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In this paper, we investigate the unit groups, the 2-class groups, the 2-class field towers and the structures of the second 2-class groups of some multiquadratic number fields of degree 8 and 16. [ABSTRACT FROM AUTHOR]

We study the structure of the Galois group of the maximal unramified 2-extension of some family of number fields of large degree. Especially, we show that for each positive integer n, there exist infinitely many number fields with large degree, for which the defined Galois group is quaternion of order 2n. [ABSTRACT FROM AUTHOR]

We describe a class of fixed polyominoes called k -omino towers that are created by stacking rectangular blocks of size k × 1 on a convex base composed of these same k -omino blocks. By applying a partition to the set of k -omino towers of fixed area k n , we give a recurrence on the k -omino towers therefore showing the set of k -omino towers is enumerated by a Gauss hypergeometric function. The proof in this case implies a more general hypergeometric identity with parameters similar to those given in a classical result of Kummer. [ABSTRACT FROM AUTHOR]

Let denote the discriminant of a real quadratic field. For all bicyclic biquadratic fields , having a -class group of type , the possibilities for the isomorphism type of the Galois group of the second Hilbert -class field of are determined. For each coclass graph , , in the sense of Eick, Leedham-Green, Newman and O'Brien, the roots of even branches of exactly one coclass tree and, in the case of even coclass , additionally their siblings of depth and defect , turn out to be admissible. The principalization type of -classes of in its four unramified cyclic cubic extensions is given by for , and by for . The theory is underpinned by an extensive numerical verification for all fields with values of in the range , which supports the assumption that all admissible vertices will actually be realized as Galois groups for certain fields , asymptotically. [ABSTRACT FROM AUTHOR]

Let l be a prime number, K a finite extension of Ql, and D a finite-dimensional central division algebra over K. We prove that the profinite group G = Dx/Kx is finitely sliceable, i.e. G has finitely many closed subgroups H1,..., Hn of infinite index such that G = Uin=i HiG. Here, HiG = {hg | h e Hi, g e G}. On the other hand, we prove for l ≠ 2 that no open subgroup of GL2(Zl) is finitely sliceable and we give an arithmetic interpretation to this result, based on the possibility of realizing GL2(Zl) as a Galois group over Q. Nevertheless, we prove that G = GL2(Zl) has an infinite slicing, that is G = Ui=1∞ HiG, where each Hi is a closed subgroup of G of infinite index and Hi A Hj has infinite index in both Hi and Hj if i ≠ j. [ABSTRACT FROM AUTHOR]

Let p1 ≡ p2 ≡ -q ≡ 1 ( 4) be primes such that and . Put and d = p1p2q, then the bicyclic biquadratic field has an elementary Abelian 2-class group of rank 3. In this paper we determine the nilpotency class, the coclass, the generators and the structure of the non-Abelian Galois group of the second Hilbert 2-class field of 핂, we study the 2-class field tower of 핂, and we study the capitulation problem of the 2-classes of 핂 in its fourteen abelian unramified extensions of relative degrees two and four. [ABSTRACT FROM AUTHOR]

Let v be a henselian valuation of arbitrary rank of a field K with value group G(K) and residue field R(K) and be the unique extension of v to a fixed algebraic closure of K with value group . It is known that a complete distinguished chain for an element θ belonging to with respect to (K, v) gives rise to several invariants associated to θ, including a chain of subgroups of , a tower of fields, together with a sequence of elements belonging to which are the same for all K-conjugates of θ. These invariants satisfy some fundamental relations. In this paper, we deal with the converse: Given a chain of subgroups of containing G(K), a tower of extension fields of R(K), and a finite sequence of elements of satisfying certain properties, it is shown that there exists a complete distinguished chain for an element associated to these invariants. We use the notion of lifting of polynomials to construct it. [ABSTRACT FROM AUTHOR]

In this work, we give sufficient conditions in order to have finite ramification locus in sequences of function fields defined by different kind of Kummer extensions. These conditions can be easily implemented in a computer to generate several examples. We present some new examples of asymptotically good towers of Kummer type and we show that many known examples can be obtained from our general results. [ABSTRACT FROM AUTHOR]

The p -group generation algorithm is used to verify that the Hilbert 3-class field tower has length 3 for certain imaginary quadratic fields K with 3-class group Cl 3 ( K ) ≅ [ 3 , 3 ] . Our results provide the first examples of finite p -class towers of length >2 for an odd prime p . [ABSTRACT FROM AUTHOR]

Let T= T be the rooted binary tree, Aut( T) = $\mathop {\lim }\limits_ \leftarrow $Aut( T) its automorphism group and Ψ: Aut( T)→Aut( T) the restriction maps to the first n levels of the tree. If L is the the n level of the tree then Aut( T) < Sym( L) can be identified with the 2-Sylow subgroup of the symmetric group on 2 points. Consider a random subgroup Γ:= 〈 a〉= 〈 a, a,..., a〉 ∈ Aut( T) generated by m independent Haar-random tree automorphisms. Theorem A. The following hold, almost surely, for every non-cyclic subgroup Δ < Γ: [ABSTRACT FROM AUTHOR]

Let $$p_1$$ and $$p_2$$ be primes such that $$p_1\equiv p_2\equiv 5 \pmod 8$$ , $$i=\sqrt{-1}$$ , $$d=2p_1p_2$$ , $$\mathbb K =\mathbb Q (\sqrt{d},i)$$ , $$\mathbb K _2^{(1)}$$ be the Hilbert 2-class field of $$\mathbb K $$ , $$\mathbb K _2^{(2)}$$ be the Hilbert 2-class field of $$\mathbb K _2^{(1)}$$ , $$G$$ be the Galois group of $$\mathbb K _2^{(2)}/\mathbb K $$ and $$\mathbb K ^{(*)}=\mathbb Q (\sqrt{p_1},\sqrt{p_2},\sqrt{2}, i)$$ be the genus field of $$\mathbb K $$ . The 2-part $$\mathbf C _{\mathbb{K },2}$$ of the class group of $$\mathbb K $$ is of type $$(2, 2, 2)$$ . Our goal is to study the 2-class field tower of $$\mathbb K $$ and to calculate the order of $$G$$ . [ABSTRACT FROM AUTHOR]

Abstract: Let K be a one variable function field over a number field. Fix a prime number ℓ. We give examples of Jacobian varieties defined over K whose Mordell–Weil groups have uniformly bounded ranks along -towers of function fields over K. [Copyright &y& Elsevier]

Let K be a number field and dK the absolute value of the discriminant of K/Q. We consider the root discriminant ... of extensions L/K. We show that for any N > 0 and any positive integer n, the set of length n solvable extensions of K with root discriminant less than N is finite. The result is motivated by the study of class field towers. [ABSTRACT FROM AUTHOR]

Abstract: Given a smooth compact Riemannian n-manifold , we consider the equation , where h is a -function on M, the exponent is the critical Sobolev exponent, and ε is a small positive real parameter such that . We prove the existence of blowing-up families of sign-changing solutions which develop bubble towers at some point where the function h is greater than the Yamabe potential . [Copyright &y& Elsevier]

Abstract: This paper gives examples of function fields over a finite field of p power order ramified only at one finite regular prime over , which admit infinite Hilbert p-class field towers. Such a can be taken as an extension of a cyclotomic function field for a certain regular prime in . [Copyright &y& Elsevier]

Abstract: In Burger et al. (2002) and Goldfeld et al. (2004) it was conjectured that if H is a simple Lie group of real rank at least 2, then the number of conjugacy classes of (arithmetic) lattices in H of covolume at most x is where is an explicit constant computable from the (absolute) root system of H. In this paper we prove that this conjecture is false. In fact, we show that the growth is at rate . A crucial ingredient of the proof is the existence of towers of field extensions with bounded root discriminant which follows from the seminal work of Golod and Shafarevich on class field towers. [Copyright &y& Elsevier]

We find a countable partition P on a Lebesgue space, labeled {1,2,3,…}, for any non-periodic measure-preserving transformation T such that P generates T and, for the T,P process, if you see an n on time −1 then you only have to look at times −n,1−n,…−1 to know the positive integer i to put at time 0 . We alter that proof to extend every non-periodic T to a uniform martingale (i.e. continuous g function) on an infinite alphabet. If T has positive entropy and the weak Pinsker property, this extension can be made to be an isomorphism. We pose remaining questions on uniform martingales. In the process of proving the uniform martingale result we make a complete analysis of Rokhlin towers which is of interest in and of itself. We also give an example that looks something like an independent identically distributed process on ℤ2 when you read from right to left but where each column determines the next if you read left to right. [ABSTRACT FROM PUBLISHER]

Abstract: In this paper, we find several equations of recursive towers of function fields over finite fields corresponding to sequences of elliptic modular curves. This is a continuation of the work of Noam D. Elkies on modular equations of higher degrees. [Copyright &y& Elsevier]

The minimum depth d(B,A) of a subring B ⊆ A introduced in the work of Boltje, Danz and Külshammer (2011) is studied and compared with the tower depth of a Frobenius extension. We show that d(B,A) < ∞ if A is a finite-dimensional algebra and Be has finite representation type. Some conditions in terms of depth and QF property are given that ensure that the modular function of a Hopf algebra restricts to the modular function of a Hopf subalgebra. If A ⊇ B is a QF extension, minimum left and right even subring depths are shown to coincide. If A ⊇ B is a Frobenius extension with surjective Frobenius, homomorphism, its subring depth is shown to coincide with its tower depth. Formulas for the ring, module, Frobenius and Temperley-Lieb structures are noted for the tower over a Frobenius extension in its realization as tensor powers. A depth 3 QF extension is embedded in a depth 2 QF extension; in turn certain depth n extensions embed in depth 3 extensions if they are Frobenius extensions or other special ring extensions with ring structures on their relative Hochschild bar resolution groups. [ABSTRACT FROM AUTHOR]

Infinite pillars of quadratic transformations are used to describe residue fields of subrings of finitely generated ring extensions of the ring of integers. Towers whose underlying quadratic transformations are finite pillars or nonpillars are employed for the construction of basic dicritical divisors. [ABSTRACT FROM AUTHOR]

In this article we investigate the p-rank of function fields in several good towers. To do this we first recall and establish some properties of the behaviour of the p-rank under extensions. Then we compute the p-ranks of function fields in several optimal towers over a quadratic field % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC % vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz % ZbItLDhis9wBH5garqqr1ngBPrgifHhDYfgasaacH8srps0lbbf9q8 % WrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0RYxir-J % bba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaae % qabaWaaqaafaaakeaatuuDJXwAK1uy0HMmaeXbfv3ySLgzG0uy0Hgi % uD3BaGGbaiab-vi8gnaaBaaaleaaieGacqGFXbqCdaahaaadbeqaaG % qaaiab9jdaYaaaaSqabaaaaa!4B2B! $$ \mathbb{F}_{q^2 } $$, as well as for a specific good tower over a cubic field % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC % vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz % ZbItLDhis9wBH5garqqr1ngBPrgifHhDYfgasaacH8srps0lbbf9q8 % WrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0RYxir-J % bba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaae % qabaWaaqaafaaakeaatuuDJXwAK1uy0HMmaeXbfv3ySLgzG0uy0Hgi % uD3BaGGbaiab-vi8gnaaBaaaleaaieGacqGFXbqCdaahaaadbeqaaG % qaaiab9ndaZaaaaSqabaaaaa!4B2D! $$ \mathbb{F}_{q^3 } $$, which was introduced by Bassa, Garcia and Stichtenoth. [ABSTRACT FROM AUTHOR]

We introduce functional norms for hyperbolic Young towers which allow us to directly study the transfer operator on the full tower. By eliminating the need for secondary expanding towers commonly employed in this context, this approach simplifies and expands the analysis of this class of Markov extensions and the underlying systems for which they are constructed. As an example, we prove large-deviation estimates with a uniform rate function for a large class of non-invariant measures and show how to translate these to the underlying system. [ABSTRACT FROM AUTHOR]

In this paper, we study certain asymptotic properties of global fields. We consider the set of Tsfasman-Vlăduţ invariants of infinite global fields and answer some natural questions arising from their work. In particular, we prove the existence of infinite global fields having finitely many strictly positive invariants at given places, and the existence of infinite number fields with certain prescribed invariants being zero. We also give precisions on the deficiency of infinite global fields and on the primes decomposition in those fields. [ABSTRACT FROM AUTHOR]

Abstract: An account of the interpolation and the root-finding steps of list decoding of one-point codes is given. The interpolation step is reduced to the problem of finding the minimal element of the Gröbner basis of a submodule of a free module over a polynomial ring of one variable. The procedure for root-finding of the interpolation polynomial going modulo a large degree place is described from the tower point of view. [Copyright &y& Elsevier]

Abstract: Let K be a number field, p a prime, and let be the T-ramified, S-split p-class field tower of K, i.e., the maximal pro-p-extension of K unramified outside T and totally split on S, where T and S are disjoint finite sets of places of K. Using a theorem of Tate on nilpotent quotient groups, we give (Theorem 2 in Section 3) an elementary characterisation of the finite extensions , with a normal closure of degree prime to p, such that the analogous p-class field tower of L is equal to the compositum . This N.S.C. only depends on classes and units of L. Some applications and examples are given. [Copyright &y& Elsevier]

Abstract: Let be a small parameter. In this paper, we study existence of multiple multi-bump positive solutions for the semilinear Schrödinger equation where , if , if or , , for , and . We also study existence of multiple multi-tower positive solutions for the prescribed scalar curvature equation where , , for , , and . [Copyright &y& Elsevier]

Abstract: We present a new bound on the number of -rational places in an algebraic function field. It uses information about the generators of the Weierstrass semigroup related to a rational place. As we demonstrate, the bound has implications to the theory of towers of function fields. [Copyright &y& Elsevier]

Abstract: Let W be a finite Coxeter group. We define its Hecke-group algebra by gluing together appropriately its group algebra and its 0-Hecke algebra. We describe in detail this algebra (dimension, several bases, conjectural presentation, combinatorial construction of simple and indecomposable projective modules, Cartan map) and give several alternative equivalent definitions (as symmetry preserving operator algebra, as poset algebra, as commutant algebra, …). In type A, the Hecke-group algebra can be described as the algebra generated simultaneously by the elementary transpositions and the elementary sorting operators acting on permutations. It turns out to be closely related to the monoid algebras of respectively nondecreasing functions and nondecreasing parking functions, the representation theory of which we describe as well. This defines three towers of algebras, and we give explicitly the Grothendieck algebras and coalgebras given respectively by their induction products and their restriction coproducts. This yields some new interpretations of the classical bases of quasi-symmetric and noncommutative symmetric functions as well as some new bases. [Copyright &y& Elsevier]

Abstract: We discuss the phenomenology of -mode adjoint scalars in the framework of two Universal Extra Dimensions. The Kaluza–Klein (KK) towers of these adjoint scalars arise in the 4-dimensional effective theory from the 6th component of the gauge fields after compactification. Adjoint scalars can have KK-number conserving as well as KK-number violating interactions. We calculate the KK-number violating operators involving these scalars and two Standard Model fields. Decay widths of these scalars into different channels have been estimated. We have also briefly discussed pair-production and single production of such scalars at the Large Hadron Collider. [Copyright &y& Elsevier]

The article focuses on the process of measuring the amount energy and carbon being exchanged between the biosphere and the atmosphere. A survey related to the status of this approach indicates that this method often miscarries everytime standard analysis routines are implemented to single towers data into composite flows. In order to clarify complex flows and to prompt a systematic method of correcting single-tower results. Several paper reviews related to the cause and magnitude of flow-based problems, as well as field experiments dealing with the mechanism of complex flows, are also presented.

The subject of the paper is the collapse of towers and highscrapers, particularly, the collapse of the World Trade Center towers in New York on September 11, 2001. The deduced equations of progressive collapse are used to refute the generally accepted opinion of experts about progressive collapse of the WTC towers in the free-fall regime, which is the official version of the US government. It is proved that progressive collapse is much slower than free fall. The critical floors where collapses started from are estimated using the well-established fact of the free-fall time of all WTC collapses. To this end, the most comprehensive “hybrid” analysis is advanced taking into account that collapses could start on several floors simultaneously, not on one floor as suggested before. According to this “hybrid” model, at the first stage, several floors collapsed simultaneously as a result of fracture waves causing a dust cloud and, at the second stage, the lower part of tower being intact in the first stage collapsed in the regime of progressive failure. Five different collapse types are studied, including the fastest and slowest collapses, and then the hybrid mode is examined with initial collapse of several floors followed by the “domino-effect” of the remaining floors. It is established that the floors where the WTC collapses started from were located significantly lower than the floors hit by the terrorists and subjected to fire. This conclusion confirms the same former result obtained by using the simple official theory of pure progressive collapse. [ABSTRACT FROM AUTHOR]

We prove that there are groups in the constructible universe whose automorphism towers are highly malleable by forcing. This is a consequence of the fact that, under a suitable diamond hypothesis, there are sufficiently many highly rigid non-isomorphic Souslin trees whose isomorphism relation can be precisely controlled by forcing. [ABSTRACT FROM AUTHOR]

Abstract: Let K be a real quadratic field with 2-class rank equal to 4 or 5 and 4-class rank equal to 3. This paper computes density information for such fields to have infinite Hilbert 2-class field towers. [Copyright &y& Elsevier]

In this article we derive .strong conditions on the defining equations of asymptotically good Artin-Schreier towers. We will show that at most three kinds of defining equations can give rise to a recursively defined good tower, if we restrict ourselves to prime degrees. [ABSTRACT FROM AUTHOR]

In this paper, we construct an infinite family of real quadratic fields k such that the maximal unramified pro-2-extension of the cyclotomic Z2-extension of k is a finite non-abelian extension. [Copyright &y& Elsevier]

Ihara defined the quantity A(q), which is the lim sup as g approaches ∞ of the ratio Nq(g)/g, where Nq(g) is the maximum number of rational points a curve of genus g defined over a finite field Fq may have. A(q) is of great relevance for applications to algebraic–geometric codes. It is known that A(q)&les;√q−1 and equality holds when q is a square. By constructing class field towers with good parameters, in this paper we present improvements of lower bounds of A(q) for q an odd power of a prime. [Copyright &y& Elsevier]