Mathematics

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https://hdl.handle.net/20.500.14540/131

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A study of fuzzy logic and fuzzy sequence with their application to the real world
(Institute of Science and Technology, 2024-06) Paudel, Gyan Prasad; Prof. Dr. Narayan prasad Pahari
Sequence space and difference sequence spaces play an important role in many areas of analysis, such as the Schauer basis, summability, fixed point theory, non-linear analysis, and structural theory of topological vector space. Fuzzy logic is the study of uncertainty and vagueness. It is a flexible, uncertainty-based reasoning method for rational decision making that addresses vague or incomplete information and solves specific problems. The fuzzy set theory has been successfully applied in a wide range of mathematical fields. Fuzzy sequence analysis offers a robust framework for handling uncertainty and imprecision in sequence-based data, enhancing practicality and effectiveness. This dissertation deals with the fundamental topological properties of sequence space and the difference sequence spaces of fuzzy real numbers. To study the basic topological properties of the classes l_F (X,λ ̅,p ̅ ) and l_F (X,λ ̅,p ̅,L) we use the Orlicz and paranorm function. Moreover, linearity, completeness, solidity, and some inclusion properties of a class S(X,M,P,A) of difference sequence and classes F_∞ (ρ,M,p,A), F_c (ρ,M,p,A) and F_o (ρ,M,p,A) of generalized difference sequences. We also study some topological properties classes Z_F (M,λ\,ξ) where, Z_F=l_∞^F,C^F,C_o^F f double sequences of fuzzy real numbers. Additionally, this thesis also includes the generalized form of the P-bounded variation bV_p^F of fuzzy real numbers. In addition this thesis further explores the practical implementation of fuzzy real numbers in various real-world scenarios. Specifically, it examines how fuzzy sets and fuzzy logic are employed in decision-making processes, particularly in selecting the best option using the Bellmen-Zadeh max-min method. Furthermore, this thesis delves into the field of healthcare and addresses Sanchez’s medical condition, utilizing a case study to illustrate the application of fuzzy arithmetic-based methods in identifying and assessing medical issues with a case study. Moreover, the thesis extends its exploration to the domain of insurance fraud detection. It presents a fuzzy model designed to assist internal auditors in identifying potentially fraudulent claims during the claim-settlement process. Additionally, the thesis examines the utilization of machine learning techniques in the detection of cardiovascular diseases. It outlines how a fuzzy model is developed to classify and assess the risk of cardiovascular disease based on various input factors.
Advection-Dispersion Equation For Pollutant Concentration
(Department of Mathematics, 2022) Paudel, Keshav
The advection-dispersion-reaction equation is used to describe the dispersion process. Here, we solve one-dimensional steady advection-dispersion equation numerically by using nite di erence method. Also, we formulate the model to minimize the cost of wastewater treatment. Analytical solution to unsteady advection-dispersion equation using Laplace transformation technique is derived to describe the pollutant concentration C(x; t). We have obtained analytic unsteady solution by taking the water velocity u in the x-direction as a linear function of x and dispersion coe cient D as zero in case of concentration of pollutant in one region. Numerical studies show variation of C with time t. If the added pollutant rate along the river q is very small amount, the variation of C along the river at di erent times coincide to each other. In case of concentration of pollutant in two regions, analytical solutions are determined by taking dispersion coe cient D as non-zero. A coupled system of advection-dispersion equations based water pollution model is presented that incorporates di erent parameters. We have proposed analytical solution for mathematical model. One dimensional model is used to observe the concentrations by taking dimension along the length of river. By considering the removal of pollutant by aeration, event of steady states is investigated. In this model, coupled advection-dispersion equations are solved by taking dispersion coe cient as zero and non-zero, respectively. Keywords: Pollutant, Concentration, Laplace transformation, Dispersion, Analytical solution, Dissolved oxygen.
Bayesian Modelling Approaches on some Issues of Agro-Food Production and Quality Control
(Central Department of Statistics, 2010) Khatiwada, Ram Prasad
Not Available
Bottleneck Just-in-Time Sequencing for Mixed-Model Production Systems
(Department of Mathematics, 2008) Poudyal, Chudamani
Due to today’s competitive automotive industrial challenges of providing a variety of products at a very low cost by smoothing productions on a flexible transfer line, one of the most important and fertile research topic in industrial mathematics is to penalize jobs both for being early and for being tardy. A problem is to determine a production sequence for producing different types of products on the line. Just-in-Time (JIT) mixedmodel production system is used to address this problem, which involves producing only the right products of different models of a common base product in evenly balanced sequences in the exact quantities, at the right times, at the right place. Sequencing JIT production system can be formulated as a challenging nonlinear integer programming problem. The goal of such system is to balance the rate of production of products. Minimization of the variation in demand rates for outputs of supplying processes is the output rate variation problem (ORVP) and minimization of the variation in the rate at which different products are produced on the line is the product rate variation problem (PRVP). The problem for minimizing of deviations between actual and desired production for PRVP can be solved efficiently in pseudo-polynomial time complexity. However, the ORVPs for two or more levels are strongly NP-hard. Heuristic algorithms and dynamic programming to solve such NP-hard problems are summarized. But ORVPs with pegging assumption are solvable by reducing them to the corresponding weighted PRVPs. The cyclic sequences are optimal for both sum and max deviation PRVPs. For the bottleneck PRVP, a binary search technique is used to test the existence of a perfect matching and thereby to get optimal sequence. A feasible sequence always exists such that, at all times, the deviation of actual production from the desired level of production for every product is never more than one unit for the max-absolute and maxsquared PRVPs. An elegant algebraic concept of balanced words is used to deal the bottleneck PRVP. The max-absolute PRVP is shown to be Co-NP with leaving its general complexity open. In this thesis, we study several interesting algebraic structures, properties, existence of cyclic solutions and two applications of bottleneck PRVP. An optimal sequence for an instance of max-absolute PRVP is obtained. With considering two min-sum and maxabsolute objectives, a bicriterion objective for balancing the sequence is analyzed. A comparative study of different objectives is also summarized. Moreover, several directions for further research are also explored including some emerged conjectures.
Dynamic Network Contraflow Evacuation Planning Problem
(Department of Mathematics, 2020) Bhandari, Phanindra Prasad
Evacuation planning problem gives effcient way-out on existing road network that attempts to shift evacuees from risk zone to safer in minimum time with minimum casualty during disasters. Its domain based on network ow problems has been ourished with models and solutions with various network attributes. A common feature on almost all of these models is that the ow function obeys conservation constraints at each intermediate vertex. In particular, maximum dynamic ow (MDF) problem, earliest arrival ow (EAF) problem and quickest ow (QF) problem have great applicability in evacuation planning problems. Contra ow approach recon gures the network identifying ideal direction and reallocating available capacity for each arc to improve ow egress time and/or improve the number of ow units from source to sink. This thesis sketches a brief survey of models and results on contra ow evacuation planning problems. Continuous time model for maximum dynamic contra ow (MDCF) problem is studied with its e cient solution. Thesis also extends contra ow model for multi-network. Network modi cation strategy is applied to give polynomial time algorithms to solve the problems; namely, MDCF problem and earliest arrival contra ow (EACF) problem based on extended model with discrete as well as continuous time setting. The former problems are considered in general networks whereas the latter problems in two terminal series parallel (TTSP) networks. Arc reversibility is allowed only once at time zero in each of the cases. Evacuation models with intermediate temporary shelters could be extra bene t while implementing them. This thesis formulates, as another contribution, ow model for network with capacitated vertices of given priority order in which ow conservation may be violated. This violation makes possible for ow units to be held at intermediate vertices which turns out to be applicable in modeling an evacuation planning problem with intermediate holding of evacuees at temporary shelters despite sending them into the sink. Based on this model, maximum ow problem is considered and proposed a polynomial solution for static case and pseudo-polynomial solution for dynamic case. Also, polynomial solutions for MDF problem and QF problem modeled on uniform path length (UPL) network and for EAF problem modeled on UPL-TTSP network are proposed. As the nal contribution, contra ow approach is linked to evacuation problems with capacitated prioritized vertices. Keywords: Network ow models, Contra ow, Capacitated vertices, Evacuation planning problem, Disaster management.
Dynamics of Transmission and Control of Measles
(Institute of Sciecne and Technology, 2024-08) Pokharel, Anjana; Prof.Dr. Kedar Nath Uprety
The emergence and re-emergence of infectious diseases have become a global problem. Measles is a highly contagious human viral disease whose outbreaks frequently occur in many countries, including Nepal, despite the availability of vaccines partly due to the lack of compliance with vaccination. While the National Immunization Program is in place in Nepal, the frequent occurrence of measles in Nepal remains a major cause of child morbidity and mortality. Mathematical modeling for infectious diseases aids in forecasting and comprehending the dynamics of such diseases, facilitating the deployment of effective public health interventions and the allocation of resources. In this work, we developed a novel transmission dynamics model in the form of system of nonlinear ordinary differential equations to evaluate the effects of monitored vaccination programs on individuals who have skipped the regular vaccination program, aiming to control and eliminate measles properly. Our model was validated by Nepal's yearly incidence case data from 2000 to 2019. We calculated the vaccinated reproduction number, Rv, using the Next Generation Matrix method. We also computed the effective reproduction number of measles in Nepal. We performed model analyses to establish the global asymptotic stability of the disease-free equilibrium point for Rv<1 and the uniform persistence of the disease for Rv>1. Moreover, we performed model simulations to identify monitored vaccination strategies for successfully controlling measles in Nepal. Additionally, using the model, we analyzed the long-term dynamics of the epidemic. Our model demonstrates that the monitored vaccination programs can help control the potential resurgence of the disease. Due to the nationwide lockdown enforced by the government of Nepal during the COVID-19 pandemic, the scheduled immunization program was disrupted. As a result, amid the ongoing lockdown, measles outbreaks, including fatal cases, have been reported in several districts of Nepal. Moreover, measles cases in adult groups, albeit small in number, indicate that the previously neglected adult group may need to be brought into vaccine coverage to achieve the WHO’s goal of measles eradication around the world. To examine the role of measles-infected adult groups and to evaluate combined adult-child vaccination programs for eradication, we develop a further extended transmission dynamics model describing measles cases in adults and children. We validated the new model using measles outbreak cases in Nepal from November 24, 2022, to March 10, 2023. Detailed analyses of our model provide the vaccination reproduction number, conditions for measles eradication or persistence, and the role of contact network size. Furthermore, our results highlight that while children are the primary targets for measles outbreaks, a small infection in adults may act as a reservoir for measles, posing obstacles to eradication. Moreover, our model predicts that while impactful controls can be achieved by children-focused vaccines, a combined adult-child vaccination program may help ensure the eradication of the disease.
Effect of Changing the Dimension of Initial Debris Mass in the Dynamics of Landslide Generated Tsunami
(Department of Mathematics, 2021) Acharya, Grishma
Debris ow is a traveling mass of loose mud, soil, air, water and sand that moves down a slope caused due to gravity. When debris ows, landslides, or any gravitational mass ows hit closed or partially open water sources such as seas, oceans, fjords, hydraulic reservoirs, mountain lakes, bays and landslide dams, it results in tsunami (impulse water waves) by transforming their impact energy to water body, potentially causing damages of infrastructures and human casualties both near eld and the distant coastlines. The degree of hazard depends on the scale, types, location and process of the landslide. Volume or size of the initial debris mass that fails in the slope, is one of the dominant factors in accelerating the splash strength or intensity, the propagation and amplitudes of the subsequent water waves and potential dam breach or water spill over. Here, we numerically integrate the two-phase mass ow model [61] for quasi three dimensional, high-resolution simulation results with variation of size of the two-phase initial landslide or debris both longitudinally and laterally. In our numerical experimental results, we observe fundamentally di erent solid and uid wave structures in the reservoir, and the dynamics of submarine mass ow for di erent volumes of the release mass by extending or contracting the base area along down-slope and/or cross-slope directions. The simulation results show that tsunami amplitudes and run out extents are rapidly increased when the volume of initial release mass in the form of a triangular wedge is enlarged by increasing the base area through the increment of the length and breadth of the release base. This study can be an instructive tool to develop and implement tsunami hazard mitigation measures to enhance public safety and reduce potential loss.
Efficient Dynamic Flow Algorithms for Evacuation Planning
(Department of Mathematics, 2020) Dhungana, Ram Chandra
The large scale calamities caused by different natural or human-created disasters are challenging issues to protect life and their surroundings. A great loss of people and socio-economic damages of our society on such disasters is due to the lack of proper planning and their implementation rather than the disaster itself. These issues draw increasing attention of the researchers towards different aspects of disaster management. It is a complex task to develop a signiﬁcant and universally accepted solution strategy to handle such issues. During such disasters, the primary concern is to protect the life, property, and their surroundings with a minimum loss as far as possible. There are different solution approaches to have a signiﬁcant solution for an evacuation planning problem. Contraﬂow, the lane reversal strategy, is one of the widely accepted solution approaches for evacuation planning as it maximizes the outbound capacities of roads by reversing the required road directions and makes the trafﬁc smooth. This signiﬁcantly increases the ﬂow value and decreases the evacuation time. The abstract ﬂow model deals with the ﬂow paths (routes) that satisﬁes the switching property. This concept can be embedded in the contraﬂow technique to have the mathematical formulation on abstract contraﬂow models with efﬁcient algorithms for solving such abstract contraﬂow problems. In this work, different efﬁcient solution procedures are presented for maximum dynamic, lexicographically maximum, and earliest arrival abstract contraﬂow problems. This approach maximizes the ﬂow value in a given time and seeks to eliminate the crossing conﬂicts. The earliest arrival ﬂow problem is one of the most important aspects of evacuation planning with a given capacity and travel time. The objective of the problem is to send the maximum number of evacuees from the given sources to the sinks as quickly as possible. It maximizes the ﬂow value at each time instances simultaneously. Here, we study the earliest arrival ﬂow problem with the contraﬂow approach having supplies and demands in abstract network. v During the evacuation planning problem, one of the essential components is the facility location as it correlates the pre- and post-disaster management. Appropriate facility locations and transportation facilities play a vital role in the solution of evacuation planning problems. Here, the network facility location and the contraﬂow approach are incorporated into the ﬂow models and some efﬁcient algorithms are presented to locate the facility with an objective of minimum ﬂow loss on the evacuation network. Our facility location contraﬂow solutions obtain optimal plans concerning the given and as well as arbitrary locations. With limited resources, it is not an easy task to develop a universally accepted model to handle different aspects and challenges of the evacuation planning problem. However, the budget-constrained network ﬂow improvement approach plays a signiﬁcant role to evacuate the maximum number of people within the given time horizon for the budget provided. We consider an evacuation planning problem that aims to shift the maximum number of evacuees from a danger area to a safe zone in limited time under the budget constraints for network modiﬁcation. In this work, different ﬂow improvement strategies for ﬁxed switching costs will be investigated namely integral, rational, and either to increase the full capacity of an arc or not at all. A solution technique on a static network is extended to the dynamic one. Moreover, we introduce the static and dynamic maximum ﬂow problems with lane reversal strategy and also propose efﬁcient algorithms for their solutions. Here, the contraﬂow approach reverses the direction of arcs concerning the lane reversal costs to increase the ﬂow value. As an implementation of an evacuation plan may demand a large cost, the solutions proposed in this thesis with budget-constrained problems play an important role in practice. In this thesis, the contraﬂow models and their solutions strategies have been established and investigated in an abstract network topology. To allocate the facility during the evacuation process FlowLoc problems and their solution have been introduced in the evacuation network. The arc switching costs have been considered for the ﬁrst time in the evacuation network. These optimization methods play signiﬁcant roles in maximizing the ﬂow and minimizing the evacuation time, and also have the great support for logistics and emergency vehicle movements in disasters.
Evacuation Optimization with Minimum Clearance Time
(Department of Mathematics, 2020) Adhikari, Iswar Mani
We are under the threats of natural or human-created disasters. Disasters are unavoidable and are mostly uncertain to happen. If occurs, the situation becomes vulnerable, effects badly the society, and its socio-economic status. Its direct impact is on the trafﬁc systems. On the other hand, the increasing number of complex trafﬁc networks brought difﬁculty in managing the rush hour trafﬁc as well as the large events in urban areas. The optimal use of the vehicles and their assignments to the appropriate shelters from the disastrous zones are highly complicated in emergency situations. The maximum efﬁciency and effectiveness of the evacuation planning can be achieved by the appropriate assignment of the transit-vehicles during pre- and post-disaster operations. The evacuation planning problem deals with sending the maximum number of evacuees from the danger zones to the safe zones in minimum time, as efﬁciently as possible. It can be further classiﬁed into microscopic and macroscopic planning. The microscopic planning deals with the individual evacuee’s behavior in which some probabilistic laws for individual evacuees movement are presented and mainly based on the simulation approaches. But in macroscopic planning, it is principally based on optimization approaches where the evacuees are treated as the homogeneous group and only the common characteristics are considered. Optimization approaches on such macroscopic evacuation planning can further be classiﬁed as a heuristic approach, population optimization, modeling as ﬂuid dynamics, mathematical programmings, trafﬁc management, optimal evacuation destination, and network ﬂow formulation. Among them, the dynamic network ﬂow formulation has been found suitable evacuation optimization approach with the variants of ﬂow maximization and/or time minimization problems. In such formulations, time can be considered as discrete or continuous. Evacuation planning problems are handled with different prospectives, namely, the transit-based, car-based, and pedestrian movements depending upon the movement of the evacuees on the evacuation scenarios. The transit-based planning problems are to vi minimize the duration of evacuation by routing and scheduling a ﬂeet of vehicles, say buses, as the bus-based evacuation planning problem. Such a problem is an important variant of the vehicle routing problem. Trafﬁc route guidance, destination optimization, and optimal route choice are some of the approaches to accelerate the evacuation planning process. Their effectiveness depends upon the evacuee arrival patterns at the pickup locations and their appropriate assignment to the transit-vehicles in the available evacuation network. An embedded network is composed of two constituent sub-networks, namely, the primary and the secondary sub-networks. Evacuees are to be collected at the pickup locations of the primary sub-network from the danger zone(s) and are to be assigned to transit-vehicles in the secondary sub-network. For time minimization evacuation planning problems, evacuees are to be collected in the earliest arrival ﬂow pattern at zero transit times and is to be followed by dominant vehicle assignments. Transit-vehicles are provided from the bus depot in the secondary sub-network. Pickup locations are taken as the sources for the subsequent process to minimize the overall network clearance time from the danger zone to safety. In our work, we have proposed an integrated optimization approach in such an embedding to achieve the minimum clearance time. The earliest arrival pattern respects the partial lane reversal strategy, whereas the better assignments are based on the dominance relations concerning the evacuation duration. We use the quickest transshipment partial arc reversal strategy to collect the evacuees in minimum time from the disaster zones to the pickup locations of the primary prioritized sub-network. By treating such pickup locations as sources, the available set of transit-buses is assigned simultaneously in the secondary sub-network to shift the evacuees ﬁnally to the sinks with minimum clearance time. The lane reversal strategy signiﬁcantly reduces the evacuation time and maximizes the ﬂow of evacuees, whereas reversing them only partially has an additional beneﬁt that the unused road capacities can be used for supplying emergency logistics and allocating facilities as well.
The Interplay Between Measure Theory and Topology
(Department of Mathematics, 2016) Rana, Jit Bahadur
A connection between measure theory and topology is established when a eld F is de ned in terms of topological properties. More precisely, we de ned F as the smallest eld containing all the open sets of a topological space , then there are interesting interrelation between measure theory and topology. We study the interrelation between topological space, open sets and continuous functions in one hand and measure space, measurable set and measurable function on the other.
Modeling and Analysis of Dynamics of Malaria Transmission with Control Measures: Imported Cases
(Institute of Science & Technology, 2024-08) Gautam, Ramesh; Prof.Dr. Kedar Nath Uprety
Many countries, including low and middle-income countries like Nepal, are facing many challenges in pursuing malaria elimination. Despite progress in reducing the malaria burden significantly, these countries still struggle with low levels of malaria transmission, making complete elimination difficult. Mathematical modeling of diseases like malaria provides essential insights into disease dynamics. These insights help develop effective public health strategies for allocating resources efficiently, and support evidence-based policies to reduce endemic diseases like malaria. In the global effort to eliminate malaria, human mobility and the relapse of Plasmodium Vivax and Plasmodium Ovale malaria pose significant challenges. In this thesis, first, we develop a mathematical model of malaria transmission, integrating the cross-border mobility of migrant workers from low-endemic countries like Nepal to high-endemic countries like India. The model describes how migrant workers become infectious abroad and bring malaria back to their home country as imported cases. Despite complicated features with eight-dimensional nonlinear non-homogeneous systems, we were able to derive three disease-free equilibria and three epidemic thresholds, R0, R1, and R2, which establish their local stability. In addition, we established the theorems for global stability and uniform persistence. Our model simulations show that among Insecticide Treated Nets (ITN), Indoor Residual Spraying (IRS), Border Screening and Isolation (BSI), and Migration Reduction (MR), MR is the most effective strategy at low mosquito biting rates, whereas ITN is the most effective at high mosquito biting rates for malaria control and elimination. Second, we conducted a thorough bifurcation analysis of our model to examine whether migration can cause backward bifurcation phenomena, demonstrating bistability for threshold values less than one. Along with theoretical derivations, we developed MATLAB code to obtain different types of bifurcation diagrams associated with various modes of mobility. Our backward bifurcation analysis revealed three major results considering three different mobility conditions based on policies implemented at home and abroad: (a) If the mobility of migrants is completely restricted, the home country becomes free from malaria when the threshold R0 < 1 and the disease-induced death rate falls below some threshold. (b) If the mobility of migrants continues with complete protection from malaria transmission abroad, then both the home country and migrants abroad are free from malaria when the threshold R1 < 1, provided the mobility of infectious migrants is below the certain threshold. However, a backward bifurcation occurs if the mobility exceeds the threshold. (c) If there is mobility of migrants without protection abroad, the home country can only reduce the malaria burden with local control strategies and by reducing the mobility rate of migrants below some levels. However, elimination is only possible if the abroad region is free from malaria. Third, we developed a model incorporating delay in relapses to address the role of Plasmodium vivax and Plasmodium Ovale malaria relapses in malaria elimination programs in low-endemic countries like Nepal. Our model analyses and simulations predict that in the absence of imported cases, with less than 50% initial relapse rate of malaria and less than 14% subsequent relapses within five months, malaria can potentially be eliminated by 2025. However, the initial relapse rate above 28% and the subsequent relapses above 25% stand as the obstacle to eliminating malaria by 2025. Also, shortening the relapse interval to two months under an initial relapse rate below 50% enables malaria elimination by 2024, while extending it to six months will cause a delay in elimination beyond 2025. Furthermore, periodic outbreaks are observed via Hopf bifurcation when the reproduction number exceeds unity. Our study has made two major policy recommendations to ensure the successful elimination of malaria in Nepal by 2026: (1) Implementing a comprehensive awareness program to protect migrants from malaria transmission abroad, coupled with rigorous border screening and isolation of infectious migrants, to maintain the mobility rate of infectious migrants below the threshold. (2) Establishing and enforcing a comprehensive radical cure treatment protocol, along with a structured follow-up program, to keep relapse proportions below a critical level. नेपालजस्ता न्यून र मध्यम आय भएका देश लगायत धेरै देशले औलो उन्मूलनका लागि धेरै चुनौतीहरूको सामना गरिरहेका छन् । औलोको बोझलाई उल्लेखनीय रूपमा घटाउनमा प्रगति भएता पनि, यी देशहरू अझै पनि औलो प्रसारणको न्यून स्तरसँग संघर्ष गरिरहेका छन् र पूर्ण रूपमा उन्मूलन गर्न गाह्रो भइरहेको छ । औलो जस्ता रोगहरूको गणितीय मोडेलिङले रोगको गतिशीलतामा आवश्यक अन्तरदृष्टि प्रदान गर्दछ । यी अन्तर्दृष्टिहरूले प्रभावकारी रूपमा स्रोतहरू विनियोजन गर्न प्रभावकारी सार्वजनिक स्वास्थ्य रणनीतिहरू विकास गर्न मद्दत गर्दछ, र औलो जस्ता लामोसमयसम्म रहने रोगहरू कम गर्न र उन्मूलन गर्नका लागी प्रमाण-आधारित नीति निर्माण गर्नका लागी सहयोग गर्दछ। औलो उन्मूलन गर्ने विश्वव्यापी प्रयासमा, मानव गतिशीलता र प्लाज्मोडियम भाइभ्याक्स र ओभेल मलेरियाको पुनरावृत्तिले महत्त्वपूर्ण चुनौतीहरू खडा गरेको छ । यस थीसिसमा, सर्वप्रथम, हामी नेपाल जस्ता न्यून-स्थानीय देशहरूबाट भारत जस्ता उच्च-स्थानीय देशहरूमा आप्रवासी कामदारहरूको सीमापार गतिशीलतालाई एकीकृत गर्दै औलो प्रसारणको गणितीय मोडेल विकास गर्छौं । मोडेलले प्रवासी कामदारहरू विदेशमा कसरी संक्रामक हुन्छन् र औलो लाई आयातित केसहरूको रूपमा आफ्नो देशमा फिर्ता ल्याउँदछ भनेर वर्णन गर्दछ । आठ-आयामी ननलिनयर ननहोमोजीनियस प्रणालीहरूसँगको जटिलताको बावजुद पनि, हामीले तीन औलो मुक्त सन्तुलन र तीन महामारी मापक थ्रेसहोल्डहरू, R0, R1, र R2 प्राप्त गर्न सक्षम भयौं, जसले तिनीहरूको स्थानीय स्थिरता स्थापना गर्दछ । थप रूपमा, हामीले विश्वव्यापी स्थिरता र समान दृढताका लागि प्रमेयहरू स्थापना गरेका छौं । हाम्रो मोडेल सिमुलेशनहरूले देखाउँदछ कि कीटनाशक-उपचारित जालहरू (ITN), भित्री अवशिष्ट स्प्रेइ (IRS), सीमा जाँच र अलगाव (BSI), र माइग्रेसन रिडक्सन (MR), हरूमा MR कम लामखुट्टे टोक्ने दरहरूमा सबैभन्दा प्रभावकारी रणनीति र ITN औलो नियन्त्रण र उन्मूलनको लागि उच्च लामखुट्टेले टोक्ने दरमा सबैभन्दा प्रभावकारी पुष्टी भय। दोस्रो, हामीले माइग्रेसनले गर्दा महामारी मापक थ्रेसहोल्ड एक भन्दा कम भयर मात्रै औलो मुक्त हुन नसक्ने र थ्रेसहोल्ड मानहरूको लागि बिस्टेबिलिटी प्रदर्शन गर्दै पछाडिको विभाजन घटना निम्त्याउन सक्छ कि भनेर जाँच्नको लागि मोडेलको विस्तृत विभाजन विश्लेषण गरीयको छ । सैद्धान्तिक व्युत्पन्नहरूको साथमा, हामीले गतिशीलताका विभिन्न मोडहरूसँग सम्बन्धित विभिन्न प्रकारका विभाजन रेखाचित्रहरू प्राप्त गर्न MATLAB कोड विकास गर्यौं । हाम्रो पछाडी विभाजन विश्लेषणले स्वदेश र विदेशमा लागू हुनसक्ने नीतिहरु का कारण तीनवटा प्रमुख नतिजाहरू पत्ता लगाएको छ जुन तीन फरक गतिशीलता अवस्थाहरूमा आधारित छ: (क) यदि आप्रवासीहरूको गतिशीलता पूर्ण रूपमा प्रतिबन्धित गर्न सकियो भने, थ्रेसहोल्ड R0 <1 र रोग-प्रेरित मृत्यु दर केही थ्रेसहोल्ड भन्दा तल झर्दा गृह देश औलो बाट मुक्त हुन्छ । (ख) विदेशमा औलोको संक्रमणबाट पूर्ण सुरक्षाका साथ प्रवासीहरूको आवतजावत जारी छ भने, थ्रेसहोल्ड R1 <1 र संक्रामक आप्रवासीहरूको गतिशीलता निश्चित थ्रेसहोल्डभन्दा कम भएमा स्वदेश र विदेशमा बस्नेहरू दुवै औलोबाट मुक्त हुन्छन् । यद्यपि, यदि गतिशीलता थ्रेसहोल्ड भन्दा बढी छ भने पछाडि विभाजन हुन्छ र उन्मुलनको लागी थप प्रयास जरुरी पर्दछ । (ग) विदेशमा सुरक्षा बिना आप्रवासीहरूको गतिशीलता भएमा, स्वदेशले स्थानीय नियन्त्रण रणनीतिहरू र केही तहभन्दा तल आप्रवासीहरूको गतिशीलता दर घटाएर मात्र औलोको बोझ कम गर्न सक्छ । तर, विदेश क्षेत्र औलोमुक्त भए मात्रै उन्मूलन सम्भव हुन्छ । तेस्रो, हामीले पुन: आवृत्तिमा ढिलाइलाई समावेश गर्दै एक मोडल विकास गर्यौं जसले कम संक्रमण दर भएका देशहरू जस्तै नेपालमा औलो को उन्मूलन कार्यक्रमहरूमा प्लाज्मोडियम विवाक्स र ओभाले मलेरियाका पुन: आवृत्तिहरूको भूमिकालाई सम्बोधन गर्दछ । हाम्रो मोडल विश्लेषणहरू र सिमुलेशनहरूले भविष्यवाणी गर्छन् कि आयातित केसहरूको अभावमा, मलेरियाको प्रारम्भिक पुन: आवृत्ति दर ५०% भन्दा कम र पाँच महिनाभित्र दोस्रो पुन: आवृत्ति १४% भन्दा कम हुने अवस्थामा, २०२५ सम्ममा मलेरिया उन्मूलन गर्न सकिन्छ । तर, प्रारम्भिक पुन: आवृत्ति दर २८% भन्दा माथि र दोस्रो पुन: आवृत्ति २५% भन्दा माथि हुने अवस्थामा २०२५ सम्ममा मलेरियाको उन्मूलनमा बाधा पुग्नेछ । साथै, महामारी थ्रेसहोल्ड R0 एक भन्दा माथि पुगेमा हप्फ बिफरकेसनको माध्यमबाट समय-समयमा प्रकोपहरू देखिन्छन् । हाम्रो अध्ययनले २०२६ सम्ममा नेपालमा मलेरियाको सफल उन्मूलन सुनिश्चित गर्न दुई प्रमुख नीतिगत सिफारिसहरू गरेको छ: (१) विदेशमा मलेरिया संक्रमणबाट आप्रवासीहरूलाई जोगाउन व्यापक चेतना कार्यक्रम लागू गर्दै, संक्रमण भएका आप्रवासीहरूको गति दरलाई थ्रेसहोल्डभन्दा तल राख्न कडा सीमा स्क्रिनिङ र संक्रमित आप्रवासीहरूको अलगाव गर्न । (२) पुन: आवृत्ति अनुपातलाई एक महत्वपूर्ण स्तरभन्दा तल राख्न व्यापक उपचार विधि र संरचित अनुगमन कार्यक्रमको स्थापना र कार्यान्वयन गर्न ।
Multi-Commodity Dynamic Flow Problems With Intermediate Storage and Varying Transit Times
(Institute of Science & Technology, 2023-09) Khanal, Durga Prasad
Network flow problems, with single or multiple commodity, are commonly used to transship the objects from the source to the destination. In single commodity flow problem, objects are considered to be uniform and are send from a source to a sink (in case of multiple source-sink, it can be reduced to single source-sink by assigning virtual source and sink) whereas in multi-commodity flow problem, different commodities are transshipped from respective sources to corresponding sinks. Similarly, flow with intermediate storage is a network flow problem in which flow from the source is not only sent to the sink but also at appropriate intermediate shelters so that total flow out from the source is maximized. On the other hand, contraflow is very well known and commonly used technique of flow increment in two-way network topology in which oppositely directed anti-parallel arcs are reversed towards the destination. As an extension of the flow with intermediate storage in multi-commodity flow (MCF), we solve the maximum static MCF problem in polynomial time and maximum dynamic MCF problem in pseudo-polynomial time. For the polynomial time approximation, we present priority based maximum dynamic MCF which can be useful in disaster management. Similarly, we provide the approximate solutions to maximum and quickest MCF problems by sharing the capacity in bundle (common) arcs using proportional capacity sharing technique in polynomial time and flow-dependent capacity sharing technique in pseudo-polynomial time. We also discuss the polynomial time approximations of inflow-dependent quickest MCF problem with partial contraflow configuration using length bound and ∆-condense approaches. Besides the different applications of network flow models, our main concern is to relate our problems to the evacuation scenarios. So, we consider source/s as the danger zone/s, sink/s as the safe zone/s and intermediate shelters comparatively safer than the source/s. As single commodity flow problem is a special case of multi-commodity flow problem, we solve the single commodity maximum dynamic flow (MDF) and earliest arrival flow (EAF) problems with intermediate storage in general network and series-parallel network, respectively, by using temporal repetition of the flow in polynomial time complexity. Similarly, to solve the contraflow problem with asymmetric transit times in anti-parallel arcs, we introduce anti-parallel path decomposition technique. For the implementation of temporally repeated solution to MDF with intermediate storage and anti-parallel path decomposition to asymmetric contraflow network, we apply our solution strategies to the real road network of Kathmandu, Nepal as the case illustrations. For the sequential development of the thesis, we start with single commodity flow problem and turn to the multiple commodity case. Abstract network flow concerns with shifting of the flow not in node-arc form but in element-path form in which paths must satisfy the switching property. We incorporate the flow with intermediate storage in abstract network and solve the static, lexicographic static and dynamic flow problems in polynomial time complexity. It helps to eliminate the congestion by diverting the flow in non-crossing sides and storing the excess flow at intermediate shelters (elements). To improve the flow in abstract network, we propose the partial switching technique and solve maximum and quickest flow problems in polynomial time. The facility allocation problem is another important area of network flow problem whose objective is to maximize the flow transmission along with placement of the facilities at appropriate locations. We give the bi-level formulation of the problem in which upper level problem searches an appropriate location for the placement of the facility and lower level problem finds the optimal solution of maximum flow problem. A naive approach and KarushKuhn-Tucker (KKT) transformation with big-M constant and ϵ bound method are solution approaches used to solve the problem. Keywords: Intermediate storage, asymmetric contraflow, anti-parallel path decomposition, multi-commodity, commodity priority, proportional and flow-dependent capacity sharing, facility allocation . सामान्यतया कमोडिटीहरू (एकल वा बहु-कमोडिटी) लाई स्रोतबाट गन्तव्यमा पठाउनको लागि प्रयोग गरिने सञ्जाल (Network) संग सम्बन्धित प्रवाह समस्याहरू लाई सञ्जाल प्रवाह भनिन्छ। एकल कमोडिटी प्रवाह समस्यामा कमोडिटीहरूलाई समान मानिन्छ र एक स्रोतबाट गन्तव्यमा पठाइन्छ (बहु स्रोत-गन्तव्यको अवस्थामा यसलाई भर्चुअल स्रोत र गन्तव्य प्रदान गरेर एकल स्रोत-गन्तव्यमा घटाउन सकिन्छ) जबकि बहु-कमोडिटी प्रवाह समस्यामा विभिन्न कमोडिटीहरू सम्बन्धित स्रोतहरूबाट सम्बन्धित गन्तव्यहरूमा पठाइन्छ। त्यसै गरी, मध्यवर्ती भण्डारणसहितको प्रवाह एक सञ्जाल प्रवाह समस्या हो जसमा स्रोतबाट प्रवाहित कमोडिटीहरू गन्तव्यमा मात्र नभइ उपयुक्त मध्यवर्ती आश्रयहरूमा पनि पठाइन्छ ताकि स्रोतबाट कुल प्रवाह अधिकतम होस। अर्कोतर्फ, कन्ट्राफ्लो दुई-तर्फी सञ्जालमा प्रवाह वृद्धिको लागी प्रयोग हुने विधि हो जसमा विपरीत दिशाका समानान्तर आर्कहरू गन्तव्यतर्फ उल्टाइन्छ। बहु-कमोडिटी प्रवाह (Multi-commodity flow) मा मध्यवर्ती भण्डारणको साथ प्रवाहको विस्तारको रूपमा हामी पोलिनोमीयल (Polynomial) समयमा अधिकतम स्थिर (Maximum static) बहु-कमोडिटी प्रवाह समस्या र सुडो-पोलिनोमीयल (Pseudo-polynomial) समयमा अधिकतम गतिशील (Maximum dynamic) बहु-कमोडिटी प्रवाह समस्या समाधान गर्छौं। पोलिनोमीयल समय अनुमानको लागि हामी प्राथमिकतामा आधारित अधिकतम गतिशील बहु-कमोडिटी प्रवाह प्रस्तुत गर्दछौं जुन विपद् व्यवस्थापनमा उपयोगी हुन सक्छ। त्यसैगरी, हामी पोलिनोमीयल समयमा समानुपातिक क्षमता साझेदारी प्रविधि र सुडो-पोलिनोमीयल समयमा प्रवाह-निर्भर क्षमता साझेदारी विधि प्रयोग गरेर बन्डल (साझा) आर्कहरूमा क्षमता साझेदारी गरेर अधिकतम (Maximum) र द्रुत (Quickest) बहु-कमोडिटी प्रवाह समस्याहरूको समाधान प्रदान गर्दछौं। हामी Length bound र Delta-condensed विधिहरू प्रयोग गरेर आंशिक कन्ट्राफ्लोको साथ प्रवाह-निर्भर द्रुत बहु-कमोडिटी प्रवाह समस्याको पोलिनोमीयल समयमा समाधान गर्छौं। सञ्जाल प्रवाह मोडेलहरूको विभिन्न अनुप्रयोगहरू बाहेक हाम्रो मुख्य लक्ष्य भनेको हाम्रा समस्याहरूलाई विपद् व्यवस्थापन परिदृश्यहरूसँग सम्बन्धित गर्नु हो। त्यसकारण हामी स्रोत/हरूलाई खतरा क्षेत्र/हरू, गन्तव्य/हरूलाई सुरक्षित क्षेत्र/हरू र मध्यवर्ती आश्रयहरूलाई स्रोत/हरू भन्दा तुलनात्मक रूपमा सुरक्षित मान्दछौं। एकल कमोडिटी प्रवाह समस्या बहु-कमोडिटी प्रवाह समस्या को एक विशेष परिस्थिति हो। हामी एकल कमोडिटी अधिकतम गतिशील प्रवाह (Maximum Dynamic Flow (MDF)) र प्रारम्भिक आगमन प्रवाह (Earliest arrival flow (EAF)) समस्याहरू क्रमशः सामान्य सञ्जाल र श्रृंखला-समानान्तर (Series-parallel) सञ्जालमा मध्यवर्ती भण्डारणको साथ पोलिनोमीयल समयमा प्रवाहको पुनरावृत्ति (Temporally repeated) प्रयोग गर्दै समाधान गर्छौं। त्यसैगरी, विपरीत-समानान्तर आर्कहरूमा सममित (Asymmetric) ट्रान्जिट समयहरूसँग कन्ट्राफ्लो समस्या समाधान गर्न हामी विपरीत-समानान्तर पथ गठन (Anti-parallel path decomposition) प्रविधिको विकास गर्छौं। मध्यवर्ती भण्डारण सहितको अस्थायी पुनरावृत्ति सममित र कन्ट्राफ्लो सञ्जालमा विपरीत-समानान्तर पथ गठन विधि प्रयोग गर्दै MDF को मामला अध्ययन दृष्टान्त (Case illustration) को लागि हाम्रो समाधान रणनीतिलाई काठमाडौंको सडक सञ्जालमा लागू गर्छौं। शोध प्रबन्धको क्रमिक विकासको लागि, हामी एकल कमोडिटी प्रवाह समस्याबाट सुरु गर्छौं र बहु-कमोडिटी मामलामा फर्कन्छौं। नोड-आर्क रुपमा नभई पथहरूले स्विच गर्ने गुण सहित एलिमेन्ट-पथ रुपमा कमोडिटी प्रवाह लाई अमूर्त (Abstract) सञ्जाल प्रवाह भनिन्छ। हामी अमूर्त (Abstract) सञ्जालमा मध्यवर्ती भण्डारणको साथ प्रवाहलाई पोलिनोमीयल समयमा स्थिर (Static), लेकशीकोग्राफिक स्थिर (Lexicographic static) र गतिशील (Dynamic) प्रवाह समस्याहरू समाधान गर्छौं। यसले गैर-क्रसिङ पक्षहरूमा पथहरूलाई मोड्दै र मध्यवर्ती आश्रयहरू (एलिमेन्टहरू) मा अतिरिक्त प्रवाह भण्डारण गरेर भीड हटाउन मद्दत गर्दछ। अमूर्त सञ्जालमा प्रवाह सुधार गर्न हामी आंशिक स्विचिंग (Partial switching) विधि प्रस्ताव गर्दछौं र पोलिनोमीयल समयमा अधिकतम र द्रुत प्रवाह समस्याहरू समाधान गर्दछौं। सुविधा बाँडफाँड (Facility allocation) समस्या सञ्जाल प्रवाह समस्याको अर्को महत्त्वपूर्ण क्षेत्र हो जसको उद्देश्य उपयुक्त स्थानहरूमा सुविधाहरूको प्रतिस्थापनसँगै प्रवाह प्रसारणलाई अधिकतम बनाउनु हो। हामी समस्याको द्वि-स्तरीय सूत्र दिन्छौं जसमा माथिल्लो तहले सुविधाको स्थानको लागि उपयुक्त स्थान खोज्छ र तल्लो तहले अधिकतम प्रवाह समस्याको इष्टतम समाधान फेला पार्छ। Big-M र epsilon-Bound विधिको साथ Karush-Kuhn-Tucker (KKT) रूपान्तरण समस्या समाधान गर्न प्रयोग गरिने समाधान उपायहरू हुन्।
Newton Type Iterative Methods for Solving Nonlinear Equations
(Faculty of Mathematics, 2017) Jnawali, Jivandhar
Solving single variable nonlinear equations efficiently is an important consideration in numerical analysis and has wide range of applications in all elds of science and engineering. Finding the analytic solutions of such equations is not always possible. Newton's method is the most widely used numerical method for solving such equations. In this thesis, we have developed several new Newton type iterative methods for solving nonlinear equations of a single variable. To obtain these methods, we used different techniques such as: (i) amalgamation of existing methods; (ii) amalgamation of existing and our investigated methods with the secant method; (iii) amalgamation of existing methods and modi ed secant method; (iv) idea of integral approximation; and (v) use of inverse function methods. The work done in this thesis is inspired by the work of Potra and Pt ak, Kasturiarachi, Jain, Weerakon and Fernando, Ozban, Dhegain and Hajarian, Ujevi c, Erceg and Laki c, Amit and Basqular, Hasanov, Ivanov and Nedzhibov as well as recent work of McDaugall and Wotherspoon. For each method obtained in this thesis, the order of convergence has been calculated and compared with that of the similar existing methods. Also, most of the methods are supported by numerical examples.
Numerical approaches for solving special functions in fractional calculus
(Institute of Science & Technology, 2024-05) Pariyar, Shankar; Dr.Jeevan Kafle
The study uses systemic diagrams to compare fractional equations at  values 0 <  < 1 with non-fractional equations at  = 1. Under Caputo Fabrizio's fractional differential equation, it looks at the identity, sine, and cosine functions. While avoiding non-zero fractional derivatives of constant functions, the Grunwald-Letinikov (G-L) numerical solution assures precision. Ensuring numerical results from fractional and traditional calculus remain comparable is the goal of this research. It is necessary to use non-linear operators because deviations arise when fractional orders diverge. Ideas such as curve smoothness and value fluctuations are made clear by illustrative examples. Additionally, utilizing MLF at different values of , , and  within the range 0 <  < 1, the study examines the numerical solution of one, two, and three parameters. It is shown how Magnus Gosta (M.G.) Mittag-Leffler's computations are used in scientific and practical fields. In order to familiarize with the latest research trends and MLF consequences, this study provides a thorough overview of the several types of Mittag-Leffler functions (MLF) that can be found in the literature. This research significantly enhances the knowledge of fractional derivatives and the integration of transcendental forms of functions. Using the Lagrange interpolation approach, we will demonstrate some numerical strategies, including the L1 scheme for 0 <  < 1, the L2 scheme for 1<  < 2, and the L1 method for the Caputo Fabrizio derivative.
Numerical modeling of influence of source in heat transformation: An application in blacksmithing metal heating
(Department of mathematics, 2022) Kandel, Hari Prapanna
Partial differential equations (PDEs) are used to mimic a variety of real-world physical issues. A standard parabolic PDE of the form u ; ( > 0) is an 1D heat equation. In a regular form of domain, the heat equation has an analytical solution. Computing an analytical solution becomes challenging, if not impossible, any time the domain of such modeled issues has an uneven shape. In this case, numerical methods can be used to ﬁnd the numerical solution of these PDEs. Through the domain’s discretization into a limited number of areas. One of the numerical techniques used to determine the numerical solutions of PDEs is the ﬁnite difference method (FDM). Here, the FTCSSfor the one-dimensional heat equation and the numerical computation of its solution using FTCSS are discussed. Furthermore, numerical solution and analytic solution of heat equation has been compared and analyzed. Additionally, the 1D heat equation with variable starting conditions (ICs) and numerous initial conditions (ICs)has been solved numerically using FDMs. Blacksmiths heated the parts at various temperatures and locations to mold different metals into the necessary shapes. The numerical solution method for the 1D heat problem given here can be used to solve heat equations used in engineering and scientiﬁc disciplines. t = u xx
Numerical Solutions of 2D Incompressible Navier-Stokes Equations in Variable Viscosity Case
(Faculty of Mathematics, 2016) Thapa, Dhak Bahadur
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On the Study of Distribution of Primes and Twin Prime Conjecture
(Faculty of Mathematics, 2016) Adhikari, Khagendra
The distribution of primes, mainly focusing on the Tchebycheff estimates of prime counting function, Mertens Theorem which are most signiﬁcant results for distribution of primes have beeen studied in this thesis. Distribution of Twin Primes, Twin Prime Conjecture and some developments towards the Twin Prime Conjecture is also studied. The alternative approaches for the Twin Prime Conjecture has also been studied in this thesis.
Optimization Models and Algorithms for Evacuation Planning
(Faculty of Mathematics, 2020) Nath, Hari Nandan
An important strategy, to save a life from natural or human-created disasters, is to evacuate the population from the disastrous zones to safe places. Intelligent evacuation planning requires a carefully designed trafﬁc plan with optimal use of the facilities available. Reversing the direction of the trafﬁc ﬂow in the appropriate road segments, known as a contraﬂow approach, has been an important strategy in evacuation planning. To avoid unnecessary arc reversals, we introduce partial contraﬂow approach and design strongly polynomial algorithms to solve maximum static, maximum dynamic, and quickest ﬂow problems with partial arc reversals, and polynomial-time algorithms to solve maximum static/dynamic abstract partial contraﬂow problems. Sometimes the travel time on an arc may change when it is reversed. We propose a network transformation that helps convert the existing contraﬂow algorithms to the ones with orientation-dependent transit times. To address the dependency of transit times in the ﬂow, we extend the contraﬂow approach to inﬂow-dependent transit times and load-dependent transit times. Realizing the NP-hardness of such problems, we propose strongly polynomial (2 + )-approximation algorithms to solve the corresponding contraﬂow/partial contraﬂow problems. If facilities are to be adjusted on arcs to facilitate evacuation, there may be an increase in the evacuation time. We introduce the quickest FlowLoc model to address such a problem. We prove that the single facility case of the problem can be solved in strongly polynomial time. Proving NP-hardness of the multi-facility case, we propose two heuristics. Taking the case of a Kathmandu road network, the faster heuristic has an average optimality gap of 3.48% and an average running time of 0.17 seconds. The corresponding values for the slower heuristic are 0.18% and 1.02 seconds. The algorithms for quickest FlowLoc problem with arc reversals are also designed. With an objective to maximize the static/dynamic ﬂows, and minimize quickest ﬂow, the problem of choosing a single shelter location are modeled as MaxStatic, MaxDynamic, Quickest sink location problems respectively. We establish that each of such a problem can be solved in strongly polynomial time with or without arc reversals. By reversing the direction of the trafﬁc ﬂow towards the sink, a contraﬂow conﬁguration may obstruct the paths towards the source. We model the problem of maximizing the dynamic contraﬂow saving a path not exceeding a given length as a mixed binary integer linear programming problem. The analogous problem of minimizing the evacuation time is modeled as a mixed binary integer programming problem with a fractional objective. A linearization strategy is suggested so that the algorithms to solve the mixed integer linear programming problems can be used. The problem of minimizing the path length and maximizing the dynamic contraﬂow has been modeled as a bicriteria optimization problem. A procedure using -constrained method is constructed to obtain efﬁcient solutions. The solutions, using available software solvers, considering a road network of Kathmandu city can be obtained within 1 second. We also model the problem of minimizing the path length and evacuation time as a bicriteria problem and construct a procedure to solve it. We model a path saving model maximizing the dynamic contraﬂow to optimize a general objective, as a bilevel program. To solve the problem, we replace the lower level problem by Karush-Kuhn-Tucker (KKT) conditions converting it to a single level non-linear binary integer program. We linearize it using a big M method and also suggest a procedure to tune M.
Optimization models with exclusive bus lanes
(Department of Mathematics, 2022) Chand, Gaurab
Exclusively reserved lane for public buses in arterial road of the city is called exclusive bus lane (EBL). In this research study, we survey network optimization EBL models, then we review min-max dynamic optimization EBL model with three modes of vehicles. Major upgraded terms on reviewed model have been taken prior origin count of the bus travel time, bureau of public road (BPR) constraint to the car mode and maximum number of motorcycle rider constraint. Among them, BPR constraint has impacted signi cantly over objective function as well as planning of EBL on the transportation network. Tra c data related to the motorcycle mode had been estimated using statistical tool by increasing the capacity of arcs and without loss of generality with original data of buses and cars. We prefer parallel genetic algorithm (PGA) for the solution of the revised problem and proved that complexity is NP-hard. A numerical example is revealed as a reviewed optimization network model to achieve the feasibility and therefore yield optimal solution.
Orthogonality in Normed Linear Spaces
(Department of Mathematics, 2022) Ojha, Bhuwan Prasad
This thesis deals with the orthogonality in normed linear spaces. The goal is to investigate and study different notions of orthogonality in normed spaces. By utilizing the 2-HH norm and bounded linear operators, some notions of orthogonality are introduced and then, different properties of orthogonality in relation to these orthogonalities are studied. We generalize the Robert, Birkhoff-James, and a new orthogonality in terms of the 2HH norm, and study the main properties of orthogonality. We prove that the Birkoff and Robert orthogonality in terms of the 2-HH norm are equivalent if the underlying space is real inner product space. Further, we prove that the isosceles orthogonality is homogeneous if and only if it is additive. Additionally, we prove that the orthogonality relation of type (I) in terms of 2-HH norm satisﬁes non-degeneracy, simpliﬁcation, continuity, and uniqueness properties. Moreover, we prove that the Carlsson orthogonality in terms of bounded linear operators also satisﬁes non-degeneracy, simpliﬁcation, and continuity properties. In the case of norm attaining bounded linear operator with disjoint support in a Hilbert space H, we prove that two operators are orthogonal in the sense of Pythagoras if and only if they are orthogonal in the sense of isosceles. In terms of buonded linear operators, we prove that the Pythagorean orthogonality and orthogonality relation of type (I), imply the Birkhoff-James orthogonality, but the converse may not be true. Under the restriction of an element belonging to the norm attainment set, we prove that the orthogonality of images also implies the orthogonality of operators in the Carlsson as well as Robert’s sense. Finally, as applications, we prove that the Pythagorean orthogonality implies the best approximation, and the best approximation (resp. best approximation) and Birkhoff orthogonality ( resp. Birkhoff orthogonality) are equivalents. Keywords: Normed linear spaces, Inner-product space, Birkhoff-James orthogonality, Pythagorean orthogonality, p-HH norm, Best approximation

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