In this article, we prove the generalized Hyers–Ulam stability of the following Cauchy additive functional equation

While functional regression models have received increasing attention recently, most existing approaches assume both a linear relationship and a scalar response variable. We suggest a new method which extends the usual linear regression model to situations involving both functional responses, Xj(t), and functional predictors, Y (t). Our approach uses a penalized least squares optimization crite...

A classical question in the theory of functional equations is “when is it true that a mapping, which approximately satisfies a functional equation, must be somehow close to an exact solution of the equation?” Such a problem, called a stability problem of the functional equation, was formulated by Ulam 1 in 1940. In the next year, Hyers 2 gave a partial solution of Ulam’s problem for the case of...

In 1940, Ulam [1] gave a talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of unsolved problems. Among these was the following question concerning the stability of homomorphisms. We are given a group G and a metric group G′ with metric ρ(·,·). Given > 0, does there exist a δ > 0 such that if f :G→G′ satisfies ρ( f (xy), f (x) f (y)) < δ for all x, y ...

Katsaras 1 defined a fuzzy norm on a vector space to construct a fuzzy vector topological structure on the space. Some mathematicians have defined fuzzy norms on a vector space from various points of view 2–4 . In particular, Bag and Samanta 5 , following Cheng and Mordeson 6 , gave an idea of fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michálek type 7 . T...

The main purpose of this paper is to prove the Hyers-Ulam stability of the additive functional equation for a large class of unbounded domains. Furthermore, by using the theorem, we prove the stability of Jensen's functional equation for a large class of restricted domains. 1. Introduction. The starting point of studying the stability of functional equations seems to be the famous talk of Ulam ...

The functional generalized additive model (FGAM) was recently proposed in McLean et al. (2012) as a more flexible alternative to the common functional linear model (FLM) for regressing a scalar on functional covariates. In this paper, we develop a Bayesian version of FGAM for the case of Gaussian errors with identity link function. Our approach allows the functional covariates to be sparsely ob...

1 Department of Mathematics, Islamic Azad University-Ayatollah Amoli Branch, Amol, P.O. Box 678, Iran 2 Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju 660-701, South Korea 3 Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, South Korea 4 Dipartimento di Matematica ed Applicazioni, Università degli Stu...

The topic of this presentation is the use of additive-factor designs in combination with functional MRI to assess multisensory integration. Unisensory and multisensory stimuli were presented across two different pairings of sensory systems, audio-visual (AV) and visuohaptic (VH). In addition to stimulus modality, signal-to-noise ratio (SNR) was included as an additive factor. Previous research ...

We present a novel additive process, which allows the spatially controlled integration of nanoparticles (NPs) inside silicon surfaces. The NPs are placed between a conductive stamp and a silicon surface; by applying a bias voltage a SiO(2) layer grows underneath the stamp protrusions, thus embedding the particles. We report the successful nanoembedding of CoFe(2)O(4) nanoparticles patterned in ...