Nonzero Real Numbers Research Articles

Four Ways to Determine Equivalent Ratios

Ratios and proportions are important concepts that occur in real life, but they are difficult to learn and complicated to teach (Lamon 2007). In general, proportional reasoning in the middle grades is an area in need of attention because of its connection to later concepts (Ojose 2015). In this article, we explain the meanings of ratios, proportions, and equivalent ratios and then provide useful methods to determine equivalent ratios using students' examples. Because multiple definitions exist, it is necessary to determine definitions to avoid possible confusion. Johnson (2010) defines a ratio as a pair of positive nonzero real numbers such that there are a units for every b units, written a:b; read a to b; and represented as a fraction, a/b. We follow definitions suggested by the writers of the Common Core (CCSSI 2010): Fraction representation is known as the associated unit rate or the value of a ratio A:B and is found by dividing B into A, where “equivalent ratios have the same unit rate” (McCallum, Zimba, and Daro 2011). Using both contextual and noncontextual proportion tasks, we saw various ways that students found the equivalency of ratios as well as misconceptions they have.

Results of Diophantine approximation by unlike powers of primes

Let k be an integer with k ≥ 6: Suppose that λ1, λ2,..., λ5 be nonzero real numbers not all of the same sign, satisfying that λ1/λ2 is irrational, and suppose that η is a real number. In this paper, for any e > 0; we consider the inequality |λ1p1 + λ2p 2 2 + λ3p 3 3 + λ4p 4 4 + λ5p 5 k + η | < (max pj)-σ(k)+e has infinitely many solutions in prime variables p1, p2,...,p5, where σ(k) depends on k. Our result gives an improvement of the recent result. Furthermore, using the similar method in this paper, we can refine some results on Diophantine approximation by unlike powers of primes, and get the related problem.

Diophantine approximation with mixed powers of primes

Let [Formula: see text] be an integer with [Formula: see text], and [Formula: see text] be any real number. Suppose that [Formula: see text] are nonzero real numbers, not all the same sign and [Formula: see text] is irrational. It is proved that the inequality [Formula: see text] has infinitely many solutions in primes [Formula: see text], where [Formula: see text], and [Formula: see text] for [Formula: see text]. This generalizes earlier results. As application, we get that the integer parts of [Formula: see text] are prime infinitely often for primes [Formula: see text].

A note on Diophantine approximation by unlike powers of primes

It is proved that if [Formula: see text] are nonzero real numbers, not all of the same sign and [Formula: see text] is irrational, then for given real numbers [Formula: see text] and [Formula: see text], [Formula: see text], the inequality [Formula: see text] has infinitely many solutions in prime variables [Formula: see text]. This result constitutes an improvement upon that of Liu for the range [Formula: see text].

Spectral analysis of the Euler‐Bernoulli beam model with fully nonconservative feedback matrix

The Euler‐Bernoulli beam model with fully nonconservative boundary conditions of feedback control type is investigated. The output vector (the shear and the moment at the right end) is connected to the observation vector (the velocity and its spatial derivative on the right end) by a 2 × 2 matrix (the boundary control matrix), all entries of which are nonzero real numbers. For any combination of the boundary parameters, the dynamics generator, , of the model is a non–self‐adjoint matrix differential operator in the state Hilbert space. A set of 4 self‐adjoint operators, defined by the same differential expression as on different domains, is introduced. It is proven that each of these operators, as well as , is a finite‐rank perturbation of the same self‐adjoint dynamics generator of a cantilever beam model. It is also shown that the non–self‐adjoint operator, , shares a number of spectral properties specific to its self‐adjoint counterparts, such as (1) boundary inequalities for the eigenfunctions, (2) the geometric multiplicities of the eigenvalues, and (3) the existence of real eigenvalues. These results are important for our next paper on the spectral asymptotics and stability for the multiparameter beam model.

Subspace Lang conjecture and some remarks on a transcendental criterion

Let $$b\ge 2$$ be an integer and $$\alpha $$ is a non-zero real number written in b-ary expansion. Adamczewski et al. (C. R. Acad. Sci. Paris 339 (2004) 11–14) provided a criterion for an irrational number to be a transcendental number using b-ary expansion. In this paper, we make some remarks on this criterion and, under the assumption of Subspace Lang Conjecture, we extend this criterion for a much wider class of irrational numbers.

Extremal graphs with respect to generalized [formula omitted] index

The generalized ABC index of a graph G, denoted by ABCα(G), is defined as the sum of weights (di+dj−2didj)α over all edges vivj of G, where α is an arbitrary non-zero real number, and di is the degree of vertex vi of G. In this paper, we first prove that the generalized ABC index of a connected graph will increase with addition of edge(s) if α<0 or 0<α≤1∕2, which provides a useful tool for the study of extremal properties of the generalized ABC index. By means of this result, we then characterize the graphs having the maximal ABCα value for α<0 among all connected graphs with given order and vertex connectivity, edge connectivity, or matching number. Our work extends some previously known results.

Skew-signings of positive weighted digraphs

An arc-weighted digraph is a pair (D,ω) where D is a digraph and ω is an arc-weight function that assigns to each arc uv of D a nonzero real number ω(uv). Given an arc-weighted digraph (D,ω) with vertices v1,…,vn, the weighted adjacency matrix of (D,ω) is defined as the n×n matrix A(D,ω)=[aij] where aij=ω(vivj) if vivj is an arc of D, and 0otherwise. Let (D,ω) be a positive arc-weighted digraph and assume that D is loopless and symmetric. A skew-signing of (D,ω) is an arc-weight function ω′ such that ω′(uv)=±ω(uv) and ω′(uv)ω′(vu)<0 for every arc uv of D. In this paper, we give necessary and sufficient conditions under which the characteristic polynomial of A(D,ω′) is the same for all skew-signings ω′ of (D,ω). Our main theorem generalizes a result of Cavers et al. (2012) about skew-adjacency matrices of graphs.

Lower bounds for the energy of weighted digraphs

ABSTRACTLet D be a weighted digraph with weights taken from the set of non-zero real numbers and let be its eigenvalues. The energy of D is defined as , where denotes the real part of the complex number . In this paper, we obtain lower bounds for the energy of weighted digraphs in terms of weights of directed cycles of length 2. We study normal weighted digraphs and give spectral and structural characterizations of normal weighted digraphs. Using these results, we determine unicyclic and bicyclic normal weighted digraphs. For normal weighted digraphs, we obtain an improved and sharp lower bound for the energy.

English

Suppose that λ1, λ2, λ3, λ4 are nonzero real numbers, not all negative, δ > 0, V is a well-spaced set, and the ratio λ1/λ2 is algebraic and irrational. Denote by E(V,N, δ) the number of v ∈ V with v ≤ N such that the inequality $$\left| {{\lambda _1}p_1^2 + {\lambda _2}p_2^3 + {\lambda _3}p_3^4 + {\lambda _4}p_4^5 - \upsilon } \right| < {\upsilon ^{ - \delta }}$$ has no solution in primes p1, p2, p3, p4. We show that $$E\left( {\upsilon ,N,\delta } \right) \ll {N^{1 + 2\delta - 1/72 + \varepsilon }}$$ for any ɛ > 0.

Fractional dynamic inequalities harmonized on time scales

We present here some symmetric fractional Rogers-Hölder’s inequalities using Riemann–Liouville integral on time scales. We consider and impose different conditions on three non-zero real numbers p, q and r when 1p+1q+1r=0.

Generalized Rough Cesaro and Lacunary Statistical Triple Difference Sequence Spaces in Probability of Fractional Order Defined by Musielak-Orlicz Function

We generalized the concepts in probability of rough Ces$\grave{a}$ro and lacunary statistical by introducing the difference operator $\Delta^{\alpha}_{\gamma}$ of fractional order, where $\alpha$ is a proper fraction and $\gamma=\left(\gamma_{mnk}\right)$ is any fixed sequence of nonzero real or complex numbers. We study some properties of this operator involving lacunary sequence $\theta$ and arbitrary sequence $p=\left(p_{rst}\right)$ of strictly positive real numbers and investigate the topological structures of related with triple difference sequence spaces. The main focus of the present paper is to generalized rough Ces$\grave{a}$ro and lacunary statistical of triple difference sequence spaces and investigate their topological structures as well as some inclusion concerning the operator $\Delta^{\alpha}_{\gamma}.$

Piecewise ⁎-homomorphisms and Jordan maps on C⁎-algebras and factor von Neumann algebras

We investigate maps between C⁎-algebras that are well behaved with respect to mutually commuting elements. We contribute to the Mackey–Gleason problem by showing that any continuous bijection between self-adjoint parts of C⁎-algebras that preserves triple product (a,b)→aba, and is linear on commutative subspaces, is already linear. This allows us to describe such maps as direct differences of linear Jordan isomorphisms. We shall show that any weak⁎-continuous bijection between positive invertible elements of von Neumann factors (of dimension at least 9) that preserves products of commuting elements in both directions is of the form a→eψ(log⁡a)θ(ac), where θ is a linear Jordan ⁎-isomorphism, c nonzero real number and ψ is a hermitian continuous functional. In a similar way we describe the same type of bicontinuous maps between unitary groups of von Neumann factors. General form of the above mentioned maps on C⁎-algebras is also presented.

Zero-Sum 6-Flows in 5-Regular Graphs

Let G be a graph. A zero-sum flow of G is an assignment of nonzero real numbers to the edges of G such that the sum of the values of all edges incident with each vertex is zero. Let k be a natural number. A zero-sum k-flow is a flow with values from the set $$\{\pm 1, \ldots , \pm (k - 1)\}$$ . In this paper, we prove that every 5-regular graph admits a zero-sum 6-flow.

Diophantine approximation by unlike powers of primes

Let [Formula: see text] be nonzero real numbers not all of the same sign, satisfying that [Formula: see text] is irrational, and [Formula: see text] be a real number. In this paper, we prove that for any [Formula: see text] [Formula: see text] has infinitely many solutions in prime variables [Formula: see text].

The values of cubic forms at prime arguments

Let λ1, λ2, λ3, λ4, λ5 be non-zero real numbers, not all negative. Let V be a well-spaced sequence, δ>0. If λ1/λ2 is irrational and algebraic, then we prove that E(V,X,δ)≪X17/18+2δ+ε, where E(V,X,δ) denotes the number of v∈V with v≤X such that the inequality |λ1p13+λ2p23+λ3p33+λ4p43+λ5p53−v|<v−δ has no solution in primes p1, p2, p3, p4, p5. Further, we assume that except for one, all other the ratios λk/λl (1≤k<l≤5) are irrational and algebraic, then 17/18 can be replaced by 11/12. These improve the earlier results.

Subdivision of the Spectra for the Generalized Difference Operator Δ_(a,b) on the Sequence Space l_p " " (1&lt;p&lt;∞)

El-Shabrawy has introduced the generalized difference operator denoted by . Let nonzero real numbers sequences and be convergent sequences such that and The generalized difference operator is with . In this paper we study the approximate point spectrum, the defect spectrum and the compression spectrum of the operator on the p-absolutely summable sequences space .

Diophantine approximation with one prime and three squares of primes

Let λ1,…,λ4 be non-zero real numbers, not all of the same sign, with λ1/λ2 irrational, and ϖ be a real number. We prove that for any ε>0, the inequality|λ1p1+λ2p22+λ3p32+λ4p42+ϖ|≤(max⁡{p1,p22,p32,p42})−114+ε has infinitely many solutions in prime variables p1,…,p4. If we further assume that λ1/λ3 is also irrational and λ1/λ2, λ1/λ3 are both algebraic, then we may replace the exponent by −340+ε.

Quantum teleportation and information splitting via four-qubit cluster state and a Bell state

Quantum teleportation provides a `bodiless' way of transmitting the quantum state from one object to another, at a distant location, using a classical communication channel and a previously shared entangled state. In this paper, we present a tripartite scheme for probabilistic teleportation of an arbitrary single qubit state, without losing the information of the state being teleported, via a four-qubit cluster state of the form $\left.|{\phi}\right\rangle_{1234}=\left.\alpha|0000\right\rangle+\left.\beta|1010\right\rangle+\left.\gamma|0101\right\rangle-\eta\left.|1111\right\rangle$, as the quantum channel, where the nonzero real numbers $\alpha$, $\beta$, $\gamma$, and $\eta$ satisfy the relation $|\alpha|^2+|\beta|^2+|\gamma|^2+|\eta|^2=1$. With the introduction of an auxiliary qubit with state $\left|0\right\rangle$, using a suitable unitary transformation and a positive-operator valued measure (POVM), the receiver can recreate the state of the original qubit. An important advantage of the teleportation scheme demonstrated here is that, if the teleportation fails, it can be repeated without teleporting copies of the unknown quantum state, if the concerned parties share another pair of entangled qubit. We also present a protocol for quantum information splitting of an arbitrary two-particle system via the aforementioned cluster state and a Bell-state as the quantum channel. Problems related to security attacks were examined for both the cases and it was found that this protocol is secure. This protocol is highly efficient and easy to implement.

On a theorem of Mahler

Let b ≥ 2 b\geq 2 be an integer and α \alpha a non-zero real number written in base b b . In 1973, Mahler proved the following result: Let α \alpha be an irrational number written in base b b and let n ≥ 1 n\geq 1 be a given integer. Let B = b 0 b 1 … b n − 1 B = b_0b_1\ldots b_{n-1} be a given block of digits in base b b of length n n . Then, there exists an integer X X with 1 ≤ X &gt; b 2 n + 1 1\leq X &gt; b^{2n+1} such that B B occurs infinitely often in the base b b representation of the fractional part of X α X\alpha . In this short note, we deal with some conditional quantitative version of this result.