A denjoy-type integral for Banach-valued functions

In this paper, we shall show that theHL ϕ-integral and the Denjoy ϕ-integral, defined in [2] are equivalent.     

Connected Q-integral graphs with maximum edge-degree less than or equal to 8

Graphs with integral signless Laplacian spectrum are called Q-integral graphs. The number of adjacent edges to an edge is defined as the edge-degree of that edge. The Q-spectral radius of a graph is the largest eigenvalue of its signless Laplacian. In 2019, Park and Sano [16] studied connected Q-integral graphs with the maximum edge-degree at most six. In this article, we extend their result and study the connected Q-integral graphs with maximum edge-degree less than or equal to eight. Further, we give an upper bound and a lower bound for the maximum edge-degree of a connected Q-integral graph with respect to its Q-spectral radius. As a corollary, we show that the Q-spectral radius of the connected edge-non-regular Q-integral graph with maximum edge-degree five is six, which we anticipate to be a key for solving the unsolved problem of characterizing such graphs.     

Adjoint classes of functions in the H<Subscript>1</Subscript> sense

Using the concept of the H₁-integral, we consider a similarly defined Stieltjes integral. We prove a Riemann-Lebesgue type theorem for this integral and give examples of adjoint classes of functions.     

Comparison of Two Generalized Trigonometric Integrals

The P ²-integral of James is compared with the T ²-integral. A trigonometric series convergent almost everywhere to a function which is T ²-integrable but not P ²-integrable is constructed.     

关于复合材料单层板裂纹尖端的J积分

该文采用复变函数方法,通过将裂纹尖端的应力和位移代入J积分的一般公式,推出了线弹性正交异性复合材料单层板受对称载荷作用的非弹性主方向的裂纹尖端犑积分的复形式－ 复变函数积分的实部,证明了该J积分的路径无关性,得到了它的具体计算公式     

Remarks on Q-integral complete multipartite graphs

A graph is Q-integral if the spectrum of its signless Laplacian matrix consists entirely of integers. In their study of Q-integral complete multipartite graphs, [Zhao et al., Q-integral complete r-partite graphs, Linear Algebra Appl. 438 (2013) 1067–1077] posed two questions on the existence of such graphs. We resolve these questions and present some further results characterizing particular classes of Q-integral complete multipartite graphs.     

Chebyshev type inequalities for pseudo-integrals

Chebyshev type inequalities for two classes of pseudo-integrals are shown. One of them concerning the pseudo-integrals based on a function reduces on the g-integral where pseudo-operations are defined by a monotone and continuous function g. Another one concerns the pseudo-integrals based on a semiring ([a,b],max,⊙), where ⊙ is generated. Moreover, a strengthened version of Chebyshev’s inequality for pseudo-integrals is proved.     

Asplund Operators and p-Integral Polynomials

We introduce the Banach ideals of p-integral and of p-nuclear polynomials for 1 ≤ p ≤ + ∞, extending to the polynomial setting the well known notions of p-integral and p-nuclear operators. For p = 1, we recover the Pietsch integral and nuclear polynomials, respectively. Given a Banach space E, let K be a compact Hausdorff space such that there is an embedding h : E → C(K). Let R _h be the polynomial from E into C(K) given by R _h (x) : = h(x) ^m for all ${x \epsilon E}$ . We prove that a polynomial is p-integral (1 ≤ p ≤ + ∞) if and only if it factors through a polynomial of the form R _h followed by the canonical inclusion of C(K) into L _p (K, μ) for some finite measure μ. We also prove that a polynomial P is p-integral if and only if we may write ${P = T \circ R_{h}}$ where T is a p-integral operator on a C(K) space. We show that P is ∞-integral if and only if it factors in the form ${P = T \circ R_{h}}$ where T is a weakly compact operator on a C(K) space. Analogous results are true if we replace C(K) by L _∞ (Ω, μ) for some finite measure space (Ω, Σ, μ). It is proved that a polynomial ${P \epsilon \mathcal{P}(^{m}E, F)}$ is p-integral if and only if its linearization is well defined and p-integral on ${\bigotimes ^{m}_{{\epsilon}_{s}}, s^{E}}$ . It is also shown that a p-integral polynomial may be extended to a p-integral polynomial on every larger space, and the extension has the same p-integral norm. We give a factorization theorem for p-nuclear polynomials. Finally, we prove that a polynomial P is p-nuclear if and only if it may be written in the form ${P = Q \circ A}$ where A is a compact operator and Q is a p-integral polynomial, if and only ${P = Q^{\prime} \circ H}$ with H an Asplund operator and Q′ a p-integral polynomial. This extends a result obtained by C. Cardassi in the linear case.     

关于C1-积分

本文利用Riemann和的Moore-Smith极限来定义并研究C1-积分.利用C1-积分的有关性质,讨论了C1-积分与C-积分之间的关系,并给出了一个零测度集特征函数C1-可积的充要条件,从而推广了文献[5]的有关结论.     

Comparison of Some Trigonometric Integrals

Mathematical Notes - We show that Preiss–Thomson’s AS-integral is inconsistent with Burkill’s SCPintegral, James’ P2-integral, and Denjoy’s totalization T2s over the...     

Expression of the Lebesgue Δ-integral on time scales as a usual Lebesgue integral; application to the calculus of Δ-antiderivatives

This paper is devoted to the study of the Lebesgue $\Delta$-integral on time scales. We give a formula in which such an integral is obtained as a sum of adequate real Lebesgue integrals. By using the relationship between these kinds of integrals and the Riemann ones, we rewrite the given expression to obtain the $\Delta$-antiderivatives of functions defined on time scales. The results obtained are illustrated with some examples.     

Hilbert subsets ands-integral points

A classical tool for studying Hilbert's irreducibility theorem is Siegel's finiteness theorem forS-integral points on algebraic curves. We present a different approach based ons-integral points rather thanS-integral points. Given an integers>0, an elementt of a fieldK is said to bes-integral if the set of placesv ∈M _K for which |t|_v > l is of cardinality ≤s (instead of contained inS for “S-integral”). We prove a general diophantine result fors-integral points (Th.1.4). This result, unlike Siegel's theorem, is effective and is valid more generally for fields with the product formula. The main application to Hilbert's irreducibility theorem is a general criterion for a given Hilbert subset to contain values of given rational functions (Th.2.1). This criterion gives rise to very concrete applications: several examples are given (§2.5). Taking advantage of the effectiveness of our method, we can also produce elements of a given Hilbert subset of a number field with explicitely bounded height (Cor.3.7). Other applications, including the case thatK is of characteristicp>0, will be given in forthcoming papers ([8], [9]).     

The Durrmeyer type modification of the q-Baskakov type operators with two parameter α and β

In this paper, we are dealing with q analogue of Durrmeyer type modified the Baskakov operators with two parameter α and β, which introduces a new sequence of positive linear q-integral operators. We show that this sequence is an approximation process in the polynomial weighted space of continuous function defined on the interval [0, ∞). We study moments, weighted approximation properties, the rate of convergence using a weighted modulus of smoothness, asymptotic formula and better error estimation for these operators.     

The M α and C-integrals

In this paper, we define the ${M_\alpha }$ -integral of real-valued functions defined on an interval [a, b] and investigate important properties of the ${M_\alpha }$ -integral. In particular, we show that a function f: [a, b] → R is ${M_\alpha }$ -integrable on [a, b] if and only if there exists an $AC{G_\alpha }$ function F such that F′ = f almost everywhere on [a, b]. It can be seen easily that every McShane integrable function on [a, b] is ${M_\alpha }$ -integrable and every ${M_\alpha }$ -integrable function on [a, b] is Henstock integrable. In addition, we show that the ${M_\alpha }$ -integral is equivalent to the C-integral.     

Higher integrality conditions, volumes and Ehrhart polynomials

A polytope is integral if all of its vertices are lattice points. The constant term of the Ehrhart polynomial of an integral polytope is known to be 1. In previous work, we showed that the coefficients of the Ehrhart polynomial of a lattice-face polytope are volumes of projections of the polytope. We generalize both results by introducing a notion of k-integral polytopes, where 0-integral is equivalent to integral. We show that the Ehrhart polynomial of a k-integral polytope P has the properties that the coefficients in degrees less than or equal to k are determined by a projection of P, and the coefficients in higher degrees are determined by slices of P. A key step of the proof is that under certain generality conditions, the volume of a polytope is equal to the sum of volumes of slices of the polytope.     

L2空间中的均方随机积分

本文在Ｌ²（ｄｕ）空间中定义了比Ｒｉｃｍａｎｎ均方积分更为广泛的一种均方随机积分，并讨论了这种积分的性质及随机积分可交换顺序定理。     

Right closure of martingale sequences in the sense of theA-integral

In this paper new sufficient (necessary and sufficient for martingales of special form) conditions for the martingale closure from the right in the sense of theA-integral are given. These results follow from the theorem about passing to the limit under theA-integral. The theorem is established using the criterion for transposing iterated limits with respect to the base. It is shown that the sufficient conditions thus obtained are stronger than those previously known. Translated fromMatematicheskie Zametki, Vol.68, No. 1, pp. 98–104, July, 2000.     

Remarks on D-integral complete multipartite graphs

A graph is called distance integral (or D-integral) if all eigenvalues of its distance matrix are integers. In their study of D-integral complete multipartite graphs, Yang and Wang (2015) posed two questions on the existence of such graphs. We resolve these questions and present some further results on D-integral complete multipartite graphs. We give the first known distance integral complete multipartite graphs \({K_{{p_1},{p_2},{p_3}}}\) with p₁ < p₂ < p₃, and \({K_{{p_1},{p_2},{p_3},{p_4}}}\) with p₁ < p₂ < p₃ < p₄, as well as the infinite classes of distance integral complete multipartite graphs \({K_{{a_1}{p_1},{a_2}{p_2},...,{a_s}{p_s}}}\) with s = 5, 6.     

Holomorphic curves and integral points off divisors

We deal with the distributions of holomorphic curves and integral points off divisors. We will simultaneously prove an optimal dimension estimate from above of a subvariety W off a divisor D which contains a Zariski dense entire holomorphic curve, or a Zariski dense D-integral point set, provided that in the latter case everything is defined over a number field. Then, if the number of components of D is large, the estimate leads to the constancy of such a holomorphic curve or the finiteness of such an integral point set. At the beginning, we extend logarithmic Bloch-Ochiai's Theorem to the K?hler case. Received: 10 January 2000 / Published online: 18 January 2002     

Estimateur crible de l'opérateur d'un processus ARB(1)

We consider the sieve estimator of the operator of a Banach autoregressive process. We show the almost sure convergence when the operator is 2-summing, strictly 2-integral, afterwards 2-nuclear for the adequate norms. To cite this article: F. Rachedi, T. Mourid, C. R. Acad. Sci. Paris, Ser. I 336 (2003).