On the rank function of a differential poset

We study $r$-differential posets, a class of combinatorial objects introduced in 1988 by the first author, which gathers together a number of remarkable combinatorial and algebraic properties, and generalizes important examples of ranked posets, including the Young lattice. We first provide a simple bijection relating differential posets to a certain class of hypergraphs, including all finite projective planes, which are shown to be naturally embedded in the initial ranks of some differential poset. As a byproduct, we prove the existence, if and only if $r\geq 6$, of $r$-differential posets nonisomorphic in any two consecutive ranks but having the same rank function. We also show that the Interval Property, conjectured by the second author and collaborators for several sequences of interest in combinatorics and combinatorial algebra, in general fails for differential posets. In the second part, we prove that the rank function $p_n$ of any arbitrary $r$-differential poset has nonpolynomial growth; namely, $p_n\gg n^ae^{2\sqrt{rn}},$ a bound very close to the Hardy-Ramanujan asymptotic formula that holds in the special case of Young's lattice. We conclude by posing several open questions.Comment: A few minor revisions/updates. Published in the Electron. J. Combin. (vol. 19, issue 2, 2012

Random Matrices and Chaos in Nuclear Physics: Nuclear Reactions

The application of random-matrix theory (RMT) to compound-nucleus (CN) reactions is reviewed. An introduction into the basic concepts of nuclear scattering theory is followed by a survey of phenomenological approaches to CN scattering. The implementation of a random-matrix approach into scattering theory leads to a statistical theory of CN reactions. Since RMT applies generically to chaotic quantum systems, that theory is, at the same time, a generic theory of quantum chaotic scattering. It uses a minimum of input parameters (average S-matrix and mean level spacing of the CN). Predictions of the theory are derived with the help of field-theoretical methods adapted from condensed-matter physics and compared with those of phenomenological approaches. Thorough tests of the theory are reviewed, as are applications in nuclear physics, with special attention given to violation of symmetries (isospin, parity) and time-reversal invariance.Comment: 50 pages, 26 figure

C-DIFFERENTIALS AND GENERALIZED CRYPTOGRAPHIC PROPERTIES OF VECTORIAL BOOLEAN AND P-ARY FUNCTIONS

This dissertation investigates a newly defined cryptographic differential, called a c-differential, and its relevance to the nonlinear substitution boxes of modern symmetric block ciphers. We generalize the notions of perfect nonlinearity, bentness, and avalanche characteristics of vectorial Boolean and p-ary functions using the c-derivative and a new autocorrelation function, while capturing the original definitions as special cases (i.e., when c=1). We investigate the c-differential uniformity property of the inverse function over finite fields under several extended affine transformations. We demonstrate that c-differential properties do not hold in general across equivalence classes typically used in Boolean function analysis, and in some cases change significantly under slight perturbations. Thus, choosing certain affine equivalent functions that are easy to implement in hardware or software without checking their c-differential properties could potentially expose an encryption scheme to risk if a c-differential attack method is ever realized. We also extend the c-derivative and c-differential uniformity into higher order, investigate some of their properties, and analyze the behavior of the inverse function's second order c-differential uniformity. Finally, we analyze the substitution boxes of some recognizable ciphers along with certain extended affine equivalent variations and document their performance under c-differential uniformity.Commander, United States NavyApproved for public release. Distribution is unlimited

Exploring non-Hermitian physics in mechanical metamaterials

One of the postulates of quantum mechanics demands observables to be real, and as a consequence Hamiltonians, representing the energy of a system, to be Hermitian. In fact, this constraint is unnecessarily strong, since also a non-Hermitian Hamiltonian can feature a real spectrum, for example, in the case that they satisfy e.g. both parity and time-reversal (PT) symmetries, as obtained by Bender and Boettcher in 1998. Following the wave of interest towards this topic in the last decades, I will review the consequences of lifting the Hermiticity condition for Hamiltonians described by time-dependent parameters. I will use this to investigate effects related to the adiabatic geometric phase in PT-symmetric non-Hermitian systems. This will be investigated both theoretically, and with our experimental collaborators, in a non-Hermitian dimer model realized by a mechanical metamaterial platform. Metamaterials consist of an arrangement of “meta-atoms”, artificially designed units, in this case corresponding to classical harmonic oscillators, whose interactions are then engineered in order to obtain the desired effective Hamiltonian. We will exploit the controllability of this platform to investigate not only the non-Hermitian geometric phase, but then to study the dynamic properties of the interacting so-called Hatano-Nelson dimer, where the interplay of non-Hermiticity and interactions leads to the co-existence of stable and unstable population dynamics. This sets the ground for the final part of this thesis, in which I will theoretically investigate a larger system with three sites with periodic boundary conditions: a triangular plaquette, pierced by a magnetic flux, that might in turn be of interest for the investigation of topological phenomena in the future

Analysis, classification and construction of optimal cryptographic Boolean functions

Modern cryptography is deeply founded on mathematical theory and vectorial Boolean functions play an important role in it. In this context, some cryptographic properties of Boolean functions are defined. In simple terms, these properties evaluate the quality of the cryptographic algorithm in which the functions are implemented. One cryptographic property is the differential uniformity, introduced by Nyberg in 1993. This property is related to the differential attack, introduced by Biham and Shamir in 1990. The corresponding optimal functions are called Almost Perfect Nonlinear functions, shortly APN. APN functions have been constructed, studied and classified up to equivalence relations. Very important is their classification in infinite families, i.e. constructing APN functions that are defined for infinitely many dimensions. In spite of an intensive study of these maps, many fundamental problems related to APN functions are still open and relatively few infinite families are known so far. In this thesis we present some constructions of APN functions and study some of their properties. Specifically, we consider a known construction, L1(x^3)+L2(x^9) with L1 and L2 linear maps, and we introduce two new constructions, the isotopic shift and the generalised isotopic shift. In particular, using the two isotopic shift constructing techniques, in dimensions 8 and 9 we obtain new APN functions and we cover many unclassified cases of APN maps. Here new stands for inequivalent (in respect to the so-called CCZ-equivalence) to already known ones. Afterwards, we study two infinite families of APN functions and their generalisations. We show that all these families are equivalent to each other and they are included in another known family. For many years it was not known whether all the constructed infinite families of APN maps were pairwise inequivalent. With our work, we reduce the list to those inequivalent to each other. Furthermore, we consider optimal functions with respect to the differential uniformity in fields of odd characteristic. These functions, called planar, have been valuable for the construction of new commutative semifields. Planar functions present often a close connection with APN maps. Indeed, the idea behind the isotopic shift construction comes from the study of isotopic equivalence, which is defined for quadratic planar functions. We completely characterise the mentioned equivalence by means of the isotopic shift and the extended affine equivalence. We show that the isotopic shift construction leads also to inequivalent planar functions and we analyse some particular cases of this construction. Finally, we study another cryptographic property, the boomerang uniformity, introduced by Cid et al. in 2018. This property is related to the boomerang attack, presented by Wagner in 1999. Here, we study the boomerang uniformity for some known classes of permutation polynomials.Doktorgradsavhandlin

Systematic Data Extraction in High-Frequency Electromagnetic Fields

The focus of this work is on the investigation of billiards with its statistical eigenvalue properties. Specifically, superconducting microwave resonators with chaotic characteristics are simulated and the eigenfrequencies that are needed for the statistical analysis are computed. The eigenfrequency analysis requires many (in order of thousands) eigenfrequencies to be calculated and the accurate determination of the eigenfrequencies has a crucial significance. Consequently, the research interests cover all aspects from accurate numerical calculation of many eigenvalues and eigenvectors up to application development in order to get good performance out of the programs for distributed-memory and shared-memory multiprocessors. Furthermore, this thesis provides an overview and detailed evaluation of the used numerical approaches for large-scale eigenvalue calculations with respect to the accuracy, the computational time, and the memory consumption. The first approach for an accurate eigenfrequency extraction takes into consideration the evaluated electric field computations in Time Domain (TD) of a superconducting resonant structure. Upon excitation of the cavity, the electric field intensity is recorded at different detection probes inside the cavity. Thereafter, Fourier analysis of the recorded signals is performed and by means of signal-processing and fitting techniques, the requested eigenfrequencies are extracted by finding the optimal model parameters in the least squares sense. The second numerical approach is based on a numerical computation of electromagnetic fields in Frequency Domain (FD) and further employs the Lanczos method for the eigenvalue determination. Namely, when utilizing the Finite Integration Technique (FIT) to solve an electromagnetic problem for a superconducting cavity, which enclosures excited electromagnetic fields, the numerical solution of a standard large-scale eigenvalue problem is considered. Accordingly, if the numerical solution of the same problem is treated by the Finite Element Method (FEM) based on curvilinear tetrahedrons, it yields to the generalized large-scale eigenvalue problem. Afterward, the desired eigenvalues are calculated with the direct solution of the large (generalized) eigenvalue formulations. For this purpose, the implemented Lanczos solvers combine two major ingredients: the Lanczos algorithm with polynomial filtering on the one hand and its parallelization on the other

Relativistic Quantum Chaos

Generations of Ph.D. students at Arizona State University and later at Lanzhou University were involved in research on relativistic quantum chaos. They are: Dr. Ryan Yang, Dr. Xuan Ni, Dr. Guanglei Wang, Dr. Lei Ying, Dr. Rui Bao, Mr. Pei Yu, Mr. Ziyuan Li and Mr. Chengzhen Wang. We thank them. We would also like to thank Dr. Arje Nachman at the Air Force Office of Scientific Research and Dr. Michael Shlesinger from the Office of Naval Research for their great support over the years - without which the works on relativistic quantum chaos would not have been possible. YCL is currently supported by the Pentagon Vannevar Bush Faculty Fellowship program sponsored by the Basic Research Office of the Assistant Secretary of Defense for Research and Engineering and funded by the Office of Naval Research through Grant No. N00014-16-1-2828. LH was supported by NNSF of China under Grants No. 11422541 and No. 11775101.Peer reviewedPostprin

Double and Multiple Stellar Systems: Observational Techniques, Data Administration and Scientific Results

This dissertation, written as a compendium of research articles, was proposed and supervised by J.A. Docobo, Full Professor in Astronomy and Director of the Ramon María Aller Astronomical Observatory of the University of Santiago de Compostela. The focus is on the practical application of speckle interferometry techniques, including the initiation and development of several observational campaigns employing OARMA's eMCCD speckle camera attached to the 2.6m telescope at BAO. Additionally, we present 26 orbits of accessible binaries of the Southern hemisphere based on SOAR speckle data. Also, the orbital information of the double-line spectroscopic binaries, HD 183255, HD 114882, and HD 30712, together with new speckle measurements performed using large telescopes, allowed us to determine the main physical parameters of these systems