Nearly General Septic Functional Equation

If a mapping can be expressed by sum of a septic mapping, a sextic mapping, a quintic mapping, a quartic mapping, a cubic mapping, a quadratic mapping, an additive mapping, and a constant mapping, we say that it is a general septic mapping. A functional equation is said to be a general septic functional equation provided that each solution of that equation is a general septic mapping. In fact, there are a lot of ways to show the stability of functional equations, but by using the method of G a ˘ vruta, we examine the stability of general septic functional equation ∑ i = 0 8 C 8 i − 1 8 − i f x + i − 4 y = 0 which considered. The method of G a ˘ vruta as just mentioned was given in the reference Gavruta (1994).

Ulam Stabilities and Instabilities of Euler–Lagrange-Rassias Quadratic Functional Equation in Non-Archimedean IFN spaces

In this paper, we use direct and fixed-point techniques to examine the generalised Ulam–Hyers stability results of the general Euler–Lagrange quadratic mapping in non-Archimedean IFN spaces (briefly, non-Archimedean Intuitionistic Fuzzy Normed spaces) over a field.

Approximation of the multi-m-Jensen-quadratic mappings and a fixed point approach

In this article, by using a new form of multi-quadratic mapping, we define multi-m-Jensen-quadratic mappings and then unify the system of functional equations defining a multi-m-Jensen-quadratic mapping to a single equation. Using a fixed point theorem, we study the generalized Hyers-Ulam stability of multi-quadratic and multi-m-Jensen-quadratic functional equations. As a consequence, we show that every multi-m-Jensen-quadratic functional equation (under some conditions) can be hyperstable.

Type 2 Vague Events and Their Applications

[1] discovered a mapping formula for Type 1 Vague events, and presented an alternative problem as an example of its application. Since it is well known that the alternative problem results in sequential Bayesian inference, the subsequent research flow is to make the mapping formula multidimensional, to derive the Markov (decision) process by introducing the concept of time, and so on. Furthermore, the stochastic differential equation from which it is derived was formulated. [2] This paper refers to Type 2 Vague events based on the secondary mapping formula. This quadratic mapping formula gives a certain rotation to a non-mapping function by transforming it with a relationship between the two mapping functions. Furthermore, here we refer to the derivation of the Type 2 Vague Markov process and the initial and stop conditions for its rotation.

Type 2 Vague Events and Their Applications

[1] discovered a mapping formula for Type 1 Vague events, and presented an alternative problem as an example of its application. Since it is well known that the alternative problem results in sequential Bayesian inference, the subsequent research flow is to make the mapping formula multidimensional, to derive the Markov (decision) process by introducing the concept of time, and so on. Furthermore, the stochastic differential equation from which it is derived was formulated. [2] This paper refers to Type 2 Vague events based on the secondary mapping formula. This quadratic mapping formula gives a certain rotation to a non-mapping function by transforming it with a relationship between the two mapping functions. Furthermore, here we refer to the derivation of the Type 2 Vague Markov process and the initial and stop conditions for its rotation.

A New Approach to Hyers-Ulam Stability of r -Variable Quadratic Functional Equations

In this paper, we investigate the general solution of a new quadratic functional equation of the form ∑ 1 ≤ i < j < k ≤ r ϕ l i + l j + l k = r − 2 ∑ i = 1 , i ≠ j r ϕ l i + l j + − r 2 + 3 r − 2 / 2 ∑ i = 1 r ϕ l i . We prove that a function admits, in appropriate conditions, a unique quadratic mapping satisfying the corresponding functional equation. Finally, we discuss the Ulam stability of that functional equation by using the directed method and fixed-point method, respectively.

GENERALIZED STABILITY OF AN ADDITIVE-QUADRATIC FUNCTIONAL EQUATION IN VARIOUS SPACES

ABSTRACT. In this paper, using the direct and fixed point methods, we have established the generalized Hyers-Ulam stability of the following additive-quadratic functional equation in non-Archimedean and intuitionistic random normed spaces. AMS 2010 Subject Classification: 39B82, 39B52, 46S40. Keywords. generalized Hyers-Ulam stability; additive mapping; quadratic mapping; non-Archimedean random normed spaces; intuitionistic random normed spaces; fixed point.

Further Observations on SIMON and SPECK Block Cipher Families

SIMON and SPECK families of block ciphers are well-known lightweight ciphers designed by the NSA. In this note, based on the previous investigations on SIMON, a closed formula for the squared correlations and differential probabilities of the mapping ϕ ( x ) = x ⊙ S 1 ( x ) on F 2 n is given. From the aspects of linear and differential cryptanalysis, this mapping is equivalent to the core quadratic mapping of SIMON via rearrangement of coordinates and EA -equivalence. Based on the proposed explicit formula, a full description of DDT and LAT of ϕ is provided. In the case of SPECK, as the only nonlinear operation in this family of ciphers is addition mod 2 n , after reformulating the formula for linear and differential probabilities of addition mod 2 n , straightforward algorithms for finding the output masks with maximum squared correlation, given the input masks, as well as the output differences with maximum differential probability, given the input differences, are presented. By the aid of the tools given in this paper, the process of the search for linear and differential characteristics of SIMON and SPECK families of block ciphers could be sped up, and the complexity of linear and differential attacks against these ciphers could be reduced.