Lattice Method Calculator

[Bran Bast]

Inthe materialized views usage research field there are two fundamental types of problems: finding which views are best to materialize and rewriting the original query to use these views. This article focuses on the first problem.

But why should we care about a Lattice in general if the psc root view can answer the same query set? In this particular example, if 95% of user queries can be answered with s and only 5% require accessing psc, there will be a 1 - (0.01*0.95 + 0.05*6) / 6 = 94.8% waiting time reduction if the s view is used along with psc.

Given that the original Data Cube Lattice problem in general can be formulated as: what is the optimal query set from the Data Cube Lattice that should be materialized given the query set that needs to be answered.

The paper shows that the benefit of the greedy algorithm is at least 63% of the optimal algorithm benefit. This makes this simple algorithm a very powerful method of finding candidate solutions. However, there are several practical considerations we found to be important for real world use cases.

Another thing is that all queries for a greedy algorithm are equal. Even those that will never be queried by users. To mitigate with this bias, the greedy algorithm should be assisted with query weights or probabilities or even a lazy lattice evaluation should be considered based on user querying history to improve Data Cube Lattice performance. Such approaches used by Dremio Aggregation Reflections for example. It's worth mentioning the Genetic Selection Algorithm here as well.

Cube.js uses a pre-aggregation layer, which is a set of materialized views, as a first caching level. It provides a significant performance boost, especially on dynamic dashboards with a lot of filters. You can check out this example dashboard to see how the performance could be improved by enabling pre-aggregations.

Cube.js provides a way to manually declare which materialized views to build. But, as the complexity of the application grows, it becomes tedious and inefficient to declare all of them manually. Cube.js lets you automatically build all the necessary pre-aggregations with the Auto Rollup feature.It heavily relies on the mentioned techniques to solve the Data Cube Lattice optimization problem. It uses querying history to predict optimal queries to materialize and adapt it over time.

A consequence of this property is that the root query is always a union of all queries from the querying history. For the p, s querying history we would have ps as a root query. Obviously the ps query answers both the p and s queries. These properties allow us to formulate a querying history constrained greedy algorithm.

The lattice energies of 47 crystal structures of organic compounds spanning a wide range of chemical functionalities are calculated using simple atom-atom potential energy functions, using coulombic terms with point-charge parameters, and using the PIXEL formulation, which is based on integral sums over the molecular electron density to obtain coulombic, polarization, dispersion and repulsion lattice energies. Comparisons among the different formulations, and the sensitivity and significance of the results against convenience, ease of application and number of parameters, are discussed. Improvements in the treatment of overlap repulsion in PIXEL are described, as well as a scheme for the minimization of the crystal lattice energy, based on the Symplex algorithm, which although computationally demanding, is shown to be feasible even with comparatively modest computing resources. The reproduction of experimental heats of sublimation is only marginally better with the PIXEL method, which however has great advantages in its being generally applicable in principle throughout the periodic table, at the expense of a minimal number of parameters, and in the fact that it sees the intermolecular interaction as the effect of the whole molecular electron density, in a physically more justifiable approach. This latter view in turn suggests a transition from a consideration of atom-atom bonds to a consideration of molecule-molecule bonding, opening a new view of packing factors in molecular crystals.

This can be a sequence of three numbers foran orthorhombic unit cell or three by three numbers for a generalunit cell (a sequence of three sequences of three numbers) or six numbers(three legths and three angles in degrees). The default value is[0,0,0] which is the same as[[0,0,0],[0,0,0],[0,0,0]] or [0,0,0,90,90,90] meaning that none of thethree lattice vectors are defined.

Here is the full list of the get/set methods operating on all theatoms at once. The get methods return an array of quantities, one foreach atom; the set methods take similar arrays.E.g. get_positions() return N * 3 numbers,get_atomic_numbers() return N integers.

The set_pbc() method specifies whetherperiodic boundary conditions are to be used in the directions of thethree vectors of the unit cell. A slab calculation with periodicboundary conditions in x and y directions and free boundaryconditions in the z direction is obtained through

It is also possible to work directly with the attributespositions, numbers,pbc and cell. Herewe change the position of the 2nd atom (which has count number 1because Python starts counting at zero) and the type of the firstatom:

In case of a DFT calculator, it is up to the user to check exactly whatthe get_potential_energy() method returns. Forexample it may be the result of a calculation with a finitetemperature smearing of the occupation numbers extrapolated to zerotemperature. More about this can be found for the differentase.calculators.

Note that the del method can be used with the more powerful numpy-style indexing, as in the second example above. This can be combined with python list comprehension in order to selectively delete atoms within an ASE Atoms object. For example, the below code creates an ethanol molecule and subsequently strips all the hydrogen atoms from it:

The Atoms object can represent an isolated molecule, or aperiodically repeated structure. It has a unit cell andthere may be periodic boundary conditions along any of the threeunit cell axes.Information about the atoms (atomic numbers and position) isstored in ndarrays. Optionally, there can be information abouttags, momenta, masses, magnetic moments and charges.

Items in the info attribute survives copy and slicing and canbe stored in and retrieved from trajectory files given that thekey is a string, the value is JSON-compatible and, if the value is auser-defined object, its base class is importable. One shouldnot make any assumptions about the existence of keys.

If any cell vectors are zero, the corresponding coordinatesare evaluated as if the cell were completed usingcell.complete(). This means coordinates will be Cartesianas long as the non-zero cell vectors span a Cartesian axis orplane.

By default, the random number generator always uses the same seed (42)for repeatability. You can provide your own seed (an integer), or if youwant the randomness to be different each time you run a script, thenprovide \(rng=numpy.random\). For a parallel calculation, it is importantto use the same seed on all processors!

If mask is not None, all the atoms described in mask(read: the entire subgroup) are moved. Alternatively to the mask,the indices of the atoms to be rotated can be supplied. If bothmask and indices are given, indices overwrites mask.

Set the distance between atoms a0 and a1 to distance.By default, the center of the two atoms will be fixed. Usefix=0 to fix the first atom, fix=1 to fix the secondatom and fix=0.5 (default) to fix the center of the bond.

Employee stock option (ESO) valuation: Standard Black-Scholes and lattice pricing models cannot be used to value ESOs due to vesting requirements, the impact of staff turnover rates, and other ESO-specific factors which are not a part of standard option pricing. For tools which specifically handle IFRS 2 and FASB 123R-compliant ESO valuation see ESO valuation.

You can use the on-line options pricing analysis calculators to see, in tabular form and graphically, how changing each of the Black-Scholes variables impacts the option price, time value and the derived "Greeks".

You can also examine how changes in the Black-Scholes variables affect the probability of the option being in the money (ITM) at expiration. In calculating the probability of closing ITM a lognormal distribution of stock prices with the stock earning a rate of return equal to the risk free interest rate is assumed.

For simplicity, dividends are ignored in the first pricing calculator and the calculator applies European-style pricing. This provides an easy-to-use way to see how the Black-Scholes pricing model works.

Note that the pricing in both calculators is for European style options so don't use the model to try to accurately price equity options, particularly those with an ex-dividend date very close to expiration. If you want accurate prices for American options then either use the on-line binomial or trinomial pricing calculators, the Options Strategy Evaluation Todel, or the Finance Add-in for Excel.

This calculator calculates the value of standard barrier options (up and out, down and out, up and in, down and in) using a trinomial lattice. Whilst analytic formulas exist for some European options (eg with continuous dividends) most barrier options --eg American barrier options with discrete dividends -- have no analytic solution and thus numerical methods, such as the trinomial lattice, must be used.

This set of calculators lets you examine how a discrete dividend paid during the life of American options, and the timing of the dividend payment, impacts option prices by using the dividend impact analysis calculator.

For European options, the timing of the ex-dividend date has relatively little effect on the price of the option -- whether the ex-dividend date is close to today's date, or close to the expiration date generally has only a slight effect.

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