Robust dynamics in minimal hybrid models of genetic networks

Many gene-regulatory networks necessarily display robust dynamics that are insensitive to noise and stable under evolution. We propose that a class of hybrid systems can be used to relate the structure of these networks to their dynamics and provide insight into the origin of robustness. In these systems, the genes are represented by logical functions, and the controlling transcription factor protein molecules are real variables, which are produced and destroyed. As the transcription factor concentrations cross thresholds, they control the production of other transcription factors. We discuss mathematical analysis of these systems and show how the concepts of robustness and minimality can be used to generate putative logical organizations based on observed symbolic sequences. We apply the methods to control of the cell cycle in yeast

Multi-Colour Braid-Monoid Algebras

We define multi-colour generalizations of braid-monoid algebras and present explicit matrix representations which are related to two-dimensional exactly solvable lattice models of statistical mechanics. In particular, we show that the two-colour braid-monoid algebra describes the Yang-Baxter algebra of the critical dilute A-D-E models which were recently introduced by Warnaar, Nienhuis, and Seaton as well as by Roche. These and other solvable models related to dense and dilute loop models are discussed in detail and it is shown that the solvability is a direct consequence of the algebraic structure. It is conjectured that the Yang-Baxterization of general multi-colour braid-monoid algebras will lead to the construction of further solvable lattice models.Comment: 32 page

Product Wave Function Renormalization Group: construction from the matrix product point of view

We present a construction of a matrix product state (MPS) that approximates the largest-eigenvalue eigenvector of a transfer matrix T, for the purpose of rapidly performing the infinite system density matrix renormalization group (DMRG) method applied to two-dimensional classical lattice models. We use the fact that the largest-eigenvalue eigenvector of T can be approximated by a state vector created from the upper or lower half of a finite size cluster. Decomposition of the obtained state vector into the MPS gives a way of extending the MPS, at the system size increment process in the infinite system DMRG algorithm. As a result, we successfully give the physical interpretation of the product wave function renormalization group (PWFRG) method, and obtain its appropriate initial condition.Comment: 8 pages, 8 figure

Colored Vertex Models, Colored IRF Models and Invariants of Trivalent Colored Graphs

We present formulas for the Clebsch-Gordan coefficients and the Racah coefficients for the root of unity representations ($N$-dimensional representations with $q^{2N}=1$) of $U_q(sl(2))$. We discuss colored vertex models and colored IRF (Interaction Round a Face) models from the color representations of $U_q(sl(2))$. We construct invariants of trivalent colored oriented framed graphs from color representations of $U_q(sl(2))$.Comment: 39 pages, January 199

Dilute Birman--Wenzl--Murakami Algebra and Dn+1(2)D^{(2)}_{n+1} models

A ``dilute'' generalisation of the Birman--Wenzl--Murakami algebra is considered. It can be ``Baxterised'' to a solution of the Yang--Baxter algebra. The $D^{(2)}_{n+1}$ vertex models are examples of corresponding solvable lattice models and can be regarded as the dilute version of the $B^{(1)}_{n}$ vertex models.Comment: 11 page

Disorder Effect on the Vortex Pinning by the Cooling Process Control in the Organic Superconductor κ\kappa-(BEDT-TTF)2_2Cu[N(CN)2_2]Br

We investigate the influence of disorders in terminal ethylene groups of BEDT-TTF molecules (ethylene-disorders) on the vortex pinning of the organic superconductor $\kappa$-(BEDT-TTF)$_2$Cu[N(CN)$_2$]Br. Magnetization measurements are performed under different cooling-processes. The second peak in the magnetization hysteresis curve is observed for all samples studied, and the hysteresis width of the magnetization becomes narrower by cooling faster. In contradiction to the simple pinning effect of disorder, this result shows the suppression of the vortex pinning force by introducing more ethylene-disorders. The ethylene-disorder domain model is proposed for explaining the observed result. In the case of the system containing a moderate number of the ethylene-disorders, the disordered molecules form a domain structure and it works as an effective pinning site. On the contrary, an excess number of the ethylene-disorders may weaken the effect of the domain structure, which results in the less effective pinning force on the vortices.Comment: 6 pages, 6 figure

Impurity Effect on the In-plane Penetration Depth of the Organic Superconductors κ\kappa-(BEDT-TTF)2X_2X (XX = Cu(NCS)2_2 and Cu[N(CN)2_2]Br)

We report the in-plane penetration depth $\lambda_{\parallel}$ of single crystals $\kappa$-(BEDT-TTF)$_2X$ ($X=$ Cu(NCS)$_2$ and Cu[N(CN)$_2$]Br) by means of the reversible magnetization measurements under the control of cooling-rate. In $X$ = Cu(NCS)$_2$, $\lambda_{\parallel}(0)$ as an extrapolation toward $T$ = 0 K does not change by the cooling-rate within the experimental accuracy, while $T_{\textrm{c}}$ is slightly reduced. On the other hand, in $X$ = Cu[N(CN)$_2$]Br, $\lambda_{\parallel}(0)$ indicates a distinct increase by cooling faster. The different behavior of $\lambda_{\parallel}(0)$ on cooling-rate between the two salts is quantitatively explained in terms of the local-clean approximation (London model), considering that the former salt belongs to the very clean system and the later the moderate clean one. The good agreement with this model demonstrates that disorders of ethylene-group in BEDT-TTF introduced by cooling faster increase the electron(quasiparticle)-scattering, resulting in shorter mean free path.Comment: 8 pages, 9 figure

Evidence for structural and electronic instabilities at intermediate temperatures in κ\kappa-(BEDT-TTF)2_{2}X for X=Cu[N(CN)2_{2}]Cl, Cu[N(CN)2_{2}]Br and Cu(NCS)2_{2}: Implications for the phase diagram of these quasi-2D organic superconductors

We present high-resolution measurements of the coefficient of thermal expansion $\alpha (T)=\partial \ln l(T)/\partial T$ of the quasi-twodimensional (quasi-2D) salts $\kappa$-(BEDT-TTF)$_2$X with X = Cu(NCS)$_2$, Cu[N(CN)$_2$]Br and Cu[N(CN)$_2$]Cl. At intermediate temperatures (B), distinct anomalies reminiscent of second-order phase transitions have been found at $T^\ast = 38$ K and 45 K for the superconducting X = Cu(NCS)$_2$ and Cu[N(CN)$_2$]Br salts, respectively. Most interestingly, we find that the signs of the uniaxial pressure coefficients of $T^\ast$ are strictly anticorrelated with those of $T_c$. We propose that $T^\ast$ marks the transition to a spin-density-wave (SDW) state forming on minor, quasi-1D parts of the Fermi surface. Our results are compatible with two competing order parameters that form on disjunct portions of the Fermi surface. At elevated temperatures (C), all compounds show $\alpha (T)$ anomalies that can be identified with a kinetic, glass-like transition where, below a characteristic temperature $T_g$, disorder in the orientational degrees of freedom of the terminal ethylene groups becomes frozen in. We argue that the degree of disorder increases on going from the X = Cu(NCS)$_2$ to Cu[N(CN)$_2$]Br and the Cu[N(CN)$_2$]Cl salt. Our results provide a natural explanation for the unusual time- and cooling-rate dependencies of the ground-state properties in the hydrogenated and deuterated Cu[N(CN)$_2$]Br salts reported in the literature.Comment: 22 pages, 7 figure

Delta-Function Bose Gas Picture of S=1 Antiferromagnetic Quantum Spin Chains Near Critical Fields

We study the zero-temperature magnetization curve (M-H curve) of the S=1 bilinear-biquadratic spin chain, whose Hamiltonian is given by $H=\sum_{i} S_i S_{i+1}+\beta (S_iS_{i+1})^2 with$0 \leq \beta <1$. We focus on validity of the delta-function bose-gas picture near the two critical fields: the saturation field$H_s$and the lower critical field$H_c$associated with the Haldane gap. Near$H_s$, we take ``low-energy effective S-matrix'' approach, which gives correct effective bose-gas coupling constant$c$, different from the spin-wave value. Comparing the M-H curve of the bose gas with the product-wavefunction renormalization group (PWFRG) calculation, excellent agreement is seen. Near$H_c$, comparing the PWFRG result with the bose-gas prediction, we find that there are two distinct regions of$\beta$separated by a critical value$\beta_c(\approx 0.41)$. In the region$0<\beta<\beta_c$, the effective coupling$c$is positive but rather small. The small value of$c$makes the ``critical region'' of the square-root behavior$M\sim \sqrt{H-H_c}$very narrow. Further, we find that in the$\beta \to \beta_c-0$, the square-root behavior transmutes to a different one,$M\sim (H-H_c)^{1/4}$. In the region$\beta_c<\beta <1$, the square-root behavior is rather distinct, but the effective coupling$c$ becomes negative.Comment: 6 pages, RevTeX, 8 ps figure

Middle-Field Cusp Singularities in the Magnetization Process of One-Dimensional Quantum Antiferromagnets

We study the zero-temperature magnetization process (M-H curve) of one-dimensional quantum antiferromagnets using a variant of the density-matrix renormalization group method. For both the S=1/2 zig-zag spin ladder and the S=1 bilinear-biquadratic chain, we find clear cusp-type singularities in the middle-field region of the M-H curve. These singularities are successfully explained in terms of the double-minimum shape of the energy dispersion of the low-lying excitations. For the S=1/2 zig-zag spin ladder, we find that the cusp formation accompanies the Fermi-liquid to non-Fermi-liquid transition.Comment: 4 pages, RevTeX, 3 figures, some mistakes in references are correcte