Open boundary Quantum Knizhnik-Zamolodchikov equation and the weighted enumeration of Plane Partitions with symmetries

We propose new conjectures relating sum rules for the polynomial solution of the qKZ equation with open (reflecting) boundaries as a function of the quantum parameter $q$ and the $\tau$-enumeration of Plane Partitions with specific symmetries, with $\tau=-(q+q^{-1})$. We also find a conjectural relation \`a la Razumov-Stroganov between the $\tau\to 0$ limit of the qKZ solution and refined numbers of Totally Symmetric Self Complementary Plane Partitions.Comment: 27 pages, uses lanlmac, epsf and hyperbasics, minor revision

Sum rules for the ground states of the O(1) loop model on a cylinder and the XXZ spin chain

The sums of components of the ground states of the O(1) loop model on a cylinder or of the XXZ quantum spin chain at Delta=-1/2 (of size L) are expressed in terms of combinatorial numbers. The methods include the introduction of spectral parameters and the use of integrability, a mapping from size L to L+1, and knot-theoretic skein relations.Comment: final version to be publishe

Loop model with mixed boundary conditions, qKZ equation and alternating sign matrices

The integrable loop model with mixed boundary conditions based on the 1-boundary extended Temperley--Lieb algebra with loop weight 1 is considered. The corresponding qKZ equation is introduced and its minimal degree solution described. As a result, the sum of the properly normalized components of the ground state in size L is computed and shown to be equal to the number of Horizontally and Vertically Symmetric Alternating Sign Matrices of size 2L+3. A refined counting is also considered

Quantum Knizhnik-Zamolodchikov equation: reflecting boundary conditions and combinatorics

We consider the level 1 solution of quantum Knizhnik-Zamolodchikov equation with reflecting boundary conditions which is relevant to the Temperley--Lieb model of loops on a strip. By use of integral formulae we prove conjectures relating it to the weighted enumeration of Cyclically Symmetric Transpose Complement Plane Partitions and related combinatorial objects

High concordance between trained nurses and gastroenterologists in evaluating recordings of small bowel video capsule endoscopy (VCE)

Background & Aims: The video capsule endoscopy (VCE) is an accurate and validated tool to investigate the entire small bowel mucosa, but VCE recordings interpretation by the gastroenterologist is time-consuming. A pre-reading of VCE recordings by an expert nurse could be accurate and cost saving. We assessed the concordance between nurses and gastroenterologists in detecting lesions on VCE examinations. Methods: This was a prospective study enrolling consecutive patients who had undergone VCE in clinical practice. Two trained nurses and two expert gastroenterologists participated in the study. At VCE pre-reading the nurses selected any abnormalities, saved them as “thumbnails” and classified the detected lesions as a vascular abnormality, ulcerative lesion, polyp, tumor mass, and unclassified lesion. Then, the gastroenterologist evaluated and interpreted the selected lesions and, successively, reviewed the entire video for potential missed lesions. The time for VCE evaluation was recorded. Results: A total of 95 VCE procedures performed on consecutive patients (M/F: 47/48; mean age: 63 ± 12 years, range: 27−86 years) were evaluated. Overall, the nurses detected at least one lesion in 54 (56.8%) patients. There was total agreement between nurses and gastroenterologists, no missing lesions being discovered at a second look of the entire VCE recording by the physician. The pre-reading procedure by nurse allowed a time reduction of medical evaluation from 49 (33-69) to 10 (8-16) minutes (difference:-79.6%). Conclusions: Our data suggest that trained nurses can accurately identify and select relevant lesions in thumbnails that subsequently were faster reviewed by the gastroenterologist for a final diagnosis. This could significantly reduce the cost of VCE procedure

Polynomial solutions of qKZ equation and ground state of XXZ spin chain at Delta = -1/2

Integral formulae for polynomial solutions of the quantum Knizhnik-Zamolodchikov equations associated with the R-matrix of the six-vertex model are considered. It is proved that when the deformation parameter q is equal to e^{+- 2 pi i/3} and the number of vertical lines of the lattice is odd, the solution under consideration is an eigenvector of the inhomogeneous transfer matrix of the six-vertex model. In the homogeneous limit it is a ground state eigenvector of the antiferromagnetic XXZ spin chain with the anisotropy parameter Delta equal to -1/2 and odd number of sites. The obtained integral representations for the components of this eigenvector allow to prove some conjectures on its properties formulated earlier. A new statement relating the ground state components of XXZ spin chains and Temperley-Lieb loop models is formulated and proved.Comment: v2: cosmetic changes, new section on refined TSSCPPs vs refined ASM

The Razumov-Stroganov conjecture: Stochastic processes, loops and combinatorics

A fascinating conjectural connection between statistical mechanics and combinatorics has in the past five years led to the publication of a number of papers in various areas, including stochastic processes, solvable lattice models and supersymmetry. This connection, known as the Razumov-Stroganov conjecture, expresses eigenstates of physical systems in terms of objects known from combinatorics, which is the mathematical theory of counting. This note intends to explain this connection in light of the recent papers by Zinn-Justin and Di Francesco.Comment: 6 pages, 4 figures, JSTAT News & Perspective

Inhomogeneous loop models with open boundaries

We consider the crossing and non-crossing O(1) dense loop models on a semi-infinite strip, with inhomogeneities (spectral parameters) that preserve the integrability. We compute the components of the ground state vector and obtain a closed expression for their sum, in the form of Pfaffian and determinantal formulas.Comment: 42 pages, 31 figures, minor corrections, references correcte

Parameterized thermal macromodeling for fast and effective design of electronic components and systems

We present a parameterized macromodeling approach to perform fast and effective dynamic thermal simulations of electronic components and systems where key design parameters vary. A decomposition of the frequency-domain data samples of the thermal impedance matrix is proposed to improve the accuracy of the model and reduce the number of the computationally costly thermal simulations needed to build the macromodel. The methodology is successfully applied to analyze the impact of layout variations on the dynamic thermal behavior of a state-of-the-art 8-finger AlGaN/GaN HEMT grown on a SiC substrate

A_k Generalization of the O(1) Loop Model on a Cylinder: Affine Hecke Algebra, q-KZ Equation and the Sum Rule

We study the A_k generalized model of the O(1) loop model on a cylinder. The affine Hecke algebra associated with the model is characterized by a vanishing condition, the cylindric relation. We present two representations of the algebra: the first one is the spin representation, and the other is in the vector space of states of the A_k generalized model. A state of the model is a natural generalization of a link pattern. We propose a new graphical way of dealing with the Yang-Baxter equation and $q$-symmetrizers by the use of the rhombus tiling. The relation between two representations and the meaning of the cylindric relations are clarified. The sum rule for this model is obtained by solving the q-KZ equation at the Razumov-Stroganov point.Comment: 43 pages, 22 figures, LaTeX, (ver 2) Introduction rewritten and Section 4.3 adde