2021-12-05T17:57:49Z
https://ijnaa.semnan.ac.ir/?_action=export&rf=summon&issue=43
International Journal of Nonlinear Analysis and Applications
IJNAA
2015
7
1
Relative order and type of entire functions represented by Banach valued Dirichlet series in two variables
Dibyendu
Banerjee
Nilkanta
Mondal
In this paper, we introduce the idea of relative order and type of entire functions represented by Banach valued Dirichlet series of two complex variables to generalize some earlier results. Proving some preliminary theorems on the relative order, we obtain sum and product theorems and we show that the relative order of an entire function represented by Dirichlet series is the same as that of its partial derivative.
Banach valued Dirichlet series
relative order
relative type
Entire function
2016
02
11
1
14
https://ijnaa.semnan.ac.ir/article_288_5d4b6ff0b04ebc85198867c84f85d85c.pdf
International Journal of Nonlinear Analysis and Applications
IJNAA
2015
7
1
On the quadratic support of strongly convex functions
S.
Abbaszadeh
M
Eshaghi Gordji
In this paper, we first introduce the notion of $c$-affine functions for $c> 0$. Then we deal with some properties of strongly convex functions in real inner product spaces by using a quadratic support function at each point which is $c$-affine. Moreover, a Hyers–-Ulam stability result for strongly convex functions is shown.
strongly convex function
Hahn-Banach theorem
$c$-affine functions
quadratic support
2015
12
11
15
20
https://ijnaa.semnan.ac.ir/article_273_368395c1469c8c8c42fa81b1d2a2a883.pdf
International Journal of Nonlinear Analysis and Applications
IJNAA
2015
7
1
Nonexpansive mappings on complex C*-algebras and their fixed points
Davood
Alimohammadi
A normed space $mathfrak{X}$ is said to have the fixed point property, if for each nonexpansive mapping $T : E longrightarrow E $ on a nonempty bounded closed convex subset $ E $ of $ mathfrak{X} $ has a fixed point. In this paper, we first show that if $ X $ is a locally compact Hausdorff space then the following are equivalent: (i) $X$ is infinite set, (ii) $C_0(X)$ is infinite dimensional, (iii) $C_0 (X)$ does not have the fixed point property. We also show that if $A$ is a commutative complex $ mathsf{C}^star$--algebra with nonempty carrier space, then the following statements are equivalent: (i) Carrier space of $ A $ is infinite, (ii) $ A $ is infinite dimensional, (iii) $ A $ does not have the fixed point property. Moreover, we show that if $ A $ is an infinite complex $ mathsf{C}^star$--algebra (not necessarily commutative), then $ A $ does not have the fixed point property.
Banach space
C*-algebra
Fixed point property
Nonexpansive mapping
normed linear space
2015
12
10
21
29
https://ijnaa.semnan.ac.ir/article_289_75ca5b7bd96a777bf6f51352b152a680.pdf
International Journal of Nonlinear Analysis and Applications
IJNAA
2015
7
1
On the fine spectrum of generalized upper triangular double-band matrices $Delta^{uv}$ over the sequence spaces $c_o$ and $c$
Javad
Fathi
The main purpose of this paper is to determine the fine spectrum of the generalized upper triangular double-band matrices uv over the sequence spaces c0 and c. These results are more general than the spectrum of upper triangular double-band matrices of Karakaya and Altun[V. Karakaya, M. Altun, Fine spectra of upper triangular doubleband matrices, Journal of Computational and Applied Mathematics, 234(2010) 1387-1394].
Spectrum of an operator
matrix mapping
sequence space
2016
01
21
31
43
https://ijnaa.semnan.ac.ir/article_290_35d0cd244cf09dff6eaf8a53cd56341d.pdf
International Journal of Nonlinear Analysis and Applications
IJNAA
2015
7
1
Common fixed point theorem for nonexpansive type single valued mappings
Pankaj
Jhade
A S
Saluja
The aim of this paper is to prove a common fixed point theorem for nonexpansive type single valued mappings which include both continuous and discontinuous mappings by relaxing the condition of continuity by weak reciprocally continuous mapping. Our result is generalize and extends the corresponding result of Jhade et al. [P.K. Jhade, A.S. Saluja and R. Kushwah, Coincidence and fixed points of nonexpansive type multivalued and single valued maps, European J. Pure Appl. Math., 4 (2011) 330-339].
Nonexpansive mapping
Common fixed point
reciprocal continuous
weak reciprocal continuous
2016
01
27
45
51
https://ijnaa.semnan.ac.ir/article_293_565f06624c411fcd198bd19c2a9d1e4a.pdf
International Journal of Nonlinear Analysis and Applications
IJNAA
2015
7
1
Some fixed point theorems for weakly subsequentially continuous and compatible of type (E) mappings with an application
Said
Beloul
In this paper, we will establish some xed point results for two pairs of self mappings satisfying generalized contractive condition by using a new concept as weak subsequential continuity with compatibility of type (E) in metric spaces, as an application the existence of unique common solution for a system of functional equations arising in system programming is proved.
generalized contractive condition
weakly subsequentially continuous
compatible of type (E)
functional equation
2016
02
01
53
62
https://ijnaa.semnan.ac.ir/article_294_616d183be61dda4a9cc57edf91853d69.pdf
International Journal of Nonlinear Analysis and Applications
IJNAA
2015
7
1
Approximate a quadratic mapping in multi-Banach spaces, a fixed point approach
Sattar
Alizadeh
Fridoun
Moradlou
Using the fixed point method, we prove the generalized Hyers-Ulam-Rassias stability of the following functional equation in multi-Banach spaces:begin{equation} sum_{ j = 1}^{n}fBig(-2 x_{j} + sum_{ i = 1, ineq j}^{n} x_{i}Big) =(n-6) fBig(sum_{ i = 1}^{n} x_{i}Big) + 9 sum_{ i = 1}^{n}f(x_{i}).end{equation}
Fixed point method, Hyers-Ulam-Rassias stability
Multi-Banach spaces, Quadratic mapping
2016
02
04
63
75
https://ijnaa.semnan.ac.ir/article_295_c27b0b230605db7769e5251a845fed26.pdf
International Journal of Nonlinear Analysis and Applications
IJNAA
2015
7
1
On the generalization of Trapezoid Inequality for functions of two variables with bounded variation and applications
Huseyin
Budak
Mehmet
Sarikaya
In this paper, a generalization of trapezoid inequality for functions of two independent variables with bounded variation and some applications are given.
Bounded Variation
Ostrowski type inequalities
Riemann-Stieltjes
Trapezoid Inequality
2016
02
18
77
85
https://ijnaa.semnan.ac.ir/article_296_24c7771b70f4d911a2cd1e8f6b47c36b.pdf
International Journal of Nonlinear Analysis and Applications
IJNAA
2015
7
1
Generalized solution of Sine-Gordon equation
Lalla Saadia
Chadli
Said
Melliani
Abdelaziz
Moujahid
Mhamed
Elomari
In this paper, we are interested to study the Sine-Gordon equation in generalized functions theory introduced by Colombeau, in the first we give result of existence and uniqueness of generalized solution with initial data are distributions (elements of the Colombeau algebra). Then we study the association concept with the classical solution.
Algebra Colombeau
Generalized functions theory
Sine-Gordon equation
2016
02
27
87
92
https://ijnaa.semnan.ac.ir/article_297_7593898e083c1954e8404529bfff0844.pdf
International Journal of Nonlinear Analysis and Applications
IJNAA
2015
7
1
Titchmarsh theorem for Jacobi Dini-Lipshitz functions
Mustapha
Boujeddaine
Said
Fahlaoui
Radouan
Daher
Our aim in this paper is to prove an analog of Younis's Theorem on the image under the Jacobi transform of a class functions satisfying a generalized Dini-Lipschitz condition in the space $mathrm{L}_{(alpha,beta)}^{p}(mathbb{R}^{+})$, $(1< pleq 2)$. It is a version of Titchmarsh's theorem on the description of the image under the Fourier transform of a class of functions satisfying the Dini-Lipschitz condition in $L^{p}$.
Dini-Lipschitz functions
Jacobi operator
Jacobi transform
2016
03
01
93
101
https://ijnaa.semnan.ac.ir/article_298_98bc5a10d20c0d2f6d2e2df1b9b479f4.pdf
International Journal of Nonlinear Analysis and Applications
IJNAA
2015
7
1
Some new results using Hadamard fractional integral
Sabrina
Taf
Kamel
Brahim
Fractional calculus is the field of mathematical analysis which deals with the investigation and applications of integrals and derivatives of arbitrary order. The purpose of this work is to use Hadamard fractional integral to establish some new integral inequalities of Gruss type by using one or two parameters which ensues four main results . Furthermore, other integral inequalities of reverse Minkowski's type are obtained for positive functions resulting in two theorems.
Hadamard fractional integral
Fractional integral inequalities
Minkowski's inequality
2015
12
16
103
109
https://ijnaa.semnan.ac.ir/article_299_8f090a9047f90a63156afc736973f47f.pdf
International Journal of Nonlinear Analysis and Applications
IJNAA
2015
7
1
Some common fixed point theorems for four $(psi,varphi)$-weakly contractive mappings satisfying rational expressions in ordered partial metric spaces
R.A.
Rashwan
S.M.
Saleh
The aim of this paper is to prove some common fixed point theorems for four mappings satisfying $(\psi,\varphi)$-weak contractions involving rational expressions in ordered partial metric spaces. Our results extend, generalize and improve some well-known results in the literature. Also, we give two examples to illustrate our results.
Common fixed point
rational contractions
ordered partial metric spaces
dominating and dominated mappings
2015
11
17
111
130
https://ijnaa.semnan.ac.ir/article_300_ee2c2f17e0cefae6067c770ab7ee52a6.pdf
International Journal of Nonlinear Analysis and Applications
IJNAA
2015
7
1
A determinant inequality and log-majorisation for operators
Seyed Mahmoud
Manjegani
Let $\lambda_1,\dots,\lambda_n$ be positive real numbers such that $\sum_{k=1}^n \lambda_k=1$. In this paper, we prove that for any positive operators $a_1,a_2,\ldots, a_n$ in semifinite von Neumann algebra $M$ with faithful normal trace that $\t(1)<\infty$, $$\prod_{k=1}^n(\det a_k)^{\lambda_k}\,\le\,\det (\sum_{k=1}^n \lambda_k a_k),$$where $\det a=exp(\int_0^{\t(1)} \mu_a(t)\,dt)$. If furthermore $\t(a_i)<\infty$ for every $1\le i\le n$ and $ \prod_{k=1}^n(\det a_k)^{\lambda_k}\neq 0$, then equality holds if and only if $a_1=a_2=\cdots =a_n$. A log-majorisation version of Young inequality are given as well.
Singular values
Semifinite trace
Majorisation
log-majorisation
2016
01
07
131
140
https://ijnaa.semnan.ac.ir/article_301_f16f986333a145f2dd7a7128e1029da9.pdf
International Journal of Nonlinear Analysis and Applications
IJNAA
2015
7
1
On the $s^{th}$ derivative of a polynomial
Abdullah
Mir
For every $1\leq s< n$, the $s^{th}$ derivative of a polynomial $P(z)$ of degree $n$ is a polynomial $P^{(s)}(z)$ whose degree is $(n-s)$. This paper presents a result which gives generalizations of some inequalities regarding the $s^{th}$ derivative of a polynomial having zeros outside a circle. Besides, our result gives interesting refinements of some well-known results.
Polynomial
Zeros
$s^{th}$ derivative
2015
11
01
141
145
https://ijnaa.semnan.ac.ir/article_130_55c76fca4b5e4d033c72d893fe0a47b7.pdf
International Journal of Nonlinear Analysis and Applications
IJNAA
2015
7
1
Vector ultrametric spaces and a fixed point theorem for correspondences
Kourosh
Nourouzi
In this paper, vector ultrametric spaces are introduced and a fixed point theorem is given for correspondences. Our main result generalizes a known theorem in ordinary ultrametric spaces.
Vector ultra metric space
Correspondence
Fixed point
2016
01
15
147
153
https://ijnaa.semnan.ac.ir/article_302_8a23d5e13714c2c3f04be5ce3ed78c8c.pdf
International Journal of Nonlinear Analysis and Applications
IJNAA
2015
7
1
Variational inequalities on Hilbert $C^*$-modules
Hedayat
Fathi
S.A.R.
Hosseinioun
We introduce variational inequality problems on Hilbert $C^*$-modules and we prove several existence results for variational inequalities defined on closed convex sets. Then relation between variational inequalities, $C^*$-valued metric projection and fixed point theory on Hilbert $C^*$-modules is studied.
variational inequality
Hilbert $C^*$-module
metric projection
Fixed point
2015
11
30
155
165
https://ijnaa.semnan.ac.ir/article_303_379b7e5db0cd9d58ab8583a7f784939c.pdf
International Journal of Nonlinear Analysis and Applications
IJNAA
2015
7
1
Approximately generalized additive functions in several variables via fixed point method
R.
Farokhzad Rostami
S.A.R.
Hoseinioun
In this paper, we obtain the general solution and the generalized Hyers-Ulam-Rassias stability in random normed spaces, in non-Archimedean spaces and also in $p$-Banach spaces and finally the stability via fixed point method for a functional equation\begin{align*}&D_f(x_{1},.., x_{m}):= \sum^{m}_{k=2}(\sum^{k}_{i_{1}=2}\sum^{k+1}_{i_{2}=i_{1}+1}... \sum^{m}_{i_{m-k+1}=i_{m-k}+1}) f(\sum^{m}_{i=1, i\neq i_{1},...,i_{m-k+1} } x_{i}-\sum^{m-k+1}_{ r=1} x_{i_{r}})\\& \hspace {2.8cm}+f(\sum^{m}_{ i=1} x_{i})-2^{m-1} f(x_{1})=0\end{align*}where $m \geq 2$ is an integer number.
Additive function
$p$-Banach spaces
Random normed spaces
Non-Archimedean spaces
Fixed point method
generalized Hyers-Ulam stability
2016
03
19
167
181
https://ijnaa.semnan.ac.ir/article_304_9d50c3cecad33f485b18a635d7221af3.pdf
International Journal of Nonlinear Analysis and Applications
IJNAA
2015
7
1
Coupled coincidence point in ordered cone metric spaces with examples in game theory
Alireza
Naeimi Sadigh
Samaneh
Ghods
In this paper, we prove some coupled coincidence point theorems for mappings with the mixed monotone property and obtain the uniqueness of this coincidence point. Then we providing useful examples in Nash equilibrium.
Coupled fixed point
Coupled coincidence Fixed point
Partially ordered sets
Cone metric space
game theory
Nash equilibrium
2016
02
06
183
194
https://ijnaa.semnan.ac.ir/article_305_50df6bdc73b5dc783d4dfd0d2c6a6513.pdf
International Journal of Nonlinear Analysis and Applications
IJNAA
2015
7
1
On a class of paracontact Riemannian manifold
Mahmood
Parchetalab
We classify the paracontact Riemannian manifolds that their Riemannian curvature satisfies in the certain condition and we show that this classification is hold for the special cases semi-symmetric and locally symmetric spaces. Finally we study paracontact Riemannian manifolds satisfying R(X, ξ).S = 0, where S is the Ricci tensor.
paracontact structure
Einstein structure
parasasakian
2016
03
25
195
205
https://ijnaa.semnan.ac.ir/article_306_7b5dec458ce15e057e2a3d1addca1f0b.pdf
International Journal of Nonlinear Analysis and Applications
IJNAA
2015
7
1
Existence of solution and solving the integro-differential equations system by the multi-wavelet Petrov-Galerkin method
Mohsen
Rabbani
In this paper, we discuss about existence of solution for integro-differential system and then we solve it by using the Petrov-Galerkin method. In the Petrov-Galerkin method choosing the trial and test space is important, so we use Alpert multi-wavelet as basis functions for these spaces. Orthonormality is one of the properties of Alpert multi-wavelet which helps us to reduce computations in the process of discretizing and we drive a system of algebraic equations with small dimension which it leads to approximate solution with high accuracy. We compare the results with similar works and it guarantees the validity and applicability of this method.
System of Integro-differential equations
Multi-wavelet
Petrov-Galerkin
Regular pairs
Trial space
Test space
2016
02
26
207
218
https://ijnaa.semnan.ac.ir/article_307_74c3fcefd0d66fde79c657652f29cbcd.pdf
International Journal of Nonlinear Analysis and Applications
IJNAA
2015
7
1
On fixed points of fundamentally nonexpansive mappings in Banach spaces
Mohammad
Moosaei
We first obtain some properties of a fundamentally nonexpansive self-mapping on a nonempty subset of a Banach space and next show that if the Banach space is having the Opial condition, then the fixed points set of such a mapping with the convex range is nonempty. In particular, we establish that if the Banach space is uniformly convex, and the range of such a mapping is bounded, closed and convex, then its the fixed points set is nonempty, closed and convex.
Fixed point
fundamentally nonexpansive mappings
nonexpansive mappings
Opial's condition
uniformly convex Banach spaces
2016
03
05
219
224
https://ijnaa.semnan.ac.ir/article_308_db07694ccc7a387cc590bac3f64bd749.pdf
International Journal of Nonlinear Analysis and Applications
IJNAA
2015
7
1
Common fixed point of multivalued graph contraction in metric spaces
Masoud
Hadian Dehkordi
Masoud
Ghods
In this paper, we introduce the (G-$\psi$) contraction in a metric space by using a graph. Let $F,T$ be two multivalued mappings on $X.$ Among other things, we obtain a common fixed point of the mappings $F,T$ in the metric space $X$ endowed with a graph $G.$
Fixed point
multivalued
common(G-$psi$) contraction
directed graph
2016
02
12
225
230
https://ijnaa.semnan.ac.ir/article_309_2b668c7b7359b2f90973027b71b332c3.pdf
International Journal of Nonlinear Analysis and Applications
IJNAA
2015
7
1
Dynamical behavior of a stage structured prey-predator model
Shashi
Kant
Vivek
Kumar
In this paper, a new stage structured prey-predator model with linear functional response is proposed and studied. The stages for prey have been considered. The proposed mathematical model consists of three nonlinear ordinary differential equations to describe the interaction among juvenile prey, adult prey and predator populations. The model is analyzed by using linear stability analysis to obtain the conditions for which our model exhibits stability around the possible equilibrium points. Besides this a rigorous global stability analysis has been performed for our proposed model by using Li and Muldowney approach (geometric approach). Global stability conditions for the proposed model are described in the form of theorem. This is not a case study, hence the real parameters are not available for this model. However, model may be simulated by using hypothetical set of parameters. Investigation of real parameters for the proposed model is an open problem.
Stage Structured Population
Global Stability
Local Stability
2016
04
08
231
241
https://ijnaa.semnan.ac.ir/article_311_ea84e1c2cacf1562a1872a4dda1ad2b9.pdf
International Journal of Nonlinear Analysis and Applications
IJNAA
2015
7
1
An analog of Titchmarsh's theorem for the Bessel transform in the space $mathrm{L}_{p,alpha}(mathbb{R}_{+})$
Mohamed
El Hamma
R.
Daher
M.
Boujeddaine
Using a Bessel generalized translation, we obtain an analog of Titchmarsh's theorem for the Bessel transform for functions satisfying the Lipschitz condition in the space $\mathrm{L}_{p,\alpha}(\mathbb{R}_{+})$, where $\alpha>-\frac{1}{2}$ and $1<p\leq 2$.
Bessel operator
Bessel transform
Bessel generalized translation
2016
01
04
243
248
https://ijnaa.semnan.ac.ir/article_312_6d7a608b8b31cc91bf4c1e508fe5d3ab.pdf
International Journal of Nonlinear Analysis and Applications
IJNAA
2015
7
1
On a Hardy-Hilbert-Type Inequality with a General Homogeneous Kernel
Michael Th.
Rassias
Bicheng
Yang
By the method of weight coefficients and techniques of real analysis, a Hardy-Hilbert-type inequality with a general homogeneous kernel and a best possible constant factor is given. The equivalent forms, the operator expressions with the norm, the reverses and some particular examples are also considered.
Hardy-Hilbert-type inequality, weight coefficient, equivalent form, reverse
operator
2016
04
08
249
269
https://ijnaa.semnan.ac.ir/article_323_fdd0e926c44dc6c795dd27dfb4bef204.pdf
International Journal of Nonlinear Analysis and Applications
IJNAA
2015
7
1
On the shadowing property of nonautonomous discrete systems
Hossein
Rasuoli
In this paper we study shadowing property for sequences of mappings on compact metric spaces, i.e. nonautonomous discrete dynamical systems. We investigate the relation of weak contractions with shadowing and h-shadowing property.
Nonautonomous discrete system
nonautonomos difference equation
shadowing property
2016
05
01
271
277
https://ijnaa.semnan.ac.ir/article_291_3a018c19284bae1aec6ba9b21f64092c.pdf
International Journal of Nonlinear Analysis and Applications
IJNAA
2015
7
1
Regularized fractional derivatives in Colombeau algebra
Mohsen
Alimohammady
Fariba
Fattahi Amirdehi
The present study aims at indicating the existence and uniqueness result of system in extended colombeau algebra. The Caputo fractional derivative is used for solving the system of ODEs. In addition, Riesz fractional derivative of Colombeau generalized algebra is considered. The purpose of introducing Riesz fractional derivative is regularizing it in Colombeau sense. We also give a solution to a nonlinear heat equation illustrating the application of the theory.
2016
02
18
279
287
https://ijnaa.semnan.ac.ir/article_147_ab2bc88c6c15434692b835d36280d41a.pdf
International Journal of Nonlinear Analysis and Applications
IJNAA
2015
7
1
Functionally closed sets and functionally convex sets in real Banach spaces
Madjid
Eshaghi
Hamidreza
Reisi Dezaki
Alireza
Moazzen
Let $X$ be a real normed space, then $C(\subseteq X)$ is functionally convex (briefly, $F$-convex), if $T(C)\subseteq \Bbb R $ is convex for all bounded linear transformations $T\in B(X,R)$; and $K(\subseteq X)$ is functionally closed (briefly, $F$-closed), if $T(K)\subseteq \Bbb R $ is closed for all bounded linear transformations $T\in B(X,R)$. We improve the Krein-Milman theorem on finite dimensional spaces. We partially prove the Chebyshev 60 years old open problem. Finally, we introduce the notion of functionally convex functions. The function $f$ on $X$ is functionally convex (briefly, $F$-convex) if epi $f$ is a $F$-convex subset of $X\times \mathbb{R}$. We show that every function $f : (a,b)\longrightarrow \mathbb{R}$ which has no vertical asymptote is $F$-convex.
Convex set
Chebyshev set
Krein-Milman theorem
2016
04
28
289
294
https://ijnaa.semnan.ac.ir/article_340_0cfc0fd433839779402cad7c246e5cc6.pdf
International Journal of Nonlinear Analysis and Applications
IJNAA
2015
7
1
On intermediate value theorem in ordered Banach spaces for noncompact and discontinuous mappings
Ali
Farajzadeh
In this paper, a vector version of the intermediate value theorem is established. The main theorem of this article can be considered as an improvement of the main results have been appeared in [\textit{On fixed point theorems for monotone increasing vector valued mappings via scalarizing}, Positivity, 19 (2) (2015) 333-340] with containing the uniqueness, convergent of each iteration to the fixed point, relaxation of the relatively compactness and the continuity on the map with replacing topological interior of the cone by the algebraic interior. Moreover, by applying Ascoli-Arzela's theorem an example in order to show that the main theorem of the paper [\textit{An intermediate value theorem for monotone operators in ordered Banach spaces}, Fixed point theory and applications, 2012 (1) (2012) 1-4] may fail, is established.
intermediate value theorem
Fixed point
increasing mapping
algebraic interior
normal cone
2016
04
04
295
300
https://ijnaa.semnan.ac.ir/article_341_ffd5a9c8d2472ec439f9cf564da35d43.pdf
International Journal of Nonlinear Analysis and Applications
IJNAA
2015
7
1
Completely continuous Banach algebras
Bahman
Hayati
For a Banach algebra $\fA$, we introduce ~$c.c(\fA)$, the set of all $\phi\in \fA^*$ such that $\theta_\phi:\fA\to \fA^*$ is a completely continuous operator, where $\theta_\phi$ is defined by $\theta_\phi(a)=a\cdot\phi$~~ for all $a\in \fA$. We call $\fA$, a completely continuous Banach algebra if $c.c(\fA)=\fA^*$. We give some examples of completely continuous Banach algebras and a sufficient condition for an open problem raised for the first time by J.E Gal\`{e}, T.J. Ransford and M. C. White: Is there exist an infinite dimensional amenable Banach algebra whose underlying Banach space is reflexive? We prove that a reflexive, amenable, completely continuous Banach algebra with the approximation property is trivial.
Amenability
Completely continuous
Banach algebra
2016
06
01
301
308
https://ijnaa.semnan.ac.ir/article_383_3bb2ce040cb0b5b1b2133ec62d0d7465.pdf