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In this paper the explicit necessary and sufficient conditions for the existence of Luenberger reduced order observer are established. In particular, it is proven that for the given linear time-invariant system of order n, having p linearly independent outputs and m inputs, a Luenberger observer of order (n - p) can be constructed if and only if the given system is detectable. Furthermore, a procedure is given for the construction of the observer. Our approach is based on the properties of real and polynomial matrices.

In this paper, we study the Dirichlet problem of Hessian quotient equations of the form $S_k(D^2u)/S_l(D^2u)=g(x)$ in exterior domains. For $g\\equiv \\mbox {const.}$ , we obtain the necessary and sufficient conditions on the existence of radially symmetric solutions. For g being a perturbation of a generalized symmetric function at infinity, we obtain the existence of viscosity solutions by Perron’s method. The key technique we develop is the construction of sub- and supersolutions to deal with the non-constant right-hand side g.

The first part of this work provides explicit solutions for two integral equations; both are solved by means of Fourier transform. In the second part of this paper, sufficient conditions for the existence and uniqueness of the solutions satisfying sandwich constraints for two types of full moment problems are provided. The only given data are the moments of all positive integer orders of the solution and two other linear, not necessarily positive, constraints on it. Under natural assumptions, all the linear solutions are continuous. With their value in the subspace of polynomials being given by the moment conditions, the uniqueness follows. When the involved linear solutions and constraints are positive, the sufficient conditions mentioned above are also necessary. This is achieved in the third part of the paper. All these conditions are written in terms of quadratic expressions.

In this article we consider an optimal control problem of a semi-linear elliptic equation, with bound constraints on the control. Our aim is to characterize local quadratic growth for the cost function JJ in the sense of strong solutions. This means that the function JJ grows quadratically over all feasible controls whose associated state is close enough to the nominal one, in the uniform topology. The study of strong solutions, classical in the Calculus of Variations, seems to be new in the context of PDE optimization. Our analysis, based on a decomposition result for the variation of the cost, combines Pontryagin’s principle and second order conditions. While these two ingredients are known, we use them in such a way that we do not need to assume that the Hessian of the Lagrangian of the problem is a Legendre form, or that it is uniformly positive on an extended set of critical directions.

For a locally optimal solution to the nonlinear semidefinite programming problem, under Robinson's constraint qualification, the following conditions are proved to be equivalent: the strong second-order sufficient condition and constraint nondegeneracy; the nonsingularity of Clarke's Jacobian of the Karush-Kuhn-Tucker system; the strong regularity of the Karush-Kuhn-Tucker point; and others.

We consider the optimality conditions for the DC (difference of two convex functions) optimization problem with the objective and constraint functions given as DC functions. Adopting convexification technique, the local and global KKT type conditions for this optimization problem are defined. By using properties of the subdifferentials of the involved functions, some sufficient and/or necessary conditions for these two types of optimality conditions are provided. 2010Mathematics Subject Classification: 90C26, 90C46. Key words and phrases: Local KKT condition, Global KKT condition, DC infinite optimization problems.

This paper considers the following problem: for what value , a function that is univalent in the unit disk and convex in the disk becomes starlike in . The number is called the radius of convexity sufficient for starlikeness in the class of univalent functions. Several related problems are also considered.

In this study we utilize the notion of learner-generated examples, suggesting that examples generated by students mirror their understanding of particular mathematical concepts. In particular, we explore examples generated by a group of prospective secondary school teachers for a definition of a square. Our framework for analysis includes the categories of accessibility and correctness, richness, and generality. Results shed light on participants' understanding of what a mathematical definition should entail and, moreover, contrast their pedagogical preferences with mathematical considerations.

In this paper, the explicit necessary and sufficient conditions are established for the existence of proportional-integral observer for the state estimation of linear time-invariant continuous-time systems. In particular, it is proven that for a given linear time-invariant continuous-time system of order n, having m inputs and p linearly independent outputs, a proportional-integral observer of order n can be constructed if and only if the given system is detectable. Furthermore a simple procedure is given for the construction of proportional-integral observer. Our approach is based on properties of real and polynomial matrices.

In the first part of the study, we proposed that logicians adopt from linguists their detailed distinction of conditions on a scale of varying degrees of satisfiability from factually true via potentially satisfiable to absolutely irreal. On this basis, we have proposed a suitable distinction for the logico-semantic investigation of conditionals of the following kinds: Factual, hypothetical (agnostic), counterfactual, and counterpossible. We consider the syntactic-semantic distinction between potentially and absolutely irreal conditions to be crucial. Therefore, we think that the conditional mode in the form of antepreterite is an appropriate syntactic signal for identifying the absolute irreality of the condition and, especially in scientific and artistic style, we consider it functional. On the other hand, the distinction between sufficient and necessary conditions could be inspiring for linguists. In classical logic, these are the basic kinds of conditions and are associated with the rules of deductive reasoning modus ponens and modus tollens, respectively. These rules are central to the application of logic. It turns out that these tools are not sufficiently established in Slovak linguistics, as there is no systematic distinction between sufficient and necessary conditions. The situation in czech dictionaries is similar. Finally, we have put forward a hypothesis why the conjunction is considered in Slovak linguistics as a double – in our opinion wrongly – of the conjunction : There is a “forgotten” ellipsis behind it, which is not brought to the reader’s attention.