We show existence and regularity of solutions in R to nonlinear elliptic equations of the form −div A(x, Du) + g(x, u) = f , when f is just a locally integrable function, under appropriate growth conditions on A and g but not on f . Roughly speaking, in the model case −∆p(u) + |u|s−1u = f , with p > 2 − (1/N), existence of a nonnegative solution in R is guaranteed for every nonnegative f ∈ Lloc...

In graph clustering methods, MinMax Cut tends to provide more balanced clusters as compared to Ratio Cut and Normalized Cut. The traditional approach used spectral relaxation to solve the graph cut problem. The main disadvantage of this approach is that the obtained spectral solution has mixed signs, which could severely deviate from the true solution and have to resort to other clustering meth...

Nonnegativity constraints arise frequently in statistical learning and pattern recognition. Multiplicative updates provide natural solutions to optimizations involving these constraints. One well known set of multiplicative updates is given by the ExpectationMaximization algorithm for hidden Markov models, as used in automatic speech recognition. Recently, we have derived similar algorithms for...

Let a,b (b > a) be nonnegative integers. We define the discrete interval [a,b] = {a,a + 1, . . . ,b}. All other intervals will carry its standard meaning, for example, [0,∞) denotes the set of nonnegative real numbers. The symbol Δ denotes the forward difference operator with step size 1, that is, Δu(k) = u(k + 1)− u(k). Furthermore for a positive m, Δm is defined as Δmu(k)= Δm−1(Δu(k)). In thi...

We study the following quasilinear problem with nonlinear boundary conditions −∆pu = λa(x)|u|p−2u+ k(x)|u|q−2u− h(x)|u|s−2u, in Ω, |∇u|p−2∇u · η + b(x)|u|p−2u = 0 on ∂Ω, where Ω is an unbounded domain in RN with a noncompact and smooth boundary ∂Ω, η denotes the unit outward normal vector on ∂Ω, ∆pu = div(|∇u|p−2∇u) is the p-Laplacian, a, k, h and b are nonnegative essentially bounded functions...

We establish the existence of a nonnegative fully nontrivial solution to non-variational weakly coupled competitive elliptic system. show that this kind solutions belong topological manifold Nehari-type, and apply degree-theoretical argument on derive existence.

Constrained least squares problems arise in a variety of applications, and many iterative methods are already available to compute their solutions. This paper proposes a new efficient approach to solve nonnegative linear least squares problems. The associated KKT conditions are leveraged to form an adaptively preconditioned linear system, which is then solved by a flexible Krylov subspace metho...

New properties of the first‐order Riemann–Liouville fractional integrals are proved and derivatives for equivalence class functions in analyzed. Solutions, normal solutions, generalized solutions initial value problems (IVPs) nonlinear differential equations (FDEs) with fraction introduced. The equivalences between FDEs corresponding (Volterra) integral established based on solutions. results e...

We study the Dirichlet problem for an elliptic equation involving $1$-Laplace operator and a reaction term, namely: $$ \left\{\begin{array}{ll} \displaystyle -\Delta_1 u =h(u)f(x)&\hbox{in }\Omega\,,\\ u=0&\hbox{on }\partial\Omega\,, \end{array}\right. where $ \Omega \subset \mathbb{R}^N$ is open bounded set having Lipschitz boundary, $f\in L^1(\Omega)$ nonnegative, $h$ continuous real function...

Nonnegative matrix factorization (NMF) has become a very popular technique in machine learning because it automatically extracts meaningful features through a sparse and part-based representation. However, NMF has the drawback of being highly ill-posed, that is, there typically exist many different but equivalent factorizations. In this paper, we introduce a completely new way to obtaining more...