Decomposition of Schramm–Loewner evolution along its curve

We show that, for $\kappa \in (0,8)$, the integral of the laws of two-sided radial SLE${}_{\kappa }$ curves through different interior points against a measure with SLE${}_{\kappa }$ Green’s function density is the law of a chordal SLE${}_{\kappa }$ curve, biased by the path’s natural length. We also show that, for $\kappa >0$, the integral of the laws of extended SLE${}_{\kappa }(?8)$ curves through different interior points against a measure with a closed formula density restricted in a bounded set is the law of a chordal SLE${}_{\kappa }$ curve, biased by the path’s capacity length restricted in that set. Another result is that, for $\kappa \in (4,8)$, if one integrates the laws of two-sided chordal SLE${}_{\kappa }$ curves through different force points on $\mathbb{R}$ against a measure with density on $\mathbb{R}$, then one also gets a law that is absolutely continuous w.r.t. that of a chordal SLE${}_{\kappa }$ curve. To obtain these results, we develop a framework to study stochastic processes with random lifetime, and improve the traditional Girsanov’s Theorem.     

ℓ1-symmetric vector random fields

This paper studies the properties of ${?}_1$-symmetric vector random fields in ${\mathbb{R}}^d$, whose direct/cross covariances are functions of ${?}_1$-norm. The spectral representation and a turning bands expression of the covariance matrix function are derived for an ${?}_1$-symmetric vector random field that is mean square continuous. We also establish an integral relationship between an ${?}_1$-symmetric covariance matrix function and an isotropic one. In addition, a simple but efficient approach is proposed to construct the ${?}_1$-symmetric random field in ${\mathbb{R}}^d$, whose univariate marginal distributions may be taken as arbitrary infinitely divisible distribution with finite variance.     

Identities from representation theory

We give a new Jacobi–Trudi-type formula for characters of finite-dimensional irreducible representations in type $C_n$ using characters of the fundamental representations and non-intersecting lattice paths. We give equivalent determinant formulas for the decomposition multiplicities for tensor powers of the spin representation in type $B_n$ and the exterior representation in type $C_n$. This gives a combinatorial proof of an identity of Katz and equates such a multiplicity with the dimension of an irreducible representation in type $C_n$. By taking certain specializations, we obtain identities for $q$-Catalan triangle numbers, a slight modification of the $q,t$-Catalan number of Stump, $q$-triangle versions of Motzkin and Riordan numbers, and generalizations of Touchard’s identity. We use (spin) rigid tableaux and crystal base theory to show some formulas relating Catalan, Motzkin, and Riordan triangle numbers.     

On arithmetical completeness of the logic of proofs

In this paper, we establish a stronger version of Artemov's arithmetical completeness theorem of the Logic of Proofs ${\mathsf{LP}}_0$. Moreover, we prove a version of the uniform arithmetical completeness theorem of ${\mathsf{LP}}_0$.     

Context-free grammars,generating functions and combinatorial arrays

Let ${\left[R_{n,k}\right]}_{n,k\ge 0}$ be an array of nonnegative numbers satisfying the recurrence relation $R_{n,k}=(a_{1}n+a_{2}k+a_3)R_{n-1,k}+(b_{1}n+b_{2}k+b_3)R_{n-1,k-1}+(c_{1}n+c_{2}k+c_3)R_{n-1,k-2}$ with $R_{0,0}=1$ and $R_{n,k}=0$ unless $0\le k\le n$. In this paper, we first prove that the array ${\left[R_{n,k}\right]}_{n,k\ge 0}$ can be generated by some context-free Grammars, which gives a unified proof of many known results. Furthermore, we present criteria for real rootedness of row-generating functions and asymptotical normality of rows of ${\left[R_{n,k}\right]}_{n,k\ge 0}$. Applying the criteria to some arrays related to tree-like tableaux, interior and left peaks, alternating runs, flag descent numbers of group of type $B$, and so on, we get many results in a unified manner. Additionally, we also obtain the continued fraction expansions for generating functions related to above examples. As results, we prove the strong $q$-log-convexity of some generating functions.     

Fq-linear codes over Fq-algebras

Huffman (2013) [12] studied ${\mathbb{F}}_q$-linear codes over ${\mathbb{F}}_{q^m}$ and he proved the MacWilliams identity for these codes with respect to ordinary and Hermitian trace inner products. Let S be a finite commutative ${\mathbb{F}}_q$-algebra. An ${\mathbb{F}}_q$-linear code over S of length n is an ${\mathbb{F}}_q$-submodule of $S^n$. In this paper, we study ${\mathbb{F}}_q$-linear codes over S. We obtain some bounds on minimum distance of these codes, and some large classes of MDR codes are introduced. We generalize the ordinary and Hermitian trace products over ${\mathbb{F}}_q$-algebras and we prove the MacWilliams identity with respect to the generalized form. In particular, we obtain Huffman's results on the MacWilliams identity. Among other results, we give a theory to construct a class of quantum codes and the structure of ${\mathbb{F}}_q$-linear codes over finite commutative graded ${\mathbb{F}}_q$-algebras.