We propose a novel analytic approach for the comparative statics analysis of operations management problems on the capacity investment decision and the influenza (flu) vaccine composition decision. Our approach involves exploiting the properties of the underlying mathematical models, and linking those properties to the concept of stochastic orders relationship. The use of stochastic orders allows us to establish our main results without restriction to a specific distribution. A major strength of our approach is that it is "scalable," i.e., it applies to capacity investment decision problem with any number of non-independent (i.e., demand or resource sharing) products and resources, and to the influenza vaccine composition problem with any number of candidate strains, without a corresponding increase in computational effort. This is unlike the current approaches commonly used in the operations management literature, which typically involve a parametric analysis followed by the use of the implicit function theorem. Providing a rigorous framework for comparative statics analysis, which can be applied to other problems that are not amenable to traditional parametric analysis, is our main contribution. We demonstrate this approach on two problems: (1) Capacity investment decision, and (2) influenza vaccine composition decision. A comparative statics analysis is integral to the study of these problems, as it allows answers to important questions such as, "does the firm acquire more or less of the different resources available as demand uncertainty increases? does the firm benefit from an increase in demand uncertainty? how does the vaccine composition change as the yield uncertainty increases?" Using our proposed approach, we establish comparative statics results on how the newsvendor's expected profit and optimal capacity decision change with demand risk and demand dependence in multi-product multi-resource newsvendor networks; and how the societal vaccination benefit, the manufacturer's profit, and the vaccine output change with the risk of random yield of strains. / Ph. D.

Monistic substantivalists believe that material objects and regions of space-time are not two distinct kinds of fundamental of entities. For the monist, objects either are identical with regions or are somehow derivative from them. Dualistic substantivalists view regions and objects as distinct kinds of fundamental entities. One virtue monists claim over dualists is that their view is more ideologically parsimonious than dualism because monists can do without a primitive notion of location. In this paper I provide an argument that undercuts some of the theoretical edge that monists claim over dualists. The assumption that the monist can provide a reduction of location unique to her position rests on a false assumption about the possible structures spacetime can have. If it is metaphysically possible for two distinct regions to coincide with respect to all their significant spatiotemporal properties and relations (call these 'coincident regions'), then analyses of location unique to monism will turn out to be inadequate. / Master of Arts / You, I, a rock on the ground, electrons, and galaxies all have something in common: we are all material objects. Material objects are often defined as the things that have locations within spacetime. But what is it to have a location within spacetime? Some authors, monists, believe that to have a location in spacetime is to be no more than a bit of the spatiotemporal manifold. Others, dualists, think of spacetime like a box that objects get placed into. For them having a location is to “take up” part of the room in this box. As the debate currently stands, many philosophical considerations look to point in favor of monism over dualism. In this paper I discuss a novel argument that this assessment does not stand up to scrutiny. The argument makes use of contemporary theories in physics and advanced geometry to argue that distinct parts of spacetime can be located at one another. This is shown to undermine many of the considerations which are thought to favor monism over dualism.

京都大学 / 新制・課程博士 / 博士(理学) / 甲第25097号 / 理博第5004号 / 新制||理||1714(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 望月 新一, 教授 大木谷 耕司, 准教授 星 裕一郎 / 学位規則第4条第1項該当 / Doctor of Agricultural Science / Kyoto University / DGAM

This paper introduces a novel methodology that integrates Machine Learning (ML), Operations Research (OR), and Game Theory (GT) to develop an interpretable heuristic for principal-agent models (PAM). We extract solution patterns from ensemble tree models trained on solved instances of a PAM. Using these patterns, we develop a hierarchical tree-based approach that forms an interpretable ML-based heuristic to solve the PAM. This method ensures the interpretability, feasibility, and generalizability of ML predictions for game-theoretic models. The predicted solutions from this ensemble model-based heuristic are consistently high quality and feasible, significantly reducing computational time compared to traditional optimization methods to solve PAM. Specifically, the computational results demonstrate the generalizability of the ensemble heuristic in varying problem sizes, achieving high prediction accuracy with optimality gaps between 1--2% and significant improvements in solution times. Our ensemble model-based heuristic, on average, requires only 4.5 out of the 9 input features to explain its predictions effectively for a particular application. Therefore, our ensemble heuristic enhances the interpretability of game-theoretic optimization solutions, simplifying explanations and making them accessible to those without expertise in ML or OR. Our methodology adds to the approaches for interpreting ML predictions while also improving numerical tractability of PAMs. Consequently, enhancing policy design and operational decisions, and advancing real-time decision support where understanding and justifying decisions is crucial. / Master of Science / This paper introduces a new method that combines Machine Learning (ML) with Operations Research (OR) to create a clear and understandable approach for solving a principal-agent model (PAM). We use patterns from a group of decision trees to form an ML-based strategy to predict solutions that greatly reduces the time to solve the problem compared to traditional optimization techniques. Our approach works well for different sizes of problems, maintaining high accuracy with very small differences in objective function value from the best possible solutions (1-2%). The solutions predicted are consistently high quality and practical, significantly reducing the time needed compared to traditional optimization methods. Remarkably, our heuristic typically uses only 4.5 out of 9 input features to explain its predictions, making it much simpler and more interpretable than other methods. The results show that our method is both efficient and effective, with faster solution times and better accuracy. Our method can make complex game-theoretic optimization solutions more understandable, even for those without expertise in ML or OR. By improving the interpretability making PAMs analytically explainable, our approach supports better policy design and operational decision-making, advancing real-time decision support where clarity and justification of decisions are essential.

Master of Science / Division of Biology / Martha E. Mather / Understanding how and why fish distribution is related to specific habitat characteristics underlies many ecological patterns and is crucial for effective research and management. Blue Catfish, Ictalurus furcatus, are an important concern for many fisheries agencies; however, lack of information about their distribution and habitat use remains a hindrance to proper management. Here, over all time periods and across months, I quantified Blue Catfish distribution and environmental correlates of distribution in Milford Reservoir, the largest reservoir in Kansas. I tested relationships among acoustically tagged Blue Catfish and three groups of variables postulated to influence Blue Catfish distribution in the literature (i. localized microhabitat variables, ii. larger-scale mesohabitat variables, iii. biotic variables). Blue Catfish were consistently aggregated in two locations of the reservoir across five months during summer and fall, 2013. Using multiple linear regression and an information theoretic model selection approach, consistent correlates of distribution included localized, microhabitat variables (i.e., dissolved oxygen, slope) larger-scale, mesohabitat variables (i.e., distance to channel, river kilometer from the dam) and a biotic variable (i.e., Secchi depth). This research identified which 5 of the 12 variables identified in the literature were most influential in determining Blue Catfish distribution. As a guide for future hypothesis generation and research, I propose that Blue Catfish distribution was driven by three ecologically-relevant tiers of influence. First, Blue Catfish avoided extremely low dissolved oxygen concentrations that cause physiological stress. Second, Blue Catfish aggregated near the channel, an area of bathymetric heterogeneity that may offer a foraging advantage. Third, Blue Catfish aggregated near low Secchi depths, shown here to be associated with increased productivity and prey abundance. Building on my results, future research into the distribution and habitat use of Blue Catfish should incorporate aggregated distributions of fish into research designs, focus on how both small and large scale relationships interact to produce patterns of distribution, and explore further the mechanisms, consequences, and interactions among the three tiers of influence identified here.

This thesis concerns the use of perverse sheaves with coefficients in commutative rings and in particular, fields of positive characteristic, in the study of modular representation theory. We begin by giving a new geometric interpretation of classical connections between the representation theory of the general linear groups and symmetric groups. We then survey work, joint with D. Juteau and G. Williamson, in which we construct a class of objects, called parity sheaves. These objects share many properties with the intersection cohomology complexes in characteristic zero, including a decomposition theorem and a close relation to representation theory. The final part of this document consists of two computations of IC stalks in the nilpotent cones of sl₃and sl₄. These computations build upon our calculations in sections 3.5 and 3.6 of (31), but utilize slightly more sophisticated techniques and allow us to compute the stalks in the remaining characteristics. / text

Efficient implementation of the number theoretic transform(NTT), also known as the discrete Fourier transform(DFT) over a finite field, has been studied actively for decades and found many applications in digital signal processing. In 1971 Schonhage and Strassen proposed an NTT based asymptotically fast multiplication method with the asymptotic complexity O(m log m log log m) for multiplication of $m$-bit integers or (m-1)st degree polynomials. Schonhage and Strassen's algorithm was known to be the asymptotically fastest multiplication algorithm until Furer improved upon it in 2007. However, unfortunately, both algorithms bear significant overhead due to the conversions between the time and frequency domains which makes them impractical for small operands, e.g. less than 1000 bits in length as used in many applications. With this work we investigate for the first time the practical application of the NTT, which found applications in digital signal processing, to finite field multiplication with an emphasis on elliptic curve cryptography(ECC). We present efficient parameters for practical application of NTT based finite field multiplication to ECC which requires key and operand sizes as short as 160 bits in length. With this work, for the first time, the use of NTT based finite field arithmetic is proposed for ECC and shown to be efficient. We introduce an efficient algorithm, named DFT modular multiplication, for computing Montgomery products of polynomials in the frequency domain which facilitates efficient multiplication in GF(p^m). Our algorithm performs the entire modular multiplication, including modular reduction, in the frequency domain, and thus eliminates costly back and forth conversions between the frequency and time domains. We show that, especially in computationally constrained platforms, multiplication of finite field elements may be achieved more efficiently in the frequency domain than in the time domain for operand sizes relevant to ECC. This work presents the first hardware implementation of a frequency domain multiplier suitable for ECC and the first hardware implementation of ECC in the frequency domain. We introduce a novel area/time efficient ECC processor architecture which performs all finite field arithmetic operations in the frequency domain utilizing DFT modular multiplication over a class of Optimal Extension Fields(OEF). The proposed architecture achieves extension field modular multiplication in the frequency domain with only a linear number of base field GF(p) multiplications in addition to a quadratic number of simpler operations such as addition and bitwise rotation. With its low area and high speed, the proposed architecture is well suited for ECC in small device environments such as smart cards and wireless sensor networks nodes. Finally, we propose an adaptation of the Itoh-Tsujii algorithm to the frequency domain which can achieve efficient inversion in a class of OEFs relevant to ECC. This is the first time a frequency domain finite field inversion algorithm is proposed for ECC and we believe our algorithm will be well suited for efficient constrained hardware implementations of ECC in affine coordinates.

Based on Cohn and Umans’ group-theoretic method, we embed matrix multiplication into several group algebras, including those of cyclic groups, dihedral groups, special linear groups and Frobenius groups. We prove that SL2(Fp) and PSL2(Fp) can realize the matrix tensor ⟨p, p, p⟩, i.e. it is possible to encode p × p matrix multiplication in the group algebra of such a group. We also find the lower bound for the order of an abelian group realizing ⟨n, n, n⟩ is n3. For Frobenius groups of the form Cq Cp, where p and q are primes, we find that the smallest admissible value of q must be in the range p4/3 ≤ q ≤ p2 − 2p + 3. We also develop an algorithm to find the smallest q for a given prime p.

In this thesis we study Ax-Schanuel type inequalities for abstract differential equations. A motivating example is the exponential differential equation. The Ax-Schanuel theorem states positivity of a predimension defined on its solutions. The notion of a predimension was introduced by Hrushovski in his work from the 1990s where he uses an amalgamation-with-predimension technique to refute Zilber's Trichotomy Conjecture. In the differential setting one can carry out a similar construction with the predimension given by Ax-Schanuel. In this way one constructs a limit structure whose theory turns out to be precisely the first-order theory of the exponential differential equation (this analysis is due to Kirby (for semiabelian varieties) and Crampin, and it is based on Zilber's work on pseudo-exponentiation). One says in this case that the inequality is adequate. Thus, by an Ax-Schanuel type inequality we mean a predimension inequality for a differential equation. Our main question is to understand for which differential equations one can find an adequate predimension inequality. We show that this can be done for linear differential equations with constant coefficients by generalising the Ax-Schanuel theorem. Further, the question turns out to be closely related to the problem of recovering the differential structure in reducts of differentially closed fields where we keep the field structure (which is quite an interesting problem in its own right). So we explore that question and establish some criteria for recovering the derivation of the field. We also show (under some assumptions) that when the derivation is definable in a reduct then the latter cannot satisfy a non-trivial adequate predimension inequality. Another example of a predimension inequality is the analogue of Ax-Schanuel for the differential equation of the modular j-function due to Pila and Tsimerman. We carry out a Hrushovski construction with that predimension and give an axiomatisation of the first-order theory of the strong Fraïssé limit. It will be the theory of the differential equation of j under the assumption of adequacy of the predimension. We also show that if a similar predimension inequality (not necessarily adequate) is known for a differential equation then the fibres of the latter have interesting model theoretic properties such as strong minimality and geometric triviality. This, in particular, gives a new proof for a theorem of Freitag and Scanlon stating that the differential equation of j defines a trivial strongly minimal set.

One of the most prevalent inherited diseases is cystic fibrosis. This disease is caused by a mutation in a membrane protein, the cystic fibrosis transmembrane conductance regulator (CFTR). CFTR is known to function as a chloride channel that regulates the viscosity of mucus that lines the ducts of a number of organs. Generally, most of the prevalent mutations of CFTR are located in one of two nucleotide binding domains, namely, the nucleotide binding domain 1 (NBD1). However, some mutations in nucleotide binding domain 2 (NBD2) can equally cause cystic fibrosis. In this work, a hierarchical graph is built for NBD2. Using this model for NBD2, we examine the consequence of single point mutations on NBD2. We collate the wildtype structure with eight of the most prevalent mutations and observe how the NBD2 is affected by each of these mutations.