![]() ![]() An important consequence of our result is that uniqueness of the trivial solution of the PDE is equivalent to uniqueness of the trivial solution of the corresponding ODE, which in turn is known to be equivalent to an Osgood-type integral condition on f. In this paper we provide an elementary proof of non-uniqueness for the PDE without any such concavity assumption on f. This concavity assumption has remained in place either implicitly or explicitly in all subsequent work in the literature relating to this and other, similar, non-uniqueness phenomena in parabolic equations. Theorem 3 If n > m then a homogeneous system of equations has innitely many solutions. Thus a homogeneous system of equations always either has a unique solution or an innite number of solutions. Often, solutions or examples involving the number 0 are considered trivial. ![]() from publication: The dynamics of a stage structure population model with. But I am interested in a vector different from 0. Since a homogeneous system always has a solution (the trivial solution), it can never be inconsistent. A solution or example that is ridiculously simple and of little interest. Download scientific diagram The stable trivial solution of system (5.1). In particular they showed that if the underlying ODE has non-unique solutions (as characterised via an Osgood-type condition) and the nonlinearity f satisfies a concavity condition, then the parabolic PDE also inherits the non-uniqueness property. Now Mathcad of course solves the problem But it only gives back the trivial solution, i.e.: x(0 0 0). In their (1968) paper Fujita and Watanabe considered the issue of uniqueness of the trivial solution of semilinear parabolic equations with respect to the class of bounded, non-negative solutions. ![]() Experiments on benchmarks demonstrate the proposed ensemble based DSH can improve the performance of DSH approaches significant.© 2017 Elsevier Inc. Moreover, it is very easy to parallelize the training and support incremental model learning, which are very useful for real-world applications but usually ignored by existing DSH approaches. We found out that this simple strategy is capable of effectively decorrelating different bits, making the hashcodes more informative. To tackle these problems, we propose to adopt ensemble learning strategy for deep model training. ![]() in order to have a non trivial solution the determinant of the coefficient Matrix. If 8, then rank of A and rank of (A, B) will be equal to 3.It will have unique solution. According to 1 the stability definition of the trivial solution x(/)0 of (5). Non-trivial solutions are a little more difficult to find than trivial ones. In other words, a simple solution to an equation is termed a trivial solution. They are of less importance but cannot be skipped due to the sake of completeness. It makes sense because its a trivial solution. One important reason is that it is difficult to incorporate proper constraints into the loss functions under the mini-batch based optimization algorithm. Solution : We have, x+3y-2z02x-y+4z0 . of trivial solution x()0 with respect to the vector function h(x) x. Trivial solutions are the solutions to some equations which have a simple structure. Similarly, the differential equation y y has the trivial solution y 0 and the nontrivial solution y(x) exp(x). The exact solution is y(x)cos(sqrt(2)x), while what Ive got is something like a function. In this paper, we show that the widely used loss functions, pair-wise loss and triplet loss, suffer from the trivial solution problem and usually lead to highly correlated bits in practice, limiting the performance of DSH. Hashing, Deep Learning, Neural Network Abstractĭeep supervised hashing (DSH), which combines binary learning and convolutional neural network, has attracted considerable research interests and achieved promising performance for highly efficient image retrieval. ![]()
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