Three‐Phase Barker Arrays

A 3‐phase Barker array is a matrix of third roots of unity for which all out‐of‐phase aperiodic autocorrelations have magnitude 0 or 1. The only known truly two‐dimensional 3‐phase Barker arrays have size 2 × 2 or 3 × 3. We use a mixture of combinatorial arguments and algebraic number theory to establish severe restrictions on the size of a 3‐phase Barker array when at least one of its dimensions is divisible by 3. In particular, there exists a double‐exponentially growing arithmetic function T such that no 3‐phase Barker array of size with exists for all . For example, , , and . When both dimensions are divisible by 3, the existence problem is settled completely: if a 3‐phase Barker array of size exists, then .     

Behavior settings and eco-behavioral science: a new arena for mathematical social science permitting a richer and more coherent view of human activities in social systems,part II: Relationships to established disciplines,and needs for mathematical development

Part I of this paper presented the basic concepts of behavior settings and eco-behavioral science originated by the psychologist Roger Barker, showed how they could be linked with standard economic data systems, and suggested their use as a basis for time-allocation matrices and social system accounts. Part II discusses the relationships of behavior settings and eco-behavioral science to established disciplines, describes applications of mathematics to the new concepts by Fox and associates, and points out some major areas in need of mathematical and theoretical development. These areas include representation and measurement of patterns of relationships among roles within behavior settings, relationships among behavior settings within communities and organizations, and the evolution of large, heterogeneous populations of behavior settings over time. We hope some readers will be motivated to participate in this new scientific enterprise.     

Proof of the Barker array conjecture

Using only elementary methods, we prove Alquaddoomi and Scholtz's conjecture of 1989, that no Barker array having exists except when .

     

Second order of perturbation theory correction to the radial distribution function of a liquid with the interaction potential of hard spheres plus a square-well

A liquid with the interaction potential of hard spheres plus a square-well is analyzed using the Monte-Carlo technique. Numerical results for the perturbation theory series over a square-well potential are obtained in the form of the Barker and Henderson discrete representation. Approximating expressions for the correction to a liquid radial distribution function in the second order of perturbation theory are presented. The obtained results allow us to define this correction with a root-mean-square deviation of about 0.007. It is shown that the given approach provides a complete calculation in the second order of perturbation theory, and also the determination of the third order correction to the free energy for a liquid interacting with the potential of the Lennard-Jones type.     

A construction of binary Golay sequence pairs from odd‐length Barker sequences

Binary Golay sequence pairs exist for lengths 2, 10 and 26 and, by Turyn's product construction, for all lengths of the form 2^a10^b26^c where a, b, c are non‐negative integers. Computer search has shown that all inequivalent binary Golay sequence pairs of length less than 100 can be constructed from five “seed” pairs, of length 2, 10, 10, 20 and 26. We give the first complete explanation of the origin of the length 26 binary Golay seed pair, involving a Barker sequence of length 13 and a related Barker sequence of length 11. This is the special case m=1 of a general construction for a length 16m+10 binary Golay pair from a related pair of Barker sequences of length 8m+5 and 8m+3, for integer m≥0. In the case m=0, we obtain an alternative explanation of the origin of one of the length 10 binary Golay seed pairs. The construction cannot produce binary Golay sequence pairs for m>1, having length greater than 26, because there are no Barker sequences of odd length greater than 13. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 478–491, 2009     

The role of higher-order terms in perturbation approaches to the monomer and bonding contributions in a SAFT-type equation of state for square-well chain fluids

A perturbation theory for square-well chain fluids is developed within the scheme of the (generalised) Wertheim thermodynamic perturbation theory. The theory is based on the Pavlyukhin parametrisations [Y. T. Pavlyukhin, J. Struct. Chem. 53, 476 (2012)] of their simulation data for the first four perturbation terms in the high temperature expansion of the Helmholtz free energy of square-well monomer fluids combined with a second-order perturbation theory for the contact value of the radial distribution function of the square-well monomer fluid that enters into bonding contribution. To obtain the latter perturbation terms, we have performed computer simulations in the hard-sphere reference system. The importance of the perturbation terms beyond the second-order one for the monomer fluid and of the approximations of different orders in the bonding contribution for the chain fluids in the predicted equation of state, excess energy and liquid–vapour coexistence densities is analysed.     

There are no Barker arrays having more than two dimensions

Davis, Jedwab and Smith recently proved that there are no 2-dimensional Barker arrays except of size 2 × 2. We show that the existence of a (d + 1)-dimensional Barker array implies the existence of a d-dimensional Barker array with the same number of ± 1 elements. We deduce that there are no Barker arrays having more than two dimensions, as conjectured by Dymond in 1992.      

Perturbation theory based direct correlation function for fluids with hard core potentials

Using second-order Barker–Henderson perturbation theory we are able to derive an explicit expression for the direct correlation function of fluids with hard core potentials. Using the obtained direct correlation function, one can explicitly calculate all thermodynamic properties of simple fluids with hard core potentials. Comparisons with computer simulation data show good agreement for both thermodynamic properties and the static structure factor of the hard core double Yukawa potential.     

Coded excitation of ultrasonic guided waves in long bone fracture assessment

Ultrasonic guided wave (GW) assessment of long bone fracture have conventionally been based on pulse excitation. However, the high attenuation during propagation diminishes the amplitude of received GWs and results in low signal-to-noise ratio (SNR). The Barker code (BC) excitation and the optimal binary code (OBC) excitation were utilized in this study to overcome this limitation. Both simulations and in vitro experiments were performed on the fractured cortical bone plate model, and measured signals from both the BC and OBC excitations were decoded using the finite impulse response least squares inverse filter (FIR-LSIF) and then compared with sine pulse (SP) excited signals. The results suggest the efficiency of coded excitation for amplitude and SNR improvement. Furthermore, time–frequency representation (TFR) analysis was applied to experimental signals; with increasing fracture depth, energy transformation between predominate GW modes A1 and S2 was confirmed. These results show the potential of using BC and OBC excitations to evaluate the depth of long bone fracture.