• Volume 127, Issue 1

      February 2017,   pages  1-201

    • Invariant generalized ideal classes - structure theorems for $p$-class groups in $p$-extensions

      Georges Gras

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      We give, in sections 2 and 3, an english translation of: Classes g\acute{e}n\acute{e}ralis\acute{e}es invariantes, J. Math. Soc. Japan, 46, 3 (1994), with some improvements and with notations and definitions in accordance with our book: Class Field Theory: From Theory to Practice, SMM, Springer-Verlag, 2nd corrected printing 2005. We recall, in section 4, some structure theorems for finite $\mathbb{Z}_p[G]$-modules ($G\simeq\mathbb{Z}/p\mathbb{Z}$) obtained in: Sur les$\scr l$-classes d’id\acute{e}aux dans les extensions cycliques relatives de degr\acute{e} premier $\mathcal{l}$, Annales de l’Institut Fourier, 23, 3 (1973). Then we recall the algorithm of local normic computations which allows to obtain the order and (potentially) the structure of a $p$-class group in a cyclic extension of degree $p$. In section 5, we apply this to the study of the structure of relative $p$-class groups of Abelian extensions of prime to $p$ degree, using the Thaine–Ribet–Mazur–Wiles–Kolyvagin ‘principal theorem’, and the notion of ‘admissible sets of prime numbers’ in a cyclic extension of degree $p$, from: Sur la structure des groupes de classes relatives, Annales de l’Institut Fourier, 43, 1 (1993). In conclusion, we suggest the study, in the same spirit, of some deep invariants attached to the $p$-ramification theory (as dual form of non-ramification theory) and which have become standard in a $p$-adic framework. Since some of these techniques have often been rediscovered, we give a substantial (but certainly incomplete) bibliography which may be used to have a broad view on the subject.

    • Primes of the form $x^2+dy^2$ with $x\equiv 0({\rm mod}\, N)$ or $y\equiv 0({\rm mod}\, N)$

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      In this paper, we characterize primes of the form $x^2+dy^2$ with $x\equiv 0({\rm mod}\, N)$ or $y\equiv 0({\rm mod}\, N)$ for positive integers $N$ and $d$ with $d$ being square free.

    • Some irreducibility and indecomposability results for truncated binomial polynomials of small degree

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      In this paper, we show that the truncated binomial polynomials defined by $P_{n, k}(x)=\sum^k_{j=0}({n\choose j})x^j$ are irreducible for each $k\leq 6$ and every $n\geq k+2$. Under the same assumption $n\geq k+2$, we also show that the polynomial $P_{n, k}$ cannot be expressed as a composition $P_{n, k}(x)=g(h(x))$ with $g\in\mathbb{Q}[x]$ of degree at least 2 and a quadratic polynomial $h\in\mathbb{Q}[x]$. Finally, we show that for $k\geq 2$ and $m, n\geq k+1$ the roots of the polynomial $P_{m, k}$ cannot be obtained from the roots of $P_{n, k}$, where $m\neq n$, by a linear map.

    • Projections of Veronese surface and morphisms from projective plane to Grassmannian

      A El Mazouni F Laytimi D S Nagaraj

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      In this note, we describe the image of $\mathbb{P}^2$ in ${\rm Gr}(2, \mathbb{C}^4)$ under a morphism given by a rank two vector bundle on $\mathbb{P}^2$ with Chern classes (2, 2).

    • Approximation of quadrilaterals by rational quadrilaterals in the plane

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      Many questions about triangles and quadrilaterals with rational sides, diagonals and areas can be reduced to solving certain diophantine equations. We look at a number of such questions including the question of approximating arbitrary triangles and quadrilaterals by those with rational sides, diagonals and areas. We transform these problems into questions on the existence of infinitely many rational solutions on a two parameter family of quartic curves. This is further transformed to a two parameter family of elliptic curves to deduce our main result concerning density of points on a line which are at a rational distance from three collinear points (Theorem 4). We deducefrom this a new proof of density of rational quadrilaterals in the space of all quadrilaterals (Theorem 39). The other main result (Theorem 3) of this article is on the density of rational triangles which is related to analyzing rational points on the unit circle. Interestingly, this enables us to deduce that parallelograms with rational sides and area are dense in the class of all parallelograms. We also give a criterion for density of certain sets in topological spaces using local product structure and prove the density Theorem 6 in the appendix section. An application of this proves the density of rational points as stated in Theorem 31.

    • Commutators with idempotent values on multilinear polynomials in prime rings

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      Let $R$ be a prime ring of characteristic different from 2, $C$ its extended centroid, $d$ a nonzero derivation of $R$, $f (x_1,\ldots , x_n)$ a multilinear polynomial over $C$, $\varrho$ a nonzero right ideal of $R$ and $m > 1$ a fixed integer such that $$([d(f (r_1, \ldots , r_n)), f (r_1, \ldots , _n)])^m = [d(f (r_1, \ldots , r_n)), f(r_1, \ldots , r_n)]$$ for all $r_1, \ldots , r_n \in\varrho$. Then either $[f(x_1, \ldots , x_n), x_{n+1}]x_{n+2}$ is an identity for $\varrho$ or $d(\varrho)\varrho = 0$.

    • Palindromic widths of nilpotent and wreath products

      Valeriy G Bardakov Oleg V Bryukhanov Krishnendu Gongopadhyay

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      We prove that the nilpotent product of a set of groups $A_1, \ldots , A_s$ has finite palindromic width if and only if the palindromic widths of $A_i$, $i = 1, \ldots , s$, are finite. We give a new proof that the commutator width of $F_n \wr K$ is infinite, where $F_n$ is a free group of rank $n\geq 2$ and $K$ is a finite group. This result, combining with a result of Fink [9] gives examples of groups with infinite commutator width but finite palindromic width with respect to some generating set.

    • Isometric multipliers of a vector valued Beurling algebra on a discrete semigroup

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      Let $(S, \omega)$ be a weighted abelian semigroup, let $M_\omega(S)$ be the semigroup of $\omega$-bounded multipliers of $S$, and let $\mathcal{A}$ be a strictly convex commutative Banach algebra with identity. It is shown that $T$ is an onto isometric multiplier of $\mathcal{l}^1(S, \omega, \mathcal{A})$ if and only if there exists an invertible $\sigma\in M_\omega(S)$, a unitary point $a\in\mathcal{A}$, and a $k > 0$ such that $T(f) = ka\sum_{x\in S}f(x)\delta_{\sigma (x)}$ for each $f = \sum_{x\in S}f(x)\delta_x\in\mathcal{l}^1(S, \omega, \mathcal{A})$. It is also shown that an isomorphism from $\mathcal{l}^1(S_1, \omega_1, \mathcal{A})$ onto $\mathcal{l}^1(S_2, \omega_2, \mathcal{B})$ induces an isomorphism from $M(\mathcal{l}^1(S_1, \omega_1, \mathcal{A}))$, the set of all multipliers of $\mathcal{l}^1(S_1, \omega_1, \mathcal{A})$, onto $M(\mathcal{l}^1(S_2, \omega_2, \mathcal{B}))$.

    • On Pimsner-Popa bases

      Keshab Chandra Bakshi

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      In this paper, we examine bases for finite index inclusion of ${\rm II}_1$ factors and connected inclusion of finite dimensional $C^\ast$-algebras. These bases behave nicely with respect to basic construction towers. As applications we have studied automorphisms of the hyperfinite ${\rm II}_1$ factor $R$ which are ‘compatible with respect to the Jones’ tower of finite dimensional $C^\ast$-algebras’. As a further application, in both cases we obtain a characterization, in terms of bases, of basic constructions. Finally we use these bases to describe the phenomenon of multistep basic constructions (in both the cases).

    • Quantum quaternion spheres

      Bipul Saurabh

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      For the quantum symplectic group $SP_q(2n)$, we describe the $C^\ast$-algebra of continuous functions on the quotient space $SP_q(2n)/SP_q(2n − 2)$ as an universal $C^\ast$-algebra given by a finite set of generators and relations. The proof involves a careful analysis of the relations, and use of the branching rules for representations of the symplectic group due to Zhelobenko. We then exhibit a set of generators of the $K$-groups of this $C^\ast$-algebra in terms of generators of the $C^\ast$-algebra.

    • Weak convergence of the past and future of Brownian motion given the present

      K B Athreya B Rajeev

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      In this paper, we show that for $t > 0$, the joint distribution of the past $\{W_{t−s} : 0 \leq s \leq t\}$ and the future $\{W_{t+s} : s \geq 0\}$ of a $d$-dimensional standard Brownian motion $(W_s)$, conditioned on $\{W_t\in U\}$, where $U$ is a bounded open set in $\mathbb{R}^d$, converges weakly in $C[0,\infty)\times C[0, \infty)$ as $t\rightarrow\infty$. The limiting distribution is that of a pair of coupled processes $Y + B^1$, $Y + B^2$ where $Y$, $B^1$, $B^2$ are independent, $Y$ is uniformly distributed on $U$ and $B^1$, $B^2$ are standard $d$-dimensional Brownian motions. Let $\sigma_t$, $d_t$ be respectively, the last entrance time before time $t$ into the set $U$ and the first exit time after $t$ from $U$. When the boundary of $U$ is regular, we use the continuous mapping theorem to show that the limiting distribution as $t\rightarrow\infty$ of the four dimensional vector with components $(W_{\sigma_t}, t − \sigma_t, W_{d_t}, d_t − t)$, conditioned on $\{W_t\in U\}$, is the same as that of the four dimensional vector whose components are the place and time of first exit from $U$ of the processes $Y + B^1$ and $Y + B^2$ respectively.

    • A variational approach to estimate incompressible fluid flows

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      A variational approach is used to recover fluid motion governed by Stokes and Navier–Stokes equations. Unlike previous approaches where optical flow method is used to track rigid body motion, this new framework aims at investigating incompressible flows using optical flow techniques. We formulate a minimization problem and determine conditions under which unique solution exists. Numerical results using finite element method not only support theoretical results but also show that Stokes flow forced by a potential are recovered almost exactly.

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