Mathematics

We derive closed formulas for the number of k -coloured partitions and the number of plane partitions of n in terms of the Bell polynomials.

A short survey on the properties of four graphs constructed in {0,1 } n Boolean space is presented. Flexible activation function of an artificial neuron in a sparse distributed memory model is defined on the basis of the Ugly duckling theorem. Cotan Laplacian on 2-face triangulation of n -cube has degenerate spectrum of eigenvalues corresponding to the Hamming distance distribution of {0,1 } n space. Degenerate spectrum of eigenvalues of the cotan Laplacian defined on the graph comprising 2 n 2-face triangulated n -cubes sharing common origin includes all integers from 0 to 3 n , without the eigenvalue of 3 n -1 (multiplicities of the same eigenvalues form A038717 OEIS sequence), while the multiplicities of the same eigenvalues [−n 2 – √ ,n 2 – √ ] of the adjacency matrix of 2 n -cube form trinomial triangle. The distance matrix of this graph, providing further OEIS sequences, as well as its relation with Buckminster Fuller vector equilibrium is also discussed.

Let p be an odd prime, and consider the map H_p which sends an integer x to either x/2 or (px+1)/2 depending on whether x is even or odd. The values at x=0 of arbitrary composition sequences of the maps x/2 and (px+1)/2 can be parameterized over the 2-adic integers (Z2) leading to a continuous function from Z2 to Zp which the author calls the "characteristic function" (or "numen") of H_p. Lipschitz-type estimates are given for the characteristic function when p-1 is a power of 2 and 2 is a primitive root mod p, and it is shown that the set of periodic points of H_p is equal to the set of (rational) integer values attained by the characteristic function over Z2. Additionally, although the pre-image of R under the characteristic function has zero Haar measure in the Z2, by pre-composing the characteristic function with an appropriately selected self-embedding of Z2, one can perform Fourier analysis of the aforementioned composite. Using this approach, explicit upper bounds are computed for the absolute value of a periodic point of H_p whose parity vector contains at least ceil(ln(p)/ln2)-1 zeroes between any two consecutive ones

In the literature, the left-side of Hermite--Hadamard's inequality is called a midpoint type inequality. In this article, we obtain new integral inequalities of midpoint type for Riemann--Liouville fractional integrals of convex functions with respect to increasing functions. The resulting inequalities generalize some recent classical integral inequalities and Riemann--Liouville fractional integral inequalities established in earlier works. Finally, applications of our work are demonstrated via the known special functions of real numbers.

In this paper, we propose the theory of fuzzy limit of fuzzy function depending on the Altai principle and using the representation theorem (resolution principle) to run the fuzzy arithmetic

In this paper, a new concept, the fuzzy rate of an operator in linear spaces is proposed for the very first time. Some properties and basic principles of it are studied. Fuzzy rate of an operator B which is specific in a plane is discussed. As its application, a new fixed point existence theorem is proved.

The concept of gyrogroups is a generalization of groups which do not explicitly have associativity. In this paper, the notion of fuzzy gyronorms on gyrogroups is introduced. The relations of fuzzy metrics (in the sense of George and Veeramani), fuzzy gyronorms and gyronorms on gyrogroups are studied. Also, the fuzzy metric structures on fuzzy normed gyrogroups are discussed. In the last, the fuzzy metric completion of a gyrogroup with an invariant metric are studied. We mainly show that let d be an invariant metric on a gyrogroup G and ( G ˆ , d ˆ ) is the metric completion of the metric space (G,d) ; then for any continuous t -norm ∗ , the standard fuzzy metric space ( G ˆ , M d ˆ ,∗) of ( G ˆ , d ˆ ) is the (up to isometry) unique fuzzy metric completion of the standard fuzzy metric space (G, M d ,∗) of (G,d) ; furthermore, ( G ˆ , M d ˆ ,∗) is a fuzzy metric gyrogroup containing (G, M d ,∗) as a dense fuzzy metric subgyrogroup and M d ˆ is invariant on G ˆ . Applying this result, we obtain that every gyrogroup G with an invariant metric d admits an (up to isometric) unique complete metric space ( G ˆ , d ˆ ) of (G,d) such that G ˆ with the topology introduced by d ˆ is a topology gyrogroup containing G as a dense subgyrogroup and d ˆ is invariant on G ˆ .

In the most accessible terms this paper presents a convex-geometric approach to the study of fuzzy vectors. Motivated by several key results from the theory of convex bodies, we establish a representation theorem of fuzzy vectors through support functions, in which a necessary and sufficient condition for a function to be the support function of a fuzzy vector is provided. As applications, symmetric and skew fuzzy vectors are postulated, based on which a Mareš core of each fuzzy vector is constructed through convex bodies and support functions, and it is shown that every fuzzy vector over the n-dimensional Euclidean space has a unique Mareš core if, and only if, the dimension n=1.

This paper presents a novel direct elementary proof for Fermat's Last Theorem. We use algebra, modular math, and binomial series to develop inherent mathematical relationships hidden within Fermat's Last Theorem. With these derived relationships, we are able to develop general pattern applicable for all positive integers of n. Finally, we are able to confirm and complete the direct proof for Fermat's Diophantine equation for all n.

In this paper we present a direct formula for the solution of the general second order linear ordinary differential equation as our main result such that the parameters required for the formula are determined using another differential equation, which happens to be a Riccati Equation. We also derive other results based on the main result which include special cases for the concerned differential equation with variable coefficients, formula for solution of concerned differential equation with constant coefficients and formula for the solution of the concerned differential equation with one complimentary solution known.