In mathematical optimization theory, the linear complementarity problem plays a vital role in many areas such as bimatrix game theory, market equilibrium, computational complexity, and many more. The nature of the solution of the linear complementarity problem can be discussed with the help of the matrix A involved in the problem. Sign reversing is a property of matrices along with a given vector. An n × n matrix A reverses the sign of a vector x in an n-dimensional real vector space if it satisfies x_i(Ax)_i≤ 0 for all index i. The concept of sign reversing is a useful tool to identify and characterize certain matrix classes in linear complementarity problems. The sign reversing set of a matrix A is defined as {x: x_i(Ax)_i ≤ 0, for all i} . In this paper, we have defined the concept of sign reversing set for operators on separable Hilbert spaces relative to a given orthonormal basis and we have observed that an operator can have different sign reversing sets relative to different orthonormal bases. With the help of sign reversing property, we generalized the sufficient matrix classes to operators in infinite dimensional Hilbert spaces.
The present paper aims to investigate super-biderivations and 2-local super-derivations on the n-th super Schrödinger algebra 𝒮_n , which is viewed as a semidirect product of the Lie superalgebra 𝔬𝔰𝔭(1|2) and the (3n+1) -dimensional Heisenberg Lie superalgebra. We first give a description of all super-derivations of 𝒮_n . We also show that any derivation of the n-th Schrödinger algebra (𝒮_n)_0̅ is isomorphic to the restriction of an even super-derivation on 𝒮_n to (𝒮_n)_0̅ . Then we prove that all super-biderivations (without skew-supersymmetric conditions) on 𝒮_n are inner. Based on this result, we also characterize the linear super-commuting maps and commutative post-Lie superalgebra structures on 𝒮_n . Finally, we show that every 2-local super-derivation on 𝒮_n is a super-derivation.
In this paper we introduce the concepts of α -neutralized Bowen entropy ( α >0 ) and neutralized Bowen entropy dimension for subsets, defined via neutralized Bowen open balls. We then establish a number of variational principles for these quantities. Finally, by introducing a new measure‑theoretic version of α -neutralized entropy for subsets, we prove several inverse variational principles for α -neutralized Bowen entropy and for neutralized Bowen entropy dimension.
Seymour’s second neighborhood conjecture states that every oriented graph G⃗ has a Seymour vertex, namely, G⃗ has a vertex whose second-order out-neighborhood is at least as large as its first-order out-neighborhood. In this paper, we approach the conjecture by considering an inhomogeneous random graph G, where each edge e in the complete graph K_n appears independently with probability p_n(e) . Under suitable density and regularity conditions, we show that every orientation of G contains a Seymour vertex with high probability, confirming the conjecture asymptotically. Moreover, if we consider an inhomogeneous random oriented graph G⃗ by assigning an orientation to each edge of G independently with equal probability, we prove that G⃗ contains a Seymour vertex with high probability across a broader range of regimes.
The well-known Piatetski–Shapiro primes are primes of the form [n^c] . In this paper, we prove that there are infinitely many Piatetski–Shapiro primes in arithmetic progressions for 1
A lower bound on how many times a division maximal modulus of an exact covering system in the ring of integers repeating was obtained by Berger, Felzenbaum and Fraenkel. Jiang and Deng extended Berger et al.’s result to exact covering systems in number fields. In this note we generalize above results to exact covering systems in finite principal ideal rings.
The aim of this contribution is twofold. First, by performing the quadratic decomposition of q-Dunkl-Appell sequences, a new lowering q-differential operator 𝒦_q^2 ; θ ;κ (with κ =±1/2 ) naturally emerges, as the two polynomial sequences lying in the principal diagonal are 𝒦_q^2 ; θ ;κ -Appell. Second, triggered by this result, after developing the concept of the 𝒦_q ; θ ;κ -Appell sequences, all the orthogonal 𝒦_q ; θ ;κ -Appell sequences are sought, which outcome was the Wall q-polynomials with parameter b=q^1-κ/1-(q^1/2-1)θ (resp. the Little q-Laguerre polynomials with parameter α =bq^-1 ) – they are indeed the unique ones fulfilling both properties, up to a linear transformation. This leads to a new characterization of these polynomial sequences.
Let r(1), ... , r(t) be positive integers with r(1)(-1) + center dot center dot center dot + r(t)(-1) >= 1. Recently, Chen and Xu proved that the set of positive integers that can be written as p + 2(k1r1) + center dot center dot center dot + 2(ktrt), where k(1), ... , k(t) are positive integers and p is prime, has a positive lower asymptotic density. In this paper, we generalize their result. In particular, let L = {L-n : n = 0, 1, 2, ...} be the Lucas sequence. We prove that the set of positive integers that can be written as p + L-k1r1 + center dot center dot center dot + L-ktrt has a positive lower asymptotic density. For t = r(1) = 1, we show that there is a positive proportion of all positive integers that can be uniquely represented as the sum of a prime and a Lucas number.
In this paper we introduce the notion of Archimedes invariant partition of a curve and of a family of curves. Let f:ℝ→ℝ be a strictly convex function and let n be an integer, n≥ 2 . We say that an increasing sequence t=(t_0=0,t_1,… ,t_n=1) of numbers of the interval [0; 1] defines Archimedes invariant partition of the curve ℝ∋ x ↦ (x, f(x)) if for every -∞
This paper deals with the existence and nonexistence of normalized solutions for a type of fractional Schrödinger–Choquard system with critical nonlinearity { (-Δ )^s u = λ _1 u + μ _1|u|^p-2u+(I_α *|u|^2_α ,s^*)|u|^2_α ,s^*-2u+β r_1|u|^r_1-2u|v|^r_2, (-Δ )^s v = λ _2 v + μ _2|v|^q-2v+(I_α *|v|^2_α ,s^*)|v|^2_α ,s^*-2v+β r_2|u|^r_1|v|^r_2 -2v . with the restrictions ∫ _ℝ^N|u|^2dx=a and ∫ _ℝ^N|v|^2dx=b , where a,b>0 are prescribed, 1/2≤ s<1 , 2≤ N≤ 4s , α∈ (0,N) , I_α (x):=Γ (α/2)/Γ (N-α/2)π ^N/22^N-α|x|^α , x∈ℝ^N∖{0} is the Riesz potential, μ _1 , μ _2 , β >0 , r_1,r_2>1 and 2_α ,s^*:=2N-α/N-2s . The frequencies λ _1 and λ _2 appear as Lagrange multipliers. (-Δ )^s is the fractional Laplace operator. In the literature, any (u, v) solving the above system (for some λ _1 , λ _2 ) is called a normalized solution. In the case where 2+4s/N0 , we prove the existence of a positive normalized solution. For the triple Sobolev-critical growth case p=q=r_1+r_2=2_s^* , we obtain the nonexistence of a positive normalized solution.
If a family ℱ of k-element subsets of an n-element set is intersecting, then the sum of the sizes of the pairwise intersections of any ℓ members of the family is at least ℓ ()2 . The classic result of Erdős, Ko and Rado says that under the condition 2k≤ n an intersecting family of k-element subsets of an n-element set cannot have more than n-1 ()k-1 members. Is this weaker condition for the sum of the sizes of the pairwise intersections sufficient to have the conclusion of the Erdős–Ko–Rado theorem? We will see that much more is true, the bound ℓ ()2 can be replaced by ℓ -1 ()2+1 .
The aim of this paper is to study the local existence of solutions to an abstract quasilinear coupled system of Kirchhoff and heat equations with a nonlinear inhomogeneous term. We establish local existence using a method introduced by Kato [22], combined with the multiplier method and a fixed-point argument.
Let V be the set of odd positive integers that can be represented as p+2(a2)+2(b2), where p is a prime and a, b are positive integers. In 2022, Yuchen Ding proved that V has positive lower asymptotic density. In 2024, the authors showed that n is not an element of V if n equivalent to 293(mod510). In this paper, we prove that if {kn+l:k=0,1, . . .} is an arithmetic progression of odd positive integers with no terms in V, then k >= 510 and k has at least four distinct prime factors. Furthermore, these bounds are best possible. This topic goes back to a conjecture of de Polignac from 1849.
In this paper, we find all the solutions of the Diophantine equation $$F_{k,1}^2+2F_{k,2}^2+\cdots +mF_{k,m}^2=F_{k,n}^q$$ F k , 1 2 + 2 F k , 2 2 + ⋯ + m F k , m 2 = F k , n q in positive integer variables ( m , n ), where $$F_{k,i}$$ F k , i is the $$i^{th}$$ i th term of the $$k^{th}$$ k th Fibonacci sequence, k a positive integer and $$q\in \{1,2\}$$ q ∈ { 1 , 2 } .
Let r_1, … , r_t be positive integers with r_1^-1+⋯ +r_t^-1≥ 1. Recently, Chen and Xu proved that the set of positive integers that can be written as p+2^k_1^r_1+⋯ +2^k_t^r_t, where k_1, … , k_t are positive integers and p is prime, has a positive lower asymptotic density. In this paper, we generalize their result. In particular, let ℒ={L_n: n=0,1,2,…} be the Lucas sequence. We prove that the set of positive integers that can be written as p+L_ k_1^r_1+⋯ +L_k_t^r_t has a positive lower asymptotic density. For t=r_1=1, we show that there is a positive proportion of all positive integers that can be uniquely represented as the sum of a prime and a Lucas number.
In this article, we study the approximation behavior of the linear integral operators in the settings of Morrey spaces. Under some assumptions on the kernel function, first we obtain the pointwise and uniform approximation for bounded continuous and bounded uniformly continuous functions, respectively. Next we get error estimates for these operators in terms of a K-functional. We study the regularization properties of these operators. Furthermore, using the equivalence of the K-functional and the modulus of smoothness in the Morrey space setting, we obtain a characterization of the generalized Lipschitz classes in terms of convergence of the linear integral operators. Towards the end, we provide some examples of specific kernels which satisfy the required assumptions.
Considering the fractional Brownian motion B_t as a generalized stochastic process in S'(ℝ)⊗ (S)_-1 , the fractional white noise W^H_t is defined as the distributional derivative of dB^H_t . Using the framework of white noise analysis and the integral representation of B^H_t , we explicitly calculate the coefficients of their chaos expansion and provide a recurrence formula for their effective calculation. As a novel stochastic model, we introduce the notion of fractional Brownian motion and fractional white noise with a distributed-order Hurst parameter H∈ (0,1) . Building on this construction, we derive numerical simulations of B^H_t and W^H_t by truncating the obtained chaos expansions, which allows us to approximate their sample paths and estimate the truncation error. We illustrate the results with two examples from finance and life insurance: a stock price model and a mortality model, both driven by a fractional white noise process with distributed-order Hurst parameter.
