Upper bound theorem

Strictly, shear occurs not just on the shear planes we have drawn, but on a myriad of 45° planes near the indenter. If our assumed geometry for slip is wrong it can be shown rigorously by a theorem called the upper-bound theorem that the value we get for F at yield - the so-called limit load - is always on the high side.)... 

Another way of stating the upper bound theorem is that during collapse, the work done by the external loads is equal to the work dissipated by the internal loads. The lower bound theorem states that if any internal stress distribution can be found which does not exceed the )deld stress at any point, and which is in equilibrium with the applied sustained loading, then the structure does not collapse. 

Kinematic or upper bound theorem If a kinematic admissible mechanism can be found for which the work of the external loads exceeds the internal work, then the stracture will collapse and the load factor (3ii) is greater than or equal to the failure load factor (5p). In the kinematic approach the failure load factor is determined by searching for the minimum load factor. 

With regard to kinematic approach, when the structure presents an enough number of hinges to transform it into a mechanism (Fig. 4b), the load factor obtained by equating the work of the external loads to zero corresponds to an upper bound limit (upper bound theorem). The load obtained from the kinematic approach is greater than or equal to the failure load. Different positions for the hinges on the structure can be adopted, and several mechanisms, and respective load factors, can be found. The mechanism that presents the lowest load factor corresponds to the collapse mechanism. Thus, the kinematic approach presents also several solutions. 

Random Coding. We now turn to the subsidiary developments and theorems necessary in order to prove Theorem 4-11. The first obstacle in the path of deriving an upper bound to the probability of decoding error for a particular channel is the difficulty of finding good codes. Even if we could find such codes, the problem of evaluating their error probability would be extremely tedious if M and N were large both... 

Eqs. (4-137), (4-138), and (4-139) yield an upper bound to the average probability of decoding error over an ensemble of codes involving the channel transition probabilities PJk, the ensemble input probabilities, pk, the block length N, and the code rate 22. Unfortunately this expression also involves the spurious parameters a, r, t, d, and /y to achieve the best bound, these parameters should be eliminated by mfniwiying Eq. (4-137) with respect to them. The parameters will be eliminated in the order t, d and The parameters r and 8 will then be replaced by one parameter to yield the bound on Pe expressed in Theorem 4-11. To minimize the right side of Eq. (4-137) over t, it suffices to minimize h(r,t) with respect to t. 

We shall establish an upper bound on error probability for the best code of rate B and block length N on this channel by considering a decoder that quantizes the output before decoding. Theorem 4-10 already provides a bound on error probability for such a quantized channel. We then find the limit of this bound as the quantization becomes infinitely fine. 

According to the variation theorem, the lowest root g o is an upper bound to the ground-state energy Eq Eo So- The other roots may be shown to be upper bounds for the excited-state energy levels... 

The numerical value of S is listed in Table 9.1. The simple variation function (9.88) gives an upper bound to the energy with a 1.9% error in comparison with the exact value. Thus, the variation theorem leads to a more accurate result than the perturbation treatment. Moreover, a more complex trial function with more parameters should be expected to give an even more accurate estimate. 

Here, f(t) = e2t. There is no upper bound for this function, which is in violation of the existence of a final value. The final value theorem does not apply. If we insist on applying the theorem, we will get a value of zero, which is meaningless. 

Cj is constant), is both the upper bound [32, 33, 39] and the lower [37] estimate of the exact solution. Equation (5.2.1) results from a strict mathematical theorem [32, 33] and is proved by its coincidence for d = 1 with the exact solution for both continuous [31, 37] and lattice [43] statements of the problem. 

More recently, Boley (B9) shows that the method can be extended to find minimum and maximum bounds for ablation rates. The proof is based on a uniqueness theorem deriving from a theorem due to Picone (P3) and leads to the physically obvious result that the higher the heat input rate, the higher the melting rate. The procedure consists of forming a lower bound for the known arbitrary heat input function in terms of a sequence of constant heat flux periods, for which, as noted above, the solution can be written in terms of integrals of the error function. Upper bounds are constructed in a similar manner. 

An examination of Eq. (230), which is similar in form to Eq. (81), shows that, for melting, d20i/dX2 > 0. From the mean value theorem it follows immediately that an upper bound for the temperature in melting is given by... 

Remark 4 This important lower-upper bound result for the dual-primal problems that is provided by the weak duality theorem, is not based on any convexity assumption. Hence, it is of great use for nonconvex optimization problems as long as the dual problem can be solved efficiently. 

The weak duality theorem provides the lower-upper bound relationship between the dual and the primal problem. The conditions needed so as to attain equality between the dual and primal solutions are provided by the following strong duality theorem. 

The first several variational eigenvalues are then upper bounds to the true eigenvalues, provided only that the correct number of variational eigenvalues lies below (Hylleraas-Undheim-MacDonald Theorem [24]), and the eigenvector coefficients are the optimum values of the a%jk coefficients in Eq. (3). For fixed a and / , all the eigenvalues move inexorably downward toward the exact energies as J is progressively increased. 

Theorem 1 highlights the strong relation between a decomposition of the state space into metastable subsets and a Perron cluster of dominant eigenvalues close to 1. ft states that the metastability of an arbitrary decomposition d cannot be larger than I-I-A2+... - -Ato, while it is at least 1- -K2 2+- c, which is close to the upper bound whenever the dominant eigenfunctions V2,..., Vm are almost constant on the metastable subsets Pi,..., Dm implying Kj fn 1 and c 0. The term c can be interpreted as a correction that is small whenever a 0 or Kj 1. It is demonstrated in [23] that the lower and upper bounds are sharp and asymptotically exact. 

Let us consider two cases defined in Table 1 below In case 1, we obviously have a metastable decomposition into the subsets 1,2 and 3,4 with metastability measure 0.995 while any further decomposition significantly lowers the metastability measure. Table 2 shows that the spectrum of the transfer operator exhibits a corresponding gap after the first two eigenvalues, and that the upper bound from our theorem, (Ai - - A2)/2 = 0.995, is a very good approximation of meta( l, 2, 3,4 ). The second eigenvector t>2 = (0.51,0.49,-0.49,-0.50) clearly exhibits almost constant levels on the two sets 1,2 and 3,4 of the metastable decomposition. In case 2,... 

A further important property of the Fixed node method is the existence of a variational theorem the FN-RQMC energy is an upper bound of the true ground state energy Et oo) > Eq, and the equality holds if the trial nodes coincide with the nodes of the exact ground state [6]. Therefore for fermions, even projection methods such as RQMC are variational with respect to the nodal positions the nodes are not optimized by the projection mechanism. The quality of the nodal location is important to obtain accurate results. 

Since the variational theorem proves that the energy of a Cl wavefunction is always an upper bound to the exact energy, one might start simply from the linear expansion (1) and attempt to minimize the energy by varying the Cl coefficients subject to the constraint that they remain normalized. It is easy to show63 that this method of linear variations, or the Ritz method,73 yields the matrix equation... 

The key theorem is that upper bounds, given by norms, and lower bounds, given by actually taking eigenvalues as per the definition of joint spectral radius, do in fact converge to the same value. 

This notion of exponentially small is defined by an explicit upper bound, as in previous conventional definitions. A benefit is that one can decide what error probability one is willing to tolerate and set a accordingly. A disadvantage is that the sum of two exponentially small functions need not be exponentially small in this sense, i.e., this formalization does not yield modus ponens automatically (see Section 5.4.4, General Theorems ). Hence different properties have to be defined with different error probabilities, and some explicit computations with O are needed. 

There exists a polynomial Qtau that is an upper bound on tau. Hence this formula is a special case of what has been shown in Part d) of the proof of Theorem 8.57. ... 

---