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In this paper, we consider a version of energy minimisation technique applied to images of a 2D fluid flow. The two Navier–Stokes equations describe the static flow of a 2D fluid in terms of velocity field, (u, v), pressure field, p and forcing field, f. Apart from these two Navier–Stokes equations, we have the incompressibility condition to evaluate the three parameters. While implementing this system, random noise (usually non-Gaussian) creeps into the random force field $$\underline{f}(x,y)$$ f ̲ ( x , y ) . We denote this random field by $$\delta \underline{f}(x,y)$$ δ f ̲ ( x , y ) having zero mean and non-trivial second and third moments. We assume that these two moments are known except for some unknown parameters $$\underline{\theta }$$ θ ̲ (like mean, variance, co-variance, skewness, etc.) which we wish to estimate. In the proposed technique, we first calculate the approximate shift in the average fluid energy defined as a quadratic function of the velocity field. The energy method then requires that $$\underline{\theta }$$ θ ̲ should be such that this average increases in the energy due to the random forcing component be minimised. We should, however, note that the standard statistical approach to force field estimation is to calculate the velocity field as a function of the force field and then adopt the statistical moment matching technique. Such an approach assumes spatial ergodicity of the velocity field. This approach to force field estimation is more accurate from the statistical moment matching view point but works only if velocity measurements are made. The former technique of energy minimisation does not require any velocity measurements. Both of these techniques are discussed in this paper and MATLAB simulations presented.

In this paper, we show the existence of a weak solution to the Maxwell–Stokes type equation with a potential satisfying the Dirichlet condition, under the hypothesis that the domain has no holes, using a version of the de Rham lemma that was proved in our previous paper. We also give the regularity of weak solutions.

In this paper, we introduce a cyclic subgradient extragradient algorithm and its modified form for finding a solution of a system of equilibrium problems for a class of pseudomonotone and Lipschitz-type continuous bifunctions. The main idea of these algorithms originates from several previously known results for variational inequalities. The proposed algorithms are extensions of the subgradient extragradient method for variational inequalities to equilibrium problems and the hybrid (outer approximation) method. The paper can help in the design and analysis of practical algorithms and gives us a generalization of the most convex feasibility problems.

Following the work of Gaborit et al. (in: The international workshop on coding and cryptography (WCC 13), 2013) defining LRPC codes over finite fields, Renner et al. (in: IEEE international symposium on information theory, ISIT 2020, 2020) defined LRPC codes over the ring of integers modulo a prime power, inspired by the paper of Kamche and Mouaha (IEEE Trans Inf Theory 65(12):7718–7735, 2019) which explored rank metric codes over finite principal ideal rings. In this work, we successfully extend the work of Renner et al. by constructing LRPC codes over the ring $$\mathbb {Z}_{m}$$ Z m which is not a chain ring. We give a decoding algorithm and we study the failure probability of the decoder.

The purpose of this paper is to present in linear spaces some results for new notions called A-left (resp., A-right) ascent and A-left (resp., A-right) descent of linear operators (where A is a given operator) which generalize two important notions in operator theory: ascent and descent. Moreover, if A is a positive operator, we obtain several properties of ascent and descent of an operator in semi-Hilbertian spaces. Some basic properties and many results related to the ascent and descent for a linear operator on a linear space Kaashoek (Math Ann 172:105–115, 1967), Taylor (Math Ann 163:18–49, 1966) are extended to these notions. Some stability results under perturbations by compact operators and operators having some finite rank power are also given for these notions.

In this paper, we study the global convergence of successive approximations as well as the uniqueness of the random solution of a coupled random Hilfer fractional differential system. We prove a theorem on the global convergence of successive approximations to the unique solution of our problem. In the last section, we present an illustrative example.

In this paper, we study a system of nonlinear Riemann–Liouville fractional differential equations with delays. First, we define in an appropriate way initial conditions which are deeply connected with the fractional derivative used. We introduce an appropriate generalization of practical stability which we call practical stability in time. Several sufficient conditions for practical stability in time are obtained using Lyapunov functions and the modified Razumikhin technique. Two types of derivatives of Lyapunov functions are used. Some examples are given to illustrate the introduced definitions and results.

For a fixed integer $$d\ge 4$$ d ≥ 4 , the list of groups that appear as automorphism groups of smooth plane curves whose degree is d is unknown, except for $$d=4$$ d = 4 or 5. Harui showed a certain characteristic about structures of automorphism groups of smooth plane curves. Badr and Bars began to study for certain orders of automorphisms and try to obtain exact structures of automorphism groups of smooth plane curves. In this paper, based on the result of T. Harui, we extend Badr–Bars study for different and new cases, mainly for the cases of cyclic groups that appear as automorphism groups.

A repdigit is a positive integer that has only one distinct digit in its decimal expansion, i.e., a number of the form $$a(10^m-1)/9$$ a ( 10 m - 1 ) / 9 , for some $$m\ge 1$$ m ≥ 1 and $$1 \le a \le 9$$ 1 ≤ a ≤ 9 . Let $$\left( P_n\right) _{n\ge 0}$$ P n n ≥ 0 and $$\left( E_n\right) _{n\ge 0}$$ E n n ≥ 0 be the sequence of Padovan and Perrin numbers, respectively. This paper deals with repdigits that can be written as the products of consecutive Padovan or/and Perrin numbers.

In this paper, we study the optimization problem (P) of minimizing a convex function over a constraint set with nonconvex constraint functions. We do this by given new characterizations of Robinson’s constraint qualification, which reduces to the combination of generalized Slater’s condition and generalized sharpened nondegeneracy condition for nonconvex programming problems with nearly convex feasible sets at a reference point. Next, using a version of the strong CHIP, we present a constraint qualification which is necessary for optimality of the problem (P). Finally, using new characterizations of Robinson’s constraint qualification, we give necessary and sufficient conditions for optimality of the problem (P).

Let D be an integrally closed domain, $$\{V_{\alpha }\}$$ { V α } be the set of t-linked valuation overrings of D, and $$v_c$$ v c be the star operation on D defined by $$I^{v_c} = \bigcap _{\alpha } IV_{\alpha }$$ I v c = ⋂ α I V α for all nonzero fractional ideals I of D. In this paper, among other things, we prove that D is a $$v_c$$ v c -Noetherian domain if and only if D is a Krull domain, if and only if $$v_c = v$$ v c = v and every prime t-ideal of D is a maximal t-ideal. As a corollary, we have that if D is one-dimensional, then $$v_c = v$$ v c = v if and only if D is a Dedekind domain.

This paper focuses on the problem of convex constraint nonlinear equations involving monotone operators in Euclidean space. A Fletcher and Reeves type derivative-free conjugate gradient method is proposed. The proposed method is designed to ensure the descent property of the search direction at each iteration. Furthermore, the convergence of the proposed method is proved under the assumption that the underlying operator is monotone and Lipschitz continuous. The numerical results show that the method is efficient for the given test problems.

In this paper, the Rayleigh beam system with two dynamical boundary controls is treated. Theoretically, the well-posedness of the weak solution is obtained. Later, we discretize the system by using the Implicit Euler scheme in time and the $$P^3$$ P 3 Hermite finite element in space. In addition, we show the decay of the discrete energy and we establish some a priori error estimates. Finally, some numerical simulations are presented.

In this paper, we describe the explicit structures of nonabelian tensor squares of nonabelian groups of order $$p^3q$$ p 3 q , where p and q are distinct prime numbers and $$p>2$$ p > 2 . Our method is based on determining the structures of their Schur multipliers by applying some well-known results and the presentations of groups, which leads us to obtain the orders of their nonabelian tensor squares.

In this paper, we investigate any non-invariant submanifold of a locally decomposable golden Riemannian manifold in the case that the rank of the set of tangent vector fields of the induced structure on the submanifold by the golden structure of the ambient manifold is less than or equal to the codimension of the submanifold.

The purpose of this paper to establish the semilocal convergence analysis of three-step Kurchatov method under weaker conditions in Banach spaces. We construct the recurrence relations under the assumption that involved first-order divided difference operators satisfy the $$\omega $$ ω condition. Theorems are given for the existence-uniqueness balls enclosing the unique solution. The application of the iterative method is shown by solving nonlinear system of equations and nonlinear Hammerstein-type integral equations. It illustrates the theoretical development of this study.

Let M be a module over a commutative ring R. In this paper, we continue our study about the Zariski topology-graph $$G(\tau _T)$$ G ( τ T ) which was introduced in Ansari-Toroghy et al. (Commun Algebra 42:3283–3296, 2014). For a non-empty subset T of $$\mathrm{Spec}(M)$$ Spec ( M ) , we obtain useful characterizations for those modules M for which $$G(\tau _T)$$ G ( τ T ) is a bipartite graph. Also, we prove that if $$G(\tau _T)$$ G ( τ T ) is a tree, then $$G(\tau _T)$$ G ( τ T ) is a star graph. Moreover, we study coloring of Zariski topology-graphs and investigate the interplay between $$\chi (G(\tau _T))$$ χ ( G ( τ T ) ) and $$\omega (G(\tau _T))$$ ω ( G ( τ T ) ) .

Let C be a nonempty closed convex subset of a real Hilbert space H and $$T: C\rightarrow CB(H)$$ T : C → C B ( H ) be a multi-valued Lipschitz pseudocontractive nonself mapping. A Halpern–Ishikawa type iterative scheme is constructed and a strong convergence result of this scheme to a fixed point of T is proved under appropriate conditions. Moreover, an iterative method for approximating a fixed point of a k-strictly pseudocontractive mapping $$T: C\rightarrow Prox(H)$$ T : C → P r o x ( H ) is constructed and a strong convergence of the method is obtained without end point condition. The results obtained in this paper improve and extend known results in the literature.

In this paper, a weakly dissipative viscoelastic plate equation with an infinite memory is considered. We show a general energy decay rate for a wide class of relaxation functions. To support our theoretical findings, some numerical illustrations are presented at the end. The numerical solution is computed using the popular finite element method in space, combined with time-stepping finite differences.

In this paper, a finite fractured aquifer, bounded by a stream and impervious layers on the other sides, has been considered. Variation in the level of groundwater is analyzed in confined aquifer for the unsteady flow. The governing differential equation for piezometric head involves the Caputo–Fabrizio fractional derivative operator with respect to time and is based on dual-porosity model with the assumption that the flow from fracture to block is in pseudo steady state. The obtained solutions can be used to anticipate the fluctuations in the waterlevels of the confined aquifer and for the numerical validation of a model in an aquifer.

In this paper, a new numerical technique implements on the time-space pseudospectral method to approximate the numerical solutions of nonlinear time- and space-fractional coupled Burgers’ equation. This technique is based on orthogonal Chebyshev polynomial function and discretizes using Chebyshev–Gauss–Lobbato (CGL) points. Caputo–Riemann–Liouville fractional derivative formula is used to illustrate the fractional derivatives matrix at CGL points. Using the derivatives matrices, the given problem is reduced to a system of nonlinear algebraic equations. These equations can be solved using Newton–Raphson method. Two model examples of time- and space-fractional coupled Burgers’ equation are tested for a set of fractional space and time derivative order. The figures and tables show the significant features, effectiveness, and good accuracy of the proposed method.

The aim of this paper is to explore some sufficient conditions for Aluthge transform of Toeplitz operators on the Bergman space to be unitary, average of unitaries and normal.

Zhou et al. and Huang et al. have proposed the modified shift-splitting (MSS) preconditioner and the generalized modified shift-splitting (GMSS) for non-symmetric saddle point problems, respectively. They have used symmetric positive definite and skew-symmetric splitting of the (1, 1)-block in a saddle point problem. In this paper, we use positive definite and skew-symmetric splitting instead and present new modified shift-splitting (NMSS) method for solving large sparse linear systems in saddle point form with a dominant positive definite part in (1, 1)-block. We investigate the convergence and semi-convergence properties of this method for nonsingular and singular saddle point problems. We also use the NMSS method as a preconditioner for GMRES method. The numerical results show that if the (1, 1)-block has a positive definite dominant part, the NMSS-preconditioned GMRES method can cause better performance results compared to other preconditioned GMRES methods such as GMSS, MSS, Uzawa-HSS and PU-STS. Meanwhile, the NMSS preconditioner is made for non-symmetric saddle point problems with symmetric and non-symmetric (1, 1)-blocks.

In this paper, we proved two new Riemann–Liouville fractional Hermite–Hadamard type inequalities for harmonically convex functions using the left and right fractional integrals independently. Also, we have two new Riemann–Liouville fractional trapezoidal type identities for differentiable functions. Using these identities, we obtained some new trapezoidal type inequalities for harmonically convex functions. Our results generalize the results given by İşcan (Hacet J Math Stat 46(6):935–942, 2014).

Let R a commutative ring with identity and M be a unitary R-module. In this paper, we investigate some properties of n-absorbing submodules of M as a generalization of 2-absorbing submodules. We also define the classical n-absorbing submodule, a proper submodule N of an R-module M is called a classical n-absorbing submodule if whenever $$a_1 a_2\ldots a_{n+1} m\in N$$ a 1 a 2 … a n + 1 m ∈ N for $$a_1, a_2,\ldots , a_{n+1}\in R$$ a 1 , a 2 , … , a n + 1 ∈ R and $$m \in M$$ m ∈ M , there are n of $$a_i$$ a i ’s whose product with m is in N. Furthermore, we give some characterizations of n-absorbing and classical n-absorbing submodules under some conditions.

In this paper, we first construct a suitable Boehmian space on which the Bessel–Wright transform can be defined and some desired properties are obtained in the class of Boehmians. Some convergence results are also established.

Let P(z) be a polynomial of degree n which does not vanish in $$|z| | z | < 1 . Then it was proved by Hans and Lal (Anal Math 40:105–115, 2014) that $$\begin{aligned} \Bigg |z^s P^{(s)}+\beta \dfrac{n_s}{2^s}P(z)\Bigg |\le \dfrac{n_s}{2}\Bigg (\bigg |1+\dfrac{\beta }{2^s}\bigg |+\bigg | \dfrac{\beta }{2^s}\bigg |\Bigg )\underset{|z|=1}{\max }|P(z)|, \end{aligned}$$ | z s P ( s ) + β n s 2 s P ( z ) | ≤ n s 2 ( | 1 + β 2 s | + | β 2 s | ) max | z | = 1 | P ( z ) | , for every $$\beta \in \mathbb C$$ β ∈ C with $$|\beta |\le 1,1\le s\le n$$ | β | ≤ 1 , 1 ≤ s ≤ n and $$|z|=1.$$ | z | = 1 . The $$L^{\gamma }$$ L γ analog of the above inequality was recently given by Gulzar (Anal Math 42:339–352, 2016) who under the same hypothesis proved $$\begin{aligned}&\Bigg \{\int _0^{2\pi }\Big |e^{is\theta }P^{(s)}(e^{i\theta })+\beta \dfrac{n_s}{2^s}P(e^{i\theta })\Big |^ {\gamma } \mathrm{d}\theta \Bigg \}^\frac{1}{\gamma }\\&\quad \le n_s\Bigg \{\int _0^{2\pi }\Big |\Big (1+\dfrac{\beta }{2^s}\Big )e^{i\alpha }+\dfrac{\beta }{2^s}\Big |^{\gamma } \mathrm{d}\alpha \Bigg \}^\frac{1}{\gamma }\dfrac{\Bigg \{\int _0^{2\pi }\big |P(e^{i\theta })\big |^{\gamma } \mathrm{d}\theta \Bigg \}^\frac{1}{\gamma }}{\Bigg \{\int _{0}^{2\pi }\big |1+e^{i\alpha }\big |^\gamma \mathrm{d}\alpha \Bigg \}^\frac{1}{\gamma }}, \end{aligned}$$ { ∫ 0 2 π | e i s θ P ( s ) ( e i θ ) + β n s 2 s P ( e i θ ) | γ d θ } 1 γ ≤ n s { ∫ 0 2 π | ( 1 + β 2 s ) e i α + β 2 s | γ d α } 1 γ { ∫ 0 2 π | P ( e i θ ) | γ d θ } 1 γ { ∫ 0 2 π | 1 + e i α | γ d α } 1 γ , where $$n_s=n(n-1)\ldots (n-s+1)$$ n s = n ( n - 1 ) … ( n - s + 1 ) and $$0\le \gamma 0 ≤ γ < ∞ . In this paper, we generalize this and some other related results.

This paper investigates a conditional cumulative distribution of a scalar response given by a functional random variable with an $$\alpha $$ α -mixing stationary sample using a local polynomial technique. The main purpose of this study is to establish asymptotic normality results under selected mixing conditions satisfied by many time-series analysis models in addition to the other appropriate conditions to confirm the planned prospects.

The matrix representation of operators in Hilbert spaces is a useful tool in applications. It is important to present the matrix representation by sequences other than orthonormal bases. In this paper, we extend the matrix representation of operators using g-frames and investigate their invertibility and stability.

Motivated by recent investigations, in this paper we introduce (p, q)-Szasz-beta–Stancu operators and investigate their local approximation properties in terms of modulus of continuity. We also obtain a weighted approximation and Voronovskaya-type asymptotic formula.

In this paper, we construct a forward–backward splitting algorithm for approximating a zero of the sum of an $$\alpha $$-inverse strongly monotone operator and a maximal monotone operator. The strong convergence theorem is then proved under mild conditions. Then, we add a nonexpansive mapping in the algorithm and prove that the generated sequence converges strongly to a common element of a fixed points set of a nonexpansive mapping and zero points set of the sum of monotone operators. We apply our main result both to equilibrium problems and convex programming.

In the present paper, we prove that there does not exist any $$\mathcal {P}\mathcal {R}$$-semi-slant warped product submanifolds in paracosymplectic manifolds. In addition, by presenting a non-trivial example we find that there is no proper $$\mathcal {P}\mathcal {R}$$-semi-slant warped product submanifold other than $$\mathcal {P}\mathcal {R}$$-semi-invariant warped products.

In this paper, we investigate the $${\mathcal {A}}$$-valued Cauchy–Schwarz inequality and its reverse in a 2-semi inner product $$ {\mathcal {A}}$$-module over a locally $$C^*$$-algebra $$ {\mathcal {A}}$$.

In the present paper, we first characterize real hypersurfaces in the complex quadric $$Q^{m}$$ by giving an inequality in terms of the scalar curvature and the Mean curvature vector field. We also obtain the condition under which this inequality becomes an equality. Further, we develop two extremal inequalities involving the normalized $$\delta $$-Casorati curvatures and the extrinsic generalized normalized $$\delta $$-Casorati curvatures for real hypersurfaces in $$Q^{m}$$. Finally, we derive the necessary and sufficient condition for the equality in both cases.

In this paper, we first define a new pseudo-metric d on a normed linear space X. We do this by introducing two different classes of elementary topical functions. Next, we use this pseudo-metric d to investigate the non-expansivity and some properties of topical functions. Finally, the characterizations of fixed points of topical functions are given, and a relation between the pseudo-metric d and the original norm of the normed linear space X is presented.

It is known that if $$A\in \mathscr {L}(\mathscr {X})$$ and $$B\in \mathscr {L}(\mathscr {Y})$$ are Banach operators with the single-valued extension property, SVEP, then the matrix operator $$M_\mathrm{{C}}=\begin{pmatrix} A &{} C \\ 0&{} B \\ \end{pmatrix} $$ has SVEP for every operator $$C\in \mathscr {L}(\mathscr {Y},\mathscr {X}),$$ and hence obeys generalized Browder’s theorem. This paper considers conditions on operators A, B, and $$M_0$$ ensuring generalized Weyl’s theorem and property (Bw) for operators $$M_\mathrm{{C}}$$. Moreover, certain conditions are explored on Banach space operators T and S so that $$T\oplus S$$ obeys property (gw).

Coloring the vertices of a particular graph has often been motivated by its utility to various applied fields and its mathematical interest. A dynamic coloring of a graph G is a proper coloring of the vertex set V(G) such that for each vertex of degree at least 2, its neighbors receive at least two distinct colors. A dynamic k-coloring of a graph is a dynamic coloring with k colors. A dynamic k-coloring is also called a conditional (k, 2)-coloring. The smallest integer k such that G has a dynamic k-coloring is called the dynamic chromatic number $$\chi _d(G)$$ of G. In this paper, we investigate the dynamic chromatic number for the line graph of sunlet graph and middle graph, total graph and central graph of sunlet graphs, paths and cycles. Also, we find the dynamic chromatic number for Mycielskian of paths and cycles and the join graph of paths and cycles.

In this paper we study the long-time behavior of solutions for a general class of Langevin-type fractional integro-differential equations. The involved fractional derivatives are either of Riemann–Liouville or Caputo type. Reasonable sufficient conditions under which the solutions are bounded or decay like power functions are established. For this purpose, we combine and generalize some well-known integral inequalities with some crucial estimates. Our findings are supported by examples and special cases.

In this paper, we first obtain a generalized integral identity for twice local fractional differentiable mappings on fractal sets $${\mathbb {R}}^{\alpha }\, (0

Here, we investigate the existence of upper neighbours in the lattice of $$T_{1}$$ Cech closure operators on a fixed set. In this paper, we prove that a first countable $$T_{1}$$ closure operator has no upper neighbour in the lattice of Cech closure operators.

In this paper, we first introduce an iterative process in modular function spaces and then extend the idea of a $$\lambda $$ -firmly nonexpansive mapping from Banach spaces to modular function spaces. We call such mappings as $$ \left( \lambda ,\rho \right) $$ -firmly nonexpansive mappings. We incorporate the two ideas to approximate fixed points of $$\left( \lambda ,\rho \right) $$ -firmly nonexpansive mappings using the above-mentioned iterative process in modular function spaces.

In this paper, we consider impulsive integro-differential equations in Banach space and we establish the bound on the difference between two approximate solutions. We also discuss nearness and convergence of solutions of the problem under consideration. The impulsive integral inequality of Grownwall type is used to obtain results.

WOS:000437470500002 In this paper, we use techniques and tools from time scale calculus to state and prove many refinements on the discrete Hermite-Hadamard inequality.

The aim of this paper is to give an overview of results related to nonlinear wave equations during the last half century. In this regard, we present results concerning existence, decay and blow up for classical nonlinear equations. After that, we discuss briefly some important results of the variable-exponent Lebesgue and Sobolev spaces. Results related to nonexistence and blow up for wave equations with non-standard nonlinearities (nonlinearities involving variable exponents) are given in more detail. Finally, we present some recent decay and blow up results together with their proofs.

Let G be a group and $$\omega (G)=\{o(g)|g\in G\}$$ be the set of element orders of G. Let $$k\in \omega (G)$$ and $$s_k=|\{g\in G |o(g)=k\}|$$ . Let $$nse(G)=\{s_k|k\in \omega (G) \}$$ . In this paper, we prove that if G is a group and $$G_2 (4)$$ is the Chevalley group such that $$nse(G)=nse(G_2 (4))$$ , then $$G\cong G_2 (4)$$ .

Sub-Riemannian spaces are spaces whose metric structure may be viewed as a constrained geometry, where motion is only possible along a given set of directions, changing from point to point. The simplest example of such spaces is given by the so-called Heisenberg group. The characteristic constrained motion of sub-Riemannian spaces has numerous applications in robotic control in engineering and neurobiology where it arises naturally in the functional magnetic resonance imaging (FMRI). It also arises naturally in other branches of pure mathematics as Cauchy Riemann geometry, complex hyperbolic spaces, and jet spaces. In this paper, we review the use of the relationship between Heisenberg geometry and Cauchy Riemann (CR) geometry. More precisely, we focus on the problem of the prescription of the scalar curvature using techniques related to the theory of critical points at infinity. These techniques were first introduced by Bahri, Bahri and Brezis for the Yamabe conjecture in the Riemannian settings.

This paper gives the complete proof of the Conjecture given by Hazarika and this author jointly which deals with a necessary and sufficient condition for the hyponormality of Toeplitz operator, \(T_\varphi \) on the weighted Bergman space with certain polynomial symbols under some assumptions about the Fourier coefficients of the symbol \(\varphi \). Open image in new window

In this paper, we consider the problem of existence and multiplicity of conformal metrics on a Riemannian compact 4-dimensional manifold $$(M^4,g_0)$$ with positive scalar curvature. We prove a new existence criterium which provides existence results for a dense subset of positive functions and generalizes Bahri–Coron Euler–Poincare type criterium. Our argument gives estimates of the Morse index of the founded solutions and has the advantage to extend known existence results. Moreover, it provides, for generic K Morse Inequalities at Infinity, which give a lower bound on the number of metrics with prescribed scalar curvature in terms of the topological contribution of its critical points at Infinity to the difference of topology between the level sets of the associated Euler–Lagrange functional.

This paper is concerned with prescribing the fractional Q-curvature on the unit sphere \(\mathbb {S}^{n}\) endowed with its standard conformal structure \(g_0\), \(n\ge 4\). Since the associated variational problem is noncompact, we approach this issue with techniques passed by Abbas Bahri, as the well known theory of critical points at infinity, as well as some lesser known topological invariants that appear here as criteria for existence results.

In the paper, the authors establish explicit formulas for asymptotic and power series expansions of the exponential and the logarithm of asymptotic and power series expansions. The explicit formulas for the power series expansions of the exponential and the logarithm of a power series expansion are applied to find explicit formulas for the Bell numbers and logarithmic polynomials in combinatorics and number theory.