Gröbnerbasen in Ore-Algebren

In dieser Arbeit werden grundlegende Algorithmen für Ore-Algebren in Mathematica realisiert. Dabei entsteht eine Plattform um die speziellen Beschränkungen und Möglichkeiten dieser Algebren insbesondere im Zusammenhang mit Gröbnerbasen an praktischen Beispielen auszuloten. Im Gegensatz zu den existierenden Paketen wird dabei explizit die Struktur der Ore-Algebra benutzt. Kandri-Rody und Weispfenning untersuchten 1990 Verallgemeinerungen von Gröbnerbasen auf Algebren ordnungserhaltender Art (``algebras of solvable type''). Diese verhalten sich so, dass Buchbergers Algorithmus stets eine Gröbnerbasis findet. Es wird ein Beispiel gezeigt, an dem klar wird, dass es mehr Ore-Algebren ordnungserhaltender Art gibt als die in der Literatur stets betrachteten Operator-Algebren. Für Ore-Algebren ordnungserhaltender Art werden Algorithmen zu Gröbnerbasen implementiert. Anschließend wird der Gröbner-Walk für Ore-Algebren untersucht. Der Gröbner-Walk im kommutativen Fall wird mit einem instruktiven Beispiel vorgestellt. Dann wird zum nichtkommutativen Fall übergegangen. Es wird gezeigt, dass die Eigenschaft ordnungserhaltender Art zu sein, auf der Strecke zwischen zwei Ordnungen erhalten bleibt. Eine leichte Modifikation des Walks für Ore-Algebren wird implementiert, die im Erfolgsfall die Basis konvertiert und ansonsten abbricht. Es werden Beispiele angegeben, in denen der modifizierte Walk funktioniert sowie ein Beispiel analysiert, in dem er versagt.

Elimination in Operator Algebras

A large class of special functions are solutions of systems of linear difference and differential equations with polynomial coefficients. For a given function, these equations considered as operator polynomials generate a left ideal in a noncommutative algebra called Ore algebra. This ideal with finitely many conditions characterizes the function uniquely so that Gröbner basis techniques can be applied. Many problems related to special functions which can be described by such ideals can be solved by performing elimination of appropriate noncommutative variables in these ideals. In this work, we mainly achieve the following: 1. We give an overview of the theoretical algebraic background as well as the algorithmic aspects of different methods using noncommutative Gröbner elimination techniques in Ore algebras in order to solve problems related to special functions. 2. We describe in detail algorithms which are based on Gröbner elimination techniques and perform the creative telescoping method for sums and integrals of special functions. 3. We investigate and compare these algorithms by illustrative examples which are performed by the computer algebra system Maple. This investigation has the objective to test how far noncommutative Gröbner elimination techniques may be efficiently applied to perform creative telescoping.

Dickson Polynomials that are Permutations

2000 Mathematics Subject Classification: 11T06, 13P10.

Free Resolutions from Involutive Bases

We show that the theory of involutive bases can be combined with discrete algebraic Morse Theory. For a graded k[x0 ...,xn]-module M, this yields a free resolution G, which in general is not minimal. We see that G is isomorphic to the resolution induced by an involutive basis. It is possible to identify involutive bases inside the resolution G. The shape of G is given by a concrete description. Regarding the differential dG, several rules are established for its computation, which are based on the fact that in the computation of dG certain patterns appear at several positions. In particular, it is possible to compute the constants independent of the remainder of the differential. This allows us, starting from G, to determine the Betti numbers of M without computing a minimal free resolution: Thus we obtain a new algorithm to compute Betti numbers. This algorithm has been implemented in CoCoALib by Mario Albert. This way, in comparison to some other computer algebra system, Betti numbers can be computed faster in most of the examples we have considered. For Veronese subrings S(d), we have found a Pommaret basis, which yields new proofs for some known properties of these rings. Via the theoretical statements found for G, we can identify some generators of modules in G where no constants appear. As a direct consequence, some non-vanishing Betti numbers of S(d) can be given. Finally, we give a proof of the Hyperplane Restriction Theorem with the help of Pommaret bases. This part is largely independent of the other parts of this work.

Three dimensional Sklyanin algebras and Gröbner bases

We consider Sklyanin algebras $S$ with 3 generators, which are quadratic algebras over a field $\K$ with $3$ generators $x,y,z$ given by $3$ relations $pxy+qyx+rzz=0$, $pyz+qzy+rxx=0$ and $pzx+qxz+ryy=0$, where $p,q,r\in\K$. this class of algebras has enjoyed much attention. In particular, using tools from algebraic geometry, Feigin, Odesskii \cite{odf}, and Artin, Tate and Van Den Bergh, showed that if at least two of the parameters $p$, $q$ and $r$ are non-zero and at least two of three numbers $p^3$, $q^3$ and $r^3$ are distinct, then $S$ is Artin--Schelter regular. More specifically, $S$ is Koszul and has the same Hilbert series as the algebra of commutative polynomials in 3 indeterminates (PHS). It has became commonly accepted that it is impossible to achieve the same objective by purely algebraic and combinatorial means like the Groebner basis technique. The main purpose of this paper is to trace the combinatorial meaning of the properties of Sklyanin algebras, such as Koszulity, PBW, PHS, Calabi-Yau, and to give a new constructive proof of the above facts due to Artin, Tate and Van Den Bergh. Further, we study a wider class of Sklyanin algebras, namely
the situation when all parameters of relations could be different. We call them generalized Sklyanin algebras. We classify up to isomorphism all generalized Sklyanin algebras with the same Hilbert series as commutative polynomials on
3 variables. We show that generalized Sklyanin algebras in general position have a Golod–Shafarevich Hilbert series (with exception of the case of field with two elements).

Changes In Motor Behavior During Pregnancy In Rats: The Basis For A Possible Animal Model Of Restless Legs Syndrome.

Pregnant women have a 2-3 fold higher probability of developing restless legs syndrome (RLS - sleep-related movement disorders) than general population. This study aims to evaluate the behavior and locomotion of rats during pregnancy in order to verify if part of these animals exhibit some RLS-like features. We used 14 female 80-day-old Wistar rats that weighed between 200 and 250 g. The rats were distributed into control (CTRL) and pregnant (PN) groups. After a baseline evaluation of their behavior and locomotor activity in an open-field environment, the PN group was inducted into pregnancy, and their behavior and locomotor activity were evaluated on days 3, 10 and 19 of pregnancy and in the post-lactation period in parallel with the CTRL group. The serum iron and transferrin levels in the CTRL and PN groups were analyzed in blood collected after euthanasia by decapitation. There were no significant differences in the total ambulation, grooming events, fecal boli or urine pools between the CTRL and PN groups. However, the PN group exhibited fewer rearing events, increased grooming time and reduced immobilization time than the CTRL group (ANOVA, p<0.05). These results suggest that pregnant rats show behavioral and locomotor alterations similar to those observed in animal models of RLS, demonstrating to be a possible animal model of this sleep disorder.

Kinase Inhibitor Profile For Human Nek1, Nek6, And Nek7 And Analysis Of The Structural Basis For Inhibitor Specificity.

Human Neks are a conserved protein kinase family related to cell cycle progression and cell division and are considered potential drug targets for the treatment of cancer and other pathologies. We screened the activation loop mutant kinases hNek1 and hNek2, wild-type hNek7, and five hNek6 variants in different activation/phosphorylation statesand compared them against 85 compounds using thermal shift denaturation. We identified three compounds with significant Tm shifts: JNK Inhibitor II for hNek1(Δ262-1258)-(T162A), Isogranulatimide for hNek6(S206A), andGSK-3 Inhibitor XIII for hNek7wt. Each one of these compounds was also validated by reducing the kinases activity by at least 25%. The binding sites for these compounds were identified by in silico docking at the ATP-binding site of the respective hNeks. Potential inhibitors were first screened by thermal shift assays, had their efficiency tested by a kinase assay, and were finally analyzed by molecular docking. Our findings corroborate the idea of ATP-competitive inhibition for hNek1 and hNek6 and suggest a novel non-competitive inhibition for hNek7 in regard to GSK-3 Inhibitor XIII. Our results demonstrate that our approach is useful for finding promising general and specific hNekscandidate inhibitors, which may also function as scaffolds to design more potent and selective inhibitors.

Structural and chemical basis for anticancer activity of a series of 'beta'-tubulin ligands: molecular modeling and 3D QSAR studies

An important approach to cancer therapy is the design of small molecule modulators that interfere with microtubule dynamics through their specific binding to the ²-subunit of tubulin. In the present work, comparative molecular field analysis (CoMFA) studies were conducted on a series of discodermolide analogs with antimitotic properties. Significant correlation coefficients were obtained (CoMFA(i), q² =0.68, r²=0.94; CoMFA(ii), q² = 0.63, r²= 0.91), indicating the good internal and external consistency of the models generated using two independent structural alignment strategies. The models were externally validated employing a test set, and the predicted values were in good agreement with the experimental results. The final QSAR models and the 3D contour maps provided important insights into the chemical and structural basis involved in the molecular recognition process of this family of discodermolide analogs, and should be useful for the design of new specific ²-tubulin modulators with potent anticancer activity.

Molecular basis for porcine parvovirus detection in dead fetuses

Reproductive failures are still common grounds for complaint by commercial swine producers. Porcine parvovirus (PPV) is associated with different clinical reproductive signs. The aim of the present study was to investigate PPV fetal infection at swine farms having ongoing reproductive performance problems. The presence of virus in fetal tissues was determined by nested-polymerase chain reaction assay directed to the conserved NS1 gene of PPV in aborted fetuses, mummies and stillborns. Fetuses show a high frequency of PPV infection (96.4%; N = 28). In 60.7% of the fetuses, PPV were detected in all tissue samples (lung, heart, thymus, kidney, and spleen). Viral infection differed among fetal tissues, with a higher frequency in the lung and heart (P < 0.05). Fetuses with up to 99 days of gestational age and from younger sows showed a higher frequency of PPV (P < 0.05). No significant difference in the presence of PPV was detected among the three clinical presentations. The results suggest that PPV remains an important pathogenic agent associated with porcine fetal death.

Structural basis of GC-1 selectivity for thyroid hormone receptor isoforms

Background: Thyroid receptors, TRa and TR beta, are involved in important physiological functions such as metabolism, cholesterol level and heart activities. Whereas metabolism increase and cholesterol level lowering could be achieved by TR beta isoform activation, TRa activation affects heart rates. Therefore, beta-selective thyromimetics have been developed as promising drug-candidates for treatment of obesity and elevated cholesterol level. GC-1 [ 3,5-dimethyl-4-(4'-hydroxy- 3'-isopropylbenzyl)-phenoxy acetic acid] has ability to lower LDL cholesterol with 600-to 1400-fold more potency and approximately two-to threefold more efficacy than atorvastatin (Lipitor(C)) in studies in rats, mice and monkeys. Results: To investigate GC-1 specificity, we solved crystal structures and performed molecular dynamics simulations of both isoforms complexed with GC-1. Crystal structures reveal that, in TRa Arg228 is observed in multiple conformations, an effect triggered by the differences in the interactions between GC-1 and Ser277 or the corresponding asparagine (Asn331) of TR beta. The corresponding Arg282 of TR beta is observed in only one single stable conformation, interacting effectively with the ligand. Molecular dynamics support this model: our simulations show that the multiple conformations can be observed for the Arg228 in TR alpha, in which the ligand interacts either strongly with the ligand or with the Ser277 residue. In contrast, a single stable Arg282 conformation is observed for TR beta, in which it strongly interacts with both GC-1 and the Asn331. Conclusion: Our analysis suggests that the key factors for GC-1 selectivity are the presence of an oxyacetic acid ester oxygen and the absence of the amino group relative to T(3). These results shed light into the beta-selectivity of GC-1 and may assist the development of new compounds with potential as drug candidates to the treatment of hypercholesterolemia and obesity.

Immunological Basis for the Gender Differences in Murine Paracoccidioides brasiliensis Infection

This study aimed to investigate the immunological mechanisms involved in the gender distinct incidence of paracoccidioidomycosis (pcm), an endemic systemic mycosis in Latin America, which is at least 10 times more frequent in men than in women. Then, we compared the immune response of male and female mice to Paracoccidioides brasiliensis infection, as well as the influence in the gender differences exerted by paracoccin, a P. brasiliensis component with carbohydrate recognition property. High production of Th1 cytokines and T-bet expression have been detected in the paracoccin stimulated cultures of spleen cells from infected female mice. In contrast, in similar experimental conditions, cells from infected males produced higher levels of the Th2 cytokines and expressed GATA-3. Macrophages from male and female mice when stimulated with paracoccin displayed similar phagocytic capability, while fungicidal activity was two times more efficiently performed by macrophages from female mice, a fact that was associated with 50% higher levels of nitric oxide production. In order to evaluate the role of sexual hormones in the observed gender distinction, we have utilized mice that have been submitted to gonadectomy followed by inverse hormonal reconstitution. Spleen cells derived from castrated males reconstituted with estradiol have produced higher levels of IFN-gamma (1291+/-15 pg/mL) and lower levels of IL-10 (494+/-38 pg/mL), than normal male in response to paracoccin stimulus. In contrast, spleen cells from castrated female mice that had been treated with testosterone produced more IL-10 (1284+/-36 pg/mL) and less IFN-gamma (587614 pg/mL) than cells from normal female. In conclusion, our results reveal that the sexual hormones had a profound effect on the biology of immune cells, and estradiol favours protective responses to P. brasiliensis infection. In addition, fungal components, such as paracoccin, may provide additional support to the gender dimorphic immunity that marks P. brasiliensis infection.

Transcriptomic basis for an antiserum against Micrurus corallinus (coral snake) venom

Background: Micrurus corallinus (coral snake) is a tropical forest snake belonging to the family Elapidae. Its venom shows a high neurotoxicity associated with pre- and post-synaptic toxins, causing diaphragm paralysis, which may result in death. In spite of a relatively small incidence of accidents, serum therapy is crucial for those bitten. However, the adequate production of antiserum is hampered by the difficulty in obtaining sufficient amounts of venom from a small snake with demanding breeding conditions. In order to elucidate the molecular basis of this venom and to uncover possible immunogens for an antiserum, we generated expressed sequences tags (ESTs) from its venom glands and analyzed the transcriptomic profile. In addition, their immunogenicity was tested using DNA immunization. Results: A total of 1438 ESTs were generated and grouped into 611 clusters. Toxin transcripts represented 46% of the total ESTs. The two main toxin classes consisted of three-finger toxins (3FTx) (24%) and phospholipases A(2) (PLA(2)s) (15%). However, 8 other classes of toxins were present, including C-type lectins, natriuretic peptide precursors and even high-molecular mass components such as metalloproteases and L-amino acid oxidases. Each class included an assortment of isoforms, some showing evidence of alternative splicing and domain deletions. Five antigenic candidates were selected (four 3FTx and one PLA(2)) and used for a preliminary study of DNA immunization. The immunological response showed that the sera from the immunized animals were able to recognize the recombinant antigens. Conclusion: Besides an improvement in our knowledge of the composition of coral snake venoms, which are very poorly known when compared to Old World elapids, the expression profile suggests abundant and diversified components that may be used in future antiserum formulation. As recombinant production of venom antigens frequently fails due to complex disulfide arrangements, DNA immunization may be a viable alternative. In fact, the selected candidates provided an initial evidence of the feasibility of this approach, which is less costly and not dependent on the availability of the venom.

Use of correlated potential harmonic basis functions for the description of the (4)He trimer and small clusters

A correlated many-body basis function is used to describe the (4)He trimer and small helium clusters ((4)HeN) with N = 4-9. A realistic helium dimer potential is adopted. The ground state results of the (4)He dimer and trimer are in close agreement with earlier findings. But no evidence is found for the existence of Efimov state in the trimer for the actual (4)He-(4)He interaction. However, decreasing the potential strength we calculate several excited states of the trimer which exhibit Efimov character. We also solve for excited state energies of these clusters which are in good agreement with Monte Carlo hyperspherical description. (C) 2011 American Institute of Physics. [doi:10.1063/1.3583365]

Adenosine binding to low-molecular-weight purine nucleoside phosphorylase: the structural basis for recognition based on its complex with the enzyme from Schistosoma mansoni

Schistosomes are unable to synthesize purines de novo and depend exclusively on the salvage pathway for their purine requirements. It has been suggested that blockage of this pathway could lead to parasite death. The enzyme purine nucleoside phosphorylase (PNP) is one of its key components and molecules designed to inhibit the low-molecular-weight (LMW) PNPs, which include both the human and schistosome enzymes, are typically analogues of the natural substrates inosine and guanosine. Here, it is shown that adenosine both binds to Schistosoma mansoni PNP and behaves as a weak micromolar inhibitor of inosine phosphorolysis. Furthermore, the first crystal structures of complexes of an LMW PNP with adenosine and adenine are reported, together with those with inosine and hypoxanthine. These are used to propose a structural explanation for the selective binding of adenosine to some LMW PNPs but not to others. The results indicate that transition-state analogues based on adenosine or other 6-amino nucleosides should not be discounted as potential starting points for alternative inhibitors.