Superintegrability of Kontsevich matrix model

Many eigenvalue matrix models possess a peculiar basis of observables that have explicitly calculable averages. This explicit calculability is a stronger feature than ordinary integrability, just like the cases of quadratic and Coulomb potentials are distinguished among other central potentials, and we call it superintegrability. As a peculiarity of matrix models, the relevant basis is formed by the Schur polynomials (characters) and their generalizations, and superintegrability looks like a property $$\langle character\rangle \,\sim character$$ ⟨ c h a r a c t e r ⟩ ∼ c h a r a c t e r . This is already known to happen in the most important cases of Hermitian, unitary, and complex matrix models. Here we add two more examples of principal importance, where the model depends on external fields: a special version of complex model and the cubic Kontsevich model. In the former case, straightforward is a generalization to the complex tensor model. In the latter case, the relevant characters are the celebrated Q Schur functions appearing in the description of spin Hurwitz numbers and other related contexts.

Fatigue life research of welded joints in railway girder

Progressive crack formation in welded joints of railway girders requires a reliable prediction of the fatigue life. It was found that the most common fatigue cracks are T-9 and T-10, which are formed in welded joints of vertical stiffeners to the beam web. The service life calculation of such joints, according to guideline, shows overestimated results. To reduce the error, the operational features of these joints were investigated under a moving loads. Stress-strain monitoring was carried out with the help of a small-sized automated tensometric complex «Tensor-MS». In total, 13 trussed and solid-web girders were investigated. As a result, it was found that the beam web near the welded ends of stiffeners subjected to bending deformations, but fatigue curves, obtained under tensile-compressive, are used to calculate the durability of such joints. To increase the reliability of fatigue calculation, laboratory tests were carried out for a high-cycle fatigue of welded specimens, simulating joints with T-9, T-10 cracks. The design and materials of laboratory specimens were assigned similarly to that of the main beams or the floor beams. Analysis of the actual stresses under the moving load revealed the necessary loading conditions for laboratory specimens. Totally, 42 laboratory samples were tested for bending. As a result of the tests, fatigue curves of welded joints were obtained. The results show a decrease in the joints durability by 20–50 % in comparison with the samples tested in tensile-compressive. In addition, as a result of the tests, the dependences of the fatigue cracks growth rate on the maximum cycle stresses were obtained.

Homotopy techniques for tensor decomposition and perfect identifiability

LetTbe a general complex tensor of format{(n_{1},\dots,n_{d})}. When the fraction{\prod_{i}n_{i}/[1+\sum_{i}(n_{i}-1)]}is an integer, and a natural inequality (called balancedness) is satisfied, it is expected thatThas finitely many minimal decomposition as a sum of decomposable tensors. We show how homotopy techniques allow us to find all the decompositions ofT, starting from a given one. Computationally, this gives a guess regarding the total number of such decompositions. This guess matches exactly with all cases previously known, and predicts several unknown cases. Some surprising experiments yielded two new cases of generic identifiability: formats{(3,4,5)}and{(2,2,2,3)}which have a unique decomposition as the sum of six and four decomposable tensors, respectively. We conjecture that these two cases together with the classically known matrix pencils are the only cases where generic identifiability holds, i.e., the onlyidentifiablecases. Building on the computational experiments, we use algebraic geometry to prove these two new cases are indeed generically identifiable.