Summations

As previously mentioned, a single action potential at a single synapse results in a graded potential only an EPSP or an IPSP. Therefore, generation of an action potential in the postsynaptic neuron requires the addition or summation of a sufficient number of excitatory inputs to depolarize this neuron to threshold. Two types of summation may occur  

As with temporal summation, this example has been simplified to illustrate the concept clearly. In actuality, a large number of excitatory inputs from different presynaptic neurons are necessary to depolarize the postsynaptic neuron to threshold. Because a typical neuronal cell body receives thousands of presynaptic inputs, spatial summation also occurs quite readily. The number of presynaptic neurons that are active simultaneously therefore influences the strength of the signal to the postsynaptic neuron. Under normal physiological conditions, temporal summation and spatial summation may occur concurrently. 

Throughout this book all the different types of mechanical properties are presented and reviewed. These mechanical properties include a tremendous range of different types that can usually be characterized by their stiffness, strength, and toughness. 

Stiffness The same factors that influence thermal expansion dictate the stiffness of plastics. Thus in a TS the degree of cross-linking and amount of overall flexibility are important. As an example, in a TP its crystallinity and secondary bond s strength control its stiffness. 

The crystallinity of TPs is important for their short term yield strength. Unless the crystallinity is impeded, increased molecular weight generally also increases the yield strength. However, the cross-linking of TSs increases their yield strength substantially but has an adverse effect upon toughness. 

Cross-linking produces some dimensional stability and improves toughness in a noncrystalline type of plastic above the Tg, but 

Pick an arbitrary ion near the center of the array and draw squares around it. (In a three-dimensional sum, we draw cubes around it.) We simply add the contributions from the ions in each square. For the first square, Ri = -i/l+4/2. The contribution from the second square would be P2 = 4/2 - 8/(2 +1 ) / +4/(2 +2 ) / The third term would be P3 = —4/3 + 8/(3 +1 ) — (3 + 2 ) + 4/(3 + 3 ), etc. A general term can be written as 

Slow Convergence of the Lattice Using the Direct Method  

The problem with this straightforward approach is that the convergence is extremely slow. For example, let us examine the first 10 terms and their sum shown in Table A.3.1. 

Notice how slowly the sum is converging. Even after 1000 squares, the sum has only converged to the third decimal. 

Convergence may be hastened by summing the contributions of the charges contained within each of the squares instead of the ions on the squares. For example, since only 1 /2 of the 4 face atoms and 1 /4 of the 4 comer atoms lie within the first square, the contribution from the charges within the first square is Si = 2/1 + 1. The contributions from the charges between the first and second square include the remainder of the ions outside the first square, 2/1 + 3/(1 + as well the portion of the charges that lie inside the second square, i.e., +2/2 — 4/(2 + +1 /(2 + The contributions when smnmed 

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Unfortunately, many commonly used methods for parameter estimation give only estimates for the parameters and no measures of their uncertainty. This is usually accomplished by calculation of the dependent variable at each experimental point, summation of the squared differences between the calculated and measured values, and adjustment of parameters to minimize this sum. Such methods routinely ignore errors in the measured independent variables. For example, in vapor-liquid equilibrium data reduction, errors in the liquid-phase mole fraction and temperature measurements are often assumed to be absent. The total pressure is calculated as a function of the estimated parameters, the measured temperature, and the measured liquid-phase mole fraction. 

Hess s law Sometimes called the law of constant heat summation, it states that the total heat change accompanying a chemical reaction is independent of the route taken in reactants becoming products. Hess s law is an application of the first law of thermodynamics to chemical reactions. 

At contact fatigue tests of different steel and cast iron types was used the acoustic emission technique. Processed records from the AE analyser show importance of acoustic response of tested surface continuous sensing. In graphs are obvious characteristic types of summation curves, or may be from significant changes of AE signal course identified even phases of the wear process. 

Where Ui denotes input number i and there is an implied summation over all the inputs in the expression above A, Bj, C, D, and F are polynomials in the shift operator (z or q). The general structure is defined by giving the time delays nk and the orders of the polynomials (i.e., the number of poles and zeros of the dynamic models trom u to y, as well as of the noise model from e to y). Note that A(q) corresponds to poles that are common between the dynamic model and the noise model (useful if noise enters system close to the input). Likewise Fj(q) determines the poles that are unique for the dynamics from input number i and D(q) the poles that are unique for the noise N(t). 

A 3D summation tomographic image is generated and layers in any direction may be viewed. Image quality is improved by filters. The method has been proved satisfactory on a high radiographic contrast cylindrical object and on more complex items. 

Tomographic summation images of vertical, horizontal and cylindrical layers were generated by positioning the film in suitable positions. The method has a deficiency when the films have to be positioned close one to the other. Hence for the generation of close or overlapping cuts there is a need for repeated exposures. 

After the summation of 180° rotation, the 3D voxels array was viewed from 4 different directions. These horizontal cuts (line 20, line 50 and line 80), and a vertical cut are presented in Figs. 5 and 6. 

The study presented a new tomographic method based on 3D reconstruction from a single 180 exposure. The method is based on the summation principle and the 3D voxel image is generated almost in real-time. 

One can then use the classical algorithm of tomographic reconstruction by summation of filtered back-projections ... 

Intersections of butts and seams of fabrication and section welds Throughout hull envelope, longitudinal and transverse bulkheads, inner bottom and hopper bottom The summation of checkpoint lengths (see note 2) examined at intersections is to be L where L is the overall length of the ship in metres... 

The total interaction between two slabs of infinite extent and depth can be obtained by a summation over all atom-atom interactions if pairwise additivity of forces can be assumed. While definitely not exact for a condensed phase, this conventional approach is quite useful for many purposes [1,3]. This summation, expressed as an integral, has been done by de Boer [8] using the simple dispersion formula, Eq. VI-15, and following the nomenclature in Eq. VI-19 ... 

The rotational energy of a rigid molecule is given by 7(7 + l)h /S-n- IkT, where 7 is the quantum number and 7 is the moment of inertia, but if the energy level spacing is small compared to kT, integration can replace summation in the evaluation of Q t, which becomes... 

The electronic energy, as detennined from must be added to tire ion-ion interactions to obtain the structural energies. This is a straightforward calculation for confined systems. For extended systems such as crystals, the calculations can be done using Madelimg summation techniques [2]. 

The third step follows after interchanging summation and integration and recognizing drat the N(N - l)/2 tenns in the sum are identical. The pressure follows from the relation... 

For a synnnetrical system in which the reference species are identical (e.g. hard spheres of the same size), the integral can be taken outside the summation, which then adds up to zero due to the electroneutrality condition, to yield... 

T is the free energy fiinctional, for which one can use equation (A3.3.52). The summation above corresponds to both the sum over the semi-macroscopic variables and an integration over the spatial variableThe mobility matrix consists of a synnnetric dissipative part and an antisyimnetric non-dissipative part. The syimnetric part corresponds to a set of generalized Onsager coefficients. 

The summation of pair-wise potentials is a good approximation for molecular dynamics calculations for simple classical many-body problems [27], It has been widely used to simulate hyperthennal energy (>1 eV) atom-surface scattering ... 

Gil-Villegas A, McGrother S C and Jackson G 1997 Reaction-field and Ewald summation methods in Monte Carlo simulations of dipolar liquid crystals Mol. Phys. 92 723-34... 

The summation extends over tire - 1 Rouse modes witli relaxation time i . ) = 1,. . AT> - 1, aiid c is tire... 

Savageau M A, Volt E O and Irvine D H 1987 Bioohemioal systems theory and metabolio oontrol theory 2. The role of summation and oonneotivity relationships Math. Biosci. 86 147-69... 

The symbol M represents the masses of the nuclei in the molecule, which for simplicity are taken to be equal. The symbol is the Kionecker delta. The tensor notation is used in this section and the summation convention is assumed for all repeated indexes not placed in parentheses. In Eq. (91) the NACT appears (this being a matrix in the electronic Hilbert space, whose components are denoted by labels k, m, and a vector with respect to the b component of the nuclear coordinate R). It is given by an integral over the electron coordinates... 

The summation convention for double indices, for example, k in Eq. (113), is assumed, as before. However, we no longer make distinction between covariant and contravariant sets.) We set ourselves the task to find anti-Hermitean operators Xf, such that... 

Again, the summation convention is used, unless we state otherwise. As will appear below, the same strategy can be used upon tbe Dirac Lagrangean density to obtain the continuity equation and Hamilton-Jacobi equation in the modulus-phase representation. 

Introducing the moduli ai and phases <]) for the four spinor components t]ij (i = 1,2,3,4), we note the following relations (in which no summations over i are implied) ... 

Where the summation is over all 2n pemrutations P each with parity Cp. We use a short-hand notation ... 

In the second summation, we find that, since m — u = 0 when i = m, we can safely extend the upper limit to m. Thus,... 

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