Search Results (75)

Transfer learning is the act of using the data or knowledge in a problem to help solve different but related problems. In a brain computer interface (BCI), it is important to deal with individual differences between topics and/or tasks. A kind of capsule decision neural network (CDNN) based on transfer learning is proposed. In order to solve the problem of feature distortion caused by EEG feature extraction algorithm, a deep capsule decision network was constructed. The architecture includes multiple primary capsules to form a hidden layer, and the connection between the advanced capsule and the primary capsule is determined by the neural decision routing algorithm. Unlike the dynamic routing algorithm that iteratively calculates the similarity between primary capsules and advanced capsules, the neural decision network computes the relationship between each capsule in the deep and shallow hidden layers in a probabilistic manner. At the same time, the distribution of the EEG covariance matrix is aligned in Riemann space, and the regional adaptive method is further introduced to improve the independent decoding ability of the capsule decision neural network for the subject’s EEG signals. Experiments on two motor imagery EEG datasets show that CDNN outperforms several of the most advanced transfer learning methods. Full article

The famous Riemann hypothesis (RH) asserts that all non-trivial zeros of the Riemann zeta function $\zeta \left(s\right)$ (zeros different from $s=-2m$, $m\in \mathbb{N}$) lie on the critical line $\sigma =1/2$. In this paper, combining the universality property of $\zeta \left(s\right)$ with probabilistic limit theorems, we prove that the RH is equivalent to the positivity of the density of the set of shifts $\zeta (s+i{t}_{\tau })$ approximating the function $\zeta \left(s\right)$. Here, $t_{\tau }$ denotes the Gram function, which is a continuous extension of the Gram points. Full article

This paper presents a comprehensive study of Plancherel’s theorem and inversion formulae for the Widder–Lambert transform, extending its scope to Lebesgue integrable functions, compactly supported distributions, and regular distributions with compact support. By employing the Plancherel theorem for the classical Mellin transform, we derive a corresponding Plancherel’s theorem specific to the Widder–Lambert transform. This novel approach highlights an intriguing connection between these integral transforms, offering new insights into their role in harmonic analysis. Additionally, we explore a class of functions that satisfy Salem’s equivalence to the Riemann hypothesis, providing a deeper understanding of the interplay between such equivalences and integral transforms. These findings open new avenues for further research on the Riemann hypothesis within the framework of integral transforms. Full article

In this paper, the asymptotic behavior of the modified Mellin transform ${\mathcal{Z}}_2\left(s\right)$, $s=\sigma +it$, of the fourth power of the Riemann zeta function is characterized by weak convergence of probability measures in the space of analytic functions. The main results are devoted to probability measures defined by generalized shifts ${\mathcal{Z}}_2(s+i\phi \left(\tau \right))$ with a real increasing to $+\infty$ differentiable functions connected to the growth of the second moment of ${\mathcal{Z}}_2\left(s\right)$. It is proven that the mass of the limit measure is concentrated at the point expressed as $h\left(s\right)\equiv 0$. This is used for approximation of $h\left(s\right)$ by ${\mathcal{Z}}_2(s+i\phi \left(\tau \right))$. Full article

The goal of this study is to develop numerous Hermite–Hadamard–Mercer (H–H–M)-type inequalities involving various fractional integral operators, including classical, Riemann–Liouville (R.L), k-Riemann–Liouville (k-R.L), and their generalized fractional integral operators. In addition, we establish a number of corresponding fractional integral inequalities for three-times differentiable convex functions that are connected to the right side of the H–H–M-type inequality. For these results, further remarks and observations are provided. Following that, a couple of graphical representations are shown to highlight the key findings of our study. Finally, some applications on special means are shown to demonstrate the effectiveness of our inequalities. Full article

Let G be a compact connected Lie group, $(X,\omega ,\mu )$ a Hamiltonian G-manifold with moment map $\mu$, and Z a codimension-2 Hamiltonian G-submanifold of X. We study the boundedness of the differential of symplectic vortices $(A,u)$ near Z, where A is a connection 1-form of a principal G-bundle P over a punctured Riemann surface $\mathring{\Sigma }$, and u is a G-equivariant map from P to an open cylinder model near Z. We show that if the total energy of a family of symplectic vortices on $\mathring{\Sigma }\cong [0,+\infty )\times S^1$ is finite, then the A-twisted differential $d_{A}u(r,\theta )$ is uniformly bounded for all $(r,\theta )\in [0,+\infty )\times S^1$. Full article

The existence or nonexistence of a complex structure on a differential manifold is a central problem in differential geometry. In particular, this problem on ${\mathrm{S}}^6$ was a long-standing unsolved problem, and differential geometry is an important tool. Recently, G. Clemente found a necessary and sufficient condition for almost-complex structures on a general differential manifold to be complex structures by using a covariant exterior derivative in three articles. However, in two of them, G. Clemente used a stronger condition instead of the published one. From there, G. Clemente proved the nonexistence of the complex structure on ${\mathrm{S}}^6$. We study the related differential operators and give some examples of nilmanifolds. And we prove that the earlier condition is too strong for an almost complex structure to be integrable. In another word, we clarify the situation of this problem. Full article

From the finite gap solutions of the KdV equation expressed in terms of abelian functions we construct solutions to the Schrödinger equation with a KdV potential in terms of fourfold Fredholm determinants. For this we establish a connection between Riemann theta functions and Fredholm determinants and we obtain multi-parametric solutions to this equation. As a consequence, a double Wronskian representation of the solutions to this equation is constructed. We also give quasi-rational solutions to this Schrödinger equation with rational KdV potentials. Full article

We applied the Ramsey analysis to the sets of points belonging to Riemannian manifolds. The points are connected with two kinds of lines: geodesic and non-geodesic. This interconnection between the points is mapped into the bi-colored, complete Ramsey graph. The selected points correspond to the vertices of the graph, which are connected with the bi-colored links. The complete bi-colored graph containing six vertices inevitably contains at least one mono-colored triangle; hence, a mono-colored triangle, built of the green or red links, i.e., non-geodesic or geodesic lines, consequently appears in the graph. We also considered the bi-colored, complete Ramsey graphs emerging from the intersection of two Riemannian manifolds. Two Riemannian manifolds, namely $\left(M_1,g_1\right)$ and $\left(M_2,g_2\right)$, represented by the Riemann surfaces which intersect along the curve $\left(M_1,g_1\right){\displaystyle \cap }\left(M_2,g_2\right)=ℒ$ were addressed. Curve $ℒ$ does not contain geodesic lines in either of the manifolds$\left(M_1,g_1\right)$ and $\left(M_2,g_2\right)$. Consider six points located on the $ℒ:$ $\left\{1,\dots 6\right\}\subset ℒ$. The points $\left\{1,\dots 6\right\}\subset ℒ$ are connected with two distinguishable kinds of the geodesic lines, namely with the geodesic lines belonging to the Riemannian manifold $\left(M_1,g_1\right)$/red links, and, alternatively, with the geodesic lines belonging to the manifold $\left(M_2,g_2\right)$/green links. Points $\left\{1,\dots 6\right\}\subset ℒ$ form the vertices of the complete graph, connected with two kinds of links. The emerging graph contains at least one closed geodesic line. The extension of the theorem to the Riemann surfaces of various Euler characteristics is presented. Full article

In this article, the coupled Kundu equations are analyzed using the Fokas unified method on the half-line. We resolve a Riemann–Hilbert (RH) problem in order to illustrate the representation of the potential function in the coupled Kundu equations. The jump matrix is obtained from the spectral matrix, which is determined according to the initial value data and the boundary value data. The findings indicate that these spectral functions exhibit interdependence rather than being mutually independent, and adhere to a global relation while being connected by a compatibility condition. Full article

In this study, to approximate nabla sequential differential equations of fractional order, a class of discrete Liouville–Caputo fractional operators is discussed. First, some special functions are re-called that will be useful to make a connection with the proposed discrete nabla operators. These operators exhibit inherent symmetrical properties which play a crucial role in ensuring the consistency and stability of the method. Next, a formula is adopted for the solution of the discrete system via binomial coefficients and analyzing the Riemann–Liouville fractional sum operator. The symmetry in the binomial coefficients contributes to the precise approximation of the solutions. Based on this analysis, the solution of its corresponding continuous case is obtained when the step size $p_0$ tends to 0. The transition from discrete to continuous domains highlights the symmetrical nature of the fractional operators. Finally, an example is shown to testify the correctness of the presented theoretical results. We discuss the comparison of the solutions of the operators along with the numerical example, emphasizing the role of symmetry in the accuracy and reliability of the numerical method. Full article

By the Bagchi theorem, the Riemann hypothesis (all non-trivial zeros lie on the critical line) is equivalent to the self-approximation of the function $\zeta \left(s\right)$ by shifts $\zeta (s+i\tau )$. In this paper, it is determined that the Riemann hypothesis is equivalent to the positivity of density of the set of the above shifts approximating $\zeta \left(s\right)$ with all but at most countably many accuracies $\epsilon >0$. Also, the analogue of an equivalent in terms of positive density in short intervals is discussed. Full article

In this paper, we address the problem of the divergent sums of general arithmetic functions over the set of primes. In classical analytic number theory, the sum of the logarithm of the prime numbers plays a crucial role. We consider the sums of powers of the logarithm of primes and its connection with Riemann’s zeta function (z.f.). This connection is achieved through the second Chebyshev function of order n, which can be estimated by exploiting the symmetry properties of Riemann’s zeta function. Finally, a heuristic approach to evaluating more general sums is also given. Full article

Bulk ideal flows constitute a wide class of solutions in plasticity theory. Ideal flow solutions concern inverse problems. In particular, the solution determines part of the boundary of a region where it is valid. Bulk planar ideal flows exist in the case of (i) isotropic rigid/plastic material obeying an arbitrary pressure-independent yield criterion and its associated flow rule and (ii) the double sliding and rotation model based on the Mohr–Coulomb yield criterion. In the latter case, the intrinsic spin must vanish. Both models are perfectly plastic, and the complete equation systems are hyperbolic. All available specific solutions for both models describe flows with a symmetry axis. The present paper aims at general solutions for flows with no symmetry axis. The general structure of the solutions consists of two rigid regions connected by a plastic region. The characteristic lines between the plastic and rigid regions must be straight, which partly dictates the general structure of the characteristic nets. The solutions employ Riemann’s method in regions where the characteristics of both families are curvilinear. Special solutions that do not have such regions are considered separately. In any case, the solutions are practically analytical. A numerical technique is only necessary to evaluate ordinary integrals. The solutions found determine the tool shapes that produce ideal flows. In addition, the distribution of pressure over the tool’s surface is calculated, which is important for predicting the wear of tools. Full article

The notions of strong differential subordination and its dual, strong differential superordination, have been introduced as extensions of the classical differential subordination and superordination concepts, respectively. The dual theories have developed nicely, and important results have been obtained involving different types of operators and certain hypergeometric functions. In this paper, quantum calculus and fractional calculus aspects are added to the study. The well-known q-hypergeometric function is given a form extended to fit the study concerning previously introduced classes of functions specific to strong differential subordination and superordination theories. Riemann–Liouville fractional integral of extended q-hypergeometric function is defined here, and it is involved in the investigation of strong differential subordinations and superordinations. The best dominants and the best subordinants are provided in the theorems that are proved for the strong differential subordinations and superordinations, respectively. For particular functions considered due to their remarkable geometric properties as best dominant or best subordinant, interesting corollaries are stated. The study is concluded by connecting the results obtained using the dual theories through sandwich-type theorems and corollaries. Full article