Solving Linear Tensor Equations

We develop a systematic way to solve linear equations involving tensors of arbitrary rank. We start off with the case of a rank 3 tensor, which appears in many applications, and after finding the condition for a unique solution we derive this solution. Subsequently, we generalize our result to tensors of arbitrary rank. Finally, we consider a generalized version of the former case of rank 3 tensors and extend the result when the tensor traces are also included.

Three-point functions of higher-spin spinor current multiplets in $$ \mathcal{N} $$ = 1 superconformal theory

In this paper, we study the general form of three-point functions of conserved current multiplets Sα(k) = S(α1…αk) of arbitrary rank in four-dimensional $$ \mathcal{N} $$ N = 1 superconformal theory. We find that the correlation function of three such operators $$ \left\langle {\overline{S}}_{\dot{\alpha}(k)}\left({z}_1\right){S}_{\beta \left(k+l\right)}\left({z}_2\right){\overline{S}}_{\dot{\gamma}(l)}\left({z}_3\right)\right\rangle $$ S ¯ α ̇ k z 1 S β k + l z 2 S ¯ γ ̇ l z 3 is fixed by the superconformal symmetry up to a single complex coefficient though the precise form of the correlator depends on the values of k and l. In addition, we present the general structure of mixed correlators of the form $$ \left\langle {\overline{S}}_{\dot{\alpha}(k)}\left({z}_1\right){S}_{\alpha (k)}\left({z}_2\right)L\left({z}_3\right)\right\rangle $$ S ¯ α ̇ k z 1 S α k z 2 L z 3 and $$ \left\langle {\overline{S}}_{\dot{\alpha}(k)}\left({z}_1\right){S}_{\alpha (k)}\left({z}_2\right){J}_{\gamma \dot{\gamma}}\left({z}_3\right)\right\rangle $$ S ¯ α ̇ k z 1 S α k z 2 J γ γ ̇ z 3 , where L is the flavour current multiplet and $$ {J}_{\gamma \dot{\gamma}} $$ J γ γ ̇ is the supercurrent.

Dual Pairs of Matroids with Coefficients in Finitary Fuzzy Rings of Arbitrary Rank

Infinite matroids have been defined by Reinhard Diestel and coauthors in such a way that this class is (together with the finite matroids) closed under dualization and taking minors. On the other hand, Andreas Dress introduced a theory of matroids with coefficients in a fuzzy ring which is – from a combinatorial point of view – less general, because within this theory every circuit has a finite intersection with every cocircuit. Within the present paper, we extend the theory of matroids with coefficients to more general classes of matroids, if the underlying fuzzy ring has certain properties to be specified.

Continuous automorphisms of Cremona groups

We show that if a group automorphism of a Cremona group of arbitrary rank is also a homeomorphism with respect to either the Zariski or the Euclidean topology, then it is inner up to a field automorphism of the base-field. Moreover, we show that a similar result holds if we consider groups of polynomial automorphisms of affine spaces instead of Cremona groups.

Спектральные моменты характеристик бинарных взаимодействий между линейными молекулами

A new approach to derive symmetrized expressions of leading classical moments of spectral distributions characterizing different anisotropic terms of the interaction potential for the case of two liear molecules is presented. The results allow to calculate diffuse shapes formed by transitions between continuous eigenstates of a molecular pair and open the way to account for the nonMarkov effects (due to finite collision durations) in the rotatonal relaxation matrix of an arbitrary rank. The approach is also applied to the spectral moments of vector and tensor characteristics determining the band intensities in the collision-induced spectra of linear molecules. Generally, the use of symmetrized expressions lead to considerably faster computer codes.

Brill-Noether generality of binary curves

We show that the space $G^r_{\underline d}(X)$ of linear series of certain multi-degree $\underline d=(d_1,d_2)$ (including the balanced ones) and rank r on a general genus-g binary curve X has dimension $\rho _{g,r,d}=g-(r+1)(g-d+r)$ if nonempty, where $d=d_1+d_2$ . This generalizes Caporaso’s result from the case $r\leq 2$ to arbitrary rank, and shows that the space of Osserman-limit linear series on a general binary curve has the expected dimension, which was known for $r\leq 2$ . In addition, we show that the space $G^r_{\underline d}(X)$ is still of expected dimension after imposing certain ramification conditions with respect to a sequence of increasing effective divisors supported on two general points $P_i\in Z_i$ , where $i=1,2$ and $Z_1,Z_2$ are the two components of X. Our result also has potential application to the lifting problem of divisors on graphs to divisors on algebraic curves.

Higher-rank Bohr sets and multiplicative diophantine approximation

Gallagher’s theorem is a sharpening and extension of the Littlewood conjecture that holds for almost all tuples of real numbers. We provide a fibre refinement, solving a problem posed by Beresnevich, Haynes and Velani in 2015. Hitherto, this was only known on the plane, as previous approaches relied heavily on the theory of continued fractions. Using reduced successive minima in lieu of continued fractions, we develop the structural theory of Bohr sets of arbitrary rank, in the context of diophantine approximation. In addition, we generalise the theory and result to the inhomogeneous setting. To deal with this inhomogeneity, we employ diophantine transference inequalities in lieu of the three distance theorem.

AUTOMORPHISMS OF FREE BURNSIDE GROUPS OF PERIOD 3

We have proved that any automorphism of the free Burnside group $ B(3) $ of period 3 and an arbitrary rank is induced by an automorphism of the free group of the same rank.