Search Results (39)

In this paper, we study a class of nonlocal Schrödinger–Poisson–Slater equations: $-\Delta u+u+\lambda \left(I_{\alpha }\ast {\left|u\right|}^q\right){\left|u\right|}^{q-2}u={\left|u\right|}^{p-2}u$, where $q,p>1$, $\lambda >0$, and $I_{\alpha }$ is the Riesz potential. We obtain the existence, stability, and symmetry-breaking of solutions for both radial and nonradial cases. In the radial case, we use variational methods to establish the coercivity and weak lower semicontinuity of the energy functional, ensuring the existence of a positive solution when p is below a critical threshold $\overline{p}={\displaystyle \frac{4q+2\alpha }{2+\alpha }}$. In addition, we prove that the energy functional attains a minimum, guaranteeing the existence of a ground-state solution under specific conditions on the parameters. We also apply the Pohozaev identity to identify parameter regimes where only the trivial solution is possible. In the nonradial case, we use the Nehari manifold method to prove the existence of ground-state solutions, analyze symmetry-breaking by studying the behavior of the energy functional and identifying the parameter regimes in the nonradial case, and apply concentration-compactness methods to prove the global well-posedness of the Cauchy problem and demonstrate the orbital stability of the ground state. Our results demonstrate the stability of solutions in both radial and nonradial cases, identifying critical parameter regimes for stability and instability. This work enhances our understanding of the role of nonlocal interactions in symmetry-breaking and stability, while extending existing theories to multiparameter and higher-dimensional settings in the Schrödinger–Poisson–Slater model. Full article

This paper proposes a novel fault diagnosis framework that integrates the Osprey–Cauchy–Sparrow Search Algorithm (OCSSA) optimized Variational Mode Decomposition (VMD) with a Bidirectional Temporal Convolutional Network-Attention mechanism (BiTCN-Attention). To address the limitations of empirical parameter selection in VMD, OCSSA adaptively optimizes the decomposition parameters (penalty factor $\alpha$ and mode number K) through a hybrid strategy that combines chaotic initialization, Osprey-inspired global search, and Cauchy mutation. Subsequently, the BiTCN captures bidirectional temporal dependencies from vibration signals, while the attention mechanism dynamically filters critical fault features, constructing an end-to-end diagnostic model. Experiments on the CWRU dataset demonstrate that the proposed method achieves an average accuracy of 99.44% across 10 fault categories, outperforming state-of-the-art models (e.g., VMD-TCN: 97.5%, CNN-BiLSTM: 84.72%). Full article

The isotropic Cauchy distribution is a member of the central $\alpha$-stable family that plays a role in the set of heavy-tailed distributions similar to that of the Gaussian density among finite second-moment laws. Given a sequence of n observations, we are interested in characterizing the performance of Likelihood Ratio Tests, where two hypotheses are plausible for the observed quantities: either isotropic Cauchy or isotropic Gaussian. Under various setups, we show that the probability of error of such detectors is not always exponentially decaying with n, with the leading term in the exponent shown to be logarithmic instead, and we determine the constants in that leading term. Perhaps surprisingly, the optimal Bayesian probabilities of error are found to exhibit different asymptotic behaviors. Full article

The relative energetic stability, mechanical properties, and thermodynamic behavior of B2-AlRE (RE = Sc, Y, La-Lu) second phases in Al alloys have been investigated through the integration of first-principles calculations with the quasi-harmonic approximation (QHA) model. The results demonstrate a linear increase in the calculated equilibrium lattice constant a₀ with the ascending atomic number of RE, while the enthalpy of formation ΔH_f exhibits more fluctuating variations. The lattice mismatch δ between B2-AlRE and Al matrix is closely correlated with the transferred electron et occurring between Al and RE atoms. Furthermore, the mechanical properties of the B2-AlRE phases are determined. It is observed that the calculated elastic constants C_ij, bulk modulus B_H, shear modulus G_H, and Young’s modulus E_H initially decrease with increasing atomic number from Sc to Ce and then increase up to Lu. The calculated Cauchy pressure C₁₂-C₄₄, Pugh’s ratio B/G, and Poisson’s ratio ν for all AlRE particles exhibit a pronounced directional covalent characteristic as well as uniform deformation and ductility. With the rise in temperature, the calculated vibrational entropy (S_vib) and heat capacity (C_V) of AlRE compounds exhibit a consistent increasing trend, while the Gibbs free energy (F) shows a linear decrease across all temperature ranges. The expansion coefficient (α_T) sharply increases within the temperature range of 0~300 K, followed by a slight change, except for Al, AlHo, AlCe, and AlLu, which show a linear increase after 300 K. As the atomic number increases, both S_vib and C_V increase from Sc to La before stabilizing; however, F initially decreases from Sc to Y before increasing up to La with subsequent stability. All thermodynamic parameters demonstrate similar trends at lower and higher temperatures. This study provides valuable insights for evaluating high-performance aluminum alloys. Full article

In this research paper, we consider a model of the fractional Cauchy–Euler-type equation, where the fractional derivative operator is the Caputo with order $0<\alpha <2$. The problem also constitutes a class of examples of the Cauchy problem of the Bagley–Torvik equation with variable coefficients. For proving the existence and uniqueness of the solution of the given problem, the contraction mapping principle is utilized. Furthermore, a numerical method and an algorithm are developed for obtaining the approximate solution. Also, convergence analyses are studied, and simulations on some test problems are given. It is shown that the proposed method and the algorithm are easy to implement on a computer and efficient in computational time and storage. Full article

To tackle the issue of detecting early, subtle faults in rolling bearings in the presence of noise interference, the SCSSA-VMD-MCKD method is suggested. This method optimizes the Variational Mode Decomposition (VMD) and Maximum Correlated Kurtosis Deconvolution (MCKD) by integrating the sine-cosine and Cauchy Mutation Sparrow Search Algorithm (SCSSA). The approach aims to effectively diagnose faults in rolling bearings by leveraging the strengths of VMD and MCKD in noise reduction and highlighting fault frequencies. The method utilizes the SCSSA algorithm to autonomously search for parameters in both VMD and MCKD, using the EnvelopeCrest Factor $Ec$ as a fitness function. Firstly, SCSSA is employed to optimize the decomposition mode number $K$ and penalty factor $\alpha$ in the VMD algorithm. Secondly, the parameters in the MCKD algorithm are optimized, and MCKD is used for deconvolution to enhance the impulsive characteristics of the best modal component. Finally, the signal is further analyzed after deconvolution. The results demonstrate that this algorithm can effectively identify subtle fault signals in bearing signals and diagnose fault frequencies in noisy environments. The accuracy of fault diagnosis can reach nearly 99%. Full article

We further extend the results of other researchers on existence theory to homogeneous fractional Cauchy–Euler equations ${\displaystyle \sum _{i=1}^m}{d}_i{x}^{{\alpha }_i}{D}^{{\alpha }_i}u\left(x\right)+\mu u\left(x\right)=0,{\alpha }_i>0,$ with the derivatives in Caputo or Riemann–Liouville sense. Unlike the existing works, we consider multi-term equations without any restrictions on the order of fractional derivatives. The results are based on the characteristic equations which generate the solutions. Depending on the roots of the characteristic equations (real, multiple, or complex), we construct the corresponding solutions and prove their linear independence. Full article

Let $\mu$ be a self-similar measure with compact support K. The Hausdorff dimension of K is $\alpha$. The Cauchy transform of $\mu$ is denoted by $F\left(z\right)$. For $0<\beta <1,$ we define the function $F^{\left[\beta \right]},$ which compares with the fractional derivative of F of order $\beta .$ Let $\Phi \left(z\right)=F(1/z),\left|z\right|<1$. In this paper, we prove that ${\Phi }^{\left[\beta \right]}$ belongs to $A^p$ for $0<p<1/(\beta +1)$, and ${\left({\Phi }^{\prime }\right)}^{\left[\beta \right]}$ belongs to $A^p$ for $1\le p<1/\beta \le 1/(2-\alpha )$, where $A^p$ is the Bergman space. At the same time, we give a value distribution property of F, which is similar to the big Picard theorem. Full article

In this work, we establish some characteristics for a sequence, ${\mathcal{A}}_{\alpha }(n,k)$, including recurrence relations, generating function and inversion formula, etc. Based on the sequence, we derive, by means of the generating function approach, some transformation formulas concerning certain combinatorial numbers named after Lah, Stirling, harmonic, Cauchy and Catalan, as well as several closed finite sums. In addition, the relationship between ${\mathcal{A}}_{\alpha }(n,k)$ and r-Whitney–Lah numbers is established, and some formulas for the r-Whitney–Lah numbers are obtained. Full article

In this work, our primary focus is on the numerical computation of highly oscillatory integrals involving the Airy function. Specifically, we address integrals of the form ${\int }_0^b{x}^{\alpha }f\left(x\right)\mathrm{Ai}(-\omega x)dx$ over a finite or semi-infinite interval, where the integrand exhibits rapid oscillations when $\omega \gg 1$. The inherent high oscillation and algebraic singularity of the integrand make traditional quadrature rules impractical. In view of this, we strategically partition the interval into two segments: $[0,1]$ and $[1,b]$. For integrals over the interval $[0,1]$, we introduce a Filon-type method based on a two-point Taylor expansion. In contrast, for integrals over $[1,b]$, we transform the Airy function into the first kind of Bessel function. By applying Cauchy’s integration theorem, the integral is then reformulated into several non-oscillatory and exponentially decaying integrals over $[0,+\infty )$, which can be accurately approximated by the generalized Gaussian quadrature rule. The proposed methods are accompanied by rigorous error analyses to establish their reliability. Finally, we present a series of numerical examples that not only validate the theoretical results but also showcase the accuracy and efficacy of the proposed method. Full article

We introduce the new idea of $(\alpha -{\theta }_{\sigma })$-contraction in quasi-metric spaces in this paper. For these kinds of mappings, we then prove new fixed-point theorems on left K, left M, and left $Smyth$-complete quasi-metric spaces. We also apply our results to infer the existence of a solution to a second-order boundary-value problem. Full article

We explored here the case of three-dimensional non-stationary flows of helical type for the incompressible couple stress fluid with given Bernoulli-function in the whole space (the Cauchy problem). In our presentation, the case of non-stationary helical flows with constant coefficient of proportionality α between velocity and the curl field of flow is investigated. In the given analysis for this given type of couple stress fluid flows, an absolutely novel class of exact solutions in theoretical hydrodynamics is illuminated. Conditions for the existence of the exact solution for the aforementioned type of flows were obtained, for which non-stationary helical flow with invariant Bernoulli-function satisfying to the Laplace equation was considered. The spatial and time-dependent parts of the pressure field of the fluid flow should be determined via Bernoulli-function if components of the velocity of the flow are already obtained. Analytical and numerical findings are outlined, including outstanding graphical presentations of various types of constructed solutions, in order to elucidate dynamic snapshots that show the timely development of the topological behavior of said solutions. Full article

In recent years, deep learning has been increasingly used in fault diagnosis of rotating machinery. However, the actual acquisition of rolling bearing fault signals often contains ambient noise, making it difficult to determine the optimal values of the parameters. In this paper, a sparrow search algorithm (LSSA) based on backward learning of lens imaging and Gaussian Cauchy variation is proposed. The lens imaging reverse learning strategy enhances the traversal capability of the algorithm and allows for a better balance of algorithm exploration and development. Then, the performance of the proposed LSSA was tested on the benchmark function. Finally, LSSA is used to find the optimal modal component K and the optimal penalty factor $\alpha$ in VMD-GRU, which in turn realizes the fault diagnosis of rolling bearings. The experimental results show that the model can achieve a 96.61% accuracy in rolling bearing fault diagnosis, which proves the effectiveness of the method. Full article

The classical Belinskii theorem implies that any sufficiently regular function $\mu \left(z\right)$ on the extended complex plane $\widehat{\mathbb{C}}$ with a small $C^{1+\alpha }$ norm generates via the two-dimensional Cauchy integral a quasiconformal automorphism w of $\widehat{\mathbb{C}}$ with the Beltrami coefficient $\tilde{\mu }=\mu +{O(\parallel \mu \parallel }^2)$. We consider $\mu$ supported in arbitrary bounded quasiconformal disks and show that under appropriate assumptions of $\mu$, this automorphism explicitly provides the basic curvelinear quasi-invariants associated with conformal and quasiconformal maps, advancing an old problem of quasiconformal analysis. Full article

Hyperbolic complete monotonicity property ($\mathrm{HCM}$) is a way to check if a distribution is a generalized gamma ($\mathrm{GGC}$), hence is infinitely divisible. In this work, we illustrate to which extent the Mittag-Leffler functions $E_{\alpha },\alpha \in (0,2]$, enjoy the $\mathrm{HCM}$ property, and then intervene deeply in the probabilistic context. We prove that for suitable $\alpha$ and complex numbers z, the real and imaginary part of the functions $x\mapsto E_{\alpha }\left(zx\right)$, are tightly linked to the stable distributions and to the generalized Cauchy kernel. Full article