Systematic Constructions of Bent-Negabent Functions, 2-Rotation Symmetric Bent-Negabent Functions and Their Duals

Bent-negabent functions have many important properties for their application in cryptography since they have the flat absolute spectrum under the both Walsh-Hadamard transform and nega-Hadamard transform. In this paper, we present four new systematic constructions of bent-negabent functions on $4k, 8k, 4k+2$ and $8k+2$ variables, respectively, by modifying the truth tables of two classes of quadratic bent-negabent functions with simple form. The algebraic normal forms and duals of these constructed functions are also determined. We further identify necessary and sufficient conditions for those bent-negabent functions which have the maximum algebraic degree. At last, by modifying the truth tables of a class of quadratic 2-rotation symmetric bent-negabent functions, we present a construction of 2-rotation symmetric bent-negabent functions with any possible algebraic degrees. Considering that there are probably no bent-negabent functions in the rotation symmetric class, it is the first significant attempt to construct bent-negabent functions in the generalized rotation symmetric class

MODELLING SUPERFLUID NEUTRON STARS APPLICATIONS TO PULSAR GLITCHES

In this dissertation I discuss how observations of the maximum glitch occurred in a certain pulsar provides a test for the microscopic physics of neutron star interiors, in particular the pinning forces (a parameter which effectively describes the strength of the vortex-lattice interaction at the mesoscopic scale). Conversely, by fixing the input parameters by taking estimates from recent literature, it is possible to estimate the mass of a glitching pulsar. A proof of concept of this thesis is given by constructing a quantitative model for pulsar rotational dynamics that can consistently encode state of the art models of the pinning force between vortices and ions in the crust, as well as the stratified structure of a neutron star. This point is far from being secondary as most studies on pulsar glitches are based on body-averaged models or differential models that tacitly assume a cylindrical symmetry, not consistent with the spherically layered structure

Cometric Association Schemes

The combinatorial objects known as association schemes arise in group theory, extremal graph theory, coding theory, the design of experiments, and even quantum information theory. One may think of a d-class association scheme as a (d + 1)-dimensional matrix algebra over R closed under entrywise products. In this context, an imprimitive scheme is one which admits a subalgebra of block matrices, also closed under the entrywise product. Such systems of imprimitivity provide us with quotient schemes, smaller association schemes which are often easier to understand, providing useful information about the structure of the larger scheme. One important property of any association scheme is that we may find a basis of d + 1 idempotent matrices for our algebra. A cometric scheme is one whose idempotent basis may be ordered E0, E1, . . . , Ed so that there exists polynomials f0, f1, . . . , fd with fi ◦ (E1) = Ei and deg(fi) = i for each i. Imprimitive cometric schemes relate closely to t-distance sets, sets of unit vectors with only t distinct angles, such as equiangular lines and mutually unbiased bases. Throughout this thesis we are primarily interested in three distinct goals: building new examples of cometric association schemes, drawing connections between cometric association schemes and other objects either combinatorial or geometric, and finding new realizability conditions on feasible parameter sets — using these conditions to rule out open parameter sets when possible. After introducing association schemes with relevant terminology and definitions, this thesis focuses on a few recent results regarding cometric schemes with small d. We begin by examining the matrix algebra of any such scheme, first looking for low rank positive semidefinite matrices with few distinct entries and later establishing new conditions on realizable parameter sets. We then focus on certain imprimitive examples of both 3- and 4-class cometric association schemes, generating new examples of the former while building realizability conditions for both. In each case, we examine the related t-distance sets, giving conditions which work towards equivalence; in the case of 3-class Q-antipodal schemes, an equivalence is established. We conclude by partially extending a result of Brouwer and Koolen concerning the connectivity of graphs arising from metric association schemes

Noncommutative Biology: Sequential Regulation of Complex Networks and Connected Matter

During animal development from zygote to adult, a limited set of regulatory molecules are autonomously deployed in the service of tissue-specific gene expression (reviewed in chapter 1). Inherent in the process is the tension that single cells sample heterogeneous expression states while robustly maintaining a collective final outcome. This thesis addresses theoretical issues that help resolve the paradox that one cell simultaneously contains the fate information of many. Previous models of development have likened cell fate to minima on a smooth potential energy surface. Such static pictures can be misleading because they suggest the egg knows the path it will take to the adult before it divides even once. Recognition that the potential analogy is an oversimplification has led others to propose that the surface is actually nonsmooth. Chapter 2 reviews the theoretical basis for smooth potentials and resolves these problems by appealing to the tangent space of gene expression. It is then shown that if the potential difference is sufficient to characterize the difference between egg and adult, then the tangent space controls on gene expression are one-dimensional. Furthermore, a shortcoming of models ignoring the connectivity and common origin of dividing cells is that they erect artificial barriers between alternative fates. A fundamentally different picture is sketched wherein the difference between egg and adult is schematized as the shape of the locus of equipotential fates accessible at the same point in time. The conjugacy of space and time is invoked to explain how the requirement that each fate be on a line of equipotential is the same as requiring that each alternative fate move the same distance down the surface at each step. The developmental trajectory is deterministic but not known in advance because it needs to be ascertained at each step which way cells "turn" in order to maintain their equipotential relationship. Chapters 3 and 4 refine this sequential model of collective development with specific examples. A simple solution to the problem of cell-type specific gene expression is combinatorial binding of transcription factors at promoters. It is shown in chapter 3 that such models result in substantial information bottlenecks, because all cell fate information is concentrated at the start. We explore a novel, noncommutative model of gene regulation&#8212;known as sequential logic&#8212;that spreads the information out over time. It is shown using time sequences of noncommutative controllers that targets which otherwise would have been activated together can be regulated independently. We derive scaling laws for two noncommutative models of regulation, motivated by phosphorylation/neural networks and chromosome folding, respectively, and show that they scale super-exponentially in the number of regulators. It is also shown that specificity in control is robust to loss of a regulator. Consequently, sequential logic overcomes the information bottleneck in complex problems and enables novel solutions through roundabout strategies. The theoretical results are connected to real biological networks demonstrating specificity in the context of promiscuity. Noncommutative sequential logic has improved storage capacity, but it does not specify who or what supplies the sequences of input that determine cell fate. Chapter 4 offers a solution by way of the seemingly unrelated problem of looping in twisted strings. Cells and strings obey a set of common space-time constraints, ultimately due to the conservation of energy. It is argued that the most parsimonious allocation of energy from the straight to strained string is the one in which each segment sees the same share of the total. Planar looping is shown to be a consequence of the parsimony principle and the Euler-Poincar&#233; equations for rotational motion in the presence an applied torque. We then solve the problem for the looping of a twisted string; with two strains, the Euler-Poincar&#233; predict a different answer than the classical Frenet-Serret equations. Using the results of chapter 2, it is concluded that the Frenet-Serret curvatures assigned ahead of time are not guaranteed to generate space curves that conserve energy: the predicted string has localized strains the Euler-Poincar&#233; solution lacks. Rotational dynamics of strings are connected to developing organisms by postulating conserved RNA polymerase as an analog of angular momentum, and transcriptional activity as energy. Alternative fates along a one-dimensional "string" of dividing cells are possible by finding the RNAP distribution that conserves transcriptional activity along a curve of constant developmental potential. Consequently, each alternative fate samples a different sequence of changes to the distribution as it follows a local gradient downhill from high to low developmental potential over time. In conclusion, regulation in the tangent space of gene expression resolves the paradox that development has a unique solution specified in the DNA of the egg which cannot be determined with certainty until completion of the adult. Noncommutative sequential logic generates complexity that cannot be realized at the start, while interdependent cells (and strings) require time to ensure that each fate is at the same potential difference from a common ancestor. This fundamental reimagining of the Waddington framework can be tested using new multiplexed mRNA imaging technologies that preserve the spatial context of cells in developing tissue.</p

On the information theory of clustering, registration, and blockchains

Progress in data science depends on the collection and storage of large volumes of reliable data, efficient and consistent inference based on this data, and trusting such computations made by untrusted peers. Information theory provides the means to analyze statistical inference algorithms, inspires the design of statistically consistent learning algorithms, and informs the design of large-scale systems for information storage and sharing. In this thesis, we focus on the problems of reliability, universality, integrity, trust, and provenance in data storage, distributed computing, and information processing algorithms and develop technical solutions and mathematical insights using information-theoretic tools. In unsupervised information processing we consider the problems of data clustering and image registration. In particular, we evaluate the performance of the max mutual information method for image registration by studying its error exponent and prove its universal asymptotic optimality. We further extend this to design the max multiinformation method for universal multi-image registration and prove its universal asymptotic optimality. We then evaluate the non-asymptotic performance of image registration to understand the effects of the properties of the image transformations and the channel noise on the algorithms. In data clustering we study the problem of independence clustering of sources using multivariate information functionals. In particular, we define consistent image clustering algorithms using the cluster information, and define a new multivariate information functional called illum information that inspires other independence clustering methods. We also consider the problem of clustering objects based on labels provided by temporary and long-term workers in a crowdsourcing platform. Here we define budget-optimal universal clustering algorithms using distributional identicality and temporal dependence in the responses of workers. For the problem of reliable data storage, we consider the use of blockchain systems, and design secure distributed storage codes to reduce the cost of cold storage of blockchain ledgers. Additionally, we use dynamic zone allocation strategies to enhance the integrity and confidentiality of these systems, and frame optimization problems for designing codes applicable for cloud storage and data insurance. Finally, for the problem of establishing trust in computations over untrusting peer-to-peer networks, we develop a large-scale blockchain system by defining the validation protocols and compression scheme to facilitate an efficient audit of computations that can be shared in a trusted manner across peers over the immutable blockchain ledger. We evaluate the system over some simple synthetic computational experiments and highlights its capacity in identifying anomalous computations and enhancing computational integrity