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Öğe (A(sigma))(Delta)-DOUBLE SEQUENCE SPACES VIA ORLICZ FUNCTIONS AND DOUBLE STATISTICAL CONVERGENCE(Springer International Publishing Ag, 2007) Savaş, Ekrem; Patterson, Richard F.The aim of this paper is to introduce and study a new concept of strong double (A(sigma))(Delta) convergence sequences with respect to ail Orlicz function, and some properties of the resulting sequence spaces were also examined. In addition, we define the (A(sigma))(Delta)-statistical convergence and establish some connections between the spaces of strong double (A(sigma))(Delta)-convergence sequences and the space of double (A(sigma))(Delta)-statistical convergence.Öğe (A(sigma))-double sequence spaces defined by Orlicz function and double statistical convergence(Pergamon-Elsevier Science LTD, 2008) Savaş, EkremThe purpose of this paper is to introduce and study an idea of strong double (A(sigma))-convergence sequences with respect to an Orlicz function. In addition, we define the double (A(sigma))-statistical convergence and establish some connections between the spaces of strong double (A(sigma))-converggence Sequences and the space of double (A(sigma))-statistical convergence. (C) 2007 Elsevier Ltd. All rights reserved.Öğe (A)Δ - double sequence spaces of Fuzzy numbers via orlicz function(Univ Sistan & Baluchestan, 2011) Savaş, EkremThe aim of this paper is to introduce and study a new concept of strong double (A)(Delta)-convergent sequence of fuzzy numbers with respect to an Orlicz function and also some properties of the resulting sequence spaces of fuzzy numbers are examined. In addition, we define the double (A, Delta)-statistical convergence of fuzzy numbers and establish some connections between the spaces of strong double (A)(Delta)-convergent sequence and double (A, Delta)-statistical convergent sequence.Öğe A-Cluster Points Via Ideals(Springer New York LLC, 2017) Gürdal, Mehmet; Savaş, EkremFollowing the line of the recent work by Savaş, et al., we apply the notion of ideals to A-statistical cluster points. We get necessary conditions for two matrices to be equivalent in a sense of AI-statistical convergence. In addition, we use Kolk’s idea to define and study BI-statistical convergence. © 2017, Springer Science+Business Media, LLC.Öğe A-sequence spaces in 2-normed space defined by ideal convergence and an orlicz function(Hindawi Publishing Corporation, 2011) Savaş, EkremWe study some new A-sequence spaces using ideal convergence and an Orlicz function in 2-normed space and we give some relations related to these sequence spaces.Öğe A-STATISTICAL CONVERGENCE OF ORDER alpha VIA phi - FUNCTION(Univ Belgrade, Fac Electrical Engineering, 2019) Savaş, EkremIn this paper, we introduce and examine some properties of A-statistical convergence of order alpha by using phi-function, modulus function and generalized three parametric real matrix A.Öğe An absolute double summability factor theorem(2009) Savaş, EkremIn an earlier paper Savas and Rhoades [Ekrem Savaş, B.E. Rhoades, A note on | A |k summability factors, Nonlinear Anal. 66 (2007) 1879-1883. [1]] obtained a summability factor theorem for absolute summability of order k ? 1. In this paper we extend that result to doubly infinite matrices. © 2008 Elsevier Ltd. All rights reserved.Öğe Accelerating ltv based homomorphic encryption in reconfigurable hardware(Springer Verlag, 2015) Doroz, Yarkın; Öztürk, Erdinç; Savaş, Ekrem; Sunar, BerkAfter being introduced in 2009, the first fully homomorphic encryption (FHE) scheme has created significant excitement in academia and industry. Despite rapid advances in the last 6 years, FHE schemes are still not ready for deployment due to an efficiency bottleneck. Here we introduce a custom hardware accelerator optimized for a class of reconfigurable logic to bring LTV based somewhat homomorphic encryption (SWHE) schemes one step closer to deployment in real-life applications. The accelerator we present is connected via a fast PCIe interface to a CPU platform to provide homomorphic evaluation services to any application that needs to support blinded computations. Specifically we introduce a number theoretical transform based multiplier architecture capable of efficiently handling very large polynomials. When synthesized for the Xilinx Virtex 7 family the presented architecture can compute the product of large polynomials in under 6. 25 msec making it the fastest multiplier design of its kind currently available in the literature and is more than 102 times faster than a software implementation. Using this multiplier we can compute a relinearization operation in 526 msec. When used as an accelerator, for instance, to evaluate the AES block cipher, we estimate a per block homomorphic evaluation performance of 442 msec yielding performance gains of 28. 5 and 17 times over similar CPU and GPU implementations, respectively. © International Association for Cryptologic Research 2015.Öğe Asymptotic equivalence of double sequences(2012) Patterson, Richard F.; Savaş, EkremThe goal of this paper is to present a four-dimensional matrix characterization of asymptotic equivalence of double sequences. This will be accomplished with the following notion of asymptotic equivalence of double sequences. Two double sequences are asymptotic equivalent if and only if P − limk,l xk,l yk,l = 1, where x and y are selected judicially. Using this notion necessary and sufficient conditions on the entries of a four-dimensional matrix are given to ensure that the transformation will preserve asymptotic equivalence.Öğe Asymptotically I-lambda-statistical equivalent sequences of weight g(Amer Inst Physics, 2016) Savaş, Ekrem; Ashyralyev, A; Lukashov, AThis paper presents the following definition which is a natural combination of the definition for asymptotically equivalent of weight g, I -statistically limit, and lambda- statistical convergence, where g : N -> [0,.infinity) is a function satisfying g(n) -> infinity and g(n) negated right arrow infinity. The two nonnegative sequences x = (x(k)) and y = (y(k)) are said to be asymptotically I-g- statistical equivalent of weight g to multiple L provided that for every e > 0, and delta > 0, {n is an element of N: 1/g(lambda(n))vertical bar{k is an element of I-n:vertical bar x(k)/y(k)-L vertical bar >=epsilon}vertical bar >=delta}is an element of I (denoted by x <(S-lambda(L)(1)(g))under tilde>y) and simply asymptotically I-g - statistical equivalent of weight g if L = 1. In addition, we shall also present some inclusion theorems.Öğe Asymptotically J-Lacunary statistical equivalent of order alpha for sequences of sets(Int Scientific Research Publications, 2017) Savaş, EkremThis paper presents the following definition which is a natural combination of the definition for asymptotically equivalent of order alpha, where 0 < alpha <= 1, I-statistically limit, and I-lacunary statistical convergence for sequences of sets. Let (X, rho) be a metric space and theta be a lacunary sequence. For any non-empty closed subsets A(k), B-k subset of X such that d(x, A(k)) > 0 and d(x, B-k) > 0 for each x is an element of X, we say that the sequences {A(k)} and {B-k} are Wijsman asymptotically I-lacunary statistical equivalent of order alpha to multiple L, where 0 < alpha <= 1, provided that for each epsilon > 0 and each x is an element of X, {r is an element of N : 1/h(r)(alpha)|{k is an element of I-r : |d(x; A(k),B-k)- L| >= (sic) }| >= delta} is an element of I, (denoted by {A(k)} (s theta L(Iw)alpha) similar to {B-k}) and simply asymptotically I-lacunary statistical equivalent of order alpha if L = 1. In addition, we shall also present some inclusion theorems. The study leaves some interesting open problems. (C) 2017 All rights reserved.Öğe (Aσ)Δ -double sequence spaces via Orlicz functions and double statistical convergence(Shiraz University, 2007) Savaş, Ekrem; Patterson, Richard F.The aim of this paper is to introduce and study a new concept of strong double (A?)? -convergence sequences with respect to an Orlicz function, and some properties of the resulting sequence spaces were also examined. In addition, we define the (A?) ?-statistical convergence and establish some connections between the spaces of strong double(A?)?- convergence sequences and the space of double (A?)? -statistical convergence. © Shiraz University.Öğe A category theorem for double sequences(2011) Patterson, Richard F.; Savaş, EkremThe goal of this paper is to present the following multidimensional category theorem for double sequences via four-dimensional matrix transformations. The set of second category double subsequences of the double sequence (sk,l) is summable by a four-dimensional RH-regular matrix A=(am,n,k,l) if and only if (sk,l) is P-convergent. This theorem is established using multidimensional sliding hump constructions for bounded and unbounded doubles sequences. © 2011 Elsevier Ltd. All rights reserved.Öğe Certain summability methods in intuitionistic fuzzy normed spaces(IOS Press, 2014) Savaş, Ekrem; Gürdal, MehmetIn this paper, we introduce the notion of I-statistical convergence and I-lacunary statistical convergence with respect to the intuitionistic fuzzy norm (?, v), investigate their relationship, and make some observations about these classes. © 2014-IOS Press and the authors. All rights reserved.Öğe Characterization of asymptotic statistical equivalent double and single sequences(2013) Patterson, Richard F.; Savaş, EkremThis paper presents a series of four-dimensional matrix characterizations of statistically asymptotically equivalent double and single sequences. These characterizations begins with the presentation of definitions for asymptotically equivalent and asymptotically statistical equivalent of multiple L for double and single sequences. The type of questions we provided answers for are those of the following sort. What are the necessary and sufficient conditions on the entries A that ensure the preservation of statistically asymptotically equivalent rate of convergence. © 2013 Elsevier Inc. All rights reserved.Öğe Consistent classes of double summability methods(Elsevier Ltd, 2010) Patterson, Richard F.; Savaş, EkremIn 2000 Patterson proved that if a bounded double sequence is divergent then there are RH-regular matrix methods that sum it to various values. It is now natural to ask the following question. Is there a collection ?. of RH-regular matrix methods which are consistent and such that every bounded double sequence is summable by at least one method in the collection? Similar to Goffman and Petersen's presentation we will present a class of such a collection. In addition, it is clear from the presentation here that it is extremely difficult to find all such collections. However, we have extended this class to a countable collection of RH-regular matrix methods with bounded norm. © 2010 Published by Elsevier Ltd.Öğe A custom accelerator for homomorphic encryption applications(IEEE Computer Society, 2017) Öztürk, Erdinç; Doroz, Yarkın; Savaş, Ekrem; Sunar, BerkAfter the introduction of first fully homomorphic encryption scheme in 2009, numerous research work has been published aiming at making fully homomorphic encryption practical for daily use. The first fully functional scheme and a few others that have been introduced has been proven difficult to be utilized in practical applications, due to efficiency reasons. Here, we propose a custom hardware accelerator, which is optimized for a class of reconfigurable logic, for López-Alt, Tromer and Vaikuntanathan's somewhat homomorphic encryption based schemes. Our design is working as a co-processor which enables the operating system to offload the most compute-heavy operations to this specialized hardware. The core of our design is an efficient hardware implementation of a polynomial multiplier as it is the most compute-heavy operation of our target scheme. The presented architecture can compute the product of very-large polynomials in under 6.25 ms which is 102 times faster than its software implementation. In case of accelerating homomorphic applications; we estimate the per block homomorphic AES as 442 ms which is 28.5 and 17 times faster than the CPU and GPU implementations, respectively. In evaluation of Prince block cipher homomorphically, we estimate the performance as 52 ms which is 66 times faster than the CPU implementation. © 1968-2012 IEEE.Öğe Design and implementation of a constant-Time FPGA accelerator for fast elliptic curve cryptography(Institute of Electrical and Electronics Engineers Inc., 2016) Ay, Atıl U.; Öztürk, Erdinç; Henriquez, F.R.; Savaş, EkremIn this paper we present a scalar multiplication hardware architecture that computes a constant-Time variable-base point multiplication over the Galbraith-Lin-Scott (GLS) family of binary elliptic curves. Our hardware design is especially tailored for the quadratic extension field F22n, with n = 127, which allows us to attain a security level close to 128 bits. We explore extensively the usage of digit-based and Karatsuba multipliers for performing the quadratic field arithmetic associated to GLS elliptic curves and report the area and time performance obtained by these two types of multipliers. Targeting a XILINX KINTEX-7 FPGA device, we report a hardware implementation of our design that achieves a delay of just 3.98?s for computing one scalar multiplication. This allows us to claim the current speed record for this operation at or around the 128-bit security level for any hardware or software realization reported in the literature. © 2016 IEEE.Öğe Double absolute summability factor theorems and applications(2008) Savaş, Ekrem; Rhoades, B.E.In an earlier paper the first author [Ekrem Savas, Factors for | A |k summability of infinite series, Comput. Math. Appl. 53 (7) (2007) 1045-1049. [3]] obtained a summability factor theorem for absolute summability of the order k ? 1. In this paper we extend that result to doubly infinite matrices. © 2007 Elsevier Ltd. All rights reserved.Öğe Double almost lacunary statistical convergence of order alpha(Springer International Publishing AG, 2013) Savaş, EkremIn this paper, we define and study lacunary double almost statistical convergence of order alpha. Further, some inclusion relations have been examined. We also introduce a new sequence space by combining lacunary double almost statistical convergence and Orlicz function.
