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1.
《Quaestiones Mathematicae》2013,36(5):651-663
AbstractLet G be an Abelian group with a metric d and E ba a normed space. For any f : G → E we define the generalized quadratic di?erence of the function f by the formulaQk f (x, y) := f (x + ky) + f (x ? ky) ? f (x + y) ? f (x ? y) ? 2(k2 ? 1)f (y)for all x, y ∈ G and for any integer k with k ≠ 1, ?1. In this paper, we achieve the general solution of equation Qk f (x, y) = 0, after it, we show that if Qk f is Lipschitz, then there exists a quadratic function K : G → E such that f ? K is Lipschitz with the same constant. Moreover, some results concerning the stability of the generalized quadratic functional equation in the Lipschitz norms are presented. In the particular case, if k = 0 we obtain the main result that is in [7]. 相似文献
2.
A comparative study of the functional equationsf(x+y)f(x–y)=f
2(x)–f
2(y),f(y){f(x+y)+f(x–y)}=f(x)f(2y) andf(x+y)+f(x–y)=2f(x){1–2f
2(y/2)} which characterise the sine function has been carried out. The zeros of the functionf satisfying any one of the above equations play a vital role in the investigations. The relation of the equationf(x+y)+f(x–y)=2f(x){1–2f
2(y/2)} with D'Alembert's equation,f(x+y)+f(x–y)=2f(x)f(y) and the sine-cosine equationg(x–y)=g(x)g(y) +f(x)f(y) has also been investigated. 相似文献
3.
H. Stetkær 《Aequationes Mathematicae》1997,54(1-2):144-172
Summary We produce complete solution formulas of selected functional equations of the formf(x +y) ±f(x + σ (ν)) = Σ
I
2
=1
g
l
(x)h
l
(y),x, y∈G, where the functionsf,g
1,h
1 to be determined are complex valued functions on an abelian groupG and where σ:G→G is an involution ofG. The special case of σ=−I encompasses classical functional equations like d’Alembert’s, Wilson’s first generalization of it, Jensen’s equation and
the quadratic equation. We solve these equations, the equation for symmetric second differences in product form and similar
functional equations for a general involution σ. 相似文献
4.
Summary Letf, G1 × G2 C, where G
i
(i = 1, 2) denote arbitrary groups and C denotes the set of complex numbers. The general solutions of the following functional equationsf(x
1
y
1
,x
2
y
2
) +f(x
1
y
1
,x
2
y
2
-1
) +f(x
1
y
1
-1
,x
2
y
2
) +f(x
1
y
1
-1
,x
2
y
2
-1
) =f(x
1
,x
2
)F(y
1
,y
2
) +F(x
1
,x
2
)f(y
1
,y
2
) (1) andf(x
1
y
1
,x
2
y
2
) +f(x
1
y
1
,x
2
y
2
-1
) +f(x
1
y
1
-1
,x
2
y
2
) +f(x
1
y
1
-1
,x
2
y
2
-1
) =f(x
1
,x
2
)f(y
1
,y
2
) +F(x
1
,x
2
)F(y
1
,y
2
) (2) are determined assuming thatf satisfies the conditionf(x
1y1z1, x2) = f(x1z1y1, x2), f(x1, x2y2z2) = f(x1, x2z2y2) (C) for allx
i, yi, xi Gi (i = 1, 2). The functional equations (1) and (2) are generalizations of the well known rectangular type functional equationf(x
1 + y1, x2 + y2) + f(x1 + y1, x2 – y2) + f(x1 – y1, x2 + y2) + f(x1 – y1, x2 – y2) = 4f(x1, x2) studied by J. Aczel, H. Haruki, M. A. McKiernan and G. N. Sakovic in 1968. 相似文献
5.
Summary.
Let
be a field of real or complex numbers and
denote the set of nonzero elements of
.
Let
be an abelian group. In this paper, we solve the functional equation
f
1
(x +
y) +
f
2
(x -
y) =
f
3
(x) +
f
4
(y) +
g(xy)
by modifying the domain of the unknown functions
f
3,
f
4, and
g from
to
and using a method different from [3]. Using this result,
we determine all functions
f
defined on
and taking values on
such that the difference
f(x + y) + f
(x -
y) - 2
f(x) - 2
f(y)
depends only on the product
xy for all
x and
y in
相似文献
6.
7.
Timothy J. Ford 《代数通讯》2013,41(9):3277-3298
We study algebra classes and divisor classes on a normal affine surface of the form z 2 = f(x, y). The affine coordinate ring is T = k[x, y, z]/(z 2 ? f), and if R = k[x, y][f ?1] and S = R[z]/(z 2 ? f), then S is a quadratic Galois extension of R. If the Galois group is G, we show that the natural map H1(G, Cl(T)) → H1(G, Pic(S)) factors through the relative Brauer group B(S/R) and that all of the maps are onto. Sufficient conditions are given for H1(G, Cl(T)) to be isomorphic to B(S/R). The groups and maps are computed for several examples. 相似文献
8.
Two natural extensions of Jensen’s functional equation on the real line are the equations f(xy) + f(xy
−1) = 2f(x) and f(xy) + f(y
−1
x) = 2f(x), where f is a map from a multiplicative group G into an abelian additive group H. In a series of papers (see Ng in Aequationes Math 39:85–99, 1990; Ng in Aequationes Math 58:311–320, 1999; Ng in Aequationes Math 62:143–159, 2001), Ng solved these functional equations for the case where G is a free group and the linear group
GLn(R), R=\mathbbZ,\mathbbR{{GL_n(R), R=\mathbb{Z},\mathbb{R}}} , is a quadratically closed field or a finite field. He also mentioned, without a detailed proof, in the above papers and
in (see Ng in Aequationes Math 70:131–153, 2005) that when G is the symmetric group S
n
, the group of all solutions of these functional equations coincides with the group of all homomorphisms from (S
n
, ·) to (H, + ). The aim of this paper is to give an elementary and direct proof of this fact. 相似文献
9.
Gleason [A.M. Gleason, The definition of a quadratic form, Amer. Math. Monthly 73 (1966) 1049-1066] determined all functionals Q on K-vector spaces satisfying the parallelogram law Q(x+y)+Q(x-y)=2Q(x)+2Q(y) and the homogeneity Q(λx)=λ2Q(x). Associated with Q is a unique symmetric bi-additive form S such that Q(x)=S(x,x) and 4S(x,y)=Q(x+y)-Q(x-y). Homogeneity of Q corresponds to that of S: S(λx,λy)=λ2S(x,y). The associated S is not necessarily bi-linear.Let V be a vector space over a field K, char(K)≠2,3. A tri-additive form T on V is a map of V3 into K that is additive in each of its three variables. T is homogeneous of degree 3 if T(λx,λy,λz)=λ3T(x,y,z) for all .We determine the structure of tri-additive forms that are homogeneous of degree 3. One of the keys to this investigation is to find the general solution of the functional equation
F(t)+t3G(1/t)=0, 相似文献
10.
Pierre Maréchal 《Mathematical Programming》2001,89(3):505-516
It is well known that a function f of the real variable x is convex if and only if (x,y)→yf(y
-1
x),y>0 is convex. This is used to derive a recursive proof of the convexity of the multiplicative potential function. In this
paper, we obtain a conjugacy formula which gives rise, as a corollary, to a new rule for generating new convex functions from
old ones. In particular, it allows to extend the aforementioned property to functions of the form (x,y)→g(y)f(g(y)-1
x) and provides a new tool for the study of the multiplicative potential and penalty functions.
Received: June 3, 1999 / Accepted: September 29, 2000?Published online January 17, 2001 相似文献
11.
Nicole Brillouët-Belluot 《Aequationes Mathematicae》1991,42(1):239-270
Summary LetE be a real Hausdorff topological vector space. We consider the following binary law * on ·E:(, ) * (, ) = (,
k
+ ) for(, ), (, ) × E where is a nonnegative real number,k andl are integers.In order to find all subgroupoids of ( ·E, *) which depend faithfully on a set of parameters, we have to solve the following functional equation:f(f(y)
k
x +f(x)
l
y) =f(x)f(y) (x, y E). (1)In this paper, all solutionsf: of (1) which are in the Baire class I and have the Darboux property are obtained. We obtain also all continuous solutionsf: E of (1). The subgroupoids of (* ·E, *) which dapend faithfully and continuously on a set of parameters are then determined in different cases. We also deduce from this that the only subsemigroup ofL
n
1
of the form {(F(x
2,x
3, ,x
n
),x
2,x
3, ,x
n
); (x
2, ,x
n
)
n – 1
}, where the mappingF: n – 1 * has some regularity property, is {1} ×
n – 1
.We may noitice that the Gob-Schinzel functional equation is a particular case of equation (1)(k = 0, l = 1, = 1). So we can say that (1) is of Gob—Schinzel type. More generally, whenE is a real algebra, we shall say that a functional equation is of Gob—Schinzel type if it is of the form:f(f(y)
k
x +f(x)
l
y) =F(x,y,f(x),f(y),f(xy)) wherek andl are integers andF is a given function in five variables. In this category of functional equations, we study here the equation:f(f(y)
k
x +f(x)
l
y) =f(xy) (x, y f: ). (4)This paper extends the results obtained by N. Brillouët and J. Dhombres in [3] and completes some results obtained by P. Urban in his Ph.D. thesis [11] (this work has not yet been published).Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth 相似文献
12.
Elena Prestini 《Mathematische Zeitschrift》2010,265(2):401-415
The operators S
p
f (x, y), for the sum of which we prove an L
2-estimate, act as a kind of Fourier coefficients on one variable and a kind of truncated Hilbert transforms with a phase N(x, y) on the other variable. This result is an extension to two-dimensions of an argument of almost orthogonality in Fefferman’s
proof of a.e. convergence of Fourier series, under the basic assumption N(x, y) “mainly” a function of y and the additional assumption N(x, y) non-decreasing in x, for every y fixed. 相似文献
13.
We prove the sum of squared logarithms inequality (SSLI) which states that for nonnegative vectors x, y ∈ ℝn whose elementary symmetric polynomials satisfy ek(x) ≤ ek(y) (for 1 ≤ k < n) and en(x) = en(y) , the inequality ∑i(log xi)2 ≤ ∑i(log yi)2 holds. Our proof of this inequality follows by a suitable extension to the complex plane. In particular, we show that the function f : M ⊆ ℂn → ℝ with f(z) = ∑i(log zi)2 has nonnegative partial derivatives with respect to the elementary symmetric polynomials of z. We conclude by providing applications and wider connections of the SSLI. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
14.
Endomorphisms of superelliptic jacobians 总被引:1,自引:0,他引:1
Yuri G. Zarhin 《Mathematische Zeitschrift》2009,261(3):709-707
Let K be a field of characteristic zero, n ≥ 5 an integer, f(x) an irreducible polynomial over K of degree n, whose Galois group contains a doubly transitive simple non-abelian group. Let p be an odd prime, the ring of integers in the pth cyclotomic field, C
f, p
: y
p
= f(x) the corresponding superelliptic curve and J(C
f, p
) its jacobian. Assuming that either n = p + 1 or p does not divide n(n − 1), we prove that the ring of all endomorphisms of J(C
f, p
) coincides with . The same is true if n = 4, the Galois group of f(x) is the full symmetric group S
4 and K contains a primitive pth root of unity.
An erratum to this article can be found at 相似文献
15.
Gian Luigi Forti 《Aequationes Mathematicae》1982,24(1):195-206
We consider the following problem: Let (G, +) be an abelian group,B a complex Banach space,a, bB,b0,M a positive integer; find all functionsf:G B such that for every (x, y) G ×G the Cauchy differencef(x+y)–f(x)–f(y) belongs to the set {a, a+b, a+2b, ...,a+Mb}. We prove that all solutions of the above problem can be obtained by means of the injective homomorphisms fromG/H intoR/Z, whereH is a suitable proper subgroup ofG. 相似文献
16.
17.
Ian N. Baker 《Aequationes Mathematicae》1997,54(1-2):87-101
Summary The paper determines all cases when a meromorphic functionF can be expressed both asf ⊗p andf ⊗q with the same meromorphicf and different polynomialsp andq. In all cases there are constantsk, β, a positive integerm, a root λ of unity of orderS and a polynomialr such thatp=(Lr)
m+k,q=r
m+k, whereLz=λz+β. We have eitherm=1,S arbitrary orm=2,S=2, which can occur even ifF andf are entire, or, in the remaining casesS=2, 3, 4 or 6,m dividesS andf(k+t
m) is a doubly-periodic function. 相似文献
18.
E. Preissmann 《Aequationes Mathematicae》1987,32(1):195-212
We solve independently the equations 1/θ(x)θ(y)=ψ(x)−ψ(y)+φ(x−y)/θ(x−y) and 1/θ(x)θ(y)=σ(x)−σ(y)/θ(x−y)+τ(x)τ(y), τ(0)=0. In both cases we find θ2=aθ4+bθ2+c. We deduce estimates for the spectral radius of a matrix of type(1/θ(x
r
−x
s
)) (the accent meaning that the coefficients of the main diagonal are zero) and we study the case where thex
r
are equidistant.
Dédié to à Monsieur le Professeur Otto Haupt à l'occasion de son cententiare avec les meilleurs voeux 相似文献
19.
Shuhuang Xiang 《Numerische Mathematik》2007,105(4):633-658
Based on the transformation y = g(x), some new efficient Filon-type methods for integration of highly oscillatory function òabf(x) eiwg(x) dx\int_a^bf(x)\,{\rm e}^{{\rm i}\omega g(x)}\,{\rm d}x with an irregular oscillator are presented. One is a moment-free Filon-type method for the case that g(x) has no stationary points in [a,b]. The others are based on the Filon-type method or the asymptotic method together with Filon-type method for the case that
g(x) has stationary points. The effectiveness and accuracy are tested by numerical examples. 相似文献
20.
We investigate the relationships between smooth and strongly smooth points of the unit ball of an order continuous symmetric
function space E, and of the unit ball of the space of τ-measurable operators E(M,t){E(\mathcal{M},\tau)} associated to a semifinite von Neumann algebra (M, t){(\mathcal{M}, \tau)}. We prove that x is a smooth point of the unit ball in E(M, t){E(\mathcal{M}, \tau)} if and only if the decreasing rearrangement μ(x) of the operator x is a smooth point of the unit ball in E, and either μ(∞; f) = 0, for the function f ? SE×{f\in S_{E^{\times}}} supporting μ(x), or s(x
*) = 1. Under the assumption that the trace τ on M{\mathcal{M}} is σ-finite, we show that x is strongly smooth point of the unit ball in E(M, t){E(\mathcal{M}, \tau)} if and only if its decreasing rearrangement μ(x) is a strongly smooth point of the unit ball in E. Consequently, for a symmetric function space E, we obtain corresponding relations between smoothness or strong smoothness of the function f and its decreasing rearrangement μ(f). Finally, under suitable assumptions, we state results relating the global properties such as smoothness and Fréchet smoothness
of the spaces E and E(M,t){E(\mathcal{M},\tau)}. 相似文献