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Our Journal has a We invite the entire statistical community to join us on this journey towards a more just, equitable, diverse, and inclusive future. By collectively embracing these principles, we can create a statistical landscape that reflects the richness of human experiences and contributes to a more robust and impactful field.
Published in association with the Utilitas Mathematica Academy. Offers selected original research in Pure and Applied Mathematics and Statistics. It enjoys a good reputation and popularity at the international level in terms of research papers and distribution worldwide.
https://utilitasmathematica.com/index.php/Index
Our Journal We believe that by embracing a wide range of backgrounds and viewpoints, we enrich the statistical discourse and enhance the quality and relevance of our research. Our commitment to inclusivity extends beyond the content of our publications.
https://utilitasmathematica.com/index.php/Index
Our Journal has a profession of statistics can only thrive when justice, equity, diversity, and inclusion are not just ideals but integral components of our ethos. We recognize that a diverse and inclusive community sparks creativity, innovation, and a richness of perspectives that is indispensable for advancing statistical knowledge.
1) Sorting algorithms are useful for keeping data ordered to aid other algorithms like searching. Finding the optimal order for matrix multiplication can greatly reduce computation time when transformations don't need to be performed immediately.
2) The inverse of a matrix is unique. The product of two lower triangular matrices is lower triangular. The inverse of a lower triangular matrix is also lower triangular.
3) Gaussian elimination preserves the inverse of a matrix up to row operations on the inverse. If a matrix has only real elements, then its inverse and cofactors must also be real.
Numerical Methods in Mechanical Engineering - Final ProjectStasik Nemirovsky
Final Project for the class of "Numerical Methods in Mechanical Engineering" - MECH 309.
In this project, various engineering problems were analyzed and solved using advanced numerical approximation methods and MATLAB software.
The document presents a decomposition method for solving indefinite quadratic programming problems with n variables and m linear constraints. The method decomposes the original problem into at most m subproblems, each with dimension n-1 and m linear constraints. All global minima, isolated local minima, and some non-isolated local minima of the original problem can be obtained by combining the solutions of the subproblems. The subproblems can then be further decomposed into smaller subproblems until 1-dimensional subproblems are reached, which can be solved directly.
The document discusses the gauge integral, a generalization of the Riemann integral. It defines a gauge integral as a limit of Riemann sums taken over δ-fine partitions, where δ is a gauge function. It proves some basic properties of gauge integrals, including existence of δ-fine partitions, linearity, and a Cauchy criterion for integrability. The gauge integral includes the Riemann and Lebesgue integrals as special cases and has fewer technical assumptions than the Lebesgue integral.
Some Soliton Solutions of Non Linear Partial Differential Equations by Tan-Co...IOSR Journals
Tan-cot method is applied to get exact soliton solutions of non-linear partial differential equations notably generalized Benjamin-Bona-Mahony, Zakharov-Kuznetsov Benjamin-Bona-Mahony, Kadomtsov-Petviashvilli Benjamin-Bona-Mahony and Korteweg-de Vries equations, which are important evolution equations with wide variety of physical applications. Elastic behavior and soliton fusion/fission is shown graphically and discussed physically as far as possible
Published in association with the Utilitas Mathematica Academy. Offers selected original research in Pure and Applied Mathematics and Statistics. It enjoys a good reputation and popularity at the international level in terms of research papers and distribution worldwide.
https://utilitasmathematica.com/index.php/Index
Our Journal We believe that by embracing a wide range of backgrounds and viewpoints, we enrich the statistical discourse and enhance the quality and relevance of our research. Our commitment to inclusivity extends beyond the content of our publications.
https://utilitasmathematica.com/index.php/Index
Our Journal has a profession of statistics can only thrive when justice, equity, diversity, and inclusion are not just ideals but integral components of our ethos. We recognize that a diverse and inclusive community sparks creativity, innovation, and a richness of perspectives that is indispensable for advancing statistical knowledge.
1) Sorting algorithms are useful for keeping data ordered to aid other algorithms like searching. Finding the optimal order for matrix multiplication can greatly reduce computation time when transformations don't need to be performed immediately.
2) The inverse of a matrix is unique. The product of two lower triangular matrices is lower triangular. The inverse of a lower triangular matrix is also lower triangular.
3) Gaussian elimination preserves the inverse of a matrix up to row operations on the inverse. If a matrix has only real elements, then its inverse and cofactors must also be real.
Numerical Methods in Mechanical Engineering - Final ProjectStasik Nemirovsky
Final Project for the class of "Numerical Methods in Mechanical Engineering" - MECH 309.
In this project, various engineering problems were analyzed and solved using advanced numerical approximation methods and MATLAB software.
The document presents a decomposition method for solving indefinite quadratic programming problems with n variables and m linear constraints. The method decomposes the original problem into at most m subproblems, each with dimension n-1 and m linear constraints. All global minima, isolated local minima, and some non-isolated local minima of the original problem can be obtained by combining the solutions of the subproblems. The subproblems can then be further decomposed into smaller subproblems until 1-dimensional subproblems are reached, which can be solved directly.
The document discusses the gauge integral, a generalization of the Riemann integral. It defines a gauge integral as a limit of Riemann sums taken over δ-fine partitions, where δ is a gauge function. It proves some basic properties of gauge integrals, including existence of δ-fine partitions, linearity, and a Cauchy criterion for integrability. The gauge integral includes the Riemann and Lebesgue integrals as special cases and has fewer technical assumptions than the Lebesgue integral.
Some Soliton Solutions of Non Linear Partial Differential Equations by Tan-Co...IOSR Journals
Tan-cot method is applied to get exact soliton solutions of non-linear partial differential equations notably generalized Benjamin-Bona-Mahony, Zakharov-Kuznetsov Benjamin-Bona-Mahony, Kadomtsov-Petviashvilli Benjamin-Bona-Mahony and Korteweg-de Vries equations, which are important evolution equations with wide variety of physical applications. Elastic behavior and soliton fusion/fission is shown graphically and discussed physically as far as possible
Solution of second order nonlinear singular value problemsAlexander Decker
1) The document presents a method for finding solutions to second order nonlinear singular boundary value problems using Taylor series.
2) The method modifies the differential equation to handle singularities, then obtains recurrence relations for the Taylor series coefficients by taking derivatives at the singular point.
3) Examples are provided to demonstrate the method, yielding exact solutions for problems in astronomy modeling gas spheres and other physical applications.
On the Fixed Point Extension Results in the Differential Systems of Ordinary ...BRNSS Publication Hub
This document summarizes research on extending the domain of fixed points for ordinary differential equations. It begins with definitions of fixed points and extendability. It then establishes several theorems on extending fixed points, including using Peano's theorem on existence and Picard-Lindelof theorem on uniqueness to extend fixed points over open connected domains where the vector field is continuous. The document proves that if a fixed point is bounded on its domain and the limits at the endpoints exist, then the fixed point can be extended to those endpoints. It concludes by discussing extending fixed points defined on intervals to the whole positive real line using boundedness conditions.
I am Piers L. I am a Calculus Homework Expert at mathhomeworksolver.com. I hold a Master's in Mathematics from, the University of Adelaide. I have been helping students with their homework for the past 6 years. I solve homework related to Calculus.
Visit mathhomeworksolver.com or email support@mathhomeworksolver.com.
You can also call on +1 678 648 4277 for any assistance with Calculus Homework.
This document summarizes research on extending the domain of fixed point solutions to ordinary differential equations. It begins with definitions of fixed points and extendability. The Peano and Picard-Lindelof theorems are used to prove local existence and uniqueness of fixed points. Several theorems are presented to establish conditions under which a fixed point defined on a finite interval can be extended to the endpoints of that interval or beyond. Specifically, if the vector field is bounded and limits exist at the endpoints, the fixed point can be extended. The results ensure extendability to infinite intervals under boundedness assumptions. Extendability is key to studying existence of fixed points on the half line R+.
This document summarizes research on extending the domain of fixed point solutions to ordinary differential equations. It begins with definitions of fixed points and extendability. The Peano and Picard-Lindelof theorems are used to prove local existence and uniqueness of fixed points. Several theorems are presented to establish conditions under which a fixed point defined on a finite interval can be extended to the endpoints of the interval. Specifically, if the vector field is bounded and limits exist at the endpoints, the fixed point can be extended. The results are applied to extend fixed points defined on finite intervals to the entire positive real line.
I am Piers L. I am a Calculus Assignment Expert at mathsassignmenthelp.com. I hold a Master's in Mathematics from, the University of Adelaide. I have been helping students with their assignments for the past 6 years. I solve assignments related to Calculus.
Visit mathsassignmenthelp.com or email info@mathsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Calculus Assignment.
This document presents several theorems regarding the existence, uniqueness, and extendibility of solutions to ordinary differential equation systems. It begins by introducing Peano's theorem on existence of solutions and Picard-Lindelof theorem on uniqueness of solutions locally. It then discusses extension theorems which aim to extend local solutions to open connected domains. The main results section discusses using extension theorems to show existence of solutions on the entire positive real line by extending solutions from finite intervals to infinity.
In this paper, we give several new fixed point theorems to extend results [3]-[4] ,and we apply
the effective modification of He’s variation iteration method to solve some nonlinear and linear equations are
proceed to examine some a class of integral-differential equations and some partial differential equation, to
illustrate the effectiveness and convenience of this method(see[7]). Finally we have also discussed Berge type
equation with exact solution
11.solution of a subclass of singular second orderAlexander Decker
This document presents a method for solving a subclass of singular second order differential equations of Lane-Emden type using Taylor series. The method identifies conditions where the equations can be solved analytically as a polynomial function. Several examples from literature are provided and solved using the proposed Taylor series method to demonstrate its effectiveness. The method produces exact solutions in polynomial form and is shown to perform well on both linear and nonlinear boundary value problems.
This document discusses using the Taylor series method to solve a subclass of singular second order differential equations of Lane-Emden type.
The method is applied to solve boundary value problems that arise from modeling physical phenomena. The Taylor series method produces an analytic solution in the form of a polynomial function.
Several examples from literature are considered to demonstrate the suitability and viability of the Taylor series method for solving these types of differential equations.
This document describes how to compute Fourier series and power spectra using MATLAB. It begins with an introduction to Fourier analysis and how it represents signals in the frequency domain rather than the time domain. It then provides the mathematical foundations for Fourier series, describing how to compute the coefficients to represent a signal as a sum of sines and cosines. Several examples are worked through, including representing a square wave signal. The document concludes by explaining how to apply the same techniques to discrete data using the trapezoidal rule to approximate integrals.
The document discusses solving linear differential equations with variable coefficients using power series representations. It begins by introducing series solution methods and properties of infinite series. It then discusses the Method of Frobenius for solving equations with variable coefficients that arise in cylindrical and spherical coordinate systems. The method involves finding the indicial equation to determine the index c, and then using the recurrence relation to determine the series coefficients an. An example is provided to illustrate the method where the roots of the indicial equation are distinct and not differing by an integer.
The document summarizes an analytic approach to solving the Collatz conjecture proposed by Berg and Meinardus. It introduces two linear operators, U and V, acting on the space of holomorphic functions on the open unit disk. The kernel K of U and V, defined as the set of functions where both operators equal 0, is of main interest. If it can be shown that K equals the two-dimensional space Δ2, then the Collatz conjecture would be proven true. The individual kernel KV of V is already known from prior work. The paper aims to compute U[h] for h in KV and show that K = Δ2, which would imply the truth of the Collatz conjecture.
A Numerical Method For Friction Problems With Multiple ContactsJoshua Gorinson
This document summarizes a numerical method for solving friction problems involving multiple contact surfaces. It begins by reviewing previous work on solving differential equations with discontinuities. The author then describes extending their previous method to handle problems with multiple contacts. Indicator functions are used to represent the regions of contact. Linear complementarity problems (LCPs) are solved to determine changes in the active contact surfaces. The method assumes the indicator functions and vector fields are smooth. Convergence results are proven showing the method can achieve high-order accuracy.
The document discusses the Fundamental Theorem of Calculus, which states that differentiation and integration are inverse operations. It provides examples of using the theorem to evaluate definite integrals and find the average value of functions. The Second Fundamental Theorem of Calculus describes integrals as accumulation functions and relates the integral and derivative. The Net Change Theorem applies the Fundamental Theorem to relate the net change in a function to its integral over an interval. Examples demonstrate using these theorems to solve problems in physics, chemistry, and particle motion.
This document presents a closed-form solution for a class of discrete-time algebraic Riccati equations (DTAREs) under certain assumptions. It begins with background on Riccati equations and their importance in control theory. It then provides the assumptions considered, including that the A matrix eigenvalues are distinct. The main result is a closed-form solution for the DTARE when R=1. Extensions discussed include the solution's behavior as Q approaches zero and for repeated eigenvalues. Comparisons with numerical solutions verify the closed-form solution's accuracy.
A class of boundary value methods for the complex delay differential equationijscmcj
In this paper, a class of boundary value methods (BVMs) for delay differential equations (DDEs) is considered. The delay dependent stable regions of the extended trapezoidal rules of second kind (ETR2s), which are a class of BVMs, are displayed for the test equation of DDEs. Furthermore, it is showed ETR2s cannot preserve the delay-dependent stability of the complex coefficient test equation considered. Some numerical experiments are given to confirm the theoretical results.
AMS 2000 Mathematics Subject Classification: 65L20, 65M12
11.[8 17]numerical solution of fuzzy hybrid differential equation by third or...Alexander Decker
This document presents a numerical method called the third order Runge Kutta Nystrom method to solve fuzzy hybrid differential equations. It begins with definitions of fuzzy numbers and fuzzy functions. It then describes hybrid fuzzy differential systems and how they can be modeled as piecewise differentiable systems. The document goes on to develop the third order Runge Kutta Nystrom method for solving these systems numerically. It provides a convergence result and applies the method to a numerical example to illustrate the theory.
https://jst.org.in/index.html
Our journal has academic journals form a crucial nexus. Educators leverage the latest research findings to enrich their teaching methodologies, ensuring that students are exposed to the most current and relevant information. Simultaneously, these educators contribute to the body of knowledge through their own research, creating a perpetual cycle of growth.
https://ijaast.com/index.html
Our journal has open-access nature of IJAAST fosters global collaboration. Researchers from diverse geographical locations can engage with and build upon each other's work, transcending borders to collectively address the challenges and opportunities in agricultural science and technology.
https://jst.org.in/index.html
Our journal has serves as a beacon for scholars and practitioners alike, delving into the intricacies of next-generation technologies that promise to reshape the world. From artificial intelligence and renewable energy to advanced materials and beyond, we are at the forefront of engineering's evolution.
https://ijaast.com/index.html
Our journal has increasing flow of scholarly research articles finds a dedicated channel in IJAAST. By offering a consistent platform, the journal facilitates the seamless flow of knowledge, contributing to the intellectual growth of the agricultural science and technology community.
Solution of second order nonlinear singular value problemsAlexander Decker
1) The document presents a method for finding solutions to second order nonlinear singular boundary value problems using Taylor series.
2) The method modifies the differential equation to handle singularities, then obtains recurrence relations for the Taylor series coefficients by taking derivatives at the singular point.
3) Examples are provided to demonstrate the method, yielding exact solutions for problems in astronomy modeling gas spheres and other physical applications.
On the Fixed Point Extension Results in the Differential Systems of Ordinary ...BRNSS Publication Hub
This document summarizes research on extending the domain of fixed points for ordinary differential equations. It begins with definitions of fixed points and extendability. It then establishes several theorems on extending fixed points, including using Peano's theorem on existence and Picard-Lindelof theorem on uniqueness to extend fixed points over open connected domains where the vector field is continuous. The document proves that if a fixed point is bounded on its domain and the limits at the endpoints exist, then the fixed point can be extended to those endpoints. It concludes by discussing extending fixed points defined on intervals to the whole positive real line using boundedness conditions.
I am Piers L. I am a Calculus Homework Expert at mathhomeworksolver.com. I hold a Master's in Mathematics from, the University of Adelaide. I have been helping students with their homework for the past 6 years. I solve homework related to Calculus.
Visit mathhomeworksolver.com or email support@mathhomeworksolver.com.
You can also call on +1 678 648 4277 for any assistance with Calculus Homework.
This document summarizes research on extending the domain of fixed point solutions to ordinary differential equations. It begins with definitions of fixed points and extendability. The Peano and Picard-Lindelof theorems are used to prove local existence and uniqueness of fixed points. Several theorems are presented to establish conditions under which a fixed point defined on a finite interval can be extended to the endpoints of that interval or beyond. Specifically, if the vector field is bounded and limits exist at the endpoints, the fixed point can be extended. The results ensure extendability to infinite intervals under boundedness assumptions. Extendability is key to studying existence of fixed points on the half line R+.
This document summarizes research on extending the domain of fixed point solutions to ordinary differential equations. It begins with definitions of fixed points and extendability. The Peano and Picard-Lindelof theorems are used to prove local existence and uniqueness of fixed points. Several theorems are presented to establish conditions under which a fixed point defined on a finite interval can be extended to the endpoints of the interval. Specifically, if the vector field is bounded and limits exist at the endpoints, the fixed point can be extended. The results are applied to extend fixed points defined on finite intervals to the entire positive real line.
I am Piers L. I am a Calculus Assignment Expert at mathsassignmenthelp.com. I hold a Master's in Mathematics from, the University of Adelaide. I have been helping students with their assignments for the past 6 years. I solve assignments related to Calculus.
Visit mathsassignmenthelp.com or email info@mathsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Calculus Assignment.
This document presents several theorems regarding the existence, uniqueness, and extendibility of solutions to ordinary differential equation systems. It begins by introducing Peano's theorem on existence of solutions and Picard-Lindelof theorem on uniqueness of solutions locally. It then discusses extension theorems which aim to extend local solutions to open connected domains. The main results section discusses using extension theorems to show existence of solutions on the entire positive real line by extending solutions from finite intervals to infinity.
In this paper, we give several new fixed point theorems to extend results [3]-[4] ,and we apply
the effective modification of He’s variation iteration method to solve some nonlinear and linear equations are
proceed to examine some a class of integral-differential equations and some partial differential equation, to
illustrate the effectiveness and convenience of this method(see[7]). Finally we have also discussed Berge type
equation with exact solution
11.solution of a subclass of singular second orderAlexander Decker
This document presents a method for solving a subclass of singular second order differential equations of Lane-Emden type using Taylor series. The method identifies conditions where the equations can be solved analytically as a polynomial function. Several examples from literature are provided and solved using the proposed Taylor series method to demonstrate its effectiveness. The method produces exact solutions in polynomial form and is shown to perform well on both linear and nonlinear boundary value problems.
This document discusses using the Taylor series method to solve a subclass of singular second order differential equations of Lane-Emden type.
The method is applied to solve boundary value problems that arise from modeling physical phenomena. The Taylor series method produces an analytic solution in the form of a polynomial function.
Several examples from literature are considered to demonstrate the suitability and viability of the Taylor series method for solving these types of differential equations.
This document describes how to compute Fourier series and power spectra using MATLAB. It begins with an introduction to Fourier analysis and how it represents signals in the frequency domain rather than the time domain. It then provides the mathematical foundations for Fourier series, describing how to compute the coefficients to represent a signal as a sum of sines and cosines. Several examples are worked through, including representing a square wave signal. The document concludes by explaining how to apply the same techniques to discrete data using the trapezoidal rule to approximate integrals.
The document discusses solving linear differential equations with variable coefficients using power series representations. It begins by introducing series solution methods and properties of infinite series. It then discusses the Method of Frobenius for solving equations with variable coefficients that arise in cylindrical and spherical coordinate systems. The method involves finding the indicial equation to determine the index c, and then using the recurrence relation to determine the series coefficients an. An example is provided to illustrate the method where the roots of the indicial equation are distinct and not differing by an integer.
The document summarizes an analytic approach to solving the Collatz conjecture proposed by Berg and Meinardus. It introduces two linear operators, U and V, acting on the space of holomorphic functions on the open unit disk. The kernel K of U and V, defined as the set of functions where both operators equal 0, is of main interest. If it can be shown that K equals the two-dimensional space Δ2, then the Collatz conjecture would be proven true. The individual kernel KV of V is already known from prior work. The paper aims to compute U[h] for h in KV and show that K = Δ2, which would imply the truth of the Collatz conjecture.
A Numerical Method For Friction Problems With Multiple ContactsJoshua Gorinson
This document summarizes a numerical method for solving friction problems involving multiple contact surfaces. It begins by reviewing previous work on solving differential equations with discontinuities. The author then describes extending their previous method to handle problems with multiple contacts. Indicator functions are used to represent the regions of contact. Linear complementarity problems (LCPs) are solved to determine changes in the active contact surfaces. The method assumes the indicator functions and vector fields are smooth. Convergence results are proven showing the method can achieve high-order accuracy.
The document discusses the Fundamental Theorem of Calculus, which states that differentiation and integration are inverse operations. It provides examples of using the theorem to evaluate definite integrals and find the average value of functions. The Second Fundamental Theorem of Calculus describes integrals as accumulation functions and relates the integral and derivative. The Net Change Theorem applies the Fundamental Theorem to relate the net change in a function to its integral over an interval. Examples demonstrate using these theorems to solve problems in physics, chemistry, and particle motion.
This document presents a closed-form solution for a class of discrete-time algebraic Riccati equations (DTAREs) under certain assumptions. It begins with background on Riccati equations and their importance in control theory. It then provides the assumptions considered, including that the A matrix eigenvalues are distinct. The main result is a closed-form solution for the DTARE when R=1. Extensions discussed include the solution's behavior as Q approaches zero and for repeated eigenvalues. Comparisons with numerical solutions verify the closed-form solution's accuracy.
A class of boundary value methods for the complex delay differential equationijscmcj
In this paper, a class of boundary value methods (BVMs) for delay differential equations (DDEs) is considered. The delay dependent stable regions of the extended trapezoidal rules of second kind (ETR2s), which are a class of BVMs, are displayed for the test equation of DDEs. Furthermore, it is showed ETR2s cannot preserve the delay-dependent stability of the complex coefficient test equation considered. Some numerical experiments are given to confirm the theoretical results.
AMS 2000 Mathematics Subject Classification: 65L20, 65M12
11.[8 17]numerical solution of fuzzy hybrid differential equation by third or...Alexander Decker
This document presents a numerical method called the third order Runge Kutta Nystrom method to solve fuzzy hybrid differential equations. It begins with definitions of fuzzy numbers and fuzzy functions. It then describes hybrid fuzzy differential systems and how they can be modeled as piecewise differentiable systems. The document goes on to develop the third order Runge Kutta Nystrom method for solving these systems numerically. It provides a convergence result and applies the method to a numerical example to illustrate the theory.
https://jst.org.in/index.html
Our journal has academic journals form a crucial nexus. Educators leverage the latest research findings to enrich their teaching methodologies, ensuring that students are exposed to the most current and relevant information. Simultaneously, these educators contribute to the body of knowledge through their own research, creating a perpetual cycle of growth.
https://ijaast.com/index.html
Our journal has open-access nature of IJAAST fosters global collaboration. Researchers from diverse geographical locations can engage with and build upon each other's work, transcending borders to collectively address the challenges and opportunities in agricultural science and technology.
https://jst.org.in/index.html
Our journal has serves as a beacon for scholars and practitioners alike, delving into the intricacies of next-generation technologies that promise to reshape the world. From artificial intelligence and renewable energy to advanced materials and beyond, we are at the forefront of engineering's evolution.
https://ijaast.com/index.html
Our journal has increasing flow of scholarly research articles finds a dedicated channel in IJAAST. By offering a consistent platform, the journal facilitates the seamless flow of knowledge, contributing to the intellectual growth of the agricultural science and technology community.
https://ijaast.com/index.html
Our journal has transcends traditional boundaries by embracing a multi-disciplinary approach. The journal serves as a melting pot for diverse research areas within agricultural science and technology, ensuring a holistic exploration of the subject.
https://jst.org.in/index.html
Our journal has foster a sense of community among researchers, scholars, and academics. They provide a space for intellectual exchange, allowing scholars to engage with each other's work, provide constructive feedback, and build upon existing knowledge.
https://ijaast.com/index.html
Our journal has we providing a robust platform for scholarly discourse, IJAAST contributes to the nurturing of future innovations. The journal's role in disseminating novel ideas, methodologies, and findings serves as a catalyst for pushing the boundaries of what is possible in agricultural research.
https://ijaast.com/index.html
Our journal has stands as a beacon of excellence in the realm of agricultural research. With its multi-disciplinary focus, commitment to peer-reviewed excellence, and open-access philosophy, IJAAST is not merely a journal; it is a dynamic hub driving the evolution of agricultural science and technology.
https://ijaast.com/index.html
Our journal has dynamic realm of agricultural science and technology, staying abreast of the latest research findings is crucial for professionals, scholars, and enthusiasts alike. IJAAST transcends traditional boundaries by embracing a multi-disciplinary approach.
https://jst.org.in/index.html
Our Journal has stand as pillars, providing a crucial medium for the advancement and dissemination of research results. This blog post explores the pivotal role these journals play in supporting high-level learning, teaching, and research.
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Our journal has a academic journals is the peer-review process. Before an article is published, it undergoes rigorous scrutiny by experts in the field. This ensures the quality and validity of the research, maintaining a high standard of academic integrity.
https://utilitasmathematica.com/index.php/Index
Our Journal has a exploring partnerships and initiatives to provide training and resources to researchers, reviewers, and editors on issues related to JEDI in statistics. Journal has implemented rigorous editorial practices to ensure that published research adheres to JEDI principles.
Our journal has a academic journals form a crucial nexus. Educators leverage the latest research findings to enrich their teaching methodologies, ensuring that students are exposed to the most current and relevant information. Simultaneously, these educators contribute to the body of knowledge through their own research, creating a perpetual cycle of growth.
https://utilitasmathematica.com/index.php/Index
Our journal has academic and professional communities fosters collaboration and knowledge sharing. When all voices are heard and respected, it strengthens the collective capabilities of the statistical community.
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Our journal has academic journals are not mere repositories of information; they are dynamic entities that breathe life into the academic world. They provide a stage for the drama of ideas, where researchers, educators, and learners converge to push the boundaries of knowledge.
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Our Journal has we strive to minimize barriers to access and participation, ensuring that opportunities within the statistical community are available to all, regardless of background. This includes addressing issues such as language barriers, geographical disparities, and financial constraints that may limit access to statistical education and resources.
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Our journal has stands as a beacon of excellence in the field, fostering a culture of high-quality research and unwavering commitment to academic integrity. As research continues to push the boundaries of what's possible, peer review remains an essential tool in ensuring that we continue to progress responsibly and ethically in the realms of science and technology.
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Our journal has Numbers tell stories, and in the world of research and development, mathematics is the universal language. Join us as we explore the elegant equations and mathematical models that underpin technological advancements and scientific breakthroughs.
This document discusses a study that uses GIS techniques to estimate floods in the Upper Sarada River Basin in Visakhapatnam District, India. Daily rainfall data from 23 stations in the region from 1990-2019 and discharge data from a gauge station near Anakapalle are collected. A digital elevation model of the basin is created to extract drainage characteristics. A unit hydrograph is derived from the DEM hydrological processing and used to estimate peak floods for different storms. Eight observed storm events are validated using the unit hydrograph and Thiessen polygon method.
Elevate Your Nonprofit's Online Presence_ A Guide to Effective SEO Strategies...TechSoup
Whether you're new to SEO or looking to refine your existing strategies, this webinar will provide you with actionable insights and practical tips to elevate your nonprofit's online presence.
Temple of Asclepius in Thrace. Excavation resultsKrassimira Luka
The temple and the sanctuary around were dedicated to Asklepios Zmidrenus. This name has been known since 1875 when an inscription dedicated to him was discovered in Rome. The inscription is dated in 227 AD and was left by soldiers originating from the city of Philippopolis (modern Plovdiv).
A Visual Guide to 1 Samuel | A Tale of Two HeartsSteve Thomason
These slides walk through the story of 1 Samuel. Samuel is the last judge of Israel. The people reject God and want a king. Saul is anointed as the first king, but he is not a good king. David, the shepherd boy is anointed and Saul is envious of him. David shows honor while Saul continues to self destruct.
This document provides an overview of wound healing, its functions, stages, mechanisms, factors affecting it, and complications.
A wound is a break in the integrity of the skin or tissues, which may be associated with disruption of the structure and function.
Healing is the body’s response to injury in an attempt to restore normal structure and functions.
Healing can occur in two ways: Regeneration and Repair
There are 4 phases of wound healing: hemostasis, inflammation, proliferation, and remodeling. This document also describes the mechanism of wound healing. Factors that affect healing include infection, uncontrolled diabetes, poor nutrition, age, anemia, the presence of foreign bodies, etc.
Complications of wound healing like infection, hyperpigmentation of scar, contractures, and keloid formation.
This presentation was provided by Racquel Jemison, Ph.D., Christina MacLaughlin, Ph.D., and Paulomi Majumder. Ph.D., all of the American Chemical Society, for the second session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session Two: 'Expanding Pathways to Publishing Careers,' was held June 13, 2024.
THE SACRIFICE HOW PRO-PALESTINE PROTESTS STUDENTS ARE SACRIFICING TO CHANGE T...indexPub
The recent surge in pro-Palestine student activism has prompted significant responses from universities, ranging from negotiations and divestment commitments to increased transparency about investments in companies supporting the war on Gaza. This activism has led to the cessation of student encampments but also highlighted the substantial sacrifices made by students, including academic disruptions and personal risks. The primary drivers of these protests are poor university administration, lack of transparency, and inadequate communication between officials and students. This study examines the profound emotional, psychological, and professional impacts on students engaged in pro-Palestine protests, focusing on Generation Z's (Gen-Z) activism dynamics. This paper explores the significant sacrifices made by these students and even the professors supporting the pro-Palestine movement, with a focus on recent global movements. Through an in-depth analysis of printed and electronic media, the study examines the impacts of these sacrifices on the academic and personal lives of those involved. The paper highlights examples from various universities, demonstrating student activism's long-term and short-term effects, including disciplinary actions, social backlash, and career implications. The researchers also explore the broader implications of student sacrifices. The findings reveal that these sacrifices are driven by a profound commitment to justice and human rights, and are influenced by the increasing availability of information, peer interactions, and personal convictions. The study also discusses the broader implications of this activism, comparing it to historical precedents and assessing its potential to influence policy and public opinion. The emotional and psychological toll on student activists is significant, but their sense of purpose and community support mitigates some of these challenges. However, the researchers call for acknowledging the broader Impact of these sacrifices on the future global movement of FreePalestine.
THE SACRIFICE HOW PRO-PALESTINE PROTESTS STUDENTS ARE SACRIFICING TO CHANGE T...
mathematica journal
1. UtilitasMathematica
ISSN 0315-3681 Volume 120, 2023
25
New Determinantal Inequalities for Positive Definite Matrices
Ronak Pashaei1, Mohammad Sadegh Asgari2*
1,2
Department of Mathematics, Central Tehran Branch, Islamic Azad University, Tehran, Iran.
pashai.ronak@gmail.com1
, msasgari00@gmail.com2
Abstract
R In this article, we explore some new determinant inequalities for matrices. A concavity
approach is implemented to show many sub and super-additive inequalities for the determinant.
This approach is a new direction in this type of inequality. In the end, many determinant
inequalities are presented for accretive-dissipative matrices.
Keywords: Subadditive matrix inequalities, determinant inequalities, concave function,
accretive- dissipative matrices.
1. Introduction
Matrix inequalities, which extend some of certain scalar inequalities, play an important role in
almost all branches of mathematics and other areas of science. This paper investigates the
determinant inequalities of a matrix.
In the sequel, Mn denotes the algebra of all n × n complex matrices, whose elements will be
denoted by upper case letters, in the sequel. The determinant function is a complex-valued
function defined on Mn and is denoted by det. The determinant has numerous applications and
significance in the study of matrix analysis.
In this article, we will present some new inequalities for the determinant. In our analysis, we will
use a concavity approach as a new approach to tickle determinants inequalities. Recall that a
function f : (a, b) → R is said to be concave if
f ((1 − t)α + tβ) ≥ (1 − t)f (α) + tf (β),
for all α, β ∈ (a, b) and 0 ≤ t ≤ 1. It is interesting that this inequality can be reversed as
f ((1 − t)α + tβ) ≤ (1 − t)f (α) + tf (β) + 2R .f . α + β Σ − f (α) + f (β) Σ , (1.1)
where R = max{t, 1−t}, [7]. We also refer the reader to a detailed discussion of this inequality in
[8,9,11].
The function f (A) = det n (A) is concave on the cone of positive semidefinite matrices. We will
use this concavity to find new sub and super-additivity results for the determinant. In particular,
we prove that when A, B ∈ Mn are positive definite, then
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Although this inequality can be shown using other techniques, we present a concavity proof.
Further, we
find that this latter inequality has some impact in proving many other known results for the
determinant. The significance of these results is not the results themselves only but the new
technique used to prove them. Once this has been done, we abandon concavity and start studying
the determinant of matrices having the form A + iB. Many results will be presented as an
extension of known results.
2. Determinantal inequalities via concavity
In this part of the paper, we present related determinant inequalities using a concavity approach.
In particular, we use the fact that the function f defined on the cone of positive definite matrices
in Mn by f (A) = det 1/n
(A) is a concave function. That is, if 0 ≤ t ≤ 1 and A, B ∈ Mn are positive
definite, then [1, Corollary 11.3.21]
det n ((1 − t)A + tB) ≥ (1 − t)det n (A) + tdet n (B). (2.1)
Further, noting Young’s inequality, we have
det n ((1 − t)A + tB) ≥ det n (A)det n (B),
which implies
det((1 − t)A + tB) ≥ det1−t(A)dett(B), (2.2)
showing that the function A ›→ log det A is concave, [2, Lemma II.3].
Now since f (A) = log det A is concave on positive definite matrices, then we have the inequality
where 0 ≤ t ≤ 1 and R = max{t, 1 − t}. We refer the reader to [7–10] for a detailed discussion of
this and related inequalities, as we mentioned earlier in the introduction.
The following result shows a complement of the inequality (2.2):
Theorem 2.1. Let A, B ∈ Mn be positive definite and let R = max{t, 1 − t}, 0 ≤ t ≤ 1. Then
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where S(h) is the Specht’s ratio defined by
Proof. Since f (T) = log (det (T )) is a concave function on the convex set of positive definite
matrices [2, Lemma II.3], (2.3) implies that
On the other hand,
From [12], we know that
Thus,
Whence
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Notice that replacing A by 1/1-t A and B by 1/t B with 0 < t < 1 in (2.1), we reach the super
additivity inequality [1, Corollary 11.3.21], known as the Minkowski determinant inequality,
det 1/n (A + B) ≥ det 1/n (A) + det 1/n (B). (2.4)
The following result presents a reversed version of this inequality via concavity.
The following result presents a reversed version of (2.4) via concavity.
Theorem 2.2. Let A, B ∈ Mn be positive definite and let R = max{t, 1 − t}, 0 < t < 1. Then
Proof. Since f(T) = det 1/n (T) is a concave function on the convex set of positive definite
Hermitian matrices, we infer from (1.1) that
Replacing A and B by A/(1 − t) and t/B in (2.5), respectively, we obtain the desired result.
As a consequence of this, we have the following interesting observation.
Corollary 2.1. Let A, B ∈ Mn be positive definite. Then
and
Proof. From Theorem 2.2, we have
for 0 < t < 1. In particular, this is still true when t → 0+ and t → 1−. Now, we evaluate these
limits. Notice that when t → 0+, R = 1 − t, and hence (2) becomes (when t → 0+)
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We evaluate
Thus
Consequently, (2.6) becomes
which implies the first desired inequality. The second inequality follows similarly, by taking t →
1−.
We should notice here that due to (2.4), we have
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This observation and Corollary 2.1 imply the following double-sided inequality.
Corollary 2.2. Let A, B ∈ Mn be positive definite. Then
The significance of Corollary2.2is the fact that implying that
Consequently, Corollary2.2provides an intermediate-term to the above inequality!
Further, recalling [1, Proposition II.3.20] stating that
which has been shown differently in Corollary2.1.
Another observation that follows from Corollary2.1is when B = ∈I, for ∈ > 0. By Corollary2.1,
we have
Letting s → 0+ implies the well known inequality for the positive definite matrix A that
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The last observation we would like to pay the attention of the reader to is how we can use
Corollary 2.1 to prove [1, Proposition II.3.20]. This discussion acknowledges the concavity
discussion of the function Concavity of this function implies the superadditivity
inequality (2.4) and Corollary2.1.
These two consequences imply
This implies that if A ∈ Mn is positive definite, then for any other positive definite matrix B with
determinant 1; one has
Since this is true for any positive B with det(B) = 1, we may write
Since det replacing B in the above inequality by implies an
identity, which proves the well known identity [1, Proposition II.3.20]
Therefore, the concavity approach we adopted in the above results provides an alternative way to
prove some well-known determinant identities.
For the rest of this section, we present sub and super additive determinantal inequalities for
matrices of the form A + iB, where A, B ∈ Mn are positive (i.e., accretive-dissipative matrices)
and i2 = −1. Although the approach differs from the above approach, we find it convenient to add
these results as they resemble the same theme of this paper. The following two lemmas will be
needed.
Lemma 2.1. ( [6]) Let A, B ∈ Mn be positive definite. Then
Lemma 2.2. ( [5]) (Fan’s determinant inequality) Let H, K ∈ Mn be Hermitian and let A = H +
iK, If H is positive definite, then
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with equality if and only if all the eigenvalues of H−1K have the same absolute value.
The following lemma is shown in [13, p. 48], but here we present its extension to complex
matrices.
Lemma 2.3. Let A, B ∈ Mn be positive definite. Then
The first result in this direction is a simple proof of Fan’s determinant inequality.
Theorem 2.3. Let A, B ∈ Mn be positive definite. Then
Proof. We have
which completes the proof.
The following proposition aims to present an upper bound for |det (A + iB)| .
Proposition 2.1. Let A, B ∈ Mn be positive definite. Then
Proof. From Lemma 2.3, we have
where we have used Young’s inequality to obtain the last inequality. This completes the proof.
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Remark 2.1. We would like to mention that the inequality (2.8) can be regarded as an
improvement the left side of (2.7). When B ≤ 2A, we get
Let A, B ∈ Mn be positive definite. It is well known that (e.g., [5, p. 511])
det(A + B) ≥ det A + det B. (2.9)
Haynsworth obtained the following refinement of this inequality.
Theorem 2.4. [4, Theorem 3] Suppose A, B ∈ Mn are positive definite. Let Ak and Bk , k = 1, 2,
..., n−1, denote the k-th principal submatrices of A and B respectively. Then
Hartel [3] proved an improvement of (2.10) as follows:
As a direct result, Hartel also presented the following inequality
det(A + B) ≥ det A + det B + (2n − 2) √det AB. (2.11)
Now, we present the complex version of (2.11).
Theorem 2.5. Let A, B ∈ Mn be positive definite. Then
|det (A + iB)|2
≥ det2
A + det2
B + (2n
− 2) det AB.
Proof. By Lemma 2.3 and (2.11), we have
The desired conclusion follows.
Remark 2.2. If n ≥ 2, then (2n − 2) ≥ 2. Therefore by Theorem2.5,
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This implies the superadditive inequality
|det (A + iB)| ≥ (det A + det B),
which is also an improvement of (2.9).
In the following result, we will present a Haynsworth-Hartfiel-type result for A + iB. For this, we
will need the following two lemmas [13].
Lemma 2.4. Let A ∈ Mn be positive definite and Ai be a principal submatrix of A. Then
(A-1
)i > (Ai)-1
.
Lemma 2.5. Let A ∈ Mn be positive definite. Then for any B ∈ Mn,
Now we are ready for the following main result.
Theorem 2.6. Let A, B ∈ Mn be positive definite. Then
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REFERENCES 9
Proof. We can write
Acknowledgements
The authors work was partially supported by the Central Tehran Branch of Islamic Azad
University.
Compliance with Ethical Standards
Conflict of interest
All authors declare that they have no conflict of interest.
Ethical approval
This article does not contain any studies with human participants or animals performed by any
authors.
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