Finite Numbers Research Articles

Performance Analysis of Microwave Photonic Spectral Filters based on Optical Microcombs

AbstractMicrowave transversal filters, which are implemented based on the transversal filter structure in digital signal processing, offer a high reconfigurability for achieving a variety of signal processing functions without changing hardware. When implemented using microwave photonic (MWP) technologies, also known as MWP transversal filters, they provide competitive advantages over their electrical counterparts, such as large operation bandwidth, strong immunity to electromagnetic interference, and low loss when processing signals at high frequencies. Recent advances in high‐performance optical microcombs provide compact and powerful multi‐wavelength sources for MWP transversal filters that require a larger number of wavelength channels to achieve high performance, allowing for the demonstration of a diverse range of filter functions with improved performance and new features. Here, a comprehensive performance analysis for microcomb‐based MWP spectral filters based on the transversal filter approach is presented. First, the theoretical limitations are investigated in the filter spectral response induced by finite tap numbers. Next, the distortions are analyzed in the filter spectral response resulting from experimental error sources. Finally, the influence of input signal's bandwidth on the filtering errors is assessed. These results provide a valuable guide for the design and optimization of microcomb‐based MWP transversal filters for a variety of applications.

Mesh objective characteristic element length for higher-order finite beam elements

The use of fracture energy regularization techniques can effectively mitigate the mesh dependency of numerical solutions caused by the strain softening behavior of quasi-brittle materials. However, the successful regularization depends on the correct estimation of the crack bandwidth in Finite Element solutions. This paper aims to present an enhanced crack band formulation to overcome the strain localization instability especially for the higher-order elements developed in the framework of Carrera Unified Formulation (CUF). Besides, a modified Mazars damage method incorporating fracture energy regularization is employed to describe the nonlinear damage behavior of the concrete. To evaluate the efficiency of the proposed crack band formulation, three experimental concrete benchmarks are selected for the numerical damage analysis. By comparing numerical and experimental results, the proposed method can guarantee mesh objectivity despite varying finite element numbers and orders, indicating perseved fracture energy consumption within proposed higher-order beam models.

The ring of finite algebraic numbers and its application to the law of decomposition of primes

In this paper, we develop an explicit method to express finite algebraic numbers (in particular, certain idempotents among them) in terms of linear recurrent sequences, and give applications to the characterization of the splitting primes in a given finite Galois extension over the rational field.

Infinite multisets: Basic properties and cardinality

This research work presents the topic of infinite multisets, their basic properties and cardinality from a somewhat different perspective. In this work, a new property of multisets, ‘m-cardinality’, is defined using multiset functions. M-cardinality unifies and generalizes the definitions of cardinality, injection, bijection, and surjection to apply to multisets. M-cardinality takes into account both the number of distinct elements in a multiset and the number of copies of each element (i.e., the multiplicity of the elements). Based on m-cardinality, ‘m-cardinal numbers’ are defined as a generalization of cardinal numbers in the context of multisets. Some properties of m-cardinal numbers associated with finite and infinite msets have been researched. Concrete examples of transfinite m-cardinal numbers are given, corresponding to infinite msets which are less than ℵ0 (the cardinality of the countably infinite set). It has been established that between finite numbers and ℵ0 there exist hierarchies of transfinite m-cardinals, corresponding to infinite msets. Furthermore, there are examples of infinite msets with negative multiplicity that have a cardinality less than zero. We prove that there is a decreasing sequence of transfinite m-cardinal numbers, corresponding to infinite msets with negative multiplicity, and in this sequence, there is not a smallest transfinite m-cardinal number.

Solutions of the Schrödinger equation with velocity-dependent potentials:a study on Kratzer, Mie and Hulthen potentials

The effect of isotropic velocity-dependent potentials on the bound state energy eigenvalues of the Kratzer, Mie and Hulthen potentials is obtained for any quantum states in the presence of constant form factor εr=γρ0. The corresponding energy eigenvalues and eigenfunctions are determined and tabulated for a set of finite quantum numbers n and l in the framework of the well-known Nikiforov-Uvarov method.

Impact of a floating flexible plate on the stability of double-layered falling flow

The hydrodynamic stability behavior of a two-layer falling film is explored with a floating flexible plate on the top layer surface. The stress balance at the surface is modeled using a modified membrane equation. There is an insoluble surfactant at the liquid-liquid interface of the flow system. The linear instability of perturbation waves is captured by numerically solving the generalized Orr-Sommerfeld eigenvalue problem using the Chebyshev spectral collocation technique. Four different types of linear stability modes are identified: surface mode (SM), interface mode (IM), interface surfactant mode (ISM), and shear mode (SHM). The floating flexible plate has an inferior impact on all the existing modes in the longwave zone. However, a significant influence is noticed for the finite wave numbers. The surface and interface wave instabilities can be suppressed in the smaller wavenumber zone by imposing higher structural rigidity and uniform thickness of the flexible plate. The stabilizing nature of the surface mode becomes more powerful when the top layer viscosity is dominant. A new interfacial instability emerges when the top layer is less viscous than the lower one. At moderate Reynolds numbers, the behavior of interface mode is different in two different zones mr<1 and mr>1, where m and r denote the viscosity and density ratio, respectively. Further, the unstable shear modes induced by the top and bottom layers are detected under the low inclination angles with a strong inertia force. The occurrence of the shear modes requires the viscosity and density of the lower layer to be much higher than those of the upper layer. The influence of characteristic parameters of the flexible plate on the lower layer shear mode is not very sensitive. Finally, the competition between the different modes for dominance of stability boundaries is also discussed.

L-functions and meromorphic functions satisfying the same Riemann-type functional equation and sharing sets

In 2023, Li–Du–Yi [Complex Var. Elliptic Equ., 68(10)2023, 1653–1677] proved that if two L-functions L 1 and L 2 in the extended Selberg class S # have positive degrees, satisfy the same functional equation with a ( 1 ) = 1 and E L 1 ( S ) = E L 2 ( S ) for a finite set S = { c 1 , c 2 , c 3 } , where c 1 , c 2 and c 3 are three distinct finite complex values, then L 1 = L 2 . We prove that if two L-functions L 1 and L 2 in the extended Selberg class S # have positive degrees, satisfy the same functional equation with a ( 1 ) = 1 and share a set { α 1 , α 2 , … , α t } CM, where t ≥ 1 is a positive integer, and α 1 , α 2 , ··· , α t are t distinct finite complex numbers, then L 1 = L 2 . The result also improves the corresponding results from Khoai–An [The Ramanujan J., 58(1)(2022),253-267]. We also prove that if a non-constant meromorphic function f with finitely many poles in the complex plane shares a set { α 1 , α 2 , … , α t } CM and some finite value c IM with a non-constant L-function L having a positive degree, then f = L. Some examples are provided to show that the main results obtained in this paper, in a sense, are best possible.

Corrections to long wavelength approximation of peristalsis viscous fluid flow within a wavy channel

This research paper delves into the pumping phenomenon of peristaltic motion within a wavy, straight channel. Our aim is to provide analytical solutions for the well-established long wavelength and small Reynolds number approximation. To achieve this, we derive third-order corrections to the ideal leading-order solutions. These closed-form solutions encompass the axial and transverse velocity components, pressure rise and axial pressure gradients, and drag force along the wavy channel. Additionally, we elucidate the trapping phenomena of viscous flowing boluses through stream function contours. Notably, these exact solutions enable the analysis of peristalsis with finite wave numbers and moderate Reynolds numbers, eliminating the need for numerical simulations of the full Navier–Stokes equations at such physical parameters. Our simulations reveal that the pumping region is significantly influenced by short wavelength and moderate Reynolds number corrections. Further, the corrections enhance the drag force, and cell divisions within the boluses are observed at higher Reynolds numbers. The analytical derivations provided save researchers from the need for full simulations of the complex equations to acquire the key insights and contributions to capture the essence of the physical peristalsis mechanism inside wavy channels.

Derivatives and indefinite integrals of single valued neutrosophic functions

&lt;abstract&gt;&lt;p&gt;With the continuous development of the fuzzy set theory, neutrosophic set theory can better solve uncertain, incomplete and inconsistent information. As a special subset of the neutrosophic set, the single-valued neutrosophic set has a significant advantage when the value expressing the degree of membership is a set of finite discrete numbers. Therefore, in this paper, we first discuss the change values of single-valued neutrosophic numbers when treating them as variables and classifying these change values with the help of basic operations. Second, the convergence of sequences of single-valued neutrosophic numbers are proposed based on subtraction and division operations. Further, we depict the concept of single-valued neutrosophic functions (SVNF) and study in detail their derivatives and differentials. Finally, we develop the two kinds of indefinite integrals of SVNF and give the relevant examples.&lt;/p&gt;&lt;/abstract&gt;

Topography-Generated Submesoscale Coherent Vortices in the Kuroshio–Oyashio Extension Region from High-Resolution Simulations

Abstract A variety of submesoscale coherent vortices (SCVs) in the Kuroshio Extension region have been reported by recent observational studies, and the preliminary understanding of their properties, spatial distribution, and possible origins has progressively improved. However, due to relatively sparse in situ observations, the generation mechanisms of these SCVs and associated dynamic processes remain unclear. In this study, we use high-resolution model simulations to fill the gaps of the in situ observations in terms of the three-dimensional structures and life cycles of SCVs. Vortex detection and tracking algorithms are adopted and the characteristics of warm-core and cold-core SCVs are revealed. These vortices have finite Rossby numbers (0.25–0.4), and their horizontal structures can be well described by the Taylor vortex model in terms of the gradient wind balance. The vertical velocity field is characterized by a distinct dipole pattern with upwelling and downwelling cells at the vortex edge. It is very likely that both types of SCVs are generated along the eastern Japan coast through flow–topography interactions, and the Izu–Ogasawara Ridge and Hokkaido slope are found to be two important generation sites where topography friction produces extremely low potential vorticity. After leaving the boundary, SCVs can propagate over long distances and trap a water volume of ∼1011 m3.

Functorial Fast-Growing Hierarchies

Abstract We prove an isomorphism theorem between the canonical denotation systems for large natural numbers and large countable ordinal numbers, linking two fundamental concepts in Proof Theory. The first one is fast-growing hierarchies. These are sequences of functions on $\mathbb {N}$ obtained through processes such as the ones that yield multiplication from addition, exponentiation from multiplication, etc. and represent the canonical way of speaking about large finite numbers. The second one is ordinal collapsing functions, which represent the best-known method of describing large computable ordinals. We observe that fast-growing hierarchies can be naturally extended to functors on the categories of natural numbers and of linear orders. The isomorphism theorem asserts that the categorical extensions of binary fast-growing hierarchies to ordinals are isomorphic to denotation systems given by cardinal collapsing functions. As an application of this fact, we obtain a restatement of the subsystem $\Pi ^1_1$ - ${\mathsf {CA_0}}$ of analysis as a higher-type well-ordering principle asserting that binary fast-growing hierarchies preserve well-foundedness.

Exploring the Quantum Spectral Curve for AdS3/CFT2

Despite the rich and fruitful history of the integrability approach to string theory on the AdS3 × S3 × T4 background, it has not been possible to extract many concrete predictions from integrability, except in a strict asymptotic regime of large quantum numbers, due to the severity of wrapping effects. The situation changed radically with two independent and identical proposals for the Quantum Spectral Curve (QSC) for this system in a background of pure Ramond-Ramond flux. In other integrable superstring backgrounds there is compelling evidence that this formulation captures all wrapping effects exactly and describes the full planar spectrum. This great success motivates us to study the new proposed QSC and develop methods to extract from it concrete predictions for spectral data. The AdS3 × S3 × T4 case presents a significant novel feature and challenge compared to its higher-dimensional analogues — massless modes. It has been conjectured that these manifest themselves in a new property of this QSC: the non-quadratic nature of the branch-cut singularities of the QSC Q-functions. This feature implies new technical challenges in solving the QSC equations as compared to the well-studied case of N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 SYM. In this paper we resolve these difficulties and obtain the first ever predictions for unprotected string excitations in the planar limit with finite quantum numbers and RR flux. We explain how to extract a systematic expansion around the analogue of the weak ’t Hooft coupling limit in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 SYM and also obtain high-precision numerical results. These concrete data and others obtainable from the QSC could help to identify the so-far mysterious dual CFT.

On Horadam finite operator hybrid numbers

In this paper, we introduce and study a new hybrid number sequence with Horadam numbers and a finite operator, called Horadam finite operator hybrid numbers. We derive recurrence relation, Binet-like formula, ordinary generating function, exponential generating function, Poisson generating function, and summation formula for Horadam finite operator hybrid numbers. Moreover, we give matrix representation and Cassini's identities for these numbers.

Three-Dimensional Simulation of a High-Velocity Body Motion in a Tube with Rarefied Gas

Flow around a body moving at a high subsonic velocity in a tube filled with rarefied gas is studied. This aerodynamic problem is considered as applied to the task of designing a high-speed vacuum transport at finite Knudsen numbers. Parameters that are close to target characteristics of such systems are chosen, more precisely, speed of about 1000 km/h, significant transverse size of the body, and nitrogen–oxygen mixture (air) as the filling gas are chosen. The problem was solved in a three-dimensional statement.

A Compositional Dissipativity Approach for Data-Driven Safety Verification of Large-Scale Dynamical Systems

This work is concerned with a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">compositional data-driven approach</i> for formal safety verification of large-scale continuous-time dynamical systems with unknown models. The proposed framework enjoys the interconnection matrix and joint dissipativity-type properties of subsystems, described by the notion of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">stochastic storage certificates</i> . In the first part of the paper, we cast the required conditions for constructing storage certificates as a robust optimization program (ROP). Since the proposed ROP is not tractable due to the unknown model appearing in one of its constraints, we propose a scenario optimization program (SOP) corresponding to the original ROP by collecting finite numbers of data from trajectories of each subsystem. By establishing a probabilistic relation between the optimal value of SOP and that of ROP, we construct a storage certificate for each unknown subsystem based on the number of data and a required level of confidence. We accordingly propose a compositional technique based on <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">dissipativity reasoning</i> to construct <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">stochastic barrier certificates</i> of interconnected systems based on storage certificates of individual subsystems. By leveraging the acquired barrier certificate, we quantify a lower bound on the probability that an interconnected system never reaches a certain unsafe region in finite time horizons with an a-priori guaranteed confidence. We also propose an auxiliary compositional approach without requiring any compositionality condition but at the cost of providing a potentially conservative safety guarantee. In the second part of the paper, we propose our approaches for <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">deterministic</i> continuous-time systems with unknown dynamics. We verify our results over an unknown room temperature network containing 100 rooms.

On a result of Bao-Qin Li concerning Dirichlet series and shared values

Li (2011) proved that if two L-functions \(L_1\) and \(L_2\) in the extended Selberg class \({\mathcal{S}}^{\sharp}\) satisfy the same functional equation with \(a(1)=1\) and \(L_1^{-1}(c_{j})=L_2^{-1}(c_{j})\) with \(j\in\{1,2\}\) for two distinct finite complex numbers \(c_{1}\) and \(c_{2}\), then \(L_{1}=L_{2}\). Later on, Gonek--Haan--Ki (2014) proved that if two L-functions \(L_1\) and \(L_2\) in the extended Selberg class \({\mathcal {S}}^{\sharp}\) have the positive degrees and \(L_1^{-1}(c)=L_2^{-1}(c)\) for a finite non-zero complex number \(c\), then \(L_1=L_2\). This implies that if two L-functions \(L_1\) and \(L_2\) in the extended Selberg class \({\mathcal {S}}^{\sharp}\) have the positive degrees and \(L_1^{-1}(c_j)=L_2^{-1}(c_j)\) with \(j\in\{1,2\}\) for two distinct finite complex numbers \(c_1\) and \(c_2\), then \(L_1=L_2\). In this paper, we prove that if two L-functions \(L_1\) and \(L_2\) in the extended Selberg class \({\mathcal{S}}^{\sharp}\) have the zero degrees and satisfy \(L_1^{-1}(c_{j})=L_2^{-1}(c_{j})\) with \(j\in\{1,2\}\) for two distinct finite complex numbers \(c_{1}\) and \(c_{2}\), and if \(a_1(1)=a_2(1)\) or \(\lim_{r\rightarrow+\infty}\frac{T(r,L_2)}{T(r,L_1)}=1\), then \(L_{1}= L_{2}\). The main results obtained in this paper improve Theorem 1 from Li (2011) when the L-functions in the extended Selberg class \({\mathcal {S}}^{\sharp}\) have the zero degrees. Some examples are provided to show that the results obtained in this paper, in a sense, are best possible.

WKB analysis of the bifurcation for a three-dimensional neo-Hookean cylindrical tube under restricted compression

We study the bifurcation of a three-dimensional neo-Hookean elastic cylindrical tube under axial compression, where the movement of the outer surface is restricted. For the first time, the WKB method is applied to the three-dimensional eigenvalue problem and the bifurcation conditions are calculated for thick and thin-walled cylinders, separately. In this paper, two WKB expansions are considered for the two cases (i.e. for large axial or circumferential mode number) and the asymptotic expansion of the eigenvalue is obtained for each case separately. It is found that the changes of the axial stretch relative to the thickness of the cylinder have a boundary layer structure. The dependency of the axial stretch on the mode numbers, length and the wall thickness is assessed. The critical stretches occur for the finite critical mode numbers and are an decreasing function of the thickness of the tube, so thick cylinders are easier to buckle. Also, transitions between axial and circumferential modes occur in three points for mode numbers of 10,20 and at a point for mode number of zero. Using the compound matrix method to verify the analytical results, the comparison of outcomes of the numerical and analytical data shows a good agreement.

On Helly numbers of exponential lattices

Given a set S⊆R2, define the Helly number ofS, denoted by H(S), as the smallest positive integer N, if it exists, for which the following statement is true: for any finite family F of convex sets in R2 such that the intersection of any N or fewer members of F contains at least one point of S, there is a point of S common to all members of F.We prove that the Helly numbers of exponential lattices{αn:n∈N0}2 are finite for every α>1 and we determine their exact values in some instances. In particular, we obtain H({2n:n∈N0}2)=5, solving a problem posed by Dillon (2021).For real numbers α,β>1, we also fully characterize exponential lattices L(α,β)={αn:n∈N0}×{βn:n∈N0} with finite Helly numbers by showing that H(L(α,β)) is finite if and only if logα(β) is rational.

Finite Froude and Rossby numbers counter-rotating vortex pairs

We investigate the nonlinear evolution of pairs of three-dimensional, equal-sized and opposite-signed vortices at finite Froude and Rossby numbers. The two vortices may be offset in the vertical direction. The initial conditions stem from relative equilibria obtained numerically in the quasi-geostrophic regime, for vanishing Froude and Rossby numbers. We first address the linear stability of the quasi-geostrophic opposite-signed pairs of vortices, and show that for all vertical offsets, the vortices are sensitive to an instability when close enough together. In the nonlinear regime, the instability may lead to the partial destruction of the vortices. We then address the nonlinear interaction of the vortices for various values of the Rossby number. We show that as the Rossby number increases, destructive interactions, where the vortices break into pieces, may occur for a larger separation between the vortices, compared to the quasi-geostrophic case. We also show that for well-separated vortices, the interaction is non-destructive, and ageostrophic effects lead to the deviation of the trajectory of the pair of vortices, as the anticyclonic vortex dominates the interaction. Finally, we show that the flow remains remarkably close to a balanced state, emitting only waves containing negligible energy, even when the interaction leads to the destruction of the vortices.

Precise description of medium-induced emissions

We study jet fragmentation via final-state parton splittings in the medium. These processes are usually calculated theoretically by invoking the large-Nc limit. In this paper we perform the first computation of a 1 → 2 parton splitting in a thermal medium at finite numbers of colors Nc, for arbitrary momentum-sharing fraction z and with full transverse dynamics. We show how the problem can be transformed into a system of coupled Schrödinger equations, that we solve numerically. The novel numerical results are used to estimate the accuracy of several widely used approximations. We check the error introduced while going from finite Nc (i.e. Nc = 3) to the large-Nc limit, which we find to be small. For unbalanced splittings, e.g. when z → 0, only one of the partons is affected by transverse momentum exchanges with the medium. The emission process then separates into a term responsible for the 1 → 2 splitting and the subsequent independent broadening of the daughter partons. This is also referred to as the factorizable term. For finite z, further contributions arise that are responsible for the coherent color dynamics of the two-parton system, and these are referred to as non-factorizable terms. These were argued to be small for soft (unbalanced) splittings and for large media. In this work we therefore determine the accuracy of keeping only the factorizable term of the large-Nc solution. We find that the error is insignificant at a small splitting fraction z ~ 0, but can be sizable in a more balanced splitting with z ~ 0.5. Finally, we also examine the eikonal approximation, which amounts to approximating the partons’ paths through the medium as straight lines. We find that it is associated with a substantial error for the parameter values we explored in this work.