Our goal in this set of lecture notes is to provide students with a strong foundation in mathemat... more Our goal in this set of lecture notes is to provide students with a strong foundation in mathematical analysis. Such a foundation is crucial for future study of deeper topics of analysis. Students should be familiar with most of the concepts presented here after completing the calculus sequence. However, these concepts will be reinforced through rigorous proofs.
In this paper we first consider the class of minimal time functions in the general setting of loc... more In this paper we first consider the class of minimal time functions in the general setting of locally convex topological vector (LCTV) spaces. The results obtained in this framework are based on a novel notion of closedness of target sets with respect to constant dynamics. Then we introduce and investigate a new class of signed minimal time functions, which are generalizations of the signed distance functions. Subdifferential formulas for the signed minimal time and distance functions are obtained under the convexity assumptions on the given data.
A bilevel hierarchical clustering model is commonly used in designing optimal multicast networks.... more A bilevel hierarchical clustering model is commonly used in designing optimal multicast networks. In this paper we will consider two different formulations of the bilevel hierarchical clustering problem -- a discrete optimization problem which can be shown to be NP-hard. Our approach is to reformulate the problem as a continuous optimization problem by making some relaxations on the discreteness conditions. This approach was considered by other researchers earlier, but their proposed methods depend on the square of the Euclidian norm because of its differentiability. By applying the Nesterov smoothing technique and the DCA -- a numerical algorithm for minimizing differences of convex functions -- we are able to cope with new formulations that involve the Euclidean norm instead of the squared Euclidean norm. Numerical examples are provided to illustrate our method
The smallest intersecting ball problem asks for the smallest radius necessary to intersect a coll... more The smallest intersecting ball problem asks for the smallest radius necessary to intersect a collection of m closed sets. Formally, we write min x∈Rn D(x) = max{dist(x,Ωi) ∣∣i = 1, 2, ...,m} This research explores various methods of finding the solution, and some tools of convex optimization which facilitate these methods. The max distance function is non-smooth and convex, which lends itself to minimization by the classical subgradient method. A second approach uses a log-exponential smoothing approximation of the max distance function, coupled with distance majorization and Nesterov acceleration. Two original algorithms are presented: The first method expands the sets and finds their smallest non-empty intersection, in which the optimal solution is proven to lie. The other–weighted projections– searches for the optimal solution as a convex combination of projections onto each set, with coefficients iteratively updated based on which set is most distant. Each algorithm is implement...
International Review of Economics & Finance, 2021
Abstract This research argues that the inconclusive evidence on the board gender diversity–firm p... more Abstract This research argues that the inconclusive evidence on the board gender diversity–firm performance relationship across nations may be due to the moderating effect of national governance quality. Applying a multi-hierarchical modeling technique on a dataset containing 15,051 firm-year observations from 2931 companies in 46 countries, the results generally confirm this Hypothesis. Specifically, board gender diversity only seems to positively affect the performance of companies operating in countries with above-average levels of national governance quality. The effect of gender diversity on firm performance decreases and turns negative as national governance quality drops. The results are robust to different measures of national governance quality and firm performance, changes in methods of estimation, changes in sample structure, and tokenism.
The majority of strains of Candida albicans can switch frequently and reversibly between two or m... more The majority of strains of Candida albicans can switch frequently and reversibly between two or more general phenotypes, a process now considered a putative virulence factor in this species.Candida albicans WO-1 switches frequently and reversibly between a white and an opaque phase, and this phenotypic transition is accompanied by the differential expression of white-phase-specific and opaque-phase-specific genes. In the opaque phase, cells differentially express the gene OP4, which encodes a putative protein 402 amino acids in length that contains a highly hydrophobic amino-terminal sequence and a carboxy-terminal sequence with a pI of 10.73. A series of deletion constructs fused to the Renilla reniformisluciferase was used to functionally characterize the OP4promoter in order to investigate how this gene is differentially expressed in the white-opaque transition. An extremely strong 17-bp transcription activation sequence was identified between −422 and −404 bp. This sequence cont...
Atherosclerotic coronary artery disease (CAD) results from build-up of cholesterol-rich plaques i... more Atherosclerotic coronary artery disease (CAD) results from build-up of cholesterol-rich plaques in the walls of the coronary arteries and is a leading cause of death. Inflammation is central to atherosclerosis. Uncontrolled inflammation makes coronary plaques “unstable” and vulnerable to rupture or erosion, leading to thrombosis and myocardial infarction (MI). As multiple inflamed plaques often co-exist in the coronary system, patients are at risk of repeated atherothrombotic cardiovascular events after MI, with rates of 10–12% at one year and 18–20% at three years. This is largely because current therapies for CAD, such as lipid-lowering statins, do not adequately control plaque inflammation. New anti-atherosclerotic agents are therefore needed, especially those that better target inflammation. The recent positive results for the anti-interleukin-1-beta (IL-1β) monoclonal antibody, Canakinumab, in the Canakinumab Anti-inflammatory Thrombosis Outcome Study (CANTOS) clinical trial ha...
We deal with a class of integral equations on the unit circle in the complex plane with a regular... more We deal with a class of integral equations on the unit circle in the complex plane with a regular part and with rotations of the form (*) x(t) + a(t)(T x)(t) = b(t), where T = M n1,k1. .. M nm,km and M nj ,kj are of the form (3) below. We prove that under some assumptions on analytic continuation of the given functions, (*) is a singular integral equation for m odd and is a Fredholm equation for m even. Further, we prove that T is an algebraic operator with characteristic polynomial P T (t) = t 3 − t. By means of the Riemann boundary value problems, we give an algebraic method to obtain all solutions of equation (*) in closed form.
In this paper, we study bornological generalized differential properties of sets with nonsmooth b... more In this paper, we study bornological generalized differential properties of sets with nonsmooth boundaries, nonsmooth functions, and set-valued mappings in smooth Banach spaces. We establish a fuzzy intersection rule for bornological normal cones and develop fuzzy calculus for bornological generalized differential constructions as well as exact calculus for the limiting counterparts of these constructions.
This paper is a continuation of our effort in using mathematical optimization involving DC progra... more This paper is a continuation of our effort in using mathematical optimization involving DC programming in clustering and multifacility location. We study a penalty method based on distance functions and apply it particularly to a number of problems in clustering and multifacility location in which the centers to be found must lie in some given set constraints. We also provide numerical examples to test our method.
This paper continues our effort initiated in [19] to study Multicast Communication Networks, mode... more This paper continues our effort initiated in [19] to study Multicast Communication Networks, modeled as bilevel hierarchical clustering problems, by using mathematical optimization techniques. Given a finite number of nodes, we consider two different models of multicast networks by identifying a certain number of nodes as cluster centers, and at the same time, locating a particular node that serves as a total center so as to minimize the total transportation cost through the network. The fact that the cluster centers and the total center have to be among the given nodes makes this problem a discrete optimization problem. Our approach is to reformulate the discrete problem as a continuous one and to apply Nesterov smoothing approximation technique on the Minkowski gauges that are used as distance measures. This approach enables us to propose two implementable DCA-based algorithms for solving the problems. Numerical results and practical applications are provided to illustrate our approach.
Journal of Optimization Theory and Applications, 2017
In this paper we develop algorithms to solve generalized Fermat-Torricelli problems with both pos... more In this paper we develop algorithms to solve generalized Fermat-Torricelli problems with both positive and negative weights and multifacility location problems involving distances generated by Minkowski gauges. We also introduce a new model of clustering based on squared distances to convex sets. Using the Nesterov smoothing technique and an algorithm for minimizing differences of convex functions called the DCA introduced by Tao and An, we develop effective algorithms for solving these problems. We demonstrate the algorithms with a variety of numerical examples.
International Journal of Hybrid Intelligent Systems, 2014
ABSTRACT Obstacle detection is a fundamental issue of robot navigation and there have been severa... more ABSTRACT Obstacle detection is a fundamental issue of robot navigation and there have been several proposed methods for this problem. In this paper, we propose a new approach to find out obstacles on Depth Camera streams. The proposed approach consists of three stages. First, preprocessing stage is for noise removal. Second, different depths in a frame are clustered based on the Interval Type-2 Fuzzy Subtractive Clustering algorithm. Third, the objects of interest are detected from the obtained clusters. Beside that, it gives an improvement in the Interval Type-2 Fuzzy Subtractive Clustering algorithm to reduce the time consuming. In theory, it is at least 3700 times better than the original one, and approximate 980100 in practice on our depth frames. The results conducted on frames demonstrate that the distance from the camera to objects retrieved is exact enough for indoor robot navigation problems.
ABSTRACT In this paper, we study the weak and strong convergence of the proximal point algorithm ... more ABSTRACT In this paper, we study the weak and strong convergence of the proximal point algorithm for equilibrium problems of pseudo-monotone type in Hilbert spaces. We prove the weak convergence of the generated sequence to a common solution of two equilibrium problems and some strong convergence results with additional assumptions on pseudo-monotone bifunctions. Then we study a regularization of Halpern-type and prove the strong convergence of the generated sequence to an equilibrium point of two pseudo-monotone bifunctions without any additional assumption on bifunctions. Finally, some examples of pseudo-monotone bifunctions from pseudo-monotone operators and Nash-Cournot oligopolistic equilibrium models are also presented. Our results extend some similar results in the literature for monotone and pseudo-monotone equilibrium problems and also the related results for variational inequalities associated with monotone and pseudo-monotone operators.
2010 5th International Conference on Critical Infrastructure (CRIS), 2010
This paper deals with the aspects of stability assessment for the VSC HVDC transmission systems t... more This paper deals with the aspects of stability assessment for the VSC HVDC transmission systems that are especially relevant for this technology when applied on a large scale. It focuses on the control strategies for the VSC HVDC transmission necessary to reach a full integration of large offshore wind farms into the power system operation while minimizing the critical situations
Morphological characteristics of Cordyceps species and its allies collected in Korea were clarifi... more Morphological characteristics of Cordyceps species and its allies collected in Korea were clarified. Through the survey conducted from June 1999 to October 2002 in 19 mountains in Korea, 667 samples of entomogenous fungi were collected. Cordyceps and its allies of 17 species of 5 genera were identified as Cordyceps gracilioides, C. japonica, C. longissima, C. martialis, C. militaris, C. myrmecophila, C. nutans, C. pruinosa, C. sphecocephala, C. tricentri, Hirsutella nutans, Paecilomyces cicadae, P. farinosus, P. tenuipes, Paecilomyces sp., Shimizuomyces paradoxa, Tilachlidiopsis nigra. The fungi with insect hosts have been collected mainly in the place of shade or mosses near brooks and streams that had high humidity. Overall the frequenct of fungal infection in natural ecosystem was relatively low as few as 10 collections per each species. However, many species were found in terms of the few number of colleciton sites with seasonal limitations. Occurrence of the fungi in Jeju island remote from inland of the Korean peninsula were diverse in their species due to the varied weather of vertical distribution following the altitude. Three most common species were C. nutans, P. tenuipes and C. militaris, mainly found early in August when the relative humidity and temperature were high, of which C. nutans occupied the highest frequency consisting of 65% in total collections. Neither variation in ascomata arrangement in stromata nor developement of secondary spores was recognizable, while the number, shape and colour of stromata varied with insect hosts and weather conditions.
The dissertation is mainly devoted to the study of general classes of marginal/value functions, i... more The dissertation is mainly devoted to the study of general classes of marginal/value functions, in particular distance functions, which play a significant role in optimization, variational analysis, and their applications. A characteristic feature of marginal functions is that they are intrinsically nonsmooth and thus require appropriate tools of generalized differentiation for their study and various applications, particularly to sensitivity analysis
Our goal in this set of lecture notes is to provide students with a strong foundation in mathemat... more Our goal in this set of lecture notes is to provide students with a strong foundation in mathematical analysis. Such a foundation is crucial for future study of deeper topics of analysis. Students should be familiar with most of the concepts presented here after completing the calculus sequence. However, these concepts will be reinforced through rigorous proofs.
In this paper we first consider the class of minimal time functions in the general setting of loc... more In this paper we first consider the class of minimal time functions in the general setting of locally convex topological vector (LCTV) spaces. The results obtained in this framework are based on a novel notion of closedness of target sets with respect to constant dynamics. Then we introduce and investigate a new class of signed minimal time functions, which are generalizations of the signed distance functions. Subdifferential formulas for the signed minimal time and distance functions are obtained under the convexity assumptions on the given data.
A bilevel hierarchical clustering model is commonly used in designing optimal multicast networks.... more A bilevel hierarchical clustering model is commonly used in designing optimal multicast networks. In this paper we will consider two different formulations of the bilevel hierarchical clustering problem -- a discrete optimization problem which can be shown to be NP-hard. Our approach is to reformulate the problem as a continuous optimization problem by making some relaxations on the discreteness conditions. This approach was considered by other researchers earlier, but their proposed methods depend on the square of the Euclidian norm because of its differentiability. By applying the Nesterov smoothing technique and the DCA -- a numerical algorithm for minimizing differences of convex functions -- we are able to cope with new formulations that involve the Euclidean norm instead of the squared Euclidean norm. Numerical examples are provided to illustrate our method
The smallest intersecting ball problem asks for the smallest radius necessary to intersect a coll... more The smallest intersecting ball problem asks for the smallest radius necessary to intersect a collection of m closed sets. Formally, we write min x∈Rn D(x) = max{dist(x,Ωi) ∣∣i = 1, 2, ...,m} This research explores various methods of finding the solution, and some tools of convex optimization which facilitate these methods. The max distance function is non-smooth and convex, which lends itself to minimization by the classical subgradient method. A second approach uses a log-exponential smoothing approximation of the max distance function, coupled with distance majorization and Nesterov acceleration. Two original algorithms are presented: The first method expands the sets and finds their smallest non-empty intersection, in which the optimal solution is proven to lie. The other–weighted projections– searches for the optimal solution as a convex combination of projections onto each set, with coefficients iteratively updated based on which set is most distant. Each algorithm is implement...
International Review of Economics & Finance, 2021
Abstract This research argues that the inconclusive evidence on the board gender diversity–firm p... more Abstract This research argues that the inconclusive evidence on the board gender diversity–firm performance relationship across nations may be due to the moderating effect of national governance quality. Applying a multi-hierarchical modeling technique on a dataset containing 15,051 firm-year observations from 2931 companies in 46 countries, the results generally confirm this Hypothesis. Specifically, board gender diversity only seems to positively affect the performance of companies operating in countries with above-average levels of national governance quality. The effect of gender diversity on firm performance decreases and turns negative as national governance quality drops. The results are robust to different measures of national governance quality and firm performance, changes in methods of estimation, changes in sample structure, and tokenism.
The majority of strains of Candida albicans can switch frequently and reversibly between two or m... more The majority of strains of Candida albicans can switch frequently and reversibly between two or more general phenotypes, a process now considered a putative virulence factor in this species.Candida albicans WO-1 switches frequently and reversibly between a white and an opaque phase, and this phenotypic transition is accompanied by the differential expression of white-phase-specific and opaque-phase-specific genes. In the opaque phase, cells differentially express the gene OP4, which encodes a putative protein 402 amino acids in length that contains a highly hydrophobic amino-terminal sequence and a carboxy-terminal sequence with a pI of 10.73. A series of deletion constructs fused to the Renilla reniformisluciferase was used to functionally characterize the OP4promoter in order to investigate how this gene is differentially expressed in the white-opaque transition. An extremely strong 17-bp transcription activation sequence was identified between −422 and −404 bp. This sequence cont...
Atherosclerotic coronary artery disease (CAD) results from build-up of cholesterol-rich plaques i... more Atherosclerotic coronary artery disease (CAD) results from build-up of cholesterol-rich plaques in the walls of the coronary arteries and is a leading cause of death. Inflammation is central to atherosclerosis. Uncontrolled inflammation makes coronary plaques “unstable” and vulnerable to rupture or erosion, leading to thrombosis and myocardial infarction (MI). As multiple inflamed plaques often co-exist in the coronary system, patients are at risk of repeated atherothrombotic cardiovascular events after MI, with rates of 10–12% at one year and 18–20% at three years. This is largely because current therapies for CAD, such as lipid-lowering statins, do not adequately control plaque inflammation. New anti-atherosclerotic agents are therefore needed, especially those that better target inflammation. The recent positive results for the anti-interleukin-1-beta (IL-1β) monoclonal antibody, Canakinumab, in the Canakinumab Anti-inflammatory Thrombosis Outcome Study (CANTOS) clinical trial ha...
We deal with a class of integral equations on the unit circle in the complex plane with a regular... more We deal with a class of integral equations on the unit circle in the complex plane with a regular part and with rotations of the form (*) x(t) + a(t)(T x)(t) = b(t), where T = M n1,k1. .. M nm,km and M nj ,kj are of the form (3) below. We prove that under some assumptions on analytic continuation of the given functions, (*) is a singular integral equation for m odd and is a Fredholm equation for m even. Further, we prove that T is an algebraic operator with characteristic polynomial P T (t) = t 3 − t. By means of the Riemann boundary value problems, we give an algebraic method to obtain all solutions of equation (*) in closed form.
In this paper, we study bornological generalized differential properties of sets with nonsmooth b... more In this paper, we study bornological generalized differential properties of sets with nonsmooth boundaries, nonsmooth functions, and set-valued mappings in smooth Banach spaces. We establish a fuzzy intersection rule for bornological normal cones and develop fuzzy calculus for bornological generalized differential constructions as well as exact calculus for the limiting counterparts of these constructions.
This paper is a continuation of our effort in using mathematical optimization involving DC progra... more This paper is a continuation of our effort in using mathematical optimization involving DC programming in clustering and multifacility location. We study a penalty method based on distance functions and apply it particularly to a number of problems in clustering and multifacility location in which the centers to be found must lie in some given set constraints. We also provide numerical examples to test our method.
This paper continues our effort initiated in [19] to study Multicast Communication Networks, mode... more This paper continues our effort initiated in [19] to study Multicast Communication Networks, modeled as bilevel hierarchical clustering problems, by using mathematical optimization techniques. Given a finite number of nodes, we consider two different models of multicast networks by identifying a certain number of nodes as cluster centers, and at the same time, locating a particular node that serves as a total center so as to minimize the total transportation cost through the network. The fact that the cluster centers and the total center have to be among the given nodes makes this problem a discrete optimization problem. Our approach is to reformulate the discrete problem as a continuous one and to apply Nesterov smoothing approximation technique on the Minkowski gauges that are used as distance measures. This approach enables us to propose two implementable DCA-based algorithms for solving the problems. Numerical results and practical applications are provided to illustrate our approach.
Journal of Optimization Theory and Applications, 2017
In this paper we develop algorithms to solve generalized Fermat-Torricelli problems with both pos... more In this paper we develop algorithms to solve generalized Fermat-Torricelli problems with both positive and negative weights and multifacility location problems involving distances generated by Minkowski gauges. We also introduce a new model of clustering based on squared distances to convex sets. Using the Nesterov smoothing technique and an algorithm for minimizing differences of convex functions called the DCA introduced by Tao and An, we develop effective algorithms for solving these problems. We demonstrate the algorithms with a variety of numerical examples.
International Journal of Hybrid Intelligent Systems, 2014
ABSTRACT Obstacle detection is a fundamental issue of robot navigation and there have been severa... more ABSTRACT Obstacle detection is a fundamental issue of robot navigation and there have been several proposed methods for this problem. In this paper, we propose a new approach to find out obstacles on Depth Camera streams. The proposed approach consists of three stages. First, preprocessing stage is for noise removal. Second, different depths in a frame are clustered based on the Interval Type-2 Fuzzy Subtractive Clustering algorithm. Third, the objects of interest are detected from the obtained clusters. Beside that, it gives an improvement in the Interval Type-2 Fuzzy Subtractive Clustering algorithm to reduce the time consuming. In theory, it is at least 3700 times better than the original one, and approximate 980100 in practice on our depth frames. The results conducted on frames demonstrate that the distance from the camera to objects retrieved is exact enough for indoor robot navigation problems.
ABSTRACT In this paper, we study the weak and strong convergence of the proximal point algorithm ... more ABSTRACT In this paper, we study the weak and strong convergence of the proximal point algorithm for equilibrium problems of pseudo-monotone type in Hilbert spaces. We prove the weak convergence of the generated sequence to a common solution of two equilibrium problems and some strong convergence results with additional assumptions on pseudo-monotone bifunctions. Then we study a regularization of Halpern-type and prove the strong convergence of the generated sequence to an equilibrium point of two pseudo-monotone bifunctions without any additional assumption on bifunctions. Finally, some examples of pseudo-monotone bifunctions from pseudo-monotone operators and Nash-Cournot oligopolistic equilibrium models are also presented. Our results extend some similar results in the literature for monotone and pseudo-monotone equilibrium problems and also the related results for variational inequalities associated with monotone and pseudo-monotone operators.
2010 5th International Conference on Critical Infrastructure (CRIS), 2010
This paper deals with the aspects of stability assessment for the VSC HVDC transmission systems t... more This paper deals with the aspects of stability assessment for the VSC HVDC transmission systems that are especially relevant for this technology when applied on a large scale. It focuses on the control strategies for the VSC HVDC transmission necessary to reach a full integration of large offshore wind farms into the power system operation while minimizing the critical situations
Morphological characteristics of Cordyceps species and its allies collected in Korea were clarifi... more Morphological characteristics of Cordyceps species and its allies collected in Korea were clarified. Through the survey conducted from June 1999 to October 2002 in 19 mountains in Korea, 667 samples of entomogenous fungi were collected. Cordyceps and its allies of 17 species of 5 genera were identified as Cordyceps gracilioides, C. japonica, C. longissima, C. martialis, C. militaris, C. myrmecophila, C. nutans, C. pruinosa, C. sphecocephala, C. tricentri, Hirsutella nutans, Paecilomyces cicadae, P. farinosus, P. tenuipes, Paecilomyces sp., Shimizuomyces paradoxa, Tilachlidiopsis nigra. The fungi with insect hosts have been collected mainly in the place of shade or mosses near brooks and streams that had high humidity. Overall the frequenct of fungal infection in natural ecosystem was relatively low as few as 10 collections per each species. However, many species were found in terms of the few number of colleciton sites with seasonal limitations. Occurrence of the fungi in Jeju island remote from inland of the Korean peninsula were diverse in their species due to the varied weather of vertical distribution following the altitude. Three most common species were C. nutans, P. tenuipes and C. militaris, mainly found early in August when the relative humidity and temperature were high, of which C. nutans occupied the highest frequency consisting of 65% in total collections. Neither variation in ascomata arrangement in stromata nor developement of secondary spores was recognizable, while the number, shape and colour of stromata varied with insect hosts and weather conditions.
The dissertation is mainly devoted to the study of general classes of marginal/value functions, i... more The dissertation is mainly devoted to the study of general classes of marginal/value functions, in particular distance functions, which play a significant role in optimization, variational analysis, and their applications. A characteristic feature of marginal functions is that they are intrinsically nonsmooth and thus require appropriate tools of generalized differentiation for their study and various applications, particularly to sensitivity analysis
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