International Journal of Modern Engineering Research (IJMER) is Peer reviewed, online Journal. It serves as an international archival forum of scholarly research related to engineering and science education.
This document discusses the inverse Laplace transform. It defines the inverse Laplace transform and provides some key properties and theorems. It then gives examples of taking the inverse Laplace transform of various functions and using Laplace transforms to solve initial value problems involving differential equations. It also defines the unit step function and discusses its properties. Finally, it presents a convolution theorem relating the inverse Laplace transform of a product of functions to a convolution integral.
Generalized Laplace - Mellin Integral TransformationIJERA Editor
The main propose of this paper is to generalized Laplace-Mellin Integral Transformation in between the positive regions of real axis. We have derived some new properties and theorems .And give selected tables for Laplace-Mellin Integral Transformation.
This document contains the statement and solution of a translation property problem in Laplace transforms. The statement shows that if L(F(t)) = f(s) and G(t) = F(t-a) for t > a, then the Laplace transform of G(t) is e^-as f(s). An example problem applies this, finding that if G(t) = sin^2(t - π/5) for t > π/5, then the Laplace transform of G(t) is 2e^-π/5s / (s(s^2 + 4)).
Partial fraction decomposition for inverse laplace transformVishalsagar657
This document discusses partial fraction decomposition for inverse Laplace transforms. It begins with an introduction to partial fraction decomposition and why it is useful for integration. It then covers various cases for partial fraction decomposition of inverse Laplace transforms, including when the denominator is a quadratic with two real roots, a double root, or complex conjugate roots. It also covers the case when the denominator is a cubic with one real and two complex conjugate roots. The goal is to decompose the function into simpler forms that can be easily inverted using the Laplace transform table.
Evaluate functions & fundamental operations of functionsAjayQuines
1. The document discusses key concepts related to functions including relations, domains, ranges, and different ways of representing functions.
2. It provides examples and explanations of fundamental operations on functions including addition, subtraction, multiplication, division, and composition of functions.
3. The examples demonstrate how to perform each operation by substituting values, distributing terms, and combining like terms according to specific rules for exponents.
New Method for Finding an Optimal Solution of Generalized Fuzzy Transportatio...BRNSS Publication Hub
The document summarizes an extension of Calderón's transfer principle to prove vector-valued inequalities in ergodic theory using vector-valued inequalities in harmonic analysis. Specifically, it proves that Calderón's transfer principle can be extended to the vector-valued setting. It then applies this extended transfer principle to a theorem of Fefferman and Stein to prove vector-valued strong Lp norm inequalities and weak type (1,1) inequality for the ergodic maximal function.
1. The document discusses Laplace transforms and provides definitions, properties, and examples. Laplace transforms take a function of time and transform it into a function of a complex variable s.
2. Key properties discussed include linearity, shifting theorems, and Laplace transforms of common functions like 1, t, e^at, sin(at), etc. Explicit formulas for the Laplace transforms of these functions are given.
3. Examples of calculating Laplace transforms of various functions are provided.
Some Properties of Determinant of Trapezoidal Fuzzy Number MatricesIJMERJOURNAL
ABSTRACT: The fuzzy set theory has been applied in many fields such as management, engineering, matrices and so on. In this paper, some elementary operations on proposed trapezoidal fuzzy numbers (TrFNs) are defined. We also defined some operations on trapezoidal fuzzy matrices (TrFMs). The notion of Determinant of trapezoidal fuzzy matrices are introduced and discussed. Some of their relevant properties have also been verified.
This document discusses the inverse Laplace transform. It defines the inverse Laplace transform and provides some key properties and theorems. It then gives examples of taking the inverse Laplace transform of various functions and using Laplace transforms to solve initial value problems involving differential equations. It also defines the unit step function and discusses its properties. Finally, it presents a convolution theorem relating the inverse Laplace transform of a product of functions to a convolution integral.
Generalized Laplace - Mellin Integral TransformationIJERA Editor
The main propose of this paper is to generalized Laplace-Mellin Integral Transformation in between the positive regions of real axis. We have derived some new properties and theorems .And give selected tables for Laplace-Mellin Integral Transformation.
This document contains the statement and solution of a translation property problem in Laplace transforms. The statement shows that if L(F(t)) = f(s) and G(t) = F(t-a) for t > a, then the Laplace transform of G(t) is e^-as f(s). An example problem applies this, finding that if G(t) = sin^2(t - π/5) for t > π/5, then the Laplace transform of G(t) is 2e^-π/5s / (s(s^2 + 4)).
Partial fraction decomposition for inverse laplace transformVishalsagar657
This document discusses partial fraction decomposition for inverse Laplace transforms. It begins with an introduction to partial fraction decomposition and why it is useful for integration. It then covers various cases for partial fraction decomposition of inverse Laplace transforms, including when the denominator is a quadratic with two real roots, a double root, or complex conjugate roots. It also covers the case when the denominator is a cubic with one real and two complex conjugate roots. The goal is to decompose the function into simpler forms that can be easily inverted using the Laplace transform table.
Evaluate functions & fundamental operations of functionsAjayQuines
1. The document discusses key concepts related to functions including relations, domains, ranges, and different ways of representing functions.
2. It provides examples and explanations of fundamental operations on functions including addition, subtraction, multiplication, division, and composition of functions.
3. The examples demonstrate how to perform each operation by substituting values, distributing terms, and combining like terms according to specific rules for exponents.
New Method for Finding an Optimal Solution of Generalized Fuzzy Transportatio...BRNSS Publication Hub
The document summarizes an extension of Calderón's transfer principle to prove vector-valued inequalities in ergodic theory using vector-valued inequalities in harmonic analysis. Specifically, it proves that Calderón's transfer principle can be extended to the vector-valued setting. It then applies this extended transfer principle to a theorem of Fefferman and Stein to prove vector-valued strong Lp norm inequalities and weak type (1,1) inequality for the ergodic maximal function.
1. The document discusses Laplace transforms and provides definitions, properties, and examples. Laplace transforms take a function of time and transform it into a function of a complex variable s.
2. Key properties discussed include linearity, shifting theorems, and Laplace transforms of common functions like 1, t, e^at, sin(at), etc. Explicit formulas for the Laplace transforms of these functions are given.
3. Examples of calculating Laplace transforms of various functions are provided.
Some Properties of Determinant of Trapezoidal Fuzzy Number MatricesIJMERJOURNAL
ABSTRACT: The fuzzy set theory has been applied in many fields such as management, engineering, matrices and so on. In this paper, some elementary operations on proposed trapezoidal fuzzy numbers (TrFNs) are defined. We also defined some operations on trapezoidal fuzzy matrices (TrFMs). The notion of Determinant of trapezoidal fuzzy matrices are introduced and discussed. Some of their relevant properties have also been verified.
This document provides an introduction to Laplace transforms. It defines the Laplace transform, lists some of its key properties including how it transforms derivatives and functions, and demonstrates how to use Laplace transforms to solve ordinary differential equations (ODEs). The document contains examples of taking Laplace transforms, applying properties like linearity and shifting, performing inverse Laplace transforms using tables and techniques like partial fractions, and solving a sample ODE using Laplace transforms. It also introduces concepts like the step function, Dirac delta function, and convolution as related topics.
This document defines relations and functions. Relations are rules that connect input and output numbers. A relation is a set of ordered pairs. A function is a special type of relation where each input has exactly one output. The document discusses types of relations like reflexive, symmetric, and transitive relations. It also discusses types of functions like one-to-one, onto, and bijective functions. Examples are provided to illustrate relations and functions.
This document summarizes key concepts about inverse Laplace transformations:
1. Inverse Laplace transformations involve using partial fraction expansions and the method of residues to determine the inverse of rational functions with various types of poles.
2. Simple poles, complex conjugate poles, and repeated poles each have specific inverse Laplace transform pairs and procedures.
3. The finger method provides a visual way to apply the method of residues for simple poles.
4. The initial and final value theorems allow determining initial and steady-state conditions without fully computing the inverse Laplace transform.
5. Laplace transforms can be used to solve differential equations by including initial conditions in the solution.
The document discusses identifying the domain and range of functions. The domain is the set of all x-coordinates in a relation, while the range is the set of all y-coordinates. A relation is a function if each element in the domain is mapped to only one element in the range - in other words, if each x-value has a single, unique y-value. The document provides examples of stating the domain and range of relations and determining whether they represent functions.
The document discusses the inverse Laplace transform and related topics. It provides three main cases for performing partial fraction expansions when taking the inverse Laplace transform: 1) non-repeated simple roots, 2) complex poles, and 3) repeated poles. It also discusses the convolution integral and how it relates the time domain convolution of two functions to the multiplication of their Laplace transforms. An example uses the convolution integral to find the output of a system given its impulse response and input.
The document discusses the Laplace transform, which transforms a signal from the time domain to the frequency domain. It defines the Laplace transform and inverse Laplace transform. Important properties include: linearity, shifting, scaling. Common Laplace transform pairs are presented in a table. Theorems allow taking derivatives and integrals of signals in the Laplace domain. Partial fraction expansion can be used to simplify rational functions.
The document discusses the Laplace transform and its applications. The Laplace transform maps functions defined in the time domain to functions defined in the complex frequency domain. It makes solving differential equations easier by converting calculus operations into algebra. Some key properties include: the Laplace transform of derivatives can be obtained algebraically instead of using calculus rules, and the transform allows shifting between time and complex frequency domains. Examples are provided to illustrate definitions, properties, and how to use Laplace transforms to solve initial value problems for ordinary differential equations.
This document defines and discusses functions. It begins by defining a relation and function, noting that a function is a special type of relation where each input is mapped to exactly one output. It introduces function notation and discusses the domain, codomain, and range of a function. Examples are provided to illustrate determining if a relation defines a function. The document also covers identifying functions from equations or graphs, and the vertical line test. It concludes with a discussion of function notation and classwork assignments.
This document discusses the inverse Laplace transform, which finds the original function given its Laplace transform. It defines the inverse Laplace transform and proves it is unique. The key points are:
1. The inverse Laplace transform of a function F(s) is the function f(t) whose Laplace transform is F(s).
2. The uniqueness theorem proves there is only one function f(t) that corresponds to a given F(s).
3. The inverse is only defined for t ≥ 0, as the Laplace transform only uses information from the positive t-axis.
This document defines relations and functions in mathematics. A relation is a set of ordered pairs where the domain is the set of all x values and the range is the set of all y values. A function assigns each element in the domain (set of x values) to exactly one element in the range (set of y values). Functions are commonly represented by letters like f(x), where f denotes the name of the function and x is the variable. The left side of a function equation tells us the name and variable of the function, not that the function is being multiplied.
The document discusses Cartesian products, domains, ranges, and co-domains of relations and functions through examples and definitions. It explains that the Cartesian product of sets A and B, written as A×B, is the set of all ordered pairs (a,b) where a is an element of A and b is an element of B. It also defines what constitutes a relation between two sets and provides examples of relations and functions, discussing their domains and ranges. Arrow diagrams are presented to illustrate various functions along with questions and their solutions related to relations and functions.
This document discusses different types of relations and functions. It defines equivalence relations, identity relations, empty relations, universal relations, one-to-one functions, onto functions, bijective functions, composition of functions, and invertible functions. It provides examples to illustrate these concepts.
The document provides an outline for a lesson on functions and relations. It includes:
- A review of functions as machines, tables of values, graphs, and the vertical line test.
- How functions can represent real-life situations, including piecewise functions.
- An example of using a piecewise function to model the temperature of water as heat is added.
- The lesson aims to represent real-life situations using functions and solve problems involving functions.
The document discusses Fourier series, integrals, and transforms. Fourier series are used to represent periodic functions as infinite sums of sines and cosines. The Fourier series coefficients are defined using integrals over a period. Fourier integrals extend the idea to non-periodic functions using integrals from negative to positive infinity. Fourier transforms result in a complex representation and are useful in solving partial differential equations. Examples are provided to demonstrate calculating Fourier series for various periodic functions.
This document provides an introduction to Laplace transforms for engineers. It begins with definitions of Laplace transforms and motivates their use by explaining how they can help solve differential equations involving discontinuous functions. It then discusses three methods for finding Laplace transforms: directly from the definition, using properties of transforms, and looking transforms up in lists. The document explains how to use partial fractions to find inverse Laplace transforms and solve ordinary differential equations (ODEs) and systems of ODEs using Laplace transform methods. It concludes with an example of using Laplace transforms to handle an impulse problem involving differential equations.
The document presents a condition under which unbounded unions of languages can be learned from positive data using refinement operators. Specifically, it introduces two theorems:
1) Theorem 1 states that a concept class (C,R,L) is learnable if it admits a refinement operator satisfying properties [A-1] to [A-3].
2) Theorem 2 (the contribution of the paper) states that the union concept class (C*,R*,L) is learnable if (C,R,L) admits a refinement operator satisfying [A-1] to [A-3] and additional properties [C-1] and [C-2]. This allows learning of unbounded unions of languages.
The document defines relations and functions. A relation is a set of ordered pairs where each element in the domain (set of x-values) is paired with an element in the range (set of y-values). A function is a special type of relation where each element of the domain is mapped to exactly one element in the range. The document provides examples of relations that are and are not functions based on this one-to-one mapping property. It also discusses using function notation and evaluating functions for different inputs. Finally, it explains how to determine the domain of a function by identifying values that would result in illegal operations like division by zero.
This document discusses one-to-one functions and logarithmic functions. It defines one-to-one functions as those where each element of the domain corresponds to exactly one element in the range. It provides examples of functions that are and aren't one-to-one. The document also discusses inverse functions and how the inverse of an exponential function is a logarithmic function. It provides examples of evaluating logarithms and using properties of exponents and logarithms to simplify expressions.
Laplace transform: UNIT STEP FUNCTION, SECOND SHIFTING THEOREM, DIRAC DELTA F...saahil kshatriya
The document discusses the unit step function (also called the Heaviside function) and provides its definition and Laplace transform. It also discusses properties related to the Laplace transform of the unit step function, including:
1) The Laplace transform of the unit step function u(t-a) is 1/s when t ≥ a and 0 when t < a.
2) Using the shifting property, the Laplace transform of f(t)u(t-a) is e-asL[f(t+a)], where L[f(t)] is the Laplace transform of f(t).
3) An example calculates the Laplace transform of t2u(t-2) using the
This document discusses techniques for taking the inverse Laplace transform using partial fraction expansion. It covers:
1) Expanding fractions with distinct real roots, repeated real roots, and complex roots into terms with forms in the Laplace transform table.
2) A second method for complex roots that uses a second order polynomial without complex numbers.
3) Examples that combine multiple expansion methods or involve fractions where the numerator polynomial is not of lower order than the denominator.
Improvement of Surface Roughness of Nickel Alloy Specimen by Removing Recast ...IJMER
Abstract: In this investigation, experimental work and computational work are combined to obtain improvement in the surface roughness of nickel alloy specimen, the machining is carried out by means of CNC wire electric discharge machining (WEDM). Brass wire is used as the tool electrode and nickel alloy (Inconel600) is used as the work piece material. The machining parameters such as Pulse-On time (Ton), Pulse-Off time (Toff), Peak Current (Ip), and Bed speed are considered as input parameters for this project. Surface roughness and Recast layer are considered the output parameters. The experiments
with the pre-planned set of input parameters are designed based on Taguchi’s orthogonal array. The surface roughness is measured using stylus type roughness tester and the thickness of the Recast layer is measured using Scanning Electron Microscope (SEM). The results obtained from the experiments are fed to the Minitab software and optimum input parameters for the desired output parameters are identified. The software uses the concept of analysis of variance (ANOVA) and indicates the nature of effect of input parameters on the output parameters and confirmation is done by validation
experiments. Once the recast layer thickness is obtained Chemical Etching and abrasive blasting is performed in order to remove the recast layer and again the surface roughness is measured by using stylus type roughness tester. Finally from the obtained results it was found that there was significant improvement in the Surface roughness of the nickel alloy material. In addition using regression analysis this work is stimulated by computational method and the results are obtained
Review of Intrusion and Anomaly Detection Techniques IJMER
Intrusion detection is the act of detecting actions that attempt to compromise the
confidentiality, integrity or availability of a resource. With the tremendous growth of network-based
services and sensitive information on networks, network security is getting more and more importance
than ever. Intrusion poses a serious security threat in a huge network environment. The increasing use of
internet has dramatically added to the growing number of threats that inhabit within it. Intrusion
detection does not, in general, include prevention of intrusions. Now a days Network intrusion detection
systems have become a standard component in the area of security infrastructure. This review paper tries
to discusses various techniques which are already being used for intrusion detection.
This document provides an introduction to Laplace transforms. It defines the Laplace transform, lists some of its key properties including how it transforms derivatives and functions, and demonstrates how to use Laplace transforms to solve ordinary differential equations (ODEs). The document contains examples of taking Laplace transforms, applying properties like linearity and shifting, performing inverse Laplace transforms using tables and techniques like partial fractions, and solving a sample ODE using Laplace transforms. It also introduces concepts like the step function, Dirac delta function, and convolution as related topics.
This document defines relations and functions. Relations are rules that connect input and output numbers. A relation is a set of ordered pairs. A function is a special type of relation where each input has exactly one output. The document discusses types of relations like reflexive, symmetric, and transitive relations. It also discusses types of functions like one-to-one, onto, and bijective functions. Examples are provided to illustrate relations and functions.
This document summarizes key concepts about inverse Laplace transformations:
1. Inverse Laplace transformations involve using partial fraction expansions and the method of residues to determine the inverse of rational functions with various types of poles.
2. Simple poles, complex conjugate poles, and repeated poles each have specific inverse Laplace transform pairs and procedures.
3. The finger method provides a visual way to apply the method of residues for simple poles.
4. The initial and final value theorems allow determining initial and steady-state conditions without fully computing the inverse Laplace transform.
5. Laplace transforms can be used to solve differential equations by including initial conditions in the solution.
The document discusses identifying the domain and range of functions. The domain is the set of all x-coordinates in a relation, while the range is the set of all y-coordinates. A relation is a function if each element in the domain is mapped to only one element in the range - in other words, if each x-value has a single, unique y-value. The document provides examples of stating the domain and range of relations and determining whether they represent functions.
The document discusses the inverse Laplace transform and related topics. It provides three main cases for performing partial fraction expansions when taking the inverse Laplace transform: 1) non-repeated simple roots, 2) complex poles, and 3) repeated poles. It also discusses the convolution integral and how it relates the time domain convolution of two functions to the multiplication of their Laplace transforms. An example uses the convolution integral to find the output of a system given its impulse response and input.
The document discusses the Laplace transform, which transforms a signal from the time domain to the frequency domain. It defines the Laplace transform and inverse Laplace transform. Important properties include: linearity, shifting, scaling. Common Laplace transform pairs are presented in a table. Theorems allow taking derivatives and integrals of signals in the Laplace domain. Partial fraction expansion can be used to simplify rational functions.
The document discusses the Laplace transform and its applications. The Laplace transform maps functions defined in the time domain to functions defined in the complex frequency domain. It makes solving differential equations easier by converting calculus operations into algebra. Some key properties include: the Laplace transform of derivatives can be obtained algebraically instead of using calculus rules, and the transform allows shifting between time and complex frequency domains. Examples are provided to illustrate definitions, properties, and how to use Laplace transforms to solve initial value problems for ordinary differential equations.
This document defines and discusses functions. It begins by defining a relation and function, noting that a function is a special type of relation where each input is mapped to exactly one output. It introduces function notation and discusses the domain, codomain, and range of a function. Examples are provided to illustrate determining if a relation defines a function. The document also covers identifying functions from equations or graphs, and the vertical line test. It concludes with a discussion of function notation and classwork assignments.
This document discusses the inverse Laplace transform, which finds the original function given its Laplace transform. It defines the inverse Laplace transform and proves it is unique. The key points are:
1. The inverse Laplace transform of a function F(s) is the function f(t) whose Laplace transform is F(s).
2. The uniqueness theorem proves there is only one function f(t) that corresponds to a given F(s).
3. The inverse is only defined for t ≥ 0, as the Laplace transform only uses information from the positive t-axis.
This document defines relations and functions in mathematics. A relation is a set of ordered pairs where the domain is the set of all x values and the range is the set of all y values. A function assigns each element in the domain (set of x values) to exactly one element in the range (set of y values). Functions are commonly represented by letters like f(x), where f denotes the name of the function and x is the variable. The left side of a function equation tells us the name and variable of the function, not that the function is being multiplied.
The document discusses Cartesian products, domains, ranges, and co-domains of relations and functions through examples and definitions. It explains that the Cartesian product of sets A and B, written as A×B, is the set of all ordered pairs (a,b) where a is an element of A and b is an element of B. It also defines what constitutes a relation between two sets and provides examples of relations and functions, discussing their domains and ranges. Arrow diagrams are presented to illustrate various functions along with questions and their solutions related to relations and functions.
This document discusses different types of relations and functions. It defines equivalence relations, identity relations, empty relations, universal relations, one-to-one functions, onto functions, bijective functions, composition of functions, and invertible functions. It provides examples to illustrate these concepts.
The document provides an outline for a lesson on functions and relations. It includes:
- A review of functions as machines, tables of values, graphs, and the vertical line test.
- How functions can represent real-life situations, including piecewise functions.
- An example of using a piecewise function to model the temperature of water as heat is added.
- The lesson aims to represent real-life situations using functions and solve problems involving functions.
The document discusses Fourier series, integrals, and transforms. Fourier series are used to represent periodic functions as infinite sums of sines and cosines. The Fourier series coefficients are defined using integrals over a period. Fourier integrals extend the idea to non-periodic functions using integrals from negative to positive infinity. Fourier transforms result in a complex representation and are useful in solving partial differential equations. Examples are provided to demonstrate calculating Fourier series for various periodic functions.
This document provides an introduction to Laplace transforms for engineers. It begins with definitions of Laplace transforms and motivates their use by explaining how they can help solve differential equations involving discontinuous functions. It then discusses three methods for finding Laplace transforms: directly from the definition, using properties of transforms, and looking transforms up in lists. The document explains how to use partial fractions to find inverse Laplace transforms and solve ordinary differential equations (ODEs) and systems of ODEs using Laplace transform methods. It concludes with an example of using Laplace transforms to handle an impulse problem involving differential equations.
The document presents a condition under which unbounded unions of languages can be learned from positive data using refinement operators. Specifically, it introduces two theorems:
1) Theorem 1 states that a concept class (C,R,L) is learnable if it admits a refinement operator satisfying properties [A-1] to [A-3].
2) Theorem 2 (the contribution of the paper) states that the union concept class (C*,R*,L) is learnable if (C,R,L) admits a refinement operator satisfying [A-1] to [A-3] and additional properties [C-1] and [C-2]. This allows learning of unbounded unions of languages.
The document defines relations and functions. A relation is a set of ordered pairs where each element in the domain (set of x-values) is paired with an element in the range (set of y-values). A function is a special type of relation where each element of the domain is mapped to exactly one element in the range. The document provides examples of relations that are and are not functions based on this one-to-one mapping property. It also discusses using function notation and evaluating functions for different inputs. Finally, it explains how to determine the domain of a function by identifying values that would result in illegal operations like division by zero.
This document discusses one-to-one functions and logarithmic functions. It defines one-to-one functions as those where each element of the domain corresponds to exactly one element in the range. It provides examples of functions that are and aren't one-to-one. The document also discusses inverse functions and how the inverse of an exponential function is a logarithmic function. It provides examples of evaluating logarithms and using properties of exponents and logarithms to simplify expressions.
Laplace transform: UNIT STEP FUNCTION, SECOND SHIFTING THEOREM, DIRAC DELTA F...saahil kshatriya
The document discusses the unit step function (also called the Heaviside function) and provides its definition and Laplace transform. It also discusses properties related to the Laplace transform of the unit step function, including:
1) The Laplace transform of the unit step function u(t-a) is 1/s when t ≥ a and 0 when t < a.
2) Using the shifting property, the Laplace transform of f(t)u(t-a) is e-asL[f(t+a)], where L[f(t)] is the Laplace transform of f(t).
3) An example calculates the Laplace transform of t2u(t-2) using the
This document discusses techniques for taking the inverse Laplace transform using partial fraction expansion. It covers:
1) Expanding fractions with distinct real roots, repeated real roots, and complex roots into terms with forms in the Laplace transform table.
2) A second method for complex roots that uses a second order polynomial without complex numbers.
3) Examples that combine multiple expansion methods or involve fractions where the numerator polynomial is not of lower order than the denominator.
Improvement of Surface Roughness of Nickel Alloy Specimen by Removing Recast ...IJMER
Abstract: In this investigation, experimental work and computational work are combined to obtain improvement in the surface roughness of nickel alloy specimen, the machining is carried out by means of CNC wire electric discharge machining (WEDM). Brass wire is used as the tool electrode and nickel alloy (Inconel600) is used as the work piece material. The machining parameters such as Pulse-On time (Ton), Pulse-Off time (Toff), Peak Current (Ip), and Bed speed are considered as input parameters for this project. Surface roughness and Recast layer are considered the output parameters. The experiments
with the pre-planned set of input parameters are designed based on Taguchi’s orthogonal array. The surface roughness is measured using stylus type roughness tester and the thickness of the Recast layer is measured using Scanning Electron Microscope (SEM). The results obtained from the experiments are fed to the Minitab software and optimum input parameters for the desired output parameters are identified. The software uses the concept of analysis of variance (ANOVA) and indicates the nature of effect of input parameters on the output parameters and confirmation is done by validation
experiments. Once the recast layer thickness is obtained Chemical Etching and abrasive blasting is performed in order to remove the recast layer and again the surface roughness is measured by using stylus type roughness tester. Finally from the obtained results it was found that there was significant improvement in the Surface roughness of the nickel alloy material. In addition using regression analysis this work is stimulated by computational method and the results are obtained
Review of Intrusion and Anomaly Detection Techniques IJMER
Intrusion detection is the act of detecting actions that attempt to compromise the
confidentiality, integrity or availability of a resource. With the tremendous growth of network-based
services and sensitive information on networks, network security is getting more and more importance
than ever. Intrusion poses a serious security threat in a huge network environment. The increasing use of
internet has dramatically added to the growing number of threats that inhabit within it. Intrusion
detection does not, in general, include prevention of intrusions. Now a days Network intrusion detection
systems have become a standard component in the area of security infrastructure. This review paper tries
to discusses various techniques which are already being used for intrusion detection.
This document summarizes a study on the effect of aluminum variation on hardness and aluminum loss in zinc-aluminum alloys. Samples of mild steel were coated with zinc-aluminum alloys containing 2.5%, 4.5%, 6.5%, 8.5%, and 11.5% aluminum. Microstructural analysis found the coatings consisted of zinc-aluminum dendrites within a eutectic lamellar structure. Hardness measurements showed both the eutectic and dendritic structures increased in hardness with higher aluminum content. Testing also revealed aluminum loss from the coatings increased with higher initial aluminum percentages in the alloy. The study concluded that dendrite formation and coating hardness rise with aluminum
The document analyzes pores in the microstructure of two cast aluminum alloys (Al-20%wtSi and Al-20%wtCu) using fractal analysis, multi-stage random sampling, and spatial point pattern methods. Fractal analysis showed that all pores were shrinkage pores with fractal dimensions approaching 2. The multi-stage random sampling and spatial point pattern methods revealed that crack initiation for eventual failure of the Al-20%wtSi alloy would start in the worst pore found in the lower right region, as it had the lowest fractal dimension and sphericity values. This work demonstrated the effectiveness of using fractal analysis, multi-stage random sampling, and spatial point pattern methods to characterize pores in cast
This document describes a web-based application that provides medication reminders and entertainment for users. It has a notification system and easy-to-use interface. The application's target users are people on continuous medication and their caregivers. It aims to directly solve the problem of medication non-compliance. The document discusses the application's marketing strategy, competition in the space, revenue models through advertising and subscriptions, and recent achievements of the development team.
An Efficient top- k Query Processing in Distributed Wireless Sensor NetworksIJMER
Wireless Sensor Networks (WSNs) are usually defined as large-scale, ad-hoc, multi-hop and
wireless unpartitioned networks of homogeneous, small, static nodes deployed in an area of interest.
Applications of sensor networks include monitoring volcano activity, building structures or natural
habitat monitoring. In this paper, we present the problem of processing probabilistic top-k queries in a
distributed wireless sensor networks. The basic problem in top-k query processing is that, a single method
cannot be used as a solution to the problem of top-k query processing because there are many types of
top-k query processing. The method has to be based on the situation, the classification and the type of
database and the query model. Here we develop three algorithms, namely, sufficient set-based (SSB),
necessary set-based (NSB), and boundary-based (BB), for inter- cluster query processing with bounded
rounds of communications. Moreover, in responding to dynamic changes of data distribution in the
overall network, we develop an adaptive algorithm that dynamically switches among the three proposed
algorithms to minimize the transmission cost.
Application of Neuro-Fuzzy System to Evaluate Sustainability in Highway DesignIJMER
1. The document describes using an adaptive neuro-fuzzy inference system (ANFIS) to evaluate the sustainability of highway design projects in Thailand.
2. ANFIS uses 60 input variables across 14 activities associated with highway design, including geometrics, earthworks, pavement, etc. to evaluate sustainability.
3. The ANFIS model was trained on data from 50 highway design scenarios rated by a decision team. It was then tested on the remaining 15 scenarios to validate the model's ability to assess sustainability. The ANFIS approach provided reasonably accurate sustainability evaluations compared to expert ratings.
This document summarizes a research paper on clustering algorithms in data mining. It begins by defining clustering as an unsupervised learning technique that organizes unlabeled data into groups of similar objects. The document then reviews different types of clustering algorithms and methods for evaluating clustering results. Key steps in clustering include feature selection, algorithm selection, and cluster validation to assess how well the derived groups represent the underlying data structure. A variety of clustering algorithms exist and must be chosen based on the problem characteristics.
This document appears to be a set of multiple choice questions for the MGT 330 Final Exam. There are 42 questions in total related to topics like decision making, planning, ethics and organizational behavior. The questions cover concepts such as programmed vs non-programmed decisions, the formal planning process, contingency plans, ethical issues, and group decision making.
Robert Burns was a famous Scottish poet born in 1759 who came from a poor family but was inspired to succeed. He worked on his father's farm and had 14 children, though only 7 survived into adulthood. His first book of poems published in 1786 was surprisingly successful and allowed him to pursue poetry full time. Though his health deteriorated, he produced famous works like "To a Mouse" and influenced later authors. He brought freshness to poetry with works addressing Scottish cultural identity, sexuality, and poverty, and using satire and varying emotions in his writing.
This document discusses the effect of preform geometry on material behavior and densification during hot upset forging of sintered AISI 9840 steel powder metal parts. Powder blends were prepared with different compositions and compacted into preforms with varying initial aspect ratios between 0.45-0.92. The preforms were sintered and hot forged to different height strains. Results showed that lower aspect ratio preforms densified more rapidly than higher ratios. Densification curves followed a third order polynomial relationship with height strain. Preform geometry significantly affected the densification curves and Poisson's ratio with density.
This document presents a general framework for building classifiers and clustering models using hidden topics to deal with short and sparse text data. It analyzes hidden topics from a large universal dataset using LDA. These topics are then used to enrich both the training data and new short text data by combining them with the topic distributions. This helps reduce data sparseness and improves classification and clustering accuracy for short texts like web snippets. The framework is also applied to contextual advertising by matching web pages and ads based on their hidden topic similarity.
Variant Flexor Carpi Ulnaris Muscle and Variant Course of Ulnar Artery in For...IJMER
The document describes a case study of an anatomical variation observed during a routine dissection of a cadaver's right forearm. The variation involved separate humeral and ulnar heads of the flexor carpi ulnaris muscle. Notably, the bulky ulnar head separated the ulnar artery and nerve. The humeral and ulnar heads fused just before their insertion and the ulnar artery came into contact with the ulnar nerve in the lower forearm. The course and distribution of the ulnar artery and nerve were otherwise normal. This variation has clinical significance for procedures involving the forearm and hand.
Geotechnical Investigation of Soils: A Case Study of Gombe Town (Sheet 152NW)...IJMER
1. The document summarizes a study of the geotechnical properties of soils in Gombe town, Nigeria.
2. Samples were collected from 12 locations and tested for properties like moisture content, particle size, liquid limit, and compaction.
3. Based on the test results, soils from Pantame, Hamatatu, Tonde, Chongo and Kulalum were found to be clayey and unsuitable for construction, while soils from other areas like Kalshingi forest and Titi baba contained more sand and would make better subgrades.
This document summarizes a study on using electrocoagulation with aluminum electrodes to remove mercury from wastewater. The study investigated the effect of electrolyte concentration, initial mercury concentration, applied potential, pH, and agitation on mercury removal efficiency. Optimum conditions for 98.5% mercury removal within 50 minutes included a pH of 4.5, initial concentration of 50 ppm, applied potential of 9V, electrolyte concentration of 1.333 g/L, and agitation of 400 rpm. Under these conditions, aluminum ions are generated from the anode which forms hydroxide flocs that adsorb and remove mercury from the wastewater through processes like coagulation, adsorption, and precipitation.
This document summarizes research on a single-phase five-level diode-clamped multilevel inverter (DCMLI). It first describes the conventional DCMLI topology using four capacitors and eight switches to achieve five voltage levels. It then introduces the topology studied in the paper, which uses only two capacitors and eight switches by incorporating two H-bridge legs. Simulation results are presented comparing the performance of the inverter under different pulse width modulation techniques, evaluating metrics like total harmonic distortion, voltage waveform quality factors, and efficiency of utilizing the DC bus voltage. Sinusoidal pulse width modulation with phase opposition and disposition modulation is found to provide the lowest output voltage distortion.
The document summarizes a study that assessed the vulnerability of aquifers in the Imo River Basin in southeastern Nigeria to pollution. Eight locations were investigated to determine parameters like depth to water table, recharge rate, aquifer and soil properties, topography, and hydraulic conductivity. These parameters were used in the DRASTIC model to develop a vulnerability map. The map showed that areas within the Benin Formation generally have moderate vulnerability due to fine to coarse grained sandy overburden. Higher vulnerabilities were found near Aba, while lower vulnerabilities occurred around Obibiezena and Naze. The study demonstrated the usefulness of the DRASTIC model for assessing vulnerability of aquifer systems.
NMP is the most effective solvent for removing ash content from Indian coal samples based on a comparative study of 5 solvents. The maximum reduction of ash content achieved was:
1) 72% for a coal sample with initial ash content of 51.1% using a 1:10 ratio of NMP to coal at 120°C.
2) 20% for a sample with 37% initial ash using a 1:40 NMP to coal ratio.
3) 26% for a sample with 26% initial ash using a 1:10 NMP to coal ratio.
In contrast, acetic acid and toluene achieved negligible maximum reductions of only 1.5-1.9% ash
The document proposes an automated approach for data validation testing during data migration projects to improve data quality assurance. It discusses how traditional manual and sampling-based testing methods are time consuming and do not thoroughly validate all migrated data. The proposed system utilizes reusable query snippets and schedules automated testing of source and target databases to generate summary and detailed reports of any data mismatches or defects. This allows 100% of migrated data to be validated while reducing time, costs and errors compared to traditional approaches.
Further Results On The Basis Of Cauchy’s Proper Bound for the Zeros of Entire...IJMER
International Journal of Modern Engineering Research (IJMER) is Peer reviewed, online Journal. It serves as an international archival forum of scholarly research related to engineering and science education.
International Journal of Modern Engineering Research (IJMER) covers all the fields of engineering and science: Electrical Engineering, Mechanical Engineering, Civil Engineering, Chemical Engineering, Computer Engineering, Agricultural Engineering, Aerospace Engineering, Thermodynamics, Structural Engineering, Control Engineering, Robotics, Mechatronics, Fluid Mechanics, Nanotechnology, Simulators, Web-based Learning, Remote Laboratories, Engineering Design Methods, Education Research, Students' Satisfaction and Motivation, Global Projects, and Assessment…. And many more.
Utilitas Mathematica Journal original research and review articles. Utilitas Mathematica Journal commits to strengthening our professional community by making it more just, equitable, diverse, and inclusive. Algebra ,Analysis ,Geometry Offers selected original research in Pure and Applied Mathematics and Statistics.
International Refereed Journal of Engineering and Science (IRJES)irjes
1) The document presents a class of p-valent functions defined using the Hadamard product and involving the generalized hypergeometric function.
2) It introduces a linear operator Lpka,b,c involving the generalized hypergeometric function and Hadamard product.
3) Theorems regarding coefficient bounds for functions in this class to be p-valent starlike or convex of order δ are provided. Distortion theorems and fractional differential operators for these functions are also obtained.
The Laplace transform is an integral transform that converts a function of time (often a function that represents a signal) into a function of complex frequency. It has various applications in engineering for solving differential equations and analyzing linear systems. The key aspect is that it converts differential operators into algebraic operations, allowing differential equations to be solved as algebraic equations. This makes the equations much easier to manipulate and solve compared to the original differential form.
Fuzzy random variables and Kolomogrov’s important resultsinventionjournals
:In this paper an attempt is made to transform Kolomogrov Maximal inequality, Koronecker Lemma, Loeve’s Lemma and Kolomogrov’s strong law of large numbers for independent, identically distributive fuzzy Random variables. The applications of this results is extensive and could produce intensive insights on Fuzzy Random variables
This document defines logarithmic functions and explores their properties. It contains the following key points:
1. Logarithms are defined as the exponent to which a base number must be raised to equal the value. Common and natural logarithms use bases of 10 and e respectively.
2. Logarithmic and exponential functions are inverses. Basic properties include logb1=0, logbbx=x, and blogbx=x.
3. The laws of logarithms allow logarithmic expressions to be condensed using logb(uv)=logbu+logbv, logb(u/v)=logbu-logbv, and logbun=nlogbu.
IOSR Journal of Mathematics(IOSR-JM) is an open access international journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
International Journal of Engineering Research and Development (IJERD)IJERD Editor
This document presents two theorems regarding the degree of approximation of functions by generalized polynomials. Theorem 1 shows that if a function f is continuous and Lebesgue integrable on [0,1], then the generalized polynomial Uαn(f,x) approximates f with an error of at most 3/2 times the modulus of continuity w(1/n). Theorem 2 shows that if f is continuous, Lebesgue integrable, and has a bounded first derivative, then the approximation error is at most 3/4w(1/n) + o(1/n). The proofs of the theorems apply properties of modulus of continuity and utilize three lemmas regarding properties of the generalized polynomials.
In this chapter, the authors extend the theory of
the generalized difference Operator ∆L to the generalized
difference operator of the 풏
풕풉kind denoted by ∆L Where L
=푳 = {풍ퟏ,풍ퟐ,….풍풏} of positive reals풍ퟏ,풍ퟐ,….풍풏and obtain some
interesting results on the relation between the generalized
polynomial factorial of the first kind, 풏
풕풉kind and algebraic
polynomials. Also formulae for the sum of the general
partial sums of products of several powers of consecutive
terms of an Arithmetic progression in number theory are
derived.
This document presents two integrals involving the product of a generalized Riemann zeta function and an H-function. The H-function is a generalization of Fox's H-function that is defined through a Mellin-Barnes contour integral. Two integrals are derived: the first integral involves parameters m, n, p, q and the second integral involves parameters m, n, g, p, q. The integrals are proved by expressing the H-function as a Mellin-Barnes contour integral and using properties of the generalized Riemann zeta function and Gamma function. Special cases are discussed where the H-function reduces to other useful functions like Fox's H-function or a generalized Wright hypergeometric function
On Spaces of Entire Functions Having Slow Growth Represented By Dirichlet SeriesIOSR Journals
In this paper spaces of entire function represented by Dirichlet Series have been considered. A
norm has been introduced and a metric has been defined. Properties of this space and a characterization of
continuous linear functionals have been established.
Maximum Likelihood Estimation of BeetleLiang Kai Hu
- The document describes maximum likelihood estimation (MLE) of species parameters from beetle mass, length, and other character data.
- It derives EM steps to estimate species means (μ, ν), proportions (ρ), and priors (α) in the presence of missing species data.
- Running the EM algorithm for 120 iterations estimates the parameters and converges the log likelihood to 14 digits of precision with a convergence rate of approximately 1.
- It also derives steps for Gibbs sampling to estimate the missing species indicators and parameter values based on their posterior distributions.
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
On ranges and null spaces of a special type of operator named 𝝀 − 𝒋𝒆𝒄𝒕𝒊𝒐𝒏. – ...IJMER
In this article, 𝜆 − 𝑗𝑒𝑐𝑡𝑖𝑜𝑛 has been introduced which is a generalization of trijection
operator as introduced in P.Chandra’s Ph. D. thesis titled “Investigation into the theory of operators
and linear spaces” (Patna University,1977). We obtain relation between ranges and null spaces of two
given 𝜆 − 𝑗𝑒𝑐𝑡𝑖𝑜𝑛𝑠 under suitable conditions
On a Deterministic Property of the Category of k-almost Primes: A Determinist...Ramin (A.) Zahedi
In this paper based on a sort of linear function, a deterministic and simple algorithm with an algebraic structure is presented for calculating all (and only) k-almost primes (where ∃n∊ℕ, 1 ≤ k ≤ n) in certain intervals. A theorem has been proven showing a new deterministic property of the category of k-almost primes. Through a linear function that we obtain, an equivalent redefinition of the k-almost primes with an algebraic characteristic is identified. Moreover, as an outcome of our function’s property some equalities which contain new information about the k-almost primes (including primes) are presented.
Comments: Accepted and presented article in the 11th ANTS , Korea, 2014. The 11th ANTS is one of international satellite conferences of ICM 2014:The 27th International Congress of Mathematicians, Korea. (Expanded version)
Copyright: CC Attribution-NonCommercial-NoDerivs 4.0 International
License URL: https://creativecommons.org/licenses/by-nc-nd/4.0/
A Non Local Boundary Value Problem with Integral Boundary ConditionIJMERJOURNAL
This document discusses a non-local boundary value problem with an integral boundary condition for a second order differential equation. It begins by introducing the specific boundary value problem and providing relevant background information. It then establishes some preliminary definitions and results needed to prove existence and uniqueness of solutions. The key results proved are: 1) the Green's function for the corresponding homogeneous boundary value problem is derived; 2) it is shown that the unique solution can be written using this Green's function and an integral operator; and 3) an integral equation is obtained that can be used to solve for the unique solution.
The document summarizes research on Roman dominating functions and total Roman dominating functions of the corona product graph of a path with a star. It begins by defining corona product graphs and reviewing relevant concepts in graph theory domination. It then defines Roman dominating functions and minimal Roman dominating functions. The document proves that a specific function is a minimal Roman dominating function of the corona product graph. It also discusses how this function satisfies the conditions of being a Roman dominating function and is minimal.
Formal Languages and Automata Theory unit 2Srimatre K
This document provides information about regular expressions and finite automata. It includes:
1. A syllabus covering regular expressions, applications of regular expressions, algebraic laws, conversion of automata to expressions, and the pumping lemma.
2. Details of regular expressions including operators, precedence, applications, and algebraic laws.
3. How to convert between finite automata and regular expressions using Arden's theorem and state elimination methods.
4. Properties of regular languages including closure properties and how regular languages satisfy the pumping lemma.
Significance of Mathematical Analysis in Operational Methods [2014]SanjayKumar Patel
Dr Ajay Shukla from SVNIT came to Ahmedabad on 2nd August 2014,to deliver the lecture on Significance of Mathematical Analysis in Operational Methods ....
Lecture was held in St. Xavier's College,Ahmedabad under the Father Valles Lecture Series...
Presntation for the post of lecturer in MathematicsKifayat Ullah
This talk was given during the recruitment process for the post of Lecturer in Mathematics, at University of Science and technology Bannu, on the topic of "Comparison of Reiman integration and lesbegue integration theories".
Similar to A Special Type Of Differential Polynomial And Its Comparative Growth Properties (20)
A Study on Translucent Concrete Product and Its Properties by Using Optical F...IJMER
- Translucent concrete is a concrete based material with light-transferring properties,
obtained due to embedded light optical elements like Optical fibers used in concrete. Light is conducted
through the concrete from one end to the other. This results into a certain light pattern on the other
surface, depending on the fiber structure. Optical fibers transmit light so effectively that there is
virtually no loss of light conducted through the fibers. This paper deals with the modeling of such
translucent or transparent concrete blocks and panel and their usage and also the advantages it brings
in the field. The main purpose is to use sunlight as a light source to reduce the power consumption of
illumination and to use the optical fiber to sense the stress of structures and also use this concrete as an
architectural purpose of the building
Developing Cost Effective Automation for Cotton Seed DelintingIJMER
A low cost automation system for removal of lint from cottonseed is to be designed and
developed. The setup consists of stainless steel drum with stirrer in which cottonseeds having lint is mixed
with concentrated sulphuric acid. So lint will get burn. This lint free cottonseed treated with lime water to
neutralize acidic nature. After water washing this cottonseeds are used for agriculter purpose
Study & Testing Of Bio-Composite Material Based On Munja FibreIJMER
The incorporation of natural fibres such as munja fiber composites has gained
increasing applications both in many areas of Engineering and Technology. The aim of this study is to
evaluate mechanical properties such as flexural and tensile properties of reinforced epoxy composites.
This is mainly due to their applicable benefits as they are light weight and offer low cost compared to
synthetic fibre composites. Munja fibres recently have been a substitute material in many weight-critical
applications in areas such as aerospace, automotive and other high demanding industrial sectors. In
this study, natural munja fibre composites and munja/fibreglass hybrid composites were fabricated by a
combination of hand lay-up and cold-press methods. A new variety in munja fibre is the present work
the main aim of the work is to extract the neat fibre and is characterized for its flexural characteristics.
The composites are fabricated by reinforcing untreated and treated fibre and are tested for their
mechanical, properties strictly as per ASTM procedures.
Hybrid Engine (Stirling Engine + IC Engine + Electric Motor)IJMER
Hybrid engine is a combination of Stirling engine, IC engine and Electric motor. All these 3 are
connected together to a single shaft. The power source of the Stirling engine will be a Solar Panel. The aim of
this is to run the automobile using a Hybrid engine
Fabrication & Characterization of Bio Composite Materials Based On Sunnhemp F...IJMER
This document summarizes research on the fabrication and characterization of bio-composite materials using sunnhemp fibre. The document discusses how sunnhemp fibre was used to reinforce an epoxy matrix through hand lay-up methods. Various mechanical properties of the bio-composites were tested, including tensile, flexural, and impact properties. The results of the mechanical tests on the bio-composite specimens are presented. Potential applications of the sunnhemp fibre bio-composites are also suggested, such as in fall ceilings, partitions, packaging, automotive interiors, and toys.
Geochemistry and Genesis of Kammatturu Iron Ores of Devagiri Formation, Sandu...IJMER
The Greenstone belts of Karnataka are enriched in BIFs in Dharwar craton, where Iron
formations are confined to the basin shelf, clearly separated from the deeper-water iron formation that
accumulated at the basin margin and flanking the marine basin. Geochemical data procured in terms of
major, trace and REE are plotted in various diagrams to interpret the genesis of BIFs. Al2O3, Fe2O3 (T),
TiO2, CaO, and SiO2 abundances and ratios show a wide variation. Ni, Co, Zr, Sc, V, Rb, Sr, U, Th,
ΣREE, La, Ce and Eu anomalies and their binary relationships indicate that wherever the terrigenous
component has increased, the concentration of elements of felsic such as Zr and Hf has gone up. Elevated
concentrations of Ni, Co and Sc are contributed by chlorite and other components characteristic of basic
volcanic debris. The data suggest that these formations were generated by chemical and clastic
sedimentary processes on a shallow shelf. During transgression, chemical precipitation took place at the
sediment-water interface, whereas at the time of regression. Iron ore formed with sedimentary structures
and textures in Kammatturu area, in a setting where the water column was oxygenated.
Experimental Investigation on Characteristic Study of the Carbon Steel C45 in...IJMER
In this paper, the mechanical characteristics of C45 medium carbon steel are investigated
under various working conditions. The main characteristic to be studied on this paper is impact toughness
of the material with different configurations and the experiment were carried out on charpy impact testing
equipment. This study reveals the ability of the material to absorb energy up to failure for various
specimen configurations under different heat treated conditions and the corresponding results were
compared with the analysis outcome
Non linear analysis of Robot Gun Support Structure using Equivalent Dynamic A...IJMER
Robot guns are being increasingly employed in automotive manufacturing to replace
risky jobs and also to increase productivity. Using a single robot for a single operation proves to be
expensive. Hence for cost optimization, multiple guns are mounted on a single robot and multiple
operations are performed. Robot Gun structure is an efficient way in which multiple welds can be done
simultaneously. However mounting several weld guns on a single structure induces a variety of
dynamic loads, especially during movement of the robot arm as it maneuvers to reach the weld
locations. The primary idea employed in this paper, is to model those dynamic loads as equivalent G
force loads in FEA. This approach will be on the conservative side, and will be saving time and
subsequently cost efficient. The approach of the paper is towards creating a standard operating
procedure when it comes to analysis of such structures, with emphasis on deploying various technical
aspects of FEA such as Non Linear Geometry, Multipoint Constraint Contact Algorithm, Multizone
meshing .
Static Analysis of Go-Kart Chassis by Analytical and Solid Works SimulationIJMER
This paper aims to do modelling, simulation and performing the static analysis of a go
kart chassis consisting of Circular beams. Modelling, simulations and analysis are performed using 3-D
modelling software i.e. Solid Works and ANSYS according to the rulebook provided by Indian Society of
New Era Engineers (ISNEE) for National Go Kart Championship (NGKC-14).The maximum deflection is
determined by performing static analysis. Computed results are then compared to analytical calculation,
where it is found that the location of maximum deflection agrees well with theoretical approximation but
varies on magnitude aspect.
In récent year various vehicle introduced in market but due to limitation in
carbon émission and BS Séries limitd speed availability vehicle in the market and causing of
environnent pollution over few year There is need to decrease dependancy on fuel vehicle.
bicycle is to be modified for optional in the future To implement new technique using change in
pedal assembly and variable speed gearbox such as planetary gear optimise speed of vehicle
with variable speed ratio.To increase the efficiency of bicycle for confortable drive and to
reduce torque appli éd on bicycle. we introduced epicyclic gear box in which transmission done
throgh Chain Drive (i.e. Sprocket )to rear wheel with help of Epicyclical gear Box to give
number of différent Speed during driving.To reduce torque requirent in the cycle with change in
the pedal mechanism
Integration of Struts & Spring & Hibernate for Enterprise ApplicationsIJMER
This document discusses integrating the Spring, Struts, and Hibernate frameworks to develop enterprise applications. It provides an overview of each framework and their features. The Spring Framework is a lightweight, modular framework that allows for inversion of control and aspect-oriented programming. It can be used to develop any or all tiers of an application. The document proposes an architecture for an e-commerce website that integrates these three frameworks, with Spring handling the business layer, Struts the presentation layer, and Hibernate the data access layer. This modular approach allows for clear separation of concerns and reduces complexity in application development.
Microcontroller Based Automatic Sprinkler Irrigation SystemIJMER
Microcontroller based Automatic Sprinkler System is a new concept of using
intelligence power of embedded technology in the sprinkler irrigation work. Designed system replaces
the conventional manual work involved in sprinkler irrigation to automatic process. Using this system a
farmer is protected against adverse inhuman weather conditions, tedious work of changing over of
sprinkler water pipe lines & risk of accident due to high pressure in the water pipe line. Overall
sprinkler irrigation work is transformed in to a comfortableautomatic work. This system provides
flexibility & accuracy in respect of time set for the operation of a sprinkler water pipe lines. In present
work the author has designed and developed an automatic sprinkler irrigation system which is
controlled and monitored by a microcontroller interfaced with solenoid valves.
On some locally closed sets and spaces in Ideal Topological SpacesIJMER
This document introduces and studies the concept of δˆ s-locally closed sets in ideal topological spaces. Some key points:
- A subset A is δˆ s-locally closed if A can be written as the intersection of a δˆ s-open set and a δˆ s-closed set.
- Various properties of δˆ s-locally closed sets are introduced and characterized, including relationships to other concepts like generalized locally closed sets.
- It is shown that a subset A is δˆ s-locally closed if and only if A can be written as the intersection of a δˆ s-open set and the δˆ s-closure of A.
- Theore
Intrusion Detection and Forensics based on decision tree and Association rule...IJMER
This paper present an approach based on the combination of, two techniques using
decision tree and Association rule mining for Probe attack detection. This approach proves to be
better than the traditional approach of generating rules for fuzzy expert system by clustering methods.
Association rule mining for selecting the best attributes together and decision tree for identifying the
best parameters together to create the rules for fuzzy expert system. After that rules for fuzzy expert
system are generated using association rule mining and decision trees. Decision trees is generated for
dataset and to find the basic parameters for creating the membership functions of fuzzy inference
system. Membership functions are generated for the probe attack. Based on these rules we have
created the fuzzy inference system that is used as an input to neuro-fuzzy system. Fuzzy inference
system is loaded to neuro-fuzzy toolbox as an input and the final ANFIS structure is generated for
outcome of neuro-fuzzy approach. The experiments and evaluations of the proposed method were
done with NSL-KDD intrusion detection dataset. As the experimental results, the proposed approach
based on the combination of, two techniques using decision tree and Association rule mining
efficiently detected probe attacks. Experimental results shows better results for detecting intrusions as
compared to others existing methods
Natural Language Ambiguity and its Effect on Machine LearningIJMER
This document discusses natural language ambiguity and its effect on machine learning. It begins by introducing different types of ambiguity that exist in natural languages, including lexical, syntactic, semantic, discourse, and pragmatic ambiguities. It then examines how these ambiguities present challenges for computational linguistics and machine translation systems. Specifically, it notes that ambiguity is a major problem for computers in processing human language as they lack the world knowledge and context that humans use to resolve ambiguities. The document concludes by outlining the typical process of machine translation and how ambiguities can interfere with tasks like analysis, transfer, and generation of text in the target language.
Today in era of software industry there is no perfect software framework available for
analysis and software development. Currently there are enormous number of software development
process exists which can be implemented to stabilize the process of developing a software system. But no
perfect system is recognized till yet which can help software developers for opting of best software
development process. This paper present the framework of skillful system combined with Likert scale. With
the help of Likert scale we define a rule based model and delegate some mass score to every process and
develop one tool name as MuxSet which will help the software developers to select an appropriate
development process that may enhance the probability of system success.
Material Parameter and Effect of Thermal Load on Functionally Graded CylindersIJMER
The present study investigates the creep in a thick-walled composite cylinders made
up of aluminum/aluminum alloy matrix and reinforced with silicon carbide particles. The distribution
of SiCp is assumed to be either uniform or decreasing linearly from the inner to the outer radius of
the cylinder. The creep behavior of the cylinder has been described by threshold stress based creep
law with a stress exponent of 5. The composite cylinders are subjected to internal pressure which is
applied gradually and steady state condition of stress is assumed. The creep parameters required to
be used in creep law, are extracted by conducting regression analysis on the available experimental
results. The mathematical models have been developed to describe steady state creep in the composite
cylinder by using von-Mises criterion. Regression analysis is used to obtain the creep parameters
required in the study. The basic equilibrium equation of the cylinder and other constitutive equations
have been solved to obtain creep stresses in the cylinder. The effect of varying particle size, particle
content and temperature on the stresses in the composite cylinder has been analyzed. The study
revealed that the stress distributions in the cylinder do not vary significantly for various combinations
of particle size, particle content and operating temperature except for slight variation observed for
varying particle content. Functionally Graded Materials (FGMs) emerged and led to the development
of superior heat resistant materials.
Energy Audit is the systematic process for finding out the energy conservation
opportunities in industrial processes. The project carried out studies on various energy conservation
measures application in areas like lighting, motors, compressors, transformer, ventilation system etc.
In this investigation, studied the technical aspects of the various measures along with its cost benefit
analysis.
Investigation found that major areas of energy conservation are-
1. Energy efficient lighting schemes.
2. Use of electronic ballast instead of copper ballast.
3. Use of wind ventilators for ventilation.
4. Use of VFD for compressor.
5. Transparent roofing sheets to reduce energy consumption.
So Energy Audit is the only perfect & analyzed way of meeting the Industrial Energy Conservation.
An Implementation of I2C Slave Interface using Verilog HDLIJMER
This document describes the implementation of an I2C slave interface using Verilog HDL. It introduces the I2C protocol which uses only two bidirectional lines (SDA and SCL) for communication. The document discusses the I2C protocol specifications including start/stop conditions, addressing, read/write operations, and acknowledgements. It then provides details on designing an I2C slave module in Verilog that responds to commands from an I2C master and allows synchronization through clock stretching. The module is simulated in ModelSim and synthesized in Xilinx. Simulation waveforms demonstrate successful read and write operations to the slave device.
Discrete Model of Two Predators competing for One PreyIJMER
This paper investigates the dynamical behavior of a discrete model of one prey two
predator systems. The equilibrium points and their stability are analyzed. Time series plots are obtained
for different sets of parameter values. Also bifurcation diagrams are plotted to show dynamical behavior
of the system in selected range of growth parameter
Best 20 SEO Techniques To Improve Website Visibility In SERPPixlogix Infotech
Boost your website's visibility with proven SEO techniques! Our latest blog dives into essential strategies to enhance your online presence, increase traffic, and rank higher on search engines. From keyword optimization to quality content creation, learn how to make your site stand out in the crowded digital landscape. Discover actionable tips and expert insights to elevate your SEO game.
UiPath Test Automation using UiPath Test Suite series, part 6DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 6. In this session, we will cover Test Automation with generative AI and Open AI.
UiPath Test Automation with generative AI and Open AI webinar offers an in-depth exploration of leveraging cutting-edge technologies for test automation within the UiPath platform. Attendees will delve into the integration of generative AI, a test automation solution, with Open AI advanced natural language processing capabilities.
Throughout the session, participants will discover how this synergy empowers testers to automate repetitive tasks, enhance testing accuracy, and expedite the software testing life cycle. Topics covered include the seamless integration process, practical use cases, and the benefits of harnessing AI-driven automation for UiPath testing initiatives. By attending this webinar, testers, and automation professionals can gain valuable insights into harnessing the power of AI to optimize their test automation workflows within the UiPath ecosystem, ultimately driving efficiency and quality in software development processes.
What will you get from this session?
1. Insights into integrating generative AI.
2. Understanding how this integration enhances test automation within the UiPath platform
3. Practical demonstrations
4. Exploration of real-world use cases illustrating the benefits of AI-driven test automation for UiPath
Topics covered:
What is generative AI
Test Automation with generative AI and Open AI.
UiPath integration with generative AI
Speaker:
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
Digital Marketing Trends in 2024 | Guide for Staying AheadWask
https://www.wask.co/ebooks/digital-marketing-trends-in-2024
Feeling lost in the digital marketing whirlwind of 2024? Technology is changing, consumer habits are evolving, and staying ahead of the curve feels like a never-ending pursuit. This e-book is your compass. Dive into actionable insights to handle the complexities of modern marketing. From hyper-personalization to the power of user-generated content, learn how to build long-term relationships with your audience and unlock the secrets to success in the ever-shifting digital landscape.
Monitoring and Managing Anomaly Detection on OpenShift.pdfTosin Akinosho
Monitoring and Managing Anomaly Detection on OpenShift
Overview
Dive into the world of anomaly detection on edge devices with our comprehensive hands-on tutorial. This SlideShare presentation will guide you through the entire process, from data collection and model training to edge deployment and real-time monitoring. Perfect for those looking to implement robust anomaly detection systems on resource-constrained IoT/edge devices.
Key Topics Covered
1. Introduction to Anomaly Detection
- Understand the fundamentals of anomaly detection and its importance in identifying unusual behavior or failures in systems.
2. Understanding Edge (IoT)
- Learn about edge computing and IoT, and how they enable real-time data processing and decision-making at the source.
3. What is ArgoCD?
- Discover ArgoCD, a declarative, GitOps continuous delivery tool for Kubernetes, and its role in deploying applications on edge devices.
4. Deployment Using ArgoCD for Edge Devices
- Step-by-step guide on deploying anomaly detection models on edge devices using ArgoCD.
5. Introduction to Apache Kafka and S3
- Explore Apache Kafka for real-time data streaming and Amazon S3 for scalable storage solutions.
6. Viewing Kafka Messages in the Data Lake
- Learn how to view and analyze Kafka messages stored in a data lake for better insights.
7. What is Prometheus?
- Get to know Prometheus, an open-source monitoring and alerting toolkit, and its application in monitoring edge devices.
8. Monitoring Application Metrics with Prometheus
- Detailed instructions on setting up Prometheus to monitor the performance and health of your anomaly detection system.
9. What is Camel K?
- Introduction to Camel K, a lightweight integration framework built on Apache Camel, designed for Kubernetes.
10. Configuring Camel K Integrations for Data Pipelines
- Learn how to configure Camel K for seamless data pipeline integrations in your anomaly detection workflow.
11. What is a Jupyter Notebook?
- Overview of Jupyter Notebooks, an open-source web application for creating and sharing documents with live code, equations, visualizations, and narrative text.
12. Jupyter Notebooks with Code Examples
- Hands-on examples and code snippets in Jupyter Notebooks to help you implement and test anomaly detection models.
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Guest Speaker Segment with Hannah Barrington: Dive into the world of dynamic real estate marketing with Hannah, the Marketing Manager at Workspace Group. Hear firsthand how their team generates engaging descriptions for thousands of office units by integrating diverse data sources—from PDF floorplans to web pages—using FME transformers, like OpenAIVisionConnector and AnthropicVisionConnector. This use case will show you how GenAI can streamline content creation for marketing across the board.
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We’ll wrap up with a live Q&A session where you can engage with our experts on your specific use cases, and learn more about optimizing your data workflows with AI.
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Webinar Recording: https://www.panagenda.com/webinars/hcl-notes-und-domino-lizenzkostenreduzierung-in-der-welt-von-dlau/
DLAU und die Lizenzen nach dem CCB- und CCX-Modell sind für viele in der HCL-Community seit letztem Jahr ein heißes Thema. Als Notes- oder Domino-Kunde haben Sie vielleicht mit unerwartet hohen Benutzerzahlen und Lizenzgebühren zu kämpfen. Sie fragen sich vielleicht, wie diese neue Art der Lizenzierung funktioniert und welchen Nutzen sie Ihnen bringt. Vor allem wollen Sie sicherlich Ihr Budget einhalten und Kosten sparen, wo immer möglich. Das verstehen wir und wir möchten Ihnen dabei helfen!
Wir erklären Ihnen, wie Sie häufige Konfigurationsprobleme lösen können, die dazu führen können, dass mehr Benutzer gezählt werden als nötig, und wie Sie überflüssige oder ungenutzte Konten identifizieren und entfernen können, um Geld zu sparen. Es gibt auch einige Ansätze, die zu unnötigen Ausgaben führen können, z. B. wenn ein Personendokument anstelle eines Mail-Ins für geteilte Mailboxen verwendet wird. Wir zeigen Ihnen solche Fälle und deren Lösungen. Und natürlich erklären wir Ihnen das neue Lizenzmodell.
Nehmen Sie an diesem Webinar teil, bei dem HCL-Ambassador Marc Thomas und Gastredner Franz Walder Ihnen diese neue Welt näherbringen. Es vermittelt Ihnen die Tools und das Know-how, um den Überblick zu bewahren. Sie werden in der Lage sein, Ihre Kosten durch eine optimierte Domino-Konfiguration zu reduzieren und auch in Zukunft gering zu halten.
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HCL Notes und Domino Lizenzkostenreduzierung in der Welt von DLAU
A Special Type Of Differential Polynomial And Its Comparative Growth Properties
1. www.ijmer.com
International Journal of Modern Engineering Research (IJMER)
Vol. 3, Issue. 5, Sep - Oct. 2013 pp-2606-2614
ISSN: 2249-6645
A Special Type Of Differential Polynomial And Its Comparative
Growth Properties
Sanjib Kumar Datta1, Ritam Biswas2
1
2
Department of Mathematics, University of Kalyani, Kalyani, Dist.- Nadia, PIN- 741235, West Bengal, India.
Murshidabad College of Engineering and Technology, Banjetia, Berhampore, P. O.-Cossimbazar Raj, PIN-742102, West
Bengal, India
Abstract: Some new results depending upon the comparative growth rates of composite entire and meromorphic function
and a special type of differential polynomial as considered by Bhooshnurmath and Prasad[3] and generated by one of the
factors of the composition are obtained in this paper.
AMS Subject Classification (2010) : 30D30, 30D35.
Keywords and phrases: Order (lower order), hyper order (hyper lower order), growth rate, entire function, meromorphic
function and special type of differential polynomial.
I.
INTRODUCTION, DEFINITIONS AND NOTATIONS.
Let ℂ be the set of all finite complex numbers. Also let f be a meromorphic function and g be an entire function defined on
ℂ . In the sequel we use the following two notations:
𝑖 log [𝑘] 𝑥 = log(log [𝑘−1] 𝑥) 𝑓𝑜𝑟 𝑘 = 1,2,3, … ; log [0] 𝑥 = 𝑥
and
𝑖𝑖 exp[𝑘] 𝑥 = exp 𝑒𝑥𝑝 𝑘 −1 𝑥 𝑓𝑜𝑟 𝑘 = 1,2,3, … ; exp[0] 𝑥 = 𝑥.
The following definitions are frequently used in this paper:
Definition 1 The order 𝜌 𝑓 and lower order 𝜆 𝑓 of a meromorphic function f are defined as
lim
sup log 𝑇(𝑟, 𝑓)
𝜌 𝑓 = 𝑟 →∞
log 𝑟
and
log 𝑇(𝑟, 𝑓)
inf
𝜆 𝑓 = lim𝑟
.
→∞
log 𝑟
If f is entire, one can easily verify that
[2]
𝑀(𝑟, 𝑓)
lim
sup log
𝜌 𝑓 = 𝑟→∞
log 𝑟
and
log [2] 𝑀(𝑟, 𝑓)
inf
𝜆 𝑓 = lim𝑟→∞
.
log 𝑟
Definition 2 The hyper order 𝜌 𝑓 and hyper lower order 𝜆 𝑓 of a meromorphic function f are defined as follows
𝜌𝑓 =
lim
sup
𝑟 →∞
𝜆𝑓 =
lim
inf
𝑟→∞
𝜌𝑓 =
lim
sup
𝑟 →∞
log [2] 𝑇(𝑟, 𝑓)
log 𝑟
and
log [2] 𝑇(𝑟, 𝑓)
.
log 𝑟
If f is entire, then
log [3] 𝑀(𝑟, 𝑓)
log 𝑟
and
log [3] 𝑀(𝑟, 𝑓)
.
log 𝑟
Definition 3 The type 𝜍 𝑓 of a meromorphic function f is defined as follows
lim
sup 𝑇(𝑟, 𝑓)
𝜍 𝑓 = 𝑟→∞
, 0 < 𝜌 𝑓 < ∞.
𝑟𝜌𝑓
When f is entire, then
lim
sup log 𝑀(𝑟, 𝑓)
𝜍 𝑓 = 𝑟 →∞
, 0 < 𝜌 𝑓 < ∞.
𝑟𝜌𝑓
Definition 4 A function 𝜆 𝑓 (𝑟) is called a lower proximate order of a meromorphic function f of finite lower order 𝜆 𝑓 if
𝜆𝑓 =
(i)
(ii)
lim
inf
𝑟→∞
𝜆 𝑓 (𝑟) is non-negative and continuous for 𝑟 ≥ 𝑟0 , say
𝜆 𝑓 (𝑟) is differentiable for 𝑟 ≥ 𝑟0 except possibly at isolated points at which 𝜆′𝑓 (𝑟 + 0) and 𝜆′𝑓 (𝑟 − 0) exists,
www.ijmer.com
2606 | Page
2. www.ijmer.com
International Journal of Modern Engineering Research (IJMER)
Vol. 3, Issue. 5, Sep - Oct. 2013 pp-2606-2614
ISSN: 2249-6645
(iii)
lim 𝜆 𝑓 𝑟 = 𝜆 𝑓 ,
𝑟→∞
(iv)
lim 𝑟𝜆′𝑓 𝑟 log 𝑟 = 0
𝑟→∞
and
(v)
lim
inf
𝑟 →∞
𝑇(𝑟, 𝑓)
= 1.
𝑟 𝜆 𝑓 (𝑟)
Definition 5 Let ‘a’ be a complex number, finite or infinite. The Nevanlinna’s deficiency and Valiron deficiency of ‘a’ with
respect to a meromorphic function f are defined as
𝑚(𝑟, 𝑎; 𝑓)
lim
sup 𝑁(𝑟, 𝑎; 𝑓)
inf
𝛿 𝑎; 𝑓 = 1 − 𝑟→∞
= lim
𝑟→∞
𝑇(𝑟, 𝑓)
𝑇(𝑟, 𝑓)
and
𝑁(𝑟, 𝑎; 𝑓) lim
sup 𝑚(𝑟, 𝑎; 𝑓)
inf
Δ 𝑎; 𝑓 = 1 − lim
= 𝑟→∞
.
𝑟→∞
𝑇(𝑟, 𝑓)
𝑇(𝑟, 𝑓)
Let f be a non-constant meromorphic function defined in the open complex plane ℂ. Also let n0j, n1j,…, nkj (k ≥ 1) be non𝑘
negative integers such that for each j, 𝑖=0 𝑛 𝑖𝑗 ≥ 1. We call
𝑀𝑗 𝑓 = 𝐴 𝑗 (𝑓) 𝑛 0𝑗 (𝑓 (1) ) 𝑛 1𝑗 … (𝑓 (𝑘) ) 𝑛 𝑘𝑗
where 𝑇 𝑟, 𝐴 𝑗 = 𝑆(𝑟, 𝑓), to be a differential monomial generated by f. The numbers
𝑘
𝛾 𝑀𝑗 =
𝑛 𝑖𝑗
𝑖=0
and
𝑘
Γ 𝑀𝑗 =
(𝑖 + 1)𝑛 𝑖𝑗
𝑖=0
are called the degree and weight of 𝑀𝑗 [𝑓] {cf. [4]} respectively. The expression
𝑠
𝑃 𝑓 =
𝑀𝑗 [𝑓]
𝑗 =1
is called a differential polynomial generated by f. The numbers
𝛾 𝑃 = max 𝛾 𝑀 𝑗
1≤𝑗 ≤𝑠
and
Γ 𝑃 = max Γ
1≤𝑗 ≤𝑠
𝑀𝑗
are respectively called the degree and weight of P[f] {cf. [4]}. Also we call the numbers
𝛾 𝑃 = min 𝛾 𝑀 𝑗
1≤𝑗 ≤𝑠
and k ( the order of the highest derivative of f) the lower degree and the order of P[f] respectively. If 𝛾 𝑃 = 𝛾 𝑃 , P[f] is called a
homogeneous differential polynomial.
Bhooshnurmath and Prasad [3] considered a special type of differential polynomial of the form 𝐹 = 𝑓 𝑛 𝑄[𝑓] where
Q[f] is a differential polynomial in f and n = 0, 1, 2,…. In this paper we intend to prove some improved results depending
upon the comparative growth properties of the composition of entire and meromorphic functions and a special type of
differential polynomial as mentioned above and generated by one of the factors of the composition. We do not explain the
standard notations and definitions in the theory of entire and meromorphic functions because those are available in [9] and
[5].
II.
LEMMAS.
In this section we present some lemmas which will be needed in the sequel.
Lemma 1 [1] If f is meromorphic and g is entire then for all sufficiently large values of r,
𝑇 𝑟, 𝑔
𝑇 𝑟, 𝑓𝑜 𝑔 ≤ 1 + 𝑜 1
𝑇 𝑀 𝑟, 𝑔 , 𝑓 .
log 𝑀 𝑟, 𝑔
Lemma 2 [2] Let f be meromorphic and g be entire and suppose that 0 < 𝜇 < 𝜌 𝑔 ≤ ∞. Then for a sequence of values of r
tending to infinity,
𝑇(𝑟, 𝑓𝑜 𝑔) ≥ 𝑇(exp( 𝑟 𝜇 ), 𝑓).
𝑛
Lemma 3 [3] Let 𝐹 = 𝑓 𝑄[𝑓] where Q[f] is a differential polynomial in f. If n ≥ 1 then 𝜌 𝐹 = 𝜌 𝑓 and 𝜆 𝐹 = 𝜆 𝑓 .
Lemma 4 Let 𝐹 = 𝑓 𝑛 𝑄[𝑓] where Q[f] is a differential polynomial in f. If n ≥ 1 then
𝑇(𝑟, 𝐹)
lim
= 1.
𝑟→∞ 𝑇(𝑟, 𝑓)
The proof of Lemma 4 directly follows from Lemma 3.
www.ijmer.com
2607 | Page
3. International Journal of Modern Engineering Research (IJMER)
www.ijmer.com
Vol. 3, Issue. 5, Sep - Oct. 2013 pp-2606-2614
ISSN: 2249-6645
In the line of Lemma 3 we may prove the following lemma:
Lemma 5 Let 𝐹 = 𝑓 𝑛 𝑄[𝑓] where Q[f] is a differential polynomial in f. If n ≥ 1 then 𝜌 𝐹 = 𝜌 𝑓 and 𝜆 𝐹 = 𝜆 𝑓 .
Lemma 6 For a meromorphic function f of finite lower order, lower proximate order exists.
The lemma can be proved in the line of Theorem 1 [7] and so the proof is omitted.
Lemma 7 Let f be a meromorphic function of finite lower order 𝜆 𝑓 . Then for 𝛿 > 0 the function 𝑟 𝜆 𝑓 +𝛿−𝜆 𝑓 (𝑟) is an
increasing function of r.
Proof. Since
𝑑 𝜆 +𝛿−𝜆 (𝑟)
𝑓
𝑟 𝑓
= 𝜆 𝑓 + 𝛿 − 𝜆 𝑓 𝑟 − 𝑟𝜆′𝑓 log 𝑟 𝑟 𝜆 𝑓 +𝛿−𝜆 𝑓 𝑟 −1 > 0
𝑑𝑟
for sufficiently large values of r, the lemma follows.
Lemma 8 [6] Let f be an entire function of finite lower order. If there exists entire functions a i (i= 1, 2, 3,…, n; n ≤ ∞)
satisfying 𝑇 𝑟, 𝑎 𝑖 = 𝑜{𝑇(𝑟, 𝑓)} and
𝑛
𝛿(𝑎 𝑖 , 𝑓) = 1,
𝑖=1
then
lim
𝑇(𝑟, 𝑓)
1
= .
𝑀(𝑟, 𝑓)
𝜋
𝑟→∞ log
III.
THEOREMS.
In this section we present the main results of the paper.
Theorem 1 Let f be a meromorphic function and g be an entire function satisfying
𝑖 𝜆 𝑓 , 𝜆 𝑔 are both finite and
𝑖𝑖 𝑓𝑜𝑟 𝑛 ≥ 1, 𝐺 = 𝑔 𝑛 𝑄[𝑔]. Then
log 𝑇(𝑟, 𝑓𝑜 𝑔)
lim
inf
≤ 3. 𝜌 𝑓 . 2 𝜆 𝑔 .
𝑟→∞
𝑇(𝑟, 𝐺)
Proof. If 𝜌 𝑓 = ∞, the result is obvious. So we suppose that 𝜌 𝑓 < ∞. Since 𝑇 𝑟, 𝑔 ≤ log + 𝑀 𝑟, 𝑔 , in view of Lemma 1 we
get for all sufficiently large values of r that
𝑇 𝑟, 𝑓𝑜 𝑔 ≤ 1 + 𝑜 1 𝑇(𝑀 𝑟, 𝑔 , 𝑓)
i.e.,
log 𝑇(𝑟, 𝑓𝑜 𝑔) ≤ log{1 + 𝑜(1)} + log 𝑇(𝑀 𝑟, 𝑔 , 𝑓)
i.e.,
log 𝑇(𝑟, 𝑓𝑜 𝑔) ≤ 𝑜 1 + 𝜌 𝑓 + 𝜀 log 𝑀(𝑟, 𝑔)
i.e.,
log 𝑇(𝑟, 𝑓𝑜 𝑔)
log 𝑀(𝑟, 𝑔)
lim
inf
inf
≤ 𝜌 𝑓 + 𝜀 lim𝑟→∞
.
𝑟 →∞
𝑇(𝑟, 𝑔)
𝑇(𝑟, 𝑔)
Since 𝜀(> 0) is arbitrary, it follows that
log 𝑇(𝑟, 𝑓𝑜 𝑔)
log 𝑀(𝑟, 𝑔)
lim
inf
inf
≤ 𝜌 𝑓 . lim𝑟→∞
.
1
𝑟→∞
𝑇(𝑟, 𝑔)
𝑇(𝑟, 𝑔)
As by condition (v) of Definition 4
𝑇(𝑟, 𝑔)
lim
inf
= 1,
𝑟 →∞
𝑟 𝜆 𝑔 (𝑟)
so for given 𝜀(0 < 𝜀 < 1) we get for a sequence of values of r tending to infinity that
𝑇 𝑟, 𝑔 ≤ 1 + 𝜀 𝑟 𝜆 𝑔 𝑟
(2)
and for all sufficiently large values of r,
𝑇 𝑟, 𝑔 > 1 − 𝜀 𝑟 𝜆 𝑔 𝑟
(3)
Since
log 𝑀(𝑟, 𝑔) ≤ 3𝑇(2𝑟, 𝑔)
{cf. [5]}, we have by (2), for a sequence of values of r tending to infinity,
log 𝑀(𝑟, 𝑔) ≤ 3𝑇 2𝑟, 𝑔 ≤ 3 1 + 𝜀 2𝑟 𝜆 𝑔 2𝑟 .
(4)
Combining (3) and (4), we obtain for a sequence of values of r tending to infinity that
log 𝑀(𝑟, 𝑔) 3(1 + 𝜀) (2𝑟) 𝜆 𝑔 (2𝑟)
≤
.
.
𝑇(𝑟, 𝑔)
(1 − 𝜀)
𝑟 𝜆 𝑔 (𝑟)
Now for any 𝛿(> 0), for a sequence of values of r tending to infinity we obtain that
log 𝑀(𝑟, 𝑔) 3(1 + 𝜀)
(2𝑟) 𝜆 𝑔 +𝛿
1
≤
.
. 𝜆 (𝑟)
𝜆 𝑔 +𝛿−𝜆 𝑔 (2𝑟)
𝑇(𝑟, 𝑔)
(1 − 𝜀) (2𝑟)
𝑟 𝑔
i.e.,
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log 𝑀(𝑟, 𝑔) 3(1 + 𝜀) 𝜆 +𝛿
≤
.2 𝑔
(5)
𝑇(𝑟, 𝑔)
(1 − 𝜀)
is an increasing function of r by Lemma 7. Since 𝜀(> 0) and 𝛿(> 0) are arbitrary, it follows from (5)
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because 𝑟 𝜆 𝑔 +𝛿−𝜆 𝑔 (𝑟)
that
lim
inf
𝑟→∞
log 𝑀(𝑟, 𝑔)
≤ 3. 2 𝜆 𝑔 .
𝑇(𝑟, 𝑔)
(6)
Thus from (1) and (6) we obtain that
lim
inf
𝑟 →∞
log 𝑇(𝑟, 𝑓𝑜 𝑔)
≤ 3. 𝜌 𝑓 . 2 𝜆 𝑔 .
𝑇(𝑟, 𝑔)
(7)
Now in view of (7) and Lemma 3, we get that
log 𝑇(𝑟, 𝑓𝑜 𝑔) lim log 𝑇(𝑟, 𝑓𝑜 𝑔)
𝑇(𝑟, 𝑔)
lim
inf
inf
= 𝑟→∞
. lim
𝑟 →∞
𝑟→∞ 𝑇(𝑟, 𝐺)
𝑇(𝑟, 𝐺)
𝑇(𝑟, 𝑔)
≤ 3. 𝜌 𝑓 . 2 𝜆 𝑔 .
This proves the theorem.
Theorem 2 Let f be meromorphic and g be entire such that 𝜌 𝑓 < ∞, 𝜆 𝑔 < ∞ and for 𝑛 ≥ 1, 𝐺 = 𝑔 𝑛 𝑄 𝑔 . Then
log [2] 𝑇(𝑟, 𝑓𝑜 𝑔)
lim
inf
≤ 1.
𝑟 →∞
log 𝑇(𝑟, 𝐺)
Proof. Since
𝑇 𝑟, 𝑔 ≤ log + 𝑀 𝑟, 𝑔 ,
in view of Lemma 1, we get for all sufficiently large values of r that
log 𝑇(𝑟, 𝑓𝑜 𝑔) ≤ log 𝑇(𝑀 𝑟, 𝑔 , 𝑓) + log{1 + 𝑜(1)}
i.e.,
log 𝑇(𝑟, 𝑓𝑜 𝑔) ≤ 𝜌 𝑓 + 𝜀 log 𝑀(𝑟, 𝑔) + 𝑜(1)
i.e.,
log [2] 𝑇 𝑟, 𝑓𝑜 𝑔 ≤ log [2] 𝑀 𝑟, 𝑔 + 𝑂 1 .
(8)
It is well known that for any entire function g,
log 𝑀(𝑟, 𝑔) ≤ 3𝑇 2𝑟, 𝑔
𝑐𝑓. 5 .
Then for 0 < 𝜀 < 1 and 𝛿 > 0 , for a sequence of values of r tending to
infinity it follows from (5) that
log 2 𝑀 𝑟, 𝑔 ≤ log 𝑇 𝑟, 𝑔 + 𝑂 1 .
9
Now combining (8) and (9), we obtain for a sequence of values of r tending to infinity that
log 2 𝑇(𝑟, 𝑓𝑜 𝑔) ≤ log 𝑇 𝑟, 𝑔 + 𝑂(1)
i.e.,
log 2 𝑇(𝑟, 𝑓𝑜 𝑔)
≤ 1.
(10)
log 𝑇 𝑟, 𝑔
As by Lemma 4,
log 𝑇(𝑟, 𝑔)
lim
𝑟→∞ log 𝑇(𝑟, 𝐺)
exists and is equal to 1, then from (10) we get that
log [2] 𝑇(𝑟, 𝑓𝑜 𝑔) lim log [2] 𝑇(𝑟, 𝑓𝑜 𝑔)
log 𝑇(𝑟, 𝑔)
lim
inf
inf
= 𝑟→∞
. lim
𝑟→∞
𝑟→∞ log 𝑇(𝑟, 𝐺)
log 𝑇(𝑟, 𝐺)
log 𝑇(𝑟, 𝑔)
≤ 1.1 = 1.
Thus the theorem is established.
Remark 1 The condition 𝜌 𝑓 < ∞ is essential in Theorem 2 which is evident from the following example.
Example 1 Let 𝑓 = exp[2] 𝑧 and 𝑔 = exp 𝑧. Then 𝑓𝑜 𝑔 = exp[3] 𝑧 and for 𝑛 ≥ 1, 𝐺 = 𝑔 𝑛 𝑄 𝑔 . Taking 𝑛 = 1, 𝐴 𝑗 = 1, 𝑛0𝑗 =
1 and 𝑛1𝑗 = ⋯ = 𝑛 𝑘𝑗 = 0; we see that 𝐺 = exp 2𝑧 . Now we have
𝜌𝑓 =
lim
sup
𝑟 →∞
log [2] 𝑀(𝑟, 𝑓)
=
log 𝑟
lim
sup
𝑟 →∞
log [2] (exp[2] 𝑟)
=∞
log 𝑟
and
𝜆𝑔 =
lim
inf
𝑟→∞
log [2] 𝑀(𝑟, 𝑔)
=
log 𝑟
lim
inf
𝑟→∞
log 2 (exp 𝑟)
= 1.
log 𝑟
Again from the inequality
𝑇(𝑟, 𝑓) ≤ log + 𝑀(𝑟, 𝑓) ≤ 3𝑇(2𝑟, 𝑓)
{cf. p.18, [5]} we obtain that
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𝑇 𝑟, 𝐺 ≤ log 𝑀 𝑟, 𝐺 = log(exp 2𝑟)
i.e.,
log 𝑇(𝑟, 𝐺) ≤ log 𝑟 + 𝑂(1)
and
1
𝑟
1
𝑟
𝑇 𝑟, 𝑓𝑜 𝑔 ≥ log 𝑀 , 𝑓𝑜 𝑔 = exp[2] ( )
3
2
3
2
i.e.,
log [2] 𝑇 𝑟, 𝑓𝑜 𝑔 ≥
𝑟
+ 𝑂 1 .
2
Combining the above two inequalities, we have
𝑟
+ 𝑂(1)
log [2] 𝑇(𝑟, 𝑓𝑜 𝑔)
2
≥
.
log 𝑇(𝑟, 𝐺)
log 𝑟 + 𝑂(1)
Therefore
lim
inf
𝑟→∞
log [2] 𝑇(𝑟, 𝑓𝑜 𝑔)
= ∞,
log 𝑇(𝑟, 𝐺)
which is contrary to Theorem 2.
Theorem 3 Let f and g be any two entire functions such that 𝜌 𝑔 < 𝜆 𝑓 ≤ 𝜌 𝑓 < ∞ and for 𝑛 ≥ 1, 𝐹 = 𝑓 𝑛 𝑄[𝑓] and
𝐺 = 𝑔 𝑛 𝑄 𝑔 . Also there exist entire functions ai (i = 1, 2,…, n; n ≤ ∞) with
𝑖 𝑇 𝑟, 𝑎 𝑖 = 𝑜 𝑇 𝑟, 𝑔
𝑎𝑠 𝑟 → ∞ 𝑓𝑜𝑟 𝑖 = 1, 2, … , 𝑛
and
𝑛
𝑖𝑖
𝛿 𝑎 𝑖 ; 𝑔 = 1.
𝑖=1
Then
{log 𝑇(𝑟, 𝑓𝑜 𝑔)}2
= 0.
𝑟→∞ 𝑇 𝑟, 𝐹 𝑇(𝑟, 𝐺)
lim
Proof. In view of the inequality
𝑇(𝑟, 𝑔) ≤ log + 𝑀(𝑟, 𝑔)
and Lemma 1, we obtain for all sufficiently large values of r that
𝑇 𝑟, 𝑓𝑜 𝑔 ≤ {1 + 𝑜 1 }𝑇(𝑀 𝑟, 𝑔 , 𝑓)
i.e.,
log 𝑇(𝑟, 𝑓𝑜 𝑔) ≤ log{1 + 𝑜(1)} + log 𝑇(𝑀 𝑟, 𝑔 , 𝑓)
i.e.,
log 𝑇(𝑟, 𝑓𝑜 𝑔) ≤ 𝑜 1 + 𝜌 𝑓 + 𝜀 log 𝑀(𝑟, 𝑔)
i.e.,
log 𝑇(𝑟, 𝑓𝑜 𝑔) ≤ 𝑜 1 + 𝜌 𝑓 + 𝜀 𝑟 (𝜌 𝑔 +𝜀) .
Again in view of Lemma 3, we get for all sufficiently large values of r that
log 𝑇 𝑟, 𝐹 > 𝜆 𝐹 − 𝜀 log 𝑟
i.e.,
log 𝑇 𝑟, 𝐹 > 𝜆 𝑓 − 𝜀 log 𝑟
i.e.,
𝑇 𝑟, 𝐹 > 𝑟 𝜆 𝑓 −𝜀 .
Now combining (11) and (12), it follows for all sufficiently large values of r that
𝑜 1 + (𝜌 𝑓 + 𝜀)𝑟 (𝜌 𝑔 +𝜀)
log 𝑇(𝑟, 𝑓𝑜 𝑔)
≤
.
𝑇(𝑟, 𝐹)
𝑟 𝜆 𝑓 −𝜀
Since 𝜌 𝑔 < 𝜆 𝑓 , we can choose 𝜀(> 0) in such a way that
𝜌 𝑔 + 𝜀 < 𝜆 𝑓 − 𝜀.
So in view of (13) and (14), it follows that
log 𝑇(𝑟, 𝑓𝑜 𝑔)
lim
= 0.
𝑟→∞
𝑇(𝑟, 𝐹)
Again from Lemma 4 and Lemma 8, we get for all sufficiently large values of r that
𝑜 1 + 𝜌 𝑓 + 𝜀 log 𝑀(𝑟, 𝑔)
log 𝑇(𝑟, 𝑓𝑜 𝑔)
≤
𝑇(𝑟, 𝐺)
𝑇(𝑟, 𝐺)
i.e.,
lim
sup log 𝑇(𝑟, 𝑓𝑜 𝑔)
lim
sup log 𝑀(𝑟, 𝑔)
≤ (𝜌 𝑓 + 𝜀) 𝑟 →∞
𝑟→∞
𝑇(𝑟, 𝐺)
𝑇(𝑟, 𝐺)
i.e.,
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(11)
(12)
(13)
(14)
(15)
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𝑇(𝑟, 𝑔)
lim
sup log 𝑇(𝑟, 𝑓𝑜 𝑔)
lim sup log 𝑀 𝑟, 𝑔
≤ 𝜌 𝑓 + 𝜀 𝑟→∞
. lim
𝑟→∞
𝑟→∞ 𝑇(𝑟, 𝐺)
𝑇(𝑟, 𝐺)
𝑇 𝑟, 𝑔
i.e.,
lim
sup
𝑟 →∞
log 𝑇(𝑟, 𝑓𝑜 𝑔)
≤
𝑇(𝑟, 𝐺)
𝜌 𝑓 + 𝜀 . 𝜋.
Since 𝜀(> 0) is arbitrary, it follows from above that
lim
sup
𝑟→∞
log 𝑇(𝑟, 𝑓𝑜 𝑔)
≤ 𝜌 𝑓 . 𝜋.
𝑇(𝑟, 𝐺)
(16)
Combining (15) and (16), we obtain that
{log 𝑇(𝑟, 𝑓𝑜 𝑔)}2
𝑇 𝑟, 𝐹 𝑇(𝑟, 𝐺)
log 𝑇(𝑟, 𝑓𝑜 𝑔) lim log 𝑇(𝑟, 𝑓𝑜 𝑔)
sup
= lim
. 𝑟 →∞
𝑟→∞
𝑇(𝑟, 𝐹)
𝑇(𝑟, 𝐺)
lim
sup
𝑟→∞
≤ 0. 𝜋. 𝜌 𝑓 = 0,
i.e.,
{log 𝑇(𝑟, 𝑓𝑜 𝑔)}2
= 0.
𝑟→∞ 𝑇 𝑟, 𝐹 𝑇(𝑟, 𝐺)
lim
This proves the theorem.
Theorem 4 Let f and g be any two entire functions satisfying the following conditions: 𝑖 𝜆 𝑓 > 0 𝑖𝑖 𝜌 𝑓 <
∞ 𝑖𝑖𝑖 0 < 𝜆 𝑔 ≤ 𝜌 𝑔 and also let for 𝑛 ≥ 1, 𝐹 = 𝑓 𝑛 𝑄 𝑓 . Then
[2]
𝜆𝑔 𝜌𝑔
𝑇(𝑟, 𝑓𝑜 𝑔)
lim
sup log
≥ max{ , }.
𝑟 →∞
[2] 𝑇(𝑟, 𝐹)
log
𝜆𝑓 𝜌𝑓
Proof. We know that for r > 0 {cf. [8]} and for all sufficiently large values of r,
1
1
𝑟
𝑇 𝑟, 𝑓𝑜 𝑔 ≥ log 𝑀
𝑀 , 𝑔 + 𝑜 1 , 𝑓 .
(17)
3
8
4
Since 𝜆 𝑓 and 𝜆 𝑔 are the lower orders of f and g respectively then for given 𝜀(> 0) and for all sufficiently large values of r
we obtain that
log 𝑀 𝑟, 𝑓 > 𝑟 𝜆 𝑓 −𝜀
and
log 𝑀 𝑟, 𝑔 > 𝑟 𝜆 𝑔 −𝜀
where 0 < 𝜀 < min 𝜆 𝑓 , 𝜆 𝑔 . So from (17) we have for all sufficiently large values of r,
𝜆 𝑓 −𝜀
1 1
𝑟
𝑇(𝑟, 𝑓𝑜 𝑔) ≥
𝑀 , 𝑔 + 𝑜 1
3 8
4
i.e.,
1 1
𝑟
𝑇(𝑟, 𝑓𝑜 𝑔) ≥ { 𝑀( , 𝑔)} 𝜆 𝑓 −𝜀
3 9 4
i.e.,
𝑟
log 𝑇(𝑟, 𝑓𝑜 𝑔) ≥ 𝑂 1 + 𝜆 𝑓 − 𝜀 log 𝑀( , 𝑔)
4
i.e.,
𝑟
log 𝑇(𝑟, 𝑓𝑜 𝑔) ≥ 𝑂 1 + 𝜆 𝑓 − 𝜀 ( ) 𝜆 𝑔 −𝜀
4
i.e.,
log 2 𝑇 𝑟, 𝑓𝑜 𝑔 ≥ 𝑂 1 + 𝜆 𝑔 − 𝜀 log 𝑟 .
18
Again in view of Lemma 1, we get for a sequence of values r tending to infinity that
log 2 𝑇 𝑟, 𝐹 ≤ 𝜆 𝐹 + 𝜀 log 𝑟
i.e.,
log 2 𝑇 𝑟, 𝐹 ≤ 𝜆 𝑓 + 𝜀 log 𝑟 .
(19)
Combining (18) and (19), it follows for a sequence of values of r tending to infinity that
𝑂 1 + 𝜆 𝑔 − 𝜀 log 𝑟
log [2] 𝑇(𝑟, 𝑓𝑜 𝑔)
≥
.
[2] 𝑇(𝑟, 𝐹)
log
𝜆 𝑓 + 𝜀 log 𝑟
Since 𝜀(> 0) is arbitrary, we obtain that
𝜆𝑔
log [2] 𝑇(𝑟, 𝑓𝑜 𝑔)
≥ .
[2] 𝑇(𝑟, 𝐹)
log
𝜆𝑓
Again from (17), we get for a sequence of values of r tending to infinity that
lim
sup
𝑟 →∞
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(20)
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𝑟 𝜌 −𝜀
𝑔
log 𝑇(𝑟, 𝑓𝑜 𝑔) ≥ 𝑂 1 + (𝜆 𝑓 − 𝜀)( )
4
i.e.,
log 2 𝑇 𝑟, 𝑓𝑜 𝑔 ≥ 𝑂 1 + 𝜌 𝑔 − 𝜀 log 𝑟 .
Also in view of Lemma 5, for all sufficiently large values of r we have
log [2] 𝑇 𝑟, 𝐹 ≤ 𝜌 𝐹 + 𝜀 log 𝑟
i.e.,
(21)
log [2] 𝑇 𝑟, 𝐹 ≤ 𝜌 𝑓 + 𝜀 log 𝑟 .
Now from (21) and (22), it follows for a sequence of values of r tending to infinity that
𝑂 1 + 𝜌 𝑔 − 𝜀 log 𝑟
log [2] 𝑇(𝑟, 𝑓𝑜 𝑔)
≥
.
[2] 𝑇(𝑟, 𝐹)
log
𝜌 + 𝜀 log 𝑟
(22)
𝑓
As 𝜀(0 < 𝜀 < 𝜌 𝑔 ) is arbitrary, we obtain from above that
[2]
𝜌𝑔
𝑇(𝑟, 𝑓𝑜 𝑔)
lim
sup log
≥ .
𝑟 →∞
[2] 𝑇(𝑟, 𝐹)
log
𝜌𝑓
Therefore from (20) and (23), we get that
[2]
𝜆𝑔 𝜌𝑔
𝑇(𝑟, 𝑓𝑜 𝑔)
lim
sup log
≥ max{ , }.
𝑟 →∞
log [2] 𝑇(𝑟, 𝐹)
𝜆𝑓 𝜌𝑓
Thus the theorem is established.
Theorem 5 Let f be meromorphic and g be entire such that 𝑖 0 < 𝜆 𝑓 < 𝜌 𝑓 , 𝑖𝑖 𝜌 𝑔 < ∞,
𝑖𝑣 𝑓𝑜𝑟 𝑛 ≥ 1, 𝐹 = 𝑓 𝑛 𝑄 𝑓 . Then
𝜆𝑔 𝜌𝑔
log [2] 𝑇(𝑟, 𝑓𝑜 𝑔)
lim
inf
≤ min{ , }.
𝑟 →∞
log [2] 𝑇(𝑟, 𝐹)
𝜆𝑓 𝜌 𝑓
Proof. In view of Lemma 1 and the inequality
𝑇 𝑟, 𝑔 ≤ log + 𝑀 𝑟, 𝑔 ,
we obtain for all sufficiently large values of r that
log 𝑇 𝑟, 𝑓𝑜 𝑔 ≤ 𝑜 1 + 𝜌 𝑓 + 𝜀 log 𝑀(𝑟, 𝑔).
Also for a sequence of values of r tending to infinity,
log 𝑀(𝑟, 𝑔) ≤ 𝑟 𝜆 𝑔 +𝜀 .
Combining (24) and (25), it follows for a sequence of values of r tending to infinity that
log 𝑇 𝑟, 𝑓𝑜 𝑔 ≤ 𝑜 1 + (𝜌 𝑓 + 𝜀)𝑟 𝜆 𝑔 +𝜀
i.e.,
log 𝑇 𝑟, 𝑓𝑜 𝑔 ≤ 𝑟 𝜆 𝑔 +𝜀 {𝑜 1 + (𝜌 𝑓 + 𝜀)}
i.e.,
log [2] 𝑇 𝑟, 𝑓𝑜 𝑔 ≤ 𝑂 1 + 𝜆 𝑔 + 𝜀 log 𝑟 .
Again in view of Lemma 5, we obtain for all sufficiently large values of r that
log 2 𝑇 𝑟, 𝐹 > 𝜆 𝐹 − 𝜀 log 𝑟
(23)
𝑖𝑖𝑖 𝜌 𝑓 < ∞ and
(24)
(25)
(26)
i.e.,
log 2 𝑇 𝑟, 𝐹 > 𝜆 𝑓 − 𝜀 log 𝑟 .
Now from (26) and (27), we get for a sequence of values of r tending to infinity that
𝑂 1 + 𝜆 𝑔 + 𝜀 log 𝑟
log [2] 𝑇 𝑟, 𝑓𝑜 𝑔
≤
.
2 𝑇 𝑟, 𝐹
log
𝜆 𝑓 − 𝜀 log 𝑟
(27)
As 𝜀(> 0) is arbitrary, it follows that
𝜆𝑔
log [2] 𝑇 𝑟, 𝑓𝑜 𝑔
≤ .
log 2 𝑇 𝑟, 𝐹
𝜆𝑓
In view of Lemma 1, we obtain for all sufficiently large values of r that
log 2 𝑇 𝑟, 𝑓𝑜 𝑔 ≤ 𝑂 1 + 𝜌 𝑔 + 𝜀 log 𝑟 .
Also by Remark 1, it follows for a sequence of values of r tending to infinity that
log 2 𝑇 𝑟, 𝐹 > 𝜌 𝐹 − 𝜀 log 𝑟
i.e.,
lim
inf
𝑟 →∞
log 2 𝑇 𝑟, 𝐹 > 𝜌 𝑓 − 𝜀 log 𝑟 .
Combining (29) and (30), we get for a sequence of values of r tending to infinity that
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(29)
30
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𝑂 1 + 𝜌 𝑔 + 𝜀 log 𝑟
log [2] 𝑇(𝑟, 𝑓𝑜 𝑔)
≤
.
log [2] 𝑇(𝑟, 𝐹)
𝜌 − 𝜀 log 𝑟
𝑓
Since 𝜀(> 0) is arbitrary, it follows from above that
lim
inf
𝑟 →∞
𝜌𝑔
log [2] 𝑇(𝑟, 𝑓𝑜 𝑔)
≤ .
[2] 𝑇(𝑟, 𝐹)
log
𝜌𝑓
(31)
Now from (28) and (31), we get that
lim
inf
𝑟 →∞
𝜆𝑔 𝜌𝑔
log [2] 𝑇(𝑟, 𝑓𝑜 𝑔)
≤ min{ , }.
[2] 𝑇(𝑟, 𝐹)
log
𝜆𝑓 𝜌 𝑓
This proves the theorem.
The following theorem is a natural consequence of Theorem 4 and Theorem 5:
Theorem 6 Let f and g be any two entire functions such that
𝑖 0 < 𝜆 𝑓 < 𝜌 𝑓 < ∞, 𝑖𝑖 0 < 𝜆 𝑓 ≤ 𝜌 𝑓 < ∞, 𝑖𝑖𝑖 0 < 𝜆 𝑔 ≤ 𝜌 𝑔 < ∞ and
𝑖𝑣 𝑓𝑜𝑟 𝑛 ≥ 1, 𝐹 = 𝑓 𝑛 𝑄 𝑓 . Then
𝜆𝑔 𝜌𝑔
𝜆𝑔 𝜌𝑔
log [2] 𝑇(𝑟, 𝑓𝑜 𝑔)
lim
inf
≤ min{ , } ≤ max{ , } ≤
𝑟→∞
[2] 𝑇(𝑟, 𝐹)
log
𝜆𝑓 𝜌𝑓
𝜆𝑓 𝜌𝑓
lim
sup
𝑟→∞
log [2] 𝑇(𝑟, 𝑓𝑜 𝑔)
.
log [2] 𝑇(𝑟, 𝐹)
Theorem 7 Let f be meromorphic and g be entire satisfying 0 < 𝜆 𝑓 ≤ 𝜌 𝑓 < ∞,𝜌 𝑔 > 0 and also let for 𝑛 ≥ 1,
𝐹 = 𝑓 𝑛 𝑄[𝑓]. Then
[2]
𝑇(exp 𝑟 𝜌 𝑔 , 𝑓𝑜 𝑔)
lim
sup log
= ∞,
𝑟 →∞
log [2] 𝑇(exp 𝑟 𝜇 , 𝐹)
where 0 < 𝜇 < 𝜌 𝑔 .
Proof. Let 0 < 𝜇 ′ < 𝜌 𝑔 . Then in view of Lemma 2, we get for a sequence of values of r tending to infinity that
log 𝑇(𝑟, 𝑓𝑜 𝑔) ≥ log 𝑇(exp(𝑟 𝜇 ) , 𝑓)
i.e.,
′
log 𝑇(𝑟, 𝑓𝑜 𝑔) ≥ 𝜆 𝑓 − 𝜀 log{exp(𝑟 𝜇 )}
i.e.,
′
log 𝑇(𝑟, 𝑓𝑜 𝑔) ≥ 𝜆 𝑓 − 𝜀 𝑟 𝜇
i.e.,
log [2] 𝑇 𝑟, 𝑓𝑜 𝑔 ≥ 𝑂 1 + 𝜇 ′ log 𝑟 .
(32)
Again in view of Lemma 3, we have for all sufficiently large values of r,
log 𝑇(exp 𝑟 𝜇 , 𝐹) ≤ 𝜌 𝐹 + 𝜀 log{exp(𝑟 𝜇 )}
i.e.,
log 𝑇(exp 𝑟 𝜇 , 𝐹) ≤ 𝜌 𝑓 + 𝜀 𝑟 𝜇
i.e.,
log 𝑇(exp 𝑟 𝜇 , 𝐹) ≤ 𝑂 1 + 𝜇 log 𝑟 .
(33)
Now combining (32) and (33), we obtain for a sequence of values of r tending to infinity that
log [2] 𝑇(exp 𝑟 𝜌 𝑔 , 𝑓𝑜 𝑔)
𝑂 1 + 𝜇′ 𝑟 𝜌 𝑔
≥
log [2] 𝑇(exp 𝑟 𝜇 , 𝐹)
𝑂 1 + 𝜇 log 𝑟
from which the theorem follows.
Remark 2 The condition 𝜌 𝑔 > 0 is necessary in Theorem 7 as we see in the following example.
Example 2 Let 𝑓 = exp 𝑧 , 𝑔 = 𝑧 and 𝜇 = 1 > 0 . Then 𝑓𝑜 𝑔 = exp 𝑧 and for 𝑛 ≥ 1, 𝐹 = 𝑓 𝑛 𝑄 𝑓 . Taking
𝑛 = 1, 𝐴 𝑗 = 1, 𝑛0𝑗 = 1 and 𝑛1𝑗 = ⋯ = 𝑛 𝑘𝑗 = 0; we see that 𝐹 = exp 2𝑧. Then
log [2] 𝑀(𝑟, 𝑓)
= 1,
log 𝑟
log [2] 𝑀(𝑟, 𝑓)
inf
𝜆 𝑓 = lim𝑟→∞
=1
log 𝑟
𝜌𝑓 =
lim
sup
𝑟→∞
𝜌𝑔 =
lim
sup
𝑟→∞
and
Also we get that
log [2] 𝑀(𝑟, 𝑔)
= 0.
log 𝑟
𝑟
𝑇 𝑟, 𝑓 = .
𝜋
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ISSN: 2249-6645
Therefore
𝑇 exp 𝑟 𝜌 𝑔 , 𝑓𝑜 𝑔 =
𝑒
𝜋
and
𝑇 exp 𝑟 𝜇 , 𝐹 =
2 exp 𝑟
.
𝜋
So from the above two expressions we obtain that
log [2] 𝑇(exp 𝑟 𝜌 𝑔 , 𝑓𝑜 𝑔)
𝑂(1)
=
[2] 𝑇(exp 𝑟 𝜇 , 𝐹)
log
log 𝑟 + 𝑂(1)
i.e.,
[2]
𝑇(exp 𝑟 𝜌 𝑔 , 𝑓𝑜 𝑔)
lim
sup log
= 0,
𝑟 →∞
[2] 𝑇(exp 𝑟 𝜇 , 𝐹)
log
which contradicts Theorem 7.
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[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
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Bergweiler, W.: On the growth rate of composite meromorphic functions, Complex Variables, Vol. 14 (1990), pp. 187-196.
Bhooshnurmath, S. S. and Prasad, K. S. L. N.: The value distribution of some differential polynomials, Bull. Cal. Math. Soc., Vol.
101, No. 1 (2009), pp. 55- 62.
Doeringer, W.: Exceptional values of differential polynomials, Pacific J. Math., Vol. 98, No. 1 (1982), pp. 55-62.
Hayman, W. K.: Meromorphic functions, The Clarendon Press, Oxford, 1964.
Lin, Q. and Dai, C.: On a conjecture of Shah concerning small functions, Kexue Tongbao (English Ed.), Vol. 31, No. 4 (1986),
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Lahiri, I.: Generalised proximate order of meromorphic functions, Mat. Vensik, Vol. 41 (1989), pp. 9-16.
Niino, K. and Yang, C. C.: Some growth relationships on factors of two composite entire functions, Factorization Theory of
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Valiron, G.:
Lectures on the general theory of integral functions, Chelsea Publishing Company, 1949.
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