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This is one in a series of papers studying measures of information in the so-called “mixed” theory of information (i.e. considering the events as well as their probabilities) on the “open” domain (i.e. without empty sets and zero probabilities). In this paper we find allβ-recursive, 3-semisymmetric inset entropies on the open domain. We do this by solving the fundamental equation of inset information of degreeβ (β∈ℝ) on the open domain.

The cocycle functional equation $$F(x,y) + F(xy, z) = F(x,yz) + F(y,z)$$has a long and rich history, with important roles in contexts from homological algebra to polyhedral algebra to information theory. This paper deals with the problem of finding explicit forms of symmetric cocycles on periodic semigroups. A semigroup Sis periodic if each of its elements has finite order, that is if the cyclic subsemigroup $\langle a \rangle = \{ a^k \mid k = 1,2,3,\ldots \}$generated by each element aof Sis finite. For several classes of abelian semigroups, including idempotent semigroups, cancellative semigroups, and certain types of topological semigroups, it is known that every symmetric cocycle Fis a coboundary (in other words, a Cauchy difference), that is $$F(x,y) = f(x) + f(y) - f(xy).$$We show that in fact every cocycle has this form on periodic semigroups.

Summary Motivated by results on the classical cocycle equation, we solved the more general equationF1(x + y, z) + F2(y + z, x) + F3(z + x, y) + F4(x, y) + F5(y, z) + F6(z, x) = 0 for six unknown functions mapping ordered pairs from an abelian group into a vector space over the rationals.

Summary In 1970, Halina Swiatak considered the functional equation $$\begin{gathered} f(xyz) + f(xyz^{ - 1} ) + f(xy^{ - 1} z) + f(x^{ - 1} yz) \hfill \\ = 4\{ f(x) + f(y) + f(z)\} + 2\{ g(x)g(y) + g(y)g(z) + g(x)g(z)\} , \hfill \\ \end{gathered} $$ wheref andg are functions defined on an arbitrary abelian group and taking values in an arbitrary commutative ring without proper zero divisors. She determined the general solution in the caseg(e) ? 0, wheree is the unit element of the group, and asked what the general solution is. In this paper, we determine the general solution of the above equation and some related functional equations. Further, we do not assume the group to be abelian but we assume thatf satisfies the Kannappan condition.

Summary This paper deals with the problem of characterizing higher order Cauchy differences of mappings on groups and semigroups. Symmetric, first order Cauchy differencesf(x + y)−f(x)−f(y) for mapsf between groups were characterized by Jessen, Karpf, and Thorup [8] through the use of first partial Cauchy differences. Our results are similar and extend their result to higher order differences. Our results also extend those of Heuvers [6] for mappings between vector spaces over the rationals.

Abstract: The general solution of the generalized multidimensional fundamental equation of information of multiplicative type on the open domain is obtained.

In the characterization of multidimensional sum form information measures the two functional equations % MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A!$$f(pq) + f(p(1 - q)) = f(p)\lbrace f(q) + f(1 - q)\rbrace \ \ \ p,q,\in I,$$% MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A!$$f(pq) + f(p(1 - q)) = f(p)\lbrace M(q) + M(1 - q)\rbrace \ \ \ p,q,\in I,$$arise. For the one-dimensional case, these equations were studied by Maksa [2] and Kannappan and Sahoo [1], respectively. This paper extends their results to the n-dimensional case.

In this paper, we determine all functions ƒ, defined on a field K (belonging to a certain class) and taking values in an abelian group, such that the quadratic difference ƒ(x + y) + ƒ(x - y) - 2ƒ(x) - 2ƒ(y) depends only on the product xy for all x, y ? K. Using this result, we find the general solution of the functional equation ƒ1(x + y) + ƒ2(x - y) = ƒ3(x) + ƒ4(y) + g(xy).

The forms of all semisymmetric, branching, multidimensional measures of inset information on open domains are determined. This is done for both the complete and the possibly incomplete partition cases. The key to these results is to find the general solution of the functional equation $$(*)\Delta \left( {\begin{array}{*{20}c} {E,F} \\ {p,q} \\ \end{array} } \right) + \Delta \left( {\begin{array}{*{20}c} {E \cup F,G} \\ {p + q,r} \\ \end{array} } \right) = \Delta \left( {\begin{array}{*{20}c} {E,G} \\ {p,r} \\ \end{array} } \right) + \Delta \left( {\begin{array}{*{20}c} {E \cup G,F} \\ {p + r,q} \\ \end{array} } \right)$$ for all(p, q, r) ∈ D3:={(p1, p2, p3)∣pi∈]0, 1[k(i = 1,2,3),p1+ p2+ p3∈]0, 1[k}and all (E, F, G) for which there exists anHwith (E, F, G, H) ∈ ℒ4:={(E1,⋯,E4)∣Ei∈ ℛ\{Ø},E1∩Ej= Ø for alli ≠ j}. Here, ℛ is a ring of subsets of a given universal set. Functional equation (*) is solved by use of the general solution of the generalized characteristic equation of branching information measures.

Summary Let B be a ring of sets, and letI be thek-dimensional open unit interval. The functional equation $$\begin{gathered} \varphi (E \cup F,G;p) + \mu (1 - p)\varphi \left( {E,F;\frac{q}{{1 - p}}} \right) \hfill \\ = \varphi (E \cup G,F;q) + \mu (1 - q)\varphi \left( {E,G;\frac{p}{{1 - q}}} \right), \hfill \\ \end{gathered} $$ for all disjoint triplesE, F, G of nonvoid sets in B and all pairsp, q inI withp + q ? I, is solved for ? and multiplicative µ. This problem was posed by Aczél in Aequationes Math.26 (1984), 255–260. Our solution to this problem leads to an axiomatic characterization of measures of inset informationIn(E1,?,En; 1,?, n) which have the representations $$\begin{gathered} f(E_1 ) + \sum\limits_2^n {g(E_i ) - f\left( {\bigcup\limits_1^n {E_i } } \right) + l(\bar p_1 ) + \sum\limits_1^n {\lambda (E_i ,\bar p_i )} } (for \mu = 1) \hfill \\ f(E_1 )\mu (\bar p_1 ) + \sum\limits_2^n {g(E_i )\mu (\bar p_i ) - f\left( {\bigcup\limits_1^n {E_i } } \right) + \sum\limits_1^n {l(\bar p_1 )\mu (\bar p_i )} } (for \mu additive) \hfill \\ f( \hfill \\ \end{gathered} $$ Herel is logarithmic, ? is additive in the events and logarithmic in the probabilities, andf andg are arbitrary functions.

We determine the general solution of the functional equation fxy,

We determine the general solution of the functional equation % MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A!$$\psi(x+y)=g(xy)+h(x-y),\ \ \ x,y\in\ {\rm \bf K}$$for ?,g,h: K? G, where Kis a field belonging to a certain class, and Gis an abelian group. This functional equation was one of the several treated by Abel in his 1823 manuscript. Recently, this equation was solved by Aczél and also independently by Lajko without any regularity assumption when K= G= R (reals). We consider also the conditional Cauchy equation % MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A!$$G(x^{2}- y^{2})=G(x^{2})- G(y^{2})$$for G: K?> G.For a large class of fields K, this forces Gto be a morphism from the additive group of Kinto G.For us, additional motivation for studying these functional equations came from a characterization problem related to generalized quadratic differences of functions.

We investigate a number of methods of measuring convergence rates of sequences. We clarify the relationships between these different measures of convergence rates. We present eight such measurement methods and demonstrate almost all relations among them (with two open problems remaining). We also present the notion of convergence-order profiles for sequences.

Our main goal is to determine the general solution of the functional equation

\begin{displaymath}\begin{array}{*{20}{c}} {{f_1}(xy) + {f_2}(x{y^{ - 1}}) = {f_... ...,} \\ {{f_i}(txy) = {f_i}(tyx)\qquad (i = 1,2)} \\ \end{array} \end{displaymath} where are complex-valued functions defined on a group. This equation contains, among others, an equation of H. Swiatak whose general solution was not known until now and an equation studied by K.S. Lau in connection with a characterization of Rao's quadratic entropies. Special cases of this equation also include the Pexider, quadratic, d'Alembert and Wilson equations.