78 research outputs found
Free Energy of the Eight Vertex Model with an Odd Number of Lattice Sites
We calculate the bulk contribution for the doubly degenerated largest
eigenvalue of the transfer matrix of the eight vertex model with an odd number
of lattice sites N in the disordered regime using the generic equation for
roots proposed by Fabricius and McCoy. We show as expected that in the
thermodynamic limit the result coincides with the one in the N even case.Comment: 11 pages LaTeX New introduction, Method change
Improving design safety of tractor-trailers by upgrading towing couplers
One major way of increasing efficiency in production, construction and agriculture industries is to use motor- and tractor-trailers for transportation needs. At the same time, a tractor-trailer is a complex and less maneuverable machine in comparison with a single vehicle. Exceeding the velocity regime, especially in the case of tractor-trailers, leads to a loss of stability during linear motion, on curvilinear sections of the trajectory or in complex traffic conditions. Speed limit is a standard prevention method; however, it does not guarantee any acceptable result. Other ways of resolving this issue have almost exhausted their potential. It should be noted that solving this problem is further complicated by a lack of a unified road-movement theory for tractor-trailers - thus, mainly restrictive measures are used. The authors suggested a hypothesis on possible improvement of tractor-trailer safety by means of increasing their motion stability via upgraded design characteristics of towing tractor-trailer couplers. The higher stability would be achieved by raising the damping factor of transverse oscillations by applying lateral disturbing forces. The authors proposed over 55 technical solutions, all protected by certificates of authorship in engineering, of upgrading the towing couplers and providing the increase of tractor-trailer operation safety in various road conditions. © Published under licence by IOP Publishing Ltd
Exact expressions for correlations in the ground state of the dense O(1) loop model
Conjectures for analytical expressions for correlations in the dense O
loop model on semi infinite square lattices are given. We have obtained these
results for four types of boundary conditions. Periodic and reflecting boundary
conditions have been considered before. We give many new conjectures for these
two cases and review some of the existing results. We also consider boundaries
on which loops can end. We call such boundaries ''open''. We have obtained
expressions for correlations when both boundaries are open, and one is open and
the other one is reflecting. Also, we formulate a conjecture relating the
ground state of the model with open boundaries to Fully Packed Loop models on a
finite square grid. We also review earlier obtained results about this relation
for the three other types of boundary conditions. Finally, we construct a
mapping between the ground state of the dense O loop model and the XXZ
spin chain for the different types of boundary conditions.Comment: 25 pages, version accepted by JSTA
Ground-state properties of a supersymmetric fermion chain
We analyze the ground state of a strongly interacting fermion chain with a
supersymmetry. We conjecture a number of exact results, such as a hidden
duality between weak and strong couplings. By exploiting a scale free property
of the perturbative expansions, we find exact expressions for the order
parameters, yielding the critical exponents. We show that the ground state of
this fermion chain and another model in the same universality class, the XYZ
chain along a line of couplings, are both written in terms of the same
polynomials. We demonstrate this explicitly for up to N = 24 sites, and provide
consistency checks for large N. These polynomials satisfy a recursion relation
related to the Painlev\'e VI differential equation, and using a scale-free
property of these polynomials, we derive a simple and exact formula for their
limit as N goes to infinity.Comment: v2: added more information on scaling function, fixed typo
Auxiliary matrices on both sides of the equator
The spectra of previously constructed auxiliary matrices for the six-vertex
model at roots of unity are investigated for spin-chains of even and odd
length. The two cases show remarkable differences. In particular, it is shown
that for even roots of unity and an odd number of sites the eigenvalues contain
two linear independent solutions to Baxter's TQ-equation corresponding to the
Bethe ansatz equations above and below the equator. In contrast, one finds for
even spin-chains only one linear independent solution and complete strings. The
other main result is the proof of a previous conjecture on the degeneracies of
the six-vertex model at roots of unity. The proof rests on the derivation of a
functional equation for the auxiliary matrices which is closely related to a
functional equation for the eight-vertex model conjectured by Fabricius and
McCoy.Comment: 22 pages; 2nd version: one paragraph added in the conclusion and some
typos correcte
Fluctuations and skewness of the current in the partially asymmetric exclusion process
We use functional Bethe Ansatz equations to calculate the cumulants of the
total current in the partially asymmetric exclusion process. We recover known
formulas for the first two cumulants (mean value of the current and diffusion
constant) and obtain an explicit finite size formula for the third cumulant.
The expression for the third cumulant takes a simple integral form in the limit
where the asymmetry scales as the inverse of the square root of the size of the
system, which corresponds to a natural separation between weak and strong
asymmetry.Comment: 21 pages, 3 figure
Baxter operators for the quantum sl(3) invariant spin chain
The noncompact homogeneous sl(3) invariant spin chains are considered. We
show that the transfer matrix with generic auxiliary space is factorized into
the product of three sl(3) invariant commuting operators. These operators
satisfy the finite difference equations in the spectral parameters which follow
from the structure of the reducible sl(3) modules.Comment: 20 pages, 4 figures, references adde
Analysis of model for assessing the road train movement stability
In this paper, we conduct a mathematical analysis of the model of ensuring the road trains movement stability by changing the design of coupling devices to determine the critical characteristic parameters of the road trains, which result in the loss of en-route directional stability under external action. The concept of the model was to separate the process of yawing of the road trains and its elements (due to external perturbing action) on the highway into several typical stages. The main parameters of the stages (the displacement amplitude and rotation angle of the road trains elements in relation to the driving direction) were determined based on the initial conditions of the road trains movement, the force and duration of the external action. The most dangerous areas of external action application to the road trains were determined in this paper. The maximum permissible exposure limit should not exceed 0.5-1.0% of the road trains trailer momentum, with duration having the greater effect than the amount of impact. The results obtained can be used in mechanical engineering to improve the road trains performance. © Published under licence by IOP Publishing Ltd
Model for assessing the road train stability movement
Road trains (RT) are the main transport in agriculture. However, their accident rate is quite high. In the vehicles movement research their design characteristics and operating modes are mainly improved. The main way to reduce the accident rate remains the speed limit. The authors proposed methods (confirmed by the issued patents) for ensuring the stability of the RT movement by changing the design of trailer devices. The purpose of this work is to form a mathematical model for assessing the stability of the RT movement in the presence of side effects (modeling of sharp turns, impacts). Methods of analysis and synthesis, mathematical analysis, and vector geometry were applied. The process of RT yawing on the highway is split into a number of stages, for which the laws of changing its geometry and kinematics are formulated. Side effect leads to increase in the RT impulse (lateral oscillations increase the speed), or to a violation of stability (the movement amplitude goes beyond the highway or tipping). This model can be used to analyze the stability of the RT movement and evaluate methods to influence it in order to reduce the probability of accidents. © 2020 Published under licence by IOP Publishing Ltd
Quantum Knizhnik-Zamolodchikov equation, generalized Razumov-Stroganov sum rules and extended Joseph polynomials
We prove higher rank analogues of the Razumov--Stroganov sum rule for the
groundstate of the O(1) loop model on a semi-infinite cylinder: we show that a
weighted sum of components of the groundstate of the A_{k-1} IRF model yields
integers that generalize the numbers of alternating sign matrices. This is done
by constructing minimal polynomial solutions of the level 1 U_q(\hat{sl(k)})
quantum Knizhnik--Zamolodchikov equations, which may also be interpreted as
quantum incompressible q-deformations of fractional quantum Hall effect wave
functions at filling fraction nu=1/k. In addition to the generalized
Razumov--Stroganov point q=-e^{i pi/k+1}, another combinatorially interesting
point is reached in the rational limit q -> -1, where we identify the solution
with extended Joseph polynomials associated to the geometry of upper triangular
matrices with vanishing k-th power.Comment: v3: misprint fixed in eq (2.1
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