Tractable Relaxations of Composite Functions

In this paper, we introduce new relaxations for the hypograph of composite functions assuming that the outer function is supermodular and concave extendable. Relying on a recently introduced relaxation framework, we devise a separation algorithm for the graph of the outer function over P, where P is a special polytope to capture the structure of each inner function using its finitely many bounded estimators. The separation algorithm takes [Formula: see text] time, where d is the number of inner functions and n is the number of estimators for each inner function. Consequently, we derive large classes of inequalities that tighten prevalent factorable programming relaxations. We also generalize a decomposition result and devise techniques to simultaneously separate hypographs of various supermodular, concave-extendable functions using facet-defining inequalities. Assuming that the outer function is convex in each argument, we characterize the limiting relaxation obtained with infinitely many estimators as the solution of an optimal transport problem. When the outer function is also supermodular, we obtain an explicit integral formula for this relaxation.

Monotonicity properties and solvability of dominated best approximation problem in Orlicz spaces equipped with s-norms

In the paper, Wisła (J Math Anal Appl 483(2):123659, 2020, 10.1016/j.jmaa.2019.123659), it was proved that the classical Orlicz norm, Luxemburg norm and (introduced in 2009) p-Amemiya norm are, in fact, special cases of the s-norms defined by the formula $$\left\| x\right\| _{\Phi ,s}=\inf _{k>0}\frac{1}{k}s\left( \int _T \Phi (kx)d\mu \right) $$ x Φ , s = inf k > 0 1 k s ∫ T Φ ( k x ) d μ , where s and $$\Phi $$ Φ are an outer and Orlicz function respectively and x is a measurable real-valued function over a $$\sigma $$ σ -finite measure space $$(T,\Sigma ,\mu )$$ ( T , Σ , μ ) . In this paper the strict monotonicity, lower and upper uniform monotonicity and uniform monotonicity of Orlicz spaces equipped with the s-norm are studied. Criteria for these properties are given. In particular, it is proved that all of these monotonicity properties (except strict monotonicity) are equivalent, provided the outer function s is strictly increasing or the measure $$\mu $$ μ is atomless. Finally, some applications of the obtained results to the best dominated approximation problems are presented.

A common range problem for model spaces

We refine a result of [J. E. McCarthy, Common range of co-analytic Toeplitz operators, J. Amer. Math. Soc. 3 1990, 4, 793–799] and explore the common range of the co-analytic Toeplitz operators on a model space. The tools used to do this also yield information about when one can interpolate with an outer function.

Methodology for Implementing the State Estimation in Renewable Energy Management Systems

This paper describes a methodology for implementing the state estimation and enhancing the accuracy in large-scale power systems that partially depend on variable renewable energy resources. To determine the actual states of electricity grids, including those of wind and solar power systems, the proposed state estimation method adopts a fast-decoupled weighted least square approach based on the architecture of application common database. Renewable energy modeling is considered on the basis of the point of data acquisition, the type of renewable energy, and the voltage level of the bus-connected renewable energy. Moreover, the proposed algorithm performs accurate bad data processing using inner and outer functions. The inner function is applied to the largest normalized residue method to process the bad data detection, identification and adjustment. While the outer function is analyzed whether the identified bad measurements exceed the condition of Kirchhoff’s current law. In addition, to decrease the topology and measurement errors associated with transformers, a connectivity model is proposed for transformers that use switching devices, and a transformer error processing technique is proposed using a simple heuristic method. To verify the performance of the proposed methodology, we performed comprehensive tests based on a modified IEEE 18-bus test system and a large-scale power system that utilizes renewable energy.

On kernels of Toeplitz operators

We apply the theory of de Branges–Rovnyak spaces to describe kernels of some Toeplitz operators on the classical Hardy space $$H^2$$ H 2 . In particular, we discuss the kernels of the operators $$T_{{\bar{f}}/ f}$$ T f ¯ / f and $$T_{{\bar{I}}{\bar{f}}/ f}$$ T I ¯ f ¯ / f , where f is an outer function in $$H^2$$ H 2 and I is inner such that $$I(0)=0$$ I ( 0 ) = 0 . We also obtain a result on the structure of de Branges–Rovnyak spaces generated by nonextreme functions.

Multilevel Monte Carlo methods and lower–upper bounds in initial margin computations

The multilevel Monte Carlo (MLMC) method developed by M. B. Giles [Multilevel Monte Carlo path simulation, Oper. Res. 56 2008, 3, 607–617] has a natural application to the evaluation of nested expectations {\mathbb{E}[g(\mathbb{E}[f(X,Y)|X])]}, where {f,g} are functions and {(X,Y)} a couple of independent random variables. Apart from the pricing of American-type derivatives, such computations arise in a large variety of risk valuations (VaR or CVaR of a portfolio, CVA), and in the assessment of margin costs for centrally cleared portfolios. In this work, we focus on the computation of initial margin. We analyze the properties of corresponding MLMC estimators, for which we provide results of asymptotic optimality; at the technical level, we have to deal with limited regularity of the outer function g (which might fail to be everywhere differentiable). Parallel to this, we investigate upper and lower bounds for nested expectations as above, in the spirit of primal-dual algorithms for stochastic control problems.

On Closed Ideals in a Certain Class of Algebras of Holomorphic Functions

We recently introduced a weighted Banach algebra of functions that are holomorphic on the unit disc D, continuous up to the boundary, and of the class C(n) at all points where the function G does not vanish. Here, G refers to a function of the disc algebra without zeros on D. Then we proved that all closed ideals in with at most countable hull are standard. In this paper, on the assumption that G is an outer function in C(n) having infinite roots in and countable zero set h0(G), we show that all the closed ideals I with hull containing h0(G) are standard.

Application of “Planar Muscle” with Soft Skin-Like Outer Function Suitable for Musculoskeletal Humanoid

In recent years, human-like robots have received a lot of attentions. A musculoskeletal humanoid is an effective approach for making a human-like robot, and many musculoskeletal humanoids have been developed. However, none have been equipped with really human-like bones and muscles, especially shapes and alignments. Formaking really human-like musculoskeletal humanoids, we thought of the “planar muscle” as the key. A planar muscle is an enhanced wiredriven system in which a motor winds a wire. In a prior system, one motor controlled one wire and the linear muscle needed one motor per wire and complex control systems. We therefore developed the planar muscle that controlled several wires simultaneously by using two moving pulley bars and one motor. The planar muscle is suited to musculoskeletal humanoids because they need a lot of motors and complex control systems in the case of using linear muscles. Furthermore, planar muscles are useful for soft skin-like outers that protect external shocks and sense touch. Using the planar muscles, we are developing a new musculoskeletal humanoid that has human-like bones and muscles. In this paper, we show the planarmuscle concept, especially its soft skin-like outer functions, and evaluate its motion with a body trunk model having multiple vertebrae that we developed.