This paper provides a complete classification of 4-dimensional complex transposed Poisson algebras. The classification yields 120 distinct isomorphism classes, comprising 48 isolated algebras and 72 parametric families.
Let F(x) be the zero-balanced Gaussian hypergeometric function. In this paper, by introducing two classes of logarithmic-type functions and proving the monotonicity and absolute monotonicity of certain functions involving F(x) , we establish several sharp functional inequalities for F(x) . In particular, the best possible parameters in the corresponding two-sided bounds are determined. These results extend the corresponding sharp bounds for the complete elliptic integral of the first kind.
The weighted majorization inequality is extended to ϵ -approximately convex functions. Using this weighted majorization tool, we prove that the Landau–Kolmogorov type inequality ∏ _i=0^m‖ f^(i)‖ _∞^q_i≤ C(i;p,q)∏ _i=0^m‖ f^(i)‖ _∞^p_i holds for all f∈ L_∞ ,∞^m , if and only if i_q≺ _pi for non-negative p, q. As the constant estimate C(i;p,q)<(π/2)^∑ _i=0^mq_i is not sharp, we obtain the sharp constant in some easy cases. Under the assumption i_q≺ _pi and some simplifications, the constant C(i; p, q) is ∏ _i=0^m-1(K_m-i)^q_i/∏ _i=0^m-1(K_m-i)^p_i (where K_j are Favard constants) and is the best, if ∑ _i≥ lp_i≤∑ _i≥ lq_i, l=0,...,m'. Since this is not the necessary condition, an algorithm is provided as the necessary and sufficient condition. We also consider the best constant in some more difficult cases when the sharp constant is not the situations mentioned above, which are some of the simplest examples of difficult cases.
We study finite sums weighted by generalized hyperharmonic-type terms of the form ∑ _n=1^N α _n· h_n^(d)(β _i), where (α _n) , (β _i) are arbitrary sequences. Our main result is a general transformation formula that reduces such sums to finite linear combinations of simpler double sums with polynomial weights. This provides a unified framework for the reduction of a broad class of hyperharmonic-type sums and series. As applications, we derive explicit formulas for Dirichlet-type sums with powers of linear denominators, together with their alternating and odd variants, and recover several known evaluations as special cases. In particular, the method yields new derivations of classical results for series with hyperharmonic numbers, extends them to more general hyperharmonic-type terms, and produces closed-form evaluations in terms of zeta values, Euler-type double sums, and polylogarithmic constants. We also obtain finite summation identities involving Bernoulli and Euler polynomials.
Let f be a transcendental entire function with hyper-order strictly less than 1. If f and Δ ^n f , or f' and f(z+1) , share a function CM, then the exact form of f is determined, and the relationship between the corresponding pair of functions is established. These results are used to investigate the sharing value question provided f and Δ ^n f , or f' and f(z+1) , share a set, which partially solve Liu’s question raised in [22]. Furthermore, several examples are presented to demonstrate these results.
In this paper, we get the dichotomy of fiber topological pressure of the fiber irregular set I_ψ(ω ) in the skew product systems with the fiber specification driven by an ergodic system. That is, for ℙ- a.e. ω∈Ω , I_ψ(ω ) is either empty or carries full fiber topological pressure. As an application, we prove that for a class of random conformal repellers with the fiber topological mixing property, the fiber irregular set has full fiber BS dimension and Hausdorff dimension.
This paper proposes a new approach, centered on the introduction of a characteristic tuple, denoted τ _Δ , which serves as the mathematical foundation for the calculus of type-2 interval-valued functions (T2-IVFs). By establishing a consistent order relation based on τ _Δ -monotonicity, we overcome the disadvantage of many preceding studies to define the concept of generalized Hukuhara difference (gH-difference) that guarantees the operational closure of the difference. Based on this approach, some important calculus concepts of the space of T2-IVFs are systematically developed and especially, the concept of gH-differentiability defined by τ _Δ -monotonicity is a key foundation to determine the concepts of Riemann-Liouville (RL) and Caputo fractional derivative operators for T2-IVFs. Finally, we establish the existence of a unique type-2 interval-valued solution to the initial value problem for the T2-IVF fractional differential systems, employing the characteristic tuple τ _Δ on a suitable weighted metric space of continuous T2-IVFs.
Let A be a unital C^* -algebra. An A-multiplier cover is a C^* -algebra E together with a faithful non-degenerate * -homomorphism from A to M(E). We preorder such covers by A-preserving unital completely positive maps between their multiplier algebras. We prove that Hamana’s injective envelope I(A) is a greatest cover in this preorder and that the maximal rigid covers are precisely those whose multiplier algebra is canonically * -isomorphic to I(A). Consequently, a maximal rigid cover is greatest, rather than merely maximal among rigid covers. For A=C(X) , we further classify these covers: after the canonical identification with C(G(X)), where G(X) is the Gleason cover, their underlying ideals are exactly the algebras C_0(U) for dense open C^* -embedded subsets U of G(X). Dense cozero subsets provide an important special case.
We introduce a generator-counting refinement of algebrability for abelian C^* -algebras and related Banach algebras. Given an abelian C^* -algebra A, we define (C^*) -genalgebrability in terms of the minimal possible cardinality of a generating set, encoded by the invariants gen _C^*(A) and gen (A) . Using compact multipliers and the ideal K(A) of compact elements, we develop embedding results into ℓ _∞ whose ranges avoid c_0 (except for the zero vector), and we obtain a universal (ℓ _∞∖ c_0) -embeddability phenomenon under the assumption K(A)={0} . As an application, we construct a a^* -isomorphic copy of ℓ _∞ inside (ℓ _∞∖ c_0)∪{0} and transfer the results to Calkin-type settings such as (B(ℓ _2)∖𝒦(ℓ _2))∪{0} and their unitizations. We also establish a generator-counting theorem for abelian C^* -algebras: gen _C^*(A) equals the smallest cardinal n for which the spectrum Δ (A) embeds into ℝ^n , and we derive topological formulas for gen _C^*(A) in the non-finitely generated case. Finally, we provide a complete classification of the pairs (d,κ ) for which (ℓ _∞∖ c_0)∪{0} is (d,κ ) - (C^*) -genalgebrable, and we discuss the connection with classical algebrability.
In this paper, we demonstrate that a class of high-dimensional stochastic partial differential equations (SPDEs) can be approximated by residual neural networks (ResNet). Under our numerical scheme, the complexity of our proposed ResNet is controllable in the case of high dimensions, growing at most polynomially. We use the splitting-up method to separate the SPDE into deterministic partial differential equation (PDE) and stochastic differential equation (SDE). In each time interval, we associate the solution of PDE with SDE by Feynman–Kac formula. Since the right endpoint of PDE and SDE for each time interval in the splitting-up method are generated by the two ResNets, the ResNet constructed by us in the proof requires repeated cross-iteration, but the complexity is still well controlled.
Abstract This paper establishes lower and upper bounds for the radial eigenvalues of nonlinear elliptic systems defined in appropriate annular domains of $$\mathbb {R}^N$$ R N . For the problem, we will prove a result in the sense of a conjecture proposed by Nápoli and Pinasco in the paper Estimates for eigenvalues of quasilinear elliptic systems , J. Diff. Eq. 227 (2006), 102–115, when we are in certain domains of $$\mathbb {R}^N$$ R N . Moreover, we obtain a hyperbolic type function defining a region that contains all the generalized eigenvalues (whether variational or not), and with general hypotheses, the proof is carried out via ABP estimate for the p-Laplacian operator.
Abstract How reasonable is the conjecture that $$\ell _\infty /C[0,1]$$ ℓ ∞ / C [ 0 , 1 ] separably injective? Formulated in homological terms this amounts to asking whether $$\operatorname {Ext}^2(S, C[0,1])=0$$ Ext 2 ( S , C [ 0 , 1 ] ) = 0 for every separable space. We wheel around this question and obtain new connections between $$\operatorname {Ext}^2$$ Ext 2 -results and classical Banach space theorems of Johnson–Zippin, Kalton and Lindenstrauss–Pełczyński.
We prove that the family of Lebesgue integrable functions which are nowhere Riemann integrable is strongly 2^𝔠 -algebrable. We show that there is a free algebra 𝒜 with 𝔠 generators consisting of continuous functions such that for any nonzero g∈𝒜 there is a free algebra ℬ_g with 2^𝔠 generators consisting of measurable functions such that f∘ g is nonmeasurable on any interval for any nonzero f∈ℬ_g . It turns out that such ℬ cannot be found for any linear space of continuous g’s. These give the best possible improvements of results obtained by Azagra, Fernández-Sánchez, Muñoz-Fernández, Rodríguez-Vidanes, Sánchez, Seoane-Sepúlveda and Trutschnig.
Let P and Q 0 be real numbers such that P^2-4Q>0 . The Lucas sequence {U_n}_n ≥ 0 is defined by the recurrence relation U_n+2=P U_n+1- QU_n (n ≥ 0) with U_0=0 and U_1=1 . Kamano introduced the Lucas zeta function Φ ^(P,Q)(s):=∑ ^∞_n=11/U_n^s, σ =Re s>0, and proved that the Lucas zeta function Φ ^(P,Q)(s) can be meromorphically continued to the whole s-plane except for some simple poles. Moreover, he discussed the values of Φ ^(P,-1)(s) at negative integers and found that -m with m≥ 0 , m ≡ 2 ( 4) are its integral zeros. In this paper we provide its zero-free regions in the half-plane σ =Re s >0 and consider the zero-free regions in the half-plane σ <0 with the assumption Q=± 1 . In the case Q=1 , we derive that the whole negative real axis is the zero-free interval of Φ ^(P,1)(s) for P≥ 3 and that all the compact subsets of H_-2,0∪ H_-4,-2∪ H_-6,-4 are the zero-free regions of Φ ^(P,1)(s) if P is large enough, where H_a,b= { s= σ +it: a< σ < b}. And in the other case Q=-1 , we derive that [-1,0)∪⋃ _ n ∈ℕ_+ [ -4n-1,-4n+1 ] is the zero-free interval of Φ ^(P,-1)(s) for P≥ 2 and that all the compact subsets of H_-2,0∪ H_-4,-2 are the zero-free regions of Φ ^(P,-1)(s) if P is large enough.
In this paper, by virtue of the beta polynomials, we introduce and study a new special word family referred to as the beta-based words. In order to construct the beta-based words and their generation algorithm, we inspire from the definition of the beta polynomials. By making implementation in the Wolfram language, we obtain some tables for these words and their decimal equivalents. We also show that the decimal equivalents of the beta-based words are associated with the Mersenne numbers. For their visual illustrations, we also present plots designed by black-white superimposed square blocks emerging from the first few beta-based words. Moreover, we investigate some properties of the beta-based words, involving length, height, slope, symmetry and complement properties. Additionally, in this paper, we construct a generalization of harmonic numbers. We also derive some combinatorial sums related to the beta polynomials, the Bernstein polynomials and the generalized harmonic numbers.
Here we expose multi-composite multivariate quantitative approximations of Banach space valued continuous multivariate functions on a box or ℝ ^N, N∈ℕ , by the multi-composite multivariate normalized, quasi-interpolation, Kantorovich type and quadrature type neural network operators. We treat also the case of approximation by multi-composite iterated multi-layer operators of the last four types. These approximations are derived by establishing multi-composite multidimensional Jackson type inequalities involving the multivariate modulus of continuity of the engaged function or its high order Fréchet derivatives. Our multivariate operators are defined by using a multi-composite multidimensional density function induced by a multi-composite general sigmoid activation function. The approximations are pointwise and uniform. The related feed-forward neural network is with one hidden layer.
In this paper, we establish a family of weighted logarithmic Hardy–Rellich inequalities on connected Lie groups equipped with Hörmander systems of left-invariant vector fields. In the setting of graded Lie groups, we obtain refinements in terms of homogeneous Sobolev norms associated with Rockland operators. As a consequence, on stratified (Carnot) groups we derive Gross-type logarithmic Hardy inequalities with respect to a product measure that is Gaussian on the first stratum and Lebesgue on the higher strata. We also present a logarithmic Poincaré inequality on stratified groups and a fractional logarithmic Hardy inequality for the fractional p-sub-Laplacian on homogeneous groups. Several of the resulting weighted logarithmic Hardy–Rellich inequalities appear to be new already in ℝ^n .
