Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year.
This document summarizes research on obtaining inequalities for the Hausdorff dimension in the Ricker population model. It defines Hausdorff measure and dimension, and outlines two propositions that are used to calculate the Hausdorff dimension of the attractor for the Ricker model. The main results derive upper and lower bounds for the dimension by taking the maximum and minimum values of the inverse functions in the model. The conclusion states that considering fixed points may further constrain the bounds and provide a better estimate of the Hausdorff dimension.
A tutorial on the Frobenious Theorem, one of the most important results in differential geometry, with emphasis in its use in nonlinear control theory. All results are accompanied by proofs, but for a more thorough and detailed presentation refer to the book of A. Isidori.
This research statement summarizes Susovan Pal's postdoctoral research in two areas: 1) Regularity and asymptotic conformality of quasiconformal minimal Lagrangian diffeomorphic extensions of quasisymmetric circle homeomorphisms. This focuses on proving these extensions are asymptotically conformal if the boundary maps are symmetric. 2) Discrete geometry of left conformally natural homeomorphisms of the unit disk from a discrete viewpoint. This constructs homeomorphisms between polygons in the disk that preserve a weighted minimal distance property. The goal is to show these homeomorphisms converge to a continuous one.
This document discusses continuity and the Intermediate Value Theorem (IVT) in mathematics. It defines continuity, examines examples of continuous and discontinuous functions, and establishes theorems about continuity. The IVT states that if a function is continuous on a closed interval and takes on intermediate values within its range, there exists at least one value in the domain where the function value is intermediate. An example proves the existence of the square root of two using the IVT and bisection method.
This document discusses partial derivatives of functions with multiple variables. It defines partial derivatives as derivatives of a function where all but one variable is held constant. For a function z=f(x,y), the partial derivatives with respect to x and y are defined. Higher order partial derivatives and partial derivatives of functions with more than two variables are also introduced. Examples are provided to demonstrate calculating first and second order partial derivatives.
Continuity of functions by graph (exercises with detailed solutions)Tarun Gehlot
The document contains 11 exercises analyzing the continuity of various functions. It begins by verifying the continuity of square root and rational functions at specific points. Later exercises involve determining the domains of piecewise functions and studying their continuity by analyzing limits. Graphs are drawn to illustrate discontinuity points for functions involving floor, trigonometric, and fractional expressions. The solutions find appropriate definitions or parameters to extend functions to continuous forms at boundaries between pieces.
Lesson 20: Derivatives and the Shapes of Curves (slides)Matthew Leingang
This document contains lecture notes on derivatives and the shapes of curves from a Calculus I class taught by Professor Matthew Leingang at New York University. The notes cover using derivatives to determine the intervals where a function is increasing or decreasing, classifying critical points as maxima or minima, using the second derivative to determine concavity, and applying the first and second derivative tests. Examples are provided to illustrate finding intervals of monotonicity for various functions.
The document discusses using the derivative to determine whether a function is increasing or decreasing over an interval. It provides examples of using the sign of the derivative to determine if a function is increasing or decreasing. It also discusses using the second derivative test to determine if a stationary point is a relative maximum or minimum. Specifically:
- The sign of the derivative indicates whether the function is increasing or decreasing over an interval. Positive derivative means increasing, negative means decreasing.
- Stationary points where the derivative is zero require the second derivative test to determine if it is a relative maximum or minimum. Positive second derivative means a relative minimum, negative means a maximum.
- Examples demonstrate finding stationary points, using the first and second derivative
This document discusses the concept of the derivative and differentiation. It begins by explaining how the slope of a curve changes at different points, unlike the constant slope of a line. It then defines the derivative of a function f at a point x0 as the limit of the difference quotient as h approaches 0. If this limit exists, then f is said to be differentiable at x0. The derivative f'(x) then represents the slope of the curve y=f(x) at each point x and is a measure of how steeply the curve is rising or falling at that point. Several examples are provided to illustrate how to compute derivatives using this limit definition.
A tutorial on the Frobenious Theorem, one of the most important results in differential geometry, with emphasis in its use in nonlinear control theory. All results are accompanied by proofs, but for a more thorough and detailed presentation refer to the book of A. Isidori.
This research statement summarizes Susovan Pal's postdoctoral research in two areas: 1) Regularity and asymptotic conformality of quasiconformal minimal Lagrangian diffeomorphic extensions of quasisymmetric circle homeomorphisms. This focuses on proving these extensions are asymptotically conformal if the boundary maps are symmetric. 2) Discrete geometry of left conformally natural homeomorphisms of the unit disk from a discrete viewpoint. This constructs homeomorphisms between polygons in the disk that preserve a weighted minimal distance property. The goal is to show these homeomorphisms converge to a continuous one.
This document discusses continuity and the Intermediate Value Theorem (IVT) in mathematics. It defines continuity, examines examples of continuous and discontinuous functions, and establishes theorems about continuity. The IVT states that if a function is continuous on a closed interval and takes on intermediate values within its range, there exists at least one value in the domain where the function value is intermediate. An example proves the existence of the square root of two using the IVT and bisection method.
This document discusses partial derivatives of functions with multiple variables. It defines partial derivatives as derivatives of a function where all but one variable is held constant. For a function z=f(x,y), the partial derivatives with respect to x and y are defined. Higher order partial derivatives and partial derivatives of functions with more than two variables are also introduced. Examples are provided to demonstrate calculating first and second order partial derivatives.
Continuity of functions by graph (exercises with detailed solutions)Tarun Gehlot
The document contains 11 exercises analyzing the continuity of various functions. It begins by verifying the continuity of square root and rational functions at specific points. Later exercises involve determining the domains of piecewise functions and studying their continuity by analyzing limits. Graphs are drawn to illustrate discontinuity points for functions involving floor, trigonometric, and fractional expressions. The solutions find appropriate definitions or parameters to extend functions to continuous forms at boundaries between pieces.
Lesson 20: Derivatives and the Shapes of Curves (slides)Matthew Leingang
This document contains lecture notes on derivatives and the shapes of curves from a Calculus I class taught by Professor Matthew Leingang at New York University. The notes cover using derivatives to determine the intervals where a function is increasing or decreasing, classifying critical points as maxima or minima, using the second derivative to determine concavity, and applying the first and second derivative tests. Examples are provided to illustrate finding intervals of monotonicity for various functions.
The document discusses using the derivative to determine whether a function is increasing or decreasing over an interval. It provides examples of using the sign of the derivative to determine if a function is increasing or decreasing. It also discusses using the second derivative test to determine if a stationary point is a relative maximum or minimum. Specifically:
- The sign of the derivative indicates whether the function is increasing or decreasing over an interval. Positive derivative means increasing, negative means decreasing.
- Stationary points where the derivative is zero require the second derivative test to determine if it is a relative maximum or minimum. Positive second derivative means a relative minimum, negative means a maximum.
- Examples demonstrate finding stationary points, using the first and second derivative
This document discusses the concept of the derivative and differentiation. It begins by explaining how the slope of a curve changes at different points, unlike the constant slope of a line. It then defines the derivative of a function f at a point x0 as the limit of the difference quotient as h approaches 0. If this limit exists, then f is said to be differentiable at x0. The derivative f'(x) then represents the slope of the curve y=f(x) at each point x and is a measure of how steeply the curve is rising or falling at that point. Several examples are provided to illustrate how to compute derivatives using this limit definition.
This document defines and provides examples of several concepts relating to metric spaces and functions between metric spaces:
- A metric space is a non-empty set with a distance function satisfying four properties.
- A set is countable if it is finite or equivalent to the natural numbers, and uncountable otherwise.
- A function between metric spaces is continuous if small changes in the input result in small changes in the output.
- A function is uniformly continuous if it is continuous with respect to all inputs simultaneously.
- A metric space is connected if it cannot be represented as the union of two disjoint open sets.
This document contains notes from a calculus class lecture on evaluating definite integrals. It discusses using the evaluation theorem to evaluate definite integrals, writing derivatives as indefinite integrals, and interpreting definite integrals as the net change of a function over an interval. The document also contains examples of evaluating definite integrals, properties of integrals, and an outline of the key topics covered.
This document defines and explains partial derivatives. It begins by defining a partial derivative as the rate of change of a function with respect to one variable, holding other variables fixed. It then covers notation, calculating partial derivatives, interpreting them geometrically and as rates of change, higher derivatives, and applications to partial differential equations.
PRESENTATION ON INTRODUCTION TO SEVERAL VARIABLES AND PARTIAL DERIVATIVES Mazharul Islam
This document provides an introduction to partial derivatives and several examples of calculating them. It begins by defining partial derivatives as the rate of change of a function with respect to one variable, holding other variables constant. Several examples are then provided of calculating partial derivatives of multivariable functions. The document concludes by stating the chain rule for partial derivatives, which relates the derivative of a composite function to its constituent partial derivatives.
A factorization theorem for generalized exponential polynomials with infinite...Pim Piepers
The document presents a factorization theorem for a class of generalized exponential polynomials called polynomial-exponent exponential polynomials (pexponential polynomials). The theorem states that if a pexponential polynomial F(x) has infinitely many integer zeros belonging to a finite union of arithmetic progressions, then F(x) can be factorized into a product of factors corresponding to the zeros in each progression multiplied by a pexponential polynomial with only finitely many integer zeros. The proof relies on two lemmas showing that certain polynomial sums in the components of F(x) vanish for integers in the progressions.
The document discusses evaluating definite integrals. It begins by reviewing the definition of the definite integral as a limit. It then discusses estimating integrals using the midpoint rule and properties of integrals such as integrals of nonnegative functions being nonnegative and integrals being "increasing" if one function is greater than another. An example is worked out using the midpoint rule to estimate an integral. The document provides an outline of topics and notation for integrals.
The concept of limit formalizes the notion of closeness of the function values to a certain value "near" a certain point. Limits behave well with respect to arithmetic--usually. Division by zero is always a problem, and we can't make conclusions about nonexistent limits!
On Extendable Sets in the Reals (R) With Application to the Lyapunov Stabilit...BRNSS Publication Hub
This document summarizes a research article that defines extendable sets in the real numbers (R) and applies this concept to the Lyapunov stability comparison principle of ordinary differential equations. It begins with the author's own definition of extension on R and a basic result called the basic extension fact for R. It then reviews existing definitions and theorems on extension, including Urysohn's lemma and Tietze's extension theorem. The document concludes by extensively applying these results to prove some important results relating to the comparison principle of Lyapunov stability theory in ordinary differential equations.
1) The document provides an overview of continuity, including defining continuity as a function having a limit equal to its value at a point.
2) It discusses several theorems related to continuity, such as the sum of continuous functions being continuous and various trigonometric, exponential, and logarithmic functions being continuous on their domains.
3) The document also covers inverse trigonometric functions and their domains of continuity.
We define the definite integral as a limit of Riemann sums, compute some approximations, then investigate the basic additive and comparative properties
The document provides an overview of the EM algorithm and Jensen's inequality. It can be summarized as follows:
1) The EM algorithm is an approach for maximum likelihood estimation involving latent variables. It alternates between estimating the latent variables (E-step) and maximizing the likelihood based on those estimates (M-step).
2) Jensen's inequality states that for a convex function f, the expected value of f(X) is greater than or equal to f of the expected value of X.
3) The EM algorithm derives a lower bound on the log-likelihood using Jensen's inequality, with the latent variable distribution Q chosen to make the bound tight. It then alternates between tightening the bound
The document discusses the Divergence Theorem and provides examples of using it to calculate the outward flux of a vector field across the boundary of a region. Specifically:
- The Divergence Theorem states that the total outward flux of a vector field F across the boundary of a region equals the triple integral of the divergence of F over the region.
- Example A calculates the outward flux of F(x,y,z)=1k across a unit cube to be 0 using both approaches.
- Example B calculates the outward flux of F(x,y,z)=xi+yj+zk across a sphere to be 4πr^3 using the divergence approach.
- Example C calculates
At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.
The document discusses an introductory calculus class and provides announcements about homework due dates and a student survey. It also outlines guidelines for written homework assignments, the grading rubric, and examples of what to include and avoid in written work. The document aims to provide students information about course policies and expectations for written assignments.
This presentation discusses applications of derivatives including: extreme values of functions, the mean value theorem, monotonic functions, and concavity. It defines maximum and minimum values and explains how the mean value theorem states that between two points a and b on a continuous function, there exists a point c where the slope of the tangent line is equal to the slope of the secant line between a and b. Monotonic functions are defined as increasing or decreasing functions based on the mean value theorem. Concavity is defined based on whether the derivative of a function is increasing or decreasing, which determines if the graph is concave up or down.
There are various reasons why we would want to find the extreme (maximum and minimum values) of a function. Fermat's Theorem tells us we can find local extreme points by looking at critical points. This process is known as the Closed Interval Method.
Lesson 14: Derivatives of Logarithmic and Exponential Functions (slides)Matthew Leingang
The exponential function is pretty much the only function whose derivative is itself. The derivative of the natural logarithm function is also beautiful as it fills in an important gap. Finally, the technique of logarithmic differentiation allows us to find derivatives without the product rule.
1) The document discusses uncertainty principles and how precisely controlling one variable leads to less control over its transform, and vice versa. It uses the example of a "box" function and its Fourier transform to illustrate this tradeoff.
2) It shows how keeping the integral of the box function constant forces the function to approach a Dirac delta, making its Fourier transform very spread out. Keeping the height of the box constant forces its Fourier transform to have a constant integral.
3) The document generalizes these ideas to discuss how gaining precision in one domain leads to loss in another, and examines implications for probability distributions and their transforms.
This document provides an outline for a calculus lecture on basic differentiation rules. It includes objectives to understand key rules like the constant multiple rule, sum rule, and derivatives of sine and cosine. Examples are worked through to find the derivatives of functions like squaring, cubing, square root, and cube root using the definition of the derivative. Graphs and properties of derived functions are also discussed.
This document summarizes an experimental study on the effect of granite powder on the strength properties of concrete. Some key points:
- Granite powder was used to partially replace river sand in concrete mixtures at levels of 0%, 25%, 50%, 75%, and 100%. Cement was also partially replaced with silica fume (7.5%), fly ash (10%), and slag (10%).
- Tests were conducted to determine the compressive, split tensile, and flexural strengths as well as modulus of elasticity and water absorption of the concrete mixtures.
- The results showed that concrete mixtures with 25% granite powder replacement together with the admixtures achieved the highest strength. Therefore, granite powder can
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
This document defines and provides examples of several concepts relating to metric spaces and functions between metric spaces:
- A metric space is a non-empty set with a distance function satisfying four properties.
- A set is countable if it is finite or equivalent to the natural numbers, and uncountable otherwise.
- A function between metric spaces is continuous if small changes in the input result in small changes in the output.
- A function is uniformly continuous if it is continuous with respect to all inputs simultaneously.
- A metric space is connected if it cannot be represented as the union of two disjoint open sets.
This document contains notes from a calculus class lecture on evaluating definite integrals. It discusses using the evaluation theorem to evaluate definite integrals, writing derivatives as indefinite integrals, and interpreting definite integrals as the net change of a function over an interval. The document also contains examples of evaluating definite integrals, properties of integrals, and an outline of the key topics covered.
This document defines and explains partial derivatives. It begins by defining a partial derivative as the rate of change of a function with respect to one variable, holding other variables fixed. It then covers notation, calculating partial derivatives, interpreting them geometrically and as rates of change, higher derivatives, and applications to partial differential equations.
PRESENTATION ON INTRODUCTION TO SEVERAL VARIABLES AND PARTIAL DERIVATIVES Mazharul Islam
This document provides an introduction to partial derivatives and several examples of calculating them. It begins by defining partial derivatives as the rate of change of a function with respect to one variable, holding other variables constant. Several examples are then provided of calculating partial derivatives of multivariable functions. The document concludes by stating the chain rule for partial derivatives, which relates the derivative of a composite function to its constituent partial derivatives.
A factorization theorem for generalized exponential polynomials with infinite...Pim Piepers
The document presents a factorization theorem for a class of generalized exponential polynomials called polynomial-exponent exponential polynomials (pexponential polynomials). The theorem states that if a pexponential polynomial F(x) has infinitely many integer zeros belonging to a finite union of arithmetic progressions, then F(x) can be factorized into a product of factors corresponding to the zeros in each progression multiplied by a pexponential polynomial with only finitely many integer zeros. The proof relies on two lemmas showing that certain polynomial sums in the components of F(x) vanish for integers in the progressions.
The document discusses evaluating definite integrals. It begins by reviewing the definition of the definite integral as a limit. It then discusses estimating integrals using the midpoint rule and properties of integrals such as integrals of nonnegative functions being nonnegative and integrals being "increasing" if one function is greater than another. An example is worked out using the midpoint rule to estimate an integral. The document provides an outline of topics and notation for integrals.
The concept of limit formalizes the notion of closeness of the function values to a certain value "near" a certain point. Limits behave well with respect to arithmetic--usually. Division by zero is always a problem, and we can't make conclusions about nonexistent limits!
On Extendable Sets in the Reals (R) With Application to the Lyapunov Stabilit...BRNSS Publication Hub
This document summarizes a research article that defines extendable sets in the real numbers (R) and applies this concept to the Lyapunov stability comparison principle of ordinary differential equations. It begins with the author's own definition of extension on R and a basic result called the basic extension fact for R. It then reviews existing definitions and theorems on extension, including Urysohn's lemma and Tietze's extension theorem. The document concludes by extensively applying these results to prove some important results relating to the comparison principle of Lyapunov stability theory in ordinary differential equations.
1) The document provides an overview of continuity, including defining continuity as a function having a limit equal to its value at a point.
2) It discusses several theorems related to continuity, such as the sum of continuous functions being continuous and various trigonometric, exponential, and logarithmic functions being continuous on their domains.
3) The document also covers inverse trigonometric functions and their domains of continuity.
We define the definite integral as a limit of Riemann sums, compute some approximations, then investigate the basic additive and comparative properties
The document provides an overview of the EM algorithm and Jensen's inequality. It can be summarized as follows:
1) The EM algorithm is an approach for maximum likelihood estimation involving latent variables. It alternates between estimating the latent variables (E-step) and maximizing the likelihood based on those estimates (M-step).
2) Jensen's inequality states that for a convex function f, the expected value of f(X) is greater than or equal to f of the expected value of X.
3) The EM algorithm derives a lower bound on the log-likelihood using Jensen's inequality, with the latent variable distribution Q chosen to make the bound tight. It then alternates between tightening the bound
The document discusses the Divergence Theorem and provides examples of using it to calculate the outward flux of a vector field across the boundary of a region. Specifically:
- The Divergence Theorem states that the total outward flux of a vector field F across the boundary of a region equals the triple integral of the divergence of F over the region.
- Example A calculates the outward flux of F(x,y,z)=1k across a unit cube to be 0 using both approaches.
- Example B calculates the outward flux of F(x,y,z)=xi+yj+zk across a sphere to be 4πr^3 using the divergence approach.
- Example C calculates
At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.
The document discusses an introductory calculus class and provides announcements about homework due dates and a student survey. It also outlines guidelines for written homework assignments, the grading rubric, and examples of what to include and avoid in written work. The document aims to provide students information about course policies and expectations for written assignments.
This presentation discusses applications of derivatives including: extreme values of functions, the mean value theorem, monotonic functions, and concavity. It defines maximum and minimum values and explains how the mean value theorem states that between two points a and b on a continuous function, there exists a point c where the slope of the tangent line is equal to the slope of the secant line between a and b. Monotonic functions are defined as increasing or decreasing functions based on the mean value theorem. Concavity is defined based on whether the derivative of a function is increasing or decreasing, which determines if the graph is concave up or down.
There are various reasons why we would want to find the extreme (maximum and minimum values) of a function. Fermat's Theorem tells us we can find local extreme points by looking at critical points. This process is known as the Closed Interval Method.
Lesson 14: Derivatives of Logarithmic and Exponential Functions (slides)Matthew Leingang
The exponential function is pretty much the only function whose derivative is itself. The derivative of the natural logarithm function is also beautiful as it fills in an important gap. Finally, the technique of logarithmic differentiation allows us to find derivatives without the product rule.
1) The document discusses uncertainty principles and how precisely controlling one variable leads to less control over its transform, and vice versa. It uses the example of a "box" function and its Fourier transform to illustrate this tradeoff.
2) It shows how keeping the integral of the box function constant forces the function to approach a Dirac delta, making its Fourier transform very spread out. Keeping the height of the box constant forces its Fourier transform to have a constant integral.
3) The document generalizes these ideas to discuss how gaining precision in one domain leads to loss in another, and examines implications for probability distributions and their transforms.
This document provides an outline for a calculus lecture on basic differentiation rules. It includes objectives to understand key rules like the constant multiple rule, sum rule, and derivatives of sine and cosine. Examples are worked through to find the derivatives of functions like squaring, cubing, square root, and cube root using the definition of the derivative. Graphs and properties of derived functions are also discussed.
This document summarizes an experimental study on the effect of granite powder on the strength properties of concrete. Some key points:
- Granite powder was used to partially replace river sand in concrete mixtures at levels of 0%, 25%, 50%, 75%, and 100%. Cement was also partially replaced with silica fume (7.5%), fly ash (10%), and slag (10%).
- Tests were conducted to determine the compressive, split tensile, and flexural strengths as well as modulus of elasticity and water absorption of the concrete mixtures.
- The results showed that concrete mixtures with 25% granite powder replacement together with the admixtures achieved the highest strength. Therefore, granite powder can
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
This document provides a syllabus for an AP U.S. History course. The course is designed to provide a college-level experience and prepare students for the AP exam in May. It meets multiple times per week for class sessions. The course will examine the evolution of American history and identity from pre-Columbian societies to the present day. Students will develop skills in critical thinking, document analysis, and essay writing. They will study themes of American politics, economics, and foreign policy over time. The syllabus outlines course objectives, topics, assignments, and assessments that will be covered each period of American history.
1) The document describes a method developed for implementing a mass customization framework in small and medium manufacturing companies. It involves developing a strategic vision and training material to educate personnel across key functions like production, product development, and supply chain management.
2) The method was tested through pilot projects with two manufacturing companies. It began by analyzing each company's current "mass customization as-is status" and identifying challenges. A "mass customization mountain" model was used to visualize the framework and guide development.
3) Training materials like a video animation and digital learning content were created to explain how mass customization affects different business functions and the overall process. The goal was to help employees understand the framework and how their
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
This document proposes a doubly secured authentication scheme using the RKO technique of visual cryptography. It involves splitting a user's photo and signature image into shares during registration. These shares are sent to the user's email and stored in a database. During login, the user submits their shares which are overlapped with the bank's shares using XOR operation. If the reconstructed photo and signature match the originals, access is granted. The scheme improves security over password-based methods by requiring two biometric factors and preventing unauthorized login using mismatched shares. It was implemented using Java and the RKO technique achieved perfect reconstruction quality without data loss. The proposed scheme enhances authentication security for applications like online banking.
This document discusses research on heat source/sink effects on magnetohydrodynamic mixed convection boundary layer flow over a vertical permeable plate embedded in a porous medium saturated with a nanofluid. Numerical solutions are obtained for the governing similarity equations using a shooting method. Results show that imposition of suction increases velocity profiles and delays boundary layer separation, while injection decreases velocity profiles. Dual solutions exist for opposing flow with different nanoparticles, with upper branch solutions being physically stable. Suction also delays flow separation compared to impermeable or injection cases.
This document describes a proposed tool called Warehouse Creator that can automatically generate data warehouses from heterogeneous data sources within an enterprise. The tool extracts data from various data sources like databases and files, integrates the data by generating dimension and fact tables, and provides a web interface for users to search and retrieve information from the warehouse without needing direct access to the underlying data sources. The tool aims to address issues like the need for users to have detailed knowledge of different data sources and query languages by providing a centralized warehouse that integrates data from multiple sources.
Similar to Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year.
The document discusses membrane harmonics and the Helmholtz equation. It begins by considering the one-dimensional Helmholtz equation on an interval, finding the eigenfunctions and eigenvalues. It then extends this to the two-dimensional case on a rectangle using separation of variables, obtaining eigenfunctions that are products of sine waves and eigenvalues that are sums of the one-dimensional eigenvalues.
This document discusses counterexamples to classical calculus theorems like the Inverse and Implicit Function Theorems in the context of Scale Calculus. Scale Calculus generalizes multivariable calculus to infinite-dimensional spaces and is foundational to Polyfold Theory. The authors construct examples of scale-differentiable and scale-Fredholm maps whose differentials are discontinuous when the base point changes, disproving naive extensions of the classical theorems. However, they show the scale-Fredholm notion in Polyfold Theory ensures continuity of differentials in specific coordinates, justifying its technical complexity and allowing for a perturbation theory without the classical theorems.
On Analytic Review of Hahn–Banach Extension Results with Some GeneralizationsBRNSS Publication Hub
The useful Hahn–Banach theorem in functional analysis has significantly been in use for many years ago. At this point in time, we discover that its domain and range of existence can be extended point wisely so as to secure a wider range of extendibility. In achieving this, we initially reviewed the existing traditional Hahn–Banach extension theorem, before we carefully and successfully used it to generate the finite extension form as in main results of section three.
This document provides an overview and proofs of several theorems related to the Hahn-Banach theorem. It begins with an introduction to linear functionals and the Hahn-Banach theorem. It then presents two main theorems - the Hahn-Banach theorem and the topological Hahn-Banach theorem. The document provides proofs of these theorems and several related theorems using the Hahn-Banach extension lemma. It also discusses consequences of the Hahn-Banach extension form and provides proofs of the theorems using the lemma.
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Partial derivatives are used in vector calculus and differential geometry.
The document discusses concepts related to partial differentiation and its applications. It covers topics like tangent planes, linear approximations, differentials, Taylor expansions, maxima and minima problems, and the Lagrange method. Specifically, it defines the tangent plane to a surface at a point using partial derivatives, describes how to find the linear approximation of functions, and explains how to find maximum and minimum values of functions using critical points and the second derivative test.
This document outlines the contents of a Mathematics II course, including five units: vector calculus, Fourier series and Fourier transforms, interpolation and curve fitting, solutions to algebraic/transcendental equations and linear systems of equations, and numerical integration and solutions to differential equations. It lists three textbooks and four references used in the course. It then provides examples and explanations of key concepts from the first two units, including vector differential operators, gradient, divergence, curl, and Fourier series representations of functions.
MA500-2: Topological Structures 2016
Aisling McCluskey, Daron Anderson
[email protected], [email protected]
Contents
0 Preliminaries 2
1 Topological Groups 8
2 Morphisms and Isomorphisms 15
3 The Second Isomorphism Theorem 27
4 Topological Vector Spaces 42
5 The Cayley-Hamilton Theorem 43
6 The Arzelà-Ascoli theorem 44
7 Tychonoff ’s Theorem if Time Permits 45
Continuous assessment 30%; final examination 70%. There will be a weekly
workshop led by Daron during which there will be an opportunity to boost
continuous assessment marks based upon workshop participation as outlined in
class.
This module is self-contained; the notes provided shall form the module text.
Due to the broad range of topics introduced, there is no recommended text.
However General Topology by R. Engelking is a graduate-level text which has
relevant sections within it. Also Undergraduate Topology: a working textbook by
McCluskey and McMaster is a useful revision text. As usual, in-class discussion
will supplement the formal notes.
1
0 PRELIMINARIES
0 Preliminaries
Reminder 0.1. A topology τ on the set X is a family of subsets of X, called
the τ-open sets, satisfying the three axioms.
(1) Both sets X and ∅ are τ-open
(2) The union of any subfamily is again a τ-open set
(3) The intersection of any two τ-open sets is again a τ-open set
We refer to (X,τ) as a topological space. Where there is no danger of ambi-
guity, we suppress reference to the symbol denoting the topology (in this case,
τ) and simply refer to X as a topological space and to the elements of τ as its
open sets. By a closed set we mean one whose complement is open.
Reminder 0.2. A metric on the set X is a function d: X×X → R satisfying
the five axioms.
(1) d(x,y) ≥ 0 for all x,y ∈ X
(2) d(x,y) = d(y,x) for x,y ∈ X
(3) d(x,x) = 0 for every x ∈ X
(4) d(x,y) = 0 implies x = y
(5) d(x,z) ≤ d(x,y) + d(y,z) for all x,y,z ∈ X
Axiom (5) is often called the triangle inequality.
Definition 0.3. If d′ : X × X → R satisfies axioms (1), (2), (3) and (5) but
maybe not (4) then we call it a pseudo-metric.
Reminder 0.4. Every metric on X induces a topology on X, called the metric
topology. We define an open ball to be a set of the form
B(x,r) = {y ∈ X : d(x,y) < r}
for any x ∈ X and r > 0. Then a subset G of X is defined to be open (wrt the
metric topology) if for each x ∈ G, there is r > 0 such that B(x,r) ⊂ G. Thus
open sets are arbitrary unions of open balls.
Topological Structures 2016 2 Version 0.15
0 PRELIMINARIES
The definition of the metric topology makes just as much sense when we are
working with a pseudo-metric. Open balls are defined in the same manner, and
the open sets are exactly the unions of open balls. Pseudo-metric topologies are
often neglected because they do not have the nice property of being Hausdorff.
Reminder 0.5. Suppose f : X → Y is a function between the topological
spaces X and Y . We say f is continuous to mean that whenever U is open in
Y ...
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Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year.
1. RESEARCH INVENTY: International Journal of Engineering and Science
ISSN: 2278-4721, Vol. 1, Issue 7(November 2012), PP 42-45
www.researchinventy.com
Some Inequalities on the Hausdorff Dimension In
The Ricker Population Model
Dr. Nabajyoti Das ,
Assistant Professor, Department of Mathematics, Jawaharlal Nehru College,
Boko,Kamrup 781 123, Assam, India
Abstract : In this paper, we consider the Ricker population model , where r is the control
parameter and k is the carrying capacity, and obtain some interesting inequalities on the Hausdorff dimension. Our
idea can be extended to Higher dimensional models for further investigation in our research field.
Key Words: Hausdorff dimension / Ricker’s model / Fractals 2010subject classification : 37 G 15, 37 G 35 ,37 C
45
I. Introduction
One natural question with a dynamical system is: how chaotic is a system’s chaotic behavior ? To give a
quantitative answer to that question , the dimension theory plays a crucial role . Why is dimensionality important ?
One possible answer is that the dimensionality of the state space is closely related to dynamics. The dimensionality
is important in determin ing the range of possible dynamical behavior. Similarly, the dimensionality of an
attractor tells about the actual long- term dynamics. A mong various dimensions, Hausdorff dimension plays a very
important role as a fractal d imension and measure of chaos , [1,2,5,6,9].
We first define Hausdorff Measure as follo ws:
If U is any nonempty subset of n- dimensional Euclidean space, Rn , the diameter of U is defined as
U sup x y : x, y U , i.e. the greatest distance apart of any pair of points in U . If {Ui } is a countable
[or fin ite] collection of sets of distance at most that covers F , i.e. F i 1U i with 0 U i for each i
, we say that {Ui } is a cover of F Rn and < 1 ,
Suppose that F is a subset of Rn and s is a nonnegative number. For any > 0 , we define
s
H s ( F ) inf{ U i : {U i } is a - cover of F } (1.1 )
i 1
As decreases, the class of permissible covers of F in ( 1.1) is reduced. Therefore, the infimu m H s (F )
increases, and so approaches a limit as → 0 . We write
lim H F . We call
s s s
H (F) = H (F) the s- dimensional Hausdorff Measure of F . It is clear that for any
0
given set F , H s (F ) is non-increasing with s , so Hs (F) is also non- increasing. In fact, if t > s and {Ui } is a -
cover of F we have
U Ui U i t s U i
t t s s s
i
i i i
So, taking infima , H (F )
y t s
H s . Letting → 0 , we see that H s (F) < ∞ . Thus, Hs (F) = 0 fo r t > s .
Hence a graph of Hs (F) against s shows that there is a critical value of s at which Hs (F) ju mps from ∞ to 0 .
This crit ical value is called the Hausdorff dimension of F and written as dimH (F). Formally,
DimH (F) = inf { s ≥ 0 : Hs (F) =0 } = sup { s : Hs (F) = ∞ } so that
Hs (F) = ∞ if 0 ≤ s < dimH (F) and 0 if s > d imH (F) .
42
2. Some Inequalities On The Hausdorff Dimension In…
If s = dimH (F), then Hs (F) = may be zero or in fin ite , or may satisfy 0 < Hs (F) < ∞ .
Fig 1 : Graph of Hs (F) against s for a set F . The Haus dorff di mension is the val ue of s at which the
“jump” from ∞ to 0 occurs .
Some alternative equivalent definitions for Hausdorff dimension and the method of calculation are
available in [6] . It is important to note that most dimension calculations involve an upper estimate and a lower
estimate, which are hopefully equal. Each of these estimates usually involves a geometric observa tion follo wed by a
calculation , [3,6,7,10]
II. The Main Results:
CALCULATION OF HAUS DORFF DIMENS ION: [ 5,6,8]
Let us consider the model . We take the parameter value and the constant value in such
a way that the function becomes contracting. So as shown in the graph we can take x1 and x2 such that the function
becomes
Fig 2
. Being a unimodel function, if we consider the inverse of f it will
give us two values in that particular range. Let us consider the two inverse functions as s 1 (x) and s 2 (x) ,
where and . [fro m the figure 2 we can see the
domain and range.]
43
3. Some Inequalities On The Hausdorff Dimension In…
To get the Hausdroff dimension we use the following two propositions heavily:
A.Proposition 9.6 in the book fractal geometry by Kennith Falconer,[6] states:
Let F be the attractor of an IFS consists of contractions {s 1 .s2 ,….,s m} on a closed subset D of Rn such that
(x,y ) with o<ci <1 for each i. Then dimH F s , where
B.Proposition 9.7 in the book fractal geomet ry by Kennith Falconer,[] states:
Let F be the attractor of an IFS consists of contractions {s 1 .s2 ,….,s m} on a closed subset D of Rn such that
(x,y ) with o< b i <1 for each i.Then dimH F s , where
So we shall try to use these two theorems to calculate the Hausdroff dimension.
By their properties we have f(s i (x)) = x for i =1,2. So we have
(1.2) , [as f(x)=x er(1-x/k)] for i =1,2. Solving this equation we may get the
functions s i (x). However solving this type of nonlinear equation analytically is not so easy.
We differentiate equation (1.2) w.r.t x and we get
,fo r i=1,2
If we again d ifferentiate we have
… ( 1.3)
The maximu m or min imu m value occurs at s(x)= 2k/r, which is the end point of the definition of our function (if we
consider 2k/r as the end point). Clearly >0 for s 1 (x) (as s 1 (x) < k/r , we can see this fro m the above figure). So
s 1/ (x) is an increasing function. Hence the bounds are given by as follows
where s means s 1 .Again we have x1 >0 .Fro m this we get
… ( 1.4)
Again we consider the equation (1.4) for s = s 2.
As k/r<s 2 (x)<2k/r
That is , 1 < r s 2 (x)/k < 2
That is, -1>- r s 2 (x)/k >- 2 or, 0>1- r s 2 (x)/k >- 1
Hence for s 2 (x) .So s 2 / (x) is a decreasing function. Again we see s 2 (x) is also decreasing function. Hence
the bounds are given by
(1.5)
Where s means s 2 .
If 2k/r lies in the range of s 2 then the bounds becomes
Max and min of { ,-e(2- r)} (1.6)
The Hausdorff dimension of our model lies in between the maximu m and the min imu m values obtained from (1.6) .
III. Conclusion
We select the upper and lower bound for where s means s1 and s2 from propositions (A) and (B), say
c1 and c2 then the upper and lower bounds of the Hausdroff dimension will be {a1,a2} where 2.c1 a1 =1 and 2.c2a2
=1 . If we are not satisfied with the bounds then we may go for the composit e mappings like s 1s1 ,s1s 2 ,s 2s2,s 2s1.As s1
is an increasing function so s 1s 1, s 1s2. Moreover, as s2 is a decreasing function so is s 2s 2,s2 s1. Again s 1s2
:[s 2 (x2 ),s 2 (x1 )] [s 1s2 x2 ,s 1s 2 x1 ].For all these we need [s1 x1 ,s 1 x2 ] [x1 ,x2 ] unfortunately these does not happen all the
time. But if we take x1 and x2 as the fixed points then there will be no problem, again in our case 0 is one obvious
fixed point if we consider 0 then our function will no longer be contracting so we have to see other fixed points
other than 0 . So if we go with the fixed points then proceeding with the above procedure exactly in the same way
we may further contract the bounds, and hence get a better Hausdroff dimension.
44
4. Some Inequalities On The Hausdorff Dimension In…
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