Type Formula Research Articles

The Basso-Dixon formula and Calabi-Yau geometry

We analyse the family of Calabi-Yau varieties attached to four-point fishnet integrals in two dimensions. We find that the Picard-Fuchs operators for fishnet integrals are exterior powers of the Picard-Fuchs operators for ladder integrals. This implies that the periods of the Calabi-Yau varieties for fishnet integrals can be written as determinants of periods for ladder integrals. The representation theory of the geometric monodromy group plays an important role in this context. We then show how the determinant form of the periods immediately leads to the well-known Basso-Dixon formula for four-point fishnet integrals in two dimensions. Notably, the relation to Calabi-Yau geometry implies that the volume is also expressible via a determinant formula of Basso-Dixon type. Finally, we show how the fishnet integrals can be written in terms of iterated integrals naturally attached to the Calabi-Yau varieties.

A Bourgain-Brezis-Mironescu type result for the fractional relativistic seminorm

We establish a version of the Bourgain-Brezis-Mironescu type formula on the limit as s → 1 − s\rightarrow 1^- of the fractional relativistic seminorm.

Quadrature formulas on combinatorial graphs

The goal of the paper is to establish quadrature formulas on combinatorial graphs. Three types of quadrature formulas are developed. Quadrature formulas of the first type are obtained through interpolation by variational splines. This set of formulas is exact on spaces of variational splines on graphs. Since bandlimited functions can be obtained as limits of variational splines we obtain quadrature formulas which are approximately exact on spaces of bandlimited functions. Accuracy of this type of quadrature formulas is given in terms of geometry of the set of nodes of splines and in terms of smoothness of functions which is measured by means of the combinatorial Laplace operator. Quadrature formulas of the second type are obtained through point-wise sampling for bandlimited functions and based on existence of certain frames in appropriate subspaces of bandlimited functions. The third type quadrature formulas are based on the average sampling over subgraphs. Our quadrature formulas which are based on sampling are exact on a relevant subspaces of bandlimited functions. The results of the paper have potential applications to problems that arise in data mining.

Inversion formulas for Toeplitz-plus-Hankel matrices

The main aim of the present paper is to establish inversion formulas of Gohberg-Semencul type for Toeplitz-plus-Hankel matrices. In particular, it is shown how the inverse of such a structured matrix An of order n is computed by means of their first two and last two columns or rows under the additional assumption that a certain 2×2 matrix is nonsingular. Moreover, a formula for the inverse of the Toeplitz-plus-Hankel matrix An−2 of order n−2 in the center of An is established and sufficient invertibility criteria for both An and An−2 are obtained. Hereby a main tool is to use known inversion formulas involving not only rows or columns of the inverse but also solutions of equations the right hand sides of which depend on entries of the Toeplitz-plus-Hankel matrix itself.

A Molev-Sagan type formula for double Schubert polynomials

We give a Molev-Sagan type formula for computing the product Su(x;y)Sv(x;z) of two double Schubert polynomials in different sets of coefficient variables where the descents of u and v satisfy certain conditions that encompass Molev and Sagan's original case and conjecture positivity in the general case. Additionally, we provide a Pieri formula for multiplying an arbitrary double Schubert polynomial Su(x;y) by a factorial elementary symmetric polynomial Ep,k(x;z). Both formulas remain positive in terms of the negative roots when we set y=z, so in particular this gives a new equivariant Littlewood-Richardson rule for the Grassmannian, and more generally a positive formula for multiplying a factorial Schur polynomial sλ(x1,…,xm;y) by a double Schubert polynomial Sv(x1,…,xp;y) such that m≥p. An additional new result we present is a combinatorial proof of a conjecture of Kirillov of nonnegativity of the coefficients of skew Schubert polynomials, and we conjecture a weight-preserving bijection between a modification of certain diagrams used in our formulas and RC-graphs/pipe dreams arising in formulas for double Schubert polynomials.

Partial Fourier Transform Method for Solution Formula of Stokes Equation with Robin Boundary Condition in Half-space

The area of applied science known as fluid dynamics studied how gases and liquids moved. The motion of the fluid in the liquid and vapour phases is described by a special system of partial differential equations. The research purpose of this article investigated the solution formula of incompressible Stokes equation with the Robin boundary condition in half-space case. The solution formula for Stokes equation was calculated using the partial Fourier transform. This calculation was carried out over the Weis’s multipliers theorem. Our calculation showed that the solution formula of Stokes equation with Robin boundary condition in half-space for velocity and pressure were contained multipliers as due to work Shibata & Shimada. Due to our consideration of the half-space situation, the partial Fourier transform approach is the most appropriate one to use to get the velocity and pressure for the Stokes equation with Robin boundary condition. Furthermore, research methods in this article, in the first stage, we use the resolvent problem of the model. Secondly, we apply the partial Fourier transform to the model problem and finally, we use inverse partial Fourier transform to get the solution formula of the incompressible type of Stokes equation for velocity and pressure. This result indicates that Weis' multiplier theorem also allows us to find the local well-posedness of the model problem in addition to the maximal Lp-Lq regularity class (Gerard-Varet et al., 2020).

On Crofton’s type formulas and the solid angle of convex sets

AbstractHere we analyze three dimensional analogues of the classic Crofton formula for planar compact convex sets. In this formula a fundamental role is played by the visual angle of the convex set from an exterior point. A generalization of the visual angle to convex sets in the Euclidean space is the visual solid angle. This solid angle, being an spherically convex set in the unit sphere, has perimeter, area and other geometric quantities to be considered. The main goal of this note is to express invariant quantities of the original convex set depending on volume, surface area and mean curvature integral by means of integrals of functions related to the solid angle.

Высокоточная схема для уравнения переноса в задаче нейтронной защиты

The paper considers the problems of calculating the stationary transport of high-energy neutrons from an external source in a hollow metal structure. The spatial grid for calculation consists of tetrahedra, which allows to model complicated geometric shapes of objects. The calculated area is highly heterogeneous, since it consists of metal parts with strong absorption and scattering and practically non-absorbing internal parts. A scheme based on a minimal stencil, within a single tetrahedron for the calculation. The scheme has a high approximation order and is based on characteristic approaches. For a smooth solution and a constant absorption coefficient in each cell, the scheme has a third order of convergence. The cells are traversed using the graph topological sorting algorithm. A new version of the parallelization of the calculation for the OpenMP standard has been proposed. For discretization by an angular variable, a cubature formula of the direct product type over two angles with an algebraic order of accuracy of nine is used. The angular grid is constructed in such a way as to correctly calculate the moments for narrowly directed beams of external radiation. The standard 26-group approximation of the BNAB library is used for energy sampling. A decrease in the neutron flux inside the structure is shown for groups with an energy above 1 MeV. Graphs of the angular distribution inside the structure are given.

A study of the multipoint boundary value problem for nonlinear differential equations of arbitrary fractional order based on the fixed point theorem for composite nonlinear operators

Abstract The existence and distinctiveness of solutions, as well as the iterative approximations of the distinctive answer to the initial boundary value issue of nonlinear differential equations, may be solved with the help of the theory of nonlinear operators, which offers a strong theoretical guarantee and a fundamental tool. In this work, we examine the fixed point theory for composites nonlinear operators, which may offer a fresh approach to solving linear differential equations having arbitrary order fractional multipoint boundary value problems. The existing result of the boundary value issue’s solution is confirmed by the fixed point theorem for composites nonlinear operators, and the four-point boundary value problem for fractional-order linear equations of the Riemann-Liouville type is examined. A group of fractional-order nonlinear differential equations with deviating quantities are examined to examine the presence of an unusual positive solution for this boundary value problem. The existence of this unique positive solution is determined by the use of both the mixed monotone operator and the combined nonlinear operator fixed point theorem. Ultimately, the nonlinear Bagley-Torvik equation with a fractional order variable coefficient four-point boundary value problem is converted into a Fredholm-Hammerstein integral the formula of the second type with weakly singular or continuous kernels, and the fixed point principle is used to demonstrate the uniqueness of the solution in the space of continuous functions. Approximate solutions are provided for the second class of nonlinear Fredholm-Hammerstein integral equations with weakly singular kernels. The outcomes demonstrate the applicability and validity of the fixed point theorem to composite nonlinear operators, offering a fresh approach to solving the multipoint boundary value issue for nonlinear differential equations of any fractional order.

On The Generalized Leonardo Quaternions and Associated Spinors

In this paper, we introduce and study a new family of sequences called the generalized Leonardo spinors by defining a linear correspondence between the generalized Leonardo quaternions and spinors. We start with defining the generalized Leonardo quaternions and then present their some important properties such as Binet type formula, Catalan’s identity, d’Ocagne’s identity, series sums, etc. We give some interrelations of these quaternions with the Fibonacci and Lucas quaternions. Then, we present the generating functions, sum formulae, various well-known identities, etc. for the Leonardo spinors and show their connection with the Fibonacci and Lucas spinors.

The geometric constraints on Filippov algebroids

<abstract><p>Filippov $ n $-algebroids are introduced by Grabowski and Marmo as a natural generalization of Lie algebroids. On this note, we characterized Filippov $ n $-algebroid structures by considering certain multi-input connections, which we called Filippov connections, on the underlying vector bundle. Through this approach, we could express the $ n $-ary bracket of any Filippov $ n $-algebroid using a torsion-free type formula. Additionally, we transformed the generalized Jacobi identity of the Filippov $ n $-algebroid into the Bianchi-Filippov identity. Furthermore, in the case of rank $ n $ vector bundles, we provided a characterization of linear Nambu-Poisson structures using Filippov connections.</p></abstract>

Hermite-Hadamard, Fejér and trapezoid type inequalities using Godunova-Levin Preinvex functions via Bhunia's order and with applications to quadrature formula and random variable.

Convex and preinvex functions are two different concepts. Specifically, preinvex functions are generalizations of convex functions. We created some intriguing examples to demonstrate how these classes differ from one another. We showed that Godunova-Levin invex sets are always convex but the converse is not always true. In this note, we present a new class of preinvex functions called $ (\mathtt{h_1}, \mathtt{h_2}) $-Godunova-Levin preinvex functions, which is extensions of $ \mathtt{h} $-Godunova-Levin preinvex functions defined by Adem Kilicman. By using these notions, we initially developed Hermite-Hadamard and Fejér type results. Next, we used trapezoid type results to connect our inequality to the well-known numerical quadrature trapezoidal type formula for finding error bounds by limiting to standard order relations. Additionally, we use the probability density function to relate trapezoid type results for random variable error bounds. In addition to these developed results, several non-trivial examples have been provided as proofs.

Q-Deformed and L-parametrized hyperbolic tangent function relied complex valued multivariate trigonometric and hyperbolic neural network approximations

Here we study the multivariate quantitative approximation of complex valued continuous functions on a box of RN , N ∈ N, by the multivariate normalized type neural network operators. We investigate also the case of approximation by iterated multilayer neural network operators. These approximations are achieved by establishing multidimen-sional Jackson type inequalities involving the multivariate moduli of continuity of the en- gaged function and its partial derivatives. Our multivariate operators are defined by using a multidimensional density function induced by a q-deformed and λ-parametrized hyper-bolic tangent function, which is a sigmoid function. The approximations are pointwise and uniform. The related feed-forward neural network are with one or multi hidden layers. The basis of our theory are the introduced multivariate Taylor formulae of trigonometric and hyperbolic type.

Reliability analysis and comparison of ring-PUF based on probabilistic models

—To address the issue of maladaptive reliability theory models in delay-based physical unclonable functions (PUF) of ring classes, this paper begins by analyzing the relationship between four key transistor parameters (channel length, channel width, threshold voltage, and gate oxide thickness) and delay sensitivity. Then establishes reliability quantification formulas for various PUF types, including ring oscillator PUF, k-out-of-1 PUF, Loop PUF, transient effect ring oscillator PUF, and duty cycle PUF. This analysis and comparison are conducted using the inverter delay model and probabilistic statistical methods to ensure fairness at the theoretical level. Finally, the proposed model is validated through simulation experiments, along with further analysis and comparison of ring-PUF. The results demonstrate that delay sensitivity can be utilized to assess the extent of delay impact when process parameters fluctuate. In terms of reliability, the ring-PUF can be ranked as follows: k-out-of-1 PUF, Loop PUF, ring oscillator PUF, transient effect ring oscillator PUF, and duty cycle PUF. This paper offers extensive technical guidance and theoretical support for ring PUF designers.

Initial boundary value problem for 1D scalar balance laws with strictly convex flux

A Lax-Oleĭnik type explicit formula for 1D scalar balance laws has been recently obtained for the pure initial value problem by Adimurthi et al. in [2]. In this article, by introducing a suitable boundary functional, we establish a Lax-Oleĭnik type formula for the initial-boundary value problem. For the pure initial value problem, the solution for the corresponding Hamilton-Jacobi equation turns out to be the minimizer of a functional on the set of curves known as h-curves. In the present situation, part of the h-curve joining any two points in the quarter plane may cross the boundary x=0. This phenomenon breaks the simplicity of the minimization process through the boundary functional compared to the case of conservation laws. Moreover, this complicates the verification of the boundary condition in the sense of Bardos, le Roux, and Nédélec [4]. To verify the boundary condition, the boundary points are classified into three types depending on the structure of the minimizers at those points. Finally, by introducing characteristic triangles, we construct generalized characteristics and show that the explicit solution is entropy admissible.

The Gardner Problem on the Lock-In Range of Second-Order Type 2 Phase-Locked Loops

Phase-locked loops (PLLs) are nonlinear automatic control circuits widely used in telecommunications, computer architecture, gyroscopes, and other applications. One of the key problems of nonlinear analysis of PLL systems has been stated by Floyd M. Gardner as being “to define exactly any unique lock-in frequency.” The lock-in range concept describes the ability of PLLs to reacquire a locked state without cycle slipping and its calculation requires nonlinear analysis. The present work analyzes a second-order type 2 phase-locked loop with a sinusoidal phase detector characteristic. Using the qualitative theory of dynamical systems and classical methods of control theory, we provide stability analysis and suggest analytical lower and upper estimates of the lock-in range based on the exact lock-in range formula for a second-order type 2 PLL with a triangular phase detector characteristic, obtained earlier. Applying phase plane analysis, an asymptotic formula for the lock-in range which refines the existing formula is obtained. The analytical formulas are compared with computer simulation and engineering estimates of the lock-in range. The comparison shows that engineering estimates can lead to cycle slipping in the corresponding PLL model and cannot provide a reliable solution for the Gardner problem, whereas the lower estimate presented in this article guarantees frequency reacquisition without cycle slipping for all parameters, which provides a solution to the Gardner problem.

Numerical treatment of time-fractional sub-diffusion equation using p-fractional linear multistep methods

In this paper, a kind of the differential equation including a time-fractional sub-diffusion equation is considered. Through this memorandum, a well-known technique, in the time direction is adopted by the p-fractional linear multistep method (p-FLMM) according to the q-fractional backward difference formula (q-FBDF) of implicit type for q = 1, 2, 3, and the spatial direction is approximated by the second-order central difference method. The stability properties of the proposed method can be investigated in combination with the Fourier technique and Z -transformation and also its convergence is studied by using the truncated error maximum. It is shown that the method is unconditionally stable and the orders of convergence are O ( τ p + h 2 ) for 1 ≤ p ≤ 4 , in which p is the order of accuracy in the time direction and τ and h determine temporal and spatial stepsizes, respectively. Some numerical experiments are included to demonstrate the validity and applicability of the scheme.

Approximation by exponential-type polynomials

In this paper, a family of exponential-type polynomials is introduced and studied. Both the uniform and the Lp convergence are established in suitable function spaces. In the Lp-case, some estimates are also achieved using an exponentially weighted version of the p-norm. Further, a Voronovskaja type formula is proved, finding the exact order of pointwise approximation in case of continuous functions having second derivative at some points. Finally, quantitative estimates for the order of approximation are established in both the continuous and the Lp-cases in terms of the modulus of continuity and K-functionals of the involved functions. In the latter estimate, a crucial role is played by the so-called Hardy-Littlewood maximal function.

Impact of venting, caloric density, and formula type on flow rates from bottle nipples

PurposeTo determine whether flow rate from the bottle nipple is impacted by a venting system, caloric density of the infant formula, or by milk type of the formula (i.e., standard cow's milk, hypoallergenic cow's milk, or soy protein). MethodFlow rates were tested of six nipple types with and without a venting system. Separately, five nipple types were tested for flow rate with formula at caloric densities of 20, 22, 24, and 30 calories per ounce. Then, two nipple types were tested with six infant formulas of different milk types. ResultsFor two nipple types, flow rate was faster with the venting system compared to without. For three of the nipple types, flow rate increased as caloric density increased. There were no significant differences in flow by milk type. ConclusionsVenting systems and caloric density may impact flow rate from the bottle nipple, but not consistently across nipple types.

A new diagonal and Toeplitz splitting preconditioning method for solving time-dependent Riesz space-fractional diffusion equations

The initial–boundary value problem of the Riesz space-fractional diffusions equation is an important class of equations arising in many application fields. In this paper, we apply the Grünwald–Letnikov type formulas to discretize the time-dependent Riesz space-fractional diffusion equations, and obtain a system of linear equations from the discretization results. A new diagonal and Toeplitz splitting (NDTS) iteration method is constructed for this linear system. Based on the NDTS iteration method, an NDTSτ preconditioner is proposed and the generalized minimal residual (GMRES) method combined with this preconditioner is applied to solve the linear system. We theoretically show that the eigenvalues of the NDTSτ preconditioned matrix are clustered. Numerical experiments illustrate the efficiency of the proposed method.