which commute with the flow defined by (0.1) (we set τ1 = z and τ2 = τ). The important fact is that the right hand sides Pn of equations (0.2) are polynomials in v and its derivatives, i.e., differential polynomials in v. This fact allows us to treat the KdV hierarchy (0.2) in the following way. Consider the ring R = C[v]n≥0 of polynomials in the variables v (n) (we shall write v for v(0)). Let...

Let V be an l-dimensional Euclidean space. Let G ⊂ O(V ) be a finite irreducible orthogonal reflection group. Let A be the corresponding Coxeter arrangement. Let S be the algebra of polynomial functions on V. For H ∈ A choose αH ∈ V ∗ such that H = ker(αH). For each nonnegative integer m, define the derivation module D(A) = {θ ∈ DerS | θ(αH) ∈ SαmH}. The module is known to be a free S-module of...

A logical error in the usual derivation of the energy conservation law is analyzed, and a way to avoid the error is presented. In earlier papers [1], [2] we identified a logical error adopted by repetition in textbooks on classical electrodynamics when the laws of conservation of energy and momentum are derived for a system consisting of electromagnetic fields and charged particles. In this pap...

BACKGROUND The recently proposed nomogram of Barcelona Clinic Liver Cancer (BCLC) lacks predictive accuracy for patients with stage D hepatocellular carcinoma (HCC). Tumor burden is crucial in prognostic prediction but is not included in the criteria of stage D HCC. This study aims to develop a nomogram with tumor burden as the core element for BCLC stage D patients. METHODS A total of 386 pa...

Let R be an associative prime ring, U a Lie ideal such that u2 ∈ U for all u ∈ U . An additive function F : R→ R is called a generalized derivation if there exists a derivation d : R→ R such that F(xy)= F(x)y + xd(y) holds for all x, y ∈ R. In this paper, we prove that d = 0 or U ⊆ Z(R) if any one of the following conditions holds: (1) d(x) ◦F(y)= 0, (2) [d(x),F(y) = 0], (3) either d(x) ◦ F(y) ...

We characterize the derivation d : A→ ΩDer(A) by a universal property introducing a new class of bimodules. Abridged English Version In this Note, A denotes an associative algebra over K = R or C with a unit 1l. In the Note [5], a graded differential algebra ΩDer(A) with Ω 0 Der(A) = A was introduced with the notation ΩD(A). It is one of the aims of this Note to show that, by introducing a new ...

In [17] Lee and Shiue showed that if R is a non-commutative prime ring, I a nonzero left ideal of R and d is a derivation of R such that [d(x)x, x]k = 0 for all x ∈ I, where k,m, n, r are fixed positive integers, then d = 0 unless R ∼= M2(GF (2)). Later in [1] Argaç and Demir proved the following result: Let R be a non-commutative prime ring, I a nonzero left ideal of R and k,m, n, r fixed posi...

We prove that every J-holomorphic variational vector field can be realized as derivation $$\frac{d}{dt}_{|t=0}f_t$$ where $$(f_t)$$ is a one parametric family of discs. Furthermore, we discuss properness an extremal disc in bounded pseudoconvex domain.

We study algebraic varieties X over a universal iterative differential field (K, ∂) (typically of positive characteristic), together with an extension of ∂ to an iterative derivation D of the structure sheaf of X. Our work is motivated by the conjecture that if X is projective then the pair (X,D) is isotrivial, namely isomorphic over K to a pair (Y, D0) where Y is defined over the constants C o...