Some finite generalizations of Euler’s pentagonal number theorem

Euler’s pentagonal number theorem was a spectacular achievement at the time of its discovery, and is still considered to be a beautiful result in number theory and combinatorics. In this paper, we obtain three new finite generalizations of Euler’s pentagonal number theorem.     

Hyers–Ulam stability of Euler’s equation

We prove that Euler’s equation $x_1\frac{?u}{?x_1}+x_2\frac{?u}{?x_2}+?+x_n\frac{?u}{?x_n}=\alpha u$, characterising homogeneous functions, is stable in Hyers–Ulam sense if and only if $\alpha \in \mathbb{R}?\left\{0\right\}$.     

Hyers–Ulam Stability of Euler’s Differential Equation

We obtain a result on generalized Hyers–Ulam stability for Euler’s differential equation in Banach spaces. Our result extends and improves some recent results of Mortici, Jung and Rassias concerning the stability of Euler’s equation on a bounded domain.     

Jackson’s theorem in Q_p spaces

A Jackson type inequality in Q p spaces is established, i.e., for any f (z) = Σ∞ j=0 ajzj ∈ Qp , 0≤p ∞, a 1, and k-1 ∈ N,where ω(1/k, f, Q p ) is the modulus of continuity in Q p spaces and C(a) is an absolute constant depending only on the parameter a.     

Rees–Shishikura’s theorem for geometrically finite rational maps

In this paper, we generalize Rees–Shishikura’s theorem to the class of geometrically finite rational maps.     

Periods and Young’s diagram of Fermat–Euler’s geometrical progressions of residues

The Gibbs phenomenon is described for the Fourier series of a function at its jump, the function being defined along the finite circle ℤ/pℤ.     

Euler’s Exponential Formula for Semigroups

The aim of this paper is to show that Eulers exponential formula $\lim_{n\rightarrow\infty}\linebreak[4] (I-tA/n)^{-n}x = e^{tA}x$, well known for $C_0$ semigroups in a Banach space $X\ni x$, can be used for semigroups not of class $C_0$, the sense of the convergence being related to the regularity of the semigroup for $t>0$. Although the strong convergence does not hold in general for not strongly continuous semigroups, an integrated version is stated for once integrated semigroups. Furthermore by replacing the initial topology on $X$ by some (coarser) locally convex topology $\tau$, the strong $\tau$-convergence takes place provided the semigroup is strongly $\tau$-continuous; in particular this applies to the class of bi-continuous semigroups. More generally if a $k$-times integrated semigroup $S(t)$ in a Banach space $X$ is strongly $k$-times $\tau$-differentiable, then Eulers formula holds in this topology with limit $S^{(k)}(t)$. On the other hand, for bounded holomorphic semigroups not necessarily of class $C_0$, Eulers formula is shown to hold in operator norm, with the error bound estimate ${\cal O}(\ln n/n)$, uniformly in $t>0$. All these results also concern degenerate semigroups.     

Complex Euler’s groups and values of Euler’s function at complex integer Gauss points

The complex Euler group is defined associating to an integer complex number z the multiplicative group of the complex integers residues modulo z, relatively prime to z. This group is calculated for z=(3+0i) ⁿ : it is isomorphic to the product of three cyclic group or orders (8, 3 ⁿ⁻¹ and 3 ⁿ⁻¹).     

A generalization of Euler’s constant

The purpose of this paper is to evaluate the limit γ(a) of the sequence , where a ∈ (0, + ∞ ).      

Jackson’s integral of the Hurwitz zeta function

We give a Jacksonq-integral analogue of Euler’s logarithmic sine integral established in 1769 from several points of view, specifically from the one relating to the Hurwitz zeta function. Partially supported by Grant-in-Aid for Exploratory Research No. 15654003. Partially supported by Grant-in-Aid for Scientific Research (B) No. 15340003. Partially supported by Grant-in-Aid for Scientific Research (B) No. 15340012.     

Generalizations of Arnold’s version of Euler’s theorem for matrices

A recent result, conjectured by Arnold and proved by Zarelua, states that for a prime number p, a positive integer k, and a square matrix A with integral entries one has ${\textrm tr}(A^{p^k}) \equiv {\textrm tr}(A^{p^{k-1}}) ({\textrm mod}{p^k})${\textrm tr}(A^{p^k}) \equiv {\textrm tr}(A^{p^{k-1}}) ({\textrm mod}{p^k}). We give a short proof of a more general result, which states that if the characteristic polynomials of two integral matrices A,  B are congruent modulo p ^k then the characteristic polynomials of A ^p and B ^p are congruent modulo p ^k+1, and then we show that Arnold’s conjecture follows from it easily. Using this result, we prove the following generalization of Euler’s theorem for any 2 × 2 integral matrix A: the characteristic polynomials of A ^Φ(n) and A ^Φ(n)-ϕ(n) are congruent modulo n. Here ϕ is the Euler function, ?_i=1^l p_i^a_i\prod_{i=1}^{l} p_i^{\alpha_i} is a prime factorization of n and $\Phi(n)=(\phi(n)+\prod_{i=1}^{l} p_i^{\alpha_i-1}(p_i+1))/2$\Phi(n)=(\phi(n)+\prod_{i=1}^{l} p_i^{\alpha_i-1}(p_i+1))/2.     

Vibration analysis of Euler–Bernoulli nanobeams by using finite element method

In the present study, an efficient finite element model for vibration analysis of a nonlocal Euler–Bernoulli beam has been reported. Nonlocal constitutive equation of Eringen is proposed. Equations of motion for a nonlocal Euler–Bernoulli are derived based on varitional statement. The finite element method is employed to discretize the model and obtain a numerical approximation of the motion equation. The model has been verified with the previously published works and found a good agreement with them. Vibration characteristics, such as fundamental frequencies, are illustrated in graphical and tabulated form. Numerical results are presented to figure out the effects of nonlocal parameter, slenderness ratios, rotator inertia, and boundary conditions on the dynamic characteristics of the beam. The above mention effects play very important role on the dynamic behavior of nanobeams.