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.
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.
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।
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.
Quantum Calculus: New Development and Applications
(Institue of Science & Technology, 2024-06) Tiwari, Pitamber; Prof. Dr. Prof. Dr. Chet Raj Bhatta
The investigation of the calculus without the notion of limits is quantum calculus or q-calculus. This calculus is applied to estimate the discrete phenomena as well. The famous result on the theory of convex functions is the Hermite-Hadamard type integral inequality which provides a lower and an upper bounds for the integral average of any convex function defined on a compact interval involving the mid-point and the end points of the domain having a wider application for generalized means, information measures, quadrature rules, and many more. In this thesis, some new convexities related to harmonic convexity have been defined and for each new convexity, a corresponding Hermite-Hadamard inequality has been proved. The q-analogues of these inequalities have also been obtained. Apart from this, in this thesis we have generalized harmonically convex function to m-harmonically convex function and m-harmonically P-function and have obtained some Hermite-Hadamard type integral inequalities whose first order derivatives are m-harmonically convex functions. The integral mean of a convex function is connected with the Hermite-Hadamard inequality. From the results on the products of two classical convex functions as given by B. G. Pachpatte, in this thesis, we have been able to establish some new results on the products of generalized geometric convex functions, and these results are further extended to q-analogues which is a new paradigm in convexity theory and integral inequality. Diagnosing the idea of the classical convex functions, it is enhanced into m- convex functions which have helped to extend a convex function's equality of quantum estimate into an m-convex function. Using the resulted information, a few novel Hermite-Hadamard integral inequalities are established whose first order q-derivatives are m-convex functions, and these results are presented in q-analogues too. In this thesis, Hermite-Hadamard's inequalities via Riemann-Liouville fractional integral for the case of harmonically convex function as well as the products of two harmonically convex functions via Riemann-Liouville fractional integrals are also established. सिमा विनाको आधारमा खोज गरिएको calculus लाई quantum calculus वा q-calculus भनिन्छ । यो calculus को माध्यमबाट discrete (खण्डित) घट्नाहरूको पनि अनुमान गर्न मद्दत मिल्छ । Theory of convex functions र Theory of Inequality मा Hermite-Hadamard प्रकारको integral inequality एक अभिन्न inequality हो । यो inequality को मद्दतबाट कुनै पनि convex function को लागि माथिल्लो र तल्लो सिमा निर्धारण गर्न सकिन्छ । यस किसिमको inequality हरू generalized means, information measures, quadratures rules इत्यादिमा प्रयोग गरिन्छ । यो Thesis मा केहि नयां प्रकारका convexities लाई generalized form मा परिभाषित गरि तिनिहरूको Hermite-Hadamard प्रकारका integral inequality प्रमाणित गरिएका छन् । यसरी प्रमाणित गरिएका inequality लाई quantum calculus को स्वरूपमा पनि प्रमाणित गरिएका छन् । B. G. Pachpatte ले प्रस्तुत गरेको product of classical convex functions नतिजाको आधारमा अन्य प्रकारका convex functions जस्तै geometrically- arithmetic, harmonically – arithmetic, geometrically- geometric convex function हरूलाई generalized form मा प्रस्तुत गरी तिनिहरूका product form को नतिजाहरू प्रमाणित गरिएको छ । यसरी प्राप्त नतिजाहरूलाई q-calculus मा extension गरिएका छन् । यो Thesis मा harmonically- convex functions को लागि Riemann-Liouville fractional integral को माध्यमबाट Hermite-Hadamard integral inequality प्रमाणित गरी तिनिहरूको product form को नतिजा पनि प्रस्तुत गरिएका छन् । भविष्यमा अन्य अनुसन्धानकर्ताहरूले Theory of convex functions र Theory of inequalities मा बिभिन्न प्रकारका convex functions को Hermite- Hadamard प्रकारका integral inequality र तिनिहरूको quantum version को domain मा काम गर्न सक्नेछन् ।
Regularity of 2D Surface Quai-Geostrophic (SQG) Equations
(Institute of Science & Technology, 2023-02) Shrestha, Pawan
In this research, we delve into three distinct topics within the realm of nonlinear fluid dynamics, namely the generalized Korteweg-de Vries (KdV)-type equation, the regularity of solutions in the $2$D Surface Quasi-Geostrophic (SQG) equation, and the behavior of water waves under indefinite boundary constraints. Firstly, we undertake an analytical and numerical examination of the following generalized KdV-type equation ut+aux+2buux+cuxxx- duxx=0, u(x,0)=u0(x) where a, b, c, d are real parameters. Our study involves allowing the coefficients a, b, c , and d to approach zero in the limiting sense, while contrasting the outcomes with the scenario in which each coefficient is precisely zero. By analyzing this nonlinear partial differential equation in one dimension, we trace the impact of the nonlinear term on the solution. Furthermore, we extend our findings to a two-dimensional equation with structures comparable to those in the 2D SQG equation. Secondly, we focus on the regularity of solutions in the following 2D SQG equation where κ ≥ 0 and α > 0 are parameters, conducting a thorough analysis that addresses a notable gap in analytical and numerical research. The SQG equation exhibits numerous characteristics similar to the 3D Euler equation and the Navier-Stokes equation, with the regularity of the latter being recognized as one of the Clay Institute of Mathematics' millennium problems. To bridge this gap, we concentrate on various aspects of the SQG equation, exploring both inviscid and dissipative instances. In the dissipative case, we categorize the instances as subcritical, critical, and supercritical. Analytical solutions have recently been derived for the subcritical and critical scenarios, while the question of regularity in the supercritical case remains unresolved. Our research focuses on numerical calculations of the inviscid and supercritical SQG equations, with particular attention to the proximity of level curves, the L2 norm, and the expansion of the quantity. We meticulously examine the nature of the solution, particularly in the region where α =. Finally, we turn our attention to the study of the following water waves where u is the velocity, P is the pressure, and g is the acceleration due to gravity, which are typically modeled using Euler equations with unit density. We address an outstanding open problem concerning the existence of closed orbits for water waves under indefinite boundary constraints. Our investigation begins with a discussion of advancements in water wave structure under finite bottom conditions. We then shift our focus to the behavior of water waves at the kinematic barrier of infinite depth. By employing the Crandall-Rabinowitz theorem to construct water wave profiles for scenarios with zero and
Some common fixed point results in probabilistic metric space and its applications
(2023) Chaudhary, Ajay Kumar; Kanhaiya Jha
Available in full text
Some Sequence Spaces and Matrix Transformation with Vedic Relations
(Institute of Science & Technology, Mathematics, 2021) Ray, Suresh
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“A Study of Topological Structures of Linear Spaces of Generalized Sequences
(Institute of Science & Technology, 2023-02) Ghimire, Jhavi Lal
This dissertation deals with the sequence spaces and applications. The various topological and algebraic properties of different sequence spaces defined by Orlicz function have been studied. We introduce and study the sequence spaces that are the generalization of classical sequence spaces of null, convergent, and bounded type. We introduce and study a class c_0 (M,(X,||.||),(a,) ̅α ̅ ) of vector valued difference sequences of null type with the help of Orlicz function. It is the generalization of classical null sequence space. We prove some linear structures and prove some inclusion and equality relations in terms of different parameters a ̅ and α ̅. In the similar fashion, we study the sequence space of bounded type l_∞ (M,(X,||.||),(a,) ̅α ̅ ) of normed space valued difference sequences using Orlicz function M. The containment relations on different parameters are established. The class l_∞ (M,X,(Y,||.||) of Banach space Y - valued functions is introduced as the generalization of bounded complex sequences. The different topological structures have been studied when topologized it with the suitable natural norm. The difference sequence spaces W_0 (∆,f),W(∆,f) and W_∞ (∆,f) defined by non-negative Φ-function on R are introduced and studied their different topological properties endowed by paranormed structure on these spaces. Infinite series and sequences played important role in the development of Calculus and other branches of mathematics. But the mathematicians were facing the problems of calculating the limits of infinite sequences and series, in particular with those having divergent in behaviour. Then the mathematicians developed the various types of convergence to assign a limit in some sense to divergent sequences and series. We also introduce and investigate sequence spaces defined by ideal convergence and Orlicz function in 2-normed space. The theory of sequence space and frame theory are interconnected as frame theory makes the use of sequence space. The sequence spaces are used as the vector spaces in frame theory. Some of the application of frame theory that makes the use of sequence spaces are image processing, signal processing, error correction, data compression etc. The atomic decomposition in a non-locally convex Banach space is defined and discussed its existence. It is also proved that if a p-Banach space has an atomic decomposition then the space is isomorphic to its associated p-Banach sequence space. The necessary and sufficient condition for an atomic decomposition in p-Banach space is given. Certain properties associated with Schauder frames in Banach space have been defined and studied.
Transmission Dynamics of COVID-19: Mathematical Models for Effective Controls
(Institue of science & Technology, T.U., 2023-12) Adhikari, Khagendra; Prof. Dr. Kedarnath Upreti
The emergence of a pandemic disease often presents unforeseen challenges to the global healthcare system, particularly affecting low and middle-income nations like Nepal. Mathematical modeling of infectious diseases helps to predict and understand the dynamics of the diseases enabling the implementation of efficient public health interventions and resources allocation. This contributes to evidence-based policy decisions to mitigate a pandemic. Despite worldwide efforts and vaccine development, the COVID-19 pandemic has had devastating global impacts varying significantly from one country to another, making country-specific studies essential for a deeper understanding of the disease and its control strategies. This dissertation presents novel mathematical models designed to comprehensively analyze COVID-19 transmission dynamics. Our models have been rigorously validated with multiple datasets, enhancing their reliability and validity. Our mathematical model for the first wave of COVID-19 in higher dimensional systems, characterized by non-linear ordinary differential equations, possesses a significant capability to assess the count of returnees, particularly those crossing the open border between Nepal and India. To estimate the temporal pattern of the returnees, we enhance the system by introducing non-autonomous features. By using the Next Generation Matrix Method, we calculate the Basic Reproduction Number (R_0) and Effective Reproduction number which successfully predict the bifurcating nature of diseases trajectories. By taking advantage of this model, we evaluate the effectiveness of various control measures implemented during the first wave of the pandemic in Nepal. We specifically investigate the impact of three key intervention policies enacted during the first wave of COVID-19 in Nepal: the 1st Lockdown, the 2nd Border Screening and Quarantine, and the 3rd Detection and Isolation. Our findings uncover their effectiveness in the mitigation of COVID-19 transmission in Nepal. In addition, we focus on the Delta variant dominated second wave of COVID-19 in Nepal. We shed light on the transmission dynamics and seroprevalence associated with this highly transmissible variant. Furthermore, we estimate the expected burden on medical resources, including ICU beds and ventilators, in Nepal, providing crucial insights for healthcare preparedness. Additionally, we investigate vaccination programs and the gradual relaxation of lockdown measures as prospective pandemic control methods, which are especially important in resource-limited country like Nepal, where good healthcare management is critical. Our work demonstrates that our mathematical model successfully predicted the seroprevalence during the Delta surge with estimates closely matching the results obtained by the government through a nationwide seroprevalence survey. This finding demonstrates the model's ability, reliability and effectiveness in tracking and understanding disease dynamics, which is crucial for public health planning and response during infectious disease outbreaks. It also highlights the potential for mathematical modeling to complement and authenticate real-world data collection efforts, improving our ability to assess and manage public health crises. We also develop the data-driven models for the estimation of real-time risk of infection and hospitalization during a pandemic. Our probabilistic model for estimation of the risk of infection incorporates susceptible populations, active infectious cases, contact patterns of people, and the effective reproduction number. It offers a more precise description of pandemic's transmission patterns that can be achieved solely through the reproduction number. We also use Maximum Likelihood Function to construct the mathematical model for estimating the rate of the temporal pattern of hospitalization during a pandemic. These models are applied to unique datasets of new COVID-19 cases and hospitalization cases in Nepal including its seven provinces, enabling us to assess disease transmission and efficiently manage healthcare resources to minimize the pandemic's burden. These data-driven models introduce innovative techniques and yield exciting results that advance our understanding of the risk of infection and hospitalization during a pandemic. These findings hold the potential to inform guidelines and strategies for pandemic control, particularly in the face of catastrophic outbreaks. Our biologically realistic models, data integration methods, and probabilistic approaches contribute to the broader scientific fields encompassing life sciences, mathematics, and computational science.

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