Baker's Method Research Articles

Extrema of polynomials with real roots and Diophantine equations

There are many results in the literature concerning polynomial values and (shifted) power values of polynomials with consecutive integer roots, or more generally, with roots forming an arithmetic progression. It is an interesting question that how far one can ‘disturb’ the structure of the roots such that the finiteness results still remain valid. Also there are many results into this direction, with adding or removing one or more terms (roots).In this paper we study a case where (part of) the symmetric root structure is preserved, however, we allow (possibly large) increasing gaps between the roots. We prove that the finiteness of the solutions can also be guaranteed under these generalized circumstances. In our proofs we combine Baker's method and the Bilu-Tichy theorem with a new result providing an increasing property of the extremal values of polynomials with distinct real roots satisfying certain symmetry and increasing gap properties.

PENCILS OF NORM FORM EQUATIONS AND A CONJECTURE OF THOMAS

We introduce a new method to deal with families of norm form equations. These generalize the Thue equations studied first by Thomas using Baker's Method (which, however, we do not use here). We show that for all large integer values of the parameter t, every solution over Z arises from specializing a solution over Z [ T ] by T = t . The results are completely effective.

Tubular approaches to Baker’s method for curves and varieties

Baker's method, relying on estimates on linear forms in logarithms of algebraic numbers, allows one to prove in several situations the effective finiteness of integral points on varieties. In this article, we give a generalisation of results of Levin regarding Baker's method for varieties, and explain how, quite surprisingly, it mixes (under additional hypotheses) with Runge's method to improve some known estimates in the case of curves by bypassing (or more generally reducing) the need for linear forms in $p$-adic logarithms.

On the Diophantine equation (x + 1)k + (x + 2)k + ... + (2x)k = yn

In this work, we give upper bounds for n on the title equation. Our results depend on assertions describing the precise exponents of 2 and 3 appearing in the prime factorization of Tk(x)=(x+1)k+(x+2)k+...+(2x)k. Further, on combining Baker's method with the explicit solution of polynomial exponential congruences (see e.g. [6]), we show that for 2≤x≤13,k≥1,y≥2 and n≥3 the title equation has no solutions.

On the equation 1k + 2k + ⋯ + xk = yn for fixed x

We provide all solutions of the title equation in positive integers x,k,y,n with 1≤x<25 and n≥3. For these values of the parameters, our result gives an affirmative answer to a related, classical conjecture of Schäffer. In our proofs we combine several tools: Baker's method (in particular, sharp bounds for the linear combinations of logarithms of two algebraic numbers), polynomial-exponential congruences and computational methods.

Protein-protein interface detection using the energy centrality relationship (ECR) characteristic of proteins.

Specific protein interactions are responsible for most biological functions. Distinguishing Functionally Linked Interfaces of Proteins (FLIPs), from Functionally uncorrelated Contacts (FunCs), is therefore important to characterizing these interactions. To achieve this goal, we have created a database of protein structures called FLIPdb, containing proteins belonging to various functional sub-categories. Here, we use geometric features coupled with Kortemme and Baker's computational alanine scanning method to calculate the energetic sensitivity of each amino acid at the interface to substitution, identify hotspots, and identify other factors that may contribute towards an interface being FLIP or FunC. Using Principal Component Analysis and K-means clustering on a training set of 160 interfaces, we could distinguish FLIPs from FunCs with an accuracy of 76%. When these methods were applied to two test sets of 18 and 170 interfaces, we achieved similar accuracies of 78% and 80%. We have identified that FLIP interfaces have a stronger central organizing tendency than FunCs, due, we suggest, to greater specificity. We also observe that certain functional sub-categories, such as enzymes, antibody-heavy-light, antibody-antigen, and enzyme-inhibitors form distinct sub-clusters. The antibody-antigen and enzyme-inhibitors interfaces have patterns of physical characteristics similar to those of FunCs, which is in agreement with the fact that the selection pressures of these interfaces is differently evolutionarily driven. As such, our ECR model also successfully describes the impact of evolution and natural selection on protein-protein interfaces. Finally, we indicate how our ECR method may be of use in reducing the false positive rate of docking calculations.

On the Diophantine equation [formula omitted

Recently, mixed polynomial–exponential equations similar to the one in the title have been considered by many authors. In these results a certain non-coprimality condition plays an important role. In this paper we completely solve the title equation for odd positive integers x with x<50. Since we avoid the mentioned non-coprimality condition, this can be considered as a partial completion of the above mentioned results. It seems that the deep effective tools (such as Baker's method) alone are not capable to handle the problem. We combine local arguments and Baker's method to prove our results.

FINDING SINGULAR MODULI ON A COMPLEX LINE

Let (α, β, γ) be three non-zero coprime integers. Using Baker's method and classical technique of reduction, we describe an algorithm which computes every point (x1, x2) on the ℂ-line αX + βY + γ = 0 such that x1 and x2 are both singular moduli. We illustrate our algorithm by writing down some examples, in particular we prove that there are non-complex multiplication points on a line such that |α|, |β| and |γ| are less than 10.

Minorations simultanées de formes linéaires de logarithmes de nombres algébriques

This work falls within the theory of linear forms in logarithms over a commutative linear group defined over a number field. We give lower bounds for simultaneous linear forms in logarithms of algebraic numbers, treating both the archimedean and $p$-adic cases. The proof includes Baker's method, Hirata's reduction, Chudnovsky's process of variable change. The novelty is that we integrated into the proof the modern tools of adelic slope theory, using also a new small values Siegel's lemma.

On Ramachandra's Contributions to Transcendental Number Theory

The first part of this paper is a survey on Ramachandra's contribution to transcendental number theory included in his 1968 paper in Acta Arithmetica. It includes a discussion of pseudo-algebraic points of algebraically additive functions. The second part deals with applications to density statements related with a conjecture due to B.~Mazur. The next part is a survey of other contributions of Ramachandra to transcendence questions (on the numbers $2^{\pi^k}$, a note on Baker's method, an easy transcendence measure for $e$). Finally, related open questions are raised.

On $(a,b)$-balancing numbers

Year: 2010 Vol.: 77 Fasc.: 3-4 Title: On (a; b)-balancing numbers Author(s): TAunde Kov¶acs , K¶alm¶an Liptai and P¶eter Olajos A positive integer n is called a balancing number if 1+: : : +(n i 1) = (n + 1) + ¢ ¢ ¢ + (n + r) for some positive integer r. Balancing numbers and their generaliza- tions have been investigated by several authors, from many aspects. In this paper we introduce the concept of balancing numbers in arithmetic progressions, and prove several e®ective ¯niteness and explicit results about them. In the proofs of our results, among others, we combine Baker's method, the modular method developed by Wiles and others, a result of Bennett about the diophantine equation jaxn i bynj = 1, the Chabauty method and the theory of elliptic curves. Address: TAunde Kov¶acs Institute of Mathematics University of Debrecen H-4010 Debrecen, P.O. Box 12 Hungary Address: K¶alm¶an Liptai Institute of Mathematics and Informatics Eszterh¶azy K¶aroly College H-3300 Eger, Eszterh¶azy t¶er 1 Hungary Address: P¶eter Olajos Department of Applied Mathematics University of Miskolc H-3515 Miskolc-Egyetemv¶aros Hungary

Endoscopic CO2 Laser-Assisted Capsulotomy (Minimally Invasive Method for Treatment of Capsular Contracture)

Introduction: Capsular contracture is a common problem faced by cosmetic/plastic surgeons and their patients. This retrospective study looks at the effectiveness of and overall patient satisfaction with endoscopic laser-assisted capsulotomy and suggests it is a safe and effective method of treating capsular contracture while minimizing morbidity and scarring and allowing patients to return to their daily activities almost immediately. Materials and Methods: The authors present their experience with 56 patients over a 7-year period. The operation is performed through a small (5–7 mm) periareolar access incision into the capsule, followed by horizontal and radial release of the capsular contracture using a CO2 laser visualized with a 4-mm endoscope with a specialized sleeve designed and built by the senior surgeon. Each patient was assessed preoperatively and postoperatively. Capsular contracture was graded based on Baker's classification method. The procedure was performed on patients with silicone and saline implants. Postoperative complications and patient satisfaction were noted. Results: Of 56 patients in the study, 6 (10.7%) were excluded: 2 had prior radiation therapy, 1 had reconstructive breast surgery for cancer prior to having the implant put in, 2 had a deflation 3 months postprocedure, and 1 had the procedure to treat an implant that had not adequately settled. Many of the patients had been previously treated elsewhere 1 or more times with recurrent capsule formation. One of the 50 patients (2%) presented with Baker II capsular contracture but was lost to follow-up. According to the chart review, the patient was happy with the results postprocedure, and Baker I score was achieved. Thirty-six of 50 patients (72%) presented initially with a Baker III capsular contracture, and of these, 20 (55%) were lost to follow-up, and thus long-term results of the procedure could not accurately be established. Of 16 patients with long-term follow-up, 6 (38%) achieved a score of Baker I and were pleased with the results. Ten patients (62%) experienced improvement initially but recurred shortly after (2–6 months). Thirteen of 50 patients (26%) presented initially with a Baker IV capsular contracture of which 7 (54%) were lost to long-term follow-up, and long-term results could not accurately be established. Two of 6 patients (34%) followed long term achieved a score of Baker I and were pleased with the results. Four patients (66%) recurred shortly after the procedure (2–6 months). Complications included 1 infection postprocedure requiring removal of the implant (in the patient excluded from the study) and 2 drain placements (both patients were lost to follow-up). The mean follow-up was 21 months.

A system for quantitative evaluation of fixatives for light microscopy using paraffin sections of kidney and brain

A numerical scoring system is presented for evaluating structural fixation of certain mammalian tissues for light microscopy. Small pieces of rat's kidney and brain, tissues for which artifacts of fixation are well documented, were fixed in various fluids. Random code numbers hid their identities, and paraffin sections were stained to show nuclear chromatin, cytoplasm and extracellular materials. Two sections of each specimen were examined and awarded scores according to described schemes, for microanatomical and cytological fixation. The assessment was confined to preservation of structure; chemical changes were not taken into account. When the code was broken, the scores for both sections of each specimen from each fixative were added. The scores obtained (24 best to eight worst) are generally comparable to the grades (I–V) given for traditional fixatives by JR Baker. The criteria for quality of fixation were defined more explicitly than those for Baker's grading method, which used mouse testis as the test object. Assessments are presented for several traditional fixatives and for four zinc-formaldehyde mixtures. The scoring system should be useful for evaluating newly developed fixatives for animal tissues for light microscopic examination of paraffin sections. In an evaluation of four traditional fixatives and four zinc-formaldehyde mixtures, variation among three different observers was only ±1 point on either side of the median score for microanatomical or cytological preservation by any of the eight fixatives. This approach has certain limitations, notably that the criteria for cytological fixation do not include the preservation of chromosomes or specific cytoplasmic organelles.

Effect of Irradiation on the Development of Immature Hippocampal Neurons In Vitro

X-ray irradiation is known to cause mental retardation. Last year, we reported very low dose irradiation (0.5 Gy) would not affect neural cell death and its development, using Baker's method which enable to exclude the secondly effects on neuron via injuries of blood vessels and glial cells. Thereafter, we investigated the effect of middle-high dose of X-ray irradiation on development of immature primary hippocampal neuronal culture without contact of glial cells.

Validation of the World Health Organization's Rapid Assessment Method for Urinary Schistosomiasis in Southeastern Nigeria

Urinary schistosomiasis is a helminth disease that causes high morbidity in endemic areas of tropical and subtropical regions. Efforts are being made to evolve a cost-effective method for diagnosing the infection in large populations. A study supported by the World Health Organization (WHO) has established the Rapid Assessment (Questionnaire) method in which diagnosis is based on the respondent's ability to answer yes to the presence of hematuria. This method has been validated in some African countries and elsewhere. The aim of the present study was to validate the Rapid Assessment method in a community in southeastern Nigeria where the disease is endemic. A survey was carried out using both the parasitological diagnosis of the presence of the characteristic egg of Schistosoma haematabium in urine samples and the WHO Rapid Assessment method. Positive results in the 2 methods were calculated as percentages, and a correlation analysis of the percentages was done using product moment statistics. This gave a significant value of r = 2.9435 (P < 0.05). Sex-related prevalence was observed at significant correlation values of r = 1.0011 and r = 1.574 (P < 0.05). The diagnostic performance of the Rapid Assessment method was calculated using Baker's procedure method. A high sensitivity of 93.4%, specificity of 99%, positive predictive value of 96.6%, and negative predictive value of 99.4% were calculated. The consistent high correction performance values confirm that the Rapid Assessment method may be a useful alternative to the parasitological tests for use in schools and community surveys in identifying high-risk individuals for urinary schistosomiasis in southeastern Nigeria.

On the power values of power sums

Using several currently available techniques, including Baker's method, Frey curves and modular forms, we prove that for odd values of k with 1 ⩽ k < 170 , the equation 1 k + 2 k + ⋯ + x k = y 2 n in positive integers x , y , n with n > 2 has only the trivial solution ( x , y ) = ( 1 , 1 ) .

Mesures de transcendance pour les points algébriques de fonctions modulaires de Siegel

We give an effective version of a result of Cohen, Shiga and Wolfart, which is a generalisation to the case of Siegel spaces of arbitrary degree, of the classical theorem of Schneider on the modular invariant j ( τ ) . Given a point τ of the Siegel space parameterizing a principally polarised Abelian variety A defined over Q ¯ , we obtain a lower bound for the distance between τ and algebraic points β of the Siegel space, in terms of the geometrical data of the problem. To achieve this, we establish a simultaneous measure of linear independence for periods of Abelian integrals, using Baker's method. To cite this article: E. Villani, C. R. Acad. Sci. Paris, Ser. I 344 (2007).

Quantitative results of algebraic independence and Baker's method

Quantitative results of algebraic independence and Baker's method

Endoscopic CO2 Laser-Assisted Capsulotomy—A New Therapeutic Method for Correcting Capsular Contracture

Introduction: Capsular contracture presents as a common and frustrating complication of breast augmentation surgery. Various strategies have been recommended for resolving this problem, including both closed and open techniques. However, morbidity associated with previously described procedures has left the clinician without a fully satisfactory solution. In this retrospective study, we report our experience with the endoscopic laser-assisted capsulotomy and suggest that it provides a safe, effective method of treating capsular contracture while minimizing morbidity. Materials and Methods: The endoscopic, laser-assisted capsulotomy is a technique that has been utilized for 10 years by the senior author and is presented here with our experience in 28 consecutive affected breasts treated over a 5- year period. The operation consists of a small periareolar access incision into the capsule space, followed by capsular release utilizing a CO2 Ultrapulse laser visualized with a 4-mm endoscope and specialized sleeve developed by the senior author. In this retrospective study, each patient was assessed by the senior author both pre- and postoperatively. Capsular contracture was graded according to Baker's classification method. Postoperative complications and their respective frequencies were also noted. Results: Of the 11 patients that initially presented with a Baker score of IV, 9 patients (82%) achieved an acceptable result after laser-assisted endoscopic capsulotomy. One patient (9%) had a final Baker score of III, but was pleased with the result and did not seek further treatment. One patient saw no change in Baker score and underwent a second endoscopic capsulotomy. A final Baker score of I was then achieved. Of the 16 patients that began with a Baker score of III, 15 (94%) achieved an acceptable result. One patient (6%) saw no change in Baker score but elected to forego further treatment. One patient that presented with an objectionable Baker grade II contracture achieved a Baker score of I following treatment. Complications noted included: intraoperative bleeding requiring drain placement in 3 patients, scar hypopigmentation in 1 patient, superficial skin infection in 1 patient that responded to oral antibiotics, and asymmetry between the operated and unoperated breast in 2 patients that did not require correction. Late-term complications included: 4 patients presented with recurrent capsule formation. All underwent a second capsulotomy procedure with acceptable results. Conclusion: To our knowledge, this is the first report of use of the CO2 Ultrapulse laser for this purpose. The capsuloscope sleeve used was developed by the senior author. The technique was effective in correcting capsular contracture, while minimizing complications, scarring, and patient morbidity.

Mesure d'indépendance linéaire de logarithmes dans un groupe algébrique commutatif

We obtain some new results in the theory of linear forms in logarithms on a commutative algebraic group, defined over Q . We generalize recent progress of S. David and N. Hirata [1]. In particular, we achieve an optimal linear independence measure in the height of the linear form and a measure more precise than Hirata's [4] for parameters related to the logarithms. The proof rests on Baker's method and a new argument of arithmetical nature.