mathbb{N}\) , the immediate goal is to determine a binary operation that will yield a group. The key is to use modular arithmetic. Let \(n\) be a positive integer. To calculate the sum (respectively, product) of two integers modulo \(n\) (we say “mod \(n\) " for short), add (respectively, multiply) the two numbers and then find the remainder after dividing the sum (respectively, product) by \(n\) . For example, \(4+9\) is \(3\) mod \(5\) since \(13\) has remainder 3 when divided by 5. Similarly, \(4\cdot 9\) is 1 mod \(5\) since 36 has remainder 1 when divided by 5. The hope is that these two operations turn \(\mathbb{Z}_n\) and \(U_n\) into groups.
We write \(i\equiv j\pmod n\) , and say “ \(i\) is equivalent to \(j\) modulo \(n\) " or “ \(i\) is equal to \(j\) modulo \(n\) ", if \(i\) and \(j\) both have the same remainder when divided by \(n\) . It is common to abbreviate “modulo" as “mod". It is also common to write \(i\equiv_n j\) , or even \(i=j\) if the context is perfectly clear.
It is well-known, and not too hard to prove, that \(\equiv_n\) is an equivalence relation on \(\mathbb{Z}\) . The corresponding equivalence classes are called congruence classes. The elements of a single congruence class are the integers that all have the same remainder when divided by \(n\) . According to the Division Algorithm, there are \(n\) congruence classes modulo \(n\) , one for each of the remainders \(0,1,\ldots, n-1\) . We can think of \(\mathbb{Z}_n\) as the set of canonical representatives of these equivalence classes.
Let \(n\) be a positive integer and let \(i,j\in\mathbb{Z}\) . Then \(i\equiv j\pmod n\) if and only if \(n\) divides \(i-j\) .
The next result follows immediately from Theorems \(\PageIndex{14}\) and \(\PageIndex{9}\).
Suppose \(G\) is a group and let \(g\in G\) such that \(|g|=n\) . Then \(g^i=g^j\) if and only if \(i\equiv j\pmod n\) .
The set \(\mathbb{Z}_n\) is a cyclic group under addition mod \(n\) .*
- *
-
There are two things to prove here. First, you need to prove that \(\mathbb{Z}_n\) is a group under addition mod \(n\) , and then you need to argue that the group is cyclic.
The set \(U_n\) is an abelian group under multiplication mod \(n\) .
- *
-
Like the previous theorem, there are two things to prove. First, prove that \(U_n\) is a group under multiplication mod \(n\) , and then argue that the group is abelian.
Consider \(\mathbb{Z}_4\) .
- Find the group table for \(\mathbb{Z}_4\) .
- Is \(\mathbb{Z}_4\) cyclic? If so, list elements of \(\mathbb{Z}_4\) that individually generate \(\mathbb{Z}_4\) . If \(\mathbb{Z}_4\) is not cyclic, explain why.
- Is \(\mathbb{Z}_4\) isomorphic to either of \(R_4\) or \(V_4\) ? Justify your answer.
- Draw the subgroup lattice for \(\mathbb{Z}_4\) .
The next two problems illustrate that \(U_n\) may or may not be cyclic.
Consider \(U_{10}=\{1,3,7,9\}\) .
- Find the group table for \(U_{10}\) .
- Is \(U_{10}\) cyclic? If so, list elements of \(U_{10}\) that individually generate \(U_{10}\) . If \(U_{10}\) is not cyclic, explain why.
- Is \(U_{10}\) isomorphic to either of \(R_4\) or \(V_4\) ? Justify your answer.
- Is \(U_{10}\) isomorphic to \(\mathbb{Z}_4\) ? Justify your answer.
- Draw the subgroup lattice for \(U_{10}\) .
Consider \(U_{12}=\{1,5,7,11\}\) .
- Find the group table for \(U_{12}\) .
- Is \(U_{12}\) cyclic? If so, list elements of \(U_{12}\) that individually generate \(U_{12}\) . If \(U_{12}\) is not cyclic, explain why.
- Is \(U_{12}\) isomorphic to either of \(R_4\) or \(V_4\) ? Justify your answer.
- Draw the subgroup lattice for \(U_{12}\) .
The upshot of the next theorem is that for \(n\geq 3\) , \(\mathbb{Z}_n\) is just the set of exponents in the set \(R_n=\{e,r,r^2,\ldots,r^{n-1}\}\) (where \(e=r^0\) ).
\mathbb{Z}_n\) Isomorphic to \(R_n\)
For \(n\geq 3\) , \(\mathbb{Z}_n\cong R_n\) . Moreover, \(\mathbb{Z}_2\cong S_2\) and \(\mathbb{Z}_1\) is isomorphic to the trivial group.
The next result can be thought of as a repackaging of Theorems \(\PageIndex{11}\) and \(\PageIndex{13}\).
Let \(G\) be a cyclic group. If the order of \(G\) is infinite, then \(G\) is isomorphic to \(\mathbb{Z}\) . If \(G\) has finite order \(n\) , then \(G\) is isomorphic to \(\mathbb{Z}_n\) .
Now that we have a complete description of the cyclic groups, let’s focus our attention on subgroups of cyclic groups.
Suppose \(G\) is a cyclic group. If \(H\leq G\) , then \(H\) is also cyclic.
It turns out that for proper subgroups, the converse of Theorem \(\PageIndex{19}\) is not true.
Provide an example of a group \(G\) such that \(G\) is not cyclic, but all proper subgroups of \(G\) are cyclic.
The next result officially settles Problem 3.1.11(d) and also provides a complete description of the subgroups of infinite cyclic groups up to isomorphism.
The subgroups of \(\mathbb{Z}\) are precisely the groups \(n\mathbb{Z}\) for \(n\in \mathbb{Z}\) .
Let’s further explore finite cyclic groups.
If \(G\) is a finite cyclic group with generator \(g\) such that \(|G|=n\) , then for all \(m\in\mathbb{Z}\) , \(\displaystyle |g^m|=\dfrac{n}{\gcd(n,m)}\) .*
- *
-
By Corollary \(\PageIndex{2}\), the order of \(g^m\) is the smallest positive exponent \(k\) such that \((g^m)^k=e\) . First, verify that \(k=\dfrac{n}{\gcd(n,m)}\) has the desired property and then verify that it is the smallest such exponent.
If \(G\) is a finite cyclic group with generator \(g\) such that \(|G|=n\) , then \(\langle g^m\rangle=\langle g^k\rangle\) if and only if \(\gcd(m,n)=\gcd(k,n)\).*
- *
-
Use Theorem \(\PageIndex{20}\) for the forward implication. For the reverse implication, first prove that for all \(m\in\mathbb{Z}\) , \(\langle g^m\rangle=\langle g^{\gcd(m,n)}\rangle\) by proving two set containments. To show \(\langle g^m\rangle\subseteq \langle g^{\gcd(m,n)}\rangle\) , use the fact that there exists an integer \(q\) such that \(m=q\cdot \gcd(m,n)\) . For the reverse containment, you may freely use a fact known as Bezout’s Lemma, which states that \(\gcd(m,n)=nx+my\) for some integers \(x\) and \(y\). *
Suppose \(G\) is a cyclic group of order 12 with generator \(g\) .
- Find the orders of each of the following elements: \(g^2\) , \(g^7\) , \(g^8\) .
- Which elements of \(G\) individually generate \(G\) ?
Suppose \(G\) is a finite cyclic group with generator \(g\) such that \(|G|=n\) . Then \(\langle g\rangle=\langle g^k\rangle\) if and only if \(n\) and \(k\) are relatively prime. That is, \(g^k\) generates \(G\) if and only if \(n\) and \(k\) are relatively prime.
Theorem \(\PageIndex{20}\), Theorem \(\PageIndex{21}\), and Corollary \(\PageIndex{6}\) are written using multiplicative notation. Rewrite both of these results using additive notation.
Consider \(\mathbb{Z}_{18}\) .
- Find all of the elements of \(\mathbb{Z}_{18}\) that individually generate all of \(\mathbb{Z}_{18}\) .
- Draw the subgroup lattice for \(\mathbb{Z}_{18}\) . For each subgroup, list the elements of the corresponding set. Moreover, circle the elements in each subgroup that individually generate that subgroup. For example, \(\langle 2\rangle=\{0,2,4,6,8,10,12,14,16\}\) . In this case, we should circle 2, 4, 8, 10, 14, and 16 since each of these elements individually generate \(\langle 2\rangle\) and none of the remaining elements do. I’ll leave it to you to figure out why this is true.
Repeat the above exercise, but this time use \(\mathbb{Z}_{12}\) instead of \(\mathbb{Z}_{18}\) .
If \(G\) is a finite cyclic group such that \(|G|=p\) , where \(p\) is prime, then \(G\) has no proper nontrivial subgroups.
If there is exactly one group up to isomorphism of order \(n\) , then to what group are all the groups of order \(n\) isomorphic?
Suppose \(G\) is a group and \(x,y\in G\) such that \(|x|=m\) and \(|y|=n\) . Is it true that \(|xy|=mn\) ? If this is true, provide a proof. If this is not true, then provide a counterexample.
The punchline of the next two theorems is Theorem \(\PageIndex{22}\).
Suppose \(G\) is a finite abelian group and let \(x,y\in G\) such that \(|x|=m\) and \(|y|=n\) . If \(\gcd(m,n)=1\) , then \(|xy|=mn\) .*
- *
-
Hint: First, verify that \((xy)^{mn}=e\) . Now, suppose \(|xy|=k\) . What do you immediately know about the relationship between \(k\) and \(mn\) . Next, consider \((xy)^{kn}\) . Argue that \(m\) divides \(kn\) and then argue that \(m\) divides \(k\) . Similarly, \(n\) divides \(k\) . Ultimately, conclude that \(mn=k\) .
Suppose \(G\) is a finite abelian group. If \(n\) is the maximal order among all elements in \(G\) , then the order of every element in \(G\) divides \(n\).*
- *
-
Hint: Suppose \(g\in G\) such that \(|g|=n\) . Let \(h\) be an arbitrary element in \(G\) such that \(|h|=m\) . You need to show that \(m\) divides \(n\) . For sake of a contradiction, assume otherwise. Then there exists a prime \(p\) whose multiplicity as a factor of \(m\) exceeds that of \(n\) . Let \(p^a\) be the highest power of \(p\) in \(m\) and \(p^b\) be the highest power of \(p\) in \(n\) , so \(a>b\) . Consider the elements \(g^{p^a}\) and \(h^{m/p^b}\) .
Recall that every cyclic group is abelian (see Theorem \(\PageIndex{1}\)). However, we know that not every abelian group is cyclic (see Problem \(\PageIndex{5}\)). The next theorem tells us that abelian groups with some additional properties are cyclic.
If \(G\) is a finite abelian group with at most one subgroup of any order, then \(G\) is cyclic.
- *
-
Hint: Let \(n\) be the maximal order among the elements of \(G\) and let \(g\in G\) be an element with order \(n\) . Prove that \(G=\langle g\rangle\) .
Is the converse of Theorem \(\PageIndex{22}\) true for finite groups? That is, if \(G\) is a finite cyclic group, does that imply that \(G\) contains at most one subgroup of each order? If the answer is yes, then prove it. Otherwise, provide a counterexample.
We conclude this section with a couple interesting counting problems involving the number of generators of certain cyclic groups.
Let \(p\) and \(q\) be distinct primes. Find the number of generators of \(\mathbb{Z}_{pq}\) .
Let \(p\) be a prime. Find the number of generators of \(\mathbb{Z}_{p^r}\) , where \(r\) is an integer greater than or equal to 1.


20.2: Banking
Example \(\PageIndex{1}\): Going around a banked curve
Roadway engineers often bank a curve, especially if it is a very tight turn, so the cars will not have to rely on friction alone to provide the required centripetal force. The picture shows a car going around such a curve, which we can model as an arc of a circle of radius \(r\). In terms of \(r\), the bank angle \(\theta\), and the coefficient of static friction, find the maximum safe speed around the curve.
The figure shows the appropriate choice of axes for this problem. The criterion is, again, to choose the axes so that one of them will coincide with the direction of the acceleration. In this case, the acceleration is all centripetal, that is to say, pointing, horizontally, towards the center of the circle on which the car is traveling.
It may seem strange to see the force of static friction pointing down the slope, but recall that for a car turning on a flat surface it would have been pointing inwards (towards the center of the circle), so this is the natural extension of that. In general, you should always try to imagine which way the object would slide if friction disappeared altogether: \(\vec F^s\) must point in the direction opposite that. Thus, for a car traveling at a reasonable speed, the direction in which it would skid is up the slope, and that means \(\vec F^s\) must point down the slope. But, for a car just sitting still on the tilted road, \(\vec F^s\) must point upwards, and we shall see in a moment that in general there is a minimum velocity required for the force of static friction to point in the direction we have chosen.
Apart from this, the main difference with the flat surface case is that now the normal force has a component along the direction of the acceleration, so it helps to keep the car moving in a circle. On the other hand, note that we now lose (for centripetal purposes) a little bit of the friction force, since it is pointing slightly downwards. This, however, is more than compensated for by the fact that the normal force is greater now than it would be for a flat surface, since the car is now, so to speak, “driving into” the road somewhat.
The dashed blue lines in the free-body diagram are meant to indicate that the angle \(\theta\) of the bank is also the angle between the normal force and the positive \(y\) axis, as well as the angle that \(\vec F^s\) makes below the positive \(x\) axis. It follows that the components of these two forces along the axes shown are:
\begin{align}
&F_{x}^{n}=F^{n} \sin \theta \nonumber \\
&F_{y}^{n}=F^{n} \cos \theta \label{eq:8.55}
\end{align}
and
\begin{align}
&F_{x}^{s}=F^{s} \cos \theta \nonumber \\
&F_{y}^{n}=-F^{s} \sin \theta \label{eq:8.56}
\end{align}
The vertical force equation is then:
\[ 0=m a_{y}=F_{y}^{n}+F_{y}^{s}-F^{G}=F^{n} \cos \theta-F^{s} \sin \theta-m g \label{eq:8.57} .\]
This shows that \(F^n = (mg + F^s \sin \theta)/ \cos \theta \) is indeed greater than just \(mg\) for this problem, and must increase as the angle \(\theta\) increases (since \(\cos \theta\) decreases with increasing \(\theta\)). The horizontal equation is:
\[ m a_{x}=F_{x}^{n}+F_{x}^{s}=F^{n} \sin \theta+F^{s} \cos \theta=\frac{m v^{2}}{r} \label{eq:8.58} \]
where I have already substituted the value of the centripetal acceleration for \(a_x\). Equations (\ref{eq:8.57}) and (\ref{eq:8.58}) form a system that needs to be solved for the two unknowns \(F^n\) and \(F^s\). The result is:
\begin{align}
&F^{n}=m g \cos \theta+\frac{m v^{2}}{r} \sin \theta \nonumber \\
&F^{s}=-m g \sin \theta+\frac{m v^{2}}{r} \cos \theta \label{eq:8.59}.
\end{align}
Note that the second equation would have \(F^s\) becoming negative if \( v^{2}<g r \tan \theta \). This means that below that speed, the force of static friction must actually point up the slope, as discussed above. We can call this particular speed, for which \(F^s\) becomes zero, \(v_{no \: friction}\):
\[ v_{\text {no friction }}=\sqrt{g r \tan \theta} \label{eq:8.60} .\]
What this means is that it is possible to arrange the banking angle so that a car going at a specific speed would not have to rely on friction at all in order to make the curve: the normal force would be just right to provide the required centripetal acceleration. A car going at that speed would not feel either pulled down or pushed up the slope. However, a car going faster than that would tend to “fly off”, and the static friction force would be required to pull it in and keep it on the curve, whereas a car moving more slowly would tend to slide down and would have to be pushed up by the friction force. Friction, therefore, provides a range of safe speeds to drive in this case, just as it did in the flat surface case.
We can calculate the maximum safe speed as we did before, recalling that we must always have \(F^{s} \leq \mu_{s} F^{n}\). Substituting Eqs. (\ref{eq:8.59}) in this expression, and solving for \(v\), we get the condition
\[ v_{\max }=\sqrt{g r} \sqrt{\frac{\mu_{s}+\tan \theta}{1-\mu_{s} \tan \theta}} \label{eq:8.61} .\]
This reproduces our result ( 8.4.5 ) for \(\theta\) = 0 (a flat road), as it should.
To put some numbers into this, suppose the curve has a radius of 20 m, and the coefficient of static friction between the tires and the road is \(\mu_s\) = 0.7. Then, for a flat surface, we get \(v_{max}\) = 11.7 m/s, or about 26 mph, whereas for a bank angle of \(\theta\) = 10\(^{\circ}\) (the angle chosen for the figure above) we get \(v_{max}\) = 14 m/s, or about 31 mph.
Equation (\ref{eq:8.61}) actually indicates that the maximum velocity would “become infinite” for a finite bank angle, namely, if \(1 − \mu_s \tan \theta = 0\), or \(\tan \theta = 1/ \mu_s\) (if \(\mu_s\) = 0.7, this corresponds to \(\theta\) = 55\(^{\circ}\)). This is mathematically correct, but of course we cannot take it literally: it assumes that there is no limit to how large a normal force the roadway may exert without sustaining damage, and also that \(F^s\) can become arbitrarily large as long as it stays below the bound \(F^{s} \leq \mu_{s} F^{n} \). Neither of these assumptions would hold in real life for very large speeds. Also, the angle \(\theta=\tan ^{-1}\left(1 / \mu_{s}\right) \) is much too steep: recall that, according to Equation ( 8.3.11 ), the force of friction will only be able to keep an object (initially at rest) from sliding down the slope if \(\tan \theta \leq \mu_{s}\), which for \(\mu_s\) = 0.7 means \(\theta \leq 35^{\circ}\). So, with a bank angle of 55\(^{\circ}\) you might drive on the curve, provided you were going fast enough, but you could not park on it—the car would slide down! Bottom line, use Equation (\ref{eq:8.61}) only for moderate values of \(\theta\)... and do not exceed \(\theta=\tan ^{-1} \mu_{s}\) if you want a car to be able to drive around the curve slowly without sliding down into the ditch.
Example \(\PageIndex{2}\): Staying on track
(This example studies a situation that you could easily setup experimentally at home (you can use a whole sphere instead of a half-sphere!), although to get the numbers to work out you really need to make sure that the friction between the surface and the object you choose is truly negligible. Essentially the same mathematical approach could be used to study the problem of a skier going over a mogul, or a car losing contact with the road if it is going too fast over a hill.)
A small object is placed at the top of a smooth (frictionless) dome shaped like a half-sphere of radius \(R\), and given a small push so it starts sliding down the dome, initially moving very slowly \(\left(v_{i} \simeq 0\right)\), but picking up speed as it goes, until at some point it flies off the surface.
- At that point, when the object loses contact with the surface, what is the angle that its position vector (with origin at the center of the sphere) makes with the vertical?
- How far away from the sphere does the object land?
Solution
(a) As we saw in Section 8.4 , in order to get an object to move along an arc of a circle, a centripetal force of magnitude \(mv^2/r\) is required. As long as our object is in contact with the surface, the forces acting on it are the normal force (which points along the radial direction, so it makes a negative contribution to the centrifugal force) and gravity, which has a component \(mg \cos \theta\) along the radius, towards the center of the circle (see Figure \(\PageIndex{1}\)(b), the dashed light blue line). So, the centripetal force equation reads
\[ \frac{m v^{2}}{R}=m g \cos \theta-F^{n} \label{eq:8.46} .\]
The next thing we need to do is find the value of the speed v for a given angle \(\theta\). If we treat the object as a particle, its only energy is kinetic energy, and \(\Delta K = W_{net}\) (Equation ( 7.2.8 )), where \(W_{net}\) is the work done on the particle by the net force acting on it. The normal force is always perpendicular to the displacement, so it does no work, whereas gravity is always vertical and does work \(W_{grav} = −mg\Delta y\) (taking upwards as positive, so \(\Delta y\) is negative). In fact, from Figure \(\PageIndex{1}\)(a) (follow the dashed blue line) you can see that for a given angle \(\theta\), the height of the object above the ground is \(R \cos \theta\), so the vertical displacement from its initial position is
\[ \Delta y=-(R-R \cos \theta) \label{eq:8.47} \]
Hence we have, for the change in kinetic energy,
\[ \frac{1}{2} m v^{2}-\frac{1}{2} m v_{i}^{2}=m g R-m g R \cos \theta \label{eq:8.48} .\]
Assuming, as we are told in the text of the problem, that \(v_{i} \simeq 0\), we get \(v^2 \simeq 2gR − 2gR \cos \theta \), and using this in Equation (\ref{eq:8.46})
\[ 2 m g-2 m g \cos \theta=m g \cos \theta-F^{n} \label{eq:8.49} \]
or \(F^n = 3mg \cos \theta −2mg\). This shows that \(F^n\) starts out (when \(\theta\) = 0) having its usual value of \(mg\), and then it becomes progressively smaller as the object slides down. The point where the object loses contact with the surface is when \(F^n\) = 0, and that happens for
\[ 3 \cos \theta_{max} = 2 \label{eq:8.50} \]
or \(\theta_{\max }=\cos ^{-1}(2 / 3)=48.2^{\circ} \).
Recalling that \(\Delta y = −(R − R \cos \theta)\), we see that when \( \cos \theta\) = 2/3, the object has fallen a distance \(R\)/3; put otherwise, its height above the ground at the time it flies off is 2\(R\)/3, or 2/3 of the initial height.
(b) This is just a projectile problem now. We just have to find the values of the initial conditions (\(x_i\), \(y_i\), \(v_{x,i}\) and \(v_{y,i}\)) and substitute in the equations ( 16.2.5 ). By inspecting the figure, you can see that, at the time the object flies off,
\begin{align}
&x_{i}=R \sin \theta_{\max }=0.745 R \nonumber \\
&y_{i}=R \cos \theta_{\max }=0.667 R \label{e:8.51}.
\end{align}
Also, we found above that \(v^{2} \simeq 2 g R-2 g R \cos \theta\), and when \(\theta = \theta_{max}\) this gives \(v^2 = 0.667gR\), or \(v=0.816 \sqrt{g R} \). The projection angle in this case is \(−\theta_{max}\); that is, the initial velocity of the projectile (dashed red arrow in Figure \(\PageIndex{1}\)(a)) is at an angle 48.2\(^{\circ}\) below the positive \(x\) axis, so we have:
\begin{align}
&v_{x, i}=v_{i} \cos \theta_{\max }=0.544 \sqrt{g R} \nonumber \\
&v_{y, i}=-v_{i} \sin \theta_{\max }=-0.609 \sqrt{g R} \label{eq:8.52}
\end{align}
Now we just use these results in Eqs. ( 16.2.5 ). Specifically, we want to know how long it takes for the object to reach the ground, so we use the last equation ( 16.2.5 ) with \(y\) = 0 and solve for \(t\):
\[ 0=y_{i}+v_{y, i} t-\frac{1}{2} g t^{2} \label{eq:8.53} \]
The result is \(t=0.697 \sqrt{R / g}\). (You do not need to carry the “\(g\)” throughout; it would be OK to substitute 9.8 m/s 2 for it. I have just kept it in symbolic form so far to make it clear that the quantities we derive will have the right units.) Substituting this in the equation for \(x\), we get
\[ x=x_{i}+v_{x, i} t=0.745 R+0.544 \sqrt{g R} \times 0.697 \sqrt{R / g}=1.125 R \label{eq:8.54} \]
(Note how the \(g\) cancels, so we would get the same result on any planet!) Since the sphere has a radius \(R\), the object falls a distance \(0.125R\) away from the sphere.


9.2: Planning a Diet
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The definition of diet is anything that is consumed by a particular person or people on a regular basis. That means if someone routinely drinks coffee in the morning, that is part of his/her diet. If a person consistently eats a Big Mac from McDonald’s, that is part of his/her diet. However, it is clear that food choices influence short-term and long-term health. That is why it is so important to make wise choices in what one eats on a regular basis. If a person chooses to have a diet high in calories without balancing energy use, that person can expect to put on unhealthy weight. A diet that is high in fiber, with the appropriate amount of calories and proper amounts of the macronutrients, will contribute to a healthy body.
When people discuss “going on a diet,” they are actually talking about changing their existing dietary habits in order to change their body shape. All people are “on a diet” because everyone eats! Many times, the term diet is thought of as a method to lose weight or to change body shape. However, it is important to focus on the nutritional concepts listed below, so long-term health can be achieved.
Decisions about nutrition can be difficult. Knowing and using scientific research can lead to better health. Over time public health organizations have developed tools based on nutritional science to help people design healthy diets. These tools should be used as guidelines for each individual with the awareness that everyone is different and therefore has different needs. Everyone, regardless of age, size, shape, physique, can benefit from learning and utilizing the following tools:
Acceptable Macronutrient Distribution Range (AMDR)
The AMDR describes the proportions of daily caloric intake that should be carbohydrates, lipids, and proteins. Basically the AMDR provides guidelines on how many macronutrient calories one should consume a day.
According to the AMDR, the range of caloric intake in a daily diet should be:
- Carbohydrates: 45-65%
- Lipids: 20-35%
- Proteins: 10-35%
Dietary Reference Intakes (DRI)
The DRI’s are reference values of nutrient intake that help with nutrition planning and assessment of healthy individuals. There are four measures that together comprise the DRI:
- Recommended Dietary Allowance (RDA): the average daily dietary intake level that is sufficient to meet the nutrient requirement of nearly all (about 97%) healthy individuals in a group. This is the basic quantity of a nutrient recommended.
- Adequate Intake (AI): a value based on observed or experimentally determined approximations of nutrient intake by a group (or groups) of healthy people—used when an RDA cannot be determined. This is the minimum amount of a nutrient needed for maintaining health.
- Tolerable Upper Intake Level (UL): the highest level of daily nutrient intake that is likely to pose no risk of adverse health effects to almost all individuals in the general population. As intake increases above the UL, the risk of adverse effects increases. This is the maximum that would be consumed prior to developing negative effects of eating too much. This is not a level that is met, but rather one that is avoided to prevent a decrease in health.
- Estimated Average Requirement (EAR): a nutrient intake value that is estimated to meet the requirement of half the healthy individuals in a group. These nutrient values should be used as goals for dietary intake for health.
4 Key Concepts for Personalizing a Healthy Diet
Personalizing meal plans can be extremely beneficial psychologically as well as physically. Knowing that one is eating healthy reduces some of the subconscious doubts about doing what needs to be done to be well. However, as with every healthy practice, there can be pitfalls. To help avoid these, there are 4 approaches that can be taken:
- Assessing and changing your diet
- Staying committed to a healthy diet
- Try additions and substitutions to bring your current diet closer to your goals
- Plan ahead for challenging situations
Planning Healthy Meals
Individual requirements for nutrients vary considerably depending on factors such as age and gender. Other relevant factors are size, metabolic rate, and occupation. A farmer would have a different dietary need than someone in a sedentary occupation. The body also has stores of certain nutrients (fat-soluble vitamins, for example) so that variations in daily intake of such nutrients can be accommodated. Thus it could be misleading to recommend a particular daily intake level.
When considering dietary needs, various techniques have been established by health officials to assist people in choosing foods and food amounts wisely. Choose My Plate is a graphic representation of what a healthy plate of food might look like. Other tools, such as meal planning guides have also been established.
Nutrition Labels
Perhaps one of the most effective tools provided to consumers is the nutrition label that is by law on all food packaging. This information can be useful for evaluating the nutrient content of food and planning healthy meals.
Understanding Food Labels
In the United States, the Food and Drug Administration (FDA) requires packaged foods to have a label that helps consumers make educated decisions about the foods they purchase. The label provides caloric, macronutrient, and some micronutrient content of the food. Labels also indicate ingredients and manufacturer information.
Understanding the information on food labels can lead to healthy choices about food. This page will outline how to interpret food labels.
FYI: Food labels do not provide all nutritional information; they just include the basics to help consumers make healthy food decisions.
Information On Food Labels
- Name of product: Sometimes the name of the product includes important information. For example some brands are vegetarian, kosher, gluten-free, etc. It is also important to compare/contrast ingredients in generic and brand name foods
- Serving Size: It’s important to pay attention to how many servings a package contains. Many packages contain multiple servings.
Figure \(\PageIndex{1}\). Using your hand to determine serving size
- Calories: Pay attention to whether the caloric content of the food is per serving or per package. Also, some food labels indicate calories before or after preparation.
- Fats: The food label includes all fats. Note that the label indicates different types of fats. A later chapter will address different fats and how they are important in human nutrition.
- Cholesterol: Dietary cholesterol is a major factor in cardiovascular health. Limiting the intake of cholesterol can prevent heart disease.
- Sodium: Another major factor in promoting good health is limiting the amount of sodium intake.
- Carbohydrates: The food label includes simple and complex carbohydrates. Note that the label indicates different types of carbohydrates; a later chapter will address these and how they are important in human nutrition.
- Proteins: Protein intake needs to be carefully monitored because over or under consumption of protein can cause severe issues.
- Vitamins and Minerals: There are four vitamins and minerals (Vitamin D, Calcium, Iron, and Potassium) that are required on food labels; however, the label might include more than these four.
- Ingredients: The ingredients are listed in order of their content per volume. If sugar is listed as the first ingredient, there is more sugar in the food than any other ingredient. The last ingredient has the least amount in the food.
- Name of manufacturer: In addition to the Nutrition Facts, the food label includes the name and contact information for the manufacturer as required by law.
- Allergens: Food manufacturers are required to draw specific attention to common allergens such as nuts, milk products, soy, etc. There is no specific location for allergen information, but it should be someplace on the packaging.
Figure \(\PageIndex{2}\). Nutrition Facts
To learn additional details about all of the information contained within the Nutrition Facts panel, see the following website:www.fda.gov/Food/ResourcesFor.../ucm274593.htm
Finding Quality Nutrition Research
“New study shows that margarine contributes to arterial plaque.” “Asian study reveals that two cups of coffee per day can have detrimental effects on the nervous system.” How do you react when you read news of this nature? Do you boycott margarine and coffee? When reading nutrition-related claims, articles, websites, or advertisements always remember that one study does not substantiate a fact. One study neither proves nor disproves anything. Readers who may be looking for complex answers to nutritional dilemmas can quickly misconstrue such statements and be led down a path of misinformation. Listed below are ways that you can develop discerning eyes when reading nutritional news.
- The scientific study under discussion should be published in a peer-reviewed journal, such as the Journal of the International Society of Sports Nutrition . Question studies that come from less trustworthy sources (such as non peer-reviewed journals or websites) or that are not published.
- The report should disclose the methods used by the researcher(s). Did the study last for three or thirty weeks? Were there ten or one hundred participants? What did the participants actually do? Did the researcher(s) observe the results themselves or did they rely on self -reports from program participants?
- Who were the subjects of this study? Humans or animals? If human, are any traits/characteristics noted? You may realize you have more in common with certain program participants and can use that as a basis to gauge if the study applies to you.
- Credible reports often disseminate new findings in the context of previous research. A single study on its own gives you very limited information, but if a body of literature supports a finding, it gives you more confidence in it.
- Peer-reviewed articles deliver a broad perspective and are inclusive of findings of many studies on the exact same subject.
- When reading such news, ask yourself, “Is this making sense?” Even if coffee does adversely affect the nervous system, do you drink enough of it to see any negative effects? Remember, if a headline professes a new remedy for a nutrition-related topic, it may well be a research-supported piece of news, but more often than not it is a sensational story designed to catch the attention of an unsuspecting consumer. Track down the original journal article to see if it really supports the conclusions being drawn in the news report.
When reading information on websites, remember the following criteria for discerning if the site is valid:
- Who sponsors the website?
- Are names and credentials disclosed?
- Is an editorial board identified?
- Does the site contain links to other credible informational websites? Even better, does it reference peer-reviewed journal articles? If so, do those journal articles actually back up the claims being made on the website?
- How often is the website updated?
- Are you being sold something at this website?
- Does the website charge a fee?
Trustworthy Sources
Now let’s consider some reputable organizations and websites from which you can obtain valid nutrition information.
Organizations Active in Nutrition Policy and Research
- US Department of Agriculture Food and Nutrition Information Center. The USDA site fnic.nal.usda.gov has more than twenty-five hundred links to dietary, nutrition, diet and disease, weight and obesity, food-safety and food-labeling, packaging, dietary supplement and consumer questions sites. Using this interactive site, you can find tips and resources on how to eat a healthy diet, my Foodapedia, and a food planner, among other sections.
- The Academy of Nutrition and Dietetics (AND). The AND promotes scientific evidenced-based, research-supported food and nutrition related information on its website, http://www.eatright.org . It is focused on informing the public about recent scientific discoveries and studies, weightloss concerns, food safety topics, nutrition issues, and disease prevention.
- Department of Health and Human Services. The HHS website, HealthFinder.gov, provides credible information about healthful lifestyles and the latest in health news. A variety of online tools are available to assist with food-planning, weight maintenance, physical activity, and dietary goals. You can also find healthful tips for all age groups, tips for preventing disease, and on daily health issues in general.
- Centers for Disease Control and Prevention. The Centers for Disease Control and Prevention ( http://www.cdc.gov ) distributes an online newsletter called CDC Vital Signs. This newsletter is a valid and credible source for up-to-date public health information and data regarding food, nutrition, cholesterol, high blood pressure, obesity, teenage drinking, and tobacco usage.
- Dietitians of Canada. Dietitians of Canada, http://www.dietitians.ca/ , is the national professional association for dietitians. It provides trusted nutrition information to Canadians and health professionals.
Genetically Modified Foods
Genetically modified foods (also known as GM or GMO foods), are plants or animals that have undergone some form of genetic engineering. In the United States, much of the soybean, corn, and canola crop is genetically modified. The process involves the alteration of an organism’s DNA, which allows farmers to cultivate plants with desirable characteristics.
For example, scientists could extract a gene that produces a chemical with antifreeze properties from a fish that lives in an arctic region (such as a flounder). They could then splice that gene into a completely different species, such as a tomato, to make it resistant to frost, which would enable farms to grow that crop year-round.
Certain modifications can be beneficial in resisting pests or pesticides, improving the ripening process, increasing the nutritional content of food, or providing resistance to common viruses. Although genetic engineering has improved productivity for farmers, it has also stirred up debate about consumer safety and environmental protection. Possible side effects related to the consumption of GM foods include an increase in allergenicity, or tendencies to provoke allergic reactions. There is also some concern related to the possible transfer of the genes used to create genetically engineered foods from plants to people. This could influence human health if antibiotic-resistant genes are transferred to the consumer. Therefore, the World Health Organization (WHO) and other groups have encouraged the use of genetic engineering without antibiotic-resistance genes. Genetically modified plants may adversely affect the environment as well and could lead to the contamination of non-genetically engineered organisms.
Genetically modified foods fall under the purview of the EPA, the USDA, and the FDA. Each agency has different responsibilities and concerns in the regulation of GM crops. The EPA ensures that pesticides used for GM plants are safe for the environment. The USDA makes sure genetically engineered seeds are safe for cultivation prior to planting. The FDA determines if foods made from GM plants are safe to eat. Although these agencies act independently, they work closely together and many products are reviewed by all three.
Organic food products
Organically produced foods have been cultivated or raised without synthetic pesticides, antibiotics, or genetic engineering. Certified organic foods can be identified by the USDA’s stamp and is usually more expensive than conventionally produced foods. There have been a lot of questions about the nutritional superiority of organic foods over those traditionally produced. These questions do not have simple answers. Most studies differ substantially in their methodology and involve many different aspects of the food being studied. The USDA makes no claims that organic food is safer or more nutritious than conventionally produced food, and indeed many organic foods - e.g., milk, butter, ice cream, meat - are likely to match their conventional counterparts for fat and calories.
There are, however, specific health reasons that motivate shoppers to buy natural or organic, foods. For instance, people with food allergies, chemical allergies or intolerance to preservatives can substitute organic foods, personal care products and clothing. To reduce fat and cholesterol in their diets, consumers can replace meat with products made from organic soy, wheat or vegetables.
Some organic foods also have significantly higher levels of cancer- fighting antioxidants, according to a study of corn, strawberries and marionberries, which was published in Feb.26, 2003, in the Journal of Agriculture and Food Chemistry, a peer-reviewed publication of the American Chemical Society ("Bitter or Harsh Phenalics Guard the Plant Against These Pests").
Some officials say, however, that organic foods may at times 'be less safe than conventional foods. In October 2002, USDA's undersecretary for food safety warned that organic foods' lack of preservatives makes them vulnerable to bacteria and parasites. The following month., the Institute of Food Technologies issued a release stating that organics "have the potential for greater pathogen contamination." Also in 2002, records from the U.S. Food and Drug Administration and Health Canada revealed that organic and all-natural products are eight times more likely to be recalled for safety problems, including bacterial contamination. Organic products are as safe as conventional ones, according to the ` Organic Trade Association. "Certified organic growers follow strict guidelines for safe and hygienic food production. As with all food producers, they must comply with local, state and federal health standards.
Dietary Food Trends
Hundreds of years ago, when food was less accessible and daily life required much more physical activity, people worried less about obesity and more about simply getting enough to eat. In today’s industrialized nations, conveniences have solved some problems and introduced new ones, including the hand-in-hand obesity and diabetes epidemics. Fad diets gained popularity as more North Americans struggled with excess pounds. However, new evidence-based approaches that emphasize more holistic measures are on the rise. These new dietary trends encourage those seeking to lose weight to eat healthy, whole foods first, while adopting a more active lifestyle. These sound practices put dietary choices in the context of wellness and a healthier approach to life.
Functional Foods
Many people seek out foods that provide the greatest health benefits. This trend is giving rise to the idea of functional foods , which not only help meet basic nutritional needs but also are reported to fight illness and aging. According to the Academy of Nutrition and Dietetics (AND), formerly known as the The Academy of Nutrition and Dietetics, functional foods may reduce the risk of disease or promote optimal health. The AND recognizes four types of functional foods. They are: conventional foods, modified foods, medical foods, and special dietary use foods.
The first group, conventional foods , represents the simplest form of functional foods. They are whole foods that have not been modified. Examples include whole fruits and vegetables (which are abundant in phytochemicals and antioxidants), yogurt and kefir (which contain natural probiotic bacteria that can help maintain digestive system health), and dark chocolate (which contains antioxidants).
- Modified foods have been fortified, enriched, or enhanced with additional nutrients or bioactive compounds. Foods are modified using biotechnology to improve their nutritional value and health attributes. Examples of modified foods include calcium-fortified orange juice, breads enriched with B vitamins, iodized salt, cereals fortified with vitamins and minerals, margarine enhanced with plant sterols, and energy drinks that have been enriched with herbs (ginseng or guarana) or amino acids (taurine). It is important to consider that the health claims of some modified foods may be debatable, or entirely fraudulent. Check with a health professional regarding the effects of modified foods on your health.
- Medical foods are designed for enteric administration under the guidance of a medical professional. (During enteric administration, food is treated so that it goes through the stomach undigested. Instead, the food is broken down in the intestines only.) Medical foods are created to meet very specific nutritional requirements. Examples of medical foods include liquid formulas for people with kidney disease, liver disease, diabetes, or other health issues. Medical food is also given to comatose patients through a gastronomy tube because they cannot eat by mouth.
- Special dietary use foods do not have to be administered under a doctor’s care and can be found in a variety of stores. Similar to medical foods, they address special dietary needs and meet the nutritional requirements of certain health conditions. For example, a bottled oral supplement administered under medical supervision is a medical food, but it becomes a special dietary use food when it is sold to retail customers. Examples of special dietary use foods include gluten-free foods, lactose-free dairy products, and formulas and shakes that promote weight loss.


2.2: Historical Context
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As both a historical and literary age, the Victorian era dates chronologically with the reign of Queen Victoria. Some critics, though, see it as beginning with Alfred, Lord Tennyson’s (1809-1892) publication of Poems, Chiefly Lyrical and the opening of the Liverpool-Manchester railroad, both of which occurred in 1830.
The Victorian Age can be divided into two sections, with the fulcrum occurring around 1870. The first part was characterized by optimism in material, cultural, and social progress. The optimistic saw great progress occurring in this era. The Crimean War (1854-56), though filled with military miscalculations and deaths and achieved no victory, did not diminish this optimism. The second part, however, was affected by the Depression of 1873, which continued until the end of the century. England in the 1860s was at its zenith as a world power, followed by a slow decline over the next 100 years.
The paramount characteristic of the Victorian Age was rapid change and concomitant conflict. It was a complex age, an age of great wealth and extreme poverty, of the family as sacred center and a burgeoning of prostitution, of morality and fraud, of belief in the Bible and in scientific determinism. In the 1830s, railroad expansion transformed England as it spurred immense material progress and economic growth. The Victorians came to think of progress as natural and tied progress to wealth and prosperity.
Britain exceeded the exports of competing nations like the United States by three to four times in number. The British Empire grew in pursuit of resources and markets for its exports. At its height of Empire, Britain ruled about a quarter of the world, including Ceylon, India, Australia, New Zealand, the Sudan, and South Africa. The British saw themselves as the leaders of the world, assuming the “white man’s burden” and spreading civilization and religion to the so-called dark places on earth. They overlooked the commercial exploitation, racism, and moral degradation that they also spread. A measure of Britain’s place in the world was The Crystal Palace (1851) which collected in one massive glass and iron structure inventions and artifacts taken from all over the British Empire.
During this economic and social transformation, England’s aristocracy and the rising middle class, comprising industrialists, businessmen, trade leaders, and workers, vied for political power, with the middle class incrementally overtaking the aristocracy. The Reform Bill of 1832 began the process of extending the franchise, ultimately reaching the worker. During this struggle, England shifted from an agrarian to an industrial society. Industrialization wrought a grim physical change on the landscape and in the growth of urban slums around factories. Farmers migrated from the country to the city. The population in London doubled in a matter of a few years. A dramatic increase in the population overall led to an urban concentration, in London and in northern industrial cities like Liverpool, Leeds, and Manchester.
The laboring masses of the poor, though, had little power. Men, women, and children lived in abysmal conditions, working six days a week for up to sixteen hours a day in factories and mines at a time when there were no minimum wage or age limits. These conditions were partially improved through various acts, including the Factory Act of 1833 that improved conditions in textile factories; the Poor Law Amendment Act of 1834 that created the Workhouse, a place intended for the desperate destitute as it separated families and forced able-bodied people to work alongside the lunatic and the ill; and the Mine Act of 1842 that excluded women and children from working underground. Nevertheless, women in particular remained considerably disadvantaged.
A large number of women worked as domestic servants and governesses. Prostitution was an option for the desperate, and the number of prostitutes increased tremendously. The Victorian connection of virtue with prosperity allowed what now seems a remarkable hypocrisy in this growth, as Victorians thought that the poor and the prostitute were “no better than they should be” and suffered poverty because of their moral laxity. The other side of this view was the Victorian feminine ideal that viewed women as the standard bearers of morality, as the protectors of virtue in the home, their natural sphere. A woman’s duty first was to her husband; a woman’s privilege was her freedom from the stress of the public sphere. A married woman’s property was in her husband’s hands up until the 1870s and 1880s. Despite their apparent spiritual and moral elevation, women were considered inferior to men intellectually, physically, and temperamentally, and the position of women became a concern for social change. Charlotte Bronte’s Jane Eyre (1847) spoke for women’s individuality, and John Stuart Mill’s The Subjection of Women (1869) advocated for women’s education and options for occupations outside the domestic sphere.
Change and reform also occurred in religious life. Since the 1560s, England was a Protestant nation, with the Church of England supported by the government. Protestant evangelicals and dissenters of the church pushed strongly against this unquestioned authority, demanding strong Christian ethics and increased social welfare. Known for their religious fervor, evangelicals stressed the authority of the Bible, considering it to be the direct authority of God. Besides the Evangelical Movement there occurred in the 1870s the Oxford Movement that increased the size and power of the Roman Catholic Church. This movement was led by John Henry Newman (1801-1890) who published his spiritual biography Apologia Pro Vita Sua (1864).
These challenges to the established church reflected a shift in religion and philosophy caused by scientific and higher critical studies, including Sir Charles Lyell’s (1797-1875) Principles of Geology (1830-33) that used geology to measure the Earth’s age; Charles Darwin’s (1809-1882) On the Origins of Species (1859) that saw life as biologically determined; and August Renan’s (1823-1894) Life of Jesus (1863), a biography of Jesus as human rather than divine. Additionally, the Empire exposed the British to unexpected diversity of cultures and religion.
Missionaries intent on spreading the word of God to the so-called heathen were taken by surprise at the rational skepticism of some of their intended converts. John William Colenso (1814-1883), the Bishop of Natal, described the impact of such skepticism in his controversial work of biblical criticism, The Pentateuch and Book of Joshua Critically Examined (1862).