Search

A novel chaotic map based on Euler number namely Euler Map (EMAP) is suggested for image encryption algorithms (IEAs). The chaotic behavior of EMAP is appreciated with bifurcation analysis and Lyapunov Exponent (LE). The LE is the best due to the ultimate and most stable characteristic in case compared with the literature. A two-stage IEA, permutation and diffusion, is used to demonstrate the successful performance of EMAP, and this is how the success of EMAP is verified. The IEA with EMAP is examined in secure cryptanalyses and computational complexity. Some results are validated with the state of the arts. In the comparison, the suggested EMAP-based IEA stands out due to the diversity and complexity performance. It achieves a cropped image having minimum loss, a good operation time of 0.2297s and a low computational complexity. Therefore, the proposed EMAP can be applied to realistic systems.

The Euler number is a topological number recently debuted in the topological physics. It is defined for a set of bands unlike the Chern number defined for a single band. We propose a simple model realizing the topological Euler insulator. We utilize the fact that the Euler number in a three-band model in two dimensions is reduced to the Pontryagin number. A skyrmion structure appears in momentum phase, yielding a nontrivial Euler number. Topological edge states emerge when the Euler number is nonzero. We discuss how to realize this model in electric circuits. We show that topological edge states are well signaled by impedance resonances.

In a September 1976 PRL Eguchi and Freund considered two topological invariants: the Pontryagin number $P \sim \int d^4x \sqrt{g}R^* R$ and the Euler number $\chi \sim \int d^4x \sqrt{g}R^* R^*$ and posed the question: to what anomalies do they contribute? They found that $P$ appears in the integrated divergence of the axial fermion number current, thus providing a novel topological interpretation of the anomaly found by Kimura in 1969 and Delbourgo and Salam in 1972. However, they found no analogous role for $\chi$. This provoked my interest and, drawing on my April 1976 paper with Deser and Isham on gravitational Weyl anomalies, I was able to show that for Conformal Field Theories the trace of the stress tensor depends on just two constants: \[ g^{\mu\nu}\langle T_{\mu\nu}\rangle=\frac{1}{(4\pi)^2}(cF-aG)\] where $F$ is the square of the Weyl tensor and $\int d^4x\sqrt{g} G/(4\pi)^2$ is the Euler number. For free CFTs with $N_s$massless fields of spin $s$ \[ 720c=6N_0 + 18N_{1/2} + 72 N_1~~~~ 720a=2N_0 + 11N_{1/2} + 124N_1 \], Comment: Published version, minor corrections and improvements, added references

We define Riordan matrix functions associated with Riordan arrays and study their algebraic properties. We also give their applications in the construction of new classes of Bernoulli and Euler polynomials and Bernoulli and Euler numbers, referred to as the duals and conjugate Bernoulli and Euler polynomials and dual and conjugate Bernoulli and Euler numbers, respectively.

We prove some results about the structure of all Lucas numbers whose Euler function is a repdigit in base 10. For example, we show that if Ln is such a Lucas number, then n < 10111 is of the form p or p2, where p3 | 10p-1 -1.

We discuss the multiple integral of a multivariate exponential taken with respect either to the Lebesgue measure or to the discrete uniform Bernoulli measure. In the first case the integral is linked to Euler's everywhere divergent power series and its generalizations, while in the second case the integral is linked to a one-dimensional model of spin systems as encountered in physics. Obravnavamo večkratni integral multivariatnega eksponenciala, računanega glede na Lebesgueovo mero ali glede na diskretno enakomerno Bernoullijevo mero. V prvem primeru je integral povezan z Eulerjevo povsod divergentno potenčno vrsto in njenimi posplošitvami, v drugem primeru pa z enodimenzionalnim modelom spinskih sistemov, na katere naletimo v fiziki.

By giving some new treatments we can improve a classical result of Walfisz (1963) on the asymptotic formula of Euler's function.

We use the hyperbolic cotangent function to deduce another proof of Euler's formula for ζ(2n).

Aerodynamic resistance of a brush seal was mainly studied. The velocity distribution along three specified lines was presented. By considering the pressure differential, Reynolds number and Euler number (Eu) were modified. The effect of geometric arrangements and pressure differentials on Eu and leakage were analyzed. Two correlations were fitted based on the numerical results. The results reveal the velocity distribution is almost flat, asymmetric along the specified lines. The velocity increases and decreases almost linearly at centerlines. Eu decreases gradually less with the increase of pressure differential and trends towards a fixed value. A larger Eu indicates stronger resistance but not necessarily less leakage. Finally, two fitted correlations are developed and one is exponential to the row number fits better. © 2018 Elsevier Ltd

Experiments are usually necessary to solve scientific and technologic problems because they can provide theoretical basis and criterion. The theoretical fundamentals for experimental design and evaluation are similitude and dimensional analysis. This chapter includes similitude, dimensional analysis, and its application. The learning goals are as follows. First, we should understand the concepts of mechanics similitude and similarity criterion. Then we have to master the applications of approximate similitude, the π theorem and dimensional analysis. Finally, three important approximate models should be grasped including Froude model, Reynolds model, and Euler model.

In this paper we present a new family of identities for Euler sums and integrals of polylogarithms by using the methods of generating function and integral representations of series. Then we apply it to obtain the closed forms of all quadratic Euler sums of weight ten. Furthermore, we also establish some relations between multiple zeta (star) values and nonlinear Euler sums. As applications of these relations, we give new closed form representations of several cubic Euler sums through single zeta values and linear sums. Finally, with the help of numerical computations of Mathematica or Maple, we evaluate several other Euler sums of weight ten.

We give some properties of the periodic Bernoulli functions and study the Fourier series of the periodic Euler functions which are derived periodic functions from the Euler polynomials. And we derive the relations between the periodic Bernoulli functions and those from Euler polynomials by using the Fourier series.

In this paper we consider the following Euler sums where p and q are positive integers, z and zi(i=1, 2, … ) are complex numbers on the unit circle or points within the unit circle except the origin and x is a complex number with Re x>0. We introduce a kernel function with a complex parameter and use it to obtain some identities of the linear extended Euler sums with parameters and some new Euler sums, which generalize the identities of the standard Euler sums. The linear extended Euler sums discussed in this paper can be expressed in terms of the Lerch transcendent. In some cases they can also be expressed by Riemann zeta function and Hurwitz zeta function.

Many formulas involving Euler’s constant have been proved by different authors. Some of these expressions concern the expansion of Euler’s constant in terms of the Zeta function evaluated at integers. In this paper, using the Pade approximation, we prove a general formula depending on three parameters which contains as particular cases Euler’s, Euler–Stieltjes, Flajolet–Vardi and some new formulas. The last section contains similar formulas for generalized Euler constants.

Two new versions of the Euler-Poincaré formula are proposed considering two new defined cuboids: the tetra-voxel and the octo-voxel, without losing information on the number of vertices and edges. The well-known relationship between contact and enclosing surface concepts, as well as the relationships between vertices, edges and enclosing surfaces, allowed us to compute an innovative algorithm for obtaining alternative versions of the Euler-Poincaré formula. This is a very important topological descriptor of 3D binary images. We considered not only topological but geometric aspects. Our method was compared to other proposals, obtaining that our proposed contact surface-based method offers more advantages.

The definition for the $p$-adic Hurwitz-type Euler zeta functions has been given by using the fermionic $p$-adic integral on $\mathbb Z_p$. By computing the values of this kind of $p$-adic zeta function at negative integers, we show that it interpolates the Euler polynomials $p$-adically. Many properties are provided for the $p$-adic Hurwitz-type Euler zeta functions, including the convergent Laurent series expansion, the distribution formula, the functional equation, the reflection formula, the derivative formula, the $p$-adic Raabe formula and so on. The definition for the $p$-adic Euler $L$-functions has also been given by using the $p$-adic Hurwitz-type Euler zeta functions., 37 Pages. arXiv admin note: text overlap with arXiv:1010.4440

In the paper, by a very simple approach, the author establishes an expression in terms of a lower Hessenberg determinant for the Euler polynomials. By the determinantal expression, the author finds a recurrence relation for the Euler polynomials. By the way, the author derives the corresponding expression and recurrence relation for the Euler numbers.

In order to accurately predict the complex nonlinear relationships between the cyclone performance parameters (The Euler and Stokes numbers) and the four significant geometrical dimensions (the inlet section height and width, the vortex finder diameter and the cyclone total height), the support vector machines approach has been used. Two support vector regression surrogates (SVR) have been trained and tested by CFD datasets. The result demonstrates that SVR can offer an alternative and powerful approach to model the performance parameters. The SVR model parameters have been optimized to obtain the most accurate results from the cross validation steps. SVR (with optimized parameters) can offer an alternative and powerful approach to model the performance parameters better than Kriging. SVR surrogates have been employed to study the effect of the four geometrical parameters on the cyclone performance. The genetic algorithms optimization technique has been applied to obtain a new geometrical ratio for minimum Euler number and for minimum Euler and Stokes numbers. New cyclones over-perform the standard Stairmand design performance. Pareto optimal solutions have been obtained and a new correlation between the Euler and Stokes numbers is fitted.

In this paper, we find a new recurrence formula fo the Euler zeta functions.

The equal-tempered 10-tone scale e n/10 (n=0,±1,±2, …), using the Euler number e=2.71828… as a pseudo-octave is shown to approximate well the prime number harmonics 2, 3, 5, and 11. Equal-tempered scales simultaneously approximating certain frequency ratios, shall be called tonal scales.1 Some of the properties of the Euler scale and its relation to other tonal scales are explored. The general mathematical problem of identifying tonal scales can be solved by investigating integer relations, using the μEuclidean algorithm, a modification of the PSLQ algorithm. If restricted to two numbers, the μEuclidean algorithm goes over identically into the ancient Euclidean algorithm, contrary to the PSLQ algorithm. The μEuclidean is able to solve a certain class of higher dimensional integer relations where the PSLQ (not the PPSLQ) algorithm breaks down. In general, the μEuclidean algorithm finds smaller integer relations than the (P)PSLQ algorithm. In an appendix, a simple alternative procedure is presented for dete...

We give an account of the most important results obtained by Euler in number theory, including the main contribution of Euler, application of analysis to problems of number theory. We note an important role played in modern number theory by the function that was introduced by Euler and is called the Riemann zeta function. We also discuss Euler’s works in other fields of science such as function theory and theory of music, as well as the relationship between music and mathematics.

We consider different generalizations of the Euler formula and discuss the properties of the associated trigonometric functions. The problem is analyzed from different points of view and it is shown that it can be formulated in a natural way in algebraic and geometric terms., Comment: 10 pages

We discuss Euler characteristics for finitely generated modules over Iwasawa algebras. We show that the Euler characteristic of a module is well-defined whenever the 0 0 th homology group is finite if and only if the relevant compact p p -adic Lie group is finite-by-nilpotent and that in this case all pseudo-null modules have trivial Euler characteristic. We also prove some other results relating to the triviality of Euler characteristics for pseudo-null modules as well as some analogous results for the Akashi series of Coates et al.

A new class of sequences convergent to Euler's constant is investigated. Special choices of parameters show that the class includes the original sequence defined by Euler, as well as more recently defined sequences due to DeTemple [1] and Vernescu [9]. It is shown how the rate of convergence of the sequences can be improved by computing optimal values of the parameters.

In this paper, we study on ( h , q ) -Euler-zeta and l -functions. We give relations between twisted ( h , q ) -partial zeta function, twisted ( h , q ) -Euler-zeta and twisted two variable ( h , q ) -Euler l -functions. We also give the value of twisted two variable ( h , q ) -Euler l -function at s = 0 . The main purpose of this paper is to construct two-variable twisted p -adic Euler ( h , q ) - l -function, which interpolates twisted ( h , q ) -Euler polynomials at negative integers. We give p -adic fermionic integral representation of this functions.

We define a topological space over the p-adic numbers, in which Euler products and Dirichlet series converge. We then show how the classical Riemann zeta function has a (p-adic) Euler product structure at the negative integers. Finally, as a corollary of these results, we derive a new formula for the non-Archimedean Euler–Mascheroni constant.

We present a computer algebra approach to proving identities on Bernoulli polynomials and Euler polynomials by using the extended Zeilberger's algorithm given by Chen, Hou and Mu. The key idea is to use the contour integral definitions of the Bernoulli and Euler numbers to establish recurrence relations on the integrands. Such recurrence relations have certain parameter free properties which lead to the required identities without computing the integrals., 22 pages; Final version. References updated and a typo in (8.1) corrected

In this paper we study that the $q$-Euler numbers and polynomials are analytically continued to $E_q(s)$. A new formula for the Euler's $q$-Zeta function $\zeta_{E,q}(s)$ in terms of nested series of $\zeta_{E,q}(n)$ is derived. Finally we introduce the new concept of the dynamics of analytically continued $q$-Euler numbers and polynomials., Comment: 5 pages

Summary An unbiased estimator of the Euler-Poincare characteristic (Euler number) of an arbitrary object or a random structure, respectively, is given. The estimator is based on joint observations of pairs of parallel section profiles. Thus the present paper extends the use of Sterio's disector from counting particles to determining the Euler number for a wide class of probes. The correctness of the given formulae is proved with mathematical strictness. Furthermore, the feasibility of the method is illustrated by an example from materials sciences.

In this paper, we first investigate several further interesting properties of the multiple Hurwitz–Lerch Zeta function Φ n ( z , s , a ) which was introduced recently by Choi et al. [J. Choi, D.S. Jang, H.M. Srivastava, A generalization of the Hurwitz–Lerch Zeta function, Integral Transform. Spec. Funct., 19 (2008)]. We then introduce and investigate some q -extensions of the multiple Hurwitz–Lerch Zeta function Φ n ( z , s , a ), the Apostol–Bernoulli polynomials B k ( n ) ( x ; λ ) of order n , and the Apostol–Euler polynomials E k ( n ) ( x ; λ ) of order n . Relevant connections of the results presented here with those obtained in earlier works are also indicated precisely.

Closed expressions are obtained for sums of products of Kronecker's double series of the form ∑ ( n j 1 , … , j N ) B j 1 ( x 1 ′ , x 1 ; τ ) ⋯ B j N ( x N ′ , x N ; τ ) , where the summation ranges over all nonnegative integers j 1 , … , j N with j 1 + ⋯ + j N = n . Corresponding results are derived for functions which are an elliptic analogue of the periodic Euler polynomials. As corollaries, we reproduce the formulas for sums of products of Bernoulli numbers, Bernoulli polynomials, Euler numbers, and Euler polynomials, which were given by K. Dilcher.

One purpose of this paper is to construct twisted q-Euler numbers by using p-adic invariant integral on Zp in the sense of fermionic. Finally, we consider twisted Euler q-zeta function and q-l-series which interpolate twisted q-Euler numbers and polynomials., 8 pages

Carlitz has introduced an interesting q-analogue of Frobenius–Euler numbers in [L. Carlitz, q-Bernoulli numbers and polynomials, Duke Math. J. 15 (1948) 987–1000; L. Carlitz, q-Bernoulli and Eulerian numbers, Trans. Amer. Math. Soc. 76 (1954) 332–350]. He has indicated a corresponding Stadudt–Clausen theorem and also some interesting congruence properties of the q-Euler numbers. A recent author's study of more general q-Euler and Genocchi numbers can be found in previous publication [T. Kim, L.C. Jang, H.K. Pak, A note on q-Euler and Genocchi numbers, Proc. Japan Acad. Ser. A Math. Sci. 77 (2001) 139–141]. In this paper we give a new construction of q-Euler numbers, which are different from Carlitz's q-extension and author's q-extension in previous publication (see [T. Kim, L.C. Jang, H.K. Pak, A note on q-Euler and Genocchi numbers, Proc. Japan Acad. Ser. A Math. Sci. 77 (2001) 139–141]). By using our q-extension of Euler numbers, we can also consider a new q-extension of Genocchi numbers and obtain some interesting relations between q-extension of Euler numbers and q-extension of Genocchi numbers.

A notion of (average) fractal Euler number for subsets of ℝd with infinite singular complexes is introduced by means of rescaled Euler numbers of infinitesimal e -neighbourhoods. For certain classes of self-similar sets we calculate the associated Euler exponent and the (average) fractal Euler number with the help of the Renewal theorem. Examples like the Sierpinski gasket or carpet are provided. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Recently, T. Kim considered Euler zeta function which interpolates Euler polynomials at negative integer (see [3]). In this paper, we study degenerate Euler zeta function which is holomorphic function on complex s-plane associated with degenerate Euler polynomials at negative integers., 6 bpages

This research presents a formula to calculate Euler number using gamma function. The representation is somewhat similar to Taylor series expansion of e. The number e is presented as sum of an integral and decimal part. But it is well known that e is an irrational number, and therefore such an expression for e does not exist in principle, that is why we are approximately representing it for use in computation. The approximation of our formula increases as we increase the value of index in the summation formula. We also analyse the approximation of our formula by both numerical and graphical methods. KEYWORDS: Euler Number, Gamma Function, Approximation, Computing, L’Hospital's Rule

Let E E be an elliptic curve over Q \mathbb {Q} and let ϱ \varrho be an odd, irreducible two-dimensional Artin representation. This article proves the Birch and Swinnerton-Dyer conjecture in analytic rank zero for the Hasse-Weil-Artin L L -series L ( E , ϱ , s ) L(E,\varrho ,s) , namely, the implication \[ L ( E , ϱ , 1 ) ≠ 0 ⇒ ( E ( H ) ⊗ ϱ ) G a l ( H / Q ) = 0 , L(E,\varrho ,1) \ne 0\quad \Rightarrow \quad (E(H)\otimes \varrho )^{\mathrm {Gal}(H/\mathbb {Q})} = 0, \] where H H is the finite extension of Q \mathbb {Q} cut out by ϱ \varrho . The proof relies on p p -adic families of global Galois cohomology classes arising from Beilinson-Flach elements in a tower of products of modular curves.

We introduce Euler(p, q) processes as an extension of the Euler(p) processes for purposes of obtaining more parsimonious models for non stationary processes whose periodic behavior changes approximately linearly in time. The discrete Euler(p, q) models are a class of multiplicative stationary (M-stationary) processes and basic properties are derived. The relationship between continuous and discrete mixed Euler processes is shown. Fundamental to the theory and application of Euler(p, q) processes is a dual relationship between discrete Euler(p, q) processes and ARMA processes, which is established. The usefulness of Euler(p, q) processes is examined by comparing spectral estimation with that obtained by existing methods using both simulated and real data.

We establish an asymptotic formula for the number of positive integers n⩽x for which φ(n) is free of kth powers.

In this short paper, we give several new formulas for ζ ( n ) when n is an odd positive integer. The method is based on a recent proof, due to H. Tsumura, of Euler's classical result for even n. Our results illuminate the similarities between the even and odd cases, and may give some insight into why the odd case is much more difficult.

Euler Number is one of the most important characteristics in topology. In two- dimension digital images, the Euler characteristic is locally computable. The form of Euler Number formula is different under 4-connected and 8-connected conditions. Based on the definition of the Foreground Segment and Neighbor Number, a formula of the Euler Number computing is proposed and is proved in this paper. It is a new idea to locally compute Euler Number of 2D image.

As a natural generalization of the classical Euler sum defined by [Formula: see text] we change the upper limit of the inner summation into kr, a fixed rational multiple of k, and obtain countable families of new sums which we call the extended Euler sums. We shall develop a systematic new method to evaluate these extended Euler sums as well as corresponding alternating sums in terms of values at non-negative integers of cosine and sine parts of the periodic zeta function when the weight p + q is odd.

Ever since the time of Euler, the so-called Euler sums have been evaluated in many different ways. We give here a (presumably) new proof of the classical Euler sum. We show that several interesting analogues of the Euler sums can be evaluated by systematically analyzing some known summation formulas involving hypergeometric series. Many other identities related to the Euler sums are also presented.