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FeCo/PZT/FeCo cylindrical heterostructures were prepared by electroless deposition. Scanning electron microscopy shows that the magnetostrictive FeCo layers are in contact with piezoelectric PZT directly. To control the resonant frequency at which the magnetoelectric coupling shows an obvious improvement, the height of cylinder needs to be taken into consideration. In the present work, three hollow cylinders with different height were polarized along the radial direction. It was shown that with the increase of cylinder height the magnetoelectric coefficient increased. Both experimental and calculated results show that the resonant frequency of FeCo/PZT/FeCo cylindrical layered composites decreases with increasing the cylinder height. Proper resonant frequency and strong ME effect could be obtained by optimizing the configuration.

We introduce a family of polynomial invariants for knotoids. Via the corresponding Gauss diagrams for knotoids, we investigate the method to calculate these polynomial invariants. Also, we give some properties and examples.

The height of a knotoid is a measure of how far it is from being a knot. Here, we define the positive and negative parts of the height, and we prove that they determine the unsigned height. Some polynomial invariants provide lower bounds for the signed heights. We also study a set of sequences associated to a knotoid.

Our main goal in this paper is to set the general frame for studying the dimension theory of tensor products of algebras over an arbitrary ring $R$. Actually, we translate the theory initiated by Grothendieck and Sharp and subsequently developed by Wadsworth on Krull dimension of tensor products of algebras over a field $k$ into the general setting of algebras over an arbitrary ring $R$. For this sake, we introduce and study the notion of a fibered AF-ring over a ring $R$. This concept extends naturally the notion of AF-ring over a field introduced by Wadsworth in [The Krull dimension of tensor products of commutative algebras over a field, J. London Math. Soc.19 (1979) 391–401.] to algebras over arbitrary rings. We prove that Wadsworth theorems express local properties related to the fiber rings of tensor products of algebras over a ring. Also, given a triplet of rings $(R,A,B)$ consisting of two $R$-algebras $A$ and $B$ such that $A{\otimes }_{R}B\ne \{0\}$, we introduce the inherent notion to $(R,A,B)$ of a $B$-fibered AF-ring which allows to compute the Krull dimension of all fiber rings of the considered tensor product $A{\otimes }_{R}B$. As an application, we provide a formula for the Krull dimension of $A{\otimes }_{R}B$ when either $R$ or $A$ is zero-dimensional as well as for the Krull dimension of $A{\otimes }_{\mathbb{Z}}B$ when $A$ is a fibered AF-ring over the ring of integers $\mathbb{Z}$ with nonzero characteristic and $B$ is an arbitrary ring. This enables us to answer a question of Jorge Martinez on evaluating the Krull dimension of $A{\otimes }_{\mathbb{Z}}B$ when $A$ is a Boolean ring. Actually, we prove that if $A$ and $B$ are rings such that $A{\otimes }_{\mathbb{Z}}B$ is not trivial and $A$ is a Boolean ring, then dim$(A{\otimes }_{\mathbb{Z}}B)=\text{dim}(\frac{B}{2B})$.

Height is associated with postural balance performance. However, there has been no study on whether the test-retest reliability of the balance performance variables is related to height. The goal of our study was to analyze the effect of body height on the test-retest reliability of center of pressure (COP) variables during natural and feet-together stances in healthy young subjects. Twenty normal young men were classified into the short height group (height range, 163–174cm) and the tall height group (height range, 176–184cm) with 10 subjects each to investigate the influence of height on test-retest reliability. All subjects performed the static balance test on a commercial force plate in feet-together and natural stance conditions three times. Four variables were calculated from the COP data. COP distance, COP area, COP velocity, and COP mean frequency were selected as the postural balance variables. Intraclass correlation coefficient (ICC) was evaluated for an analysis of the test-retest reliability of each COP variable. In both stance conditions, the COP distance and the COP area in the short height group (ICC = 0.717–0.902) showed better test-retest reliability compared to those in the tall height group (ICC = 0.146–0.688). COP velocity had the highest ICC value in both stance conditions (ICC = 0.744–0.970). Additionally, the short height group showed slightly better reliability in the feet-together stance condition, whereas the tall height group showed slightly higher ICC value in the natural stance condition. These results suggest that height should be considered for a more reliable assessment of repetitive static balance performance. Our findings might be useful for the development of an optimal static balance test protocol regardless of the effect of height.

Let H be an abelian group written additively and k be a positive integer. Let G(H, k) denote the digraph whose set of vertices is just H, and there exists a directed edge from a vertex a to a vertex b if b = ka. In this paper we give a necessary and sufficient condition for G(H, k₁) ≃ G(H, k₂). We also discuss the problem when G(H₁, k) is isomorphic to G(H₂, k) for a given k. Moreover, we give an explicit formula of G(H, k) when H is a p-group and gcd(p, k)=1.

We use Fourier-analytic methods to give a new proof of Bilu's theorem on the complex equidistribution of small points on the one-dimensional algebraic torus. Our approach yields a quantitative bound on the error term in terms of the height and the degree.

What is the probability for a number field of composite degree d to have a nontrivial subfield? As the reader might expect the answer heavily depends on the interpretation of probability. We show that if the fields are enumerated by the smallest height of their generators the probability is zero, at least if d > 6. This is in contrast to what one expects when the fields are enumerated by the discriminant. The main result of this paper is an estimate for the number of algebraic numbers of degree d = en and bounded height which generate a field that contains an unspecified subfield of degree e. If n > max{e² + e, 10}, we get the correct asymptotics as the height tends to infinity.

Let C : Y² = a_n Xⁿ + ⋯ + a₀ be a hyperelliptic curve with the a_i rational integers, n ≥ 5, and the polynomial on the right irreducible. Let J be its Jacobian. Let S be a finite set of rational primes. We give a completely explicit upper bound for the size of the S-integral points on the model C, provided we know at least one rational point on C and a Mordell–Weil basis for J(ℚ). We use a refinement of the Mordell–Weil sieve which, combined with the upper bound, is capable of determining all the S-integral points. The method is illustrated by determining the S-integral points on the genus 2 hyperelliptic model Y² - Y = X⁵ - X for the set S of the first 22 primes.

We characterize all algebraic numbers which are roots of integer polynomials with a coefficient whose modulus is greater than or equal to the sum of moduli of all the remaining coefficients. It turns out that these numbers are zero, roots of unity and those algebraic numbers β whose conjugates over ℚ (including β itself) do not lie on the circle |z| = 1. We also describe all algebraic integers with norm B which are roots of an integer polynomial with constant coefficient B and the sum of moduli of all other coefficients at most |B|. In contrast to the above, the set of such algebraic integers is "quite small". These results are motivated by a recent paper of Frougny and Steiner on the so-called minimal weight β-expansions and are also related to some work on canonical number systems and tilings.

Over the function field of a smooth projective curve over an algebraically closed field, we investigate the set of S-integral elements in a forward orbit under a rational function by establishing some analogues of the classical Siegel theorem.

We establish an inequality comparing the height and the χ-arithmetic volume of toric metrized divisors on . This gives a partial answer to a question of Burgos, Moriwaki, Philippon and Sombra.

We study Laurent polynomials in any number of variables that are sums of at most $k$ monomials. We first show that the Mahler measure of such a polynomial is at least $h/2^{k-2}$, where $h$ is the height of the polynomial. Next, restricting to such polynomials having integer coefficients, we show that the set of logarithmic Mahler measures of the elements of this restricted set is a closed subset of the nonnegative real line, with $0$ being an isolated point of the set. In the final section, we discuss the extent to which such an integer polynomial of Mahler measure $1$ is determined by its $k$ coefficients.

Let $\rho$ be a Drinfeld ${𝔽}_q[T]$-module defined over a global function field $K.$ Let $\alpha \in K$ be a non-torsion point of $\rho$ with infinite ${\rho }_T$-orbit. For each $n\ge 1,$ write the ideal $({\rho }_{T^n}(\alpha ))={𝔄}_n{𝔅}_n^{-1}$ as a quotient of relatively prime integral ideals. We establish an analogue of the classical Zsigmondy theorem for the ideal sequence ${\{{𝔄}_n\}}_{n\ge 1},$ i.e. for all but finitely many $n\ge 1,$ there exists a prime ideal ${𝔭}_n$ such that ${𝔭}_n|{𝔄}_n$ and ${𝔭}_n\nmid {𝔄}_i$ for all $i<n.$

In this paper, as an analog of the number field case, for an elliptic curve $E$ defined over the algebraic numbers and for any subfield $F$ of algebraic numbers, we say that $E$ has the Northcott property over $F$ if there are at most finitely many $F$-rational points on $E$ of uniformly bounded height, and we say that $E$ has the property (P) over $F$ if for any infinite subset $S$ of $F$-rational points on $E$, $f(S)=S$ for an $F$-endomorphism $f$ of $E$ implies that $f$ is an automorphism. We establish some criteria for both properties and provide typical examples. We also show that the Northcott property implies the property (P), but the converse is not true.

For every algebraic number $\kappa$ on the unit circle which is not a root of unity we prove the existence of a strict sequence of algebraic numbers whose height tends to zero, such that the averages of the evaluation of $f_{\kappa }(z)=log|z-\kappa |$ at the conjugates are essentially bounded from above by $-h(\kappa )$. This completes a characterization on functions $f_{\kappa }$ initiated by Autissier and Baker–Masser, who cover the cases $\kappa =2$ and $\left|\kappa \right|\ne 1$, respectively. Using the same ideas we also prove analogues in the $p$-adic setting.

In order to study the fitting of some characteristics like low diameter at breast height (DBH), the height and height to the beginning of crown of the Oak species in two man-made and natural forests of this species in north of Guilan’s forests, Tooshi and Radarposhte forests were selected. A 100% inventory was done on a one-hectare sample plot in Tooshi man-made forests and on a two-hectare sample plot in Radarposhte natural forests. The results from this study showed that, in order to fit the diameter and the height of the Oak trees in man-made stands, Johnson’s S${}_{\mathrm{\text{B}}}$ and normal statistical distributions have more capabilities and the three-parameter gamma and the three-parameter Weibull distributions are appropriate for diameters and heights of the natural Oak trees in Radarposhte forest, respectively.

For a finite commutative ring R and a positive integer k, let G(R, k) denote the digraph whose set of vertices is R and for which there is a directed edge from a to a^k. The digraph G(R, k) is called symmetric of order M if its set of connected components can be partitioned into subsets of size M with each subset containing M isomorphic components. We primarily aim to factor G(R, k) into the product of its subdigraphs. If the characteristic of R is a prime p, we obtain several sufficient conditions for G(R, k) to be symmetric of order M.

We derive a functional upper bound for the ℤ₂-height of the ith Stiefel–Whitney class of the canonical oriented k-plane bundle over the oriented Grassmann manifold of oriented k-dimensional vector subspaces in Euclidean n-space. It turns out that is for many pairs (n, k) better than the upper bound implied by Dutta and Khare's (2002) results on the height of the second Stiefel–Whitney class of the canonical k-plane bundle over the Grassmann manifold G_{n, k} of k-dimensional vector subspaces in Euclidean n-space. In addition to this, always coincides with the exact value of the height (well known for ). We also prove that implies an upper bound which differs from the exact value of the height of the second Stiefel–Whitney class by at most one.

The virtual-height (virtual-width) of a straight line embedding of a plane graph in the xy-plane is the number of vertices with pairwise distinct y-coordinates (x-coordinates). We show that any plane graph of order n has a straight line embedding of virtual-height at most and virtual-width at most and present a polynomial time algorithm which obtains such an embedding.