Journal articles: 'Q-z curve'

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Dujella, Andrej, and Juan Carlos Peral. "High-rank elliptic curves with torsion induced by Diophantine triples." LMS Journal of Computation and Mathematics 17, no. 1 (2014): 282–88. http://dx.doi.org/10.1112/s1461157014000023.

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AbstractWe construct an elliptic curve over the field of rational functions with torsion group$\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/4\mathbb{Z}$and rank equal to four, and an elliptic curve over$\mathbb{Q}$with the same torsion group and rank nine. Both results improve previous records for ranks of curves of this torsion group. They are obtained by considering elliptic curves induced by Diophantine triples.

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Kim, Seon-Hong. "The number of connected components of certain real algebraic curves." International Journal of Mathematics and Mathematical Sciences 25, no. 11 (2001): 693–701. http://dx.doi.org/10.1155/s0161171201010481.

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For an integern≥2, letp(z)=∏k=1n(z−αk)andq(z)=∏k=1n(z−βk), whereαk,βkare real. We find the number of connected components of the real algebraic curve{(x,y)∈ℝ2:|p(x+iy)|−|q(x+iy)|=0}for someαkandβk. Moreover, in these cases, we show that each connected component contains zeros ofp(z)+q(z), and we investigate the locus of zeros ofp(z)+q(z).

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Heer, Henriette, Gary McGuire, and Oisín Robinson. "JKL-ECM: an implementation of ECM using Hessian curves." LMS Journal of Computation and Mathematics 19, A (2016): 83–99. http://dx.doi.org/10.1112/s1461157016000231.

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We present JKL-ECM, an implementation of the elliptic curve method of integer factorization which uses certain twisted Hessian curves in a family studied by Jeon, Kim and Lee. This implementation takes advantage of torsion subgroup injection for families of elliptic curves over a quartic number field, in addition to the ‘small parameter’ speedup. We produced thousands of curves with torsion$\mathbb{Z}/6\mathbb{Z}\oplus \mathbb{Z}/6\mathbb{Z}$and small parameters in twisted Hessian form, which admit curve arithmetic that is ‘almost’ as fast as that of twisted Edwards form. This allows JKL-ECM to compete with GMP-ECM for finding large prime factors. Also, JKL-ECM, based on GMP, accepts integers of arbitrary size. We classify the torsion subgroups of Hessian curves over$\mathbb{Q}$and further examine torsion properties of the curves described by Jeon, Kim and Lee. In addition, the high-performance curves with torsion$\mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/8\mathbb{Z}$of Bernsteinet al. are completely recovered by the$\mathbb{Z}/4\mathbb{Z}\oplus \mathbb{Z}/8\mathbb{Z}$family of Jeon, Kim and Lee, and hundreds more curves are produced besides, all with small parameters and base points.

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DELBOURGO, DANIEL, and ANTONIO LEI. "Non-commutative Iwasawa theory for elliptic curves with multiplicative reduction." Mathematical Proceedings of the Cambridge Philosophical Society 160, no. 1 (October 15, 2015): 11–38. http://dx.doi.org/10.1017/s0305004115000535.

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AbstractLet$E_{/{\mathbb{Q}}}$be a semistable elliptic curve, andp≠ 2 a prime of bad multiplicative reduction. For each Lie extension$\mathbb{Q}$FT/$\mathbb{Q}$with Galois groupG∞≅$\mathbb{Z}$p⋊$\mathbb{Z}$p×, we constructp-adicL-functions interpolating Artin twists of the Hasse–WeilL-series of the curveE. Through the use of congruences, we next prove a formula for the analytic λ-invariant over the false Tate tower, analogous to Chern–Yang Lee's results on its algebraic counterpart. If one assumes the Pontryagin dual of the Selmer group belongs to the$\mathfrak{M}_{\mathcal{H}}$(G∞)-category, the leading terms of its associated Akashi series can then be computed, allowing us to formulate a non-commutative Iwasawa Main Conjecture in the multiplicative setting.

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Elahi, Kashif, Ali Ahmad, and Roslan Hasni. "Construction Algorithm for Zero Divisor Graphs of Finite Commutative Rings and Their Vertex-Based Eccentric Topological Indices." Mathematics 6, no. 12 (December 4, 2018): 301. http://dx.doi.org/10.3390/math6120301.

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Chemical graph theory is a branch of mathematical chemistry which deals with the non-trivial applications of graph theory to solve molecular problems. Graphs containing finite commutative rings also have wide applications in robotics, information and communication theory, elliptic curve cryptography, physics, and statistics. In this paper we discuss eccentric topological indices of zero divisor graphs of commutative rings Z p 1 p 2 × Z q , where p 1 , p 2 , and q are primes. To enhance the importance of these indices a construction algorithm is also devised for zero divisor graphs of commutative rings Z p 1 p 2 × Z q .

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Hile, G. N., and W. E. Pfaffenberger. "Idempotents in Complex Banach Algebras." Canadian Journal of Mathematics 39, no. 3 (June 1, 1987): 625–30. http://dx.doi.org/10.4153/cjm-1987-030-1.

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The concept of the spectrum of A relative to Q, where A and Q commute and are elements in a complex Banach algebra with identity I, was developed in [1]. A complex number z is in the Q-resolvent set of A if and only if is invertible in otherwise, z is in the Q-spectrum of A, or spectrum of A relative to Q. One result from [1] was the following.THEOREM. Suppose no points in the ordinary spectrum of Q have unit magnitude. Let C be a simple closed rectifiable curve which lies in the Q-resolvent of A, and let*where P is defined asxs•

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Howe, Lawrence. "Twisted Hasse-Weil L-Functions and the Rank of Mordell-Weil Groups." Canadian Journal of Mathematics 49, no. 4 (August 1, 1997): 749–71. http://dx.doi.org/10.4153/cjm-1997-037-7.

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AbstractFollowing a method outlined by Greenberg, root number computations give a conjectural lower bound for the ranks of certain Mordell–Weil groups of elliptic curves. More specifically, for PQn a PGL2(Z/pnZ)–extension of Q and E an elliptic curve over Q, define the motive E ⊗ ρ, where ρ is any irreducible representation of Gal(PQn /Q). Under some restrictions, the root number in the conjectural functional equation for the L-function of E ⊗ ρ is easily computable, and a ‘–1’ implies, by the Birch and Swinnerton–Dyer conjecture, that ρ is found in E(PQn) ⊗ C. Summing the dimensions of such ρ gives a conjectural lower bound ofp2n–p2n–1–p–1for the rank of E(PQn).

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Vitenti, Sandro D. P., and Mariana Penna-Lima. "How well can the evolution of the scale factor be reconstructed by the current data?" Proceedings of the International Astronomical Union 10, S306 (May 2014): 219–22. http://dx.doi.org/10.1017/s1743921314011004.

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AbstractDistance measurements are currently the most powerful tool to study the expansion history of the universe without assuming its matter content nor any theory of gravitation. In general, the reconstruction of the scale factor derivatives, such as the deceleration parameter q(z), is computed using two different methods: fixing the functional form of q(z), which yields potentially biased estimates, or approximating q(z) by a piecewise nth-order polynomial function, whose variance is large. In this work, we address these two methods by reconstructing q(z) assuming only an isotropic and homogeneous universe. For this, we approximate q(z) by a piecewise cubic spline function and, then, we add to the likelihood function a penalty factor, with scatter given by σrel. This factor allows us to vary continuously between the full n knots spline, σrel → ∞, and a single linear function, σrel → 0. We estimate the coefficients of q(z) using the Monte Carlo approach, where the realizations are generated considering ΛCDM as a fiducial model. We apply this procedure in two different cases and assuming four values of σrel to find the best balance between variance and bias. First, we use only the Supernova Legacy Survey 3-year (SNLS3) sample and, in the second analysis, we combine the type Ia supernova (SNeIa) likelihood with those of baryonic acoustic oscillations (BAO) and Hubble function measurements. In both cases we fit simultaneously q(z) and 4 nuisance parameters of the supernovae, namely, the magnitudes $\mathcal{M}$1 and $\mathcal{M}$2 and the light curve parameters α and β.

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Dujella, Andrej, Ivica Gusić, and Petra Tadić. "The rank and generators of Kihara’s elliptic curve with torsion $\mathbf{Z}/4\mathbf{Z}$ over $\mathbf{Q}(t)$." Proceedings of the Japan Academy, Series A, Mathematical Sciences 91, no. 8 (August 2015): 105–9. http://dx.doi.org/10.3792/pjaa.91.105.

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BHAVE, AMALA, and LACHIT BORA. "ON THE SELMER GROUP OF A CERTAIN -ADIC LIE EXTENSION." Bulletin of the Australian Mathematical Society 100, no. 2 (February 27, 2019): 245–55. http://dx.doi.org/10.1017/s0004972719000108.

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Let $E$ be an elliptic curve over $\mathbb{Q}$ without complex multiplication. Let $p\geq 5$ be a prime in $\mathbb{Q}$ and suppose that $E$ has good ordinary reduction at $p$. We study the dual Selmer group of $E$ over the compositum of the $\text{GL}_{2}$ extension and the anticyclotomic $\mathbb{Z}_{p}$-extension of an imaginary quadratic extension as an Iwasawa module.

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HILDEN, HUGH M., MARIA TERESA LOZANO, and JOSE MARIA MONTESINOS-AMILIBIA. "ON THE ARITHMETIC 2-BRIDGE KNOTS AND LINK ORBIFOLDS AND A NEW KNOT INVARIANT." Journal of Knot Theory and Its Ramifications 04, no. 01 (March 1995): 81–114. http://dx.doi.org/10.1142/s0218216595000053.

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Let (p/q, n) denote the orbifold with singular set the two bridge knot or link p/q and isotropy group cyclic of orden n. An algebraic curve [Formula: see text] (set of zeroes of a polynomial r(x, z)) is associated to p/q parametrizing the representations of [Formula: see text] in PSL [Formula: see text]. The coordinates x, z, are trace(A2)=x, trace(AB)=z where A and B[Formula: see text] are the images of canonical generators a, b of [Formula: see text]. Let (xn, zn) be the point of [Formula: see text] corresponding to the hyperbolic orbifold (p/q, n). We prove the following result: The (orbifold) fundamental group of (p/q, n) is arithmetic if and only if the field Q(xn, zn) has exactly one complex place and ϕ(xn)<ϕ(zn)<2 for every real embedding [Formula: see text]. Consider the angle α for which the cone-manifold (p/q, α) is euclidean. We prove that 2cosα is an algebraic number. Its minimal polynomial (called the h-polynomial) is then a knot invariant. We indicate how to generalize this h-polynomial invariant for any hyperbolic knot. Finally, we compute h-polynomials and arithmeticity of (p/q, n) with p≦40, and (p/q, n) with p≦99q2≡1 mod p. We finish the paper with some open problems.

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Lei, Antonio, and R. Sujatha. "On fine Selmer groups and the greatest common divisor of signed and chromatic 𝑝-adic 𝐿-functions." Proceedings of the American Mathematical Society 149, no. 8 (May 13, 2021): 3235–43. http://dx.doi.org/10.1090/proc/15480.

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Let E / Q E/\mathbb {Q} be an elliptic curve and p p an odd prime where E E has good supersingular reduction. Let F 1 F_1 denote the characteristic power series of the Pontryagin dual of the fine Selmer group of E E over the cyclotomic Z p \mathbb {Z}_p -extension of Q \mathbb {Q} and let F 2 F_2 denote the greatest common divisor of Pollack’s plus and minus p p -adic L L -functions or Sprung’s sharp and flat p p -adic L L -functions attached to E E , depending on whether a p ( E ) = 0 a_p(E)=0 or a p ( E ) ≠ 0 a_p(E)\ne 0 . We study a link between the divisors of F 1 F_1 and F 2 F_2 in the Iwasawa algebra. This gives new insights into problems posed by Greenberg and Pollack–Kurihara on these elements.

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Frasconi, Paolo, Daniele Baracchi, Betti Giusti, Ada Kura, Gaia Spaziani, Antonella Cherubini, Silvia Favilli, Andrea Di Lenarda, Guglielmina Pepe, and Stefano Nistri. "Two-Dimensional Aortic Size Normalcy: A Novelty Detection Approach." Diagnostics 11, no. 2 (February 2, 2021): 220. http://dx.doi.org/10.3390/diagnostics11020220.

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Background: To develop a tool for assessing normalcy of the thoracic aorta (TA) by echocardiography, based on either a linear regression model (Z-score), or a machine learning technique, namely one-class support vector machine (OC-SVM) (Q-score). Methods: TA diameters were measured in 1112 prospectively enrolled healthy subjects, aging 5 to 89 years. Considering sex, age and body surface area we developed two calculators based on the traditional Z-score and the novel Q-score. The calculators were compared in 198 adults with TA > 40 mm, and in 466 patients affected by either Marfan syndrome or bicuspid aortic valve (BAV). Results: Q-score attained a better Area Under the Curve (0.989; 95% CI 0.984–0.993, sensitivity = 97.5%, specificity = 95.4%) than Z-score (0.955; 95% CI 0.942–0.967, sensitivity = 81.3%, specificity = 93.3%; p < 0.0001) in patients with TA > 40 mm. The prevalence of TA dilatation in Marfan and BAV patients was higher as Z-score > 2 than as Q-score < 4% (73.4% vs. 50.09%, p < 0.00001). Conclusions: Q-score is a novel tool for assessing TA normalcy based on a model requiring less assumptions about the distribution of the relevant variables. Notably, diameters do not need to depend linearly on anthropometric measurements. Additionally, Q-score can capture the joint distribution of these variables with all four diameters simultaneously, thus accounting for the overall aortic shape. This approach results in a lower rate of predicted TA abnormalcy in patients at risk of TA aneurysm. Further prognostic studies will be necessary for assessing the relative effectiveness of Q-score versus Z-score.

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KIM, BYOUNG DU. "THE PLUS/MINUS SELMER GROUPS FOR SUPERSINGULAR PRIMES." Journal of the Australian Mathematical Society 95, no. 2 (June 7, 2013): 189–200. http://dx.doi.org/10.1017/s1446788713000165.

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AbstractSuppose that an elliptic curve $E$ over $ \mathbb{Q} $ has good supersingular reduction at $p$. We prove that Kobayashi’s plus/minus Selmer group of $E$ over a ${ \mathbb{Z} }_{p} $-extension has no proper $\Lambda $-submodule of finite index under some suitable conditions, where $\Lambda $ is the Iwasawa algebra of the Galois group of the ${ \mathbb{Z} }_{p} $-extension. This work is analogous to Greenberg’s result in the ordinary reduction case.

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Barroero, Fabrizio. "CM RELATIONS IN FIBERED POWERS OF ELLIPTIC FAMILIES." Journal of the Institute of Mathematics of Jussieu 18, no. 5 (August 2, 2017): 941–56. http://dx.doi.org/10.1017/s1474748017000287.

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Let $E_{\unicode[STIX]{x1D706}}$ be the Legendre family of elliptic curves. Given $n$ points $P_{1},\ldots ,P_{n}\in E_{\unicode[STIX]{x1D706}}(\overline{\mathbb{Q}(\unicode[STIX]{x1D706})})$, linearly independent over $\mathbb{Z}$, we prove that there are at most finitely many complex numbers $\unicode[STIX]{x1D706}_{0}$ such that $E_{\unicode[STIX]{x1D706}_{0}}$ has complex multiplication and $P_{1}(\unicode[STIX]{x1D706}_{0}),\ldots ,P_{n}(\unicode[STIX]{x1D706}_{0})$ are linearly dependent over End$(E_{\unicode[STIX]{x1D706}_{0}})$. This implies a positive answer to a question of Bertrand and, combined with a previous work in collaboration with Capuano, proves the Zilber–Pink conjecture for a curve in a fibered power of an elliptic scheme when everything is defined over $\overline{\mathbb{Q}}$.

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ZHANG, YONGKANG, BAOYI LI, and CUIPING LI. "ABELIAN INTEGRALS FOR THE ONE-PARAMETER BOGDANOV–TAKENS SYSTEM." International Journal of Bifurcation and Chaos 21, no. 09 (September 2011): 2723–27. http://dx.doi.org/10.1142/s0218127411030052.

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An explicit upper bound Z(2, n) ≤ n + m - 1 is derived for the number of zeros of Abelian integrals M1(h) = ∮γ(h) P(x, y) dy - Q(x, y) dx on the open interval (0, 1/6), where γ(h) is an oval lying on the algebraic curve Hλ = (1/2)x2 + (1/2)y2 - (1/3)x3 - λy3 = h, P(x, y), Q(x, y) are polynomials of x and y, and max { deg P(x, y), deg Q(x, y)} = n. The proof exploits the expansion of the first order Melnikov function M1(h, λ) near λ = 0 and assume (∂m/∂λm)M1(h, λ)|λ = 0 not vanish identically.

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DELBOURGO, DANIEL, and LLOYD PETERS. "HIGHER ORDER CONGRUENCES AMONGST HASSE–WEIL -VALUES." Journal of the Australian Mathematical Society 98, no. 1 (October 14, 2014): 1–38. http://dx.doi.org/10.1017/s1446788714000445.

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AbstractFor the $(d+1)$-dimensional Lie group $G=\mathbb{Z}_{p}^{\times }\ltimes \mathbb{Z}_{p}^{\oplus d}$, we determine through the use of $p$-power congruences a necessary and sufficient set of conditions whereby a collection of abelian $L$-functions arises from an element in $K_{1}(\mathbb{Z}_{p}\unicode[STIX]{x27E6}G\unicode[STIX]{x27E7})$. If $E$ is a semistable elliptic curve over $\mathbb{Q}$, these abelian $L$-functions already exist; therefore, one can obtain many new families of higher order $p$-adic congruences. The first layer congruences are then verified computationally in a variety of cases.

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Ado, Muhammad, Awang Jusoh, Abdulhamid Usman Mutawakkil, and Tole Sutikno. "Dynamic model of A DC-DC quasi-Z-source converter (q-ZSC)." International Journal of Electrical and Computer Engineering (IJECE) 9, no. 3 (June 1, 2019): 1585. http://dx.doi.org/10.11591/ijece.v9i3.pp1585-1597.

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Two quasi-Z-source DC-DC converters (q-ZSCs) with buck-boost converter gain were recently proposed. The converters have advantages of continuous gain curve, higher gain magnitude and buck-boost operation at efficient duty ratio range when compared with existing q-ZSCs. Accurate dynamic models of these converters are needed for global and detailed overview by understanding their operation limits and effects of components sizes. A dynamic model of one of these converters is proposed here by first deriving the gain equation, state equations and state space model. A generalized small signal model was also derived before localizing it to this topology. The transfer functions (TF) were all derived, the poles and zeros analyzed with the boundaries for stable operations presented and discussed. Some of the findings include existence of right-hand plane (RHP) zero in the duty ratio to output capacitor voltage TF. This is common to the Z-source and quasi-Z-source topologies and implies control limitations. Parasitic resistances of the capacitors and inductors affect the nature and positions of the poles and zeros. It was also found and verified that rather than symmetric components, use of carefully selected smaller asymmetric components L1 and C1 produces less parasitic voltage drop, higher output voltage and current under the same conditions, thus better efficiency and performance at reduced cost, size and weight.

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Česnavičius, Kęstutis. "The Manin constant in the semistable case." Compositio Mathematica 154, no. 9 (August 13, 2018): 1889–920. http://dx.doi.org/10.1112/s0010437x18007273.

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For an optimal modular parametrization $J_{0}(n){\twoheadrightarrow}E$ of an elliptic curve $E$ over $\mathbb{Q}$ of conductor $n$, Manin conjectured the agreement of two natural $\mathbb{Z}$-lattices in the $\mathbb{Q}$-vector space $H^{0}(E,\unicode[STIX]{x1D6FA}^{1})$. Multiple authors generalized his conjecture to higher-dimensional newform quotients. We prove the Manin conjecture for semistable $E$, give counterexamples to all the proposed generalizations, and prove several semistable special cases of these generalizations. The proofs establish general relations between the integral $p$-adic étale and de Rham cohomologies of abelian varieties over $p$-adic fields and exhibit a new exactness result for Néron models.

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KIM, DOHYEONG. "p-adic L-functions over the false Tate curve extensions." Mathematical Proceedings of the Cambridge Philosophical Society 155, no. 3 (July 22, 2013): 483–98. http://dx.doi.org/10.1017/s0305004113000431.

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AbstractLet f be a primitive modular form of CM type of weight k and level Γ0(N). Let p be an odd prime which does not divide N, and for which f is ordinary. Our aim is to p-adically interpolate suitably normalized versions of the critical values L(f, ρχ,n), where n=1,2,. . .,k − 1, ρ is a fixed self-dual Artin representation of M∞ defined by (1.1) below, and χ runs over the irreducible Artin representations of the Galois group of the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$. As an application, if k ≥ 4, we will show that there are only finitely many χ such that L(f, ρχ,k/2)=0, generalizing a result of David Rohrlich. Also, we conditionally establish a congruence predicted by non-commutative Iwasawa theory and give numerical evidence for it.

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Burungale, Ashay A. "ON THE NON-TRIVIALITY OF THE -ADIC ABEL–JACOBI IMAGE OF GENERALISED HEEGNER CYCLES MODULO , II: SHIMURA CURVES." Journal of the Institute of Mathematics of Jussieu 16, no. 1 (May 7, 2015): 189–222. http://dx.doi.org/10.1017/s147474801500016x.

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Generalised Heegner cycles are associated to a pair of an elliptic newform and a Hecke character over an imaginary quadratic extension $K/\mathbf{Q}$. The cycles live in a middle-dimensional Chow group of a Kuga–Sato variety arising from an indefinite Shimura curve over the rationals and a self-product of a CM abelian surface. Let $p$ be an odd prime split in $K/\mathbf{Q}$. We prove the non-triviality of the $p$-adic Abel–Jacobi image of generalised Heegner cycles modulo $p$ over the $\mathbf{Z}_{p}$-anticyclotomic extension of $K$. The result implies the non-triviality of the generalised Heegner cycles in the top graded piece of the coniveau filtration on the Chow group, and proves a higher weight analogue of Mazur’s conjecture. In the case of weight 2, the result provides a refinement of the results of Cornut–Vatsal and Aflalo–Nekovář on the non-triviality of Heegner points over the $\mathbf{Z}_{p}$-anticyclotomic extension of $K$.

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Connes, Alain, and Caterina Consani. "Schemes over 𝔽1 and zeta functions." Compositio Mathematica 146, no. 6 (April 21, 2010): 1383–415. http://dx.doi.org/10.1112/s0010437x09004692.

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AbstractWe determine the real counting function N(q) (q∈[1,∞)) for the hypothetical ‘curve’ $C=\overline {\mathrm {Spec}\,\Z }$ over 𝔽1, whose corresponding zeta function is the complete Riemann zeta function. We show that such a counting function exists as a distribution, is positive on (1,∞) and takes the value −∞ at q=1 as expected from the infinite genus of C. Then, we develop a theory of functorial 𝔽1-schemes which reconciles the previous attempts by Soulé and Deitmar. Our construction fits with the geometry of monoids of Kato, is no longer limited to toric varieties and it covers the case of schemes associated with Chevalley groups. Finally we show, using the monoid of adèle classes over an arbitrary global field, how to apply our functorial theory of $\Mo $-schemes to interpret conceptually the spectral realization of zeros of L-functions.

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Altan, Metin, Taichi Kato, Ryoko Ishioka, Linda Schmidtobreick, Tolga Güver, Makoto Uemura, Tansel Ak, et al. "Superhump period of SDSS J214354.59+124457.8: First Z Cam star with superhumps in the standstill." Monthly Notices of the Royal Astronomical Society 489, no. 1 (August 14, 2019): 1451–62. http://dx.doi.org/10.1093/mnras/stz2247.

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Abstract The cataclysmic variable SDSS J214354.59+124457.8 (hereafter SDSS J214354) was observed photometrically on sixty one nights between 2012 July 28 and 2019 May 26. The long term variation of this object shows changes between two phases; a dwarf nova type and a novalike. This implies that the object belongs to the group of Z Cam type stars. The timing analysis of the light curve reveals a periodic signal at 0.13902(5) d, which we identify as the superhump period. However, the fractional superhump excess of 10 per cent longer than the orbital period is exceptionally large. We obtained a mass ratio of ∼0.4, which is above the accepted upper limit of q = 0.33 for the formation of superhumps. We suggest that the object contains a secondary with an evolved core. With an orbital period of 0.126 d, SDSS J214354 is situated at the upper border of the period gap. The long term light curve of SDSS J214354 is similar to those of Z Cam type stars which are characterized by recurring standstills, followed by short intervals with DN type outbursts. Therefore, we conclude that SDSS J214354 is a new member of the Z Cam type stars.

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Abdulaev, R. "The Morse Formula for Curves Which are not Locally Simple." Georgian Mathematical Journal 7, no. 4 (December 2000): 599–608. http://dx.doi.org/10.1515/gmj.2000.599.

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Abstract Let be an interior mapping of the unit disk, continuous in D2 and such that the restriction of f to the unit circle S 1 is a locally simple curve γ. Suppose that f(a) ≠ a on S 1 and denote by n(a) the number of solutions of the equation f(z) = a in D2 , by μ(f) the sum of multiplicities of the critical points of f in , by q(a) the angular order of γ with respect to a, and by τ(γ) the angular order of γ. It is proved that the Morse formula 2n(a) – μ(f) – 2q(a) + τ(γ) – 1 = 0 remains correct for a piecewise smooth curve which is not locally simple.

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Çiperiani, Mirela. "Tate–Shafarevich groups in anticyclotomic ℤp-extensions at supersingular primes." Compositio Mathematica 145, no. 2 (February 19, 2009): 293–308. http://dx.doi.org/10.1112/s0010437x08003874.

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AbstractLet E/ℚ be an elliptic curve and p a prime of supersingular reduction for E. Denote by $\mathrm {K}_\infty $ the anticyclotomic ℤp-extension of an imaginary quadratic field K which satisfies the Heegner hypothesis. Assuming that p splits in K/ℚ, we prove that ${\mbox {\textcyr {Sh}}} (\mathrm {K}_\infty , \mathrm {E})_{p^\infty }$ has trivial Λ-corank and, in the process, also show that $\mathrm {H^1_{Sel}}(\mathrm {K}_\infty , \mathrm {E}_{p^\infty })$ and $\mathrm {E}(\mathrm {K}_\infty )\otimes \mathbb {Q}_p/\mathbb {Z}_p$ both have Λ-corank two.

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de Campos, Marcos Flavio. "Determination of the Constants of Magnetocrystalline Anisotropy in Sintered Magnets with Uniaxial Texture." Materials Science Forum 498-499 (November 2005): 134–40. http://dx.doi.org/10.4028/www.scientific.net/msf.498-499.134.

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The usual process for producing the high energy magnets based on rare-earth-transition metals as for example NdFeB, SmCo5 or Sm(CoFeCuZr)z involves powder metallurgy. In many cases, it is necessary the determination of anisotropy constants (K1 – first order and K2 second order) from polycrystalline samples. This is not the ideal situation because for more accurate determinations a single crystal is necessary. Nevertheless, in many cases it is very difficult, or not possible, obtaining a single crystal. Then, for these situations, the anisotropy constants can be evaluated from polycrystalline samples with uniaxial texture. In this study, the methodology for making such determination is described. It includes the measurement of Schulz Pole figure by X-Ray diffraction in a surface perpendicular to the c-axis, the axis of easy magnetization. The measured Pole figure can be adjusted with a Gaussian distribution f(q)=exp(-q2/2s2) or with a distribution of type f(q) = cosn q. A model to evaluate the remanence from quantitative metallography is also described. From these distributions, and using the microstructural model, it is possible to estimate the initial magnetization curves for polycrystalline samples, including the effect of the 2nd order anisotropy constant (K2) which produces a curvature in initial magnetization curve. With all these data it is finally possible to estimate the initial magnetization curves for single crystal samples (theoretical), and the anisotropy constants K1 and K2. The inadequacy of Sucksmith-Thompson plots for determination of anisotropy constants from polycrystalline samples is also commented. The described method can be used either for rare-earth transition magnets or for Barium or Strontium ferrites.

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Kurihara, Masato, and Rei Otsuki. "On the Growth of Selmer Groups of an Elliptic Curve with Supersingular Reduction in the Z₂-extension of Q." Pure and Applied Mathematics Quarterly 2, no. 2 (2006): 557–68. http://dx.doi.org/10.4310/pamq.2006.v2.n2.a8.

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Vollmer, B., P. Gratier, J. Braine, and C. Bot. "Predicting HCN, HCO+, multi-transition CO, and dust emission of star-forming galaxies." Astronomy & Astrophysics 602 (June 2017): A51. http://dx.doi.org/10.1051/0004-6361/201629641.

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High-z star-forming galaxies have significantly higher gas fractions and star-formation efficiencies per molecular gas mass than local star-forming galaxies. In this work, we take a closer look at the gas content or fraction and the associated star-formation rate in main sequence and starburst galaxies at z = 0 and z ~ 1–2 by applying an analytical model of galactic clumpy gas disks to samples of local spiral galaxies, ULIRGs, submillimeter (smm), and high-z star-forming galaxies. The model simultaneously calculates the total gas mass, Hi/H2 mass, the gas velocity dispersion, IR luminosity, IR spectral energy distribution, CO spectral line energy distribution (SLED), HCN(1–0) and HCO+(1–0) emission of a galaxy given its size, integrated star formation rate, stellar mass radial profile, rotation curve, and Toomre Q parameter. The model reproduces the observed CO luminosities and SLEDs of all sample galaxies within the model uncertainties (~0.3 dex). Whereas the CO emission is robust against the variation of model parameters, the HCN and HCO+ emissions are sensitive to the chemistry of the interstellar medium. The CO and HCN mass-to-light conversion factors, including CO-dark H2, are given and compared to the values found in the literature. All model conversion factors have uncertainties of a factor of two. Both the HCN and HCO+ emissions trace the dense molecular gas to a factor of approximately two for the local spiral galaxies, ULIRGs and smm-galaxies. Approximately 80% of the molecular line emission of compact starburst galaxies originates in non-self-gravitating gas clouds. The effect of HCN infrared pumping is small but measurable (10–20%). The gas velocity dispersion varies significantly with the Toomre Q parameter. The Q = 1.5 model yields high-velocity dispersions (vdisp ≫ 10 km s-1) consistent with available observations of high-z star-forming galaxies and ULIRGs. However, we note that these high-velocity dispersions are not mandatory for starburst galaxies. The integrated Kennicutt-Schmidt law has a slope of approximately 1 for the local spirals, ULIRGs, and smm-galaxies, whereas the slope is 1.7 for high-z star-forming galaxies. The model shows Kennicutt-Schmidt laws with respect to the molecular gas surface density with slopes of approximately 1.5 for local spiral galaxies, high-z star-forming galaxies. The relation steepens for compact starburst galaxies. The model star-formation rate per unit area is, as observed, proportional to the molecular gas surface density divided by the dynamical timescale. Our relatively simple analytic model together with the recipes for the molecular line emission appears to capture the essential physics of galactic clumpy gas disks.

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29

Catanese, F. "Cayley Forms and Self-Dual Varieties." Proceedings of the Edinburgh Mathematical Society 57, no. 1 (December 19, 2013): 89–109. http://dx.doi.org/10.1017/s0013091513000928.

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AbstractGeneralized Chow forms were introduced by Cayley for the case of 3-space; their zero set on the Grassmannian G(1, 3) is either the set Z of lines touching a given space curve (the case of an ‘honest’ Cayley form), or the set of lines tangent to a surface. Cayley gave some equations for F to be a generalized Cayley form, which should hold modulo the ideal generated by F and by the quadratic equation Q for G(1, 3). Our main result is that F is a Cayley form if and only if Z = G(1, 3) ∩ {F = 0} is equal to its dual variety. We also show that the variety of generalized Cayley forms is defined by quadratic equations, since there is a unique representative F0 + QF1 of F, with F0, F1 harmonic, such that the harmonic projection of the Cayley equation is identically 0. We also give new equations for honest Cayley forms, but show, with some calculations, that the variety of honest Cayley forms does not seem to be defined by quadratic and cubic equations.

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Durán Díaz, Raúl, Luis Hernández Encinas, and Jaime Muñoz Masqué. "A Group Law on the Projective Plane with Applications in Public Key Cryptography." Mathematics 8, no. 5 (May 7, 2020): 734. http://dx.doi.org/10.3390/math8050734.

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In the context of new threats to Public Key Cryptography arising from a growing computational power both in classic and in quantum worlds, we present a new group law defined on a subset of the projective plane F P 2 over an arbitrary field F , which lends itself to applications in Public Key Cryptography and turns out to be more efficient in terms of computational resources. In particular, we give explicitly the number of base field operations needed to perform the mentioned group law. Based on it, we present a Diffie-Hellman-like key agreement protocol. We analyze the computational difficulty of solving the mathematical problem underlying the proposed Abelian group law and we prove that the security of our proposal is equivalent to the discrete logarithm problem in the multiplicative group of the cubic extension of the finite field considered. We present an experimental setup in order to show real computation times along a comparison with the group operation in the group of points of an elliptic curve. Based on current state-of-the-art algorithms, we provide parameter ranges suitable for real world applications. Finally, we present a promising variant of the proposed group law, by moving from the base field F to the ring Z / p q Z , and we explain how the security becomes enhanced, though at the cost of a longer key length.

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31

Zhang, C., C. Gao, X. Di, S. Cui, W. Liang, W. Sun, M. Yao, Q. Wang, and Z. Zheng. "THU0243 HSA_CIRC_0123190 FUNCTIONS AS A COMPETITIVE ENDOGENOUS RNA TO REGULATE APLNR EXPRESSION BY SPONGING HSA-MIR-483-3P IN LUPUS NEPHRITIS." Annals of the Rheumatic Diseases 79, Suppl 1 (June 2020): 349.2–349. http://dx.doi.org/10.1136/annrheumdis-2020-eular.4025.

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Background:Lupus nephritis (LN) is one of the most severe complications of systemic lupus erythematosus (SLE). Circular RNAs(circRNAs) can act as competitive endogenous RNAs (ceRNAs) to regulate gene transcription, which is involved in mechanism of many diseases, such as, autoimmunity diseases. However, the role of circRNA in lupus nephritis has been rarely reported.Objectives:In this study, we aim to investigate the clinical value of circRNAs and explore the mechanism of circRNA involvement in the pathogenesis of LN.Methods:Renal tissues from three untreated LN patients and three normal controls (NCs) were used to identify differently expressed circRNAs by RNA sequencing (RNA-seq). Validated assays were used by quantitative reverse transcription polymerase chain reaction (qRT-PCR). Correlation analysis and receiver operating characteristic (ROC) curve were used to reveal the clinical value of selected circRNA, miRNA and mRNA. The interactions between circRNA and miRNA, or miRNA and mRNA were further determined by luciferase reporter assay. The degrees of renal fibrosis between the two groups were compared by Masson-trichome staining and immunohistochemistry staining.Results:159 circRNAs were significantly dysregulated in LN patients compared with NC group. The expression of hsa_circ_0123190 was significantly decreased in renal tissues of patients with LN (p=0.014), as same as the sequencing results. The area under the ROC curve of hsa_circ_0123190 in renal tissues was 0.820. Bio-informatic analysis and luciferase reporter assay illustrated that hsa_circ_0123190 can act as a sponge for hsa-miR-483-3p which was also validated to interact with APLNR mRNA. APLNR mRNA expression was positively related with chronicity index (CI) of LN (R2=0.452,p=0.033). Finally, the factors of renal fibrosis, especially TGF-β (p=0.018), were more pronounced in the LN group.Conclusion:Hsa_circ_0123190 could function as a ceRNA to regulate APLNR expression involved in renal fibrosis by sponging hsa-miR-483-3p in LNReferences:[1]Aljaberi N, Bennett M, Brunner HI, Devarajan P. Proteomic profiling of urine: implications for lupus nephritis. Expert review of proteomics. 2019;16(4):303-13.[2]Zheng ZH, Zhang LJ, Liu WX, Lei YS, Xing GL, Zhang JJ, et al. Predictors of survival in Chinese patients with lupus nephritis. Lupus. 2012;21(10):1049-56.[3]Chen LL. The biogenesis and emerging roles of circular RNAs. Nature reviews Molecular cell biology. 2016;17(4):205-11.[4]Mahmoudi E, Cairns MJ. Circular RNAs are temporospatially regulated throughout development and ageing in the rat. Scientific reports. 2019;9(1):2564.[5]Liang D, Wilusz JE. Short intronic repeat sequences facilitate circular RNA production. Genes & development. 2014;28(20):2233-47.[6]Tan WL, Lim BT, Anene-Nzelu CG, Ackers-Johnson M, Dashi A, See K, et al. A landscape of circular RNA expression in the human heart. Cardiovascular research. 2017;113(3):298-309.[7]Zhao Z, Li X, Jian D, Hao P, Rao L, Li M. Hsa_circ_0054633 in peripheral blood can be used as a diagnostic biomarker of pre-diabetes and type 2 diabetes mellitus. Acta diabetologica. 2017;54(3):237-45.[8]Ouyang Q, Huang Q, Jiang Z, Zhao J, Shi GP, Yang M. Using plasma circRNA_002453 as a novel biomarker in the diagnosis of lupus nephritis. Molecular immunology. 2018;101(undefined):531-8.[9]Luan J, Jiao C, Kong W, Fu J, Qu W, Chen Y, et al. CircHLA-C Plays an Important Role in Lupus Nephritis by Sponging miR-150. Molecular therapy Nucleic acids. 2018;10(undefined):245-53.[10]Kuschnerus K, Straessler ET, Müller MF, Lüscher TF, Landmesser U, Kränkel N. Increased Expression of miR-483-3p Impairs the Vascular Response to Injury in Type 2 Diabetes. Diabetes. 2019;68(2):349-60.[11]Huang Z, Wu L and Chen L. Apelin/APJ system: A novel potential therapy target for kidney disease. Journal of cellular physiology. 2018;233(5): 3892-900.Disclosure of Interests:None declared

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32

Psaroudakis, E. G., G. E. Mylonakis, and N. S. Klimis. "Non-Linear Analysis of Axially Loaded Piles Using “t–z” and “q–z” Curves." Geotechnical and Geological Engineering 37, no. 4 (February 4, 2019): 2293–302. http://dx.doi.org/10.1007/s10706-019-00823-2.

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33

Garzón, Álvaro. "p-cycles, S2-sets and Curves with Many Points." Revista de Ciencias 21, no. 1 (April 4, 2018): 55. http://dx.doi.org/10.25100/rc.v21i1.6340.

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We construct S 2 -sets contained in the integer interval I q − 1 := [1, q − 1] with q = p^n,p a prime number and n ∈ Z +, by using the p-adic expansion of integers. Such sets comefrom considering p-cycles of length n. We give some criteria in particular cases whichallow us to glue them to obtain good S 2 -sets. After that we construct algebraic curvesover the ﬁnite ﬁeld F q with many rational points via minimal (F p , F p )-polynomials whose exponent is an S 2 -set.

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34

Nguyen Thi Thu, Hang, Son Nguyen Thanh, and Truong Vu Van. "A SECOND MAIN THEOREM FOR ENTIRE CURVES IN A PROJECTIVE VARIETY WHOSE DERIVATIVES VANISH ON INVERSE IMAGE OF HYPERSURFACE TARGETS." Journal of Science Natural Science 65, no. 6 (June 2020): 31–40. http://dx.doi.org/10.18173/2354-1059.2020-0026.

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We establish a second main theorem for algebraically nondegenerate entire curves f in a projective variety V ⊂ P n(C) and a hypersurface target {D1, D2, . . . , Dq} satisfying f∗,z = 0 for all z ∈ ∪q j=1f−1(Dj).

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35

Banerjee, Pradipto, and Ranjan Bera. "Classifying Galois groups of an orthogonal family of quartic polynomials." Notes on Number Theory and Discrete Mathematics 27, no. 2 (June 2021): 172–90. http://dx.doi.org/10.7546/nntdm.2021.27.2.172-190.

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We consider the quartic generalized Laguerre polynomials $L_{4}^{(\alpha)}(x)$ for $\alpha \in \mathbb Q$. It is shown that except $\mathbb Z/4\mathbb Z$, every transitive subgroup of $S_{4}$ appears as the Galois group of $L_{4}^{(\alpha)}(x)$ for infinitely many $\alpha \in \mathbb Q$. A precise characterization of $\alpha\in \mathbb Q$ is obtained for each of these occurrences. Our methods involve the standard use of resolvent cubics and the theory of p-adic Newton polygons. Using these, the Galois group computations are reduced to Diophantine problem of finding integer and rational points on certain curves.

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Amerik, Ekaterina, and Misha Verbitsky. "Collections of Orbits of Hyperplane Type in Homogeneous Spaces, Homogeneous Dynamics, and Hyperkähler Geometry." International Mathematics Research Notices 2020, no. 1 (February 8, 2018): 25–38. http://dx.doi.org/10.1093/imrn/rnx319.

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Abstract Consider the space M = O(p, q)/O(p) × O(q) of positive p-dimensional subspaces in a pseudo-Euclidean space V of signature (p, q), where p > 0, q > 1 and $(p,q)\neq (1,2)$, with integral structure: $V = V_{\mathbb{Z}} \otimes \mathbb{Z}$. Let Γ be an arithmetic subgroup in $G = O(V_{\mathbb{Z}})$, and $R \subset V_{\mathbb{Z}}$ a Γ-invariant set of vectors with negative square. Denote by R⊥ the set of all positive p-planes W ⊂ V such that the orthogonal complement W⊥ contains some r ∈ R. We prove that either R⊥ is dense in M or Γ acts on R with finitely many orbits. This is used to prove that the squares of primitive classes giving the rational boundary of the Kähler cone (i.e., the classes of “negative” minimal rational curves) on a hyperkähler manifold X are bounded by a number which depends only on the deformation class of X. We also state and prove the density of orbits in a more general situation when M is the space of maximal compact subgroups in a simple real Lie group.

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37

Dujella, Andrej, and Mirela Jukić Bokun. "On the rank of elliptic curves over $\mathbf{Q}(i)$ with torsion group $\mathbf{Z}/4\mathbf{Z} \times \mathbf{Z}/4\mathbf{Z}$." Proceedings of the Japan Academy, Series A, Mathematical Sciences 86, no. 6 (June 2010): 93–96. http://dx.doi.org/10.3792/pjaa.86.93.

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38

Pal, Janos, and Dana Schlomiuk. "Summing up the Dynamics of Quadratic Hamiltonian Systems With a Center." Canadian Journal of Mathematics 49, no. 3 (June 1, 1997): 582–99. http://dx.doi.org/10.4153/cjm-1997-027-0.

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AbstractIn this work we study the global geometry of planar quadratic Hamiltonian systems with a center and we sum up the dynamics of these systems in geometrical terms. For this we use the algebro-geometric concept of multiplicity of intersection Ip(P,Q) of two complex projective curves P(x, y, z) = 0, Q(x,y,z) = 0 at a point p of the plane. This is a convenient concept when studying polynomial systems and it could be applied for the analysis of other classes of nonlinear systems.

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Dieulefait, Luis V. "Modular congruences, Q-curves, and the diophantine equation $x^4 + y^4 = z^p $." Bulletin of the Belgian Mathematical Society - Simon Stevin 12, no. 3 (September 2005): 363–69. http://dx.doi.org/10.36045/bbms/1126195341.

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Mikoliūnas, Audrius, and Rimantas Kačianauskas. "STIFFNESS CHARACTERISTICS OF GEOMETRICALLY NON-LINEAR BEAM FINITE ELEMENT/GEOMETRIŠKAI NETIESINIO LENKIAMO STRYPO BAIGTINIO ELEMENTO STANDUMO RODIKLIŲ NUSTATYMAS." JOURNAL OF CIVIL ENGINEERING AND MANAGEMENT 3, no. 10 (June 30, 1997): 52–59. http://dx.doi.org/10.3846/13921525.1997.10531684.

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Two-dimensional geometrically non-linear beam element is considered in this paper. The explicit expressions of stiffness characteristics of element with three nodes are derived and tested. Among models of the geometrically non-linear beams, the elements with 2 nodes dominate [1–8]. Such elements produce constant axial force. The idea of more complex elements with tree nodes was suggested in [3]. In this paper geometrically non-linear flat bending beam element with 3 nodes for evaluating of axial force is investigated and nonlinear stiffness characteristics are derived. Basic relations of element e are derived using virtual displacement method. On the level of element e, the principle of virtual displacements is expressed by equalities (1–3). Using displacement approach, displacement functions are prescribed in the bounds of one finite element. Generalised deformations are obtained by introducing displacements approximation (4) and inserting them into non-linear geometric equations (5–6). Variation of deformation energy (3) is expressed in (7). Putting equality (7) into (1), it is possible to write equality of virtual works in terms of non-linear algebraic equations (8). Non-linear stiffness matrix is presented as the sum of 3 matrices (9). The first matrix [K 0e ] (linear matrix) is the matrix of small deflections, which is independent on deformed shape. The second matrix [K Ne ] is the matrix of large deflections. The third matrix [K Ge ] is a geometrical stiffness matrix. It reflects the second member of equality (7). Expressions of geometrically non-linear stiffness matrices are greatly dependent on the introduced assumptions and appropriate elements. Shallow beam finite element is shown in Fig 1. This finite element has 3 nodes. In the initial configuration a beam can be straight (Fig 1a), or curved (Fig 1b). The initial configuration of a beam is described by a vector z = {z1 αx1, Z2, αx2}T of a beam final nodal co-ordinates, where z i means nodes co-ordinates, αxi—initial rotations (Fig lb). However, the initial configuration is a relative statement, and is generally described by vector z. If in initial configuration the element is straight, vector z=0. Physical properties of the element are denoted with capital EA (tensional rigidity) and EI (flexural rigidity). The finite element has 7 degrees of freedom: 3 of them are defined at each end of the element (2 linear and 1 rotation) and 1 in the middle of the element Vector Ue of nodal deflections for this element is split into two parts: Ue= {u, w}T , u = {u1, u2,u3}T, w'=z+w, w={w1,Θx1, w2, Θx2}T. Deflection u3 shows the deflection of the middle beam node, which is not proportional to the final nodal deflections. To be more strict, u3 is straightened by linear law. So the linear element in the direction of longitudinal deformation expression is (11). The deflection of a point which is moved from the centre of plane surface in distance z1, deflection u (in direction x) is expressed in (12). Deformation is expressed by summarised deformations (13). So the deformed element only longitudinal deformation Δ is assigned, which is shown in (16). Evaluating earlier received expressions, it is possible to make equality of virtual work (1), where generalised vectors Θ e (x)={Δx, κ}T and Q e (x)={N, M}T. Generalised deformations Δ and κ expressed by deflections approximating expressions (4). For convenience, vertical and horizontal deflections are separated (17). By analogy with deflections, vector z and its derivatives are approximated by (24–25). Beam's curvature (15) is also expressed by nodal deflections: (26–27). Putting (20, 21, 26) into (16) and (15, 16) into (13) and expression (28) is got. Evaluating that the element work in elastic stage, expression (10) can be rewritten (29). The final stiffness matrix expression (9) is given in the form of block matrices (30). Expressions of block matrices are presented by (31–33). Having completed operations in expressions (31–33), final stiffness matrix is (34–36). After integrating, linear matrix is (37). Analogous operations are performed with matrices (35)–(38). Elements of this matrix are calculated using computer algebra. Matrix (36) consists of three parts. The first integral (39) is stiffness matrix of bending beam. If we assume that axial force in beam's length is invariable, the third integral is equal to (41). Assuming that axial force in the length of element is not constant (the axial force is calculated according to forms (28,29)), the expressions of geometrical stiffness matrix become very complicated. Analysis of geometrically non-linear system of finite elements is described by algebraic equation (42). Usually expression of non-linear deformation is investigated as a process varying in time t, where outer load F and deflections U are functions of time: F≡6F(t) and U≡6U(t). Load in the moment of time t i+1 = t i + Δt is added in portions (43). Deflections are expressed by analogy with (44). Non-linear model (42) is expressed by increments (45). Vector of residuals γ reflects solution of equation (42) inadequacy of state variables. Nowadays there exists many algorithms of different complexity for solution of non-linear problems [2–4,9,10]. The majority of methods that have already become standard uses different Newton-type variety of algorithms. Classical Newton-type algorithms are adapted to non-linear process with so called “softening” curve to model (Fig 2a). In the work there was done and realised a combined algorithm for non-linear process with “hardening” or “softening” curve to model. The illustration of algorithm is given in Fig 3. Using the algorithm in every load step, tangent stiffness matrix is counted twice. The first matrix corresponds to tangent of load step at the beginning (tangent 1), and the second one to the step at the end (tangent 2). Algorithm is implemented in the program created by the authors. A simple cantilever beam (Fig 4) is taken for the test. History of deformation was investigated. The results are given in non-dimensional quantities (Fig 5). Euler's method is realised as a particular case of implemented algorithm. The same example was also solved using program ANSYS, where beam elements are used and described only by two nodes. The results presented show obviously the advantages of three-node element and validity of proposed assumptions.

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Blatt, Simon, and Philipp Reiter. "Regularity theory for tangent-point energies: The non-degenerate sub-critical case." Advances in Calculus of Variations 8, no. 2 (April 1, 2015): 93–116. http://dx.doi.org/10.1515/acv-2013-0020.

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AbstractIn this article we introduce and investigate a new two-parameter family of knot energies ${\operatorname{TP}^{(p,\,q)}}$ that contains the tangent-point energies. These energies are obtained by decoupling the exponents in the numerator and denominator of the integrand in the original definition of the tangent-point energies. We will first characterize the curves of finite energy ${\operatorname{TP}^{(p,\,q)}}$ in the sub-critical range p ∈ (q+2,2q+1) and see that those are all injective and regular curves in the Sobolev–Slobodeckiĭ space ${W^{\scriptstyle (p-1)/q,q}(\mathbb {R}/\mathbb {Z},\mathbb {R}^n)}$. We derive a formula for the first variation that turns out to be a non-degenerate elliptic operator for the special case q = 2: a fact that seems not to be the case for the original tangent-point energies. This observation allows us to prove that stationary points of $\operatorname{TP}^{(p,2)}$ + λ length, p ∈ (4,5), λ > 0, are smooth – so especially all local minimizers are smooth.

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42

MANTUROV, VASSILY O. "VASSILIEV INVARIANTS FOR VIRTUAL LINKS, CURVES ON SURFACES AND THE JONES–KAUFFMAN POLYNOMIAL." Journal of Knot Theory and Its Ramifications 14, no. 02 (March 2005): 231–42. http://dx.doi.org/10.1142/s0218216505003804.

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We discuss the strong invariant of virtual links proposed in [23]. This invariant is obtained as a generalization of the Jones–Kauffman polynomial (generalized Kauffman's bracket) by adding to the sum some equivalence classes of curves in two-dimensional surfaces. Thus, the invariant is valued in the infinite-dimensional free module over Z[q,q-1]. We prove that this invariant can be decomposed into finite type Vassiliev invariant of virtual links (in Kauffman's sense); thus we present new infinite series of Vassiliev invariants. It is also proved that this invariant is strictly stronger than the Jones–Kauffman polynomial for virtual knots proposed by Kauffman. Some examples when the invariant can recognize virtual knots that can not be recognized by other invariants are given.

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43

Jukić Bokun, Mirela. "On the rank of elliptic curves over $\mathbf{Q}(\sqrt{-3})$ with torsion groups $\mathbf{Z}/3\mathbf{Z} \times \mathbf{Z}/3\mathbf{Z}$ and $\mathbf{Z}/3\mathbf{Z} \times \mathbf{Z}/6\mathbf{Z}$." Proceedings of the Japan Academy, Series A, Mathematical Sciences 87, no. 5 (May 2011): 61–64. http://dx.doi.org/10.3792/pjaa.87.61.

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44

Lobo Prat, D., I. Castellví, D. Castillo, S. Orozco, A. Mariscal, L. Martínez-Martínez, A. M. Millán Arciniegas, et al. "AB0666 PROGNOSTIC VALUE OF SERUM KREBS VON DEN LUNGEN-6 GLYCOPROTEIN CIRCULATING LEVELS IN COVID-19 PNEUMONIA: A PROSPECTIVE COHORT STUDY." Annals of the Rheumatic Diseases 80, Suppl 1 (May 19, 2021): 1365.1–1365. http://dx.doi.org/10.1136/annrheumdis-2021-eular.1359.

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Background:Currently, there are no biomarkers to predict respiratory worsening in patients with Coronavirus infectious disease, 2019 (COVID- 19) pneumonia.Objectives:We aimed to determine the prognostic value of Krebs von de Lungen-6 circulating serum levels (sKL-6) predicting COVID- 19 evolving trends.Methods:We prospectively analyzed the clinical and laboratory characteristics of 375 COVID- 19 patients with mild lung disease on admission. sKL-6 was obtained in all patients at baseline and compared among patients with respiratory worsening.Results:45.1% of patients developed respiratory worsening during hospitalization. Baseline sKL-6 levels were higher in patients who had respiratory worsening (median [IQR] 303 [209-449] vs. 285.5 [15.8-5724], P=0.068). The best sKL-6 cut-off point was 408 U/mL (area under the curve 0.55; 33% sensitivity, 79% specificity). Independent predictors of respiratory worsening were sKL-6 serum levels, age >51 years, time hospitalized, and dyspnea on admission. Patients with baseline sKL-6 ≥ 408 U/mL had a 39% higher risk of developing respiratory aggravation seven days after admission. In patients with serial determinations, sKL-6 was also higher in those who subsequently worsened (median [IQR] 330 [219-460] vs 290.5 [193-396]; p<0.02).Conclusion:sKL-6 has a low sensibility to predict respiratory worsening in patients with mild COVID-19 pneumonia. Baseline sKL-6 ≥ 408 U/mL is associated to a higher risk of respiratory worsening. sKL-6 levels are not useful as a screening tool to stratify patients on admission but further research is needed to investigate if serial determinations of sKL-6 may be of prognostic use.References:[1]Zhou F, Yu T, Du R, Fan G, Liu Y, Liu Z, et al. Clinical course and risk factors for mortality of adult inpatients with COVID-19 in Wuhan, China: a retrospective cohort study. Lancet. 2020;395(10229):1054-62. 5.[2]Tian W, Jiang W, Yao J, Nicholson CJ, Li RH, Sigurslid HH, et al. Predictors of mortality in hospitalized COVID-19 patients: A systematic review and meta-analysis. J Med Virol. 2020.[3]Wang D, Li R, Wang J, Jiang Q, Gao C, Yang J, et al. Correlation analysis between disease severity and clinical and biochemical characteristics of 143 cases of COVID-19 in Wuhan, China: a descriptive study. BMC Infect Dis. 2020;20(1):519.Disclosure of Interests:None declared.

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MORTON, HUGH R. "THE ALEXANDER POLYNOMIAL OF A TORUS KNOT WITH TWISTS." Journal of Knot Theory and Its Ramifications 15, no. 08 (October 2006): 1037–47. http://dx.doi.org/10.1142/s0218216506004920.

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This note gives an explicit calculation of the doubly infinite sequence Δ(p, q, 2m), m ∈ Z of Alexander polynomials of the (p, q) torus knot with m extra full twists on two adjacent strings, where p and q are both positive. The knots can be presented as the closure of the p-string braids [Formula: see text], where δp = σp-1σp-2 · σ2σ1, or equally of the q-string braids [Formula: see text]. As an application we give conditions on (p, q) which ensure that all the polynomials Δ(p, q, 2m) with |m| ≥ 2 have at least one coefficient a with |a| > 1. A theorem of Ozsvath and Szabo then ensures that no lens space can arise by Dehn surgery on any of these knots. The calculations depend on finding a formula for the multivariable Alexander polynomial of the 3-component link consisting of the torus knot with twists and the two core curves of the complementary solid tori.

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46

TORZEWSKI, ALEX. "Regulator constants of integral representations of finite groups." Mathematical Proceedings of the Cambridge Philosophical Society 168, no. 1 (September 5, 2018): 75–117. http://dx.doi.org/10.1017/s0305004118000579.

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AbstractLet G be a finite group and p be a prime. We investigate isomorphism invariants of $\mathbb{Z}_p$[G]-lattices whose extension of scalars to $\mathbb{Q}_p$ is self-dual, called regulator constants. These were originally introduced by Dokchitser–Dokchitser in the context of elliptic curves. Regulator constants canonically yield a pairing between the space of Brauer relations for G and the subspace of the representation ring for which regulator constants are defined. For all G, we show that this pairing is never identically zero. For formal reasons, this pairing will, in general, have non-trivial kernel. But, if G has cyclic Sylow p-subgroups and we restrict to considering permutation lattices, then we show that the pairing is non-degenerate modulo the formal kernel. Using this we can show that, for certain groups, including dihedral groups of order 2p for p odd, the isomorphism class of any $\mathbb{Z}_p$[G]-lattice whose extension of scalars to $\mathbb{Q}_p$ is self-dual, is determined by its regulator constants, its extension of scalars to $\mathbb{Q}_p$, and a cohomological invariant of Yakovlev.

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Shapiro, Paul R., Ira Wasserman, and Mark L. Giroux. "Cosmological HII Regions." Symposium - International Astronomical Union 117 (1987): 366. http://dx.doi.org/10.1017/s0074180900150533.

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We have generalized the classical description of ionization front propagation to the case of a point source in a uniform, cosmologically expanding gas. We present illustrative curves for the comoving radius and peculiar velocity for several turn-on redshifts, z0N, for Ωtot = 1, Ωb = 0.1, h = 1. The quantity RS is the generalized Strömgren radius [RS = RSi (1 + z0N)/(1 + z), RSi = (3Nu/4πnH,i2 α2)1/3, Nu = photoionizing number flux per source, α2 = recombination rate to n = 2, nH,i = nHo (1 + z0N)3]. The quantity T0N = 2 (1 + z0N)−3/2/(3H0). We also plot ζ, the value of (2nQoNph, Q/3HonHo) needed to ionize the IGM with overlapping QSO HII regions by redshift z0V for QSO turn-on at various z0N, where Nph, Q = ionizing photon luminosity per QSO, nQo = QSO number density (present co-moving value), nH = H density of IGM, and nH/nHo = nQ/nQo = (1 + z)3. From a recent preprint by Koo (1985), we estimate ζ ≲ 1 (for Ωb = 0.1, h = 1) for QSO's with L ∼ 1045 erg s−1. In this case, the observed QSO's cannot be the sole source of the IGM ionization that is implied by the null detection of the Gunn-Peterson effect for QSO's with z > 2.

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48

Yuan, Ge Cheng, An Chen Yang, Zhen Hua Zhu, and Yong Qi Cheng. "Modeling of Flow Stress and Grain Size Based on Z Parameter and Material Constants of Al-5.2Mg-0.6Mn Alloy during Hot Deformation." Materials Science Forum 817 (April 2015): 748–54. http://dx.doi.org/10.4028/www.scientific.net/msf.817.748.

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The flow stress curves of Al-5.2Mg-0.6Mn alloy were tested by isothermal compression method with Gleeble-1500 thermal simulator at the temperatures of 300, 350, 400, 450 and 500°C with strain rate of 0.001, 0.01, 0.1 and 1s-1, respectively. The morphologies of grains deformed were analyzed by TEM. Four material constants including structural factor (A), stress exponential (n), stress multiplier (α) and average activation energy (Q) of the alloy were calculated by linear regression processing. The models of flow stress (σ) and grain size (d) based on Zener-Hollomon parameter (Z) and the constants were established. The results show that the values of A, n, α, Q of the alloy were equal to 3.058×109s-1, 3.314, 0.0184 mm2N-1, 160.94 kJmol-1 respectively, and the models of flow stress and average sub-grains size can be described as σ=54.31ln {(0.327×10-9Z)0.302 +[(0.327×10-9Z)0.604+1]0.5} and d=(0.045lnZ-0.675)-1, respectively. The flow stress appreciably reduced but average grain size increased with decreasing Z value. The conditions of dynamic recrystallization occurring for the alloy were the temperature above or equal to 350°C and lnZ below or equal to 24.47.

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49

Lee, J. H., D. O. Kim, and K. Lee. "Compressive Deformation Behavior of Thick Micro-Alloyed HSLA Steel Plates at Elevated Temperatures." Archives of Metallurgy and Materials 62, no. 2 (June 1, 2017): 1191–96. http://dx.doi.org/10.1515/amm-2017-0175.

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Abstract The hot deformation behavior of a heavy micro-alloyed high-strength low-alloy (HSLA) steel plate was studied by performing compression tests at elevated temperatures. The hot compression tests were carried out at temperatures from 923 K to 1,223 K with strain rates of 0.002 s−1 and 1.0 s−1. A long plateau region appeared for the 0.002 s−1 strain rate, and this was found to be an effect of the balancing between softening and hardening during deformation. For the 1.0 s−1 strain rate, the flow stress gradually increased after the yield point. The temperature and the strain rate-dependent parameters, such as the strain hardening coefficient (n), strength constant (K), and activation energy (Q), obtained from the flow stress curves were applied to the power law of plastic deformation. The constitutive model for flow stress can be expressed as σ = (39.8 ln (Z) – 716.6) · ε(−0.00955ln(Z) + 0.4930) for the 1.0 s−1 strain rate and σ = (19.9ln (Z) – 592.3) · ε(−0.00212ln(Z) + 0.1540) for the 0.002 s−1 strain rate.

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50

Wang, Hong Ke, Li Wen Zhang, Sen Dong Gu, Qiu Hong Quan, and Wen Fei Shen. "Constitutive Modeling of Dynamic Recrystallization Behavior of GH80A Superalloy." Applied Mechanics and Materials 455 (November 2013): 71–76. http://dx.doi.org/10.4028/www.scientific.net/amm.455.71.

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The dynamic recrystallization (DRX) behavior of GH80A superalloy was investigated by isothermal compression tests on a Gleeble1500 thermomechanical simulator. True stress-strain curves and deformed specimens were obtained at the temperature range of 1273-1473K and the strain rate range of 0.01-5s-1. Experimental results show that the stress-strain curves at low strain rate display a typical DRX characteristic. By regression analysis of experimental results, Materials constant n, activation energy Q and Zener-Hollomon (Z) parameter were determined, and the critical strain model and austenite grain size model for dynamic recrystallization were established as a function of deformation temperature and strain rate. The dynamic recrystallization kinetic model for GH80A was established on the basis of the Avrami equation.

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