Related papers: Linear independence for $C_\ell^{(1)}$ by using $C…

A variant of the global $T(1)$ criterion to characterize the bounded Calder\'{o}n--Zygmund operators on BMO($\mathbb{R}^d$) is proved. We apply it to the certain Calder\'on commutators.

For an affine Lie algebra $\hat{\mathfrak g}$ the coefficients of certain vertex operators which annihilate level $k$ standard $\hat{\mathfrak g}$-modules are the defining relations for level $k$ standard modules. In this paper we study a…

Let $K$ be a finite extension of $\mathbb{Q}_p$ that is totally ramified over $\mathbb{Q}_p$. The set $\mathcal{M}\mathcal{F}(K)$ consists of power series in $1+zK[[z]]$ that are solutions of differential operators in $K(z)[d/dz]$ equipped…

Let $R$ be a discrete valuation domain with field of fractions $Q$ and maximal ideal generated by $\pi$. Let $\Lambda$ be an $R$-order such that $Q\Lambda$ is a separable $Q$-algebra. Maranda showed that there exists $k\in\mathbb{N}$ such…

We use difference sets to construct interesting sets of lines in complex space. Using (v,k,1)-difference sets, we obtain k^2-k+1 equiangular lines in C^k when k-1 is a prime power. Using semiregular relative difference sets with parameters…

We obtain a necessary and sufficient condition for the linear independence of solutions of differential equations for hyperlogarithms. The key fact is that the multiplier (i.e. the factor $M$ in the differential equation $dS=MS$) has only…

The grouplike elements of a coalgebra over a field are known to be linearly independent over said field. Here we prove three variants of this result. One is a generalization to coalgebras over a commutative ring (in which case the linear…

For every associative algebra $A$ and every class $\mathcal{C}$ of representations of $A$ the following question (related to nullstellensatz) makes sense: Characterize all tuples of elements $a_1,\ldots,a_n \in A$ such that vectors…

We give a purely combinatorial proof of the positivity of the stabilized forms of the generalized exponents associated to each classical root system. In finite type A_{n-1}, we rederive the description of the generalized exponents in terms…

For a complex simple Lie algebra of type A_l,B_l,C_l or D_l, given a family of elements f_\lambda\ in commutative ring Z[\Lambda], we show that f_\lambda\ is just the formal character of the Weyl module V(\lambda) if f_\lambda\ satisfy…

It is an open question of Baker whether the numbers $L(1, \chi)$ for non-trivial Dirichlet characters $\chi$ with period $q$ are linearly independent over $\mathbb{Q}$. The best known result is due to Baker, Birch and Wirsing which affirms…

A $k$-essence scalar field model having (non canonical) Lagrangian of the form $L=-V(\phi)F(X)$ where $X=1/2g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi$ with constant $V(\phi)$ is shown to be consistent with luminosity distance-redshift data…

We study nonlinear vacuum electrodynamics in a first-order formulation proposed by Pleba\'nski. By applying a Dirac constraint analysis, we derive an effective Hamiltonian, together with the equations of motion. We show that there exists a…

It was shown using perturbation theory[1] that Thouless energy Ec for a quantum system scales linearly with the conductance of the system. We derive in an alternate way in 1-D that Ec scales with the conductance in a very different way. We…

Using a generalization of forward elimination, it is proved that functions $f_1,...,f_n:X\to\mathbb{A}$, where $\mathbb{A}$ is a field, are linearly independent if and only if there exists a nonsingular matrix $[f_i(x_j)]$ of size $n$,…

For a reduced word ${\bf i}$ of the longest element in the Weyl group of $\mathrm{SL}_{n+1}(\mathbb{C})$, one can associate the string cone $C_{\bf i}$ which parametrizes the dual canonical bases. In this paper, we classify all ${\bf i}$'s…

Let $K$ be a totally real field and $\pi$ be a regular algebraic polarized cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb A_K)$. Let $\{\rho_{\pi,\lambda}:\mathrm{Gal}_K\to\mathrm{GL}_n(\overline E_\lambda)\}_\lambda$ be the…

We present a simple combinatorial model for the characters of the irreducible integrable highest weight modules for complex symmetrizable Kac-Moody algebras. This model can be viewed as a discrete counterpart to the Littelmann path model.…

In their study of Jack polynomials, Nazarov-Sklyanin introduced a remarkable new graded linear operator ${\mathcal L} \colon F[w] \rightarrow F[w]$ where $F$ is the ring of symmetric functions and $w$ is a variable. In this paper, we (1)…

We study the Hecke algebra $\H(\bq)$ over an arbitrary field $\FF$ of a Coxeter system $(W,S)$ with independent parameters $\bq=(q_s\in\FF:s\in S)$ for all generators. This algebra is always linearly spanned by elements indexed by the…