We prove Trudinger-type inequalities for variable Riesz potentials I alpha(& centerdot;),tau f of functions in Musielak-Orlicz-Morrey spaces L Phi,kappa,theta(X) over bounded non-doubling metric measure spaces X, under conditions on 'I which are weaker than those considered in the previous paper by Hurri-Syrja & uml;nen and the authors in 2023. We also discuss the case when 'I is the double-phase functional with variable exponents.
Motivated by the well-known graphical method, we give a geometric characterization of optimal linear programs. Our condition and approach rely on the tools of convex analysis. Among the applications, we present the strong duality theorem and revisit Farkas' lemma, as well. Although the auxiliary tools are well-known or follow from highly nontrivial results, we present their (independent) proof in order to keep the paper self-contained.
In this paper, we investigate a generalized Randers metric with a special it-form. Precisely, following the pullback approach to global Finsler geometry, we start with a Finsler metric (M, L) that admits a concurrent it-vector field, then we consider the Randers change Le = L + B, where B is the associated 1-form. We study some of the geometric objects and properties attached to (M, Le). We prove that the corresponding metric tensor of Le is nondegenerate without any conditions on the 1-form B. By calculating the relation between the attached geodesic sprays of L and Le, we establish that the change Le = L + B preserves the geodesics of (M, L), that is, the change is projective. Moreover, under a certain condition, this change preserves the deviation tensor, the (v)h-torsion tensor, the (h)h-curvature tensor and the curvature tensor of the Barthel connection. We provide an example of a Finsler metric with some details that admits a concurrent vector field together with the associated it-form.
Let 0 =/ D subset of R-n be a convex set. We say that the function f : D -> R is weakly k-affine if, for any affinely independent system of k+1 points x(0), x(1), ... , x(k) is an element of D, there exist lambda(0), lambda(1), ... , lambda(k) = 1 and (lambda(0)x(0 )+ lambda(1)x(1)+& centerdot;& centerdot;& centerdot;+lambda(k)x(k)) = lambda(0)f(x(0)) +lambda(1)f(x(1)) +& centerdot;& centerdot;& centerdot;+lambda(k)f(x(k)). Our main result is that any continuous, weakly 2-affine function is necessarily affine. It is a well-known result that a continuous, weakly 1-affine function is affine. However, in the paper, we will show that if continuity is replaced by several weaker regularity conditions, then this implication fails to hold. We also introduce a new concept of generalized convexity, namely the class of weakly k-convex sets, which turns out to be naturally related to weakly k-affine functions. We present results concerning a subclass of continuous, weakly k-affine functions for k >= 3, as well.
We answer two alienation questions involving the cosine and sine addition formulas on a semigroup.~The findings in both cases reveal a large degree of dependence between these two functional equations.
We prove that the minimum number of variables $\Gamma^{*}(d,K)$ which guarantees a nontrivial zero for every additive form of degree $d=4$ over the $2$-adic field $K=\mathbb{Q}_2(\sqrt{-5})$ is $9$.
A Rota-Baxter algebra AR is an algebra A equipped with a distinguished Rota-Baxter operator R on it. Rota-Baxter algebras are closely related to dendriform algebras introduced by Loday. In this paper, we first consider the non-abelian extension theory of Rota-Baxter algebras of weight zero, and classify them by introducing the non-abelian cohomology. Next, given a non-abelian extension 0-* BS-* EU-* AR-* 0 of Rota-Baxter algebras, we construct the Wells type exact sequences, and find their role in extending a Rota-Baxter automorphism beta E Aut(BS) and lifting a Rota-Baxter automorphism alpha E Aut(AR) to an automorphism in Aut(EU). We end this paper by considering a similar study for dendriform algebras.
Suppose that I is a unit square. Let A be an equilateral triangle with a side parallel to a side of I, and let {An} be a collection of the homothetic copies of A. In this note, we show that if the total sum of the areas of equilateral triangles from {An} is at least 1 + 7 root 3 12 , then these equilateral triangles can parallel cover the unit square I, and this bound is optimal.
Let $\mathcal{P}_{n}$ be the $n$-th Padovan number, $E_{n}$ be the $n$-th Perrin number, and $N_{n}$ be the $n$-th Narayana's cows number. Let $b$ be a positive integer such that $b\geq 2$. In this paper, we study the Diophantine equations $$\mathcal{P}_{n}=(b\pm 1)\cdot b^{l}\pm 1,\qquad E_{n}=(b\pm 1)\cdot b^{l}\pm 1,\qquad \mbox{and} \qquad N_{n}=(b\pm 1)\cdot b^{l}\pm 1,$$ in non-negative integers $n$, $b$, and positive integer $l$. As a result, we determine the Padovan, Perrin and Narayana's cows numbers that are Thabit and Williams numbers base $b$. Moreover, we determine all solutions of the above equations within the range $2\leq b\leq 10$.
In this note, we confirm a conjecture raised by Peth\H{o} (in 2021), which states that the absolute values of the roots of two distinct generalized Fibonacci polynomials are distinct.
Let $D$ be a nonsquare positive integer, and let $D_1$, $D_2$ be positive integers such that $D_1D_2=D$ and $\gcd(D_1,D_2)=1$. Further, let $p$ be an odd prime with $p\nmid D$. Denote by $N(D_1,D_2,p)$ the number of positive integer solutions $(x,n)$ to the generalized Ramanujan--Nagell equation $D_1x^2-D_2=p^n$. In this paper, we prove $N(D_1,D_2,p)\le4$. Moreover, we generalize the case where $D_1=1$ to give the {\it exceptional} triple $(D_1,D_2,p)$, which satisfies $$p^{\nu}=D_1a^2+\eta,\quad D_2=\dfrac{1}{D_1}\left(\dfrac{p^m-\eta}{2a}\right)^2-p^m, \qquad a,\nu,m\in\mathbb{N},\quad \nu
In this note, we prove that the discrete Riesz potential $I_{\alpha}$ defined on $\mathbb{Z}^n$ is a bounded operator $H^p(\mathbb{Z}^n)\to\ell^q(\mathbb{Z}^n)$ for $0
In this paper, we first characterize real hypersurfaces of both the K\"ahler surfaces $\mathbb{S}^2\times\mathbb{S}^2$ and $\mathbb{H}^2\times\mathbb{H}^2$ such that their shape operators have identical covariant derivatives with respect to the Levi-Civita connection and the $k$-generalized Tanaka--Webster connection for a nonzero $k\in\mathbb{R}$. Then, amongst others, we classify all real hypersurfaces of both $\mathbb{S}^2\times\mathbb{S}^2$ and $\mathbb{H}^2\times\mathbb{H}^2$ whose shape operators are parallel with respect to the $k$-generalized Tanaka--Webster connection.
In this note, we confirm a conjecture raised by Petho (in 2021), which states that the absolute values of the roots of two distinct generalized Fibonacci polynomials are distinct.
In this paper, we first characterize real hypersurfaces of both the Ka & uml;hler surfaces S2 & times; S2 and H2 & times; H2 such that their shape operators have identical covariant derivatives with respect to the Levi-Civita connection and the k-generalized Tanaka-Webster connection for a nonzero k is an element of R. Then, amongst others, we classify all real hypersurfaces of both S2 & times; S2 and H2 & times; H2 whose shape operators are parallel with respect to the k-generalized Tanaka-Webster connection.
By using the notion of pseudo-anti-commuting Ricci tensor, we have investigated a Hopf real hypersurface in the complex hyperbolic two-plane Grassmannian G & lowast;2(Cm+2) which admits a pseudo-Ricci-Bourguignon soliton. In addition to this, we have proved that a non-trivial gradient pseudo-Ricci-Bourguignon soliton (M, D f, eta, ohm, theta,gamma, g) on real hypersurfaces with isometric Reeb flow in the complex hyperbolic two-plane Grassmannian G & lowast;2(Cm+2) does not exist. In the class of contact hypersurfaces in G & lowast;2(Cm+2) except a tube with certain radius r = coth-1(root 3) over the totally geodesic and totally real quaternionic hyperbolic space HHn in G & lowast;2(Cm+2), m = 2n, it has been also proved that there does not exist a non-trivial gradient pseudo-Ricci-Bourguignon soliton in G & lowast;2(Cm+2).
Let D be a nonsquare positive integer, and let D-1, D-2 be positive integers such that D1D2 = D and gcd(D-1, D-2) = 1. Further, let p be an odd prime with p D. Denote by N(D-1, D-2, p) the number of positive integer solutions (x, n) to the generalized Ramanujan-Nagell equation D(1)x(2) D-2= p(n). In this paper, we prove N(D-1, D-2, p) <= 4. Moreover, we generalize the case where D-1 = 1 to give the exceptional triple (D-1, D-2, p), which satisfies p(nu)=D(1)a(2) +eta, D-2 = 1/D-1 (p(m)-eta/2a)(2)-( )p(m), a, nu, m is an element of N, nu < m, eta is an element of {1, -1}. In this case, it holds N(D-1, D-2, p) >= 3. We prove that if (D-1, D-2, p) is non-exceptional, and the squarefree part of D-1 is at most 1001, then N(D-1, D-2, p) <= 3.
Let N be a positive integer such that 4N(2) + 1 = q, where q is a prime. In this paper, we prove that the Diophantine D(-1)-triple of the form {1, 4N(2)+1, 1-N} cannot be extended to a quadruple in the ring Z[root-N], with a non-square integer N > 2. If N > 2 is a square, then 4N(2) + 1 is not a prime, and the set {1, 4N(2) + 1,1-N, 1+ N} is a D(-1)-quadruple in the ring Z[root-N], thus also in the ring of the Gaussian integers as well.
