Column expansion identities and quadratic spanning forest identities

Column expansion identities of determinants give a source of quadratic spanning forest polynomial identities and allow us determine the dimension of the space of certain quadratic spanning forest identities, settling a conjecture of one of us with Vlasev from 2012. Furthermore, we give a combinatorial interpretation of such spanning forest identities via an edge-swapping argument previously developed by one of us in 2019. Quadratic spanning forest polynomials identities are of particular interest because they are useful for quantum field theory calculations in four dimensions.Comment: 31 pages, added a very brief QFT motivatio

Manin matrices and Talalaev's formula

We study special class of matrices with noncommutative entries and demonstrate their various applications in integrable systems theory. They appeared in Yu. Manin's works in 87-92 as linear homomorphisms between polynomial rings; more explicitly they read: 1) elements in the same column commute; 2) commutators of the cross terms are equal: $[M_{ij}, M_{kl}]=[M_{kj}, M_{il}]$ (e.g. $[M_{11}, M_{22}]=[M_{21}, M_{12}]$). We claim that such matrices behave almost as well as matrices with commutative elements. Namely theorems of linear algebra (e.g., a natural definition of the determinant, the Cayley-Hamilton theorem, the Newton identities and so on and so forth) holds true for them. On the other hand, we remark that such matrices are somewhat ubiquitous in the theory of quantum integrability. For instance, Manin matrices (and their q-analogs) include matrices satisfying the Yang-Baxter relation "RTT=TTR" and the so--called Cartier-Foata matrices. Also, they enter Talalaev's hep-th/0404153 remarkable formulas: $det(\partial_z-L_{Gaudin}(z))$, det(1-e^{-\p}T_{Yangian}(z)) for the "quantum spectral curve", etc. We show that theorems of linear algebra, after being established for such matrices, have various applications to quantum integrable systems and Lie algebras, e.g in the construction of new generators in $Z(U(\hat{gl_n}))$ (and, in general, in the construction of quantum conservation laws), in the Knizhnik-Zamolodchikov equation, and in the problem of Wick ordering. We also discuss applications to the separation of variables problem, new Capelli identities and the Langlands correspondence.Comment: 40 pages, V2: exposition reorganized, some proofs added, misprints e.g. in Newton id-s fixed, normal ordering convention turned to standard one, refs. adde

Properties of Vacuum Arc Influenced by Electrode Diameter and Material in TMF Contact Based on Forced Current Zero

With the increase in electrical equipment in More/All Electric Aircraft, 270 V dc power supply systems will be needed. One method for DC interruption is forced current zero (FCZ). Based on FCZ technology with transverse-magnetic-field (TMF) contact, the spiral-type contacts are designed. Experiments with different currents are carried out with contact diameters being 30 mm, 40 mm, and 50 mm, and arcing surface materials Cu-W80 alloy and Cu-Cr50 alloy respectively. It is indicated by the experimental results that breaking capacity of vacuum interrupter and vacuum arc appearance are closely related to the electrode diameter and material. For the same size of electrode diameter, the breaking capacity in Cu-Cr50 is better than that in Cu-W80. With increasing electrode diameter, arc column expansion velocity and diameter increase gradually. Breaking capacity is increasing with larger contact diameter

Algebraic properties of Manin matrices 1

We study a class of matrices with noncommutative entries, which were first considered by Yu. I. Manin in 1988 in relation with quantum group theory. They are defined as "noncommutative endomorphisms" of a polynomial algebra. More explicitly their defining conditions read: 1) elements in the same column commute; 2) commutators of the cross terms are equal: $[M_{ij}, M_{kl}] = [M_{kj}, M_{il}]$ (e.g. $[M_{11},M_{22}] = [M_{21},M_{12}]$). The basic claim is that despite noncommutativity many theorems of linear algebra hold true for Manin matrices in a form identical to that of the commutative case. Moreover in some examples the converse is also true. The present paper gives a complete list and detailed proofs of algebraic properties of Manin matrices known up to the moment; many of them are new. In particular we present the formulation in terms of matrix (Leningrad) notations; provide complete proofs that an inverse to a M.m. is again a M.m. and for the Schur formula for the determinant of a block matrix; we generalize the noncommutative Cauchy-Binet formulas discovered recently [arXiv:0809.3516], which includes the classical Capelli and related identities. We also discuss many other properties, such as the Cramer formula for the inverse matrix, the Cayley-Hamilton theorem, Newton and MacMahon-Wronski identities, Plucker relations, Sylvester's theorem, the Lagrange-Desnanot-Lewis Caroll formula, the Weinstein-Aronszajn formula, some multiplicativity properties for the determinant, relations with quasideterminants, calculation of the determinant via Gauss decomposition, conjugation to the second normal (Frobenius) form, and so on and so forth. We refer to [arXiv:0711.2236] for some applications.Comment: 80 page

Cosmetic crossings and Seifert matrices

We study cosmetic crossings in knots of genus one and obtain obstructions to such crossings in terms of knot invariants determined by Seifert matrices. In particular, we prove that for genus one knots the Alexander polynomial and the homology of the double cover branching over the knot provide obstructions to cosmetic crossings. As an application we prove the nugatory crossing conjecture for twisted Whitehead doubles of non-cable knots. We also verify the conjecture for several families of pretzel knots and all genus one knots with up to 12 crossings.Comment: 16 pages, 5 Figures. Minor revisions. This version will appear in Communications in Analysis and Geometry. This paper subsumes the results of arXiv:1107.203

COMPUTING THE NUMBER OF FUZZY SUBGROUPS BY EXPANSION METHOD

Abstract: The purpose of this paper is to construct the method to find the number of fuzzy subgroups for rectangle groups. The method is formulated by using the pattern of their lattice diagram. We construct the expansion methods, raw and column expansion

Pre-explosive conduit conditions of the 1997 Vulcanian explosions at Soufrière Hills Volcano, Montserrat: II. Overpressure and depth distributions

International audienceA type example of Vulcanian eruptive dynamics is the series of 88 explosions that occurred between August and October 1997 at Soufrière Hills volcano on Montserrat Island. These explosions are interpreted to be caused by the pressurization of a conduit by a shallow highly crystalline and degassed magma plug. We test such an interpretation by combining the pressures and porosities of the pre-explosive magma column proposed by Burgisser et al. (2010, doi:10.1016/j.jvolgeores.2010.04.008) into a physical model that reconstructs a depth-referenced density profile of the column for four mechanisms of pressure buildup. Each mechanism yields a different overpressure profile: 1) gas accumulation, 2) conduit wall elasticity, 3) microlite crystallization, and 4) magma flowage. For the first three mechanisms, the three-part vertical layering of the conduit prior to explosion was spatially distributed as a dense cap atop the conduit with a thickness of a few tens of meters, a transition zone of 400–700 m with heterogeneous vesicularities, and, at greater depth, a more homogeneous, low-porosity zone that brings the total column length to ~ 3.5 km. A shorter column can be obtained with mechanism 4: a dense cap of less than a few meters, a heterogeneous zone of 200–500 m, and a total column length as low as 2.5 km. Inflation/deflation cycles linked to a periodic overpressure source offer a dataset that we use to constrain the four overpressure mechanisms. Magma flowage is sufficient to cause periodic edifice deformation through semi-rigid conduit walls and build overpressures able to trigger explosions. Gas accumulation below a shallow plug is also able to build such overpressures and can occur regardless of magma flowage. The concurrence of these three mechanisms offers the highest likelihood of building overpressures leading to the 1997 explosion series. We also explore the consequences of sudden (eruptive) overpressure release on our magmatic columns to assess the role of syn-explosive vesiculation and pre-fragmentation column expansion. We find that large shallow overpressures and efficient syn-explosive vesiculation cause the most dramatic pre-fragmentation expansion. This leads us to depict two end-member pictures of a Vulcanian explosion. The first case corresponds to the widely accepted view that the downward motion of a fragmentation front controls column evacuation. In the second case, syn-explosive column expansion just after overpressure release brings foamed-up magma up towards an essentially stationary and shallow fragmentation front