Narrow Results

This research paper aims to present closed-form analytical equations that predict the buckling behavior of bidirectional functionally graded (BDFG) nanobeams characterized by varying the material properties along axial and thickness directions. The small-scale effects inherent in nanobeams are captured by Eringen’s nonlocal elasticity theory, and the displacement field of the nanobeam is assumed by the Euler–Bernoulli beam theory. The governing equations of motion are derived by applying Hamilton’s principle and variational formulation. Analytical solutions to the governing equations are obtained by using the Mellin transform. Analytical expressions are derived as stability criteria for four distinct boundary conditions of BDFG nonlocal nanobeams. A comprehensive investigation is conducted into the effects of material constants in axial and thickness directions. To validate the proposed analytical method, critical buckling loads obtained from the presented stability criteria equations are compared with pertinent results in existing literature computed through various numerical schemes. The study reveals that variations in the material’s thickness parameter have a consistent stiffening impact on the critical buckling load across all boundary conditions and buckling modes, with an optimal magnitude closer to unity for optimal buckling stiffness. Additionally, changes in the material’s axial parameter enhance the nanobeam’s buckling stiffness, exhibiting varying increase rates across boundary conditions and buckling modes.

Multidimensional digital searching (M-d tries) is analyzed from the view point of partial match retrieval. Our first result extends the analysis of Flajolet and Puech of the average cost of retrieval under the Bernoulli model to biased probabilities of symbols occurrences in a key. The second main finding concerns the variance of the cost of the retrieval in the unbiased case. This variance is of order O(N^1−s/M) where N is the number of records stored in an M-d trie, and s is the number of specified components in a query of size M. For M=2 and s=1 we present a detailed analysis of the variance, which identifies the constant at . This analysis, which is the central part of our paper, requires certain series transformation identities which go back to Ramanujan. In the Appendix we provide a Mellin transform approach to these results.

We present the mathematical construction of the physically relevant quantum Hamiltonians for a three-body system consisting of identical bosons mutually coupled by a two-body interaction of zero range. For a large part of the presentation, infinite scattering length will be considered (the unitarity regime). The subject has several precursors in the mathematical literature. We proceed through an operator-theoretic construction of the self-adjoint extensions of the minimal operator obtained by restricting the free Hamiltonian to wave-functions that vanish in the vicinity of the coincidence hyperplanes: all extensions thus model an interaction precisely supported at the spatial configurations where particles come on top of each other. Among them, we select the physically relevant ones, by implementing in the operator construction the presence of the specific short-scale structure suggested by formal physical arguments that are ubiquitous in the physical literature on zero-range methods. This is done by applying at different stages the self-adjoint extension schemes à la Kreĭn–Višik–Birman and à la von Neumann. We produce a class of canonical models for which we also analyze the structure of the negative bound states. Bosonicity and zero range combined together make such canonical models display the typical Thomas and Efimov spectra, i.e. sequence of energy eigenvalues accumulating to both minus infinity and zero. We also discuss a type of regularization that prevents such spectral instability while retaining an effective short-scale pattern. Besides the operator qualification, we also present the associated energy quadratic forms. We structured our analysis so as to clarify certain steps of the operator-theoretic construction that are notoriously subtle for the correct identification of a domain of self-adjointness.

Let (X, 0) be the germ of a normal space of dimension n+1 with an isolated singularity at 0 and let f be a germ of holomorphic function with an isolated regularity at 0. We prove that the meromorphic extension of the current

Finding a universal method of crystal structure solution and proving the Riemann hypothesis are two outstanding challenges in apparently unrelated fields. For centro-symmetric crystals however, a connection arises as the result of a statistical approach to the inverse phase problem. It is shown that parameters of the phase distribution are related to the non-trivial Riemann zeros by a Mellin transform.

We study a 2D scalar harmonic wave transmission problem between a classical dielectric and a medium with a real-valued negative permittivity/permeability which models an ideal metamaterial. When the interface between the two media has a corner, according to the value of the contrast (ratio) of the physical constants, this non-coercive problem can be ill-posed (not Fredholm) in H¹. This is due to the degeneration of the two dual singularities which then behave like r^±iη = e^{±iη ln r} with η ∈ ℝ*. This apparition of propagative singularities is very similar to the apparition of propagative modes in a waveguide for the classical Helmholtz equation with Dirichlet boundary condition, the contrast playing the role of the wavenumber. In this work, we derive for our problem a functional framework by adding to H¹ one of these propagative singularities. Well-posedness is then obtained by imposing a radiation condition, justified by means of a limiting absorption principle, at the corner between the two media.

Interval type-2 fuzzy sets provide us with additional degrees of freedom to represent the uncertainty and the fuzziness of the real word than traditional type-1 fuzzy sets. Interval type-2 fuzzy numbers ranking has an important role in the decision making analysis. In this paper, the probatilistic mean value and variance of interval type-2 fuzzy numbers are proposed based on the Mellin transform for type-1 fuzzy numbers. The interval type-2 fuzzy number with the higher mean is ranked higher. If the mean values are equal the one with the smaller variance is judged higher rank. On this basis, some new distance measures and possibility degree formula are proposed to comparing interval type-2 fuzzy numbers based on their Mellin mean value and variance. Some benchmarking numerical examples are given, and some interpretation issues are explained.

In this paper, it will be shown that a probability measure on ℂ associated with the Jacobi–Szegö parameters of the orthogonal polynomials can be obtained by making use of the classical Mellin transform and its convolution property. We shall construct several measures on ℂ represented by the modified Bessel functions. The material in this paper gives nontrivial examples originated from the continuous dual Hahn polynomials (one of hypergeometric orthogonal polynomials), which are beyond the Meixner–Pollaczek polynomials appeared in our previous papers.^{4, 5}

We study first passage problems of a class of reflected generalized Ornstein–Uhlenbeck processes without positive jumps. By establishing an extended Dynkin's formula associated with the process, we derive that the joint Laplace transform of the first passage time and an integral functional stopped at the time satisfies a truncated integro-differential equation. Two solvable examples are presented when the driven Lévy process is a drifted-Brownian motion and a spectrally negative stable process with index α ∈ (1, 2], respectively. Finally, we give two applications in finance.

In this article, we use a Mellin transform approach to prove the existence and uniqueness of the price of a European option under the framework of a Black–Scholes model with time-dependent coefficients. The formal solution is rigorously shown to be a classical solution under quite general European contingent claims. Specifically, these include claims that are bounded and continuous, and claims whose difference with some given but arbitrary polynomial is bounded and continuous. We derive a maximum principle and use it to prove uniqueness of the option price. An extension of the put-call parity which relates the aforementioned two classes of claims is also given.

Radon transform is not only robust to noise, but also independent on the calculation of pattern centroid. In this paper, Radon–Mellin transform (RMT), which is a combination of Radon transform and Mellin transform, is proposed to extract invariant features. RMT converts any object into a closed curve. Radon–Fourier descriptor (RFD) is derived by applying Fourier descriptor to the obtained closed curve. The obtained RFD is invariant to scaling and rotation. (Generic) R-transform and some other Radon-based methods can be viewed as special cases of the proposed method. Experiments are conducted on some binary images and gray images.

In this paper, a special class of local ζ-functions is studied. The main theorem states that the functions have all zeros on the line ℜ(s) = 1/2. This is a natural generalization of the result of Bump and Ng stating that the zeros of the Mellin transform of Hermite functions have ℜ(s) = 1/2.

The Mellin transform of a summatory function involving weighted averages of Ramanujan sums is obtained in terms of Bernoulli numbers and values of the Burgess zeta function. The possible singularity of the Burgess zeta function at s = 1 is then shown to be equivalent to the evaluation of a certain infinite series involving such weighted averages. Bounds on the size of the tail of these series are given and specific bounds are shown to be equivalent to the Riemann hypothesis.

We offer two new Mellin transform evaluations for the Riemann zeta function in the region $0<\Re (s)<1$. Some discussion is offered in the way of evaluating some further Fourier integrals involving the Riemann xi function.

Many papers have been recently devoted to the study of the radial behavior as $z\to 1^{-}$ of transcendental r-Mahler functions holomorphic in the open unit disk. In particular, Bell and Coons showed in 2017 that, in a generic sense, r-Mahler functions behave like $(1+o(1))C(z)/{(1-z)}^{\rho }$ for some $\rho \in ℂ$ and $C(z)$ is a real analytic function of $z\in (0,1)$ such that $C(z)=C(z^r)$. They did not provide a formula for $C(z)$ which was made explicit only in a few examples of r-Mahler functions of orders 1 and 2, and for specific values of r. In this paper, we first provide an explicit expression of $C(z)$ as an exponential of a Fourier series in the variable $loglog(1/z)/log(r)$ for every r-Mahler function of order 1. Then, extending to a large setting a method introduced by Brent–Coons–Zudilin in 2016 to compute $C(z)$ associated to the Dilcher–Stolarsky function (a 4-Mahler function of order 2 in $\mathbb{Q}[[z]]$), we provide an explicit expression of $C(z)$ for every r-Mahler function of order 2 under mild assumptions on the coefficients in $ℝ(z)$ of the underlying r-Mahler equations. This applies in particular to the generating function of the Baum–Sweet sequence. We do the same for r-Mahler functions solutions of inhomogeneous Mahler equations of order 1.

In this paper, we have introduced and studied two-dimensional $(P,Q)$-Mellin transform which extends the known results of two-dimensional $Q$-Mellin transform. We provide its several basic properties, the appropriate convolution, the inversion formula and the Parseval-type relations. Some applications of $(P,Q)$-Mellin transform have been pointed out in solving integral equations. Finally, a Titchmarsh-type theorem has been proved.

An alternative derivation is given for the asymptotic expansion, as s → 0⁺, of the multiple integral