Showing papers on "Divisor published in 2004"

The distribution of integers with a divisor in a given interval

Abstract: We determine the order of magnitude of H(x,y,z), the number of integers n\le x having a divisor in (y,z], for all x,y and z. We also study H_r(x,y,z), the number of integers n\le x having exactly r divisors in (y,z]. When r=1 we establish the order of magnitude of H_1(x,y,z) for all x,y,z satisfying z\le x^{0.49}. For every r\ge 2, $C>1$ and $\epsilon>0$, we determine the the order of magnitude of H_r(x,y,z) when y is large and y+y/(\log y)^{\log 4 -1 - \epsilon} \le z \le \min(y^{C},x^{1/2-\epsilon}). As a consequence of these bounds, we settle a 1960 conjecture of Erdos and several related conjectures. One key element of the proofs is a new result on the distribution of uniform order statistics.

Asymptotic behavior of eigenvalues of greatest common divisor matrices

Abstract: Let $\{x_i\}_{i=1}^{\infty}$ be an arbitrary strictly increasing infinite sequence of positive integers. For an integer $n\ge 1$, let $S_n=\{x_1,\ldots,x_n\}$. Let $\varepsilon$ be a real number and $q\ge 1$ a given integer. Let \smash{$\lambda _n^{(1)}\le \cdots\le \lambda _n^{(n)}$} be the eigenvalues of the power GCD matrix $((x_i, x_j)^{\varepsilon})$ having the power $(x_i,x_j)^{\varepsilon}$ of the greatest common divisor of $x_i$ and $x_j$ as its $i,j$-entry. We give a nontrivial lower bound depending on $x_1$ and $n$ for \smash{$\lambda _n^{(1)}$} if $\varepsilon>0$. Especially for $\varepsilon>1$, this lower bound is given by using the Riemann zeta function. Let $x\ge 1$ be an integer. For a sequence \smash{$\{x_i\}_{i=1}^{\infty }$} satisfying that $(x_i, x_j)=x$ for any $i e j$ and \smash{$\sum_{i=1}^{\infty }{1\over {x_i}}=\infty$}, we show that if $0<\varepsilon\le 1$, then \smash{${\rm lim}_{n\rightarrow \infty }\lambda _n^{(1)}=x_1^{\varepsilon}-x^{\varepsilon }$}. Let $a\ge 0, b\ge 1$ and $e\ge 0$ be any given integers. For the arithmetic progression \smash{$\{x_{i-e+1}=a+bi\}_{i=e}^{\infty}$}, we show that if $0<\varepsilon\le 1$, then \smash{${\rm lim}_{n\rightarrow \infty }\lambda _n^{(q)}=0$}. Finally, we show that for any sequence \smash{$\{x_i\}_{i=1}^{\infty}$} and any \smash{$\varepsilon>0$, $\lambda_n^{(n-q+1)}$} approaches infinity when $n$ goes to infinity.

Chebyshev's bias for composite numbers with restricted prime divisors

Abstract: Let π(x;d,a) denote the number of primes p ≤ x with p ≡ a(modd). Chebyshev's bias is the phenomenon for which "more often" π(x; d, n) > π(x; d, r), than the other way around, where n is a quadratic nonresidue mod d and r is a quadratic residue mod d. If π(x; d, n) ≥ π(x; d, r) for every x up to some large number, then one expects that N(x; d, n) ≥ N(x; d, r) for every x. Here N(x; d, a) denotes the number of integers n ≤ x such that every prime divisor p of n satisfies p ≡ a(mod d). In this paper we develop some tools to deal with this type of problem and apply them to show that, for example, N(x; 4, 3) ≥ N(x; 4, 1) for every x.In the process we express the so-called second order Landau-Ramanujan constant as an infinite series and show that the same type of formula holds for a much larger class of constants.

Processing unit having decimal floating-point divider using Newton-Raphson iteration

Abstract: A decimal floating-point divider is described that implements efficient hardware-based techniques for performing decimal floating-point division. The divider uses an accurate piecewise linear approximation to obtain an initial estimate of a divisor's reciprocal. The divider improves the initial estimate of the divisor's reciprocal using a modified form of Newton-Raphson iteration. The divider multiplies the estimated divisor's reciprocal by the dividend to produce a preliminary quotient. The preliminary quotient is rounded to produce the final decimal floating-point quotient.

On the riemann zeta-function and the divisor problem

Abstract: Let Δ(x) denote the error term in the Dirichlet divisor problem, and E(T) the error term in the asymptotic formula for the mean square of $$\left| {\varsigma \left( {\tfrac{1}{2} + it} \right)} \right|$$ . If $$E^* \left( t \right) = E\left( t \right) - 2\pi \Delta ^* \left( {t / 2\pi } \right)$$ with $$\Delta ^* \left( x \right) = - \Delta \left( x \right) + 2\Delta \left( {2x} \right) - \tfrac{1}{2}\Delta \left( {4x} \right)$$ , then we obtain $$\int_0^T {\left( {E^* \left( t \right)} \right)^4 dt \ll _e T^{16/9 + \varepsilon } } $$ . We also show how our method of proof yields the bound $$\sum\limits_{r = 1}^R {\left( {\int_{tr - G}^{tr + G} {\left| {\varsigma \left( {\tfrac{1}{2} + it} \right)} \right|^2 dt} } \right)^4 \ll _e T^{2 + e} G^{ - 2} + RG^4 T^\varepsilon } $$ , where T 1/5+e≤G≪T, T

Multi-modulus programmable frequency divider

Abstract: A programmable frequency divider for dividing the frequency of a source signal according to a selectable divisor which is obtained based on a plurality of divisor signals and outputting a result signal having a divided frequency includes at least one cell of a first type. The cells of the first type are cascaded with each other. The programmable frequency divider synchronously resets all of the cells of the first type according to a reset signal in order to selectively switch each cell of the first type to perform a divide-by-two or divide-by-three operation according to a corresponding divisor signal. The last cell of the first type outputs the result signal having the divided frequency.

Dualit\'e et comparaison sur les complexes de de Rham logarithmiques par rapport aux diviseurs libres

Abstract: Let X be a complex analytic manifold and D \subset X a free divisor. Integrable logarithmic connections along D can be seen as locally free {\cal O}_X-modules endowed with a (left) module structure over the ring of logarithmic differential operators {\cal D}_X(\log D). In this paper we study two related results: the relationship between the duals of any integrable logarithmic connection over the base rings {\cal D}_X and {\cal D}_X(\log D), and a differential criterion for the logarithmic comparison theorem. We also generalize a formula of Esnault-Viehweg in the normal crossing case for the Verdier dual of a logarithmic de Rham complex.

Programmable frequency divider with symmetrical output

Abstract: A programmable frequency divider circuit with symmetrical output is disclosed. The frequency divider includes a non-symmetrical LFSR based component operated in series with a symmetrical divider component. Both the LFSR and the symmetrical divider may be programmed to provide flexibility. The frequency divider can dynamically adjust the divisor of the LFSR component to overcome limitations in the divide resolution due to the series combination of dividers, providing even and odd divisor values. The divider architecture can also provide higher level functions, including synchronization of multiple divider outputs, dynamic switching of divisor values and generation of multi-phased and spaced outputs. The linear feedback shift register (LFSR) component includes a feedback logic network decomposed into multiple stages to realize a maximum latch-to-latch operational latency of one gate delay regardless of the size of the LFSR.

Accelerating correctly rounded floating-point division when the divisor is known in advance

Abstract: We present techniques for accelerating the floating-point computation of x/y when y is known before x. The proposed algorithms are oriented toward architectures with available fused-mac operations. The goal is to get exactly the same result as with usual division with rounding to nearest. It is known that the advanced computation of 1/y allows performing correctly rounded division in one multiplication plus two fused-macs. We show algorithms that reduce this latency to one multiplication and one fused-mac. This is achieved if a precision of at least n+1 bits is available, where n is the number of mantissa bits in the target format, or if y satisfies some properties that can be easily checked at compile-time. This requires a double-word approximation of 1/y (we also show how to get it). Compilers to accelerate some numerical programs without loss of accuracy can use these techniques.

Exact Asymptotics in log log Laws for Random Fields

Abstract: Let $$\{ X,X_k ,k \in {\mathbb{N}}^r \}$$ be i.i.d. random variables, and set S n =∑ k ≤ n X k . We exhibit a method able to provide exact loglog rates. The typical result is that $${\mathop {\lim }\limits_{\varepsilon \searrow \sigma \sqrt {2r}} } \sqrt {\varepsilon ^2 - 2r\sigma ^2 } \sum\limits_n {\frac{1}{{|\,n\,|}}P(|S_n \geqslant \varepsilon \sqrt {|\,n\,|\log \log |\,n\,|} ) = \frac{{\sigma \sqrt {2r} }}{{r!}},}$$ whenever EX=0,EX 2=σ2 and E[X 2(log+ | X |) r-1] < ∞. To get this and other related precise asymptotics, we derive some general estimates concerning the Dirichlet divisor problem, of interest in their own right.

Higher Correlations of Divisor Sums Related to Primes II: Variations of the error term in the prime number theorem

Abstract: We calculate the triple correlations for the truncated divisor sum $\lambda_{R}(n)$. The $\lambda_{R}(n)$'s behave over certain averages just as the prime counting von Mangoldt function $\Lambda(n)$ does or is conjectured to do. We also calculate the mixed (with a factor of $\Lambda(n)$) correlations. The results for the moments up to the third degree, and therefore the implications for the distribution of primes in short intervals, are the same as those we obtained (in the first paper with this title) by using the simpler approximation $\Lambda_{R}(n)$. However, when $\lambda_{R}(n)$ is used the error in the singular series approximation is often much smaller than what $\Lambda_{R}(n)$ allows. Assuming the Generalized Riemann Hypothesis for Dirichlet $L$-functions, we obtain an $\Omega_{\pm}$-result for the variation of the error term in the prime number theorem. Formerly, our knowledge under GRH was restricted to $\Omega$-results for the absolute value of this variation. An important ingredient in the last part of this work is a recent result due to Montgomery and Soundararajan which makes it possible for us to dispense with a large error term in the evaluation of a certain singular series average. We believe that our results on $\lambda_{R}(n)$'s and $\Lambda_{R}(n)$'s can be employed in diverse problems concerning primes.

Logarithmic Comparison Theorem and some Euler homogeneous free divisors

Abstract: Let D, x be a free divisor germ in a complex manifold X of dimension n > 2. It is an open problem to find out which are the properties required for D,x to satisfy the so-called Logarithmic Comparison Theorem (LCT), that is, when the complex of logarithmic differential forms computes the cohomology of the complement of D,x. We give a family of Euler homogeneous free divisors which, somewhat unexpectedly, does not satisfy the LCT.

The nef cone of toroidal compactifications of ${\cal A}_4$

Abstract: The moduli space of principally polarised abelian 4-folds can be compactified in several different ways by toroidal methods. Here we consider in detail the Igusa compactification and the (second) Voronoi compactification. We describe in both cases the cone of nef Cartier divisors. The proof depends on a detailed description of the Voronoi compactification, which makes it possible to proceed by induction, using the known description of the nef cone for compactifications of ${\mathcal A}_3$. The Igusa compactification has a non-${\mathbb Q}$-factorial singularity, which is resolved by a single blow-up: this resolution is the Voronoi compactification. The exceptional divisor $E$ is a toric Fano variety (of dimension 9): the other boundary divisor, $D$, corresponds to degenerations with corank~1. After imposing a level structure in order to avoid certain technical complications, we show that the closure of $D$ in the Voronoi compactification maps to the Voronoi compactification of ${\mathcal A}_3$. The toric description of the exceptional divisor allows us to describe the map in sufficient detail to estimate the intersection numbers needed. This inductive process is only valid for the Voronoi compactification: the result for the Igusa compactification is deduced from the Voronoi compactification.

Low latency integer divider and integration with floating point divider and method

Abstract: A method and device divides a dividend by a divisor, the dividend and the divisor both being integers. The method and device determine a maximum possible number of quotient digits (NDQ) based on a number of significant digits of the divisor and the dividend, normalizes the dividend and divisor, and calculates NDQ number of quotient digits from the normalized divisor and dividend.

Linear systems on a special rational surface

Abstract: We study the Hilbert series of a family of ideals J_\phi generated by powers of linear forms in k[x_1,...,x_n]. Using the results of Emsalem-Iarrobino, we formulate this as a question about fatpoints in P^{n-1}. In the three variable case this is equivalent to studying the dimension of a linear system on a blow up of P^2. The ideals that arise have the points in very special position, but because there are only seven points, we can apply results of Harbourne to obtain the classes of the negative curves. Reducing to an effective, nef divisor and using Riemann-Roch yields a formula for the Hilbert series. This proves the n=3 case of a conjecture of Postnikov and Shapiro, which they later showed true for all n. Postnikov and Shapiro observe that for a family of ideals closely related to J_\phi a similar result often seems to hold, although counterexamples exist for n=4 and n=5. Our methods allow us to prove that for n=3 an analogous formula is indeed true. We close with a counterexample to a conjecture Postnikov and Shapiro make about the minimal free resolution of these ideals.

Smarandache zero divisors

Abstract: In this paper, we study the notion of Smarandache zero divisor in semigroups and rings. We illustrate them with examples and prove some interesting results about them.

Some improved algorithms for hyperelliptic curve cryptosystems using degenerate divisors

Abstract: Hyperelliptic curve cryptosystems (HECC) can be good alternatives to elliptic curve cryptosystems, and there is a good possibility to improve the efficiency of HECC due to its flexible algebraic structure. Recently, an efficient scalar multiplication technique for application to genus 2 curves using a degenerate divisor has been proposed. This new technique can be used in the cryptographic protocol using a fixed base point, e.g., HEC-DSA. This paper considers two important issues concerning degenerate divisors. First, we extend the technique for genus 2 curves to genus 3 curves. Jacobian variety for genus 3 curves has two different degenerate divisors: degree 1 and 2. We present explicit formulae of the addition algorithm with degenerate divisors, and then present the timing of scalar multiplication using the proposed formulae. Second, we propose several window methods using the degenerate divisors. It is not obvious how to construct a base point D such that degD = deg(aD)

Fast integer division with minimum number of iterations in substraction-based hardware divide processor

Abstract: A method and system for performing integer divisions using subtraction-based division processes in a hardware divide processor primarily dedicated for floating-point division processes. In particular, the method and system involve calculating a quotient of a dividend and a divisor, the dividend and divisor being binary coded integer values, by normalizing the divisor and the dividend, determining a number of binary digits (nV) needed to represent the divisor and a number of binary digits (nD) needed to represent the dividend, determining a number of effective binary digits (nQ) needed to represent the quotient, determining a start bit position to start a subtraction-based divide process, and performing the subtraction-based divide process only for bit positions beginning at the start bit position and at a least significant bit position. In preferred embodiments, the subtraction-based divide process is an SRT (Sweeney, Robinson, Tocher) Divide process.

High bandwidth phase locked loop (PLL)

Abstract: A phase locked loop (PLL) is provided. In one implementation, the PLL includes a feedback loop having a frequency multiplier and an integer divider to generate a divided signal. The PLL includes a re-sampling circuit operable to re-sample one or more digital pulses of the divided signal using one or more phase signals if a multiplication factor of the frequency multiplier does not divide evenly into the integer divisor.

On the Riemann zeta-function and the divisor problem III

Abstract: Let $\Delta(x)$ denote the error term in the Dirichlet divisor problem, and $E(T)$ the error term in the asymptotic formula for the mean square of $|\zeta(1/2+it)|$. If $E^*(t) = E(t) - 2\pi\Delta^*(t/2\pi)$ with $\Delta^*(x) = -\Delta(x) + 2\Delta(2x) - {1\over2}\Delta(4x)$ and we set $\int_0^T E^*(t) dt = 3\pi T/4 + R(T)$, then we obtain $$ R(T) = O_\epsilon(T^{593/912+\epsilon}), \int_0^TR^4(t) dt \ll_\epsilon T^{3+\epsilon}, $$ and $$ \int_0^TR^2(t) dt = T^2P_3(\log T) + O_\epsilon(T^{11/6+\epsilon}), $$ where $P_3(y)$ is a cubic polynomial in $y$ with positive leading coefficient.

A necessary and sufficient condition for Riemann's singularity theorem to hold on a Prym theta divisor

Abstract: Let $(P,\Xi)$ be the naturally polarized model of the Prym variety associated to the etale double cover $\pi : \tilde C\rightarrow C$ of smooth connected curves defined over an algebraically closed field k of characteristic $ e 2$ , where genus( C ) = $g \ge 3$ , Pic $^{(2g-2)}(\tilde C) \supset P = \{\mathcal L \in {\rm Pic}^{(2g-2)}(\tilde C) : {\rm Nm}(\mathcal L) = \omega_C$ and $h^0(\tilde C,\mathcal L)$ is even\} is the Prym variety, and $P \supset \Xi = \{\mathcal L \in P: h^0(\tilde C,\mathcal L) >0 \}$ is the Prym theta divisor with its reduced scheme structure. If $\mathcal L$ is any point on $\Xi$ , we prove that ‘Riemann's singularity theorem holds at $\mathcal L$ ’, i.e. mult $_{\mathcal L}(\Xi) = (1/2)h^0(\tilde C,\mathcal L)$ , if and only if $\mathcal L$ cannot be expressed as $\pi^*(\mathcal M)(B)$ where $B \ge 0$ is an effective divisor on $\tilde C$ , and $\mathcal M$ is a line bundle on C with $h^0(C,\mathcal M) >(1/2)h^0(\tilde C,\mathcal L)$ . This completely characterizes points of $\Xi$ where the tangent cone is the set theoretic restriction of the tangent cone of $\tilde {\Theta}$ , hence also those points on $\Xi$ where Mumford's Pfaffian equation defines the tangent cone to $\Xi$ .

Rational Divisors in Rational Divisor Classes

Abstract: We discuss the situation where a curve \(\mathcal{C}\), defined over a number field K, has a known K-rational divisor class of degree 1, and consider whether this class contains an actual K-rational divisor. When \(\mathcal{C}\) has points everywhere locally, the local to global principle of the Brauer group gives the existence of such a divisor. In this situation, we give an alternative, more down to earth, approach, which indicates how to compute this divisor in certain situations. We also discuss examples where \(\mathcal{C}\) does not have points everywhere locally, and where no such K-rational divisor is contained in the K-rational divisor class.

Phase- and size-adjusted CT cut-off for differentiating neoplastic lesions from normal colon in contrast-enhanced CT colonography

Abstract: A computed tomography (CT) cut-off for differentiating neoplastic lesions (polyps/carcinoma) from normal colon in contrast-enhanced CT colonography (CTC) relating to the contrast phase and lesion size is determined. CT values of 64 colonic lesions (27 polyps <10 mm, 13 polyps ≥10 mm, 24 carcinomas) were determined by region-of-interest (ROI) measurements in 38 patients who underwent contrast-enhanced CTC. In addition, the height (H) of the colonic lesions was measured in CT. CT values were also measured in the aorta (A), superior mesenteric vein (V) and colonic wall. The contrast phase was defined by % !AMS LaTeX.tdl!TeX -- AMS-LaTeX!% MathType!MTEF!2!0!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiaadg% eacqGHRaWkdaqadaqaaiaaigdacqGHsislcaWG4baacaGLOaGaayzk% aaGaamOvaaaa!3D97! $$xA + {\left( {1 - x} \right)}V$$ using x as a weighting factor for describing the different contrast phases ranging from the pure arterial phase (x=1) over the intermediate phases (x=0.9–0.1) to the pure venous phase (x=0). The CT values of the lesions were correlated with their height (H), the different phases ( % !AMS LaTeX.tdl!TeX -- AMS-LaTeX!% MathType!MTEF!2!0!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiaadg% eacqGHRaWkdaqadaqaaiaaigdacqGHsislcaWG4baacaGLOaGaayzk% aaGaamOvaaaa!3D97! $$xA + {\left( {1 - x} \right)}V$$ ) and the ratio % !AMS LaTeX.tdl!TeX -- AMS-LaTeX!% MathType!MTEF!2!0!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaaca% WG4bGaamyqaiabgUcaRmaabmaabaGaaGymaiabgkHiTiaadIhaaiaa% wIcacaGLPaaacaWGwbaacaGLBbGaayzxaaGaai4laiaadIeaaaa!4109! $${\left[ {xA + {\left( {1 - x} \right)}V} \right]}/H$$ . The CT cut-off was linearly adjusted to the imaged contrast phase and height of the lesion by the line % !AMS LaTeX.tdl!TeX -- AMS-LaTeX!% MathType!MTEF!2!0!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaiabg2% da9iaad2gadaWadaqaaiaadIhacaWGbbGaey4kaSYaaeWaaeaacaaI% XaGaeyOeI0IaamiEaaGaayjkaiaawMcaaiaadAfaaiaawUfacaGLDb% aacaGGVaGaamisaiabgUcaRiaadMhadaWgaaWcbaGaaGimaaqabaaa% aa!46C5! $$y = m{\left[ {xA + {\left( {1 - x} \right)}V} \right]}/H + y_{0} $$ . The slope m was determined by linear regression in the correlation ( % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeiBaiaabw % gacaqGZbGaaeyAaiaab+gacaqGUbGaeSipIOZaamWaaeaacaWG4bGa % amyqaiabgUcaRmaabmaabaGaaGymaiabgkHiTiaadIhaaiaawIcaca % GLPaaacaWGwbaacaGLBbGaayzxaaGaai4laiaadIeaaaa!47D1! $$ {\text{lesion}} \sim {\left[ {xA + {\left( {1 - x} \right)}V} \right]}/H $$ ) and the Y-intercept y0 by the minimal shift of the line needed to maximize the accuracy of separating the colonic wall from the lesions. The CT value of the lesions correlated best with the intermediate phase: 0.4A + 0.6V (r=0.8 for polyps ≥10 mm, r=0.6 for carcinomas, r=0.4 for polyps <10 mm). The accuracy in the differentiation between lesions and normal colonic wall increased with the height implemented as divisor, reached 91% and was obtained by the dynamic cut-off described by the formula: % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafeart1ev1aaatCvAUfeBSn0BKvguHDwzZbqefeKCPfgBGuLBPn % 2BKvginnfarmqr1ngBPrgitLxBI9gBamXvP5wqSXMqHnxAJn0BKvgu % HDwzZbqegm0B1jxALjhiov2DaeHbuLwBLnhiov2DGi1BTfMBaebbfv % 3ySLgzGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj-hEeeu0xXdbba9 % frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbP % Yxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaaiaa % bogacaqG1bGaaeiDaiaab2cacaqGVbGaaeOzaiaabAgadaqadaqaai % aadgeacaGGSaGaamOvaiaacYcacaWGibaacaGLOaGaayzkaaGaeyyp % a0JaaGymaiaac6cacaaIXaWaamWaaeaacaaIWaGaaiOlaiaaisdaca % WGbbGaey4kaSIaaGimaiaac6cacaaI2aGaamOvaaGaay5waiaaw2fa % aiaac+cacaWGibGaey4kaSIaaGOnaiaaiMdacaGGUaGaaGioaaaa!627A! $$ {\text{cut-off}}{\left( {A,V,H} \right)} = 1.1{\left[ {0.4A + 0.6V} \right]}/H + 69.8 $$ . The CT value of colonic polyps or carcinomas can be increased extrinsically by scanning in the phase in which 0.4A + 0.6V reaches its maximum. Differentiating lesions from normal colon based on CT values is possible in contrast-enhanced CTC and improves when the cut-off is adjusted (normalized) to the contrast phase and lesion size.

Method and apparatus for performing a divide instruction

Abstract: An apparatus and method to perform a division algorithm on an integer divisor and integer dividend. More particularly, embodiments of the invention relate to a technique to align integer operands such that a relatively fast division algorithm may be performed on the integer operands.

Non-rational divisors over non-degenerate cDV-points

Abstract: Let $(X,o)$ be a 3-dimensional terminal singularity of type $cD$ or $cE$ defined in $\mathbb{C}^4$ by an equation non-degenerate with respect to its Newton diagram We show that there is not more than 1 non-rational divisor $E$ over $(X,o)$ with discrepancy $a(E,X)=1$ We also describe all blowups $\sigma$ of $(X,o)$ such that $E=\Exc(\sigma)$ is non-rational and $a(E,X)=1$

Primes Generated by Recurrence Sequences

Abstract: We consider primitive divisors of terms of integer sequences defined by quadratic polynomials. Apart from some small counterexamples, when a term has a primitive divisor, that primitive divisor is unique. It seems likely that the number of terms with a primitive divisor has a natural density. We discuss two heuristic arguments to suggest a value for that density, one using recent advances made about the distribution of roots of polynomial congruences.

Method and apparatus for efficient software-based integer division

Abstract: A method and apparatus to perform efficient software-based integer division. The equivalent of a hardware-based integer division operation is enabled via a reciprocal multiplication operation that is facilitated by a minimum combination of multiplication (and/or add) and shift operations. Properties and equations are derived for determining minimum multiplication and shift instructions to perform an integer division of a variable dividend and constant divisor using reciprocal multiplication. Computer functions are disclosed for determining parameters from which the minimum multiplication and shift instructions can be derived. Software/firmware is then coded employing the minimum multiplication and shift instructions to perform software-based integer division operations via reciprocal multiplication. In one embodiment, the integer division operations are employed to determine a minimum number of cells required to store the data in a packet or frame that is processed by a network processor.

On the higher moments of the error term in the divisor problem

Abstract: Let $\Delta(x)$ denote the error term in the Dirichlet divisor problem. Our main results are the asymptotic formulas $$ \int_1^X \Delta^3(x){\rm d}x = BX^{7/4} + O_\epsilon(X^{\beta+\epsilon}) \qquad(B > 0) $$ and $$ \int_1^X \Delta^4(x){\rm d}x = CX^2 + O_\epsilon(X^{\gamma+\epsilon}) \qquad(C > 0) $$ with $\beta = 7/5, \gamma = 23/12$. This improves on the values $\beta = 47/28, \gamma = 45/23$, due to K.-M. Tsang. A result on the integrals of $\Delta^3(x)$ and $\Delta^4(x)$ in short intervals is also proved.