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In order to work with these complex numbers without drawing vectors, we first need some kind of standard mathematical notation. There are two basic forms of complex number notation: polar and rectangular .

Polar form is where a complex number is denoted by the length (otherwise known as the magnitude , absolute value , or modulus ) and the angle of its vector (usually denoted by an angle symbol that looks like this: ∠). To use the map analogy, polar notation for the vector from New York City to San Diego would be something like “2400 miles, southwest.” Here are two examples of vectors and their polar notations: (Figure nike air max 95 dynamic flywire idea
)

Vectors with polar notations.

Standard orientation for vector angles in AC circuit calculations defines 0 o as being to the right (horizontal), making 90 o straight up, 180 o to the left, and 270 o straight down. Please note that vectors angled “down” can have angles represented in polar form as positive numbers in excess of 180, or negative numbers less than 180. For example, a vector angled ∠ 270 o (straight down) can also be said to have an angle of -90 o . (Figure air jordan 11 retro bred backpacks
) The above vector on the right (7.81 ∠ 230.19 o ) can also be denoted as 7.81 ∠ -129.81 o .

The vector compass

Rectangular form, on the other hand, is where a complex number is denoted by its respective horizontal and vertical components. In essence, the angled vector is taken to be the hypotenuse of a right triangle, described by the lengths of the adjacent and opposite sides. Rather than describing a vector’s length and direction by denoting magnitude and angle, it is described in terms of “how far left/right” and “how far up/down.”

These two dimensional figures (horizontal and vertical) are symbolized by two numerical figures. In order to distinguish the horizontal and vertical dimensions from each other, the vertical is prefixed with a lower-case “i” (in pure mathematics) or “j” (in electronics). These lower-case letters do not represent a physical variable (such as instantaneous current, also symbolized by a lower-case letter “i”), but rather are mathematical operators used to distinguish the vector’s vertical component from its horizontal component. As a complete complex number, the horizontal and vertical quantities are written as a sum: (Figure nike air max 97 premium tape qs silver bullet ebay login
)

it’s all relative This is just as true in investing where you and I will experience the same market event very differently because of our biases, our personal investment history, and other ideas that have shaped our worldview.

For example, consider real U.S. stock returns by decade (including dividends):

When you look at this chart what conclusion do you draw? Yes, returns vary, but does that mean that we should just buy and hold? Does it imply that investing is mostly ? The investors from the 1930s and 1940s will likely have a very different answer than those from the 1980s and 1990s.

We can go further and compare U.S. stock and bond returns during the biggest stock market crashes of the past century. If we start at the peak before the crash, we can see how each market crash provides a different experience for the investors of that era:

However, this graphic isn’t the best we can do. We can overlay these peaks on top of each other to get an even better comparison of each one:

As you can see, each one of these periods provided a different takeaway for investors. I might say, “Bonds provide stability when stocks crash,” but the investors of 1973–1974 would say, “Not so fast.”

As you can see, each one of these periods provided a different takeaway for investors.

The point is that realizing your financial frame of reference can be helpful for understanding how you might react to future market events. For example, I know that I am biased against real estate investments nike air force 1 ultra flyknit white and black
in the housing crisis of 2007–2008. This is clearly an example of red air jordan 1s
clouding my judgement, but I honestly can’t seem to shake it though I know better.

So ask yourself: what is your financial frame of reference? Are you biased for or against any particular asset class? If so, do you have evidence for this? Studying market history can help to tame these biases, but they will always be difficult to overcome . So next time you hear someone make a comment about the market, before you agree or disagree, remember the relative lens through which they (and you) view the financial world.

Studying market history can help to tame these biases, but they will always be difficult to overcome

Absolutes in a Relative World

The research objectives of Drug Delivery and Disposition are focused on enhancing drug bioavailability of dosage forms for extravascular administration using pharmaceutical-technological approaches (new drug formulations and process technology) as well as biopharmaceutical strategies (based on knowledge of mechanisms underlying drug absorption and hepatobiliary disposition). Significant contributions have been made in the understanding of the physicochemical principles behind formulation strategies for poorly soluble drugs like amorphous solid dispersions, nanoparticles, and mesoporous drug loaded silica. The Laboratory innovates in the development of preclinical models for ADMETox profiling; these model systems include in situ intestinal perfusion in mice (enabling the use of KO and humanized mice in intestinal absorption studies) and hepatocyte-based prediction of drug-induced cholestasis.

Pieter Annaert Patrick Augustijns Thomas Bouillon Guy Van den Mooter

Intestinal drug disposition – Biopharmaceutics

Drug Delivery and Disposition has a strong track record in the biorelevant profiling of intestinal drug absorption, covering all underlying processes including dissolution, precipitation, degradation and permeation. For this purpose, a wide range of simulation models is available, including the in vitro Caco-2 cell culture system, the Ussing chamber system and the in situ intestinal perfusion system. In addition, Drug Delivery and Disposition is able and licensed to perform whole animal absorption and pharmacokinetic experiments. Physiology-based pharmacokinetic modelling (Simcyp® Simulator) is available to extrapolate experimental data to human pharmacokinetics. One of the major targets involves the biorelevant and predictive evaluation of absorption-enabling strategies, including solubilization and supersaturation of poorly soluble drugs. In this respect, Drug Delivery and Disposition has elaborated a ground-breaking approach for evaluating intraluminal drug and formulation behavior in humans, involving the aspiration and characterization of gastrointestinal fluids. All absorption studies are supported by well-developed analytical equipment (LC-UV, -fluo, -MS/MS) to assess concentrations of drugs, excipients and endogenous compounds in biological matrices.

Hepatic drug disposition and drug-induced liver injury

In the field of hepatobiliary drug disposition and drug-drug interaction assessment, drug delivery and disposition has implemented the full spectrum of non-clinical model systems of the liver including: rat/human liver microsomes, rat/human hepatocytes in suspension, sandwich-cultured hepatocytes, cell lines transfected with hepatic drug transporters, isolated perfused rat liver and in vivo. Isolation and cryopreservation of plateablerodent hepatocytes and preparation of liver subcellular fractions is performed in-house. The research group has characterized several fluorescent transporter probes for evaluation and live imaging (by confocal microscopy) of drug transport processes in hepatocytes. Drug clearance prediction in special populations (e.g. pediatric), transporter-based pharmacokinetic boosting and liver unbound concentration assessment form major research objectives. The group has also developed and validated a holistic, hepatocyte-based in vitro model for identification of drug candidates showing risk for drug-induced cholestasis . The model has been mechanistically validated by bile acid profiling in sandwich-cultured hepatocytes. Computational expertise includes in vitro-in vivo extrapolation (IVIVE) algorithms for clearance prediction (SimCYP, R), compartmental and non-compartmental pharmacokinetic data analysis, as well as population pharmacokinetic analysis (NONMEM) of clinical exposure data. The group has taken the lead in generation of large in vitro transporter inhibition data sets leading to in silico models for structure-based prediction of transporter inhibition . Bioanalytical activities include LC-UV/FLUO/MSMS for preclinical and clinical samples.

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