Explicit formulas for determinantal representations of the Drazin inverse solutions of some matrix and differential matrix equations

The Drazin inverse solutions of the matrix equations ${\rm {\bf A}}{\rm {\bf X}} = {\rm {\bf B}}$, ${\rm {\bf X}}{\rm {\bf A}} = {\rm {\bf B}}$ and ${\rm {\bf A}}{\rm {\bf X}}{\rm {\bf B}} ={\rm {\bf D}}$ are considered in this paper. We use both the determinantal representations of the Drazin inverse obtained earlier by the author and in the paper. We get analogs of the Cramer rule for the Drazin inverse solutions of these matrix equations and using their for determinantal representations of solutions of some differential matrix equations, ${\bf X}'+ {\bf A}{\bf X}={\bf B}$ and ${\bf X}'+{\bf X}{\bf A}={\bf B}$, where the matrix ${\bf A}$ is singular.Comment: 20 page

Group-like elements in quantum groups, and Feigin's conjecture

Let $A$ be an arbitrary symmetrizable Cartan matrix of rank $r$, and ${\bf n}={\bf n_+}$ be the standard maximal nilpotent subalgebra in the Kac-Moody algebra associated with $A$ (thus, ${\bf n}$ is generated by $E_1,\ldots,E_r$ subject to the Serre relations). Let $\hat U_q({\bf n})$ be the completion (with respect to the natural grading) of the quantized enveloping algebra of ${\bf n}$. For a sequence ${\bf i}=(i_1,\ldots,i_m)$ with $1\le i_k\le r$, let $P_{\bf i}$ be a skew polynomial algebra generated by $t_1,\ldots,t_m$ subject to the relations $t_lt_k=q^{C_{i_k,i_l}}t_kt_l$ ($1\le k<l\le m$) where $C=(C_{ij})=(d_ia_{ij})$ is the symmetric matrix corresponding to $A$. We construct a group-like element ${\bf e}_{\bf i}\in P_{\bf i}\bigotimes \hat U_q({\bf n})$. This element gives rise to the evaluation homomorphism $\psi_{\bf i}:{\bf C}_q[N]\to P_{\bf i}$ given by $\psi_{\bf i}(x)=x({\bf e}_{\bf i})$, where ${\bf C}_q[N]=U_q({\bf n})^0$ is the restricted dual of $U_q({\bf n})$. Under a well-known isomorphism of algebras ${\bf C}_q[N]$ and $U_q({\bf n})$, the map $\psi_{\bf i}$ identifies with Feigin's homomorphism $\Phi({\bf i}): U_q({\bf n})\to P_{\bf i}$. We prove that the image of $\psi_{\bf i}$ generates the skew-field of fractions ${\cal F}(P_{\bf i})$ if and only if ${\bf i}$ is a reduced expression of some element $w$ in the Weyl group $W$; furthermore, in the latter case, ${\rm Ker}~\psi_{\bf i}$ depends only on $w$ (so we denote $I_w:={\rm Ker}~\psi_{\bf i}$). This result generalizes the results in [5], [6] to the case of Kac-Moody algebras. We also construct an element ${\cal R}_w\in \big({\bf C}_q[N]/I_w\big)\bigotimes \hat U_q({\bf n})$ which specializes to ${\bf e}_{\bf i}$ under the embedding ${\bf C}_q[N]/I_w\hookrightarrow P_{\bf i}$. The elements ${\cal R}_w$ are closelyComment: 25 pages, plain TE

A Sherman-Morrison-Woodbury Identity for Rank Augmenting Matrices with Application to Centering

Matrices of the form $\bf{A} + (\bf{V}_1 + \bf{W}_1)\bf{G}(\bf{V}_2 + \bf{W}_2)^*$ are considered where $\bf{A}$ is a $singular$ $\ell \times \ell$ matrix and $\bf{G}$ is a nonsingular $k \times k$ matrix, $k \le \ell$. Let the columns of $\bf{V}_1$ be in the column space of $\bf{A}$ and the columns of $\bf{W}_1$ be orthogonal to $\bf{A}$. Similarly, let the columns of $\bf{V}_2$ be in the column space of $\bf{A}^*$ and the columns of $\bf{W}_2$ be orthogonal to $\bf{A}^*$. An explicit expression for the inverse is given, provided that $\bf{W}_i^* \bf{W}_i$ has rank $k$. %and $\bf{W}_1$ and $\bf{W}_2$ have the same column space. An application to centering covariance matrices about the mean is given.Comment: Better in Mathematics, Spectral Theory, General, or Numerical Analysi

Functional van den Berg-Kesten-Reimer Inequalities and their Duals, with Applications

The BKR inequality conjectured by van den Berg and Kesten in [11], and proved by Reimer in [8], states that for $A$ and $B$ events on $S$, a finite product of finite sets $S_i,i=1,\ldots,n$, and $P$ any product measure on $S$, $$P(A \Box B) \le P(A)P(B),$$ where the set $A \Box B$ consists of the elementary events which lie in both $A$ and $B$ for `disjoint reasons.' Precisely, with ${\bf n}:=\{1,\ldots,n\}$ and $K \subset {\bf n}$, for ${\bf x} \in S$ letting $[{\bf x}]_K=\{{\bf y} \in S: y_i = x_i, i \in K\}$, the set $A \Box B$ consists of all ${\bf x} \in S$ for which there exist disjoint subsets $K$ and $L$ of ${\bf n}$ for which $[{\bf x}]_K \subset A$ and $[{\bf x}]_L \subset B$. The BKR inequality is extended to the following functional version on a general finite product measure space $(S,\mathbb{S})$ with product probability measure $P$, $$E\left\{ \max_{\stackrel{K \cap L = \emptyset}{K \subset {\bf n}, L \subset {\bf n}}} \underline{f}_K({\bf X})\underline{g}_L({\bf X})\right\} \leq E\left\{f({\bf X})\right\}\,E\left\{g({\bf X})\right\},$$ where $f$ and $g$ are non-negative measurable functions, $\underline{f}_K({\bf x}) = {\rm ess} \inf_{{\bf y} \in [{\bf x}]_K}f({\bf y})$ and $\underline{g}_L({\bf x}) = {\rm ess} \inf_{{\bf y} \in [{\bf x}]_L}g({\bf y}).$ The original BKR inequality is recovered by taking $f({\bf x})={\bf 1}_A({\bf x})$ and $g({\bf x})={\bf 1}_B({\bf x})$, and applying the fact that in general ${\bf 1}_{A \Box B} \le \max_{K \cap L = \emptyset} \underline{f}_K({\bf x}) \underline{g}_L({\bf x})$. Related formulations, and functional versions of the dual inequality on events by Kahn, Saks, and Smyth [6], are also considered. Applications include order statistics, assignment problems, and paths in random graphs.Comment: BKR in title replaced by van den Berg-Kesten-Reime

The Hermite-Hadamard inequality on hypercuboid

Given any ${\bf{a}}: = \left( {a_1 ,a_2 , \ldots ,a_n } \right)$ and ${\bf{b}}: = \left( {b_1 ,b_2 , \ldots ,b_n } \right)$ in $\mathbb{R}^n$. The $\textbf{n}$-fold convex function defined on $\left[ {{\bf{a}},{\bf{b}}} \right]$, ${\bf{a}},{\bf{b}} \in \mathbb{R}^n$ with ${\bf{a}}<{\bf{b}}$ is a convex function in each variable separately. In this work we prove an inequality of Hermite-Hadamard type for $\textbf{n}$-fold convex functions. Namely, we establish the inequality \begin{align*} f\left( {\frac{{{\bf{a}} + {\bf{b}}}}{2}} \right) \le \frac{1}{{{\bf{b}} - {\bf{a}}}}\int_{\bf{a}}^{\bf{b}} {f\left( {\bf{x}} \right)d{\bf{x}}} \le \frac{1}{{2^n }}\sum\limits_{\bf{c}} {f\left( {\bf{c}} \right)}, \end{align*} where $\sum\limits_{\bf{c}} {f\left( {\bf{c}} \right)} : = \sum\limits_{\mathop {c_i \in \left\{ {a_i ,b_i } \right\}}\limits_{1 \le i \le n} } {f\left( {c_1, c_2, \ldots ,c_n } \right)}$. Some other related result are given.Comment: 12 page

On continuous Polish group actions and equivalence relations

Let $X = \left\{P \in [0,1]^{\bf N} : \left(\forall \nu \in {\bf N} \right) \left(P \left(\{\nu \} \right) > 0 \right) \wedge \sum\limits_{\nu = 0}^{\infty} P \left(\{\nu \} \right) = 1 \right\}$ be the Polish space of probability measures on ${\bf N}$, each of which assigns positive probability to every elementary event, while for any $P \in X$, let ${\Gamma}_{P} = \left\{\xi \in L^{1}({\bf N}, P) : \left(\forall \nu \in {\bf N} \right) \left(\xi (\nu) > 0 \right) \wedge \sum\limits_{\nu = 0}^{\infty} \xi (\nu) P \left(\{\nu \} \right) = 1 \right\}$ and let ${\Phi}_{P} : {\Gamma}_{P} \ni \xi \mapsto {\Phi}_{P}(\xi) \in X$ be defined by the relation $\left({\Phi}_{P}(\xi) \right) \left(\{\nu \} \right) = \xi (\nu) P \left(\{\nu \} \right)$, whenever $\nu \in {\bf N}$. If we consider the equivalence relation $E = \left\{(P,Q) \in X^{2} : \left(\exists \xi \in {\Gamma}_{P} \right) \left(Q = {\Phi}_{P}(\xi) \right) \right\}$, the Polish space ${\bf P} = \left\{{\bf x} \in {\ell}^{1} \left({\bf R} \right) : \left(\forall n \in {\bf N} \right) \left({\bf x}(n) > 0 \right) \right\}$ and the commutative Polish group ${\bf G} = \left\{{\bf g} \in (0, \infty)^{\bf N} : \lim\limits_{n \rightarrow \infty}{\bf g}(n) = 1 \right\}$, while we set $\left({\bf g} \cdot {\bf x} \right) (n) = {\bf g}(n){\bf x}(n)$, whenever ${\bf g} \in {\bf G}$, ${\bf x} \in {\bf P}$ and $n \in {\bf N}$, then $E$ is definable and it admits a strong approximation by the turbulent Polish group action of ${\bf G}$ on ${\bf P}$

Cramer's Rule for Generalized Inverse Solutions of Some Matrices Equations

By a generalized inverse of a given matrix, we mean a matrix that exists for a larger class of matrices than the nonsingular matrices, that has some of the properties of the usual inverse, and that agrees with inverse when given matrix happens to be nonsingular. In theory, there are many different generalized inverses that exist. We shall consider the Moore Penrose, weighted Moore-Penrose, Drazin and weighted Drazin inverses. New determinantal representations of these generalized inverse based on their limit representations are introduced in this paper. Application of this new method allows us to obtain analogues classical adjoint matrix. Using the obtained analogues of the adjoint matrix, we get Cramer's rules for the least squares solution with the minimum norm and for the Drazin inverse solution of singular linear systems. Cramer's rules for the minimum norm least squares solutions and the Drazin inverse solutions of the matrix equations ${\rm {\bf A}}{\rm {\bf X}} = {\rm {\bf D}}$, ${\rm {\bf X}}{\rm {\bf B}} = {\rm {\bf D}}$ and ${\rm {\bf A}}{\rm {\bf X}}{\rm {\bf B}} ={\rm {\bf D}}$ are also obtained, where ${\rm {\bf A}}$, ${\rm {\bf B}}$ can be singular matrices of appropriate size. Finally, we derive determinantal representations of solutions of the differential matrix equations, ${\bf X}'+ {\bf A}{\bf X}={\bf B}$ and ${\bf X}'+{\bf X}{\bf A}={\bf B}$, where the matrix ${\bf A}$ is singular

Average Regularity of the Solution to an Equation with the Relativistic-free Transport Operator

Let $u=u(t,{\bf x},{\bf p})$ satisfy the transport equation $\frac {\partial u}{\partial t}+\frac {{\bf p}}{p_0}\frac{\partial u}{\partial{\bf x}}=f$, where $f=f(t,\bf x,\bf p)$ belongs to $L^{p}((0,T)\times {\bf R}^{3}\times {\bf R}^{3})$ for $1<p<\infty$ and $\frac {\partial}{\partial t}+\frac {{\bf p}}{p_0}\frac{\partial}{\partial{\bf x}}$ is the relativistic-free transport operator. We show the regularity of $\int_{{\bf R}^{3}}u(t, {\bf x}, {\bf p})d{\bf p}$ using the same method as given by Golse, Lions, Perthame and Sentis. This average regularity is considered in terms of fractional Sobolev spaces and it is very useful for the study of the existence of the solution to the Cauchy problem on the relativistic Boltzmann equation

Elliptic Curves from Sextics

Let $\mathcal N$ be the moduli space of sextics with 3 (3,4)-cusps. The quotient moduli space ${\mathcal N}/G$ is one-dimensional and consists of two components, ${\mathcal N}_{torus}/G$ and ${\mathcal N}_{gen}/G$. By quadratic transformations, they are transformed into one-parameter families $C_s$ and $D_s$ of cubic curves respectively. We study the Mordell-Weil torsion groups of cubic curves $C_s$ over \bfQ and $D_s$ over \bfQ(\sqrt{-3}) respectively. We show that $C_{s}$ has the torsion group $\bf Z/3\bf Z$ for a generic $s\in \bf Q$ and it also contains subfamilies which coincide with the universal families given by Kubert with the torsion groups $\bf Z/6\bf Z$, $\bf Z/6\bf Z+\bf Z/2\bf Z$, $\bf Z/9\bf Z$ or $\bf Z/12\bf Z$. The cubic curves $D_s$ has torsion $\bf Z/3\bf Z+\bf Z/3\bf Z$ generically but also $\bf Z/3\bf Z+\bf Z/6\bf Z$ for a subfamily which is parametrized by $\bf Q(\sqrt{-3})$.Comment: A remark is added. (after replace mistake). 16 page

On Mixed Brieskorn Variety

Let f_{{\bf a},\{bf b}}({\bf z},\bar{\bf z})=z_1^{a_1+b_1}\bar z_1^{b_1}+...+z_n^{a_n+b_n}\bar z_n^{b_n} be a polar weighted homogeneous mixed polynomial with $a_j>0,b_j\ge 0$, $j=1,..., n$ and let $f_{{\bf a}}({\bf z})=z_1^{a_1}+...+z_n^{a_n}$ be the associated weighted homogeneous polynomial. Consider the corresponding link variety K_{{\bf a},{\bf b}}=f_{{\bf a},{\bf b}}\inv(0)\cap S^{2n-1} and K_{{\bf a}}=f_{{\bf a}}\inv(0)\cap S^{2n-1}. Ruas-Seade-Verjovsky \cite{R-S-V} proved that the Milnor fibrations of $f_{{\bf a},{\bf b}}$ and $f_{{\bf a}}$ are topologically equivalent and the mixed link $K_{{\bf a},{\bf b}}$ is homeomorphic to the complex link $K_{{\bf a}}$. We will prove that they are $C^\infty$ equivalent and two links are diffeomorphic. We show the same assertion for $f({\bf z},\bar{\bf z})=z_1^{a_1+b_1}\bar z_1^{b_1}z_2+...+z_{n-1}^{a_{n-1}+b_{n-1}}\bar z_{n-1}^{b_{n-1}}z_n+z_n^{a_n+b_n}\bar z_n^{b_n}$ and its associated polynomial $g({\bf z})=z_1^{a_1}z_2+...+ z_{n-1}^{a_{n-1}}z_n+z_n^{a_n}$