DIOPHANTINE APPROXIMATION IN IMAGINARY QUADRATIC FIELDS

2010 ◽  
Vol 06 (04) ◽  
pp. 731-766 ◽  
Author(s):  
L. YA. VULAKH
Keyword(s):  
Half Space ◽  
Hyperbolic Space ◽  
Lower Bounds ◽  
Diophantine Approximation ◽  
Three Dimensional ◽  
Quadratic Fields ◽  
Imaginary Quadratic Fields ◽  
Space Model ◽  
Limit Points ◽  
Group B

Let H3 be the upper half-space model of the three-dimensional hyperbolic space. For certain cocompact Fuchsian subgroups Γ of an extended Bianchi group Bd, the extremality of the axis of hyperbolic F ∈ Γ in H3 with respect to Γ implies its extremality with respect to Bd. This reduction is used to obtain sharp lower bounds for the Hurwitz constants and lower bounds for the highest limit points in the Markov spectra of Bd for some d < 1000. In particular, such bounds are found for all non-Euclidean class one imaginary quadratic fields. The Hurwitz constants for the imaginary quadratic fields with discriminants -120 and -132 are given. The second minima are also indicated for these fields.

DIOPHANTINE APPROXIMATION IN $\mathbf{Q}(\sqrt{-5})$ AND $\mathbf{Q}(\sqrt{-6})$

2006 ◽  
Vol 02 (01) ◽  
pp. 25-48 ◽  
Author(s):  
L. YA. VULAKH
Keyword(s):  
Diophantine Approximation ◽  
Hyperbolic Plane ◽  
Three Dimensional ◽  
Quadratic Fields ◽  
Imaginary Quadratic Fields ◽  
Space Model ◽  
Continuous Parameter ◽  
Markov Spectrum ◽  
Group B ◽  
Discrete Part

The complete description of the discrete part of the Lagrange and Markov spectra of the imaginary quadratic fields with discriminants -20 and -24 are given. Farey polygons associated with the extended Bianchi groups Bd, d = 5, 6, are used to reduce the problem of finding the discrete part of the Markov spectrum for the group Bd to the corresponding problem for one of its maximal Fuchsian subgroup. Hermitian points in the Markov spectrum of Bd are introduced for any d. Let H3 be the upper half-space model of the three-dimensional hyperbolic space. If ν is a hermitian point in the spectrum, then there is a set of extremal geodesics in H3 with diameter 1/ν, which depends on one continuous parameter. This phenomenon does not take place in the hyperbolic plane.

scholarly journals Estimating the critical determinants of a class of three-dimensional star bodies

2017 ◽  
Vol 25 (2) ◽  
pp. 149-157
Author(s):  
Werner Georg Nowak
Keyword(s):  
Lower Bounds ◽  
Diophantine Approximation ◽  
Positive Constant ◽  
Three Dimensional ◽  
Double Cone ◽  
Critical Determinant ◽  
Simultaneous Diophantine Approximation ◽  
Hurwitz’S Theorem ◽  
Star Bodies

Abstract In the problem of (simultaneous) Diophantine approximation in ℝ3 (in the spirit of Hurwitz’s theorem), lower bounds for the critical determinant of the special three-dimensional body K2 : (y2 + z2)(x2 + y2 + z2) ≤ 1 play an important role; see [1], [6]. This article deals with estimates from below for the critical determinant ∆ (Kc) of more general star bodies Kc : (y2 + z2)c/2(x2 + y2 + z2) ≤ 1 ; where c is any positive constant. These are obtained by inscribing into Kc either a double cone, or an ellipsoid, or a double paraboloid, depending on the size of c.

MINIMA OF BINARY INDEFINITE HERMITIAN FORMS

2010 ◽  
Vol 06 (02) ◽  
pp. 411-435 ◽  
Author(s):  
L. YA. VULAKH
Keyword(s):  
Modular Group ◽  
Quadratic Field ◽  
Three Dimensional ◽  
Hermitian Forms ◽  
Space Model ◽  
Continuous Parameter ◽  
Hilbert Modular Group ◽  
Hilbert Modular ◽  
Group B

Classification of binary indefinite primitive Hermitian forms modulo the action of the extended Bianchi group (or Hilbert modular group) Bd is given. When the discriminant of the quadratic field [Formula: see text] (and d) is negative, the results obtained can be applied to classify the maximal non-elementary Fuchsian subgroups of Bd, and to find the Hermitian points in the Markov spectrum of Bd. If ν is a Hermitian point in the spectrum, then there is a set of extremal geodesics in H3, the upper half-space model of the three-dimensional hyperbolic space, with diameter 1/ν, which depends on one continuous parameter.

CONVENTIONAL VERSUS THREE-DIMENSIONAL CONFORMAL EXTERNAL RADIOTHERAPY IN CANCER CERVIX: A COMPARATIVE STUDY FOR COMPLIANCE, RESPONSE AND TOXICITY

2016 ◽  
Vol 1 (2) ◽  
Author(s):  
Richa Gupta ◽  
Piyush Kumar ◽  
D. P. Singh ◽  
Arvind Kumar Chauhan ◽  
Kamal Sahni
Keyword(s):  
Clinical Response ◽  
Three Dimensional ◽  
Common Side Effect ◽  
P Value ◽  
High Dose ◽  
Cancer Cervix ◽  
Conventional Radiotherapy ◽  
Radiation Fields ◽  
Group A ◽  
Group B

INTRODUCTION: Cervical cancer is the second most frequent cancer among Indian women. Radiotherapy is the cornerstone of treatment in all its stages. Three-dimensional conformal radiotherapy (3DCRT) combines multiple radiation fields to deliver precise dose of radiation to the affected area. Tailoring each of the radiation fields to focus on the tumor delivers a high dose of radiation to the tumor and avoids nearby healthy tissue. The present study is done to compare conventional radiotherapy versus 3DCRT in cancer cervix for compliance, clinical response and toxicity. MATERIAL AND METHODS: Fifty patients were enrolled and randomised into two radiotherapy plans with radical intent - Group A treated by conventional radiotherapy and group B treated by 3DCRT. Concurrent cisplatin was delivered on weekly (35mg/m2) or tri-weekly (75mg/m2) basis during external beam Radiotherapy and was followed by High Dose Radiotherapy Brachytherapy. Clinical response and complication assessment were evaluated.Collected data was analyzed using standard statistical methods and softwares to calculate level of significance using “p” value by chi square test. RESULTS: In this study mean age of the patients was 48 years (26-67 years). The anemia was the most common side effect seen in both groups (96% vs 88%, p=0.29). Neutropenia was more in group B (36% vs 44%, p= 0.56). Lower GI toxicity was seen only in patients in group A (20% vs 0%, p=0.018). In follow up there were no significant early rectal and bladder reactions in both groups and 2 patients in each group had late rectal reactions of grade I and II (p= 0.312). No significant skin, bladder and small intestinal toxicity were seen in both groups. CONCLUSION: Conventional radiotherapy gives equally efficacious response though accompanied by toxicities which were acceptable.

scholarly journals Hyperbolic tessellations and generators of for imaginary quadratic fields

2021 ◽  
Vol 9 ◽  
Author(s):  
David Burns ◽  
Rob de Jeu ◽  
Herbert Gangl ◽  
Alexander D. Rahm ◽  
Dan Yasaki
Keyword(s):  
Number Field ◽  
Number Fields ◽  
Quadratic Fields ◽  
Quadratic Number ◽  
Imaginary Quadratic Fields ◽  
Quadratic Number Fields ◽  
Formula Group ◽  
Imaginary Quadratic Number ◽  
Direct Calculations

Abstract We develop methods for constructing explicit generators, modulo torsion, of the $K_3$ -groups of imaginary quadratic number fields. These methods are based on either tessellations of hyperbolic $3$ -space or on direct calculations in suitable pre-Bloch groups and lead to the very first proven examples of explicit generators, modulo torsion, of any infinite $K_3$ -group of a number field. As part of this approach, we make several improvements to the theory of Bloch groups for $ K_3 $ of any field, predict the precise power of $2$ that should occur in the Lichtenbaum conjecture at $ -1 $ and prove that this prediction is valid for all abelian number fields.

Guided Surface Waves on an Elastic Half Space

1971 ◽  
Vol 38 (4) ◽  
pp. 899-905 ◽  
Author(s):  
L. B. Freund
Keyword(s):  
Wave Propagation ◽  
Surface Wave ◽  
Surface Waves ◽  
Half Space ◽  
Body Wave ◽  
Three Dimensional ◽  
Elastic Half Space ◽  
Normal Displacement ◽  
Rigid Barrier ◽  
Wave Disturbances

Three-dimensional wave propagation in an elastic half space is considered. The half space is traction free on half its boundary, while the remaining part of the boundary is free of shear traction and is constrained against normal displacement by a smooth, rigid barrier. A time-harmonic surface wave, traveling on the traction free part of the surface, is obliquely incident on the edge of the barrier. The amplitude and the phase of the resulting reflected surface wave are determined by means of Laplace transform methods and the Wiener-Hopf technique. Wave propagation in an elastic half space in contact with two rigid, smooth barriers is then considered. The barriers are arranged so that a strip on the surface of uniform width is traction free, which forms a wave guide for surface waves. Results of the surface wave reflection problem are then used to geometrically construct dispersion relations for the propagation of unattenuated guided surface waves in the guiding structure. The rate of decay of body wave disturbances, localized near the edges of the guide, is discussed.

Complete caps in AG(3, q) from elliptic curves

2014 ◽  
Vol 13 (08) ◽  
pp. 1450050 ◽  
Author(s):  
Irene Platoni
Keyword(s):  
Elliptic Curves ◽  
Lower Bounds ◽  
Three Dimensional ◽  
Complete Caps ◽  
Odd Order ◽  
Galois Space

In a three-dimensional Galois space of odd order q, the known infinite families of complete caps have size far from the theoretical lower bounds. In this paper, we investigate some caps defined from elliptic curves. In particular, we show that for each q between 100 and 350 they can be extended to complete caps, which turn out to be the smallest complete caps known in the literature.

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