Outer Function Research Articles

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.

Cyclic Vectors, Outer Functions and Mahler Measure in Two Variables

Cyclic Vectors, Outer Functions and Mahler Measure in Two Variables

Interpolating with outer functions

The classical theorems of Mittag-Leffler and Weierstrass show that when $$(\lambda _n)_{n \geqslant 1}$$ is a sequence of distinct points in the open unit disk $$\mathbb {D}$$ , with no accumulation points in $$\mathbb {D}$$ , and $$(w_n)_{n \geqslant 1}$$ is any sequence of complex numbers, there is an analytic function $$\varphi $$ on $$\mathbb {D}$$ for which $$\varphi (\lambda _n) = w_n$$ . A celebrated theorem of Carleson [2] characterizes when, for a bounded sequence $$(w_n)_{n \geqslant 1}$$ , this interpolating problem can be solved with a bounded analytic function. A theorem of Earl [5] goes further and shows that when Carleson’s condition is satisfied, the interpolating function $$\varphi $$ can be a constant multiple of a Blaschke product. Results from [4] determine when the interpolating function $$\varphi $$ can be taken to be zero free. In this paper we explore when $$\varphi $$ can be an outer function.

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| xright| _{Phi ,s}=inf _{k>0}frac{1}{k}sleft( int _T Phi (kx)dmu right) , where s and Phi are an outer and Orlicz function respectively and x is a measurable real-valued function over a sigma -finite measure space (T,Sigma ,mu ). 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 is atomless. Finally, some applications of the obtained results to the best dominated approximation problems are presented.

Effects of D-serine treatment on outer retinal function

The role of the N-Methyl-D-Aspartate Receptor (NMDAR) in the outer retina is unclear despite expression of the NMDAR-complex and its subunits in the outer retina. The flash-electroretinogram (fERG) offers a non-invasive measurement of the retinal field potentials of the outer retina that can serve to clarify NMDAR contribution to early retinal processing. The role of the NMDAR in retinal function was assessed using a genetic mouse model for NMDAR hypofunction (SR-/-), where the absence of the enzyme serine racemase (SR) results in an 85% reduction of retinal D-serine. NMDAR hypo- and hyperfunction in the retina results in alterations in the components of the fERG. The fERG was examined after application of exogenous D-serine to the eye in order to determine whether pre- and post-topical delivery of D-serine would alter the fERG in SR-/- mice and their littermate WT controls. Amplitude and implicit time of the low-frequency components, the a- and b-wave, were conducted. Reduced NMDAR function resulted in a statistically significantly delayed a-wave and reduced b-wave in SR-/- animals. The effect of NMDAR deprivation was more prominent in male SR-/- mice. A hyperfunction of the NMDAR, through exogenous topical delivery of 5 mM D-serine, in WT mice caused a significantly delayed a-wave implicit time and reduced b-wave amplitude. These changes were not observed in female WT mice. There were temporal delays in the a-wave and amplitude and a decrease in the b-wave amplitude and implicit time in both hypo- and NMDAR hyperfunctional male mice. These results suggest that NMDAR and D-serine are involved in the retinal field potentials of the outer retina that interact based on the animal's sex. This implicates the involvement of gonadal hormones and D-serine in retinal functional integrity.

Progress on the function and regulatory mechanisms of bacterial Cpx signal transduction system

The Cpx (conjugative pilus expression) two-component signal transduction system is a complex envelope stress response system in Gram-negative bacteria, which can sense a variety of extracellular stimuli that enter the signaling pathway at different points. The phosphorylation of the CpxR, the cytoplasmic cognate response regulator of the Cpx system, can lead to changes in the expression of genes encoding proteins involved in inner and outer membrane functions. Activation of the Cpx system contributes to bacterial resistance/tolerance to certain antibiotics and acidic stress. In this review, we summarize the composition, and the mechanisms of signal detection, and the transcriptional regulation of the Cpx system, with a goal to provide guidance for the study of the regulatory network of the Cpx system and its important regulatory roles in bacterial physiology.

Neuroprotection of the Inner Retina Also Prevents Secondary Outer Retinal Pathology in a Mouse Model of Glaucoma.

PurposeWe examined structural and functional changes in the outer retina of a mouse model of glaucoma. We examined whether these changes are a secondary consequence of damage in the inner retina and whether neuroprotection of the inner retina also prevents outer retinal changes.MethodsWe used an established microbead occlusion model of glaucoma whereby intraocular pressure (IOP) was elevated. Specific antibodies were used to label rod and cone bipolar cells (BCs), horizontal cells (HCs), and retinal ganglion cells (RGCs), as well as synaptic components in control and glaucomatous eyes, to assess structural damage and cell loss. ERG recordings were made to assess outer retina function.ResultsWe found structural and functional damage of BCs, including significant cell loss and dendritic/axonal remodeling of HCs, following IOP elevation. The first significant loss of both BCs occurred at 4 to 5 weeks after microbead injection. However, early changes in the dendritic structure of RGCs were observed at 3 weeks, but significant changes in the rod BC axon terminal structure were not seen until 4 weeks. We found that protection of inner retinal neurons in glaucomatous eyes by pharmacological blockade of gap junctions or genetic ablation of connexin 36 largely prevented outer retinal damage.ConclusionsTogether, our results indicate that outer retinal impairments in glaucoma are a secondary sequalae of primary damage in the inner retina. The finding that neuroprotection of the inner retina can also prevent outer retinal damage has important implications with regard to the targets for effective neuroprotective therapy.

On the Communication Complexity of AND Functions

Log-rank conjecture is one of challenging problems in communication complexity. It is known to hold for some special function classes such as XOR functions f(x ⊕ y) when the outer function f is monotone, symmetric, AC <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sup> , low F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -degree, linear threshold or read-k. However, for AND functions f(x ∧ y), Log-rank conjecture is only known to hold when the outer function is a monotone function or a function with small alternating numbers. In this paper, we show that Log-rank conjecture holds for AND functions whose outer functions are read-once F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -degree polynomials, constant-F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -degree polynomials, and symmetric functions. For read-once and symmetric AND functions, their proof are based on the elementary analysis of Möbius sparsity of the outer functions and designing efficient protocols for computing the corresponding AND functions in terms of the Möbius sparsity of their outer functions. For a constant- F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -degree AND function f(x ∧ y), we use the concept of the variable rank to design a decision tree for computing f where the tree depth is bounded above by a polynomial in the logarithm of the Möbius sparsity of f. The communication protocol can be obtained easily from the constructed decision tree. Our results deepen the understanding of the log-rank conjecture for AND functions.

A common range problem for model spaces

Abstract 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.

In vivo evaluation of outer retinal function and structure after retrobulbar optic nerve crush by lateral orbitotomy in goats

Large animal model of optic nerve crush (ONC) plays an important role in translating novel therapeutic strategies developed in rodent model to clinical application. Due to the poor accessibility of the optic nerve (ON) in humans and large animals, lateral orbitotomy is needed to expose the retrobulbar ON. This study was to explore the effects of ONC and ON exposure with lateral orbitotomy (sham surgery) on the outer retinal function and structure in goats by using standard flash electroretinogram (FERG) and spectral-domain optical coherence tomography (SD-OCT). We found that ONC led to a transient reduction in FERG amplitudes at 1 week post injury (wpi), which recovered gradually over 2 months afterwards. Sham surgery alone also caused a similar pattern of amplitude reduction in FERG, although not as significantly as ONC did. Transient outer retinal thickening following ONC occurred at 4 wpi (when progressive thinning of the ganglion cell complex began), peaked at 8 wpi, then recovered gradually at 12 wpi. In contrast, outer retinal thickness remained unchanged statistically 3 months after sham surgery. Fundus fluorescein angiography showed that neither ONC nor ON exposure with lateral orbitotomy significantly caused any significant delay or absence of central retinal vascular filling. In summary, ONC with lateral orbitotomy affects outer retinal function and structure transiently.

(Invited) Nanotube-Based Nanohybrids: Dyes Encapsulated inside Nanotubes

The inner and outer surfaces of a nanotube made of either carbon or boron nitride material can be functionalized to tailor the chemical and optical properties of the assembly. Our studies on the optical properties of dyes encapsulated in nanotubes (dyes@CNT) have shown, for instance, enhancement effects in both Raman and fluorescence spectroscopies, which are of interest for bio-imaging and photonics. A chemical functionalization of the outer wall has also been developed to promote specific chemical interactions of the nanohybrids with a host, such as a biomarker. Here we will review some of our recent progress on the nanotube-based nanohybrids. More particularly, we will present advances towards the development of the boron-nitride nanotube nanohybrids for fluorescence imaging in the visible and near-infrared. New results on the outer functionalization to work more effectively in biological media and infer an optical response to pH will be presented. Lastly, the mechanism of the encapsulation process and the optical response due to aggregation effects of the encapsulated dyes in different conditions will be discussed.This work was done in collaboration with E. Gaufrès, C. Allard, S. K. Mal, N. Tang, R. Nascimento, C. David, L. Durate, L. Gaboury and S. Blais-Ouellette

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.

Optimal planning and operational strategy of energy storage systems in power transmission networks: An analysis of wind farms

This study formulated a bi-level mixed integer non-linear optimization planning and operation model for the optimal configuration (location, capacity, and power ratings) of energy storage systems (ESSs) in power transmission networks. The model was formulated with consideration for independent and correlated wind farms. The single objective function in the inner layer of the bi-level model includes the difference between the total daily expected operational cost of conventional generators and the energy arbitrage benefits derived when considering the operational strategies of ESSs. The outer layer is a multi-objective function composed of three objective functions to be minimized. The objective functions encompass the total daily expected planning and operational cost, total daily expected emission, and the maximum expected voltage deviation. Wind power uncertainties in independent and correlated wind farms were also examined. Multivariate model-based Clayton copulas, which represent joint power distribution amongst correlated wind farms, were discretized using a developed five-point estimation method based on the discretization. A hybrid non-dominating sorted genetic algorithm and multi-objective particle swarm optimization were used to minimize the outer layer objective function, whilst fast Tabu search that considers the probabilistic load flow represented by wind power uncertainties and the operational strategies of ESSs was adopted to minimize the inner layer objective function. An IEEE 57-bus system was subjected to a case study using the proposed two-stage model. The simulation results confirmed the advantage of considering the benefits of a peak shaving operational strategy from economic, technical, and environmental points of view.

On the Wrong Track: Alterations of Ciliary Transport in Inherited Retinal Dystrophies.

Ciliopathies are a group of heterogeneous inherited disorders associated with dysfunction of the cilium, a ubiquitous microtubule-based organelle involved in a broad range of cellular functions. Most ciliopathies are syndromic, since several organs whose cells produce a cilium, such as the retina, cochlea or kidney, are affected by mutations in ciliary-related genes. In the retina, photoreceptor cells present a highly specialized neurosensory cilium, the outer segment, stacked with membranous disks where photoreception and phototransduction occurs. The daily renewal of the more distal disks is a unique characteristic of photoreceptor outer segments, resulting in an elevated protein demand. All components necessary for outer segment formation, maintenance and function have to be transported from the photoreceptor inner segment, where synthesis occurs, to the cilium. Therefore, efficient transport of selected proteins is critical for photoreceptor ciliogenesis and function, and any alteration in either cargo delivery to the cilium or intraciliary trafficking compromises photoreceptor survival and leads to retinal degeneration. To date, mutations in more than 100 ciliary genes have been associated with retinal dystrophies, accounting for almost 25% of these inherited rare diseases. Interestingly, not all mutations in ciliary genes that cause retinal degeneration are also involved in pleiotropic pathologies in other ciliated organs. Depending on the mutation, the same gene can cause syndromic or non-syndromic retinopathies, thus emphasizing the highly refined specialization of the photoreceptor neurosensory cilia, and raising the possibility of photoreceptor-specific molecular mechanisms underlying common ciliary functions such as ciliary transport. In this review, we will focus on ciliary transport in photoreceptor cells and discuss the molecular complexity underpinning retinal ciliopathies, with a special emphasis on ciliary genes that, when mutated, cause either syndromic or non-syndromic retinal ciliopathies.

Smoothing Newton method for nonsmooth second-order cone complementarity problems with application to electric power markets

This paper is dedicated to solving a nonsmooth second-order cone complementarity problem, in which the mapping is assumed to be locally Lipschitz continuous, but not necessarily to be continuously differentiable everywhere. With the help of the vector-valued Fischer-Burmeister function associated with second-order cones, the nonsmooth second-order cone complementarity problem can be equivalently transformed into a system of nonsmooth equations. To deal with this reformulated nonsmooth system, we present an approximation function by smoothing the inner mapping and the outer Fischer-Burmeister function simultaneously. Different from traditional smoothing methods, the smoothing parameter introduced is treated as an independent variable. We give some conditions under which the Jacobian of the smoothing approximation function is guaranteed to be nonsingular. Based on these results, we propose a smoothing Newton method for solving the nonsmooth second-order cone complementarity problem and show that the proposed method achieves globally superlinear or quadratic convergence under suitable assumptions. Finally, we apply the smoothing Newton method to a network Nash-Cournot game in oligopolistic electric power markets and report some numerical results to demonstrate its effectiveness.

Can inner and outer containment counteract pulls and pushes toward delinquency? A test of Walter Reckless’s containment theory

The aim of the current study is to provide an empirical test of containment theory of Walter Reckless (1899–1988). The theory proposes that outer and inner containment hold adolescents back from delinquency even when external factors pull and push them toward it. This early control theory was ahead of its time, but never received the empirical attention it deserves. This article outlines the core theoretical concepts and the basic propositions in order to empirically examine their validity. We employed hybrid linear regression analysis using longitudinal survey data of 612 adolescents (12–18 years old) in the city of The Hague, the Netherlands. The results indicate that outer and inner containment can be meaningfully distinguished, and that several but not all propositions of the theory are supported. Inner and outer containment function as a buffer against external pulls and are able to counteract the effect of increases in environmental pulls during adolescence. We conclude that containment theory is still a promising interaction theory that can help us understand why adolescents who experience external pulls toward delinquency are able to resist these influences.

The Kolmogorov–Arnold representation theorem revisited

There is a longstanding debate whether the Kolmogorov–Arnold representation theorem can explain the use of more than one hidden layer in neural networks. The Kolmogorov–Arnold representation decomposes a multivariate function into an interior and an outer function and therefore has indeed a similar structure as a neural network with two hidden layers. But there are distinctive differences. One of the main obstacles is that the outer function depends on the represented function and can be wildly varying even if the represented function is smooth. We derive modifications of the Kolmogorov–Arnold representation that transfer smoothness properties of the represented function to the outer function and can be well approximated by ReLU networks. It appears that instead of two hidden layers, a more natural interpretation of the Kolmogorov–Arnold representation is that of a deep neural network where most of the layers are required to approximate the interior function.

Validity of Closed Ideals in Algebras of Series of Square Analytic Functions

We show the validity of a complete description of closed ideals of the algebra which is a commutative Banach algebra , that endowed with a pointwise operations act on Dirichlet space of algebra of series of analytic functions on the unit disk satisfying the Lipscitz condition of order of square sequence obtained by (Brahim Bouya, 2008), we introduce and deal with approximation square functions which is an outer functions to produce and show results in .

Variance Reduced Methods for Non-Convex Composition Optimization.

This paper explores the non-convex composition optimization consisting of inner and outer finite-sum functions with a large number of component functions. This problem arises in important applications such as nonlinear embedding and reinforcement learning. Although existing approaches such as stochastic gradient descent (SGD) and stochastic variance reduced gradient (SVRG) descent can be applied to solve this problem, their query complexities tend to be high, especially when the number of inner component functions is large. Therefore, to significantly improve the query complexity of current approaches, we have devised the stochastic composition via variance reduction (SCVR). What's more, we analyze the query complexity under different numbers of inner function and outer function. Based on different kinds of estimation of inner component function, we also present the SCVRII algorithm, though the order of query complexities are the same with SCVR. Additionally, we propose an extension to handle the mini-batch cases, which improve the query complexity under the optimal mini-batch size. The experimental results validate our proposed algorithms and theoretical analyses.

Energy Storage Sizing Optimization for Large-Scale PV Power Plant

The optimal configuration of energy storage capacity is an important issue for large scale solar systems. a strategy for optimal allocation of energy storage is proposed in this paper. First various scenarios and their value of energy storage in PV applications are discussed. Then a double-layer decision architecture is proposed in this article. Net present value, investment payback period, internal rate of return are taken as the outer objective function, energy storage capacity is the optimal variables. The investment cost of energy storage system is taken as the inner objective function, the charge and discharge strategy of the energy storage system and augmentation are the optimal variables. Finally, the effectiveness and feasibility of the proposed model and method are verified through case simulations. The results show that the comprehensive evaluation index can be aimed at the concerns of energy storage investors, comprehensively evaluate the feasibility of the energy storage project, and obtain the corresponding energy storage scale when the comprehensive evaluation index is the highest.