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1.
We obtain explicit upper bounds for the number of irreducible factors for a class of polynomials of the form f ○ g, where f,g are polynomials with integer coefficients, in terms of the prime factorization of the leading coefficients of f and g, the degrees of f and g, and the size of coefficients of f. In particular, some irreducibility results are given for this class of compositions of polynomials. 相似文献
2.
We determine the general solution of the functional equation f(x + ky) + f(x-ky) = g(x + y) + g(x-y) + h(x) + h(y) for fixed integers with k ≠ 0; ±1 without assuming any regularity conditions for the unknown functions f, g, h, and0020[(h)\tilde] \tilde{h} . The method used for solving these functional equations is elementary but it exploits an important result due to Hosszú.
The solution of this functional equation can also be obtained in groups of certain type by using two important results due
to Székelyhidi. 相似文献
3.
4.
We study additive representability of orders on multisets (of size k drawn from a set of size n) which satisfy the condition of independence of equal submultisets (IES) introduced by Sertel and Slinko (Ranking committees,
words or multisets. Nota di Laboro 50.2002. Center of Operation Research and Economics. The Fundazione Eni Enrico Mattei,
Milan, 2002, Econ. Theory 30(2):265–287, 2007). Here we take a geometric view of those orders, and relate them to certain combinatorial objects which we call discrete
cones. Following Fishburn (J. Math. Psychol., 40:64–77, 1996) and Conder and Slinko (J. Math. Psychol., 48(6):425–431, 2004), we define functions f(n,k) and g(n,k) which measure the maximal possible deviation of an arbitrary order satisfying the IES and an arbitrary almost representable
order satisfying the IES, respectively, from a representable order. We prove that g(n,k) = n − 1 whenever n ≥ 3 and (n, k) ≠ (5, 2). In the exceptional case, g(5,2) = 3. We also prove that g(n,k) ≤ f(n,k) ≤ n and establish that for small n and k the functions g(n,k) and f(n,k) coincide.
相似文献
5.
E. G. Goluzina 《Journal of Mathematical Sciences》2009,157(4):560-567
The paper continues the studies of the well-known class T of typically real functions f(z) in the disk U = {z:|z| < 1}. The region of values of the system {f(z
0), f(z
0), f(r
1), f(r
2),…, f(r
n
)} in the class T is investigated. Here, z
0 ∈ U, Im z
0 ≠ 0, 0 < r
j
< 1 for j = 1,…, n, n ≥ 2. As a corollary, the region of values of f′(z
0) in the class of functions f ∈ T with fixed values f(z
0) and f(r
j
) (j = 1,…, n) is determined. The proof is based on the criterion of solvability of the power problem of moments. Bibliography: 10 titles.
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 357, 2008, pp. 33–45. 相似文献
6.
We provide irreducibility criteria for multivariate polynomials with coefficients in an arbitrary field that extend a classical result of Pólya for polynomials with integer coefficients. In particular, we provide irreducibility conditions for polynomials of the form f(X)(Y ? f 1(X))…(Y ? f n (X)) + g(X), with f, f 1, ?, f n , g univariate polynomials over an arbitrary field. 相似文献
7.
SAUGATA BANDYOPADHYAY 《Proceedings Mathematical Sciences》2011,121(3):339-348
Let Ω ⊂ ℝ
n
be a smooth, bounded domain. We study the existence and regularity of diffeomorphisms of Ω satisfying the volume form equation
f*(g)=f, \textin W, \phi^\ast(g)=f, \quad \text{in }\Omega, 相似文献
8.
The fractional Laplacian
(-\triangle)g/2(-\triangle)^{\gamma/2} commutes with the primary coordination transformations in the Euclidean space ℝ
d
: dilation, translation and rotation, and has tight link to splines, fractals and stable Levy processes. For 0 < γ < d, its inverse is the classical Riesz potential I
γ
which is dilation-invariant and translation-invariant. In this work, we investigate the functional properties (continuity,
decay and invertibility) of an extended class of differential operators that share those invariance properties. In particular,
we extend the definition of the classical Riesz potential I
γ
to any non-integer number γ larger than d and show that it is the unique left-inverse of the fractional Laplacian
(-\triangle)g/2(-\triangle)^{\gamma/2} which is dilation-invariant and translation-invariant. We observe that, for any 1 ≤ p ≤ ∞ and γ ≥ d(1 − 1/p), there exists a Schwartz function f such that I
γ
f is not p-integrable. We then introduce the new unique left-inverse I
γ, p
of the fractional Laplacian
(-\triangle)g/2(-\triangle)^{\gamma/2} with the property that I
γ, p
is dilation-invariant (but not translation-invariant) and that I
γ, p
f is p-integrable for any Schwartz function f. We finally apply that linear operator I
γ, p
with p = 1 to solve the stochastic partial differential equation
(-\triangle)g/2 F = w(-\triangle)^{\gamma/2} \Phi=w with white Poisson noise as its driving term w. 相似文献
9.
Sofiya Ostrovska 《Numerical Algorithms》2007,44(1):69-82
Since in the case q > 1, q-Bernstein polynomials are not positive linear operators on C[0,1], the study of their approximation properties is essentially more difficult than that for 0<q<1. Despite the intensive research conducted in the area lately, the problem of describing the class of functions in C[0,1] uniformly approximated by their q-Bernstein polynomials (q > 1) remains open. It is known that the approximation occurs for functions admit ting an analytic continuation into a disc
{z:|z| < R}, R > 1. For functions without such an assumption, no general results on approximation are available. In this paper, it is shown
that the function f(x) = ln (x + a), a > 0, is uniformly approximated by its q-Bernstein polynomials (q > 1) on the interval [0,1] if and only if a ≥ 1.
相似文献
10.
A new relation between morphisms in a category is introduced—roughly speaking (accurately in the categories Set and Top), f ∥ g iff morphisms w:dom(f)→dom(g) never map subobjects of fibres of f non-constantly to fibres of g. (In the algebraic setting replace fibre with kernel.) This relation and a slight weakening of it are used to define “connectedness”
versus “disconnectedness” for morphisms. This parallels and generalises the classical treatment of connectedness versus disconnectedness
for objects in a category (in terms of constant morphisms). The central items of study are pairs (F,G)({\mathcal F},{\mathcal G}) of classes of morphisms which are corresponding fixed points of the polarity induced by the ∥-relation. Properties of such
pairs are examined and in particular their relation to (pre)factorisation systems is analysed. The main theorems characterise:
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