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This article makes a case for introducing moving averages into introductory statistics courses and contemporary modeling/data-based courses in college algebra and precalculus. The authors examine a variety of aspects of moving averages and draw parallels between them and similar topics in calculus, differential equations, and linear algebra. The article: (1) shows how to calculate the moving averages for a set of data; (2) discusses the effect of the length of the moving average cycle on the sequence of values generated; (3) discusses the mathematical properties of the moving average as a mathematical function/operator; (4) addresses the problem of how to retrieve the underlying sequence of data values from the known sequence of values of the moving averages; and (5) discusses the significance of the moving averages in the sense of smoothing out the original data to better identify the pattern in that data. [ABSTRACT FROM AUTHOR]

Developing stability analysis methods for modern dynamical system solutions has been a significant challenge in the field. This study aims to formulate a qualitative analysis approach for the monotone stability region of a specific solution to a single differential equation within a dynamical system. The system in question comprises two first-order nonlinear ordinary differential equations of a particular kind. The method proposed hinges on applying elements of combinatorics to the traditional mathematical investigation of a function with a single independent variable. This approach enables the exact determination of the different qualitative scenarios in which the desired solution changes, under the assumption that the function values monotonically diminish from a specified value down to a finite zero. This paper outlines the creation and decomposition of the monotone stability region associated with the solution under consideration. [ABSTRACT FROM AUTHOR]

We consider a liposome filled with the aqueous solution of an osmotic solute enclosed in a large water bath. Due to osmotic pressure, it works according to a sequence of cycles with two stages. In the first stage, the lipid vesicle with a certain input concentration of the osmotic substance is swelling up in an osmotic process until it reaches a size, when one pore is formed on its surface. The second stage follows, where the vesicle content is partially eliminated through the pore to induce a decrease in the vesicle volume up to its original size. If we take into account that the pore has two stages in its evolution, then the relaxing stage may be divided into two new stages. The swelling of the liposome is described by a differential equation. All the processes which contribute to the vesicle relaxing and its return to the initial size are described by three differential equations. [ABSTRACT FROM AUTHOR]

The aim of this paper is to find the exact solutions of the Schrödinger equation for the Manning–Rosen plus Graz separable potential through two different approaches to the problem. We express the irregular/Jost and physical solutions in terms of the special functions of mathematical physics. Numerical results of the phase shifts are obtained by utilising the properties of the Jost function and Fredholm determinants for nucleon–nucleon and nucleon–nucleus systems. The results obtained are in good agreement with earlier works. [ABSTRACT FROM AUTHOR]

The basic principles and approaches to mathematical modeling of conditioned reflexes in a human or animal on the basis of the developed mathematical apparatus of differential equations with a delayed argument are proposed and analyzed. This model clearly takes into account the slow interaction of a group of interconnected neurons and illustrates the dependence of the decision-making speed in real typical situations on the magnitude of the delay in the passage of a neuro signal in the CNS. The mechanism of generating a number of the so-called multilink conditioned reflexes is shown. [ABSTRACT FROM AUTHOR]

ferential equations in physics, chemistry, and engineering. Its two power series solutions are the Kummer function, M(a,b,z) often referred to as the confluent hypergeometric function of the first kind, and where a and b are parameters that appear in the differential equation. A third function, the Tricomi function, M(a, b, z), sometimes referred to as the confluent hypergeometric function of the second kind, is also a solution of the confluent hypergeometric equation that is routinely used. Contrary to common procedure, all three of these functions (and more) must be considered in a search for the two linearly independentsolutionsoftheconfluenthypergeometricequation. Therearesituations,when a and a-b are integers,where one of these functions is not defined, or two of the functions are not linearly independent, or one of the linearly independents ol utions of the differentia lequationis different from these three functions. Many of these special cases correspond precisely to cases needed to solve problems in physics. This leads to significant confusion about how to work with confluent hyper geometric equations, inspite of authoritative references such as the NIST Digital Library of Mathematical Functions. Here, we carefully describe all of the different cases one has to consider and what the explicit formulas are for the two linearly independent solutions of the confluent hypergeometric equation. The procedure to properly solve the confluent hypergeometric equation is summarized in a convenient table. As an example, we use these solutions to study the bound states of the hydrogenic atom, correcting the standard treatment in textbooks. We also briefly consider the cutoff Coulomb potential. We hope that this guide will aid physicists to properly solve problems that involve the confluent hypergeometric differential equation. [ABSTRACT FROM AUTHOR]

Analytical expression for the Hulthén off-shell T -matrix in all partial waves is constructed in its maximal reduced form. As a basic requirement the off-shell solutions and Jost function are first developed by following the differential equation approach to the problem together with judicious exploitation of the properties of certain special functions of mathematical physics. The off-shell T -matrix is found to be in order with respect to its limiting behaviours and off-shell unitarity. [ABSTRACT FROM AUTHOR]

Modeling biochemical reactions by means of differential equations often results in systems with a large number of variables and parameters. As this might complicate the interpretation and generalization of the obtained results, it is often desirable to reduce the complexity of the model. One way to accomplish this is by replacing the detailed reaction mechanisms of certain modules in the model by a mathematical expression that qualitatively describes the dynamical behavior of these modules. Such an approach has been widely adopted for ultrasensitive responses, for which underlying reaction mechanisms are often replaced by a single Hill function. Also time delays are usually accounted for by using an explicit delay in delay differential equations. In contrast, however, S-shaped response curves, which by definition have multiple output values for certain input values and are often encountered in bistable systems, are not easily modeled in such an explicit way. Here, we extend the classical Hill function into a mathematical expression that can be used to describe both ultrasensitive and S-shaped responses. We show how three ubiquitous modules (ultrasensitive responses, S-shaped responses and time delays) can be combined in different configurations and explore the dynamics of these systems. As an example, we apply our strategy to set up a model of the cell cycle consisting of multiple bistable switches, which can incorporate events such as DNA damage and coupling to the circadian clock in a phenomenological way. Author summary: Bistability plays an important role in many biochemical processes and typically emerges from complex interaction patterns such as positive and double negative feedback loops. Here, we propose to theoretically study the effect of bistability in a larger interaction network. We explicitly incorporate a functional expression describing an S-shaped input-output curve in the model equations, without the need for considering the underlying biochemical events. This expression can be converted into a functional module for an ultrasensitive response, and a time delay is easily included as well. Exploiting the fact that several of these modules can easily be combined in larger networks, we construct a cell cycle model consisting of multiple bistable switches and show how this approach can account for a number of known properties of the cell cycle. [ABSTRACT FROM AUTHOR]

Researchers often use theoretical models which provide a relatively simple, yet concise and effective way of modelling various phenomena. However, it is a well-known fact that the more complex the model, the more complex the mathematical description is. For this reason, theoretical models generally avoid large complexity and aim for the simplest possible definition, which although makes models mathematically more manageable, in practice it also often leads to sub-optimal performance. Furthermore, the data collected during the observations usually contain confounding factors, for which a simple theoretical model can not be prepared. Overall, mathematical models are usually too rigid and sophisticated, and therefore cannot really deal with sudden changes in the environment. The application of artificial intelligence, however, provides a good opportunity to develop complex models that can combine the basic capabilities of the theoretical models with the ability to learn more complex relationships. It has been shown [16] that with neural networks, we can build such models that can approximate mathematical functions. Trained artificial neural networks are thus able to behave like theoretical models developed for different fields, while still retaining their overall flexibility, which guarantees an overall better performance in a complex real-world environment. The aim of our study is to show our notion that we can create an architecture using neural networks, which is able to approximate a given theoretical model, and then further improve it with the help of real data to suit the real world and its various aspects better. In order to validate the functionality of the architecture developed by us, we have selected a simple theoretical model, namely the Kermack-McKendrick one [4] as the base of our research. This is an SIR [2] model, which is a relatively simple compartmental epidemic model, based on differential equations that can be used well for infections that spread very similarly to influenza or COVID. However, on one hand, the SIR model relies too heavily on its parameters, with slight changes in them leading to drastic overall changes of the S, I and R curves, and on the other hand, the simplicity of the SIR model distorts its accuracy in many cases. In our paper, by using the SIR model, we will show that the architecture described above can be a valid approach to modeling the spread of a given disease (such as influenza or COVID-19). To this end, we detail the accuracy of our models with different settings and configurations and show that it performs better than both a simple mathematical model and a plain neural network with randomly initialized weights. [ABSTRACT FROM AUTHOR]

We prove that among all Kollár components obtained by plt blow ups of a klt singularity 0 ∈ (X, D), there is at most one that is (log-)K-semistable. We achieve this by showing that if such a Kollár component exists, it uniquely minimizes the normalized volume function introduced in [Li18] among all divisorial valuations. Conversely, we show that any divisorial minimizer of the normalized volume function yields a K-semistable Kollar component. We also prove that for any klt singularity, the infimum of the normalized volume function is always approximated by the normalized volumes of Kollar components. [ABSTRACT FROM AUTHOR]

In this paper, we study differential equations arising from the generating functions of the r-Bell polynomials. We give explicit identities for the r-Bell polynomials. [ABSTRACT FROM AUTHOR]

The subject of fractional calculus (that is, the calculus of integrals and derivatives of any arbitrary real or complex order) has gained considerable popularity and importance during the past over four decades, due mainly to its demonstrated applications in numerous seemingly diverse and widespread fields of mathematical, physical, engineering and statistical sciences. Various operators of fractional-order derivatives as well as fractional-order integrals do indeed provide several potentially useful tools for solving differential and integral equations, and various other problems involving special functions of mathematical physics as well as their extensions and generalizations in one and more variables. The main object of this survey-cum-expository article is to present a brief elementary and introductory overview of the theory of the integral and derivative operators of fractional calculus and their applications especially in developing solutions of certain interesting families of ordinary and partial fractional "differintegral" equations. This general talk will be presented as simply as possible keeping the likelihood of non-specialist audience in mind. [ABSTRACT FROM AUTHOR]

The modified Cauchy problem is considered for systems of linear impulsive differential equations with singularities. The singularity is considered in the sense that the matrix-and vector-functions corresponding to the impulsive system are generally not integrable at the initial point. The sufficient conditions are established for the unique solvability of the problem. [ABSTRACT FROM AUTHOR]

In this paper we introduce the classes Tn (p,λ,A,B) and Kn (p,λ,µ,m,A,B) and derive distortion inequalities of the functions belonging to class Kn (p,λ,µ,m,A,B). Further we apply to the (n,δ)- neighborhoods of functios in the class Kn (p,λ,µ,m,A,B). [ABSTRACT FROM AUTHOR]

The object of the present paper is to obtain subordination, superordination and sandwich-type results related to a certain family of integral operators defined on the space of multivalent functions in the open unit disk. Also we point out relevant connections of the results presented here with those obtained in earlier. [ABSTRACT FROM AUTHOR]

To enrich any model and its dynamics introduction of delay is useful, that models a precise description of real-life phenomena. Differential equations in which current time derivatives count on the solution and its derivatives at a prior time are known as delay differential equations (DDEs). In this study, we are introducing new techniques for finding the numerical solution of fractional delay differential equations (FDDEs) based on the application of neural minimization (NM) by utilizing Chebyshev simulated annealing neural network (ChSANN) and Legendre simulated annealing neural network (LSANN). The main purpose of using Chebyshev and Legendre polynomials, along with simulated annealing (SA), is to reduce mean square error (MSE) that leads to more accurate numerical approximations. This study provides the application of ChSANN and LSANN for solving DDEs and FDDEs. Proposed schemes can be effortlessly executed by using Mathematica or MATLAB software to get explicit solutions. Computational outcomes are depicted, for various numerical experiments, numerically and graphically with error analysis to demonstrate the accuracy and efficiency of the methods. [ABSTRACT FROM AUTHOR]

Anchorage of steel bars is critical in reinforced concrete design to ensure composite action between concrete and rebars. The force–slip relationship of rebars is another factor governing the failure mode and rotation of structural joints. These two aspects can be predicted by bond–slip models and depend on bond behaviour at the reinforcement–concrete interface. However, current bond–slip models show either low accuracy and limited applications due to simplified assumptions or complicated computation due to nested iteration loops. This paper proposes analytical models derived from control field equations, which are characterised by low computational cost, high accuracy and broad application. The control field equations describe force equilibrium of embedded rebars by second-order differential equations and give accurate predictions by incorporating the local bond stress–slip relationship. The proposed models are appropriate for adequate and inadequate embedment lengths of rebars and can describe the evolution of the bond–slip distribution along embedment length. Closed-form solutions can be obtained for perfect anchorage and obviate iterative nested computations. If closed-form solutions are unavailable, owing to free-end slip for short embedment length, a mathematical function in Matlab is used to solve non-linear control field equations and complicated nested iteration is avoided. Finally, the proposed models show good agreement with published experimental results. [ABSTRACT FROM AUTHOR]

In this paper, we consider three types of functions given by sums of products of higher-order Euler functions and derive their Fourier series expansions. Moreover, we express each of them in terms of Bernoulli functions. [ABSTRACT FROM AUTHOR]

A new expression for the Hulthén off-shell Jost function in all partial waves is constructed in its maximal reduced form. As a basic requirement the on-shell solutions are first developed by following the differential equation approach to the problem together with judicious exploitation of the properties of certain special functions of mathematical physics. Utilizing the properties of the Jost function, the binding energies and phase shifts for N-N and n-d systems are computed and found excellent agreement with standard data. [ABSTRACT FROM AUTHOR]

The theory and implementation of approximate coupled-cluster (CC), in particular approximate CC singles, doubles, triples, and quadruples methods, are discussed for general single-determinant reference functions. While the extension of iterative approximate models to the non-Hartree–Fock case is straightforward, the generalization of perturbative approaches is not trivial. In contrast to the corresponding perturbative triples methods, there are additional terms required for non-Hartree–Fock reference functions, and there are several possibilities to derive approximations to these terms. As it turns out impossible to develop an approach that is consistent with the canonical Hartree–Fock-based theory, several new approximations have been implemented and their performance for total energies and heats of formation has been assessed. The numerical results show that the performance of the methods does not depend strongly on the approximations assumed. Furthermore, the new perturbative quadruples methods, when applied to canonical Hartree–Fock reference functions, outperform at least for absolute energies the existing ones without increasing the computational costs. [ABSTRACT FROM AUTHOR]

Different procedures to obtain atom condensed Fukui functions are described. It is shown how the resulting values may differ depending on the exact approach to atom condensed Fukui functions. The condensed Fukui function can be computed using either the fragment of molecular response approach or the response of molecular fragment approach. The two approaches are nonequivalent; only the latter approach corresponds in general with a population difference expression. The Mulliken approach does not depend on the approach taken but has some computational drawbacks. The different resulting expressions are tested for a wide set of molecules. In practice one must make seemingly arbitrary choices about how to compute condensed Fukui functions, which suggests questioning the role of these indicators in conceptual density-functional theory. [ABSTRACT FROM AUTHOR]

To assess the limits of single-reference coupled-cluster (CC) methods for potential-energy surfaces, several methods have been considered for the inclusion of connected quadruple excitations. Most are based upon the factorized inclusion of the connected quadruple contribution (Qf) [J. Chem. Phys. 108, 9221 (1998)]. We compare the methods for the treatment of potential-energy curves for small molecules. These include CCSD(TQf), where the initial contributions of triple (T) and factorized quadruple excitations are added to coupled-cluster singles (S) and doubles (D), its generalization to CCSD(TQf), where instead of measuring their first contribution from orders in H, it is measured from orders in H=e-(T1+T2)He(T1+T2); renormalized approximations of both, and CCSD(2) defined in [J. Chem. Phys. 115, 2014 (2001)]. We also consider CCSDT, CCSDT(Qf), CCSDTQ, and CCSDTQP for comparison, where T, Q, and P indicate full triple, quadruple, and pentuple excitations, respectively. Illustrations for F2, the double bond breaking in water, and N2 are shown, including effects of quadruples on equilibrium geometries and vibrational frequencies. Despite the fact that no perturbative approximation, as opposed to an iterative approximation, should be able to separate a molecule correctly for a restricted-Hartree–Fock reference function, some of these higher-order approximations have a role to play in developing new, more robust procedures. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of analytic functions associated with Mittag-Leffler fuction in the open unit disk. Several properties of functions belonging to this class are derived. [ABSTRACT FROM AUTHOR]

In our present investigation, we obtain several differential subordination results involving leaf-like domains. Moreover, certain sharp coefficient estimates are investigated when the class of functions lies in leaf-like domains. [ABSTRACT FROM AUTHOR]

In this paper, we study differential equations arising from the generating functions of generalized degenerate tangent polynomials. We give explicit identities for the generalized degenerate tangent polynomials arising from differential equations. [ABSTRACT FROM AUTHOR]

Starting from a second-order Fuchsian differential equation having five regular singular points, an equation obeyed by a function proportional to the first derivative of the solution of the Heun equation, we construct several expansions of the solutions of the general Heun equation in terms of Appell generalized hypergeometric functions of two variables of the first kind. Several cases when the expansions reduce to those written in terms of simpler mathematical functions such as the incomplete Beta function or the Gauss hypergeometric function are identified. The conditions for deriving finite-sum solutions via termination of the series are discussed. In general, the coefficients of the expansions obey four-term recurrence relations; however, there exist certain choices of parameters for which the recurrence relations involve only two terms, though not necessarily successive. For such cases, the coefficients of the expansions are explicitly calculated and the general solution of the Heun equation is constructed in terms of the Gauss hypergeometric functions. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we introduce a piecewise reproducing kernel method for impulsive delay differential equations with piecewise constant arguments. The method is an improved reproducing kernel method. Compared with the classical reproducing kernel method, the solutions obtained using the present method can give good approximations for a larger time interval. Some numerical examples are used to show the effectiveness and simplicity of the method. [ABSTRACT FROM AUTHOR]

Abstract We present a new smoothness result for Caputo-type fractional ordinary differential equations, which reveals that, subtracting a non-smooth function that can be obtained by the information available, a non-smooth solution belongs to Cm for some positive integer m. [ABSTRACT FROM AUTHOR]

We are concerned with the zeros of the Macdonald functions or the modified Bessel functions of the second kind with real index. By using the explicit expressions for the algebraic equations satisfied by the zeros, we describe the behavior of the zeros when the index moves. Results by numerical computations are also presented. [ABSTRACT FROM AUTHOR]

Under the assumption of prox-regularity and the presence of a tilt stable local minimum we are able to show that a VU like decomposition gives rise to the existence of a smooth manifold on which the function in question coincides locally with a smooth function. [ABSTRACT FROM AUTHOR]

Abstract Compared to G k continuity, C k continuity simplifies the construction of functions on surfaces and their refinement, e.g. to solve differential equations on the surface. The new class of free-form surfaces introduced in this paper is almost everywhere parametrically C 2. In particular, a new bi-6 construction combines a fast-contracting C 2 guided subdivision surface with a tiny multi-sided G 1 cap. The cap is chosen to be smaller than any refinement anticipated for geometric modeling or computing on surfaces. Fast contraction means that one subdivision step shrinks the remaining hole more than three steps of Catmull–Clark subdivision. This yields smooth surfaces consisting of a finite number of pieces that are suitable for engineering practice. Both the subdivision construction and the cap are guided by a reference surface that has a different structure and lower smoothness, but conveys the basic shape of a fair surface. [ABSTRACT FROM AUTHOR]

In this paper, we first generalize Gerstewitz's functions from a single positive vector to a subset of the positive cone. Then, we establish a partial order principle, which is indeed a variant of the pre-order principle [Qiu, J. H.: A pre-order principle and set-valued Ekeland variational principle. J. Math. Anal. Appl., 419, 904-937 (2014)]. By using the generalized Gerstewitz's functions and the partial order principle, we obtain a vector EVP for ε-efficient solutions in the sense of Németh, which essentially improves the earlier results by completely removing a usual assumption for boundedness of the objective function. From this, we also deduce several special vector EVPs, which improve and generalize the related known results. [ABSTRACT FROM AUTHOR]

In this article, we investigate the arithmetic behavior of the function D3(n) which counts the number of 3-regular tripartitions of n. For example, we show that for α ≥ 1 and n ≥ 0, D3(32αn+11⋅32α−1−14)≡0(mod32α+3) and D3(32αn+7⋅32α−1−14)≡0(mod32α+2). [ABSTRACT FROM AUTHOR]

In this paper first, we prove some new generalizations of Hermite-Hadamard type inequalities for the convex function f and for (s, m)-convex function f in the second sense in conformable fractional integral forms. Second, by using five new integral identities, we present some new Riemann-Liouville fractional trapezoid and midpoint type inequalities. Third, using these results, we present applications to f-divergence measures. At the end, some new bounds for special means of different positive real numbers and new error estimates for the trapezoidal and midpoint formula are provided as well. These results give us the generalizations of the earlier results. [ABSTRACT FROM AUTHOR]

In this paper, we prove the Hyers-Ulam stability of the following additive functional equations f (x+y/2 + z + w) = 1/2 f(x) + 1/2 f(y) + f(z) + f(w), f (x+y+z/3 + w) = 1/3 f(x) + 1/3 f(y) + 1/3 f(z) + f(w) in paranormed spaces. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce and investigate two new subclasses of the function class Σ of bi-univalent functions of complex order defined in the open unit disk, which are associated with the one of the orthogonal polynomial namely Generalized Meixner-Pollaczek polynomials, and satisfying subordinate conditions. Taylor-MacLaurin coefficients |α2| and |α3| were estimated for functions in new subclass. Furthermore, several known consequences are also investigated. [ABSTRACT FROM AUTHOR]

In this paper, in order to solve a class of nonlinear bilevel programming problems, we equivalently transform the nonlinear bilevel programming problems into corresponding single level nonlinear programming problems by using the Karush-Kuhn-Tucker optimality condition. Then, based on penalty function theory, we construct a smooth approximation method for obtaining optimal solutions of classic l1-exact penalty function optimality problems, which is equivalent to the single level nonlinear programming problems. Furthermore, using ∈-approximate optimal solution theory, we prove convergence of a simple ∈-approximate optimal algorithm. Finally, through adding parameters in the constraint set of objective function, we prove some perturbation convergence results for solving the nonlinear bilevel programming problems. [ABSTRACT FROM AUTHOR]

In this paper, we consider three types of functions related to higher-order Genocchi functions and derive their Fourier series expansions. In addition, we express each of them in terms of Bernoulli functions. [ABSTRACT FROM AUTHOR]

In this paper, the boundedness and compactness of the differences of generalized integration operators from the space of Cauchy integral transforms to the Bloch-type spaces and the weighted Dirichlet spaces are investigated. [ABSTRACT FROM AUTHOR]

For PM functions of height 1, the existence of continuous iterative roots of any order was obtained under the characteristic endpoints condition. A natural open question about iterative roots without that condition was raised. This question was answered partially in the case that the function is increasing on its characteristic interval. In this paper, to the opposite, we consider the decreasing case and give the existence and nonexistence results for their iterative roots. [ABSTRACT FROM AUTHOR]

Abstract In this paper we obtain a very general Gauss–Green formula for weakly differentiable functions and sets of finite perimeter. This result is obtained by revisiting Anzellotti's pairing theory and by characterizing the measure pairing (A , D u) when A is a bounded divergence measure vector field and u is a bounded function of bounded variation. [ABSTRACT FROM AUTHOR]

Abstract We show that if A is a finite set of non-negative integers then the number of zeros of the function f A (θ) = ∑ a ∈ A cos ⁡ (a θ) , in [ 0 , 2 π ] , is at least (log ⁡ log ⁡ log ⁡ | A |) 1 / 2 − ε. This gives the first unconditional lower bound on a problem of Littlewood and solves a conjecture of Borwein, Erdélyi, Ferguson and Lockhart. We also prove a result that applies to more general cosine polynomials with few distinct rational coefficients. One of the main ingredients in the proof is perhaps of independent interest: we show that if f is an exponential polynomial with few distinct integer coefficients and f "correlates" with a low-degree exponential polynomial P , then f has a very particular structure. This result allows us to prove a structure theorem for trigonometric polynomials with few zeros in [ 0 , 2 π ] and restricted coefficients. [ABSTRACT FROM AUTHOR]

Abstract Let T f : [ 0 , 1 ] → [ 0 , 1 ] be an expanding Lorenz map, this means T f x : = f (x) (mod 1) where f : [ 0 , 1 ] → [ 0 , 2 ] is a strictly increasing map satisfying inf ⁡ f ′ > 1. Then T f has two pieces of monotonicity. In this paper, sufficient conditions when T f is topologically mixing are provided. For the special case f (x) = β x + α with β ≥ 2 3 a full characterization of parameters (β , α) leading to mixing is given. Furthermore relations between renormalizability and T f being locally eventually onto are considered, and some gaps in classical results on the dynamics of Lorenz maps are corrected. [ABSTRACT FROM AUTHOR]

In the present paper, we find the general solution of the inhomogeneous Euler-Cauchy dynamic equation ... on the time scale with the constant graininess function and the linear variable graininess function, respectively. And then, we study the Ulam stability problem of the forgoing equation on different types of time scales. Our results can be viewed as a unfication and extension of the results of Mortici et al. [C. Mortici, T.M. Rassias, S.M. Jung, The inhomogeneous Euler equation and its Hyers-Ulam stability, Appl. Math. Lett. 40 (2015) 23-28]. [ABSTRACT FROM AUTHOR]

In this paper we derive the famous Fekete-Szegö inequality for the class of p-valent non-bazilevič function using differential operator. [ABSTRACT FROM AUTHOR]

We present a novel general approach for solving approximation problems on hypersurfaces drawing on standard approximation techniques in the ambient space. The method is capable of transferring approximation orders from the ambient space to the hypersurface under mild conditions. When based on tensor product B-splines of order n and knot spacing h, the resulting approximant is Cn-2, while the error decays at the optimal rate hn. Further, the method is easy to implement and, when applying quasi interpolation techniques, optimally efficient as the computational expense is linear in the number of spline coefficients. Applications include the representation of smooth surfaces of arbitrary genus. [ABSTRACT FROM AUTHOR]

We establish the unique solvability of a coupling problem for entire functions that arises in inverse spectral theory for singular second-order ordinary differential equations/two-dimensional first-order systems and is also of relevance for the integration of certain nonlinear wave equations. [ABSTRACT FROM AUTHOR]

We study upper and lower estimates as well as the asymptotic behavior of the sharp constant C = Cn(U, V) in the doubling-type condition at the origin 1/|V| ∫V f(x) dx ≤ C 1/|U| ∫U f(x) dx, where U, V ⊂ Rn are 0-symmetric convex bodies and f is a non-negative positive definite function. [ABSTRACT FROM AUTHOR]

It is known that the multiple sine function for a "rational" period satisfies an algebraic differential equation. However, for a non-"rational" period, the differential algebraicity of the multiple sine function is obscure. In this paper, we prove that, if there exists a non-real element in the set {ωj/ωi|1 ≤ i < j ≤ r}, the multiple sine function Sinr; (x,(ω1,...,ωr)) does not satisfy any algebraic differential equation. [ABSTRACT FROM AUTHOR]