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The purpose of this short paper is to clarify and present a general version of an interesting observation by [Piani and Mora, Phys. Rev. A 75 (2007) 012305], linking complete positivity of linear maps on matrix algebras to decomposability of their ampliations. Let [Formula: see text], [Formula: see text] be unital C*-algebras and let [Formula: see text] be positive linear maps from [Formula: see text] to [Formula: see text] [Formula: see text]. We obtain conditions under which any positive map [Formula: see text] from the minimal C*-tensor product [Formula: see text] to [Formula: see text], such that [Formula: see text], factorizes as [Formula: see text] for some positive map [Formula: see text]. In particular, we show that when [Formula: see text] are completely positive (CP) maps for some Hilbert spaces [Formula: see text] [Formula: see text], and [Formula: see text] is a pure CP map and [Formula: see text] is a CP map so that [Formula: see text] is also CP, then [Formula: see text] for some CP map [Formula: see text]. We show that a similar result holds in the context of positive linear maps when [Formula: see text] and [Formula: see text]. As an application, we extend IX Theorem of Ref. 4 (revisited recently by [Huber et al., Phys. Rev. Lett. 121 (2018) 200503]) to show that for any linear map [Formula: see text] from a unital C*-algebra [Formula: see text] to a C*-algebra [Formula: see text], if [Formula: see text] is decomposable for some [Formula: see text], where [Formula: see text] is the identity map on the algebra [Formula: see text] of [Formula: see text] matrices, then [Formula: see text] is CP.

The interconnetion network plays an important role in a parallel system. To avoid the edge number of the interconnect network scaling rapidly with the increase of dimension and achieve a good balance of hardware costs and properties, this paper presents a new interconnection network called exchanged [Formula: see text]-ary [Formula: see text]-cube ([Formula: see text]). Compared with the [Formula: see text]-ary [Formula: see text]-cube structures, [Formula: see text] shows better performance in terms of many metrics such as small degree and fewer links. In this paper, we first introduce the structure of [Formula: see text] and present some properties of [Formula: see text]; then, we propose a routing algorithm and obtain the diameter of [Formula: see text]. Finally, we analyze the diagnosis of [Formula: see text] and give the diagnosibility under PMC model and MM* model.

The distribution of the primes of the forms [Formula: see text] and [Formula: see text] are studied extensively, where [Formula: see text] denotes the largest integer not exceeding [Formula: see text]. In this paper, we will consider several new type problems on the distribution of the primes involving the ceiling (floor) function. For any real number [Formula: see text] with [Formula: see text], let [Formula: see text] be the number of integers [Formula: see text] with [Formula: see text] such that [Formula: see text] is prime and let [Formula: see text] be the number of primes [Formula: see text] for which there exists an integer [Formula: see text] with [Formula: see text] such that [Formula: see text], where [Formula: see text] denotes the least integer not less than [Formula: see text]. These are closely related to the number of the prime factors of the denominator of the Bernoulli polynomial [Formula: see text]. In this paper, we study asymptotic properties of [Formula: see text] and [Formula: see text]. The methods in this paper are also effective for corresponding distribution functions of the primes involving the floor function.

In a recent paper, Sun posed six conjectures on the relations between [Formula: see text] and [Formula: see text], where [Formula: see text] denotes the number of representations of [Formula: see text] as [Formula: see text], where [Formula: see text] are positive integers, [Formula: see text] are arbitrary nonnegative integers, and [Formula: see text] denotes the number of representations of [Formula: see text] as [Formula: see text], where this time [Formula: see text] are integers. In this paper, we prove Sun’s six conjectures by using Ramanujan’s theta function identities.

In this paper, we consider the following dual-gradient chemotaxis model [Formula: see text] with [Formula: see text] for [Formula: see text] and [Formula: see text] for [Formula: see text], where [Formula: see text] is a bounded domain in [Formula: see text] with smooth boundary, [Formula: see text] and [Formula: see text]. The model was proposed to interpret the spontaneous aggregation of microglia in Alzheimer’s disease due to the interaction of attractive and repulsive chemicals released by the microglia. It has been shown in the literature that, when [Formula: see text], the solution of the model with homogeneous Neumann boundary conditions either blows up or asymptotically decays to a constant in multi-dimensions depending on the sign of [Formula: see text], which means there is no pattern formation. In this paper, we shall show as [Formula: see text], the uniformly-in-time bounded global classical solutions exist in multi-dimensions and hence pattern formation can develop. This is significantly different from the results for the case [Formula: see text]. We perform the numerical simulations to illustrate the various patterns generated by the model, verify our analytical results and predict some unsolved questions. Biological applications of our results are discussed and open problems are presented.

Let [Formula: see text] be a finite sequence of integers, where [Formula: see text] and [Formula: see text] with [Formula: see text] for [Formula: see text]. A subsequence sum of [Formula: see text] is the sum of all terms of a nonempty subsequence of [Formula: see text]. Denoted by [Formula: see text] the set of all subsequence sums of [Formula: see text]. In this paper, for given [Formula: see text], we give the lower bound for [Formula: see text] with in terms of [Formula: see text] and the numbers of positive, negative integers in [Formula: see text]. We also determine the structure of the finite sequence [Formula: see text] of integers for which [Formula: see text] is minimal. This generalizes the results of Mistri, Pandey and Prakash. Moreover, we give a correction to a result of their paper.

In this paper, we deal with the problem of computing the sum of the [Formula: see text]th powers of all the elements of the matrix ring [Formula: see text] with [Formula: see text] and [Formula: see text] a finite commutative ring. We completely solve the problem in the case [Formula: see text] and give some results that compute the value of this sum if [Formula: see text] is an arbitrary finite commutative ring for many values of [Formula: see text] and [Formula: see text]. Finally, based on computational evidence and using some technical results proved in this paper, we conjecture that the sum of the [Formula: see text]th powers of all the elements of the matrix ring [Formula: see text] is always [Formula: see text] unless [Formula: see text], [Formula: see text], [Formula: see text] and the only element [Formula: see text] such that [Formula: see text] is idempotent, in which case the sum is [Formula: see text].

The consecutive linear [Formula: see text]-out-of-[Formula: see text]-from-[Formula: see text]:F system consists of [Formula: see text] linear ordered components and the consecutive circular [Formula: see text]-out-of-[Formula: see text]-from-[Formula: see text]:F system consists of [Formula: see text] circular ordered components. In this paper, we suggest, for the first time, modeling and exact reliability for these models. The linear system fails if and only if there exists a [Formula: see text]-order statistic of [Formula: see text]-consecutive [Formula: see text] [Formula: see text] of components in the failed state, [Formula: see text], [Formula: see text]; and the circular system fails if and only if there exists a [Formula: see text]-order statistic of [Formula: see text]-consecutive [Formula: see text] [Formula: see text] of components in the failed state, [Formula: see text], [Formula: see text]. In this paper, we designed an innovative algorithm to obtain the exact reliability for an extensive class of consecutive linear and circular systems. In continuation, there are the MATLAB Programs of exact reliability for consecutive linear and circular systems. In the following, we applied comparative and numerical results and calculated the exact reliability of this strategic systems. Finally, we calculated the exact reliability for two real-world practical examples.

In this paper, we study infinite families of orthogonal exponentials of some self-affine measures. The digit set [Formula: see text] and any [Formula: see text] expanding integer matrix [Formula: see text] can generate a self-affine measure [Formula: see text]. Let [Formula: see text] and [Formula: see text] be the transposed conjugate of [Formula: see text], where [Formula: see text] and the elements of [Formula: see text] come from [Formula: see text]. In this paper, we prove the following results. For [Formula: see text], [Formula: see text] is a spectral measure. For [Formula: see text], there are infinite families of orthogonal exponentials, but none of them forms an orthogonal basis in [Formula: see text].

This paper focuses on the Hölder continuity and the Box Dimension to the [Formula: see text]th Hadamard Fractional Integral (HFI) on a given interval [Formula: see text]. We use [Formula: see text] to denote it. In this paper, two different methods are used to study this problem. By using the approximation method, we obtain that for [Formula: see text] with [Formula: see text] and [Formula: see text], if [Formula: see text], then [Formula: see text] is [Formula: see text]th Hölder continuous in [Formula: see text] with [Formula: see text], and is [Formula: see text]th Hölder continuous on [Formula: see text]. Moreover, the Box Dimension of the graph of [Formula: see text] on the interval [Formula: see text] is less than or equal to [Formula: see text]. If [Formula: see text], then [Formula: see text] is locally [Formula: see text]th Hölder continuous in [Formula: see text] with the same [Formula: see text], and the Box Dimension of [Formula: see text] on [Formula: see text] is less than or equal to [Formula: see text]. By using another method, we imply that, for [Formula: see text] with [Formula: see text] and [Formula: see text], if [Formula: see text], then [Formula: see text] is [Formula: see text]th Hölder continuous, and thus the Box Dimension of the graph of [Formula: see text] is no more than [Formula: see text]; if [Formula: see text], then [Formula: see text] is locally [Formula: see text]th Hölder continuous in [Formula: see text], and is [Formula: see text]th Hölder continuous at [Formula: see text]. Then the Box Dimension of the graph to [Formula: see text] on [Formula: see text] is less than or equal to [Formula: see text]. We also give two examples to show that the above Hölder indexes given by the second method are optimal.

This paper is devoted to constructing mutually unbiased maximally entangled bases (MUMEBs) in [Formula: see text], where [Formula: see text] is a prime power. We prove that [Formula: see text] when [Formula: see text] is even, and [Formula: see text] when [Formula: see text] is odd, where [Formula: see text] is the maximal size of the sets of MUMEBs in [Formula: see text]. This highly raises the lower bounds of [Formula: see text] given in D. Xu, Quant. Inf. Process. 18(7) (2019) 213; D. Xu, Quant. Inf. Process. 19(6) (2020) 175. It should de noted that the method used in the paper is completely different from that in D. Xu, Quant. Inf. Process. 18(7) (2019) 213; D. Xu, Quant. Inf. Process. 19(6) (2020) 175.

In this paper, we consider to recover a signal which is sparse in terms of a tight frame from undersampled measurements via [Formula: see text]-minimization problem for [Formula: see text]. In [Compressed sensing with coherent tight frames via [Formula: see text]-minimization for [Formula: see text], Inverse Probl. Imaging 8 (2014) 761–777], Li and Lin proved that when [Formula: see text] there exists a [Formula: see text], depending on [Formula: see text] such that for any [Formula: see text], each solution of the [Formula: see text]-minimization problem can approximate the true signal well. The constant [Formula: see text] is referred to as the [Formula: see text]-RIP constant of order [Formula: see text] which was first introduced by Candès et al. in [Compressed sensing with coherent and redundant dictionaries, Appl. Comput. Harmon. Anal. 31 (2011) 59–73]. The main aim of this paper is to give the closed-form expression of [Formula: see text]. We show that for every [Formula: see text]-RIP constant [Formula: see text], if [Formula: see text] where [Formula: see text] then the [Formula: see text]-minimization problem can reconstruct the true signal approximately well. Our main results also hold for the complex case, i.e. the measurement matrix, the tight frame and the signal are all in the complex domain. It should be noted that the[Formula: see text]-RIP condition is independent of the coherence of the tight frame (see [Compressed sensing with coherent and redundant dictionaries, Appl. Comput. Harmon. Anal. 31 (2011) 59–73]). In particular, when the tight frame reduces to an identity matrix or an orthonormal matrix, the conclusions in our paper coincide with the results appeared in [Stable recovery of sparse signals via [Formula: see text]-minimization, Appl. Comput. Harmon. Anal. 38 (2015) 161–176].

In this paper, several explicit formulas for the number of representations of a positive integer by sums of mixed numbers are determined by employing theta function identities and the [Formula: see text]-parametrization of theta functions due to Alaca, Alaca and Williams. It is interesting that the formulas proved in this paper are linear combinations of [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text].

Let [Formula: see text] be a prime greater than [Formula: see text] and let [Formula: see text] be a rational [Formula: see text]-adic integer. In this paper, we try to determine [Formula: see text], and reveal the connection between cubic congruences and the sum [Formula: see text], where [Formula: see text] is the greatest integer not exceeding [Formula: see text]. Suppose that [Formula: see text] are rational [Formula: see text]-adic integers, [Formula: see text], [Formula: see text] and [Formula: see text]. In this paper, we show that the number of solutions of the congruence [Formula: see text] depends only on [Formula: see text]. Let [Formula: see text] be a prime of the form [Formula: see text] and so [Formula: see text] with [Formula: see text]. When [Formula: see text] and [Formula: see text], we establish congruences for [Formula: see text] and [Formula: see text] modulo p. As a consequence, when [Formula: see text] we show that [Formula: see text] has three solutions if and only if [Formula: see text] is a cubic residue of [Formula: see text].

All graphs in this paper are undirected and connected graphs. An ordered k-partition set [Formula: see text] where [Formula: see text], the representation of a vertex [Formula: see text] of [Formula: see text] with respect to [Formula: see text] is [Formula: see text] where [Formula: see text] is the distance between the vertex v and the set [Formula: see text] with [Formula: see text] for [Formula: see text]. The partition set [Formula: see text] is a local resolving partition of [Formula: see text] if [Formula: see text] for [Formula: see text] adjacent to [Formula: see text] of [Formula: see text]. The minimum local resolving partition [Formula: see text] is a local partition dimension of [Formula: see text], denoted by [Formula: see text]. In our paper, we found the sharp bounds of the local partition dimension of [Formula: see text] and determine the exact value of some special graph.

Let [Formula: see text] be a graph. A signed graph is an ordered pair [Formula: see text] where [Formula: see text] is a graph called the underlying graph of [Formula: see text] and [Formula: see text] is a function called a signature or signing function. Motivated by the innovative paper of B. D. Acharya on domination in signed graphs, we consider another way of defining the concept of domination in signed graphs which looks more natural and has applications in social science. A subset [Formula: see text] of [Formula: see text] is called a dominating set of [Formula: see text] if [Formula: see text] for all [Formula: see text]. The domination number of [Formula: see text], denoted by [Formula: see text], is the minimum cardinality of a dominating set of [Formula: see text]. Also, a dominating set [Formula: see text] of [Formula: see text] with [Formula: see text] is called a [Formula: see text]-set of [Formula: see text]. In this paper, we initiate a study on this parameter.

In this paper, we investigate the linear dynamical system [Formula: see text], where [Formula: see text] is the ring of integers modulo [Formula: see text] ([Formula: see text] is a prime). In order to facilitate the visualization of this system, we associate a graph [Formula: see text] on it, whose nodes are the points of [Formula: see text], and for which there is an arrow from [Formula: see text] to [Formula: see text], when [Formula: see text] for a fixed [Formula: see text] matrix [Formula: see text]. In this paper, the in-degree of each node in [Formula: see text] is obtained, and a complete description of [Formula: see text] is given, when [Formula: see text] is an idempotent matrix, or a nilpotent matrix, or a diagonal matrix. The results in this paper generalize Elspas’ [1959] and Toledo’s [2005].

Let [Formula: see text] be a field of [Formula: see text] with finitely many square classes. In this paper, we define a new rational valued invariant of [Formula: see text], and call it the division probability of [Formula: see text]. We compute it for all fields of elementary type. Further, we show that [Formula: see text], where [Formula: see text] is the number of Witt-equivalence classes of fields with [Formula: see text], and [Formula: see text] is the count of rational numbers that appear as division probabilities for fields [Formula: see text] of elementary type with [Formula: see text]. In the paper, we also determine [Formula: see text] for all [Formula: see text] and show that rational numbers of type [Formula: see text] always occur as division probability for a suitable field [Formula: see text].

Let us denote by [Formula: see text] the Clifford group (the circuit or operations generated by Hadamard, [Formula: see text] phase and the controlled-NOT gates) and by [Formula: see text] the set of qubit states that can be prepared by circuits from the Clifford group. In other words, a state [Formula: see text] if [Formula: see text] where [Formula: see text]. We will refer to states in [Formula: see text] as Clifford states. This paper studies the set of all three-qubit Clifford states. We prove that [Formula: see text] has 8640 states and if we define two states [Formula: see text] and [Formula: see text] in [Formula: see text] to be equivalent if [Formula: see text], with [Formula: see text] a local transformation in [Formula: see text], then the resulting quotient space has five orbits. More exactly, [Formula: see text] where the orbit [Formula: see text] is made up of states with entanglement entropy [Formula: see text]. For example, the first orbit [Formula: see text] contains the state [Formula: see text] and corresponds to the unentangled Clifford states. We say that [Formula: see text] is a real state if all its amplitudes [Formula: see text] are real numbers. We also say that an operator is real if all the entries of its matrix representation with respect to the computational basis are real numbers. In this paper, we also study the set of real Clifford 3 qubits and the way this set splits when we identify two real Clifford states [Formula: see text] and [Formula: see text] to be equivalent if [Formula: see text] where [Formula: see text] is a local real Clifford operator. An interesting aspect that follows from this study of Clifford states is the existence of two real Clifford states [Formula: see text] and [Formula: see text] that can be connected with a Clifford local transformation but they cannot be connected with a real Clifford local transformation. This is, the equation [Formula: see text] for [Formula: see text], does have a solution in the set of local transformations from [Formula: see text] but it does not have a solution among the local transformations from [Formula: see text] that are real. We go a little deeper and show that the equation [Formula: see text] does not have a solution for any local operation (not necessarily Clifford) whose entries are real numbers. Finally, we show how the CNOT gates act on the set of Clifford states and also in the set of real Clifford states.

In this paper, we study the notions of [Formula: see text]-prime and [Formula: see text]-semiprime ideals of semirings, [Formula: see text]-[Formula: see text]-system and [Formula: see text]-[Formula: see text]-system. We produce some properties and characterizations for [Formula: see text]-prime and [Formula: see text]-semiprime ideals of semirings in terms of [Formula: see text]-[Formula: see text]-system and [Formula: see text]-[Formula: see text]-system, respectively. We initiate the notion of prime semiring and semiprime semiring to set up characterizations for these two ideals of a semiring. The concept of power semiring associated to a semigroup is also studied in this paper.

In this paper, we prove an algorithmical solvability of exponential-Diophantine equations in rings represented by matrices over fields of positive characteristic. Consider the system of exponential-Diophantine equations [Formula: see text] where [Formula: see text] are constants from matrix ring of characteristic [Formula: see text], [Formula: see text] are indeterminates. For any solution [Formula: see text] of the system we construct a word (over an alphabet containing [Formula: see text] symbols) [Formula: see text] where [Formula: see text] is a [Formula: see text]-tuple [Formula: see text], [Formula: see text] is the [Formula: see text]th digit in the [Formula: see text]-adic representation of [Formula: see text]. The main result of this paper is following: the set of words corresponding in this sense to solutions of a system of exponential-Diophantine equations is a regular language (i.e., recognizable by a finite automaton). There exists an algorithm which calculates this language. This algorithm is constructed in the paper.

This paper describes the work pursued in the years 2008–2013 on improving the Standard Model prediction of selected flavor-physics observables. The latter includes: (1) [Formula: see text], that quantifies indirect CP violation in the [Formula: see text] system and (2) the very rare decay [Formula: see text], recently measured at the LHC. Concerning point (1), the paper describes our reappraisal of the long-distance contributions to [Formula: see text],[Formula: see text] that have permitted to unveil a potential tension between CP violation in the [Formula: see text]- and [Formula: see text]-system. Concerning point (2), the paper gives a detailed account of various systematic effects pointed out in Ref. 4 and affecting the Standard Model [Formula: see text] decay rate at the level of 10% — hence large enough to be potentially misinterpreted as nonstandard physics, if not properly included. The paper further describes the multifaceted importance of the [Formula: see text] decays as new physics probes, for instance how they compare with [Formula: see text]-peak observables at LEP, following the effective-theory approach of Ref. 5. Both cases (1) and (2) offer clear examples in which the pursuit of precision in Standard Model predictions offered potential avenues to discovery. Finally, this paper describes the impact of the above results on the literature, and what is the further progress to be expected on these and related observables.

The rings considered in this paper are finite commutative rings with identity, which are not fields. For any ring [Formula: see text] which is not a field and which is not necessarily finite, we denote the set of all zero-divisors of [Formula: see text] by [Formula: see text] and [Formula: see text] by [Formula: see text]. Let [Formula: see text] denote the zero-divisor graph of [Formula: see text] and for a finite ring [Formula: see text], let [Formula: see text] denote the maximum degree of [Formula: see text]. We denote [Formula: see text] by [Formula: see text]. The aim of this paper is to study some properties of [Formula: see text].

This paper seeks an answer to the following question: Let [Formula: see text] be a Noetherian ring with [Formula: see text]. When is every ideal isomorphic to a trace ideal? We prove that for a local Noetherian domain [Formula: see text] with [Formula: see text], every ideal is isomorphic to a trace ideal if and only if either [Formula: see text] is a DVR or [Formula: see text] is one-dimensional divisorial domain, [Formula: see text] is a principal ideal of [Formula: see text] and [Formula: see text] posses the property that every ideal of [Formula: see text] is isomorphic to a trace ideal of [Formula: see text]. Next, we globalize our result by showing that a Noetherian domain [Formula: see text] with [Formula: see text] has every ideal isomorphic to a trace ideal if and only if either [Formula: see text] is a PID or [Formula: see text] is one-dimensional divisorial domain, every invertible ideal of [Formula: see text] is principal and for every non-invertible maximal ideal [Formula: see text] of [Formula: see text], [Formula: see text] is a principal ideal of [Formula: see text] and every ideal of [Formula: see text] is isomorphic to a trace ideal of [Formula: see text]. We close the paper by examining some classes of non-Noetherian domains with this property to provide a large family of original examples.

The paper deals with rank, trace, eigenvalues and norms of the matrix [Formula: see text], where [Formula: see text] are ith components of any real sequence [Formula: see text]. A result in this paper is that the Euclidean and spectral norms of the matrix [Formula: see text] is [Formula: see text]. This is a generalization of the main result by Solak [Appl. Math. Comput. 232 (2014) 919–921], with the proof based on a simple property of norms of real matrices.

In this paper, we propose a new online scheduling model with linear lookahead intervals, which has the character that at any time [Formula: see text], one can foresee the jobs that will coming in the time interval [Formula: see text] in which [Formula: see text]. In this new lookahead model, the length of the lookahead intervals are variable as the time going on and the number of jobs increasing, and has the tend of steady growth. In this paper, we consider online scheduling of unit length jobs on [Formula: see text] identical parallel-batch machines under this new lookahead model to minimize makespan. The batch capacity is unbounded, that is [Formula: see text]. We present an optimal online algorithm for [Formula: see text], and provide a best possible online algorithm of competitive ratio [Formula: see text] for [Formula: see text], where [Formula: see text] is the positive root of [Formula: see text].

We first prove the following: Let [Formula: see text] and [Formula: see text]. Let [Formula: see text] and [Formula: see text] be closed, oriented, [Formula: see text]-dimensional [Formula: see text]-connected, simple submanifolds of [Formula: see text]. Then [Formula: see text] and [Formula: see text] are isotopic if and only if a Seifert matrix associated with a simple Seifert hypersurface for [Formula: see text] is [Formula: see text]-[Formula: see text]-equivalent to that for [Formula: see text]. We also discuss the [Formula: see text] case. This result implies one of our main results: Let [Formula: see text]. A 1-link [Formula: see text] is pass-equivalent to a 1-link [Formula: see text] if and only if [Formula: see text] is [Formula: see text]-pass-equivalent to [Formula: see text]. Here, [Formula: see text] means the knot product of [Formula: see text] and [Formula: see text], and [Formula: see text] means [Formula: see text]. See the body of the paper for the definition of knot products. It also implies the other main results: We strengthen the authors’ old result that two-fold cyclic suspension commutes with the performance of the twist move for spherical [Formula: see text]-knots. See the body for the precise statement. Furthermore, it implies the following: Let [Formula: see text] and [Formula: see text]. Let [Formula: see text] be a closed oriented [Formula: see text]-submanifold of [Formula: see text]. Then [Formula: see text] is a Brieskorn submanifold if and only if [Formula: see text] is [Formula: see text]-connected, simple and has a [Formula: see text]-Seifert matrix associated with a simple Seifert hypersurface that is [Formula: see text]-[Formula: see text]-equivalent to a [Formula: see text]-type (see the body of the paper for a definition). We also discuss the [Formula: see text] case.

In this paper, we study the uniqueness and local behavior at the origin of positive solutions to [Formula: see text] where [Formula: see text], [Formula: see text]) is a bounded smooth domain and [Formula: see text], and [Formula: see text] is a nonnegative continuous function over [Formula: see text]. In the two cases: (i) [Formula: see text], [Formula: see text] and [Formula: see text] on [Formula: see text]; (ii) [Formula: see text], [Formula: see text] and [Formula: see text] on [Formula: see text], we establish the uniqueness of positive solutions and the exact blow-up rate of positive solutions at the origin. This paper seems to be the first one to deal with the latter case.

The area of [Formula: see text]-calculus has attracted the serious attention of researchers. This great interest is due its application in various branches of mathematics and physics. The application of [Formula: see text]-calculus was initiated by Jackson [Jackson, On [Formula: see text]-definite integrals, Quart. J. Pure Appl. Math. 41 (1910) 193–203; On [Formula: see text]-functions and a certain difference operator, Trans. Roy. Soc. Edinburgh 46 (1908) 253–281.], who was the first to develop [Formula: see text]-integral and [Formula: see text]-derivative in a systematic way. In this paper, we make use of the concept of [Formula: see text]-calculus in the theory of univalent functions, to obtain the bounds for certain coefficient functional problems of [Formula: see text]-starlike and [Formula: see text]-convex functions. Further, we also obtain similar type of inequalities related to lemniscate of Bernoulli. The authors sincerely hope that this paper will revive this concept and encourage other researchers to work in this [Formula: see text]-calculus in the near-future in the area of complex function theory. Also, we present a direct and shortened proof for the estimates of [Formula: see text] found in [Mishra and Gochhayat, Fekete–Szego problem for [Formula: see text]-uniformly convex functions and for a class defined by Owa–Srivastava operator, J. Math. Anal. Appl. 347(2) (2008) 563–572] for [Formula: see text], [Formula: see text].

It is proved that some finite simple groups are quasirecognizable by prime graph. In [A. Mahmoudifar and B. Khosravi, On quasirecognition by prime graph of the simple groups [Formula: see text] and [Formula: see text], J. Algebra Appl. 14(1) (2015) 12pp], the authors proved that if [Formula: see text] is a prime number and [Formula: see text], then there exists a natural number [Formula: see text] such that for all [Formula: see text], the simple group [Formula: see text] (where [Formula: see text] is a linear or unitary simple group) is quasirecognizable by prime graph. Also[Formula: see text] in that paper[Formula: see text] the author posed the following conjecture: Conjecture. For every prime power [Formula: see text] there exists a natural number [Formula: see text] such that for all [Formula: see text] the simple group [Formula: see text] is quasirecognizable by prime graph. In this paper [Formula: see text] as the main theorem we prove that if [Formula: see text] is a prime power and satisfies some especial conditions [Formula: see text] then there exists a number [Formula: see text] associated to [Formula: see text] such that for all [Formula: see text] the finite linear simple group [Formula: see text] is quasirecognizable by prime graph. Finally [Formula: see text] by a calculation via a computer program [Formula: see text] we conclude that the above conjecture is valid for the simple group [Formula: see text] where [Formula: see text] [Formula: see text] is an odd number and [Formula: see text].

In rough set theory (RST), the notion of decision table plays a fundamental role. In this paper, we develop a purely mathematical investigation of this notion to show that several basic aspects of RST can be of interest also for mathematicians who work with algebraic and discrete methods.In this abstract perspective, we call decision system a sextuple [Formula: see text] [Formula: see text], where [Formula: see text], [Formula: see text], [Formula: see text] are non-empty sets whose elements are called, respectively, objects, condition attributes, values, [Formula: see text] is a (possibly empty) set whose elements are called decision attributes, [Formula: see text] and [Formula: see text] is a map.The basic tool of our analysis is the equivalence relation [Formula: see text] on [Formula: see text], depending on the choice of a condition attribute subset [Formula: see text] and defined as follows: [Formula: see text] We denote by [Formula: see text] the equivalence class of [Formula: see text] with respect to [Formula: see text]. We interpret the classical RST notions of consistency and inconsistency for a decision table in an abstract algebraic set operatorial perspective and, in such a context, we introduce and investigate a kind of local consistency in any decision system [Formula: see text]. More specifically, we fix [Formula: see text], [Formula: see text] and try to determine in what cases all objects [Formula: see text] satisfy the condition [Formula: see text]. Then, we build a formal general framework whose basic tools are two local consistency set operators and a global closure operator, the condition attribute set [Formula: see text]. This paper provides a detailed study of these set operators, of the induced set systems and of the most relevant links between them.

Let [Formula: see text] be a commutative ring with unity [Formula: see text] and let [Formula: see text] be an ideal expansion. In the first part of this paper, we extend the concept of [Formula: see text]-primary ideals to [Formula: see text]-[Formula: see text]-primary ideals, where [Formula: see text] is an ideal reduction and [Formula: see text] is an ideal expansion. We introduce some of the ideal expansion [Formula: see text] and define [Formula: see text]-[Formula: see text]-primary ideals, where [Formula: see text] is an ideal reduction. Also, we investigate ideal expansions satisfying some additional conditions and prove more properties of the generalized [Formula: see text]-[Formula: see text]-primary ideals with respect to such an ideal expansion [Formula: see text]. In the second part of this paper we investigate 2-absorbing [Formula: see text]-primary ideals which unify 2-absorbing ideals and 2-absorbing primary ideals, where [Formula: see text] is an ideal expansion. A number of results in the two parts are given.

We introduce a new topological invariant [Formula: see text] of compact manifolds-with-boundaries [Formula: see text] which is much connected with boundary-unions. A boundary-union is a kind of decomposition of compact manifolds-with-boundaries. See the body of the paper for the precise definition. Let [Formula: see text] and [Formula: see text] be [Formula: see text]-dimensional compact manifolds-with-boundaries. Let [Formula: see text] be a boundary-union of [Formula: see text] and [Formula: see text]. Then we have [Formula: see text] We define [Formula: see text] as follows: First, define an invariant of [Formula: see text]-closed manifolds. Take the maximum of the invariant of all connected-components of the boundary of each handle-body of an ordered-handle-decomposition with a fixed base [Formula: see text], where we impose the condition that the base [Formula: see text] is a (not necessarily connected) closed manifold. Take the minimum of the maximum for all ordered-handle-decompositions with the base [Formula: see text]. It is our another invariant [Formula: see text]. Take the maximum of the minimum, [Formula: see text], for all basis to satisfy the above condition. It is [Formula: see text]. See the body of the paper for the precise definition.

The Epstein ζ function ζ(Γ,s) of a lattice Γ is defined by a series which converges for any complex number s such that ℜ s > n/2, and admits a meromorphic continuation to the complex plane, with a simple pole at s = n/2. The question as to which Γ, for a fixed s, minimizes ζ(Γ,s), has a long history, dating back to Sobolev's work on numerical integration, and subsequent papers by Delone and Ryshkov among others. This was also investigated more recently by Sarnak and Strombergsson. The present paper is concerned with similar questions for positive definite quadratic forms over number fields, also called Humbert forms. We define Epstein zeta functions in that context and study their meromorphic continuation and functional equation, this being known in principle but somewhat hard to find in the literature. Then, we give a general criterion for a Humbert form to be finally ζ extreme, which we apply to a family of examples in the last section.

The paper deals with a div-curl approximation problem by weighted minimizing splines. The weighted minimizing splines are an extension of the well-known thin plate splines and are radial basis functions which allow the approximation or the interpolation of a scalar function from given scattered data. In this paper, we show that the theory of the weighted minimizing splines may also be used for the approximation or for the interpolation of a vector field controlled by the divergence and the curl of the vector field. Numerical examples are given to show the efficiency of this method.

As Lie algebra, we add the center c1 (and the outer derivation d1) to the quantum torus [Formula: see text] to give the extended torus Lie algebra [Formula: see text] (and [Formula: see text] respectively). Before the present paper, only some level 1 vertex operator representations for some [Formula: see text] (and [Formula: see text]) were constructed. In this paper, we first give vertex operator representations for [Formula: see text] where I is an arbitrary index set. By embedding some [Formula: see text] into [Formula: see text], we obtain a series of higher level vertex operator representations for [Formula: see text] and [Formula: see text]. Most of these vertex operator representations yield irreducible highest weight modules over these [Formula: see text]. Also their character formulas follow directly.

Suppose [Formula: see text] is a finite field extension of [Formula: see text] containing a primitive [Formula: see text]th root of unity. Let [Formula: see text] be a maximal [Formula: see text]-extension of [Formula: see text] with the Galois group of period [Formula: see text] and nilpotent class [Formula: see text]. In this paper, we develop formalism which allows us to study the structure of [Formula: see text] via methods of Lie theory. In particular, we introduce an explicit construction of a Lie [Formula: see text]-algebra [Formula: see text] and an identification [Formula: see text], where [Formula: see text] is a [Formula: see text]-group obtained from the elements of [Formula: see text] via the Campbell–Hausdorff composition law. In the next paper, we apply this formalism to describe the ramification filtration [Formula: see text] and an explicit form of the Demushkin relation for [Formula: see text].

In this paper, we continue to study zero-divisor properties of skew polynomial rings [Formula: see text], where [Formula: see text] is an associative ring equipped with an endomorphism [Formula: see text] and an [Formula: see text]-derivation [Formula: see text]. For an associative ring [Formula: see text], the undirected zero-divisor graph of [Formula: see text] is the graph [Formula: see text] such that the vertices of [Formula: see text] are all the nonzero zero-divisors of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are connected by an edge if and only if [Formula: see text] or [Formula: see text]. As an application of reversible rings, we investigate the interplay between the ring-theoretical properties of a skew polynomial ring [Formula: see text] and the graph-theoretical properties of its zero-divisor graph [Formula: see text]. Our goal in this paper is to give a characterization of the possible diameters of [Formula: see text] in terms of the diameter of [Formula: see text], when the base ring [Formula: see text] is reversible and also have the [Formula: see text]-compatible property. We also completely describe the associative rings all of whose zero-divisor graphs of skew polynomials are complete.

The question of a non-[Formula: see text]-type [Formula: see text] sequence wherein the fourth term shares the property [Formula: see text] with the first term has not been investigated so far. The present paper seeks to fill up the gap in this unexplored area. Let [Formula: see text] denote the set of all natural numbers and [Formula: see text] the sequence of Fibonacci numbers. Choose two integers [Formula: see text] and [Formula: see text] with [Formula: see text] such that their product increased by [Formula: see text] is a square [Formula: see text]. Certain properties of the sequence [Formula: see text] defined by the relation [Formula: see text] are established in this paper and polynomial expressions for Diophantine quadruples from the [Formula: see text] sequence [Formula: see text] are derived. The concept of a near-Diophantine quintuple is introduced and it is proved that there exist an infinite number of near-Diophantine quintuples.

A [Formula: see text]-ring [Formula: see text] is called a nil [Formula: see text]-clean ring if every element of [Formula: see text] is a sum of a projection and a nilpotent. Nil [Formula: see text]-clean rings are the [Formula: see text]-version of nil-clean rings introduced by Diesl. This paper is about the nil [Formula: see text]-clean property of rings with emphasis on matrix rings. We show that a [Formula: see text]-ring [Formula: see text] is nil [Formula: see text]-clean if and only if [Formula: see text] is nil and [Formula: see text] is nil [Formula: see text]-clean. For a 2-primal [Formula: see text]-ring [Formula: see text], with the induced involution given by[Formula: see text], the nil [Formula: see text]-clean property of [Formula: see text] is completely reduced to that of [Formula: see text]. Consequently, [Formula: see text] is not a nil [Formula: see text]-clean ring for [Formula: see text], and [Formula: see text] is a nil [Formula: see text]-clean ring if and only if [Formula: see text] is nil, [Formula: see text]is a Boolean ring and [Formula: see text] for all [Formula: see text].

Let $\ell, n$ be positive integers such that $\ell\geq n$. Let $\mathbb{G}_{n,\ell}$ be the Grassmannian which consists of the set of $n$-dimensional subspaces of $\mathbb{C}^{\ell}$. There is a $\mathbb{Z}$-graded algebra isomorphism between the cohomology $H^*(\mathbb{G}_{n,\ell},\mathbb{Z})$ of $\mathbb{G}_{n,\ell}$ and a natural $\mathbb{Z}$-form $B$ of the $\mathbb{Z}$-graded basic algebra of the type $A$ cyclotomic nilHecke algebra $H_{\ell,n}^{(0)}=\langle\psi_1,\cdots,\psi_{n-1},y_1,\cdots,y_n\rangle$. In this paper, we show that the isomorphism can be chosen such that the image of each (geometrically defined) Schubert class $(a_1,\cdots,a_{n})$ coincides with the basis element $b_{\mathbf{\lambda}}$ constructed by Jun Hu and Xinfeng Liang by purely algebraic method, where $0\leq a_1\leq a_2\leq\cdots\leq a_{n}\leq \ell-n$ with $a_i\in\mathbb{Z}$ for each $i$, $\mathbf{\lambda}$ is the $\ell$-multipartition of $n$ associated to $(\ell+1-(a_{n}+n), \ell+1-(a_{n-1}+n-1),\cdots,\ell+1-(a_1+1))$. A similar correspondence between the Schubert class basis of the cohomology of the Grassmannian $\mathbb{G}_{\ell-n,\ell}$ and the $b_{\lambda}$'s basis of the natural $\mathbb{Z}$-form $B$ of the $\mathbb{Z}$-graded basic algebra of $H_{\ell,n}^{(0)}$ is also obtained. As an application, we obtain a second version of Giambelli formula for Schubert classes., Comment: corrected a number of typos and added one corollary in the last section

Given any amalgam [Formula: see text] of inverse semigroups, we show how to construct an amalgam [Formula: see text] such that [Formula: see text] is embedded into [Formula: see text], where [Formula: see text], [Formula: see text], [Formula: see text] and, for any [Formula: see text] and [Formula: see text] with [Formula: see text] in [Formula: see text], where [Formula: see text], there exists [Formula: see text] with [Formula: see text] in [Formula: see text]; that is, [Formula: see text] is a lower bounded subsemigroup of [Formula: see text] and [Formula: see text]. A recent paper by the author describes the Schützenberger automata of [Formula: see text], for an amalgam [Formula: see text] where [Formula: see text] is lower bounded in [Formula: see text] and [Formula: see text], giving conditions for [Formula: see text] to have decidable word problem. Thus we can study [Formula: see text] by considering [Formula: see text]. As an example, we generalize results by Cherubini, Jajcayová, Meakin, Piochi and Rodaro on amalgams of finite inverse semigroups.

Denote by $G$ a finite group and by $\psi(G)$ the sum of element orders in $G$. If $t$ is a positive integer, denote by $C_t$ the cyclic group of order $t$ and write $\psi(t)=\psi(C_t)$. In this paper we proved the following Theorem A: Let $G$ be a non-cyclic group of odd order $n=qm$, where $q$ is the smallest prime divisor of $n$ and $(m,q)=1$. Then the following statements hold. (1) If $q=3$, then $\frac {\psi(G)}{\psi(|G|)}\leq \frac {85}{301}$, and equality holds if and only if $n=3\cdot 7\cdot m_1$ with $(m_1,42)=1$ and $G=(C_7\rtimes C_3)\times C_{m_1}$, with $C_7\rtimes C_3$ non-abelian. (2) If $q>3$, then $\frac {\psi(G)}{\psi(|G|)}\leq \frac {p^4+p^3-p^2+1}{p^5+1}$, where $p$ is the smallest prime bigger than $q$ and equality holds if and only if $n=qp^2m_1$ with $(m_1,p!)=1$ and $G=C_q\times C_p\times C_p \times C_{m_1}$.

WOS: 000389311800021, In [Finite presentability of Bruck-Reilly extensions of groups, J. Algebra 242 (2001) 2030], Araujo and RuAtic studied finite generation and finite presentability of Bruck Reilly extension of a group. In this paper, we aim to generalize some results given in that paper to generalized Bruck Reilly s -extension of a group. In this way, we determine necessary and sufficent conditions for generalized Bruck Reilly s-extension of a group, G R. (G; beta, gamma; u), to be finitely generated and finitely presented. Let G be a group, (3,-y: be morphisms and u is an element of H-1 (H-1(*) and H-1 are the ?-G- and ?-t-classes, respectively, contains the identity element kr of T). We prove that GE R* (G; 3, 7; u) is finitely generated if and only if there exists a finite subset Xo C G such that G is generated by ((boolean OR(i >= 0) X-0,beta(i))U(Uj >= 0X0 gamma(j)). We also prove that C superset of R.* (C; beta, gamma; u) is finitely presented if and only if G is presented by (X; R), where X is a finite set and R = (U i >= 0 R-0 beta(i)) boolean OR (U-j >= 0 R0 gamma(j)) = {w(1)beta(i) = upsilon(1)beta(i) : i >= 0, w(1) = v(1) is an element of H-1* for some finite set of relations R0 subset of X* x X*.

Doty and Giaquinto constructed a certain set denoted by [Formula: see text] in this paper, and conjectured that the set [Formula: see text] forms a [Formula: see text]-basis for the integral [Formula: see text]-Schur algebra [Formula: see text], where [Formula: see text]. In this paper, we show that the set [Formula: see text] is a [Formula: see text]-basis of [Formula: see text]-Schur algebra [Formula: see text] over [Formula: see text]. The set [Formula: see text] is a truncated form of PBW basis for quantum [Formula: see text]. Moreover, we give an example to show that the set [Formula: see text] is not [Formula: see text]-basis of [Formula: see text] in general.

A graph [Formula: see text] is said to be triangulated if it has no chordless cycles of length 4 or more. Such a graph is said to be rigid if, for a valid assignment of edge lengths, it has a unique linear layout and non-rigid otherwise. Damaschke [Point placement on the line by distance data, Discrete Appl. Math. 127(1) (2003) 53–62] showed how to compute all linear layouts of a triangulated graph, for a valid assignment of lengths to the edges of [Formula: see text]. In this paper, we extend this result to weakly triangulated graphs, resolving an open problem. A weakly triangulated graph can be constructively characterized by a peripheral ordering of its edges. The main contribution of this paper is to exploit such an edge order to identify the rigid and non-rigid components of [Formula: see text]. We first show that a weakly triangulated graph without articulation points has at most [Formula: see text] different linear layouts, where [Formula: see text] is the number of quadrilaterals (4-cycles) in [Formula: see text]. When [Formula: see text] has articulation points, the number of linear layouts is at most [Formula: see text], where [Formula: see text] is the number of nodes in the block tree of [Formula: see text] and [Formula: see text] is the total number of quadrilaterals over all the blocks. Finally, we propose an algorithm for computing a peripheral edge order of [Formula: see text] by exploiting an interesting connection between this problem and the problem of identifying a two-pair in [Formula: see text]. Using an [Formula: see text] time solution for the latter problem, we propose an [Formula: see text] time algorithm for computing its peripheral edge order, where [Formula: see text] and [Formula: see text] are respectively the number of edges and vertices of [Formula: see text]. For sparse graphs, the time complexity can be improved to [Formula: see text], using the concept of handles [R. B. Hayward, J. P. Spinrad and R. Sritharan, Improved algorithms for weakly chordal graphs, ACM Trans. Algorithms 3(2) (2007) 19pp].

The rings considered in this paper are commutative with identity which are not integral domains. Recall that an ideal [Formula: see text] of a ring [Formula: see text] is called an annihilating ideal if there exists [Formula: see text] such that [Formula: see text]. As in [M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings I, J. Algebra Appl. 10(4) (2011) 727–739], for any ring [Formula: see text], we denote by [Formula: see text] the set of all annihilating ideals of [Formula: see text] and by [Formula: see text] the set of all nonzero annihilating ideals of [Formula: see text]. Let [Formula: see text] be a ring. In [S. Visweswaran and H. D. Patel, A graph associated with the set of all nonzero annihilating ideals of a commutative ring, Discrete Math. Algorithm Appl. 6(4) (2014), Article ID: 1450047, 22pp], we introduced and studied the properties of a graph, denoted by [Formula: see text], which is an undirected simple graph whose vertex set is [Formula: see text] and distinct elements [Formula: see text] are joined by an edge in this graph if and only if [Formula: see text]. The aim of this paper is to study the interplay between the ring theoretic properties of a ring [Formula: see text] and the graph theoretic properties of [Formula: see text], where [Formula: see text] is the complement of [Formula: see text]. In this paper, we first determine when [Formula: see text] is connected and also determine its diameter when it is connected. We next discuss the girth of [Formula: see text] and study regarding the cliques of [Formula: see text]. Moreover, it is shown that [Formula: see text] is complemented if and only if [Formula: see text] is reduced.

Let [Formula: see text] be a commutative ring with identity. In this paper, we consider a simple graph associated with [Formula: see text] denoted by [Formula: see text], whose vertex set is the set of all nonzero proper ideals of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent whenever [Formula: see text] or [Formula: see text]. In this paper, we initiate the study of the graph [Formula: see text] and we investigate its properties. In particular, we show that [Formula: see text] is a connected graph with [Formula: see text] unless [Formula: see text] is isomorphic to a direct product of two fields. Moreover, we characterize all commutative rings [Formula: see text] with at least two maximal ideals for which [Formula: see text] are planar.

Let [Formula: see text] be a reduced ring containing [Formula: see text] and let [Formula: see text] be commuting locally nilpotent derivations of [Formula: see text]. In this paper, we give an algorithm to decide the local nilpotency of derivations of the form [Formula: see text], where [Formula: see text] are elements in [Formula: see text].

As one of the most fundamental networks for parallel and distributed computation, cycle is suitable for developing simple algorithms with low communication cost. A graph [Formula: see text] is called [Formula: see text]-fault-tolerant edge-pancyclic if after deleting any faulty set [Formula: see text] of [Formula: see text] vertices and/or edges from [Formula: see text], every correct edge in the resulting graph lies in a cycle of every length from [Formula: see text] to [Formula: see text], inclusively, where [Formula: see text] is the girth of [Formula: see text], the length of a shortest cycle in [Formula: see text]. The [Formula: see text]-dimensional crossed cube [Formula: see text] is an important variant of the hypercube [Formula: see text], which possesses some properties superior to the hypercube. This paper investigates the fault-tolerant edge-pancyclicity of [Formula: see text], and shows that if [Formula: see text] contains at most [Formula: see text] faulty vertices and/or edges then, for any fault-free edge [Formula: see text] and every length [Formula: see text] from [Formula: see text] to [Formula: see text] except [Formula: see text], there is a fault-free cycle of length [Formula: see text] containing the edge [Formula: see text]. The result is optimal in some senses.