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Sale 244 Action Area: Estimated Chance of One or More Large Platform erectile dysfunction needle injection order cialis professional now, Pipeline and Total Spills Occurring for the Alternatives low testosterone erectile dysfunction treatment trusted cialis professional 20 mg. Alternative Name Proposed Action and its Alternatives No Action Percent Chance of One or Percent Chance of One Percent Chance of One More Platform/ Well Spills or More Pipeline Spills or More Spills Total 5 0 17 0 221 0 Note: Based on mean spill number of 0 erectile dysfunction studies discount 40mg cialis professional mastercard. Small Refined and Crude Oil Spills: Range Assumed Showing Total Over the Life and Annual Number and Volume of Spills Over Exploration and Development and Production Activities xyzal impotence discount cialis professional 40 mg with amex. Estimated Total Estimated Total Estimated Annual Estimated Annual Activity Phase Number of Small Volume of Small Number of Small Volume of Small Spills Spills (bbl) Spills Spills (bbl) Refined Oil Spills Exploration G&G Activities Exploration & Delineation Drilling Activities Development and Production 1 0-6 0-4 0-18 0-65 Small Crude and Refined Oil Spills 0-2 0-1 0-<2 or <14 0-5 or 50 450 300 13 911 Note: Average volume over 33 years. Activity Phase Estimated Total Number of Small Spills Estimated Total Volume of Small Spills (bbl) Development and Production Total1 0- <1 bbl 0-<50 bbl 50 - <500 500-<1,000 Approximately 450 434 16 2 0 Approximately 300 10 48 252 0 Note: Total spill number or volumes are rounded to the nearest ten or hundred. Notes: Maximizes estimated flow rate to represent the Maximizes estimated flow rate to represent the largest potential discharge from one actual (known) largest potential discharge estimated from any drilling location. The model estimates discharges during mobilization, drilling, and completion of a relief well. Fate and Behavior of a Hypothetical 1,400 to 2,100-Barrel Crude Oil Spill in the Cook Inlet. Hypothetical Launch Areas and Pipelines Used in the Oil-Spill Trajectory Analysis. Kaflia, Kukak, Kuliak & Missak Bays Devils Cove, Hallo Bay Cape Chiniak, Swikshak Bay Fourpeaked Glacier Spotted Glacier, Sukoi Bay Douglas River Akumwarvik Bay, McNeil Cove, Nordyke Island Amakdedulia Cove, Bruin Bay, Chenik Head Augustine Island Rocky Cove, Tignagvik Point liamna Bay, Iniskin Bay, Ursus Cove Chinitna Point, Dry Bay Chinitna Bay Iliamna Point Chisik Island, Tuxedni Bay Redoubt Point Drift River, Drift River Terminal Kalgin Islandd Kustatan River,West Foreland Clam Gulch, Kasilof Deep Creek, Ninilchik, Ninilchik River Cape Starichkof, Happy Valley Anchor Point, Anchor River Homer, Homer Spit China Poot Bay, Gull Island Barabara Point, Seldovia Bay Nanwalek, Port Graham Elizabeth Island, Port Chatham, Koyuktolik Bay Barren Islands, Ushagat Island Amatuli Cove, East & West Amatuli Island Shuyak Island Bluefox Bay, Shuyak Island, Shuyak Strait Foul Bay, Paramanof Bay Malina Bay, Raspberry Island, Raspberry Strait Kupreanof Strait, Viekoda Bay Uganik Bay Uganik Strait, Cape Ugat Cape Kuliuk, Spiridon Bay, Uyak Bay Karluk Lagoon, Northeast Harbor, Karluk Tables A. Spill Prevention and Response: Cook Inlet Subarea Contingency Plan, Section D: Sensitive Areas. Trading Bay State Game Refuge and Redoubt Bay Critical Habitat Area Management Plan. River discharge predicts spatial distributions of beluga whales in the Upper Cook Inlet, Alaska, during early summer. Prudhoe Crude Oil in Arctic Marine Ice, Water, Degradation and Interactions with Microbial and Benthic Communities. Chemical and Biological Weathering of Oil, from the Amoco Cadiz Spillage, within the Littoral Zone. Issues and Challenges with Oil Toxicity Data and Implications for their use in Decision Making: A Quantitative Review. Proceedings of the Arctic Oilspill Program Technical Seminar, Ottawa, Ontario, Canada: Environment Canada. Large-Scale Cold Water Dispersant Effectiveness Experiments with Alaskan Crude Oils and Corexit 9500 and 9527 Dispersants. Patterns and Processes of Population Change in Selected Nearshore Vertebrate Predators. Transport and Transformation Processes Regarding Hydrocarbon and Metal Pollutants in Offshore Sedimentary Environments. Clay-Oil Flocculation and Its Effect on the Rate of Natural Cleaning in Prince William Sound Following the Exxon Valdez Oil Spill. In: Third Symposium on Environmental Toxicology and Risk Assessment: Aquatic, Plant and Terrestrial, Philadelphia, Pa. Experimental Oil Release in Broken Ice- A Large-Scale Field Verification of Results from Laboratory Studies of Oil Weathering and Ignitiability of Weathered Oil Spills. Oil Properties, Dispersibility and In Situ Burnability of Weathered Oil as a Function of Time. Climatic Atlas of the Outer Continental Shelf Waters and Coastal Regions of Alaska, Vol. Aerial Surveys of Endangered Cetaceans and other Marine Mammals in the Northwestern Gulf of Alaska and Southeastern Bering Sea. Outer Continental Shelf Environmental Assessment Program, Final Reports of Principal Investigators (June 1989). In: Outer Continental Shelf Environmental Assessment Program, Final Reports of Principal Investigators (June 1989). Numerical Simulation of Ice-Ocean Variability in the Barents Sea Region: Towards Dynamical Downscaling.

However gluten causes erectile dysfunction cheap 20mg cialis professional with visa, there have always been bureaucrats to put some limits on the enthusiasts (for which moderate users should be grateful) erectile dysfunction after radiation treatment for prostate cancer cheap cialis professional 20mg online. Because of these constraints erectile dysfunction young living cialis professional 20 mg on-line, one can get in enormous trouble if a proposed 20 hour calculation in fact requires 200 erectile dysfunction after drug use buy cialis professional 20 mg lowest price. One is library research: looking up the costs and resolution of previously published calculations for similar problems. One chooses a simple analytic function which has the same behavior as is expected of the unknown solution - boundary layers, fronts, oscillations, whatever. If the model has the same scales of variation as the true solution, then one can obtain a good estimate of how many basis functions will be needed. The "quasi-sinusoidal" rule-of-thumb is based on the observation that a wide variety of phenomena have oscillations. If we can estimate or guess the approximate wavelength - strictly speaking, we should say local wavelength since the scale of oscillation may vary from place to place - then we ought to be able to estimate how many Chebyshev polynomials are needed to resolve it. To achieve a 1% error, the second order method requires N/M > 40, the second order method needs N/M > 15, while the spectral method needs N/M > 3. The striking differences are between the spectral methods and the finite difference methods, not between different variants within the family of spectral algorithms. This is but a rule-of-thumb - an unkind wit might define "rule-of-thumb" as a synonym for "educated guess" - but it is representative of the extraordinary efficiency of pseudospectral methods versus finite difference methods for large N. The rule-of-thumb lists the needed resolution for 1% error to emphasize that the advantage of pseudospectral over finite difference methods increases very rapidly as the error decreases, even for rather moderate accuracy. There is almost no accuracy at all until N/M >, and then the error just falls off the table. The reason is that when we increase N by 1 or 2, we typically decrease the error (for a sine function) by an order-of-magnitude whether N is 10 or 10,000. This increase of N (for the sake of more accuracy) has a negligible effect on the ratio N/M when M is huge, but N/M must be significantly greater than 3 to obtain high accuracy when M is small. If we demand the truncation error = 10-d so that d is the number of decimal places of accuracy, then we obtain the following. By "width of the internal boundary layer", we mean that the spoor of the complex singularities is that the witch-of-Agnesi function rises from 1/2 to 1 and then falls to 1/2 again on the small interval [-,] as shown on the inset in. Presumably, some small-scale features in hydrodynamic flows (or whatever) are similarly associated with poles or branch points for complex x close to the center of the feature. First, singularities a distance from the real axis are most damaging to convergence when near the center of the interval. This special endpoint effect is described in the next section, but it significantly modifies (2. In this case, one can show that N (d,) should be determined from the more complex rule N + k log(N) d log(10) (2. Usually the simpler rule is best because one rarely knows the precise type or location of a complex singularity of the solution to a differential equation; one merely estimates a scale. When the boundary layer is of width where 1, it is normally necessary to increase the number of grid points as O(1/) as 0. This predicts that if a function has a peak in the interior of the interval of width 2 due to singularities at (x) =, then the error will be 10d where d = N /log(10). The errors in the Chebyshev approximation of f (x) 2 /(x2 + 2) is plotted versus N / log(10). As predicted, the curves lie approximately on top of one another (with some deviation for = 1. The inset graph illustrates the "Witch of Agnesi" function itself along with the meaning of the width parameter. A much better approach is to use a mapping, that is, a change of variable of the form y = f (x) (2. Unfortunately, there are limits: If the mapping varies too rapidly with x, it will itself introduce sharp gradients in the solution, and this will only aggravate the problem. However, the higher the order of the method, the more rapidly the transformation may vary near the boundary.

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When a Fourier cosine series is mapped so as to create a new basis set erectile dysfunction protocol book scam purchase cialis professional line, the orthogonality relationship is preserved: f cos(mx) cos(nx) dx 0 f (0) m (y) n (y) dy; f (f -1 [y]) (16 erectile dysfunction meme order cheap cialis professional online. To have the machinery for transforming derivatives is one thing; to know which mappings are useful for which problems is another impotence zinc discount cialis professional online visa. In the remainder of this chapter erectile dysfunction pills free trial purchase cialis professional, we offer a variety of illustrations of the many uses of mappings. For example, on the infinite interval y [-,], Hermite functions or sinc (Whittaker cardinal) functions are good basis sets. On the semi-infinite domain, y [0,], the Laguerre functions give exponential convergence for functions that decay exponentially as y. A second option is what we shall dub "domain truncation": solving the problem on a large but finite interval, y [-L, L] (infinite domain) or y [0, L] (semi-infinite) using Chebyshev polynomials with argument (y/L) or (L/2)[1+y]), respectively. The rationale is that if the solution decays exponentially as y, then we make only an exponentiallysmall-in-L error by truncating the interval. This method, like the three basis sets mentioned above, gives exponentially rapid convergence as the number of terms in the Chebyshev series is increased. This in turn implies that the rate of decrease of the total error with N is "subgeometric". The third option, and the only one that is strictly relevant to the theme of this chapter, is to use a mapping that will transform the unbounded interval into [-1, 1] so that we can apply Chebyshev polynomials without the artificiality of truncating the computational interval to a finite size. A wide variety of mappings are possible; an early paper by Grosch and Orszag (1977) compared y = -L log(1 - x) y = L(1 + x)/(1 - x) "Logarithmic Map" "Algebraic Map" y [0,] x [-1, 1] (16. Grosch and Orszag (1977) and Boyd (1982a) offer compelling theoretical arguments that logarithmic maps will always be inferior to algebraic maps in the asymptotic limit N. Paradoxically, however, some workers like Spalart (1984) and Boyd (unpublished) report good results with an exponential map. The resolution of the paradox is (apparently) that exponentially-mapped Chebyshev series approach their asymptotic behavior rather slowly so that for N O(50) or less, they may be just as good as algebraic maps. Nevertheless, we shall discuss only algebraic maps here; more general maps are discussed in the next chapter. The logarithmic change-of-coordinate is like a time bomb - it may not blow up your calculation, but theory shows that it is risky. However, that the Chebyshev polynomials themselves are merely cosine functions in masquerade. We can alternatively define the "rational Chebyshev functions" via T Ln L cot2 [t/2] cos(nt) (16. The pseudospectral grid points are simply the images under the map yi = L cot2 (ti /2) of the evenly spaced roots in t. We must not forget, however, that even though we may prefer to write our program to solve the trigonometric version of the problem via a Fourier cosine series, the end result is a series for u(y) as a sum of orthogonal rational functions, the T Ln (y). The key both to writing the program and to understanding the subtleties of infinite and semi-infinite domains is the change of coordinates. Stenger (1981) pointed out that a mapping of X [-1, 1] to y [-,] would heal such "weak" endpoint singularities if and only if dX/dy decays exponentially fast as y. The grid spacing in the original coordinate, X, is related to the grid spacing in y by X dX y. In startling contrast to its preimage, sech(y) is a remarkably well-behaved function. Stenger (1981) shows that the sinc expansion of sech(y) on y [-,] has subgeometric but exponential convergence with exponential index of convergence r = 1/2. Boyd (1986a) shows that a Hermite series also has r = 1/2, but with even faster convergence. Table V of Boyd (1987a) shows that the sum of the first sixteen symmetric T Bn (arctanh[X]) gives an error of no more than 1 part in 300,000 for the approximation of 1 - X 2.

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The challenging task of customizing old algorithms to new vector and parallel hardware has also (sensibly) begun with simple differences and elements erectile dysfunction san antonio order cialis professional 40 mg on line. Lastly erectile dysfunction medication for high blood pressure order line cialis professional, for weather forecasting and many other species of models erectile dysfunction treatment in urdu purchase generic cialis professional on-line, the physics is so complicated - photochemistry food erectile dysfunction causes discount cialis professional on line, radiative transfer, cloud physics, topographic effects, air-sea interaction, and finite resolution of observations - that purely numerical errors are a low priority. Another reason is that we can, with high order algorithms, explore numerical frontiers previously unreachable. As shown later, inside every low order program is a high order algorithm waiting to burst free. Given a second order finite difference or finite element boundary-value solver, one can promote the code to spectral accuracy merely by appending a single subroutine to spectrally evaluate the residual, and then calling the boundary value solver repeatedly with the spectral residual as the forcing function. Similarly, the structure and logic of an initial value solver is very much the same for both low and high order methods. The central question is simply: Will one approximate the spatial derivatives badly or well These errors are distinct from the spectral coefficients , which in turn are not the same as the terms in the series, which are coefficients multiplied by a basis function. Our first major theme is that all these quantities, though distinct, have the property of decaying to zero with increasing N at the same qualitative rate, usually exponentially. This states that the convergence of a spectral series for u(x) is controlled by the singularities of u(x) where "singularity" is a catch-all for any point in the complex x-plane where u(x) ceases to be analytic in the sense of complex variable theory. Square roots, logarithms, simple poles, step function discontinuities and infinities or abrupt discontinuities in any of the derivatives of u(x) at a point are all "singularities". This justifies the "Method of Model Functions": We can understand a lot about the success and failure of spectral methods by first understanding the spectral series of simple, explicit model functions with various types of logarithms, poles, and jumps. Several qualitatively different rates are possible: algebraic, geometric, subgeometric, and supergeometric. We will return to each of these four key themes in the middle of the chapter, though not in the same order as above. The only alteration is that the limits of integration in the coefficient integrals (2. Second note: the general Fourier series can also be written in the complex form f (x) = n=- cn exp(inx) (2. The coefficients of the two forms are related by c0 cn = a0, n=0 (an - ibn)/2, = (an + ibn)/2, n>0 n<0 Often, it is unnecessary to use the full Fourier series. In particular, if f (x) is known to have the property of being symmetric about x = 0, which means that f (x) = f (-x) for all x, then all the sine coefficients are zero. The series with only the constant and the cosine terms is known as a "Fourier cosine series". These special cases are extremely important in applications as discussed in the Chapter 8. These will allow us to develop an important theme: the smoother the function, more rapidly its spectral coefficients converge. Fourier series work best for periodic functions, and whenever possible, we will use them only when the boundary conditions are that the solution be periodic. Because all the basis functions are periodic, their sum must be periodic even if the function f (x) in the integrals is not periodic. The result is that the Fourier series converges to the so-called "saw-tooth" function. The graph of the error shows that the discontinuity has polluted the approximation with small, spurious oscillations everywhere. At any given fixed x, however, the amplitude of these oscillations decreases as O(1/N). Near the discontinuity, there is a region where (i) the error is always O(1) and (ii) the Fourier partial sum overshoots f (x) by the same amount, rising to a maximum of about 1. Fortunately, through "filtering", "sequence acceleration" and "reconstruction", it is possible to ameliorate some of these -2 - 0 2 3 Figure 2. For clarity, both the partial sums and errors have been shifted with upwards with increasing N.

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