Volume 19, Issue 1

Greedy expansions with prescribed coefficients were introduced by V. N. Temlyakov in a general case of Banach spaces. In contrast to Fourier series expansions, in greedy expansions with prescribed coefficients, a sequence of coefficients { c n } n = 1 ∞ {\left\{{c}_{n}\right\}}_{n=1}^{\infty } is fixed in advance and does not depend on an expanded element. During the expansion, only expanding elements are constructed (or, more precisely, selected from a predefined set – a dictionary). For symmetric dictionaries, V. N. Temlyakov obtained conditions on a sequence of coefficients sufficient for a convergence of a greedy expansion with these coefficients to an expanded element. In case of a Hilbert space these conditions take the form ∑ n = 1 ∞ c n = ∞ {\sum }_{n=1}^{\infty }{c}_{n}=\infty and ∑ n = 1 ∞ c n 2 < ∞ {\sum }_{n=1}^{\infty }{c}_{n}^{2}\lt \infty . In this paper, we study a possibility of relaxing the latter condition. More specifically, we show that the convergence is guaranteed for c n = o 1 n {c}_{n}=o\left(\frac{1}{\sqrt{n}}\right) , but can be violated if c n ≍ 1 n {c}_{n}\hspace{0.33em}\asymp \hspace{0.33em}\frac{1}{\sqrt{n}} .

We study the range-kernel weak orthogonality of certain elementary operators induced by non-normal operators, with respect to usual operator norm and the Von Newmann-Schatten p p -norm ( 1 ≤ p < ∞ ) \left(1\le p\lt \infty ) .

This study provides semi-analytical solutions to the Selkov-Schnakenberg reaction-diffusion system. The Galerkin method is applied to approximate the system of partial differential equations by a system of ordinary differential equations. The steady states of the system and the limit cycle solutions are delineated using bifurcation diagram analysis. The influence of the diffusion coefficients as they change, on the system stability is examined. Near the Hopf bifurcation point, the asymptotic analysis is developed for the oscillatory solution. The semi-analytical model solutions agree accurately with the numerical results.

In this paper, we call a finite group G G an N L M NLM -group ( N C M NCM -group, respectively) if every non-normal cyclic subgroup of prime order or order 4 (prime power order, respectively) in G G is contained in a non-normal maximal subgroup of G G . Using the property of N L M NLM -groups and N C M NCM -groups, we give a new necessary and sufficient condition for G G to be a solvable T T -group (normality is a transitive relation), some sufficient conditions for G G to be supersolvable, and the classification of those groups whose all proper subgroups are N L M NLM -groups.

Let X X be a non-empty set and Y Y a non-empty subset of X X . Denote the full transformation semigroup on X X by T ( X ) T\left(X) and write f ( X ) = { f ( x ) ∣ x ∈ X } f\left(X)=\{f\left(x)| x\in X\} for each f ∈ T ( X ) f\in T\left(X) . It is well known that T ( X , Y ) = { f ∈ T ( X ) ∣ f ( X ) ⊆ Y } T\left(X,Y)=\{f\in T\left(X)| f\left(X)\subseteq Y\} is a subsemigroup of T ( X ) T\left(X) and R T ( X , Y ) RT\left(X,Y) , the set of all regular elements of T ( X , Y ) T\left(X,Y) , also forms a subsemigroup of T ( X , Y ) T\left(X,Y) . Green’s ∗ \ast -relations and Green’s ∼ <mml:mpadded xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"> \sim </mml:mpadded>\hspace{0.08em} -relations (with respect to a non-empty subset U U of the set of idempotents) were introduced by Fountain in 1979 and Lawson in 1991, respectively. In this paper, we intend to present certain characterizations of these two sets of Green’s relations of the semigroup T ( X , Y ) T\left(X,Y) . This investigation proves that the semigroup T ( X , Y ) T\left(X,Y) is always a left Ehresmann semigroup, and R T ( X , Y ) RT\left(X,Y) is orthodox (resp. completely regular) if and only if Y Y contains at most two elements.

In any U -abundant semigroup with an Ehresmann transversal, two significant components R and L are introduced in this paper and described by Green’s ∼ \sim -relations. Some interesting properties associated with R and L are explored and some equivalent conditions for the Ehresmann transversal to be a quasi-ideal are acquired. Finally, a spined product structure theorem is established for a U -abundant semigroup with a quasi-ideal Ehresmann transversal by means of R and L .

The regulator problem is solvable for a linear dynamical system Σ \Sigma if and only if Σ \Sigma is both pole assignable and state estimable. In this case, Σ \Sigma is a canonical system (i.e., reachable and observable). When the ring R R is a field or a Noetherian total ring of fractions the converse is true. Commutative rings which have the property that the regulator problem is solvable for every canonical system (RP-rings) are characterized as the class of rings where every observable system is state estimable (SE-rings), and this class is shown to be equal to the class of rings where every reachable system is pole-assignable (PA-rings) and the dual of a canonical system is also canonical (DP-rings).

This paper deals with the abstract evolution equations in L s {L}^{s} -spaces with critical temporal weights. First, embedding and interpolation properties of the critical L s {L}^{s} -spaces with different exponents s s are investigated, then solvability of the linear evolution equation, attached to which the inhomogeneous term f f and its average Φ f \Phi f both lie in an L 1 / s s {L}_{1\hspace{-0.08em}\text{/}\hspace{-0.08em}s}^{s} -space, is established. Based on these results, Cauchy problem of the semi-linear evolution equation is treated, where the nonlinear operator F ( t , u ) F\left(t,u) has a growth number ρ ≥ s + 1 \rho \ge s+1 , and its asymptotic behavior acts like α ( t ) / t \alpha \left(t)\hspace{-0.1em}\text{/}\hspace{-0.1em}t as t → 0 t\to 0 for some bounded function α ( t ) \alpha \left(t) like ( − log t ) − p {\left(-\log t)}^{-p} with 2 ≤ p < ∞ 2\le p\lt \infty .

Let T {\mathcal{T}} be a triangulated category with a proper class ξ \xi of triangles and X {\mathcal{X}} be a subcategory of T {\mathcal{T}} . We first introduce the notion of X {\mathcal{X}} -resolution dimensions for a resolving subcategory of T {\mathcal{T}} and then give some descriptions of objects having finite X {\mathcal{X}} -resolution dimensions. In particular, we obtain Auslander-Buchweitz approximations for these objects. As applications, we construct adjoint pairs for two kinds of inclusion functors and characterize objects having finite X {\mathcal{X}} -resolution dimensions in terms of a notion of ξ \xi -cellular towers. We also construct a new resolving subcategory from a given resolving subcategory and reformulate some known results.

In this paper, we study the unicity of entire functions and their derivatives and obtain the following result: let f f be a non-constant entire function, let a 1 {a}_{1} , a 2 {a}_{2} , b 1 {b}_{1} , and b 2 {b}_{2} be four small functions of f f such that a 1 ≢ b 1 {a}_{1}\not\equiv {b}_{1} , a 2 ≢ b 2 {a}_{2}\not\equiv {b}_{2} , and none of them is identically equal to ∞ \infty . If f f and f ( k ) {f}^{\left(k)} share ( a 1 , a 2 ) \left({a}_{1},{a}_{2}) CM and share ( b 1 , b 2 ) \left({b}_{1},{b}_{2}) IM, then ( a 2 − b 2 ) f − ( a 1 − b 1 ) f ( k ) ≡ a 2 b 1 − a 1 b 2 \left({a}_{2}-{b}_{2})f-\left({a}_{1}-{b}_{1}){f}^{\left(k)}\equiv {a}_{2}{b}_{1}-{a}_{1}{b}_{2} . This extends the result due to Li and Yang [ Value sharing of an entire function and its derivatives , J. Math. Soc. Japan. 51 (1999), no. 7, 781–799].

One of the inverse problems of dynamics in the presence of random perturbations is considered. This is the problem of the simultaneous construction of a set of first-order Ito stochastic differential equations with a given integral manifold, and a set of comparison functions. The given manifold is stable in probability with respect to these comparison functions.

Any pentagonal quasigroup Q Q is proved to have the product x y = φ ( x ) + y − φ ( y ) xy=\varphi \left(x)+y-\varphi (y) , where ( Q , + ) \left(Q,+) is an Abelian group, φ \varphi is its regular automorphism satisfying φ 4 − φ 3 + φ 2 − φ + ε = 0 {\varphi }^{4}-{\varphi }^{3}+{\varphi }^{2}-\varphi +\varepsilon =0 and ε \varepsilon is the identity mapping. All Abelian groups of order n < 100 n\lt 100 inducing pentagonal quasigroups are determined. The variety of commutative, idempotent, medial groupoids satisfying the pentagonal identity ( x y ⋅ x ) y ⋅ x = y \left(xy\cdot x)y\cdot x=y is proved to be the variety of commutative, pentagonal quasigroups, whose spectrum is { 1 1 n : n = 0 , 1 , 2 , … } \left\{1{1}^{n}:n=0,1,2,\ldots \right\} . We prove that the only translatable commutative pentagonal quasigroup is x y = ( 6 x + 6 y ) ( mod 11 ) xy=\left(6x+6y)\left({\rm{mod}}\hspace{0.33em}11) . The parastrophes of a pentagonal quasigroup are classified according to well-known types of idempotent translatable quasigroups. The translatability of a pentagonal quasigroup induced by the group Z n {{\mathbb{Z}}}_{n} and its automorphism φ ( x ) = a x \varphi \left(x)=ax is proved to determine the value of a a and the range of values of n n .

Consider an odd prime number p ≡ 2 ( mod 3 ) p\equiv 2\hspace{0.3em}\left(\mathrm{mod}\hspace{0.3em}3) . In this paper, the number of certain type of partitions of zero in Z / p Z {\mathbb{Z}}\hspace{-0.1em}\text{/}\hspace{-0.1em}p{\mathbb{Z}} is calculated using a combination of elementary combinatorics and number theory. The focus is on the three-part partitions of 0 in Z / p Z {\mathbb{Z}}\hspace{-0.1em}\text{/}\hspace{-0.1em}p{\mathbb{Z}} with all three parts chosen from the set of non-zero quadratic residues mod p p . Such partitions are divided into two types. Those with exactly two of the three parts identical are classified as type I. The type II partitions are those with all three parts being distinct. The number of partitions of each type is given. The problem of counting such partitions is well related to that of counting the number of non-trivial solutions to the Diophantine equation x 2 + y 2 + z 2 = 0 {x}^{2}+{y}^{2}+{z}^{2}=0 in the ring Z / p Z {\mathbb{Z}}\hspace{-0.1em}\text{/}\hspace{-0.1em}p{\mathbb{Z}} . Correspondingly, solutions to this equation are also classified as type I or type II. We give the number of solutions to the equation corresponding to each type.

In this paper, we consider the long-time behavior for a class of semi-linear degenerate parabolic equations with the nonlinearity f f satisfying the polynomial growth of arbitrary p − 1 p-1 order. We establish some new estimates, i.e., asymptotic higher-order integrability for the difference of the solutions near the initial time. As an application, we obtain the ( L 2 ( Ω ) , L p ( Ω ) ) \left({L}^{2}\left(\Omega ),{L}^{p}\left(\Omega )) -global attractors immediately; moreover, such an attractor can attract every bounded subset of L 2 ( Ω ) {L}^{2}\left(\Omega ) with the L p + δ {L}^{p+\delta } -norm for any δ ∈ [ 0 , + ∞ ) \delta \in \left[0,+\infty ) .

In this article, we have developed an implicit symmetric four-step method of sixth algebraic order with vanished phase-lag and its first derivative. The error and stability analysis of this method are investigated, and its efficiency is tested by solving efficiently the one-dimensional time-independent Schrödinger’s equation. The method performance is compared with other methods in the literature. It is found that for this problem the new method performs better than the compared methods.

In this paper, we investigate the influence of sub-class sizes on a mutually permutable factorized group in which the sub-class sizes of some elements of its factors have certain quantitative properties. Some criteria for a group to be p p -nilpotent or p p -supersolvable are given.

In this paper, the regularization method of S. A. Lomov is generalized to integro-differential equations with rapidly oscillating coefficients and with a rapidly oscillating right-hand side. The main goal of the work is to reveal the influence of the oscillating components on the structure of the asymptotics of the solution of this problem. The case of coincidence of the frequencies of a rapidly oscillating coefficient and a rapidly oscillating inhomogeneity is considered. In this case, only the identical resonance is observed in the problem. Other cases of the relationship between frequencies can lead to so-called non-identical resonances, the study of which is nontrivial and requires the development of a new approach. It is supposed to study these cases in our further work.

In this study, we consider the following quasilinear Choquard equation with singularity − Δ u + V ( x ) u − u Δ u 2 + λ ( I α ∗ ∣ u ∣ p ) ∣ u ∣ p − 2 u = K ( x ) u − γ , x ∈ R N , u > 0 , x ∈ R N , \left\{\begin{array}{ll}-\Delta u+V\left(x)u-u\Delta {u}^{2}+\lambda \left({I}_{\alpha }\ast | u{| }^{p})| u{| }^{p-2}u=K\left(x){u}^{-\gamma },\hspace{1.0em}& x\in {{\mathbb{R}}}^{N},\\ u\gt 0,\hspace{1.0em}& x\in {{\mathbb{R}}}^{N},\end{array}\right. where I α {I}_{\alpha } is a Riesz potential, 0 < α < N 0\lt \alpha \lt N , and N + α N < p < N + α N − 2 \displaystyle \frac{N+\alpha }{N}\lt p\lt \displaystyle \frac{N+\alpha }{N-2} , with λ > 0 \lambda \gt 0 . Under suitable assumption on V V and K K , we research the existence of positive solutions of the equations. Furthermore, we obtain the asymptotic behavior of solutions as λ → 0 \lambda \to 0 .

In 2018, Bai, Fujita and Zhang [Discrete Math. 341 (2018), no. 6, 1523–1533] introduced the concept of a kernel by rainbow paths (for short, RP-kernel) of an arc-coloured digraph D D , which is a subset S S of vertices of D D such that ( a a ) there exists no rainbow path for any pair of distinct vertices of S S , and ( b b ) every vertex outside S S can reach S S by a rainbow path in D D . They showed that it is NP-hard to recognize whether an arc-coloured digraph has an RP-kernel and it is NP-complete to decide whether an arc-coloured tournament has an RP-kernel. In this paper, we give the sufficient conditions for the existence of an RP-kernel in arc-coloured unicyclic digraphs, semicomplete digraphs, quasi-transitive digraphs and bipartite tournaments, and prove that these arc-coloured digraphs have RP-kernels if certain “short” cycles and certain “small” induced subdigraphs are rainbow.

Many mathematicians have studied degenerate versions of quite a few special polynomials and numbers since Carlitz’s work (Utilitas Math. 15 (1979), 51–88). Recently, Kim et al. studied the degenerate gamma random variables, discrete degenerate random variables and two-variable degenerate Bell polynomials associated with Poisson degenerate central moments, etc. This paper is divided into two parts. In the first part, we introduce a new type of degenerate Bell polynomials associated with degenerate Poisson random variables with parameter α > 0 \alpha \hspace{-0.15em}\gt \hspace{-0.15em}0 , called the fully degenerate Bell polynomials. We derive some combinatorial identities for the fully degenerate Bell polynomials related to the n n th moment of the degenerate Poisson random variable, special numbers and polynomials. In the second part, we consider the fully degenerate Bell polynomials associated with degenerate Poisson random variables with two parameters α > 0 \alpha \gt 0 and β > 0 \beta \hspace{-0.15em}\gt \hspace{-0.15em}0 , called the two-variable fully degenerate Bell polynomials. We show their connection with the degenerate Poisson central moments, special numbers and polynomials.

In this paper, we study the following generalized Kadomtsev-Petviashvili equation u t + u x x x + ( h ( u ) ) x = D x − 1 Δ y u , {u}_{t}+{u}_{xxx}+{\left(h\left(u))}_{x}={D}_{x}^{-1}{\Delta }_{y}u, where ( t , x , y ) ∈ R + × R × R N − 1 \left(t,x,y)\in {{\mathbb{R}}}^{+}\times {\mathbb{R}}\times {{\mathbb{R}}}^{N-1} , N ≥ 2 N\ge 2 , D x − 1 f ( x , y ) = ∫ − ∞ x f ( s , y ) d s {D}_{x}^{-1}f\left(x,y)={\int }_{-\infty }^{x}f\left(s,y){\rm{d}}s , f t = ∂ f ∂ t {f}_{t}=\frac{\partial f}{\partial t} , f x = ∂ f ∂ x {f}_{x}=\frac{\partial f}{\partial x} and Δ y = ∑ i = 1 N − 1 ∂ 2 ∂ y i 2 {\Delta }_{y}={\sum }_{i=1}^{N-1}\frac{{\partial }^{2}}{{\partial }_{{y}_{i}}^{2}} . We get the existence of infinitely many nontrivial solutions under certain assumptions in bounded domain without Ambrosetti-Rabinowitz condition. Moreover, by using the method developed by Jeanjean [13], we establish the existence of ground state solutions in R N {{\mathbb{R}}}^{N} .

In this paper, we consider the maximal operator related to the Laplace-Bessel differential operator ( B B -maximal operator) on L p ( ⋅ ) , γ ( R k , + n ) {L}_{p\left(\cdot ),\gamma }\left({{\mathbb{R}}}_{k,+}^{n}) variable exponent Lebesgue spaces. We will give a necessary condition for the boundedness of the B B -maximal operator on variable exponent Lebesgue spaces. Moreover, we will obtain that the B B -maximal operator is not bounded on L p ( ⋅ ) , γ ( R k , + n ) {L}_{p\left(\cdot ),\gamma }\left({{\mathbb{R}}}_{k,+}^{n}) variable exponent Lebesgue spaces in the case of p − = 1 {p}_{-}=1 . We will also prove the boundedness of the fractional maximal function associated with the Laplace-Bessel differential operator (fractional B B -maximal function) on L p ( ⋅ ) , γ ( R k , + n ) {L}_{p\left(\cdot ),\gamma }\left({{\mathbb{R}}}_{k,+}^{n}) variable exponent Lebesgue spaces.

This paper is devoted to the weak and strong estimates for the linear and multilinear fractional Hausdorff operators on the Heisenberg group H n {{\mathbb{H}}}^{n} . A sharp strong estimate for T Φ m {T}_{\Phi }^{m} is obtained. As an application, we derive the sharp constant for the product Hardy operator on H n {{\mathbb{H}}}^{n} . Some weak-type ( p , q ) \left(p,q) ( 1 ≤ p ≤ ∞ ) \left(1\le p\le \infty ) estimates for T Φ , β {T}_{\Phi ,\beta } are also obtained. As applications, we calculate some sharp weak constants for the fractional Hausdorff operator on the Heisenberg group. Besides, we give an explicit weak estimate for T Φ , β → m {T}_{\Phi ,\overrightarrow{\beta }}^{m} under some mild assumptions on Φ \Phi . We extend the results of Guo et al. [ Hausdorff operators on the Heisenberg group , Acta Math. Sin. (Engl. Ser.) 31 (2015), no. 11, 1703–1714] to the fractional setting.

Let f k ( z ) = z + ∑ n = 2 k a n z n {f}_{k}\left(z)=z+{\sum }_{n=2}^{k}{a}_{n}{z}^{n} be the sequence of partial sums of the analytic function f ( z ) = z + ∑ n = 2 ∞ a n z n f\left(z)=z+{\sum }_{n=2}^{\infty }{a}_{n}{z}^{n} . In this paper, we determine sharp lower bounds for Re { f ( z ) / f k ( z ) } {\rm{Re}}\{f\left(z)\hspace{-0.08em}\text{/}\hspace{-0.08em}{f}_{k}\left(z)\} , Re { f k ( z ) / f ( z ) } {\rm{Re}}\{{f}_{k}\left(z)\hspace{-0.08em}\text{/}\hspace{-0.08em}f\left(z)\} , Re { f ′ ( z ) / f k ′ ( z ) } {\rm{Re}}\{{f}^{^{\prime} }\left(z)\hspace{-0.08em}\text{/}\hspace{-0.08em}{f}_{k}^{^{\prime} }\left(z)\} and Re { f k ′ ( z ) / f ′ ( z ) } {\rm{Re}}\{{f}_{k}^{^{\prime} }\left(z)\hspace{-0.08em}\text{/}\hspace{-0.08em}{f}^{^{\prime} }\left(z)\} , where f ( z ) f\left(z) belongs to the subclass J p , q m ( μ , α , β ) {{\mathcal{J}}}_{p,q}^{m}\left(\mu ,\alpha ,\beta ) of analytic functions, defined by Sălăgean ( p , q ) \left(p,q) -differential operator. In addition, the inclusion relations involving N δ ( e ) {N}_{\delta }\left(e) of this generalized function class are considered.

Let F F be a finite group and X X be a complex quasi-projective F F -variety. For r ∈ N r\in {\mathbb{N}} , we consider the mixed Hodge-Deligne polynomials of quotients X r / F {X}^{r}\hspace{-0.15em}\text{/}\hspace{-0.08em}F , where F F acts diagonally, and compute them for certain classes of varieties X X with simple mixed Hodge structures (MHSs). A particularly interesting case is when X X is the maximal torus of an affine reductive group G G , and F F is its Weyl group. As an application, we obtain explicit formulas for the Hodge-Deligne and E E -polynomials of (the distinguished component of) G G -character varieties of free abelian groups. In the cases G = G L ( n , C ) G=GL\left(n,{\mathbb{C}}\hspace{-0.1em}) and S L ( n , C ) SL\left(n,{\mathbb{C}}\hspace{-0.1em}) , we get even more concrete expressions for these polynomials, using the combinatorics of partitions.

Let 1 < k < 14 / 5 1\lt k\lt 14\hspace{-0.08em}\text{/}\hspace{-0.08em}5 , λ 1 , λ 2 , λ 3 {\lambda }_{1},{\lambda }_{2},{\lambda }_{3} and λ 4 {\lambda }_{4} be non-zero real numbers, not all of the same sign such that λ 1 / λ 2 {\lambda }_{1}\hspace{-0.08em}\text{/}\hspace{-0.08em}{\lambda }_{2} is irrational and let ω \omega be a real number. We prove that the inequality ∣ λ 1 p 1 + λ 2 p 2 2 + λ 3 p 3 2 + λ 4 p 4 k − ω ∣ ≤ ( max ( p 1 , p 2 2 , p 3 2 , p 4 k ) ) − ψ ( k ) + ε | {\lambda }_{1}{p}_{1}+{\lambda }_{2}{p}_{2}^{2}+{\lambda }_{3}{p}_{3}^{2}+{\lambda }_{4}{p}_{4}^{k}-\omega | \le {\left(\max \left({p}_{1},{p}_{2}^{2},{p}_{3}^{2},{p}_{4}^{k}))}^{-\psi \left(k)+\varepsilon } has infinitely many solutions in prime variables p 1 , p 2 , p 3 , p 4 {p}_{1},{p}_{2},{p}_{3},{p}_{4} for any ε > 0 \varepsilon \gt 0 , where ψ ( k ) = min 1 14 , 14 − 5 k 28 k \psi \left(k)=\min \left(\frac{1}{14},\frac{14-5k}{28k}\right) .

In this paper, we establish equivalent parameter conditions for the validity of multiple integral half-discrete Hilbert-type inequalities with the nonhomogeneous kernel G ( n λ 1 ∥ x ∥ m , ρ λ 2 ) G\left({n}^{{\lambda }_{1}}\parallel x{\parallel }_{m,\rho }^{{\lambda }_{2}}\hspace{-0.16em}) ( λ 1 λ 2 > 0 {\lambda }_{1}{\lambda }_{2}\gt 0 ) and obtain best constant factors of the inequalities in specific cases. In addition, we also discuss their applications in operator theory.

If vector-valued sublinear operators satisfy the size condition and the vector-valued inequality on weighted Lebesgue spaces with variable exponent, then we obtain their boundedness on weighted Herz-Morrey spaces with variable exponents.

The present paper aims to find some new midpoint-type inequalities for twice quantum differentiable convex functions. The consequences derived in this paper are unification and generalization of the comparable consequences in the literature on midpoint inequalities.

In this paper, we first prove an identity for twice quantum differentiable functions. Then, by utilizing the convexity of ∣ D q 2 b f ∣ | {}^{b}D_{q}^{2}\hspace{0.08em}f| and ∣ D q 2 a f ∣ | {}_{a}D_{q}^{2}\hspace{0.08em}f| , we establish some quantum Ostrowski inequalities for twice quantum differentiable mappings involving q a {q}_{a} and q b {q}^{b} -quantum integrals. The results presented here are the generalization of already published ones.

In this paper, we will assume M M to be a compact smooth manifold and f : M → M f:M\to M to be a diffeomorphism. We herein demonstrate that a C 1 {C}^{1} generic diffeomorphism f f is Axiom A and has no cycles if f f is asymptotic measure expansive. Additionally, for a C 1 {C}^{1} generic diffeomorphism f f , if a homoclinic class H ( p , f ) H\left(\hspace{0.08em}p,f) that contains a hyperbolic periodic point p p of f f is asymptotic measure-expansive, then H ( p , f ) H\left(\hspace{0.08em}p,f) is hyperbolic of f f .

A refinement of the classic equivalence relation among Cauchy sequences yields a useful infinitesimal-enriched number system. Such an approach can be seen as formalizing Cauchy’s sentiment that a null sequence “becomes” an infinitesimal. We signal a little-noticed construction of a system with infinitesimals in a 1910 publication by Giuseppe Peano, reversing his earlier endorsement of Cantor’s belittling of infinitesimals.

In 2011, Factor and Merz [Discrete Appl. Math. 159 (2011), 100–103] defined the ( 1 , 2 ) \left(1,2) -step competition graph of a digraph. Given a digraph D = ( V , A ) D=\left(V,A) , the ( 1 , 2 ) \left(1,2) -step competition graph of D , denoted C 1 , 2 ( D ) {C}_{1,2}\left(D) , is a graph on V ( D ) V\left(D) , where x y ∈ E ( C 1 , 2 ( D ) ) xy\in E\left({C}_{1,2}\left(D)) if and only if there exists a vertex z ≠ x , y z\ne x,y such that either d D − y ( x , z ) = 1 {d}_{D-y}\left(x,z)=1 and d D − x ( y , z ) ≤ 2 {d}_{D-x}(y,z)\le 2 or d D − x ( y , z ) = 1 {d}_{D-x}(y,z)=1 and d D − y ( x , z ) ≤ 2 {d}_{D-y}\left(x,z)\le 2 . They also characterized the (1, 2)-step competition graphs of tournaments and extended some results to the ( i , j ) \left(i,j) -step competition graphs of tournaments. In this paper, the definition of the (1, 2)-step competition graph of a digraph is generalized to a hypertournament and the (1, 2)-step competition graph of a k -hypertournament is characterized. Also, the results are extended to ( i , j ) \left(i,j) -step competition graphs of k -hypertournaments.

In this paper, the functions u ∈ B V φ [ 0 , 1 ] u\in B{V}_{\varphi }\left[0,1] which define compact and Fredholm multiplication operators M u {M}_{u} acting on the space of functions of bounded φ \varphi -variation are studied. All the functions u ∈ B V φ [ 0 , 1 ] u\hspace{-0.08em}\in \hspace{-0.08em}B{V}_{\varphi }\left[0,\hspace{-0.08em}1] which define multiplication operators M u : B V φ [ 0 , 1 ] → B V φ [ 0 , 1 ] {M}_{u}:B{V}_{\varphi }\left[0,1]\to B{V}_{\varphi }\left[0,1] with closed range are characterized.

In this paper, we disprove a remaining conjecture about Bohemian matrices, in which the numbers of distinct determinants of a normalized Bohemian upper-Hessenberg matrix were conjectured.

In this paper, we study the equivalent conditions for the boundedness of the commutators generated by the multilinear maximal function and the bounded mean oscillation (BMO) function on Morrey space. Moreover, the endpoint estimate for such operators on generalized Morrey spaces is also given.

In this paper, we study the inviscid and zero Froude number limits of the viscous shallow water system. We prove that the limit system is represented by the incompressible Euler equations on the whole space. Furthermore, the rate of convergence is also obtained.

In this paper, for polynomials with real coefficients P , Q P,Q satisfying ∣ P ( x ) ∣ ≤ ∣ Q ( x ) ∣ | P\left(x)| \le | Q\left(x)| for each x x in a real interval I I , we prove the bound L ( P ) ≤ c L ( Q ) L\left(P)\le cL\left(Q) between the lengths of P P and Q Q with a constant c c , which is exponential in the degree d d of P P . An example showing that the constant c c in this bound should be at least exponential in d d is also given. Similar inequalities are obtained for the heights of P P and Q Q when the interval I I is infinite and P , Q P,Q are both of degree d d . In the proofs and in the constructions of examples, we use some translations of Chebyshev polynomials.

Positive definite polynomials are important in the field of optimization. ℋ-tensors play an important role in identifing the positive definiteness of an even-order homogeneous multivariate form. In this paper, we propose an iterative scheme for identifying ℋ-tensor and prove that the algorithm can terminate within finite iterative steps. Some numerical examples are provided to illustrate the efficiency and validity of methods.

In this paper, the authors extend determinantal inequalities of the Hua-Marcus-Zhang type for positive definite matrices to the corresponding ones for quaternion matrices.

In this work, by introducing several parameters, a new kernel function including both the homogeneous and non-homogeneous cases is constructed, and a Hilbert-type inequality related to the newly constructed kernel function is established. By convention, the equivalent Hardy-type inequality is also considered. Furthermore, by introducing the partial fraction expansions of trigonometric functions, some special and interesting Hilbert-type inequalities with the constant factors represented by the higher derivatives of trigonometric functions, the Euler number and the Bernoulli number are presented at the end of the paper.

In this paper, we introduce generalizations of Quaternacci sequences (Quaternaccis), called Split Quaternacci sequences, which arose on a base of split quaternion algebras. Explicit and recurrent formulae for Split Quaternacci sequences are given, as well as generating functions. Also, matrices related to Split Quaternaccis sequences are investigated. Moreover, new identities connecting Horadam sequences with other known sequences are generated. Analogous identities for Horadam quaternions and split Horadam quaternions are proved.

The purpose of this paper is to introduce the class of split regular BiHom-Poisson color algebras, which can be considered as the natural extension of split regular BiHom-Poisson algebras and of split regular Poisson color algebras. Using the property of connections of roots for this kind of algebras, we prove that such a split regular BiHom-Poisson color algebra L L is of the form L = ⊕ [ α ] ∈ Λ / ∼ I [ α ] L={\oplus }_{\left[\alpha ]\in \Lambda \text{/} \sim }{I}_{\left[\alpha ]} with I [ α ] {I}_{\left[\alpha ]} a well described (graded) ideal of L L , satisfying [ I [ α ] , I [ β ] ] + I [ α ] I [ β ] = 0 \left[{I}_{\left[\alpha ]},{I}_{\left[\beta ]}]+{I}_{\left[\alpha ]}{I}_{\left[\beta ]}=0 if [ α ] ≠ [ β ] \left[\alpha ]\ne \left[\beta ] . In particular, a necessary and sufficient condition for the simplicity of this algebra is determined, and it is shown that L L is the direct sum of the family of its simple (graded) ideals.

The aim of this work is to study the asymptotic stability of the time-changed stochastic delay differential equations (SDDEs) with Markovian switching. Some sufficient conditions for the asymptotic stability of solutions to the time-changed SDDEs are presented. In contrast to the asymptotic stability in existing articles, we present the new results on the stability of solutions to time-changed SDDEs, which is driven by time-changed Brownian motion. Finally, an example is given to demonstrate the effectiveness of the main results.

New graph invariant, which is called a mixed metric dimension, has been recently introduced. In this paper, exact results of the mixed metric dimension on two special classes of graphs are found: flower snarks J n {J}_{n} and wheels W n {W}_{n} . It is proved that the mixed metric dimension for J 5 {J}_{5} is equal to 5, while for higher dimensions it is constant and equal to 4. For W n {W}_{n} , the mixed metric dimension is not constant, but it is equal to n n when n ≥ 4 n\ge 4 , while it is equal to 4, for n = 3 n=3 .

We study the global structure of the oscillatory perturbed bifurcation problem which comes from the stationary logarithmic Schrödinger equation − u ″ ( t ) = λ ( log ( 1 + u ( t ) ) + sin u ( t ) ) , u ( t ) > 0 , t ∈ I ≔ ( − 1 , 1 ) , u ( ± 1 ) = 0 , -{u}^{^{\prime\prime} }\left(t)=\lambda (\log \left(1+u\left(t))+\sin u\left(t)),\hspace{1.0em}u\left(t)\gt 0,\hspace{1.0em}t\in I:= \left(-1,1),\hspace{1.0em}u\left(\pm 1)=0, where λ > 0 \lambda \gt 0 is a parameter. It is known that λ \lambda is a continuous function of α > 0 \alpha \gt 0 , written as λ ( α ) \lambda \left(\alpha ) , where α \alpha is the maximum norm α = ‖ u λ ‖ ∞ \alpha =\Vert {u}_{\lambda }{\Vert }_{\infty } of the solution u λ {u}_{\lambda } associated with λ \lambda . In the field of bifurcation theory, the study of global structures of bifurcation curves is one of the main subjects of research, and it is important to investigate the influence of the oscillatory term on the global structure of bifurcation curve. Due to the effect of sin u \sin u , it is reasonable to expect that an oscillatory term appears in the second term of the asymptotic formula for λ ( α ) \lambda \left(\alpha ) as α → ∞ \alpha \to \infty (cf. [1]). Contrary to expectation, we show that the asymptotic formula for λ ( α ) \sqrt{\lambda \left(\alpha )} as α → ∞ \alpha \to \infty does not contain oscillatory terms by the third term of λ ( α ) \sqrt{\lambda \left(\alpha )} . This result implies that the oscillatory term has almost no influence on the global structure of λ ( α ) \lambda \left(\alpha ) . The result is, therefore, unexpected, new and novel, since such phenomenon as this is not known so far. For the proof, the involved time map method and stationary phase method are used.

The notions of o s {o}_{s} -convergence and S ∗ {S}^{\ast } -doubly quasicontinuous posets are introduced, which can be viewed as common generalizations of Birkhoff’s order-convergence and S ∗ {S}^{\ast } -doubly continuous posets, respectively. We first consider the relationship between o s {o}_{s} -convergence and B-topology and show that the topology induced by o s {o}_{s} -convergence according to the standard topological approach is the B-topology precisely. Then, the topological characterization for the S ∗ {S}^{\ast } -doubly quasicontinuity is presented. It is proved that a poset is S ∗ {S}^{\ast } -doubly quasicontinuous iff the poset equipped with the B-topology is locally hyperclosed iff the lattice of all B-open subsets of the poset is hypercontinuous. Finally, the order theoretical condition for the o s {o}_{s} -convergence being topological is given and the complete regularity of B-topology on S ∗ {S}^{\ast } -doubly quasicontinuous posets is explored.

More than 20 years after Fickett attempted to prove the Hyers-Ulam stability of isometries defined on bounded subsets of R n {{\mathbb{R}}}^{n} in 1981, Alestalo et al. [ Isometric approximation , Israel J. Math. 125 (2001), 61–82] and Väisälä [ Isometric approximation property in Euclidean spaces , Israel J. Math. 128 (2002), 127] improved Fickett’s theorem significantly. In this paper, we will improve Fickett’s theorem by proving the Hyers-Ulam stability of isometries defined on bounded subsets of R n {{\mathbb{R}}}^{n} using a more intuitive and more efficient approach that differs greatly from the methods used by Alestalo et al. and Väisälä.

In this paper, we discuss the solution of the inhomogeneous conformable abstract Cauchy problem. The homogeneous problem is also studied. The analysis of conformable fractional calculus and fractional semigroups is also given. Existence, uniqueness and regularity of a mild solution for the conformable abstract Cauchy problem are studied. Applications illustrating our main abstract results are also given.

In this paper, we introduce the category and the homotopy category of edge-colored digraphs and construct the functorial homology theory on the foundation of the path homology theory provided by Grigoryan, Muranov, and Shing-Tung Yau. We give the construction of the path homology theory for edge-colored graphs that follows immediately from the consideration of natural functor from the category of graphs to the subcategory of symmetrical digraphs. We describe the natural filtration of path homology groups of any digraph equipped with edge coloring, provide the definition of the corresponding spectral sequence, and obtain commutative diagrams and braids of exact sequences.

In this paper, we first obtain two new quantum Hermite-Hadamard-type inequalities for newly defined quantum integral. Then we establish several refinements of quantum Hermite-Hadamard inequalities.

We characterize seven valent symmetric graphs of order 2 p q n 2p{q}^{n} with p < q p\lt q odd primes, extending a few previous results. Moreover, a consequence partially generalizes the result of Conder, Li and Potočnik [ On the orders of arc-transitive graphs , J. Algebra 421 (2015), 167–186].

In this paper, the boundedness of certain multilinear operator related to the multiplier operator from Lebesgue spaces to Orlicz spaces is obtained.

In this paper, we consider the numerical method for solving the two-dimensional time-fractional convection-diffusion equation with a fractional derivative of order α \alpha ( 1 < α < 2 1\lt \alpha \lt 2 ). By combining the compact difference approach for spatial discretization and the alternating direction implicit (ADI) method in the time stepping, a compact ADI scheme is proposed. The unconditional stability and H 1 {H}^{1} norm convergence of the scheme are proved rigorously. The convergence order is O ( τ 3 − α + h 1 4 + h 2 4 ) O\left({\tau }^{3-\alpha }+{h}_{1}^{4}+{h}_{2}^{4}) , where τ \tau is the temporal grid size and h 1 {h}_{1} , h 2 {h}_{2} are spatial grid sizes in the x x and y y directions, respectively. It is proved that the method can even attain ( 1 + α ) \left(1+\alpha ) order accuracy in temporal for some special cases. Numerical results are presented to demonstrate the effectiveness of theoretical analysis.

In this paper, with the aid of the Faà di Bruno formula and by virtue of properties of the Bell polynomials of the second kind, the authors define a kind of notion of degenerate Narumi numbers and polynomials, establish explicit formulas for degenerate Narumi numbers and polynomials, and derive explicit formulas for the Narumi numbers and polynomials and for (degenerate) Cauchy numbers.

The intersection power graph of a finite group G G is the graph whose vertex set is G G , and two distinct vertices x x and y y are adjacent if either one of x x and y y is the identity element of G G , or ⟨ x ⟩ ∩ ⟨ y ⟩ \langle x\rangle \cap \langle y\rangle is non-trivial. In this paper, we completely classify all finite groups whose intersection power graphs are toroidal and projective-planar.

We provide explicit formulae for primitive, integral solutions to the Diophantine equation x 2 + y 2 = M {x}^{2}+{y}^{2}=M , where M M is a product of powers of Pythagorean primes, i.e., of primes of the form 4 n + 1 4n+1 . It turns out that this is a nice application of the theory of Gaussian integers.

In this paper, the authors study the poly-Bernoulli numbers of the second kind, which are defined by using polyexponential functions introduced by Kims. Also by using unipoly function, we study the unipoly Bernoulli numbers of the second kind, which are attached to an arithmetic function. We derive their explicit expressions and some identities involving poly-Bernoulli numbers of the second kind and unipoly Bernoulli numbers of the second kind.

Type 2 poly-Bernoulli polynomials were introduced recently with the help of modified polyexponential functions. In this paper, we investigate several properties and identities associated with those polynomials arising from umbral calculus techniques. In particular, we express the type 2 poly-Bernoulli polynomials in terms of several special polynomials, like higher-order Cauchy polynomials, higher-order Euler polynomials, and higher-order Frobenius-Euler polynomials.

The aim of this paper is to establish the boundedness of commutator [ b , g ˙ r ] \left[b,{\dot{g}}_{r}] generated by Littlewood-Paley g g -functions g ˙ r {\dot{g}}_{r} and b ∈ RBMO ( μ ) b\in {\rm{RBMO}}\left(\mu ) on non-homogeneous metric measure space. Under assumption that λ \lambda satisfies ε \varepsilon -weak reverse doubling condition, the author proves that [ b , g ˙ r ] \left[b,{\dot{g}}_{r}] is bounded from Lebesgue spaces L p ( μ ) {L}^{p}\left(\mu ) into Lebesgue spaces L p ( μ ) {L}^{p}\left(\mu ) for p ∈ ( 1 , ∞ ) p\in \left(1,\infty ) and also bounded from spaces L 1 ( μ ) {L}^{1}\left(\mu ) into spaces L 1 , ∞ ( μ ) {L}^{1,\infty }\left(\mu ) . Furthermore, the boundedness of [ b , g ˙ r b,{\dot{g}}_{r} ] on Morrey space M q p ( μ ) {M}_{q}^{p}\left(\mu ) and on generalized Morrey L p , ϕ ( μ ) {L}^{p,\phi }\left(\mu ) is obtained.

In this paper, we propose a family of non-stationary combined ternary ( 2 m + 3 ) \left(2m+3) -point subdivision schemes, which possesses the property of generating/reproducing high-order exponential polynomials. This scheme is obtained by adding variable parameters on the generalized ternary subdivision scheme of order 4. For such a scheme, we investigate its support and exponential polynomial generation/reproduction and get that it can generate/reproduce certain exponential polynomials with suitable choices of the parameters and reach 2 m + 3 2m+3 approximation order. Moreover, we discuss its smoothness and show that it can produce C 2 m + 2 {C}^{2m+2} limit curves. Several numerical examples are given to show the performance of the schemes.

We study the bifurcation diagrams and exact multiplicity of positive solutions for the one-dimensional prescribed mean curvature equation − u ′ 1 + u ′ 2 ′ = λ u 1 + u p , − L < x < L , u ( − L ) = u ( L ) = 0 , \left\{\begin{array}{l}-{\left(\frac{{u}^{^{\prime} }}{\sqrt{1+{u}^{^{\prime} 2}}}\right)}^{^{\prime} }=\lambda {\left(\frac{u}{1+u}\right)}^{p},\hspace{1.0em}-L\lt x\lt L,\\ u\left(-L)=u\left(L)=0,\end{array}\right. where λ \lambda is a bifurcation parameter, and L , p > 0 L,p\gt 0 are two evolution parameters. We prove that on the ( λ , ‖ u ‖ ∞ ) \left(\lambda ,\Vert u{\Vert }_{\infty }) -plane, for 0 < p ≤ 2 4 0\lt p\le \frac{\sqrt{2}}{4} , the bifurcation curve is ⊃ \supset -shaped bifurcation starting from ( 0 , 0 ) \left(0,0) . And for p = 1 , f ( u ) = u 1 + u p=1,f\left(u)=\frac{u}{1+u} is a logistic function, then the bifurcation curve is also ⊃ \supset -shaped bifurcation starting from π 2 4 L 2 , 0 \left(\frac{{\pi }^{2}}{4{L}^{2}},0\right) . While for p > 1 p\gt 1 , the bifurcation curve is reversed ε \varepsilon -like shaped bifurcation if L > L ∗ L\gt {L}^{\ast } , and is exactly decreasing for λ > λ ∗ \lambda \gt {\lambda }^{\ast } if 0 < L < L ∗ 0\lt L\lt {L}_{\ast } .

In this paper, we investigate finite p -groups G G such that whenever A , B < G A,B\lt G are non-incident, then A ∩ B ⊴ ⟨ A , B ⟩ A\cap B\hspace{0.33em}&#x22B4;\hspace{0.33em}\langle A,B\rangle . This partially solves a problem proposed by Y. Berkovich.

This paper is concerned with a chemotaxis system u t = Δ u m − ∇ ⋅ ( χ 1 ( w ) u ∇ w ) + μ 1 u ( 1 − u − a 1 v ) , x ∈ Ω , t > 0 , v t = Δ v n − ∇ ⋅ ( χ 2 ( w ) v ∇ w ) + μ 2 v ( 1 − a 2 u − v ) , x ∈ Ω , t > 0 , w t = Δ w − ( α u + β v ) w , x ∈ Ω , t > 0 , \left\{\begin{array}{ll}{u}_{t}=\Delta {u}^{m}-\nabla \cdot \left({\chi }_{1}\left(w)u\nabla w)+{\mu }_{1}u\left(1-u-{a}_{1}v),& x\in \Omega ,\hspace{0.33em}t\gt 0,\\ {v}_{t}=\Delta {v}^{n}-\nabla \cdot \left({\chi }_{2}\left(w)v\nabla w)+{\mu }_{2}v\left(1-{a}_{2}u-v),& x\in \Omega ,\hspace{0.33em}t\gt 0,\\ {w}_{t}=\Delta w-\left(\alpha u+\beta v)w,& x\in \Omega ,\hspace{0.33em}t\gt 0,\end{array}\right. under homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ R 3 \Omega \subset {{\mathbb{R}}}^{3} with smooth boundary, where μ 1 , μ 2 > 0 {\mu }_{1},{\mu }_{2}\gt 0 , a 1 , a 2 > 0 {a}_{1},{a}_{2}\gt 0 , α , β > 0 \alpha ,\beta \gt 0 , and the chemotactic sensitivity function χ i ∈ C 1 ( [ 0 , ∞ ) ) {\chi }_{i}\in {C}^{1}({[}0,\infty )) , χ i ′ ≥ 0 {\chi }_{i}^{^{\prime} }\ge 0 . It is proved that for any large initial data, for any m , n > 1 m,n\gt 1 , the system admits a global weak solution, which is uniformly bounded.

It is an interesting and difficult topic to determine the structure of a finite group by the number of elements of maximal order. This topic is related to Thompson’s conjecture, that is, if two finite groups have the same order type and one of them is solvable, then the other is solvable. In this article, we continue this work and prove that if G G is a finite group which has 4 p 2 q 4{p}^{2}q elements of maximal order, where p p , q q are primes and 7 ≤ p ≤ q 7\le p\le q , then either G G is solvable or G G has a section who is isomorphic to one of L 2 ( 7 ) {L}_{2}\left(7) , L 2 ( 8 ) {L}_{2}\left(8) or U 3 ( 3 ) {U}_{3}\left(3) .

In this paper, we discuss the existence of positive solutions to a discrete third-order three-point boundary value problem. Here, the weight function a ( t ) a\left(t) and the Green function G ( t , s ) G\left(t,s) both change their sign. Despite this, we also obtain several existence results of positive solutions by using the Guo-Krasnoselskii’s fixed-point theorem in a cone.

Let f f be a self-dual Hecke-Maass eigenform for the group S L 3 ( Z ) S{L}_{3}\left({\mathbb{Z}}) . For 1 2 < σ < 1 \frac{1}{2}\lt \sigma \lt 1 fixed we define m ( σ ) m\left(\sigma ) ( ≥ 2 \ge 2 ) as the supremum of all numbers m m such that ∫ 1 T ∣ L ( s , f ) ∣ m d t ≪ f , ε T 1 + ε , \underset{1}{\overset{T}{\int }}| L\left(s,f){| }^{m}{\rm{d}}t{\ll }_{f,\varepsilon }{T}^{1+\varepsilon }, where L ( s , f ) L\left(s,f) is the Godement-Jacquet L -function related to f f . In this paper, we first show the lower bound of m ( σ ) m\left(\sigma ) for 2 3 < σ < 1 \frac{2}{3}\lt \sigma \lt 1 . Then we establish asymptotic formulas for the second, fourth and sixth powers of L ( s , f ) L\left(s,f) as applications.

This paper is devoted to exploring the existence and the forms of entire solutions of several quadratic trinomial differential difference equations with more general forms. Some results about the forms of entire solutions for these equations are some extensions and generalizations of the previous theorems given by Liu, Yang and Cao. We also give a series of examples to explain the existence of the finite order transcendental entire solutions of such equations.

In this paper, we consider the regression model with fixed design: Y i = g ( x i ) + ε i {Y}_{i}=g\left({x}_{i})+{\varepsilon }_{i} , 1 ≤ i ≤ n 1\le i\le n , where { x i } \left\{{x}_{i}\right\} are the nonrandom design points, and { ε i } \left\{{\varepsilon }_{i}\right\} is a sequence of martingale, and g g is an unknown function. Nonparametric estimator g n ( x ) {g}_{n}\left(x) of g ( x ) g\left(x) will be introduced and its strong convergence properties are established.

In this work, we introduce the notions about the Riemann-Liouville fractional integrals for interval-valued functions on co-ordinates. We also establish Hermite-Hadamard and some related inequalities for co-ordinated convex interval-valued functions by applying the newly defined fractional integrals. The results of the present paper are the extension of several previously published results.

In this paper, we prove a quantum version of Montgomery identity and prove some new Ostrowski-type inequalities for convex functions in the setting of quantum calculus. Moreover, we discuss several special cases of newly established inequalities and obtain different new and existing inequalities in the field of integral inequalities.

In this paper, we study the eigenvalue problem of poly-drifting Laplacian on complete smooth metric measure space ( M , ⟨ , ⟩ , e − ϕ d v ) \left(M,\langle ,\rangle ,{e}^{-\phi }{\rm{d}}v) , with nonnegative weighted Ricci curvature Ric ϕ ≥ 0 {{\rm{Ric}}}^{\phi }\ge 0 for some ϕ ∈ C 2 ( M ) \phi \in {C}^{2}\left(M) , which is uniformly bounded from above, and successfully obtain several universal inequalities of this eigenvalue problem.

Weierstrass semigroups are well known along the literature. We present a new family of non-Weierstrass semigroups which can be written as an intersection of Weierstrass semigroups. In addition, we provide methods for computing non-Weierstrass semigroups with genus as large as desired.

so -metrizable spaces are a class of important generalized metric spaces between metric spaces and s n sn -metrizable spaces where a space is called an so -metrizable space if it has a σ \sigma -locally finite so -network. As the further work that attaches to the celebrated Alexandrov conjecture, it is interesting to characterize so -metrizable spaces by images of metric spaces. This paper gives such characterizations for so -metrizable spaces. More precisely, this paper introduces so -open mappings and uses the “Pomomarev’s method” to prove that a space X X is an so -metrizable space if and only if it is an so -open, compact-covering, σ \sigma -image of a metric space, if and only if it is an so -open, σ \sigma -image of a metric space. In addition, it is shown that so -open mapping is a simplified form of s n sn -open mapping (resp. 2-sequence-covering mapping if the domain is metrizable). Results of this paper give some new characterizations of so -metrizable spaces and establish some equivalent relations among so -open mapping, s n sn -open mapping and 2-sequence-covering mapping, which further enrich and deepen generalized metric space theory.

In this study, we first obtain a new identity for generalized fractional integrals which contains some parameters. Then by this equality, we establish some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals. Moreover, we show that the results proved in the main section reduce to several Simpson-, trapezoid- and midpoint-type inequalities for various values of parameters.

In this article, the concepts of cone b -norm and cone b -Banach space are given. Some new fixed point theorems in cone b -Banach spaces are established. The new results improve some fixed point theorems in cone Banach spaces. Furthermore, we also investigate the uniqueness of fixed points.

In this paper, we study the complete consistency for the estimator of nonparametric regression model based on m -END errors and obtain the convergence rates of the complete consistency under more general conditions. Finally, some simulations are illustrated to verify the validity of our results.

In this paper, we consider an approximation scheme for fractional evolution equation with variable coefficient. The space derivative is approximated by triangular finite element and the time fractional derivative is evaluated by the L 1 approximation. The main aim of this work is to provide convergence and superconvergence analysis and derive a posteriori error estimates. Some numerical examples are presented to demonstrate our theoretical results.

The concept of coupled 𝔉-orthogonal contraction mapping is introduced in this paper, and some coupled fixed point theorems in orthogonal metric spaces are proved. The obtained results generalize and extend some of the well-known results in the literature. An example is presented to support our results. Furthermore, we apply our result to obtain the existence theorem for a common solution of the integral equations: ζ ( v ) = ð ( v ) + ∫ 0 M Ξ ( v , β ) Ω ( β , ζ ( β ) , ξ ( β ) ) d β , v ∈ [ 0 , H ] , ξ ( v ) = ð ( v ) + ∫ 0 M Ξ ( v , β ) Ω ( β , ξ ( β ) , ζ ( β ) ) d β , v ∈ [ 0 , H ] , \left\{\begin{array}{ll}\zeta \left({\mathfrak{v}})=&#x00F0;\left({\mathfrak{v}})+\underset{0}{\overset{{\mathfrak{M}}}{\displaystyle \int }}\Xi \left({\mathfrak{v}},\beta )\Omega \left(\beta ,\zeta \left(\beta ),\xi \left(\beta )){\rm{d}}\beta ,& {\mathfrak{v}}\in \left[0,{\mathscr{H}}],\\ \xi \left({\mathfrak{v}})=&#x00F0;\left({\mathfrak{v}})+\underset{0}{\overset{{\mathfrak{M}}}{\displaystyle \int }}\Xi \left({\mathfrak{v}},\beta )\Omega \left(\beta ,\xi \left(\beta ),\zeta \left(\beta )){\rm{d}}\beta ,& {\mathfrak{v}}\in \left[0,{\mathscr{H}}],\end{array}\right. where (a) ð : M → R &#x00F0;:{\mathfrak{M}}\to {\mathbb{R}} and Ω : M × R × R → R \Omega :{\mathfrak{M}}\times {\mathbb{R}}\times {\mathbb{R}}\to {\mathbb{R}} are continuous; (b) Ξ : M × M \Xi :{\mathfrak{M}}\times {\mathfrak{M}} is continuous and measurable at β ∈ M , ∀ \beta \in {\mathfrak{M}},\hspace{0.33em}\forall v ∈ M {\mathfrak{v}}\in {\mathfrak{M}} ; (c) Ξ ( v , β ) ≥ 0 , ∀ v , β ∈ M \Xi \left({\mathfrak{v}},\beta )\ge 0,\hspace{0.33em}\forall {\mathfrak{v}},\beta \in {\mathfrak{M}} and ∫ 0 H Ξ ( v , β ) d β ≤ 1 , ∀ v ∈ M {\int }_{0}^{{\mathscr{H}}}\Xi \left({\mathfrak{v}},\beta ){\rm{d}}\beta \le 1,\hspace{0.33em}\forall {\mathfrak{v}}\in {\mathfrak{M}} .

In this paper, we investigate a more general category of Θ \Theta -Yetter-Drinfeld modules ( Θ ∈ Aut H ( H ) \Theta \in {\rm{Aut}}\hspace{0.33em}H\left(H) ) over a Hom-Hopf algebra, which unifies two different definitions of Hom-Yetter-Drinfeld category introduced by Makhlouf and Panaite, Li and Ma, respectively. We show that the category of Θ \Theta -Yetter-Drinfeld modules with a bijective antipode S S is a braided tensor category and some solutions of the Hom-Yang-Baxter equation and the Yang-Baxter equation can be constructed by this category. Also by the method of symmetric pairs, we prove that if a Θ \Theta -Yetter-Drinfeld category over a Hom-Hopf algebra H H is symmetric, then H H is trivial. Finally, we find a sufficient and necessary condition for a Θ \Theta -Yetter-Drinfeld category to be pseudosymmetric.

Let cd ( G ) {\rm{cd}}\left(G) be the set of irreducible complex character degrees of a finite group G G . ρ ( G ) \rho \left(G) denotes the set of primes dividing degrees in cd ( G ) {\rm{cd}}\left(G) . For any prime p , let p e p ( G ) = max { χ ( 1 ) p ∣ χ ∈ Irr ( G ) } {p}^{{e}_{p}\left(G)}=\max \left\{\chi {\left(1)}_{p}\hspace{0.08em}| \hspace{0.08em}\chi \in {\rm{Irr}}\left(G)\right\} and V ( G ) = { p e p ( G ) ∣ p ∈ ρ ( G ) } V\left(G)=\left\{{p}^{{e}_{p}\left(G)}\hspace{0.08em}| \hspace{0.1em}p\in \rho \left(G)\right\} . The degree prime-power graph Γ ( G ) \Gamma \left(G) of G G is a graph whose vertices set is V ( G ) V\left(G) , and two vertices x , y ∈ V ( G ) x,y\in V\left(G) are joined by an edge if and only if there exists m ∈ cd ( G ) m\in {\rm{cd}}\left(G) such that x y ∣ m xy| m . It is an interesting and difficult problem to determine the structure of a finite group by using its degree prime-power graphs. Qin proved that all Mathieu groups can be uniquely determined by their orders and degree prime-power graphs. In this article, we continue this topic and successfully characterize all the automorphism groups of Mathieu groups by using their orders and degree prime-power graphs.

Let R R be a ring, C C be a left R R -module and S = End R ( C ) S={{\rm{End}}}_{R}\left(C) . When C C is semidualizing, the Auslander class A C ( S ) {{\mathcal{A}}}_{C}\left(S) and the Bass class ℬ C ( R ) {{\mathcal{ {\mathcal B} }}}_{C}\left(R) associated with C C have been the subject of extensive investigations. It has been proved that these classes, also known as Foxby classes, are one of the central concepts of (relative) Gorenstein homological algebra. In this paper, we answer several natural questions which arise when we weaken the condition of C C being semidualizing: if we let C C be w-tilting (see Definition 2.1), we establish the conditions for the pair ( A C ( S ) , A C ( S ) ⊥ 1 ) \left({{\mathcal{A}}}_{C}\left(S),{{\mathcal{A}}}_{C}{\left(S)}^{{\perp }_{1}}) to be a perfect cotorsion theory and for the pair ( B C ⊥ 1 ( R ) , B C ( R ) ) \left({}^{{\perp }_{1}}B_{C}\left(R),{B}_{C}\left(R)) to be a complete hereditary cotorsion theory. This tells us when the classes of Auslander and Bass are preenveloping and precovering, which generalizes a number of results disseminated in the literature. We investigate Gorenstein flat modules relative to a not necessarily semidualizing module C C and we find conditions for the class of G C {G}_{C} -projective modules to be special precovering, the class of G C {G}_{C} -flat modules to be covering, the one of Gorenstein C C -projective modules to be precovering and that of Gorenstein C C -injective modules to be preenveloping. We also find how to recover Foxby classes from Add R ( C ) {{\rm{Add}}}_{R}\left(C) -resolutions of R R .

For each positive integer n n , we let φ n : Σ C P ∞ → Σ C P ∞ {\varphi }_{n}:\Sigma {\mathbb{C}}{P}^{\infty }\to \Sigma {\mathbb{C}}{P}^{\infty } be the self-maps of the suspension of the infinite complex projective space, or the localization of this space at a set of primes which may be an empty set. Furthermore, let [ φ m , φ n ] : Σ C P ∞ → Σ C P ∞ \left[{\varphi }_{m},{\varphi }_{n}]:\Sigma {\mathbb{C}}{P}^{\infty }\to \Sigma {\mathbb{C}}{P}^{\infty } be a commutator of self-maps φ m {\varphi }_{m} and φ n {\varphi }_{n} for any positive integers m m and n n . In the current study, we show that the image of the homomorphism [ φ ˆ m , φ ˆ n ] ∗ {\left[{\hat{\varphi }}_{m},{\hat{\varphi }}_{n}]}_{\ast } in homology induced by the adjoint [ φ ˆ m , φ ˆ n ] : C P ∞ → Ω Σ C P ∞ \left[{\hat{\varphi }}_{m},{\hat{\varphi }}_{n}]:{\mathbb{C}}{P}^{\infty }\to \Omega \Sigma {\mathbb{C}}{P}^{\infty } of the commutator [ φ m , φ n ] \left[{\varphi }_{m},{\varphi }_{n}] is both primitive and decomposable. As a further support of the above statement, we provide an example.

First, we give the concepts of G -sequence shadowing property, G -equicontinuity and G -regularly recurrent point. Second, we study their dynamical properties in the inverse limit space under group action. The following results are obtained. (1) The self-mapping f f has the G -sequence shadowing property if and only if the shift mapping σ \sigma has the G ¯ \overline{G} -sequence shadowing property; (2) The self-mapping f f is G -equicontinuous if and only if the shift mapping σ \sigma is G ¯ \overline{G} -equicontinuous; (3) R R G ¯ ( σ ) = lim ← ( R R G ( f ) , f ) R{R}_{\overline{G}}\left(\sigma )=\underleftarrow{\mathrm{lim}}\left(R{R}_{G}(f),f) . These conclusions make up for the lack of theory in the inverse limit space under group action.

In this paper, f -biharmonic submanifolds with parallel normalized mean curvature vector field in Lorentz space forms are discussed. When f f is a constant, we prove that such submanifolds have parallel mean curvature vector field with the minimal polynomial of the shape operator of degree ≤ 2 \le 2 . When f f is a function, we completely classify such pseudo-umbilical submanifolds.

In this paper, we provide some conditions for a K -woven pair of K -g-frames to be preserved under an operator and particularly, we report that applying two different operators to a K -woven pair of K -g-frames can leave them K -woven. Several new methods on the construction of K -woven pair of K -g-frames are also given. We end the paper with a new perturbation result on the weaving of K -g-frames, which shows that, under the perturbation condition involved in one known result on this topic, two K -g-frames can be K -woven in the whole space, not merely in the subspace Range ( K ).

A higher dimensional generalization of the cross product is associated with an adequate matrix multiplication. This index-free view allows for a better understanding of the underlying algebraic structures, among which are generalizations of Grassmann’s, Jacobi’s and Room’s identities. Moreover, such a view provides a higher dimensional analogue of the decomposition of the vector Laplacian, which itself gives an explicit index-free Helmholtz decomposition in arbitrary dimensions n ≥ 2 n\ge 2 .

In the present paper, we study the existence and convergence of the best proximity point for cyclic Θ \Theta -contractions. As consequences, we extract several fixed point results for such cyclic mappings. As an application, we present some solvability theorems in the topic of variational inequalities.

We investigate the additivity and multiplicativity of centrally extended higher derivations and show that every centrally extended higher derivation of a semiprime ring with no nonzero central ideals is a higher derivation. Moreover, we study preservation of the center of the ring by a centrally extended higher derivation.

Let T ( X ) T\left(X) be the full transformation semigroup on a set X X . For an equivalence E E on X X , let T E ∗ ( X ) = { α ∈ T ( X ) : ∀ x , y ∈ X , ( x , y ) ∈ E ⇔ ( x α , y α ) ∈ E } . {T}_{{E}^{\ast }}\left(X)=\left\{\alpha \in T\left(X):\forall x,y\in X,\left(x,y)\in E\iff \left(x\alpha ,y\alpha )\in E\right\}. For each nonempty subset Y Y of X X , we denote the restriction of E E to Y Y by E Y {E}_{Y} . Let T E ∗ ( X , Y ) {T}_{{E}^{\ast }}\left(X,Y) be the intersection of the semigroup T E ∗ ( X ) {T}_{{E}^{\ast }}\left(X) with the semigroup of all transformations with restricted range Y Y under the condition that ∣ X / E ∣ = ∣ Y / E Y ∣ | X\hspace{-0.1em}\text{/}E| =| Y\hspace{-0.16em}\text{/}\hspace{-0.1em}{E}_{Y}| . Equivalently, T E ∗ ( X , Y ) = { α ∈ T E ∗ ( X ) : X α ⊆ Y } {T}_{{E}^{\ast }}\left(X,Y)=\left\{\alpha \in {T}_{{E}^{\ast }}\left(X):X\alpha \subseteq Y\right\} , where ∣ X / E ∣ = ∣ Y / E Y ∣ | X\hspace{-0.1em}\text{/}\hspace{-0.1em}E| =| Y\hspace{-0.16em}\text{/}\hspace{-0.1em}{E}_{Y}| . Then T E ∗ ( X , Y ) {T}_{{E}^{\ast }}\left(X,Y) is a subsemigroup of T E ∗ ( X ) {T}_{{E}^{\ast }}\left(X) . In this paper, we characterize the natural partial order on T E ∗ ( X , Y ) {T}_{{E}^{\ast }}\left(X,Y) . Then we find the elements which are compatible and describe the maximal and minimal elements. We also prove that every element of T E ∗ ( X , Y ) {T}_{{E}^{\ast }}\left(X,Y) lies between maximal and minimal elements. Finally, the existence of an upper cover and a lower cover is investigated.

The objective of this paper is to investigate, by applying the standard Caputo fractional q -derivative of order α \alpha , the existence of solutions for the singular fractional q -integro-differential equation D q α [ k ] ( t ) = Ω ( t , k 1 , k 2 , k 3 , k 4 ) {{\mathcal{D}}}_{q}^{\alpha }\left[k]\left(t)=\Omega \left(t,{k}_{1},{k}_{2},{k}_{3},{k}_{4}) , under some boundary conditions where Ω \Omega is singular at some point 0 ≤ t ≤ 1 0\le t\le 1 , on a time scale T t 0 = { t : t = t 0 q n } ∪ { 0 } {{\mathbb{T}}}_{{t}_{0}}=\left\{t:t={t}_{0}{q}^{n}\right\}\cup \left\{0\right\} , for n ∈ N n\in {\mathbb{N}} where t 0 ∈ R {t}_{0}\in {\mathbb{R}} and q ∈ ( 0 , 1 ) q\in \left(0,1) . We consider the compact map and avail the Lebesgue dominated theorem for finding solutions of the addressed problem. Besides, we prove the main results in context of completely continuous functions. Our attention is concentrated on fractional multi-step methods of both implicit and explicit type, for which sufficient existence conditions are investigated. Finally, we present some examples involving graphs, tables and algorithms to illustrate the validity of our theoretical findings.

Let ℋ {\mathcal{ {\mathcal H} }} be an infinite dimensional complex Hilbert space and ℬ ( ℋ ) {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }}) the algebra of all bounded linear operators on ℋ {\mathcal{ {\mathcal H} }} . For an operator T ∈ ℬ ( ℋ ) T\in {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }}) , we say property ( t ) \left(t) holds for T T if σ ( T ) ⧹ σ u w ( T ) = Π 00 ( T ) \sigma \left(T)\hspace{-0.08em}\setminus \hspace{-0.08em}{\sigma }_{uw}\left(T)={\Pi }_{00}\left(T) , where σ ( T ) \sigma \left(T) and σ u w ( T ) {\sigma }_{uw}\left(T) denote the spectrum and the Weyl essential approximate point spectrum of T T , respectively, and Π 00 ( T ) = { λ ∈ iso σ ( T ) : 0 < n ( T − λ ) < ∞ } {\Pi }_{00}\left(T)=\left\{\lambda \in {\rm{iso}}\sigma \left(T):0\lt n\left(T-\lambda )\lt \infty \right\} . In this paper, we consider the stability of property ( t ) \left(t) under (small) compact perturbations. Also, we explore the relations between the stability of property ( t ) \left(t) and the stability of Weyl-type theorems. Moreover, we characterize those operators T T satisfying that property ( t ) \left(t) holds for f ( T ) f\left(T) for each function f f analytic on some neighborhood of σ ( T ) \sigma \left(T) .

By utilizing the Nevanlinna theory of meromorphic functions in several complex variables, we mainly investigate the existence and the forms of entire solutions for the partial differential-difference equation of Fermat type α ∂ f ( z 1 , z 2 ) ∂ z 1 + β ∂ f ( z 1 , z 2 ) ∂ z 2 m + f ( z 1 + c 1 , z 2 + c 2 ) n = 1 {\left(\alpha \frac{\partial f\left({z}_{1},{z}_{2})}{\partial {z}_{1}}+\beta \frac{\partial f\left({z}_{1},{z}_{2})}{\partial {z}_{2}}\right)}^{m}+f{\left({z}_{1}+{c}_{1},{z}_{2}+{c}_{2})}^{n}=1 and α ∂ f ( z 1 , z 2 ) ∂ z 1 + β ∂ f ( z 1 , z 2 ) ∂ z 2 2 + [ γ 1 f ( z 1 + c 1 , z 2 + c 2 ) − γ 2 f ( z 1 , z 2 ) ] 2 = 1 , {\left(\alpha \frac{\partial f\left({z}_{1},{z}_{2})}{\partial {z}_{1}}+\beta \frac{\partial f\left({z}_{1},{z}_{2})}{\partial {z}_{2}}\right)}^{2}+{\left[{\gamma }_{1}f\left({z}_{1}+{c}_{1},{z}_{2}+{c}_{2})-{\gamma }_{2}f\left({z}_{1},{z}_{2})]}^{2}=1, where m , n m,n are positive integers and α , β , γ 1 , γ 2 \alpha ,\beta ,{\gamma }_{1},{\gamma }_{2} are constants in C {\mathbb{C}} . We give some results about the forms of solutions for these equations, which are great improvements of the previous theorems given by Xu and Cao et al. Moreover, it is very satisfactory that we give the corresponding examples to explain the conclusions of our theorems in each case.

In this paper, we investigate the dynamics of stochastic plate equations with memory in unbounded domains. More specifically, we obtain the uniform time estimates for solutions of the problem. Based on the estimates above, we prove the existence and uniqueness of random attractors in unbounded domains.

In this paper, based on the relationship between the linear complementarity problem and its reformulated fixed-point equation, we discuss the conditions of the modulus-based type iteration methods. Moreover, we present some convergence results on the two-step modulus-based matrix splitting iteration method with an H + {H}_{+} -matrix. Finally, we give the numerical experiments.

The inverse problem of reconstruction second-order Ito stochastic differential equations by the given properties of motion is solved when control is included in the drift coefficient. By using the separation method, the form of control parameters is determined that provides sufficient conditions for the existence of a given integral manifold for the nonlinear, linear, and scalar cases.

In this paper, we consider a novel estimation for partial functional linear regression models. The functional principal component analysis method is employed to estimate the slope function and the functional predictive variable, respectively. An efficient estimation based on principal component basis function approximation is used for minimizing the proposed weighted composite quantile regression (WCQR) objective function. Since the proposed WCQR involves a vector of weights, we develop a computational strategy for data-driven selection of the optimal weights. Under some mild conditions, the theoretical properties of the proposed WCQR method are obtained. The simulation study and a real data analysis are provided to illustrate the numerical performance of the resulting estimators.

Let L L be a lattice of rank n n in an n n -dimensional Euclidean space. We show that the coincidence isometry group of L L is generated by coincidence reflections if and only if L L contains an orthogonal subset of order n n .

In this paper, we establish some sharp maximal function estimates for certain Toeplitz-type operators associated with some fractional integral operators with general kernel. As an application, we obtain the boundedness of the Toeplitz-type operators on the Lebesgue, Morrey and Triebel-Lizorkin spaces. The operators include the Littlewood-Paley operator, Marcinkiewicz operator and Bochner-Riesz operator.

We study the boundedness of the oscillatory integral T α , β f ( x , y ) = ∫ Q 2 f ( x − γ 1 ( t ) , y − γ 2 ( s ) ) e − 2 π i t − β 1 s − β 2 t − α 1 − 1 s − α 2 − 1 d t d s {T}_{\alpha ,\beta }f\left(x,y)=\mathop{\int }\limits_{{Q}^{2}}f\left(x-{\gamma }_{1}\left(t),y-{\gamma }_{2}\left(s)){e}^{-2\pi i{t}^{-{\beta }_{1}}{s}^{-{\beta }_{2}}}{t}^{{-\alpha }_{1}-1}{s}^{-{\alpha }_{2}-1}{\rm{d}}t{\rm{d}}s on Wiener amalgam spaces, where Q 2 = [ 0 , 1 ] × [ 0 , 1 ] {Q}^{2}=\left[0,1]\times \left[0,1] is the unit square in two dimensions, ( x , y ) ∈ R n × R m , γ 1 ( t ) = ( t p 1 , t p 2 , … , t p n ) , γ 2 ( s ) = ( s q 1 , s q 2 , … , s q m ) \left(x,y)\in {{\mathbb{R}}}^{n}\times {{\mathbb{R}}}^{m},{\gamma }_{1}\left(t)=\left({t}^{{p}_{1}},{t}^{{p}_{2}},\ldots ,{t}^{{p}_{n}}),{\gamma }_{2}\left(s)=\left({s}^{{q}_{1}},{s}^{{q}_{2}},\ldots ,{s}^{{q}_{m}}) are homogeneous curves on R n {{\mathbb{R}}}^{n} and R m {{\mathbb{R}}}^{m} .

In this paper, we mainly show that generalized Euler-type sums of multiple harmonic sums with reciprocal binomial coefficients can be expressed in terms of rational linear combinations of products of classical multiple zeta values (MZVs) and multiple harmonic star sums (MHSSs). Furthermore, applying the stuffle relations, we prove that the Euler-type sums involving products of generalized harmonic numbers and reciprocal binomial coefficients can be evaluated by MZVs and MHSSs.

In this paper, we introduce generalizations of rising factorials and falling factorials, respectively, and study their relations with the well-known Stirling numbers, Lah numbers, and so on. The first stage is to define poly-falling factorial sequences in terms of the polyexponential functions, reducing them to falling factorials if k = 1 k=1 , necessitating a demonstration of the relations: between poly-falling factorial sequences and the Stirling numbers of the first and second kind, respectively; between poly-falling factorial sequences and the poly-Bell polynomials; between poly-falling factorial sequences and the poly-Bernoulli numbers; between poly-falling factorial sequences and poly-Genocchi numbers; and recurrence formula of these sequences. The later part of the paper deals with poly-rising factorial sequences in terms of the polyexponential functions, reducing them to rising factorial if k = 1 k=1 . We study some relations: between poly-falling factorial sequences and poly-rising factorial sequences; between poly-rising factorial sequences and the Stirling numbers of the first kind and the power of x x ; and between poly-rising factorial sequences and Lah numbers and the poly-falling factorial sequences. We also derive recurrence formula of these sequences and reciprocal formula of the poly-falling factorial sequences.

This paper is concerned with the transition functions of symmetric Levy-type processes generated by a pseudo-differential operator with variable coefficients. We first give the general estimates of heat kernels of jump diffusion semigroups, which leads to diagonal estimates of transition function and subordination in the context of two-dimensional Cauchy semigroup. Then off-diagonal estimates of special classes of Levy-type processes where transition function can be expressed using the diagonal estimation results and related metrics are derived. Furthermore, we show geometric approximation of the general two-dimensional Levy processes, and graphical experiments have been made by freezing the coefficients of the generators.

In the present work, we are concerned with the multiple solutions for quasilinear Choquard equation with critical nonlinearity in R N {{\mathbb{R}}}^{N} . We show multiplicity results for this problem, which are characterized, respectively, by the new version of symmetric mountain-pass theorem and the mountain-pass theorem for even functionals. The novelty of our work is the appearance of the convolution terms as well as critical nonlinearities.

In the present paper, we focus on studying the bifurcations and the traveling wave solutions (TWSs) for the regularized Schamel equation. Based on the bifurcation method of a dynamical system, a complete phase portrait analysis is given in various parameter conditions and some novel TWSs with the same energy of the Hamiltonian system are discovered. Various significant results on exact expressions of TWSs, including solitary waves, periodic waves, cusp waves, weak kink waves, loop solitons, compactons in different conditions are obtained.

The definition of an almost left factorizable weakly type B semigroup is introduced by using the notion of a left factorizable weakly type B monoids, and some properties and characterizations on such semigroup are obtained. The main aim of this paper is to give characterizations of sufficient and necessary conditions for a weakly type B semigroup to be almost left factorizable. In particular, some properties of proper almost factorizable weakly type B semigroups are given.

For any positive integer n and a set of positive integers m i {m}_{i} , i = 1 , 2 , … , n + 1 i=1,2,\ldots ,n+1 , we construct a class of quadratic eigenparameter-dependent boundary Sturm-Liouville problems with n transmission conditions, which have at most ∑ i = 1 n + 1 m i + n + 5 {\sum }_{i=1}^{n+1}{m}_{i}+n+5 eigenvalues. The key to this analysis is still the division of intervals and an iterative construction of the characteristic function. Further, some examples are given for a simple explanation.

In this paper, we consider the following quasilinear Schrödinger equation: − Δ u + V ( x ) u + κ 2 Δ ( u 2 ) u = K ( x ) f ( u ) , x ∈ R N , -\Delta u+V\left(x)u+\frac{\kappa }{2}\Delta \left({u}^{2})u=K\left(x)f\left(u),\hspace{1.0em}x\in {{\mathbb{R}}}^{N}, where N ≥ 3 N\ge 3 , κ > 0 \kappa \gt 0 , f ∈ C ( R , R ) f\in {\mathcal{C}}\left({\mathbb{R}},{\mathbb{R}}) , V ( x ) V\left(x) and K ( x ) K\left(x) are positive continuous potentials. Under given conditions, by changing variables and truncation argument, the energy of ground state solutions of the Nehari type is achieved. We also prove the existence of ground state sign-changing solutions for the aforementioned equation. Our results are the generalization work of M. B. Yang, C. A. Santos, and J. Z. Zhou, Least action nodal solution for a quasilinear defocusing Schrödinger equation with supercritical nonlinearity , Commun. Contemp. Math. 21 (2019), no. 5, 1850026, https://doi.org/10.1142/S0219199718500268.

This work studies a new version of normality called epi-quasi normality, which lies between epi-normality and epi-mild normality. In this paper, we investigate this property and present some examples that illustrate the relationships between epi-quasi normality and other kinds of both normality and regularity.

The derivative of a n n -order matrix function on the complex field is usually defined as a n 2 {n}^{2} -order matrix, which is not suitable for generalizing Cauchy integral formula of matrix functions to its higher-order derivative form. In this paper, a new kind of derivative of matrix functions is defined, and the higher-order derivative form of Cauchy integral formula of matrix functions is also proved to be true under the new kind of definition of derivative. At the same time, some examples about calculating the values of matrix functions by using Cauchy integral formula of matrix functions and its higher-order derivative form are given.

Let ( X , d , μ ) \left(X,d,\mu ) denote nonhomogeneous metric measure space satisfying geometrically doubling and the upper doubling measure conditions. In this paper, the boundedness in Lebesgue spaces for two kinds of commutators, which are iterated commutators and commutators in summation form, generated by multilinear strongly singular integral operators with RBMO ( μ ) \left(\mu ) function on nonhomogeneous metric measure spaces ( X , d , μ ) \left(X,d,\mu ) is obtained.

The multi-phase systems have found their applications in many fields. We shall apply this approach to investigate the multi-phase dynamics of sea ice growth. In this paper, the weak solution existence and uniqueness of parabolic differential equations are proved. Then large-time behavior of solutions is studied, and also the existence of the global attractor is proved. The key tool in this article is the energy method. Our existence proof is only in one dimension.

In this article, we investigate the existence of positive solutions for a class of m m -point fractional differential equations whose nonlinear terms involve derivatives. By using the properties of the Green function and fixed point theorems, some new conditions for the existence of at least one, two, three, and 2 n − 1 2n-1 positive solutions are established. As verification, some simulation examples are given to illustrate the main results. It is worth mentioning that we study the m m -point fractional differential equations with nonlinear terms involving derivative and use the iterative method to simulate our examples and give the approximate solution.

In this paper, we study Bernstein-Walsh type estimates for the higher-order derivatives of an arbitrary algebraic polynomial on quasidisks.

Abundant semigroups originate from p.p. rings and are generalizations of regular semigroups. The main aim of this paper is to study the primeness and the primitivity of abundant semigroup algebras. We introduce and study D ∗ {{\mathcal{D}}}^{\ast } -graphs and Fountain matrices of a semigroup. Based on D ∗ {{\mathcal{D}}}^{\ast } -graphs and Fountain matrices, we determine when a contracted semigroup algebra of a primitive abundant semigroup is prime (respectively, semiprime, semiprimitive, or primitive). It is well known that for any algebra A {\mathcal{A}} with unity, A {\mathcal{A}} is primitive (prime) if and only if so is M n ( A ) {M}_{n}\left({\mathcal{A}}) . Our results can be viewed as some kind of generalizations of such a known result. In addition, it is proved that any contracted semigroup algebra of a locally ample semigroup whose set of idempotents is locally finite (respectively, locally pseudofinite and satisfying the regularity condition) is isomorphic to some contracted semigroup algebra of primitive abundant semigroups. Moreover, we obtain sufficient and necessary conditions for these classes of contracted semigroup algebras to be prime (respectively, semiprime, semiprimitive, or primitive). Finally, the structure of simple contracted semigroup algebras of idempotent-connected abundant semigroups is established. Our results enrich and extend the related results on regular semigroup algebras.

In this paper, a third-order ordinary differential equation coupled to three-point boundary conditions is considered. The related Green’s function changes its sign on the square of definition. Despite this, we are able to deduce the existence of positive and increasing functions on the whole interval of definition, which are convex in a given subinterval. The nonlinear considered problem consists on the product of a positive real parameter, a nonnegative function that depends on the spatial variable and a time dependent function, with negative sign on the first part of the interval and positive on the second one. The results hold by means of fixed point theorems on suitable cones.

This paper deals with the fractional calculus of zeta functions. In particular, the study is focused on the Hurwitz ζ \zeta function. All the results are based on the complex generalization of the Grünwald-Letnikov fractional derivative. We state and prove the functional equation together with an integral representation by Bernoulli numbers. Moreover, we treat an application in terms of Shannon entropy.

This paper provides a uniqueness result for positive solutions of the Neumann and periodic boundary value problems associated with the ϕ -Laplacian equation ( ϕ ( u ′ ) ) ′ + a ( t ) g ( u ) = 0 , (\phi \left(u^{\prime} ))^{\prime} +a\left(t)g\left(u)=0, where ϕ is a homeomorphism with ϕ (0) = 0, a ( t ) is a stepwise indefinite weight and g ( u ) is a continuous function. When dealing with the p -Laplacian differential operator ϕ ( s ) = ∣ s ∣ p −2 s with p  > 1, and the nonlinear term g ( u ) =  u γ with γ ∈ R \gamma \in {\mathbb{R}} , we prove the existence of a unique positive solution when γ  ∈ ]− ∞ \infty , (1 − 2 p )/( p  − 1)] ∪ ] p  − 1, + ∞ \infty [.

One of the main problems connected with neural networks is synchronization. We examine a model of a neural network with time-varying delay and also the case when the connection weights (the influential strength of the j j th neuron to the i i th neuron) are variable in time and unbounded. The rate of change of the dynamics of all neurons is described by the Caputo fractional derivative. We apply Lyapunov functions and the Razumikhin method to obtain some sufficient conditions to ensure synchronization in the model. These sufficient conditions are explicitly expressed in terms of the parameters of the system, and hence, they are easily verifiable. We illustrate our theory with a particular nonlinear neural network.

We consider quasilinear elliptic problems of the form − div ( ϕ ( ∣ ∇ u ∣ ) ∇ u ) + V ( x ) ϕ ( ∣ u ∣ ) u = f ( u ) , u ∈ W 1 , Φ ( R N ) , -{\rm{div}}\hspace{0.33em}(\phi \left(| \nabla u| )\nabla u)+V\left(x)\phi \left(| u| )u=f\left(u),\hspace{1.0em}u\in {W}^{1,\Phi }\left({{\mathbb{R}}}^{N}), where ϕ \phi and f f satisfy suitable conditions. The positive potential V ∈ C ( R N ) V\in C\left({{\mathbb{R}}}^{N}) exhibits a finite or infinite potential well in the sense that V ( x ) V\left(x) tends to its supremum V ∞ ≤ + ∞ {V}_{\infty }\le +\infty as ∣ x ∣ → ∞ | x| \to \infty . Nontrivial solutions are obtained by variational methods. When V ∞ = + ∞ {V}_{\infty }=+\infty , a compact embedding from a suitable subspace of W 1 , Φ ( R N ) {W}^{1,\Phi }\left({{\mathbb{R}}}^{N}) into L Φ ( R N ) {L}^{\Phi }\left({{\mathbb{R}}}^{N}) is established, which enables us to get infinitely many solutions for the case that f f is odd. For the case that V ( x ) = λ a ( x ) + 1 V\left(x)=\lambda a\left(x)+1 exhibits a steep potential well controlled by a positive parameter λ \lambda , we get nontrivial solutions for large λ \lambda .

Fourier approximation plays a key role in qualitative theory of deterministic and random differential equations. In this paper, we will develop a new approximation tool. For an m m -order differentiable function f f on [ 0 , 1 0,1 ], we will construct an m m -degree algebraic polynomial P m {P}_{m} depending on values of f f and its derivatives at ends of [ 0 , 1 0,1 ] such that the Fourier coefficients of R m = f − P m {R}_{m}=f-{P}_{m} decay fast. Since the partial sum of Fourier series R m {R}_{m} is a trigonometric polynomial, we can reconstruct the function f f well by the combination of a polynomial and a trigonometric polynomial. Moreover, we will extend these results to the case of random processes.

We investigate a fractional Schrödinger-Poisson system with strong singularity as follows: ( − Δ ) s u + V ( x ) u + λ ϕ u = f ( x ) u − γ , x ∈ R 3 , ( − Δ ) t ϕ = u 2 , x ∈ R 3 , u > 0 , x ∈ R 3 , \left\{\begin{array}{ll}{\left(-\Delta )}^{s}u+V\left(x)u+\lambda \phi u=f\left(x){u}^{-\gamma },& x\in {{\mathbb{R}}}^{3},\\ {\left(-\Delta )}^{t}\phi ={u}^{2},& x\in {{\mathbb{R}}}^{3},\\ u\gt 0,& x\in {{\mathbb{R}}}^{3},\end{array}\right. where 0 < s ≤ t < 1 0\lt s\le t\lt 1 with 4 s + 2 t > 3 4s+2t\gt 3 , λ > 0 \lambda \gt 0 and γ > 1 \gamma \gt 1 . When V V and f f satisfy certain conditions, existence and uniqueness of positive solution u λ {u}_{\lambda } are established via variational method and Nehari method. We also describe the asymptotic behaviour of u λ {u}_{\lambda } as λ → 0 \lambda \to 0 .

In this paper, we provide existence and uniqueness of entropy solutions to a general nonlinear parabolic problem on a general convex set with merely integrable data and in the setting of Orlicz spaces.

We deal with some impulsive Caputo-Fabrizio fractional differential equations in b b -metric spaces. We make use of α - ϕ \alpha \text{-}\phi -Geraghty-type contraction. An illustrative example is the subject of the last section.

This work intends to treat the existence of mild solutions for the Hilfer fractional hybrid differential equation (HFHDE) with linear perturbation of first and second type in partially ordered Banach spaces. First, we establish the results concerning the actuality of fixed point of sum and product of operators via the concepts of measure of noncompactness and simulation functions in partially ordered spaces. Then combining these fixed point theorems with the concepts in fractional calculus, new existence results for mild solutions of HFHDE’s are established. Furthermore, the presented fixed point results and existence results improve and extend the present state-of-art in the literature. Competent examples in support of theory are illustrated for better understanding.

We study a nonlinear system of Riemann-Liouville fractional differential equations equipped with nonseparated semi-coupled integro-multipoint boundary conditions. We make use of the tools of the fixed-point theory to obtain the desired results, which are well-supported with numerical examples.

This paper is devoted to discussing the blow-up problem of the positive solution of the following degenerate parabolic equations: ( r ( u ) ) t = div ( ∣ ∇ u ∣ p ∇ u ) + f ( x , t , u , ∣ ∇ u ∣ 2 ) , ( x , t ) ∈ D × ( 0 , T ∗ ) , ∂ u ∂ ν + σ u = 0 , ( x , t ) ∈ ∂ D × ( 0 , T ∗ ) , u ( x , 0 ) = u 0 ( x ) , x ∈ D ¯ . \left\{\begin{array}{ll}{(r\left(u))}_{t}={\rm{div}}(| \nabla u{| }^{p}\nabla u)+f\left(x,t,u,| \nabla u\hspace{-0.25em}{| }^{2}),& \left(x,t)\in D\times \left(0,{T}^{\ast }),\\ \frac{\partial u}{\partial \nu }+\sigma u=0,& \left(x,t)\in \partial D\times \left(0,{T}^{\ast }),\\ u\left(x,0)={u}_{0}\left(x),& x\in \overline{D}.\end{array}\right. Here p > 0 p\gt 0 , the spatial region D ⊂ R n ( n ≥ 2 ) D\subset {{\mathbb{R}}}^{n}\hspace{0.33em}\left(n\ge 2) is bounded, and its boundary ∂ D \partial D is smooth. We give the conditions that cause the positive solution of this degenerate parabolic problem to blow up. At the same time, for the positive blow-up solution of this problem, we also obtain an upper bound of the blow-up time and an upper estimate of the blow-up rate. We mainly carry out our research by means of maximum principles and first-order differential inequality technique.

In this paper, we study the long time decay of global solution to the 3D incompressible Navier-Stokes equations. We prove that if u ∈ C ( R + , X − 1 , σ ( R 3 ) ) u\in {\mathcal{C}}\left({{\mathbb{R}}}^{+},{{\mathcal{X}}}^{-1,\sigma }\left({{\mathbb{R}}}^{3})) is a global solution to the considered equation, where X i , σ ( R 3 ) {{\mathcal{X}}}^{i,\sigma }\left({{\mathbb{R}}}^{3}) is the Fourier-Lei-Lin space with parameters i = − 1 i=-1 and σ ≥ − 1 \sigma \ge -1 , then ‖ u ( t ) ‖ X − 1 , σ \Vert u\left(t){\Vert }_{{{\mathcal{X}}}^{-1,\sigma }} decays to zero as time goes to infinity. The used techniques are based on Fourier analysis.

We deal with the extinction properties of the weak solutions for a p -Laplacian equation with a gradient nonlinearity. The critical extinction exponent is specified and the decay estimates of the extinction solutions are given.

A nonlinear viscoelastic wave equation with Balakrishnan-Taylor damping and distributed delay is studied. By the energy method we establish the general decay rate under suitable hypothesis.

In this work, we consider the hyponormality of Toeplitz operators on the Bergman space of the annulus with a logarithmic weight. We give necessary conditions when the symbol is of the form φ + ψ ¯ \varphi +\overline{\psi } , where φ \varphi and ψ \psi are analytic on the annulus { z ∈ C ; 1 / 2 < ∣ z ∣ < 1 } \{z\in {\mathbb{C}};\hspace{0.25em}1\hspace{-0.08em}\text{/}\hspace{-0.08em}2\lt | z| \lt 1\} .

This article concerns linear one-dimensional thermoelastic Timoshenko system with memory and distributed delay terms where the Cattaneo law governs the heat flux q ( x , t ) q\left(x,t) . We proved an exponential stability result by using the energy method combined with Lyapunov functional.

We get the strong and Δ \Delta -convergence of the Picard-Krasnoselskii hybrid iteration scheme to a fixed point of a self-map endowed with the condition ( B γ , μ ) \left({B}_{\gamma ,\mu }) . We use the nonlinear context of CAT(0) spaces for establishing these results. We present a new example of a self-map endowed with ( B γ , μ ) \left({B}_{\gamma ,\mu }) condition and prove that its Picard-Krasnoselskii hybrid iterative process is more effective than the Picard and Krasnoselskii hybrid iterative processes. This improves and extends some recently announced results of the current literature.

A linear stability analysis is performed for the onset of Marangoni convection in a horizontal layer of a nanofluid heated from below and affected by rotation. The top boundary of the layer is assumed to be impenetrable to nanoparticles with their distribution being determined from a conservation condition while the bottom boundary is assumed to be a rigid surface with fixed temperature. The motion of the nanoparticles is characterized by the effects of thermophoresis and Brownian diffusion. A modification model is used in which the effects of Brownian diffusion and thermophoresis are taken into consideration by new expressions in the nanoparticle mass flux. Also, material properties of the nanofluid are modelled by non-constant constitutive expressions depending on nanoparticle volume fraction. The steady-state solution is shown to be well approximated by an exponential distribution of the nanoparticle volume fraction. The Chebyshev-Tau method is used to obtain the critical thermal and nanoparticle Marangoni numbers. Different stability boundaries are obtained using the modified model and the rotation.

This work considers the infinite multi-server Markovian queueing model with balking and catastrophes where the rates of arrivals, service, balking, and catastrophes are time dependent. The catastrophes arrive as negative customers to the system. The arrival of negative customers to a queueing system removes the positive customers. The catastrophes may come either from another service station or from outside the system. In this paper, we obtained the transient solution of this model using the approach of probability-generating function. Also, we derived an expression of transient probabilities in terms of Volterra equation of the second kind. Furthermore, we obtained a measure for time-dependent expected number of customers in the system.

The incompressible 2D stochastic Navier-Stokes equations with linear damping are considered in this paper. Based on some new calculation estimates, we obtain the existence of random attractor and the upper semicontinuity of the random attractors as ε → 0 + \varepsilon \to {0}^{+} on the two-dimensional space.

The aim of this paper is to study the Poisson random variables in relation to the Lah-Bell polynomials and the degenerate binomial and degenerate Poisson random variables in connection with the degenerate Lah-Bell polynomials. Among other things, we show that the rising factorial moments of the degenerate Poisson random variable with parameter α \alpha are given by the degenerate Lah-Bell polynomials evaluated at α \alpha . We also show that the probability-generating function of the degenerate Poisson random variable is equal to the generating function of the degenerate Lah-Bell polynomials. Also, we show similar results for the Poisson random variables. Here the n n th Lah-Bell number counts the number of ways a set of n n elements can be partitioned into non-empty linearly ordered subsets, the Lah-Bell polynomials are natural extensions of the Lah-Bell numbers and the degenerate Lah-Bell polynomials are degenerate versions of the Lah-Bell polynomials.

In this work, we initiate the study of a new class of impulsive boundary value problems consisting of mixed type fractional quantum and Hadamard derivatives. We will establish existence and uniqueness results by using tools from the functional analysis. We prove the uniqueness result via Banach’s contraction mapping principle, while we will use the Leray-Schauder nonlinear alternative to establish an existence result. We also present examples to illustrate the obtained results.

The ( p , q p,q )-th dual curvature measures and the L p {L}_{p} dual Minkowski problem were recently introduced by Lutwak, Yang, and Zhang. In this paper, we give a solution to the existence part of the L p {L}_{p} dual Minkowski problem about 0 < p < 1 0\lt p\lt 1 and q > 0 q\gt 0 for arbitrary measures. This fills up previously obtained results.