on 23-Feb-2018 (Fri)

The nabla is a triangular symbol like an inverted Greek delta: or ∇.

After defining an inner product on the set of random variables using the expectation of their product,

⟨ X , Y ⟩ := E ⁡ ( X Y ) , {\displaystyle \langle X,Y\rangle :=\operatorname {E} (XY),}

then the Cauchy–Schwarz inequality becomes

| E ⁡ ( X Y ) | 2 ≤ E ⁡ ( X 2 ) E ⁡ ( Y 2 ) . {\displaystyle |\operatorname {E} (XY)|^{2}\leq \operatorname {E} (X^{2})\operatorname {E} (Y^{2}).}

A Hilbert curve is a continuous fractal space-filling curve

Analysis of variance (ANOVA) is a collection of statistical models and their associated procedures (such as "variation" among and between groups) used to analyze the differences among group means. ANOVA was developed by statistician and evolutionary biologist Ronald Fisher. In the ANOVA setting, the observed variance in a particular variable is partitioned into components attributable to different sources of variation. In its simplest form, ANOVA provides a statistical test of whether or not the means of several groups are equal, and therefore generalizes the t-test to more than two groups. ANOVAs are useful for comparing (testing) three or more means (groups or variables) for statistical significance. It is conceptually similar to multiple two-sample t-tests, but is more conservative (results in less type I error) ^[1] and is therefore suited to a wide range of practical problems.

In calculus, foundations refers to the rigorous development of the subject from axioms and definitions.

Henri Lebesgue invented measure theory and used it to define integrals of all but the most pathological functions.

Following the work of Weierstrass, it eventually became common to base calculus on limits instead of infinitesimal quantities

The (ε, δ)-definition of limit defines the limit of a function f at point c as: whenever a point x is within δ units of c, f(x) is within ε units of L

The infinitesimal approach fell out of favor in the 19th century because it was difficult to make the notion of an infinitesimal precise. However, the concept was revived in the 20th century with the introduction of non-standard analysis and smooth infinitesimal analysis, which provided solid foundations for the manipulation of infinitesimals.

In the 19th century, infinitesimals were replaced by the epsilon, delta approach to limits. Limits describe the value of a function at a certain input in terms of its values at a nearby input. They capture small-scale behavior in the context of the real number system. In this treatment, calculus is a collection of techniques for manipulating certain limits. Infinitesimals get replaced by very small numbers, and the infinitely small behavior of the function is found by taking the limiting behavior for smaller and smaller numbers. Limits were the first way to provide rigorous foundations for calculus, and for this reason they are the standard approach.

In formal terms, the derivative is a linear operator which takes a function as its input and produces a second function as its output.

The most common symbol for a derivative is an apostrophe-like mark called prime.

The derivative is defined by taking the limit as h tends to zero, meaning that it considers the behavior of f for all small values of h and extracts a consistent value for the case when h equals zero:

The tangent line is a limit of secant lines just as the derivative is a limit of difference quotients.

In an approach based on limits, the symbol dy / dx is to be interpreted not as the quotient of two numbers but as a shorthand for the limit

Even when calculus is developed using limits rather than infinitesimals, it is common to manipulate symbols like dx and dy as if they were real numbers

Calculation of the total derivative of with respect to assumes that the other arguments too depend on t.

The total derivative of a function of several variables, e.g., , , , with respect to an exogenous argument , is the limiting ratio of the change in the function's value to the change in the exogenous argument's value, taking into account the exogenous argument's direct effect as well as indirect effects via the other arguments of the function.

The infinitesimal approach fell out of favor in the 19th century because it was difficult to make the notion of an infinitesimal precise.

derivative defined by limit considers the behavior of f for all small values of h and extracts a consistent value for the case when h equals zero

Es muy poco agradable (He/She is not very kind).

Vuelve a decirle que me llame (Tell her to call me again).

Empezamos a correr cuando vimos que llovía

Luis y Ana acaban de irse (Luis and Ana have just left).

Suelo comer en una tasca cerca del trabajo (I usually have lunch at a bar close to work).

Bioenergetics concerns only the initial and final energy states of reaction components, not the mechanism or how much time is needed for the chemical change to take place. In short, bio - energetics predicts if a process is possible, whereas kinetics measures how fast the reaction occurs.

The direction and extent to which a chemical reaction proceeds is determined by the degree to which two factors change during the reaction. These are enthalpy (ΔH, a measure of the change in heat con- tent of the reactants and products) and entropy (ΔS, a measure of the change in randomness or disorder of reactants and products, Figure 6.1). Neither of these thermodynamic quantities by itself is sufficient to determine whether a chemical reaction will proceed spontaneously in the direction it is written. However, when combined mathematically (see Figure 6.1), enthalpy and entropy can be used to define a third quantity, free energy (G), which predicts the direction in which a reaction will spontaneously proceed.

1. Negative ΔG: If ΔG is a negative number, there is a net loss of energy, and the reaction goes spontaneously as written—that is, A is converted into B (Figure 6.2A). The reaction is said to be exergonic. 2. Positive ΔG: If ΔG is a positive number, there is a net gain of energy, and the reaction does not go spontaneously from B to A (see Figure 6.2B). Energy must be added to the system to make the reaction go from B to A, and the reaction is said to be ender- gonic. 3. ΔG is zero: If ΔG = 0, the reactants are in equilibrium. [Note: When a reaction is proceeding spontaneously—that is, free energy is being lost—then the reaction continues until ΔG reaches zero and equilibrium is established.]

B. ΔG of the forward and back reactions The free energy of the forward reaction (A → B) is equal in magni- tude but opposite in sign to that of the back reaction (B → A). For example, if ΔG of the forward reaction is −5 kcal/mol, then that of the back reaction is +5 kcal/mol. [Note: ΔG can also be expressed in kilo- joules per mole or kJ/mol (1 kcal = 4.2 kJ).] C. ΔG depends on the concentration of reactants and products The ΔG of the reaction A → B depends on the concentration of the reactant and product. At constant temperature and pressure, the fol- lowing relationship can be derived: ΔG = ∆Go equation. Edit this.

A reaction with a positive ΔGo
can proceed in the forward direction
(have a negative overall ΔG) if the ratio of products to reactants
([B]/[A] ) is sufficiently small (that is, the ratio of reactants to products
is large).

2. Relationship between ΔG o and K eq : In a reaction A → B, a point of equilibrium is reached at which no further net chemical change takes place—that is, when A is being converted to B as fast as B is being converted to A. In this state, the ratio of [B] to [A] is con- stant, regardless of the actual concentrations of the two compounds:

Keq = [B]eq/[A]eq

where K eq is the equilibrium constant, and [A] eq and [B] eq are the concentrations of A and B at equilibrium. If the reaction A B is allowed to go to equilibrium at constant temperature and pres- sure, then at equilibrium the overall free energy change (ΔG) is zero. Therefore, where the actual concentrations of A and B are equal to the equi- librium concentrations of reactant and product [A] eq and [B] eq , and their ratio as shown above is equal to the K eq. Thus,

∆G = ∆Go - RTlnKeq

This equation allows for some simple predictions:

If Keq = 1, then ∆Go = 0 and A <--> B
If Keq > 1, then ∆Go < 0 and A <---->>>> B
If Keq < 1, then ∆Go > 0 and A <<<<------> B

ΔGs of a pathway are additive: This additive property of free energy changes is very important in biochemical pathways through which substrates must pass in a particular direction (for example, A → B → C → D → ...). As long as the sum of the ΔGs of the individual reactions is negative, the pathway can potentially proceed as written, even if some of the individual reactions of the pathway have a positive ΔG. The actual rate of the reactions does, of course, depend on the lowering of activation energies by the enzymes that catalyze the reactions.

Energy-rich molecules, such as glucose, are metabolized by a series of oxidation reactions ultimately yielding CO2 and water (Figure 6.6). The metabolic intermediates of these reactions donate electrons to specific coenzymes—nicotinamide adenine dinucleotide (NAD + ) and flavin adenine dinucleotide (FAD)—to form the energy-rich reduced coen- zymes, NADH and FADH 2 . These reduced coenzymes can, in turn, each donate a pair of electrons to a specialized set of electron carriers, collectively called the electron transport chain, described in this section. As electrons are passed down the electron transport chain, they lose much of their free energy. Part of this energy can be captured and stored by the production of ATP from ADP and inorganic phosphate (Pi). This process is called oxidative phosphorylation and is described on p. 77. The remainder of the free energy not trapped as ATP is used to drive ancillary reactions such as Ca 2+ transport into mitochondria, and to generate heat

Matrix of the mitochondrion: This gel-like solution in the interior of mitochondria is 50% protein. These molecules include the enzymes responsible for the oxidation of pyruvate, amino acids, fatty acids (by β-oxidation), and those of the tricarboxylic acid (TCA) cycle. The synthesis of glucose, urea, and heme occur partially in the matrix of mitochondria. In addition, the matrix contains NAD + and FAD (the oxidized forms of the two coenzymes that are required as hydrogen acceptors) and ADP and P i , which are used to produce ATP. [Note: The matrix also contains mitochondrial RNA and DNA (mtRNA and mtDNA) and mitochondrial ribosomes.]

The inner mitochondrial membrane can be disrupted into five sepa- rate protein complexes, called Complexes I, II, III, IV, and V. Complexes I–IV each contain part of the electron transport chain (Figure 6.8). Each complex accepts or donates electrons to relatively mobile electron carriers, such as coenzyme Q and cytochrome c. Each carrier in the electron transport chain can receive electrons from an electron donor, and can subsequently donate electrons to the next carrier in the chain. The electrons ultimately combine with oxygen and protons to form water. This requirement for oxygen makes the electron transport process the respiratory chain , which accounts for the greatest portion of the body’s use of oxygen.

With the exception of coenzyme Q, all members of this chain are proteins

Formation of NADH: NAD + is reduced to NADH by dehydroge- nases that remove two hydrogen atoms from their substrate. (For examples of these reactions, see the discussion of the dehydro- genases found in the TCA cycle, p. 112.) Both electrons but only one proton (that is, a hydride ion, :H – ) are transferred to the NAD + , forming NADH plus a free proton, H +

NADH dehydrogenase: The free proton plus the hydride ion car- ried by NADH are next transferred to NADH dehydrogenase , a protein complex (Complex I) embedded in the inner mitochondrial membrane. Complex I has a tightly bound molecule of flavin mono nucleotide (FMN, a coenzyme structurally related to FAD, see Figure 28.15, p. 380) that accepts the two hydrogen atoms (2e – + 2H + ), becoming FMNH 2 . NADH dehydrogenase also con- tains iron atoms paired with sulfur atoms to make iron–sulfur cen- ters (Figure 6.9). These are necessary for the transfer of the hydrogen atoms to the next member of the chain, coenzyme Q (ubiquinone)

Coenzyme Q: Coenzyme Q (CoQ) is a quinone derivative with a long, hydrophobic isoprenoid tail. It is also called ubiquinone because it is ubiquitous in biologic systems. CoQ is a mobile car- rier and can accept hydrogen atoms both from FMNH 2 , produced on NADH dehydrogenase (Complex I), and from FADH 2 , pro- duced on succinate dehydrogenase (Complex II), glycerophos- phate dehydrogenase (see p. 79), and acyl CoA dehydrogenase (see p. 192). CoQ transfers electrons to Complex III. CoQ, then, links the flavoproteins to the cytochromes

Cytochromes: The remaining members of the electron transport chain are cytochromes. Each contains a heme group (a porphyrin ring plus iron). Unlike the heme groups of hemoglobin, the cytochrome iron is reversibly converted from its ferric (Fe 3+ ) to its ferrous (Fe 2+ ) form as a normal part of its function as a reversible carrier of electrons. Electrons are passed along the chain from CoQ to cytochromes

Cytochrome a + a 3 : This cytochrome complex is the only electron carrier in which the heme iron has an available coordination site that can react directly with O 2 , and so also is called cytochrome oxidase . At this site, the transported electrons, O 2 , and free pro- tons are brought together, and O 2 is reduced to water

Free energy is released as electrons are transferred along the elec- tron transport chain from an electron donor (reducing agent or reductant) to an electron acceptor (oxidizing agent or oxidant). The electrons can be transferred as hydride ions (:H – ) to NAD + , as hydrogen atoms ( • H) to FMN, coenzyme Q, and FAD, or as elec- trons (e – ) to cytochromes.

Standard reduction potential (E o ): The E o of various redox pairs can be ordered from the most negative E o to the most positive. The more negative the E o of a redox pair, the greater the tendency of the reductant member of that pair to lose electrons. The more posi- tive the E o , the greater the tendency of the oxidant member of that pair to accept electrons. Therefore, electrons flow from the pair with the more negative E o to that with the more positive E o . The E o values for some members of the electron transport chain are shown in Figure 6.12. [Note: The components of the electron trans- port chain are arranged in order of increasingly positive E 0 values.]

ΔGo is related to ΔEο: The change in free energy is related directly to the magnitude of the change in Eo:
ΔGo = – nFΔEo
n = number of electrons transferred (1 for a cyto chrome, 2 for NADH, FADH 2 , and coenzyme Q)
F = Faraday constant (23.1 kcal/volt • mol)
ΔEo = Eo of the electron-accepting pair minus the Eo of the electron-donating pair
ΔGo = change in the standard free energy

The transfer of electrons down the electron transport chain is energeti- cally favored because NADH is a strong electron donor and molecular oxygen is an avid electron acceptor.

The chemiosmotic hypothesis (also known as the Mitchell hypothesis) explains how the free energy generated by the transport of electrons by the electron transport chain is used to produce ATP from ADP + P i

Uncoupling proteins (UCP): UCPs occur in the inner mito- chondrial membrane of mammals, including humans. These car- rier proteins create a “proton leak,” that is, they allow protons to re-enter the mitochondrial matrix without energy being captured as ATP (Figure 6.14). The energy is released as heat, and the process is called nonshivering thermogenesis

Synthetic uncouplers: Electron transport and phosphoryla- tion can also be uncoupled by compounds that increase the permeability of the inner mitochondrial membrane to protons. The classic example is 2,4-dinitrophenol, a lipophilic proton carrier that readily diffuses through the mitochondrial mem- brane. This uncoupler causes electron transport to proceed at a rapid rate without establishing a proton gradient, much as do the UCPs (see Figure 6.14). Again, energy is released as heat rather than being used to synthesize ATP. In high doses, aspirin and other salicylates uncouple oxidative phos - phorylation. This explains the fever that accompanies toxic overdoses of these drugs

The inner mitochondrial membrane is impermeable to most charged or hydrophilic substances. However, it contains numerous transport proteins that permit passage of specific molecules from the cytosol (or more correctly, the intermembrane space) to the mitochondrial matrix.

ATP-ADP transport: The inner mitochondrial membrane requires specialized carriers to transport ADP and P i from the cytosol (where ATP is used and converted to ADP in many energy- requiring reactions) into mitochondria, where ATP can be resyn- thesized. An adenine nucleotide carrier imports one molecule of ADP from the cytosol into mitochondria, while exporting one ATP from the matrix back into the cytosol

Transport of reducing equivalents: The inner mitochondrial mem- brane lacks an NADH transporter, and NADH produced in the cytosol cannot directly enter the mitochondrial matrix. However, two electrons of NADH (also called reducing equivalents) are transported from the cytosol into the matrix using substrate shut- tles. In the glycero phosphate shuttle (Figure 6.15A), two electrons are transferred from NADH to dihydroxyacetone phosphate by cytosolic glycero phosphate dehydrogenase . The glycerol 3-phos- phate produced is oxidized by the mitochondrial isozyme as FAD is reduced to FADH 2 . CoQ of the electron transport chain oxidizes FADH 2 . The glycero phos phate shuttle, therefore, results in the syn- thesis of two ATPs for each cytosolic NADH oxidized. This con- trasts with the malate-aspartate shuttle (Figure 6.15B), which produces NADH (rather than FADH 2 ) in the mitochondrial matrix and, therefore, yields three ATPs for each cytosolic NADH oxidized by malate dehydrogenase as oxaloacetate is reduced to malate. A transport protein carries malate into the matrix.

Inherited defects in oxidative phosphorylation Thirteen of the approximately 120 polypeptides required for oxida- tive phosphorylation are coded for by mtDNA and synthesized in mitochondria, whereas the remaining mitochondrial proteins are synthesized in the cytosol and transported into mitochondria. Defects in oxidative phosphorylation are more likely a result of alter- ations in mtDNA, which has a mutation rate about ten times greater than that of nuclear DNA. Tissues with the greatest ATP requirement (for example, central nervous system, skeletal and heart muscle, kidney, and liver) are most affected by defects in oxidative phospho- rylation. Mutations in mtDNA are responsible for several diseases, including some cases of mitochondrial myopathies (Figure 6.16), and Leber hereditary optic neuropathy, a disease in which bilateral loss of central vision occurs as a result of neuroretinal degeneration, including damage to the optic nerve. The mtDNA is maternally inher- ited because mitochondria from the sperm cell do not enter the fertil- ized egg.

Explain why and how the malate-aspartate shuttle moves NADH reducing equivalents from the cytosol to the mitochondrial matrix.

There is no transporter for NADH in the inner mitochondrial membrane. However, NADH can be oxidized to NAD + by the cytoplasmic isozyme of malate dehydrogenase as oxaloac- etate is reduced to malate. The malate is transported across the inner membrane, and the mitochondrial isozyme of malate dehydro- genase oxidizes it to oxaloacetate as mito- chondrial NAD + is reduced to NADH. This NADH can be oxidized by Complex I of the electron transport chain, generating three ATP through the coupled processes of respiration and oxidative phosphorylation.

Repair the common areas, where all tenants move around freely. This includes hallways, basements, pool area, clubhouse, and so on. Get those areas looking good, even if it costs you.

Audit your rent roll. Determine which leases will expire within the next 30 to 60 days. Have your manager visit these tenants,

When repairs are complete, your manager visits all tenants again and offers them a bonus to sign a new lease for another year, at the higher market rate.

keep tenant turnover at a mini- mum while you significantly increase rents.

The great opportunity to increase the value of your property quickly is to find properties with low rents.

researchers did exit polling to discover the main reasons why tenants moved out. Rising rental rates was number three on the list. Number one was failure to take care of repair requests.

When is it time to raise rents? Any time your property is at 95 per- cent occupancy or above.

ou should also raise your rents every year, even if the market has not gone up. Train your tenants to expect some annual rent increase. Perhaps it’s only $15 to $20. This is called a nuisance increase.

You have a 10-unit building, and you increase rents by $20 per month.

At a 10 Cap Rate, we divide the $2,400 by .10 to arrive at a property value increase of $24,000.

The faster you go big, the faster you become wealthy.

if you have a 100-unit apartment building and do a nuisance increase of $20 per month, your cash flow just increased by: 100 units times $20 per month = $2,000 . . . times 12 months = $24,000 per year!

$240,000 value increase divided by .2 = $1,200,000 more property you can own! All due to a nuisance increase.

if you raise a C– property to a B–, very few tenants can afford the higher rent you should be charging.

I bought the property for $58,000. Before I closed, I worked out an arrangement with the local and state police departments to stake out the property. I agreed to give them access to the property whenever they wanted. In return, they would help get rid of the drug dealers.

take this 365-unit property from a C− toaC+or even B−. The process is what you just read earlier: Do the exterior and common area repairs, and get rid of the slow-payers, non-payers, and criminals.

have the interior of all units upgraded from their current condi- tion to a grade higher. This includes all new paint, carpet, appliances, cabinet faces, counters, and lighting.

soak up the concepts, which apply to both the 6-unit and the 365-unit deals.

you have your market- ing game going, a lot of properties will be coming to you in a short period. You must have a quick and efficient way to determine whether you should take the next step and make an offer on a property.

you’ve become special only when brokers call you before deals hit the list. Until you get to that point, you must be hyper-responsive.

By being very responsive to brokers, you will start working your way up their preferred buyers list.

Very few investors respond by phone to those e-mails. A few respond by e-mail, but the vast majority don’t respond at all. You are going to stand out from the crowd because you’ll call these brokers. You’ll thank them for sending you the deal, even though you know you’re just on the mass mailing list. Then you’ll explain why the deal does or does not work.

Explain why it doesn’t work, and then explain how you do like to buy (unit size, cap rate, and so on). Then ask: Do you have any deals that fit those criteria? Depending on how far up you are on his list, how much rapport you’ve built, you might get an: Actually, I do have this one deal on the south side of town. . . . Then again, it may take more cultivating.

train your brokers that you are a professional investor,

In the commercial real estate world, the value of a property is determined by its cash flow and cap rate.