Tight Sum-of-Squares lower bounds for binary polynomial optimization problems

We give two results concerning the power of the Sum-of-Squares(SoS)/Lasserre hierarchy. For binary polynomial optimization problems of degree $2d$ and an odd number of variables $n$, we prove that $\frac{n+2d-1}{2}$ levels of the SoS/Lasserre hierarchy are necessary to provide the exact optimal value. This matches the recent upper bound result by Sakaue, Takeda, Kim and Ito. Additionally, we study a conjecture by Laurent, who considered the linear representation of a set with no integral points. She showed that the Sherali-Adams hierarchy requires $n$ levels to detect the empty integer hull, and conjectured that the SoS/Lasserre rank for the same problem is $n-1$. We disprove this conjecture and derive lower and upper bounds for the rank

Bounds on the maximum multiplicity of some common geometric graphs

We obtain new lower and upper bounds for the maximum multiplicity of some weighted and, respectively, non-weighted common geometric graphs drawn on n points in the plane in general position (with no three points collinear): perfect matchings, spanning trees, spanning cycles (tours), and triangulations. (i) We present a new lower bound construction for the maximum number of triangulations a set of n points in general position can have. In particular, we show that a generalized double chain formed by two almost convex chains admits {\Omega}(8.65^n) different triangulations. This improves the bound {\Omega}(8.48^n) achieved by the double zig-zag chain configuration studied by Aichholzer et al. (ii) We present a new lower bound of {\Omega}(12.00^n) for the number of non-crossing spanning trees of the double chain composed of two convex chains. The previous bound, {\Omega}(10.42^n), stood unchanged for more than 10 years. (iii) Using a recent upper bound of 30^n for the number of triangulations, due to Sharir and Sheffer, we show that n points in the plane in general position admit at most O(68.62^n) non-crossing spanning cycles. (iv) We derive lower bounds for the number of maximum and minimum weighted geometric graphs (matchings, spanning trees, and tours). We show that the number of shortest non-crossing tours can be exponential in n. Likewise, we show that both the number of longest non-crossing tours and the number of longest non-crossing perfect matchings can be exponential in n. Moreover, we show that there are sets of n points in convex position with an exponential number of longest non-crossing spanning trees. For points in convex position we obtain tight bounds for the number of longest and shortest tours. We give a combinatorial characterization of the longest tours, which leads to an O(nlog n) time algorithm for computing them

Randomness Quality of CI Chaotic Generators: Applications to Internet Security

Due to the rapid development of the Internet in recent years, the need to find new tools to reinforce trust and security through the Internet has became a major concern. The discovery of new pseudo-random number generators with a strong level of security is thus becoming a hot topic, because numerous cryptosystems and data hiding schemes are directly dependent on the quality of these generators. At the conference Internet`09, we have described a generator based on chaotic iterations, which behaves chaotically as defined by Devaney. In this paper, the proposal is to improve the speed and the security of this generator, to make its use more relevant in the Internet security context. To do so, a comparative study between various generators is carried out and statistical results are given. Finally, an application in the information hiding framework is presented, to give an illustrative example of the use of such a generator in the Internet security field.Comment: 6 pages,6 figures, In INTERNET'2010. The 2nd Int. Conf. on Evolving Internet, Valencia, Spain, pages 125-130, September 2010. IEEE Computer Society Press Note: Best Paper awar

Low Complexity Algorithms for Linear Recurrences

We consider two kinds of problems: the computation of polynomial and rational solutions of linear recurrences with coefficients that are polynomials with integer coefficients; indefinite and definite summation of sequences that are hypergeometric over the rational numbers. The algorithms for these tasks all involve as an intermediate quantity an integer $N$ (dispersion or root of an indicial polynomial) that is potentially exponential in the bit size of their input. Previous algorithms have a bit complexity that is at least quadratic in $N$. We revisit them and propose variants that exploit the structure of solutions and avoid expanding polynomials of degree $N$. We give two algorithms: a probabilistic one that detects the existence or absence of nonzero polynomial and rational solutions in $O(\sqrt{N}\log^{2}N)$ bit operations; a deterministic one that computes a compact representation of the solution in $O(N\log^{3}N)$ bit operations. Similar speed-ups are obtained in indefinite and definite hypergeometric summation. We describe the results of an implementation.Comment: This is the author's version of the work. It is posted here by permission of ACM for your personal use. Not for redistributio

Asymmetry and structural information in preferential attachment graphs

Graph symmetries intervene in diverse applications, from enumeration, to graph structure compression, to the discovery of graph dynamics (e.g., node arrival order inference). Whereas Erd\H{o}s-R\'enyi graphs are typically asymmetric, real networks are highly symmetric. So a natural question is whether preferential attachment graphs, where in each step a new node with $m$ edges is added, exhibit any symmetry. In recent work it was proved that preferential attachment graphs are symmetric for $m=1$, and there is some non-negligible probability of symmetry for $m=2$. It was conjectured that these graphs are asymmetric when $m \geq 3$. We settle this conjecture in the affirmative, then use it to estimate the structural entropy of the model. To do this, we also give bounds on the number of ways that the given graph structure could have arisen by preferential attachment. These results have further implications for information theoretic problems of interest on preferential attachment graphs.Comment: 24 pages; to appear in Random Structures & Algorithm

Capillary Electrochromatography-Mass Spectrometry (CEC-MS) of Surfactants

This research presents advancements in the coupling of capillary electrochromatography (CEC) to mass spectrometry (MS) for the analysis of different chemical classes of surfactants. Chapter 1 provides a brief introduction that summarizes the mechanics and fundamentals of CEC, including instrumentation and applications for CEC-MS. Chapter 2 describes the on-line hyphenation of a packed CEC column with an internally tapered tip coupled to electrospray ionization-mass spectrometry (ESI-MS) and atmospheric pressure chemical ionization-mass spectrometry (APCI-MS) for the analysis of betaine-type amphoteric or zwitterionic surfactants (Zwittergent®). The interesting aspects include CEC-MS column manufacture and charaterization, as well as a comparison between the CEC-ACPI-MS and CEC-ESI-MS ionization pattern of zwittergents. In Chapter 3, the CEC-MS of alkyltrimethyl-ammonium ions (ATMA+) with chain length ranging from C1-C18 is optimized using an internally tapered CEC-MS column packed with mixed mode C6/strong cation exchange stationary phase and coupled to an ESI source. In addition, the optimized CEC-ESI-MS protocol is applied for the challenging analysis of commercial sample Arquad S-50 ATMA+ containing cis-trans unsaturated and saturated soyabean fatty acid derivatives. In Chapter 4, a novel CEC-UV method for separation of the various Triton X-100 oligomers is presented. A systematic mobile phase tuning and comparison of monomeric vs. polymeric stationary phases was conducted. In Chapter 5, we present the first application of CEC coupled to MS for analysis of Triton X (TX-) series surfactants. A characterization from the viewpoint of the ion and adduct formation for TX-series nonionic surfactants with a variable number of ethoxy units (n=1.5-16) in the scan mode are first discussed. Next, utilizing the TX-series as model alkylphenolpolyethoxylates (APEOs), a detailed investigation of the chromatographic separation and MS detection are performed followed by analysis of very long chain TX series with n=30-70. In Chapter 6, CEC-MS utilizing full scan positive ion mode of ESI was employed to study the effect of fragmentor voltage on the in-source collision induced dissociation (IS-CID) of several APEO nonionic surfactants. Finally, in Chapter 7, the preparation and characterization of a novel liquid crystalline stationary phase suitable for separation of neutral and charged compounds in packed column CEC is evaluated

Quantum Phase Diagram for Homogeneous Bose-Einstein Condensate

We calculate the quantum phase transition for a homogeneous Bose gas in the plane of s-wave scattering length a_s and temperature T. This is done by improving a one-loop result near the interaction-free Bose-Einstein critical temperature T_c^{(0)} with the help of recent high-loop results on the shift of the critical temperature due to a weak atomic repulsion using variational perturbation theory. The quantum phase diagram shows a nose above T_c^{(0)}, so that we predict the existence of a reentrant transition above T_c^{(0)}, where an increasing repulsion leads to the formation of a condensate.Comment: Author Information under http://www.physik.fu-berlin.de/~kleinert/institution.html . Latest update of paper (including all PS fonts) at http://www.physik.fu-berlin.de/~kleinert/34