On the Solution of the Operator Equation AT-TB=S

Let H, K be two Hilbert spaces over complex field A. Let B(HI, K) denote the set of all bounded linear operators from H to K. If H=K, we write B(H) instead of B(H, K). Consider the operator defined by the equation     

|Z|~2=Z的应用技巧

公式、|z|~2=z(?)表达了共轭复数及复数模的重要性质。它沟通了复数,|z|~2与一对共轭复数zz的关系。应用它可以将复数模的问题与一对共轭复数的问题互相转化,使之在不改变问题的性质的前提下改变了问题的结构形式,有助于促成问题的解决。另外,该公式又可将关于复数z与|z|的方程转化关于     

Proof of the absence of multiplicative renormalizability of the Gross-Neveu model in the dimensional regularizationd=2+2ɛ

We prove that the simplest four-fermion Gross-Neveu model with the dimensional regularization d=2+2ɛ is not multiplicatively renormalizable. This is due to the counter-term, generated by the three-loop vertex diagrams, that is proportional to the evanescent operator . Here is the totally antisymmetric product of n γ-matrices, which is not zero in a noninteger dimension as well. Therefore, calculations of the (2+ɛ)-expansion of the critical indices η and ν in the framework of the simple Gross-Neveu model are correct only up to ε⁴ for η and to ɛ³ for ν. In higher orders, one must take into consideration the generation of other (not only V₃) evanescent operators. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 113, No. 1, pp. 85–99, October, 1997.     

On the average number of subgroups of the group $${\mathbb {Z}}_m \times {\mathbb {Z}}_n$$ Z m × Z n with k-th power sequence

The Ramanujan Journal - Let $$\mathbb {Z}_{n}$$ be the additive group of residue classes modulo n. Let s(m, n) denote the total number of subgroups of the group $$\mathbb {Z}_{m} \times...     

Explicit solutions of the equation –div z = f under the constraint ∥z ∥∞ = 1

We consider the eigenvalue problem for the 1-Laplace operator and the corresponding variational problem. Since the underlying functionals are not differentiable, the derivation of the Euler-Lagrange equation is highly nontrivial and involves methods of nonsmooth analysis. A remarkable fact is, that there are infinitely many functions f for which the equation –div z = f has to be satisfied. The purpose of the present work is the investigation of this equation. We show that in a square 2-dimensional domain for each right hand side f there are infinitely many vector fields z satisfying the Euler-Lagrange equation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The σ-algebra of pasts of a random walk over the orbits of a Bernoulli action of the group Z d

In the present paper, we study the σ-algebra of pasts Ξ = {ξ _n } _n of a random walk T over the orbits of a Bernoulli action of the group Z^d. We calculate the proper scaling and the scaling entropy of this sequence of partitions. We show that the proper scaling entropy of the σ-algebra of pasts is equal to . Bibliography: 4 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 325, 2005, pp. 103–112.     

The Continued Fraction Solution of the Matrix Equation AX-XB=C

Let K[λ] denote the polynomial ring over the number field K.Suppose thatf(λ)and g(λ)are coprime in K[λ] and deg f(λ)≥1.By the Euclidean algorithmwe obtain the continued fraction     

The structure of finite monoids satisfying the relation ℛ = ℋ

It is shown that any finite monoid S on which Green’s relations R and H coincide divides the monoid of all upper triangular row-monomial matrices over a finite group. The proof is constructive; given the monoid S, the corresponding group and the order of matrices can be effectively found. The obtained result is used to identify the pseudovariety generated by all finite monoids satisfying R = H with the semidirect product of the pseudovariety of all finite groups and the pseudovariety of all finite R-trivial monoids.     

“A=X Y Z”的故事

据说,一个爱说空话而不爱用功的年轻人,整天缠着大科学家爱因斯坦,要他们公开成功的秘诀,爱因斯坦同意了,写下了这个公式:A=X Y Z,爱因斯坦对这位年轻人解释道:“A代表成功,X代表艰苦的劳动,Y代表正确的方法,……”,“ Z代表什么呢?”年轻人迫切地追     

A note on the number of solutions of the generalized Ramanujan-Nagell equation x 2 ? D = p n

Let D be a positive integer, and let p be an odd prime with p ? D. In this paper we use a result on the rational approximation of quadratic irrationals due to M. Bauer, M.A. Bennett: Applications of the hypergeometric method to the generalized Ramanujan-Nagell equation. Ramanujan J. 6 (2002), 209–270, give a better upper bound for N(D, p), and also prove that if the equation U ² ? DV ² = ?1 has integer solutions (U, V), the least solution (u ₁, v ₁) of the equation u ² ? pv ² = 1 satisfies p ? v ₁, and D > C(p), where C(p) is an effectively computable constant only depending on p, then the equation x ² ? D = p ⁿ has at most two positive integer solutions (x, n). In particular, we have C(3) = 10⁷.     

Interior and boundary continuity of the solution of the singular equation (β(u))t=Lu

We extend some results of DiBenedetto and Vespri (Arch. Rational Mech. Anal 132(3) (1995) 247) proving the interior and boundary continuity of bounded solutions of the singular equation $\text{(β(u))}{}_{\mathrm{t}}\text{=}\text{L}\text{u}$ where $\text{L}$ is a second order elliptic operator with bounded measurable coefficients that depend both on space and time in a proper way.     

Temperature Correlators of the XXZ Heisenberg Magnet for Δ = -∞

The one-dimensional XXZ Heizenberg magnet for = - is considered, and the time-dependent temperature correlation function of the z components of local spin operators is calculated. In the thermodynamic limit, the correlation function is expressed in terms of the Fredholm determinants of linear integral operators. Bibliography: 23 titles.     

On the distinctness of modular reductions of primitive sequences over Z/(232−1)

This paper studies the distinctness of modular reductions of primitive sequences over ${\mathbf{Z}/(2^{32}-1)}$ . Let f(x) be a primitive polynomial of degree n over ${\mathbf{Z}/(2^{32}-1)}$ and H a positive integer with a prime factor coprime with 2³²?1. Under the assumption that every element in ${\mathbf{Z}/(2^{32}-1)}$ occurs in a primitive sequence of order n over ${\mathbf{Z}/(2^{32}-1)}$ , it is proved that for two primitive sequences ${\underline{a}=(a(t))_{t\geq 0}}$ and ${\underline{b}=(b(t))_{t\geq 0}}$ generated by f(x) over ${\mathbf{Z}/(2^{32}-1), \underline{a}=\underline{b}}$ if and only if ${a\left( t\right) \equiv b\left( t\right) \bmod{H}}$ for all t ≥ 0. Furthermore, the assumption is known to be valid for n between 7 and 100, 000, the range of which is sufficient for practical applications.     

Invertibility in rings of the commutator ab – ba,where aba=a and bab=b

Let ? be a ring and a,?b∈? satisfy aba?=?a and bab?=?b. We characterize when ab???ba is invertible. This study is specialized when ? has an involution and when b is the Moore–Penrose inverse of a.