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Yayın Some boundary Harnack principles with uniform constants(Springer Science and Business Media B.V., 2022-10) Barlow, Martin T.; Karlı, DenizWe prove two versions of a boundary Harnack principle in which the constants do not depend on the domain by using probabilistic methods.Yayın A multiplier related to symmetric stable processes(Hacettepe University, 2017) Karlı, DenizIn two recent papers [5] and [6], we generalized some classical results of Harmonic Analysis using probabilistic approach by means of a d- dimensional rotationally symmetric stable process. These results allow one to discuss some boundedness conditions with weaker hypotheses. In this paper, we study a multiplier theorem using these more general results. We consider a product process consisting of a d-dimensional symmetric stable process and a 1-dimensional Brownian motion, and use properties of jump processes to obtain bounds on jump terms and the Lp(Rd)-norm of a new operator.Yayın Adaptive convolution kernel for artificial neural networks(Academic Press Inc., 2021-02) Tek, Faik Boray; Çam, İlker; Karlı, DenizMany deep neural networks are built by using stacked convolutional layers of fixed and single size (often 3 × 3) kernels. This paper describes a method for learning the size of convolutional kernels to provide varying size kernels in a single layer. The method utilizes a differentiable, and therefore backpropagation-trainable Gaussian envelope which can grow or shrink in a base grid. Our experiments compared the proposed adaptive layers to ordinary convolution layers in a simple two-layer network, a deeper residual network, and a U-Net architecture. The results in the popular image classification datasets such as MNIST, MNIST-CLUTTERED, CIFAR-10, Fashion, and ‘‘Faces in the Wild’’ showed that the adaptive kernels can provide statistically significant improvements on ordinary convolution kernels. A segmentation experiment in the Oxford-Pets dataset demonstrated that replacing ordinary convolution layers in a U-shaped network with 7 × 7 adaptive layers can improve its learning performance and ability to generalize.Yayın An extension of a boundedness result for singular integral operators(Polska Akademia Nauk, 2016-03-30) Karlı, DenizWe study some operators originating from classical Littlewood–Paley the- ory. We consider their modification with respect to our discontinuous setup, where the un- derlying process is the product of a one-dimensional Brownian motion and a d-dimensional symmetric stable process. Two operators in focus are the G? and area functionals. Using the results obtained in our previous paper, we show that these operators are bounded on Lp. Moreover, we generalize a classical multiplier theorem by weakening its conditions on the tail of the kernel of singular integrals.Yayın Extension of mikhlin multiplier theorem to fractional derivatives and stable processes(Walter De Gruyter GMBH, 2018-04-25) Karlı, DenizIn this paper, we prove a new generalized Mikhlin multiplier theorem whose conditions are given with respect to fractional derivatives in integral forms with two different integration intervals. We also discuss the connection between fractional derivatives and stable processes and prove a version of Mikhlin theorem under a condition given in terms of the infinitesimal generator of symmetric stable process. The classical Mikhlin theorem is shown to be a corollary of this new generalized version in this paper.
