Student Number : 0108851W - MSc Dissertation - School of Mathematics - Faculty of Science / A brief search on www.ams.org with the keyword “Markov operator” produces some 684 papers, the earliest of which dates back to 1959. This suggests that the term “Markov operator” emerged around the 1950’s, clearly in the wake of Andrey Markov’s seminal work in the area of stochastic processes and Markov chains. Indeed, [17] and [6], the two earliest papers produced by the ams.org search, study Markov processes in a statistical setting and “Markov operators” are only referred to obliquely, with no explicit definition being provided. By 1965, in [7], the situation has progressed to the point where Markov operators are given a concrete definition and studied more directly. However, the way in which Markov operators originally entered mathematical discourse, emerging from Statistics as various attempts to generalize Markov processes and Markov chains, seems to have left its mark on the theory, with a notable lack of cohesion amongst its propagators. The study of Markov operators in the Lp setting has assumed a place of importance in a variety of fields. Markov operators figure prominently in the study of densities, and thus in the study of dynamical and deterministic systems, noise and other probabilistic notions of uncertainty. They are thus of keen interest to physicists, biologists and economists alike. They are also a worthy topic to a statistician, not least of all since Markov chains are nothing more than discrete examples of Markov operators (indeed, Markov operators earned their name by virtue of this connection) and, more recently, in consideration of the connection between copulas and Markov operators. In the realm of pure mathematics, in particular functional analysis, Markov operators have proven a critical tool in ergodic theory and a useful generalization of the notion of a conditional expectation. Considering the origin of Markov operators, and the diverse contexts in which they are introduced, it is perhaps unsurprising that, to the uninitiated observer at least, the theory of Markov operators appears to lack an overall unity. In the literature there are many different definitions of Markov operators defined on L1(μ) and/or L1(μ) spaces. See, for example, [13, 14, 26, 2], all of which manage to provide different definitions. Even at a casual glance, although they do retain the same overall flavour, it is apparent that there are substantial differences in these definitions. The situation is not much better when it comes to the various discussions surrounding ergodic Markov operators: we again see a variety of definitions for an ergodic operator (for example, see [14, 26, 32]), and again the connections between these definitions are not immediately apparent. In truth, the situation is not as haphazard as it may at first appear. All the definitions provided for Markov operator may be seen as describing one or other subclass of a larger class of operators known as the positive contractions. Indeed, the theory of Markov operators is concerned with either establishing results for the positive contractions in general, or specifically for one of the aforementioned subclasses. The confusion concerning the definition of an ergodic operator can also be rectified in a fairly natural way, by simply viewing the various definitions as different possible generalizations of the central notion of a ergodic point-set transformation (such a transformation representing one of the most fundamental concepts in ergodic theory). The first, and indeed chief, aim of this dissertation is to provide a coherent and reasonably comprehensive literature study of the theory of Markov operators. This theory appears to be uniquely in need of such an effort. To this end, we shall present a wealth of material, ranging from the classical theory of positive contractions; to a variety of interesting results arising from the study of Markov operators in relation to densities and point-set transformations; to more recent material concerning the connection between copulas, a breed of bivariate function from statistics, and Markov operators. Our goals here are two-fold: to weave various sources into a integrated whole and, where necessary, render opaque material readable to the non-specialist. Indeed, all that is required to access this dissertation is a rudimentary knowledge of the fundamentals of measure theory, functional analysis and Riesz space theory. A command of measure and integration theory will be assumed. For those unfamiliar with the basic tenets of Riesz space theory and functional analysis, we have included an introductory overview in the appendix. The second of our overall aims is to give a suitable definition of a Markov operator on Banach lattices and provide a survey of some results achieved in the Banach lattice setting, in particular those due to [5, 44]. The advantage of this approach is that the theory is order theoretic rather than measure theoretic. As we proceed through the dissertation, definitions will be provided for a Markov operator, a conservative operator and an ergodic operator on a Banach lattice. Our guide in this matter will chiefly be [44], where a number of interesting results concerning the spectral theory of conservative, ergodic, so-called “stochastic” operators is studied in the Banach lattice setting. We will also, and to a lesser extent, tentatively suggest a possible definition for a Markov operator on a Riesz space. In fact, we shall suggest, as a topic for further research, two possible approaches to the study of such objects in the Riesz space setting. We now offer a more detailed breakdown of each chapter. In Chapter 2 we will settle on a definition for a Markov operator on an L1 space, prove some elementary properties and introduce several other important concepts. We will also put forward a definition for a Markov operator on a Banach lattice. In Chapter 3 we will examine the notion of a conservative positive contraction. Conservative operators will be shown to demonstrate a number of interesting properties, not least of all the fact that a conservative positive contraction is automatically a Markov operator. The notion of conservative operator will follow from the Hopf decomposition, a fundmental result in the classical theory of positive contractions and one we will prove via [13]. We will conclude the chapter with a Banach lattice/Riesz space definition for a conservative operator, and a generalization of an important property of such operators in the L1 case. In Chapter 4 we will discuss another well-known result from the classical theory of positive contractions: the Chacon-Ornstein Theorem. Not only is this a powerful convergence result, but it also provides a connection between Markov operators and conditional expectations (the latter, in fact, being a subclass of theMarkov operators). To be precise, we will prove the result for conservative operators, following [32]. In Chapter 5 we will tie the study of Markov operators into classical ergodic theory, with the introduction of the Frobenius-Perron operator, a specific type of Markov operator which is generated from a given nonsingular point-set transformation. The Frobenius-Perron operator will provide a bridge to the general notion of an ergodic operator, as the definition of an ergodic Frobenius-Perron operator follows naturally from that of an ergodic transformation. In Chapter 6 will discuss two approaches to defining an ergodic operator, and establish some connections between the various definitions of ergodicity. The second definition, a generalization of the ergodic Frobenius-Perron operator, will prove particularly useful, and we will be able to tie it, following [26], to several interesting results concerning the asymptotic properties of Markov operators, including the asymptotic periodicity result of [26, 27]. We will then suggest a definition of ergodicity in the Banach lattice setting and conclude the chapter with a version, due to [5], of the aforementioned asymptotic periodicity result, in this case for positive contractions on a Banach lattice. In Chapter 7 we will move into more modern territory with the introduction of the copulas of [39, 40, 41, 42, 16]. After surveying the basic theory of copulas, including introducing a multiplication on the set of copulas, we will establish a one-to-one correspondence between the set of copulas and a subclass of Markov operators. In Chapter 8 we will carry our study of copulas further by identifying them as a Markov algebra under their aforementioned multiplication. We will establish several interesting properties of this Markov algebra, in parallel to a second Markov algebra, the set of doubly stochastic matrices. This chapter is chiefly for the sake of interest and, as such, diverges slightly from our main investigation of Markov operators. In Chapter 9, we will present the results of [44], in slightly more detail than the original source. As has been mentioned previously, these concern the spectral properties of ergodic, conservative, stochastic operators on a Banach lattice, a subclass of the Markov operators on a Banach lattice. Finally, as a conclusion to the dissertation, we present in Chapter 10 two possible routes to the study of Markov operators in a Riesz space setting. The first definition will be directly analogous to the Banach lattice case; the second will act as an analogue to the submarkovian operators to be introduced in Chapter 2. We will not attempt to develop any results from these definitions: we consider them a possible starting point for further research on this topic. In the interests of both completeness, and in order to aid those in need of more background theory, the reader may find at the back of this dissertation an appendix which catalogues all relevant results from Riesz space theory and operator theory.

In this project we are building a mathematical model to track the movement of spermatozoa during the process of chemotaxis. Our model is built on an off-lattice spherical biased random walk in 3-dimensional space, an extension of previous conventional deterministic 2-dimensional models. The sperm's type of movement is decided based on a comparison of the current and previous chemoattractant concentration which can be used to see whether it is approaching the egg. From the statistical analysis of the simulation results, we find that chemotaxis is an effective mechanism to increase the number of sperm reaching the egg.

by Tong Pak Yee. / Parallel title in Chinese characters. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1992. / Includes bibliographical references (leaves 12-16). / Acknowledgement --- p.i / List of Publications --- p.ii / Abstract --- p.iii / Chapter 1. --- Introduction / Chapter 1.1 --- Background --- p.1 / Chapter 1.2 --- Outline of the article --- p.5 / Chapter 1.2.1 --- Multifractal Scaling in Fractal Lattice --- p.6 / Chapter 1.2.2 --- Anomalous Multifractality in Percolation Model --- p.7 / Chapter 1.2.3 --- Anomalous Crossover Behavior in Two-Component Random Resistor Network --- p.8 / Chapter 1.2.4 --- Current Distribution in Two-Component Random Resistor Network --- p.10 / Chapter 1.2.5 --- Multif ractality in Wide Distribution Fractal Models --- p.11 / References --- p.12 / Chapter 2. --- Multifractal Analysis of Percolation Backbone and Fractal Lattices / Chapter 2.1 --- Multifractal Scaling in Fractal Lattice --- p.17 / Chapter 2.1.1 --- Multifractal Scaling in a Sierpinski Gasket --- p.18 / Chapter 2.1.2 --- Hierarchy of Critical Exponents on a Sierpinski Honeycomb --- p.38 / Chapter 2.2 --- Anomalous Multifractality in Percolation Model --- p.51 / Chapter 2.2.1 --- Anomalous Multifractality of Conductance Jumps in a Hierarchical Percolation Model --- p.52 / Chapter 2.3 --- Anomalous Crossover Behavior in Two-Component Random Resistor Network --- p.74 / Chapter 2.3.1 --- Anomalous Crossover Behaviors in the Two- Component Deterministic Percolation Model --- p.75 / Chapter 2.3.2 --- Minimum Current in the Two-Component Random Resistor Network --- p.90 / Chapter 2.4 --- Current Distribution in Two-Component Random Resistor Network --- p.105 / Chapter 2.4.1 --- Current Distribution in the Two-Component Hierarchical Percolation Model --- p.106 / Chapter 2.4.2 --- Current Distribution and Local Power Dissipation in the Two-Component Deterministic Percolation Model --- p.136 / Chapter 2.5 --- Multifractality in Wide Distribution Fractal Models --- p.174 / Chapter 2.5.1 --- Fractal Networks with a Wide Distribution of Conductivities --- p.175 / Chapter 2.5.2 --- Power Dissipation in an Exactly Solvable Wide Distribution Model --- p.193 / Chapter 3. --- Conclusion --- p.210

by Mow Wai Ho. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1991. / Bibliographies: leaves 98-104. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Channel Model and Other Basic Assumptions --- p.5 / Chapter 1.2 --- Complexity Measure --- p.8 / Chapter 1.3 --- Maximum Likelihood Sequence Estimator --- p.9 / Chapter 1.4 --- The Viterbi Algorithm ´ؤ An Implementation of MLSE --- p.11 / Chapter 1.5 --- Error Performance of the Viterbi Algorithm --- p.14 / Chapter 1.6 --- Suboptimal Viterbi-like Algorithms --- p.17 / Chapter 1.7 --- Trends of Digital Transmission and MLSE --- p.19 / Chapter 2 --- New Formulation of MLSE --- p.21 / Chapter 2.1 --- The Truncated Viterbi Algorithm --- p.21 / Chapter 2.2 --- Choice of Truncation Depth --- p.23 / Chapter 2.3 --- Decomposition of MLSE --- p.26 / Chapter 2.4 --- Lattice Interpretation of MLSE --- p.29 / Chapter 3 --- The Closest Vector Problem --- p.34 / Chapter 3.1 --- Basic Definitions and Facts About Lattices --- p.37 / Chapter 3.2 --- Lattice Basis Reduction --- p.40 / Chapter 3.2.1 --- Weakly Reduced Bases --- p.41 / Chapter 3.2.2 --- Derivation of the LLL-reduction Algorithm --- p.43 / Chapter 3.2.3 --- Improved Algorithm for LLL-reduced Bases --- p.52 / Chapter 3.3 --- Enumeration Algorithm --- p.57 / Chapter 3.3.1 --- Lattice and Isometric Mapping --- p.58 / Chapter 3.3.2 --- Enumerating Points in a Parallelepiped --- p.59 / Chapter 3.3.3 --- Enumerating Points in a Cube --- p.63 / Chapter 3.3.4 --- Enumerating Points in a Sphere --- p.64 / Chapter 3.3.5 --- Comparisons of Three Enumeration Algorithms --- p.66 / Chapter 3.3.6 --- Improved Enumeration Algorithm for the CVP and the SVP --- p.67 / Chapter 3.4 --- CVP Algorithm Using the Reduce-and-Enumerate Approach --- p.71 / Chapter 3.5 --- CVP Algorithm with Improved Average-Case Complexity --- p.72 / Chapter 3.5.1 --- CVP Algorithm for Norms Induced by Orthogonalization --- p.73 / Chapter 3.5.2 --- Improved CVP Algorithm using Norm Approximation --- p.76 / Chapter 4 --- MLSE Algorithm --- p.79 / Chapter 4.1 --- MLSE Algorithm for PAM Systems --- p.79 / Chapter 4.2 --- MLSE Algorithm for Unimodular Channel --- p.82 / Chapter 4.3 --- Reducing the Boundary Effect for PAM Systems --- p.83 / Chapter 4.4 --- Simulation Results and Performance Investigation for Example Channels --- p.86 / Chapter 4.5 --- MLSE Algorithm for Other Lattice-Type Modulation Systems --- p.91 / Chapter 4.6 --- Some Potential Applications --- p.92 / Chapter 4.7 --- Further Research Directions --- p.94 / Chapter 5 --- Conclusion --- p.96 / Bibliography --- p.104

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Thesis. 1975. Ph.D.--Massachusetts Institute of Technology. Dept. of Mathematics. / Vita. / Bibliography: leaves 132-133. / by Nguyen Quang Hien. / Ph.D.

博弈理論己貫穿於許多不同的研究領域中，成為了一種非常有用的研究工具。在演化博弈中，個體可以通過比較彼此在競爭中表現的好壞，從一個策略改變至另一個策略。當依賴於收益參數的競爭個體系統處於穩定態時，可以是一個所有個體都採用同一策略的均勻系統，或是使用不同策略的個體共同存在的非均勻系統。近年來，均勻態和混合態或是均勻相和混合相之間的轉變成為物理學家們的研究熱點。 / 在論文的第一部份(第二及第三章)中，我們研究了有空間關聯的群體在規則晶格上演化雪堆博弄(ESG) 。在文獻中的分析方法大多數是基於對近似法(pair approximation) 。但是對近似法的結果，就算是在定性上，都未能捕捉到電算模擬結果中顯現的重要特性。例如，對近似法未能給出全c 態和混合態之間以及混合態與全D 態之間的轉變特徵，其中c和D 分別表示演化雪堆博奔中的兩種策略。而這些轉變作為收益參數的函數，卻可於ESG 在不同的規則品格中的電算模擬中被觀察到。對近似法的這些缺陷引發了本論文的研究。當不同局域競爭組態的分佈形式假設為二項式分佈，並將之引入到c 個體數目隨時間演變的動態公式中，就能得出全D 態。當進一步深入考慮ESG 的演化後，我們得知在收益參數較小時，D 個體被孤立的局部組態的比例在二項式分佈假設中被高估。通過修正D 個體被孤立的局部組態比例，就能得出全C 態。通過適當調整收益參數較小時D 個體被孤立的局部組態的比例，和收益參數較大時適當調整C個體被孤立的局部組態的比例，我們就能得出與模擬結果吻合較好的結果。然後，我們還將這個方法推廣至線性鏈和六角品格結構中。該種方法比對近似法得到的結果更好。 / 在論文的第二部份(第四章)中，我們研究了有本懲罰者的存在對演化雪堆博弈在正方晶格上演化的影響。該博弈中相應地有三種策略。懲罰者願意支付額外成本以對不合作者造成顯外的損失。同時懲罰者與其他合作者之間是合作的。我們展示了在收益參數組成的空間下得出的詳盡相圖。我們從動態演化後期出現的最終生存形態(last surviving patterns) 出發，討論了相圖中的相邊界成因。 / Game theoretical methods have become a useful tool in research across many fields. In evolutionary games, agents could switch from one strategy to another based on how well they perform as compared with others. Depending on a payoff parameter, a system of competing agents may form a homogeneous system with all agents taking on the same strategy or an inhomogeneous system with the coexistence of different strategies in the steady state. The transitions between the homogeneous and mixed states or phases are of much interest to physicists in recent years. / In Part I (Chapter 2 and 3) of the thesis, the evolutionary snowdrift game (ESG) is studied in spatially structured populations on regular lattices. Analytic approaches in the literature are largely based on the pair approximation. However, results of the pair approximation cannot capture the key features in simulation results even qualitatively. For example, the pair approximation fails to give the transition between an All-C and the mixed phase and the transition between the mixed phase and an All-D phase as a function of a payoff parameter observed in ESG on different regular lattices, where C and D are the two strategies in ESG. The failure motivated the present work. By incorporating different local competing configurations within an assumption of binomial distribution into the dynamical equation for the time evolution of the number of C-agents, the All-D phase readily emerges. Further consideration of the dynamics of ESG informs us that the proportion of isolated-D local configurations is over-estimated in the binomial distribution when the payoffparameter is small. By modifying the proportion of isolated-D congurations, the All-C phase results. By suitably adjusting the weighings of isolated-D congurations for small payoff parameters and isolated-C congurations for large payoff parameters, good agreement with simulation results is obtained. The approach is then generalized to linear chain and hexagonal lattice. The present approach performs much better than the pair approximation. / In Part II (Chapter 4) of the thesis, the effects of the presence of costly punishers in ESG are studied in a square lattice. There are, thus, three strategies in the game. The punishers are willing to pay an extra cost so as to inflict an extra damage to a defector, and they cooperate with other cooperators. Detailed phase diagrams are presented in the space formed by the payoff parameters. The phase boundaries are discussed within the idea of last surviving patterns at the late stage of the dynamics. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Chan, Wan Hang = 晶格上演化雪堆博弈及相關問題的研究 / 陳運亨. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 81-86). / Abstracts also in Chinese. / Chan, Wan Hang = Jing ge shang yan hua xue dui bo yi ji xiang guan wen ti de yan jiu / Chen Yunheng. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Game Theory --- p.2 / Chapter 1.1.1 --- Prisoner Dilemma --- p.3 / Chapter 1.1.2 --- Snowdrift Game --- p.3 / Chapter 1.1.3 --- Costly Punishment --- p.4 / Chapter 1.2 --- Evolutionary Game Theory --- p.6 / Chapter 1.2.1 --- Updating Rules --- p.6 / Chapter 1.2.2 --- Updating Schemes --- p.6 / Chapter 1.2.3 --- Spatial Structures --- p.8 / Chapter 1.3 --- Analytic Approaches --- p.9 / Chapter 1.3.1 --- Mean-field Approach --- p.9 / Chapter 1.3.2 --- Last Surviving Patterns --- p.10 / Chapter 2 --- Evolutionary Snowdrift Game on Square Lattice --- p.11 / Chapter 2.1 --- Introduction --- p.11 / Chapter 2.2 --- Simulation Results --- p.12 / Chapter 2.3 --- Last Surviving Patterns --- p.15 / Chapter 2.4 --- Pair Approximation --- p.19 / Chapter 2.5 --- The Site Dynamical Equations --- p.22 / Chapter 2.5.1 --- Pure Binomial Approximation --- p.23 / Chapter 2.5.2 --- Modified Binomial Approximation --- p.24 / Chapter 2.5.3 --- Isolated Factor Correction --- p.27 / Chapter 2.5.4 --- The Zonal Correction --- p.31 / Chapter 2.6 --- Summary --- p.32 / Chapter 3 --- Evolutionary Snowdrift Game on Other Networks --- p.34 / Chapter 3.1 --- Snowdrift Game on Other Lattices --- p.34 / Chapter 3.1.1 --- Linear Chain --- p.34 / Chapter 3.1.2 --- Hexagonal Lattice --- p.40 / Chapter 3.2 --- Snowdrift Game on Small-world Network --- p.43 / Chapter 3.3 --- Summary --- p.50 / Chapter 4 --- Costly punishment in ESG on Lattices --- p.52 / Chapter 4.1 --- Introduction --- p.52 / Chapter 4.2 --- Model --- p.54 / Chapter 4.3 --- Simulation Results [1] --- p.56 / Chapter 4.3.1 --- Results with fixed α and β [1] --- p.56 / Chapter 4.3.2 --- Results with a fixed α [1] --- p.60 / Chapter 4.3.3 --- Consideration of special local structures --- p.65 / Chapter 4.3.4 --- Results of different α --- p.67 / Chapter 4.4 --- Pair Approximation Extended to Three Strategies [1] --- p.69 / Chapter 4.5 --- Summary --- p.76 / Chapter 5 --- Summary --- p.78 / Bibliography --- p.81 / Chapter A --- Uncertainty in Simulation Results --- p.87

Exotic mesons are mesons that cannot be described as a quark-antiquark pair. The number of exotic mesons has been growing every year in the charm sector and the theoretical understanding of them is often conflicted amongst the community. Some possible explanations include hybrid mesons where the quark-antiquark pair is coupled to a gluonic excitation, compact tetraquarks where four quarks are bound into a localised state and molecules which consist of pairs of extended mesons. To study exotic mesons from first principles, lattice QCD provides the framework to perform spectroscopy calculations numerically. I will give a review and describe the relevant techniques used in this thesis. After doing so, I will calculate masses of charmonium with angular momentum up to four. The results show QCD permits states with exotic quantum numbers that are not accessible by a quark-antiquark pair. I will identify states that are consistent with the quark-antiquark picture and then show that the remaining states in the extracted spectra can be interpreted to be the lightest and first excited hybrid meson supermultiplet. Whilst the mass is one quantity that can be computed, hadron spectroscopy is also concerned with the calculation of the unstable properties of resonances which can decay into meson-meson states. These meson-meson states have four quarks and could also mix with tetraquarks. I will describe how to correctly extract the energies of four quark states within lattice QCD by reviewing operators resembling meson-mesons and then constructing a general class of operators resembling tetraquarks. I will then calculate a variety of spectra in the isospin-1 hidden charm sector and the doubly charmed sector. No evidence of a bound state or narrow resonance is found in these channels. Having described how to include multi-meson states in lattice QCD, I will describe how to relate the lattice QCD spectrum to the scattering amplitudes and perform a calculation of elastic $DK$ scattering amplitudes which is relevant for the exotic $D_{s0}(2317)$. By analytically continuing the scattering amplitudes into the complex plane, I find a bound state pole near threshold which is in good agreement with what is found experimentally.

Superstring theory is one current, promising attempt at unifying gravity with the other three known forces: the electromagnetic force, and the weak and strong nuclear forces. Though this is still a work in progress, much effort has been put forth toward this goal. A set of specific tools which are used are gauge/gravity dualities. This thesis consists of a specific implementation of gauge/gravity dualities to describe k-strings of strongly coupled gauge theories as objects dual to Dpbranes embedded in confining supergravity backgrounds from low energy superstring field theory. Along with superstring theory, k-strings are also commonly investigated with lattice gauge theory and Hamiltonian methods. A k$string is a colorless combination of quark-anti-quark source pairs, between which a color flux tube develops. The two most notable terms of the k-string energy are, for large quark anti-quark separation L, the tension term, proportional to L, and the Coulombic 1/L correction, known as the Luscher term. This thesis provides an overview of superstring theories and how gauge/gravity dualities emerge from them. It shows in detail how these dualities can be used for the specific problem of calculating the k-string energy in 2+1 and 3+1 space-time dimensions as the energy of Dp-branes in the dual gravitational theory. A detailed review of k-string tension calculations is given where good agreement is found with lattice gauge theory and Hamiltonian methods. In reviewing the k-string tension, we also touch on how different representations of k-strings can be described with Dp-branes through gauge/gravity dualities. The main result of this thesis is how the Luscher term is found to emerge from the energy calculation of Dp-branes. In 2+1 space-time dimensions, we have Luscher term data to compare with from lattice gauge theory, where we find good agreement.

It is expected that the number of flavors in a gauge theory plays an important role in model building for physics beyond the standard model. We study the phase structure of the 12 flavor case through lattice simulations using a Rational Hybrid Monte Carlo (RHMC) algorithm for different masses, betas, and volumes, to investigate the question of conformality for this number of flavors. In particular, we analyze the Fisher's zeroes, in the vicinity of the endpoint of a line of first order phase transitions. This is motivated by previous studies that show how the complex renormalization group (RG) flows can be understood by looking at the zeros. The pinching of the imaginary part of these zeros with respect to increasing volume provides information about a possible unconventional continuum limit. We also study the mass spectrum of a multiflavor linear sigma model with a splitting of fermion masses. The single mass linear sigma model successfully described a light sigma in accordance to recent lattice results. The extension to two masses predicts an unusual ordering of scalar masses, providing incentive for further lattice simulations with split quark mass.