Showing papers on "Field (mathematics) published in 2016"

Fourier Analysis on Local Fields.

Abstract: This book presents a development of the basic facts about harmonic analysis on local fields and the "n"-dimensional vector spaces over these fields. It focuses almost exclusively on the analogy between the local field and Euclidean cases, with respect to the form of statements, the manner of proof, and the variety of applications.The force of the analogy between the local field and Euclidean cases rests in the relationship of the field structures that underlie the respective cases. A complete classification of locally compact, non-discrete fields gives us two examples of connected fields (real and complex numbers); the rest are local fields (p-adic numbers, p-series fields, and their algebraic extensions). The local fields are studied in an effort to extend knowledge of the reals and complexes as locally compact fields.The author's central aim has been to present the basic facts of Fourier analysis on local fields in an accessible form and in the same spirit as in Zygmund's "Trigonometric Series" (Cambridge, 1968) and in "Introduction to Fourier Analysis on Euclidean Spaces" by Stein and Weiss (1971).Originally published in 1975.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These paperback editions preserve the original texts of these important books while presenting them in durable paperback editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Review of multi-fidelity models

Abstract: Simulations are often computationally expensive and the need for multiple realizations, as in uncertainty quantification or optimization, makes surrogate models an attractive option. For expensive high-fidelity models (HFMs), however, even performing the number of simulations needed for fitting a surrogate may be too expensive. Inexpensive but less accurate low-fidelity models (LFMs) are often also available. Multi-fidelity models (MFMs) combine HFMs and LFMs in order to achieve accuracy at a reasonable cost. With the increasing popularity of MFMs in mind, the aim of this paper is to summarize the state-of-the-art of MFM trends. For this purpose, publications in this field are classified based on application, surrogate selection if any, the difference between fidelities, the method used to combine these fidelities, the field of application and the year published. Available methods of combining fidelities are also reviewed, focusing our attention especially on multi-fidelity surrogate models in which fidelities are combined inside a surrogate model. Computation time savings are usually the reason for using MFMs, hence it is important to properly report the achieved savings. Unfortunately, we find that many papers do not present sufficient information to determine these savings. Therefore, the paper also includes guidelines for authors to present their MFM savings in a way that is useful to future MFM users. Based on papers that provided enough information, we find that time savings are highly problem dependent and that MFM methods we surveyed provided time savings up to 90%. Keywords: Multi-fidelity, Variable-complexity, Variable-fidelity, Surrogate models, Optimization, Uncertainty quantification, Review, Survey

High-temperature asymptotics of supersymmetric partition functions

Abstract: We study the supersymmetric partition function of 4d supersymmetric gauge theories with a U(1) R-symmetry on Euclidean S 3 × S 1 , with S 3 the unit-radius squashed three-sphere, and β the circumference of the circle. For superconformal theories, this partition function coincides (up to a Casimir energy factor) with the 4d superconformal index. The partition function can be computed exactly using the supersymmetric localization of the gauge theory path-integral. It takes the form of an elliptic hypergeometric integral, which may be viewed as a matrix-integral over the moduli space of the holonomies of the gauge fields around S 1 . At high temperatures (β → 0, corresponding to the hyperbolic limit of the elliptic hypergeometric integral) we obtain from the matrix-integral a quantum effective potential for the holonomies. The effective potential is proportional to the temperature. Therefore the high-temperature limit further localizes the matrix-integral to the locus of the minima of the potential. If the effective potential is positive semi-definite, the leading high-temperature asymptotics of the partition function is given by the formula of Di Pietro and Komargodski, and the subleading asymptotics is connected to the Coulomb branch dynamics on R 3 × S 1. In theories where the effective potential is not positive semi-definite, the Di Pietro-Komargodski formula needs to be modified. In particular, this modification occurs in the SU(2) theory of Intriligator-Seiberg-Shenker, and the SO(N ) theory of Brodie-Cho-Intriligator, both believed to exhibit “misleading” anomaly matchings, and both believed to yield interacting superconformal field theories with c < a. Two new simple tests for dualities between 4d supersymmetric gauge theories emerge as byproducts of our analysis.

Holomorphic field realization of SH c and quantum geometry of quiver gauge theories

Abstract: In the context of 4D/2D dualities, SH c algebra, introduced by Schiffmann and Vasserot, provides a systematic method to analyse the instanton partition functions of $$ \mathcal{N}=2 $$ supersymmetricgaugetheories. Inthispaper,werewritetheSH c algebrainterms of three holomorphic fields D 0(z), D ±1(z) with which the algebra and its representations are simplified. The instanton partition functions for arbitrary $$ \mathcal{N}=2 $$ super Yang-Mills theories with A n and A (1) type quiver diagrams are compactly expressed as a product of four building blocks, Gaiotto state, dilatation, flavor vertex operator and intertwiner which are written in terms of SH c and the orthogonal basis introduced by Alba, Fateev, Litvinov and Tarnopolskiy. These building blocks are characterized by new conditions which generalize the known ones on the Gaiotto state and the Carlsson-Okounkov vertex. Consistency conditions of the inner product give algebraic relations for the chiral ring generating functions defined by Nekrasov, Pestun and Shatashvili. In particular we show the polynomiality of the qq-characters which have been introduced as a deformation of the Yangian characters. These relations define a second quantization of the Seiberg-Witten geometry, and, accordingly, reduce to a Baxter TQ-equation in the Nekrasov-Shatashvili limit of the Omega-background.

Squarefree values of polynomial discriminants I

Abstract: We determine the density of monic integer polynomials of given degree $n>1$ that have squarefree discriminant; in particular, we prove for the first time that the lower density of such polynomials is positive. Similarly, we prove that the density of monic integer polynomials $f(x)$, such that $f(x)$ is irreducible and $\mathbb Z[x]/(f(x))$ is the ring of integers in its fraction field, is positive, and is in fact given by $\zeta(2)^{-1}$. It also follows from our methods that there are $\gg X^{1/2+1/n}$ monogenic number fields of degree $n$ having associated Galois group $S_n$ and absolute discriminant less than $X$, and we conjecture that the exponent in this lower bound is optimal.

Repairing Reed-solomon codes

Abstract: A fundamental fact about polynomial interpolation is that k evaluations of a degree-(k-1) polynomial f are sufficient to determine f. This is also necessary in a strong sense: given k-1 evaluations, we learn nothing about the value of f on any k'th point. In this paper, we study a variant of the polynomial interpolation problem. Instead of querying entire evaluations of f (which are elements of a large field F), we are allowed to query partial evaluations; that is, each evaluation delivers a few elements from a small subfield of F, rather than a single element from F. We show that in this model, one can do significantly better than in the traditional setting, in terms of the amount of information required to determine the missing evaluation. More precisely, we show that only O(k) bits are necessary to recover a missing evaluation. In contrast, the traditional method of looking at k evaluations requires Omega(k log(k)) bits. We also show that our result is optimal for linear methods, even up to the leading constants. Our motivation comes from the use of Reed-Solomon (RS) codes for distributed storage systems, in particular for the exact repair problem. The traditional use of RS codes in this setting is analogous to the traditional interpolation problem. Each node in a system stores an evaluation of f, and if one node fails we can recover it by reading k other nodes. However, each node is free to send less information, leading to the modified problem above. The quickly-developing field of regenerating codes has yielded several codes which take advantage of this freedom. However, these codes are not RS codes, and RS codes are still often used in practice; in 2011, Dimakis et al. asked how well RS codes could perform in this setting. Our results imply that RS codes can also take advantage of this freedom to download partial symbols. In some parameter regimes---those with small levels of sub-packetization---our scheme for RS codes outperforms all known regenerating codes. Even with a high degree of sub-packetization, our methods give non-trivial schemes, and we give an improved repair scheme for a specific (14,10)-RS code used in the Facebook Hadoop Analytics cluster.

On parabolic induction on inner forms of the general linear group over a non-archimedean local field

Abstract: We give new criteria for the irreducibility of parabolic induction on the general linear group and its inner forms over a local non-archimedean field. In particular, we give a necessary and sufficient condition when the inducing data is of the form \(\pi \otimes \sigma \) where \(\pi \) is a ladder representation and \(\sigma \) is an arbitrary irreducible representation. As an application we simplify the proof of the classification of the unitary dual.

Finite element approximations of symmetric tensors on simplicial grids in ℝn: The lower order case

Abstract: In this paper, we construct, in a unified fashion, lower order finite element subspaces of spaces of symmetric tensors with square-integrable divergence on a domain in any dimension. These subspaces are essentially the symmetric tensor finite element spaces of order k from [Finite element approximations of symmetric tensors on simplicial grids in ℝn: The higher order case, J. Comput. Math. 33 (2015) 283–296], enriched, for each (n − 1)-dimensional simplex, by (n+1)n 2 face bubble functions in the symmetric tensor finite element space of order n + 1 from [Finite element approximations of symmetric tensors on simplicial grids in ℝn: The higher order case, J. Comput. Math. 33 (2015) 283–296] when 1 ≤ k ≤ n − 1, and by (n−1)n 2 face bubble functions in the symmetric tensor finite element space of order n + 1 from [Finite element approximations of symmetric tensors on simplicial grids in ℝn: The higher order case, J. Comput. Math. 33 (2015) 283–296] when k = n. These spaces can be used to approximate the symmetric matrix field in a mixed formulation problem where the other variable is approximated by discontinuous piecewise Pk−1 polynomials. This in particular leads to first-order mixed elements on simplicial grids with total degrees of freedom per element 18 plus 3 in 2D, 48 plus 6 in 3D. The previous record of the degrees of freedom of first-order mixed elements is, 21 plus 3 in 2D, and 156 plus 6 in 3D, on simplicial grids. We also derive, in a unified way which is completely different from those used in [D. Arnold, G. Awanou and R. Winther, Finite elements for symmetric tensors in three dimensions, Math. Comput. 77 (2008) 1229–1251; D. N. Arnold and R. Winther, Mixed finite element for elasticity, Number Math. 92 (2002) 401–419], a family of Arnold–Winther mixed finite elements in any space dimension. One example in this family is the Raviart–Thomas elements in one dimension, the second example is the mixed finite elements for linear elasticity in two dimensions due to Arnold and Winther, the third example is the mixed finite elements for linear elasticity in three dimensions due to Arnold, Awanou and Winther.

A classification of irreducible admissible mod p representations of p-adic reductive groups

Abstract: Let F be a locally compact non-archimedean field, p its residue characteristic, and G a connected reductive group over F. Let C an algebraically closed field of characteristic p. We give a complete classification of irreducible admissible C-representations of G = G(F), in terms of supercuspidal C-representations of the Levi subgroups of G, and parabolic induction. Thus we push to their natural conclusion the ideas of the third-named author, who treated the case G = GL_m, as further expanded by the first-named author, who treated split groups G. As in the split case, we first get a classification in terms of supersingular representations of Levi subgroups, and as a consequence show that supersingularity is the same as supercuspidality.

Fun with Fields

Abstract: Author(s): Johnson, William Andrew | Advisor(s): Scanlon, Thomas | Abstract: This dissertation is a collection of results in model theory, related in one way or another to fields, NIP theories, and elimination of imaginaries. The most important result is a classification of dp-minimal fields, presented in Chapter 9. We construct in a canonical fashion a non-trivial Hausdorff definable field topology on any unstable dp-minimal field. Usingthis we classify the dp-minimal pure fields and valued fields up to elementary equivalence. Furthermore we prove that every VC-minimal field is real closed or algebraically closed.In Chapter 11, we analyze the theories of existentially closed fields with several valuations and orderings, as studied by van den Dries. We show that these model complete theories are NTP2, and analyze forking, dividing, and burden in these theories. The theory of algebraically closed fields with n independent valuation rings turns out to be an example of such a theory. This provides a new and natural example of an NTP2 theory which is neithersimple nor NIP, nor even a conceptual hybrid of something simple and something NIP.In Chapter 8, we exhibit a bad failure of elimination of imaginaries in a dense o-minimal structure. We produce an exotic interpretable set which cannot be put in definable bijection with a definable set, after naming any amount of parameters. However, we show that these exotic interpretable sets are still amenable to some of the tools of tame topology: they mustadmit nice definable topologies locally homeomorphic to definable sets.Chapter 12 proves the existence of Z/nZ-valued definable strong Euler characteristics on pseudofinite fields, which measure the non-standard “size” of definable sets, mod n. The non-trivial result is that these “sizes” are definable in families of definable sets. This could probably be proven using etale cohomology, but we give a more elementary proof relying heavily on the theory of abelian varieties.We also present simplified and new proofs of several model-theoretic facts, including the definability of irreducibility and Zariski closure in ACF (Chapter 10), and elimination of imaginaries in ACVF (Chapter 6). This latter fact was originally proven by Haskell, Hrushovski, and Macpherson. We give a proof that is drastically simpler, inspired by Poizat’s proofs of elimination of imaginaries in ACF and DCF.

Matrices and Division by Zero z/0 = 0

Abstract: In this paper, a new viewpoint of the division by zero z/0 = 0 in matrices is introduced and the results will show that the division by zero is our elementary and fundamental mathematics. New and practical meanings for many mathematical and physical formulas for the denominator zero cases may be given. Furthermore, a new space idea for the point at infinity for the Eucleadian plane is also introduced.

On the Genericity of Maximum Rank Distance and Gabidulin Codes

Abstract: We consider linear rank-metric codes in $\mathbb F_{q^m}^n$. We show that the properties of being MRD (maximum rank distance) and non-Gabidulin are generic over the algebraic closure of the underlying field, which implies that over a large extension field a randomly chosen generator matrix generates an MRD and a non-Gabidulin code with high probability. Moreover, we give upper bounds on the respective probabilities in dependence on the extension degree $m$.

Matrix factorizations and cohomological field theories

Abstract: We give a purely algebraic construction of a cohomological field theory associated with a quasihomogeneous isolated hypersurface singularity W and a subgroup G of the diagonal group of symmetries of W. This theory can be viewed as an analogue of the Gromov-Witten theory for an orbifoldized Landau-Ginzburg model for W/G. The main geometric ingredient for our construction is provided by the moduli of curves with W-structures introduced by Fan, Jarvis and Ruan. We construct certain matrix factorizations on the products of these moduli stacks with affine spaces which play a role similar to that of the virtual fundamental classes in the Gromov-Witten theory. These matrix factorizations are used to produce functors from the categories of equivariant matrix factorizations to the derived categories of coherent sheaves on the Deligne-Mumford moduli stacks of stable curves. The structure maps of our cohomological field theory are then obtained by passing to the induced maps on Hochschild homology. We prove that for simple singularities a specialization of our theory gives the cohomological field theory constructed by Fan, Jarvis and Ruan using analytic tools.

The motive of a classifying space

Abstract: The Chow group of algebraic cycles generally does not satisfy the Kunneth formula. Nonetheless, there are some schemes X over a field k which satisfy the Chow Kunneth property that the product CH∗X ⊗Z CH∗Y → CH∗(X ×k Y ) is an isomorphism for all separated schemes Y of finite type over k. The Chow Kunneth property implies the weak Chow Kunneth property that CH∗X → CH∗(XF ) is surjective for every finitely generated field F over k (or, equivalently, for every extension field F of k). We characterize several properties of this type. (We also prove versions of all our results with coefficients in a given commutative ring.) Our characterizations of Kunneth properties are: first, a smooth proper scheme X over k satisfies the weak Chow Kunneth property if and only if the Chow motive of X is a summand of a finite direct sum of Tate motives (Theorem 4.1). (This is related to known results by Bloch, Srinivas, Jannsen, Kimura, and others.) A more novel result is about an arbitrary separated scheme X of finite type over k. We say that X satisfies the motivic Kunneth property if the Kunneth spectral sequence converges to the motivic homology groups of X ×k Y for all Y . (Motivic homology groups are also called higher Chow groups; they include the usual Chow groups as a special case.) We show that a k-scheme X satisfies the motivic Kunneth property if and only if the motive of X in Voevodsky’s derived category of motives is a mixed Tate motive (Theorem 7.2). (An example of a scheme with these properties is any linear scheme, as discussed in section 5.) Finally, if a smooth but not necessarily proper k-variety X satisfies the weak Chow Kunneth property, then the birational motive of X in the sense of Rost and Kahn-Sujatha is isomorphic to the birational motive of a point (Corollary 2.2). The last result cannot be strengthened to say that the motive of X is mixed Tate; one has to consider motivic homology groups to get that conclusion. For example, the complement X of a curve of genus 1 in the affine plane has the Chow Kunneth property, since CHi+2(X ×k Y ) ∼= CHiY for all separated k-schemes Y of finite type and all i; but the motive of X is not mixed Tate. As an application of these general results, we disprove the weak Chow Kunneth property for some classifying spaces BG. For an affine group scheme G of finite type over a field k, Morel-Voevodsky and I constructed BG as a direct limit of smooth k-varieties, quotients by G of open subsets of representations of G over k [41, section 4.2], [55, 57]. As a result, the Chow ring of BG makes sense. The Chow ring of BG tensored with the rationals is easy to compute; for example, if G is finite, then CH i(BG)⊗Q = 0 for i > 0. The challenge is to understand the integral or mod p Chow ring of BG. For many finite groups G and fields k, the classifying space BG over k satisfies the Chow Kunneth property that CH∗BG ⊗Z CH∗Y ∼= CH∗(BG ×k Y ) for all separated k-schemes Y of finite type. For example, an abelian p-groupG of exponent

On certain pure sextic fields related to a problem of Hasse

Abstract: An algebraic number ring is monogenic, or one-generated, if it has the form Z[α] for a single algebraic integer α. It is a problem of Hasse to characterize, whether an algebraic number ring is monogenic or not. In this note, we prove that if m is a square-free rational integer, m ≡ 1(mod 4) and m≢ ± 1(mod 9), then the pure sextic field L = Q(m6) is not monogenic. Our results are illustrated by examples.

The algebra and model theory of tame valued fields

Abstract: A henselian valued field K is called a tame field if its algebraic closure K is a tame extension, that is, the ramification field of the normal extension K|K is algebraically closed. Every algebraically maximal Kaplansky field is a tame field, but not conversely. We develop the algebraic theory of tame fields and then prove Ax–Kochen– Ershov Principles for tame fields. This leads to model completeness and completeness results relative to value group and residue field. As the maximal immediate extensions of tame fields will in general not be unique, the proofs have to use much deeper valuation theoretical results than those for other classes of valued fields which have already been shown to satisfy Ax–Kochen–Ershov Principles. The results of this paper have been applied to gain insight in the Zariski space of places of an algebraic function field, and in the model theory of large fields.

Behavior of canonical divisors under purely inseparable base changes

Abstract: Let $k$ be an imperfect field. Let $X$ be a regular variety over $k$ and set $Y$ to be the normalization of $(X \times_k k^{1/p^{\infty}})_{{\rm red}}$. In this paper, we show that $K_Y+C=f^*K_X$ for some effective divisor $C$ on $Y$. We obtain the following three applications. First, we show that a $K_X$-trivial fiber space with non-normal fibers is uniruled. Second, we prove that general fibers of Mori fiber spaces are rationally chain connected. Third, we obtain a weakening of the cone theorem for surfaces and threefolds defined over an imperfect field.

Two-dimensional RCFT’s without Kac-Moody symmetry

Abstract: Using the method of modular-invariant differential equations, we classify a family of Rational Conformal Field Theories with two and three characters having no Kac-Moody algebra. In addition to unitary and non-unitary minimal models, we find “dual” theories whose characters obey bilinear relations with those of the minimal models to give the Moonshine Module. In some ways this relation is analogous to cosets of meromorphic CFT’s. The theory dual in this sense to the Ising model has central charge $$ \frac{47}{2} $$ and is related to the Baby Monster Module.

Coregular spaces and genus one curves

Abstract: A coregular space is a representation of an algebraic group for which the ring of polynomial invariants is free. In this paper, we show that the orbits of many coregular irreducible representations where the number of invariants is at least two, over a (not necessarily algebraically closed) field k, correspond to genus one curves over k together with line bundles, vector bundles, and/or points on their Jacobians. In forthcoming work, we use these orbit parametrizations to determine the average sizes of Selmer groups for various families of elliptic curves.

Geometricity for derived categories of algebraic stacks

Abstract: We prove that the dg category of perfect complexes on a smooth, proper Deligne–Mumford stack over a field of characteristic zero is geometric in the sense of Orlov, and in particular smooth and proper. On the level of triangulated categories, this means that the derived category of perfect complexes embeds as an admissible subcategory into the bounded derived category of coherent sheaves on a smooth, projective variety. The same holds for a smooth, projective, tame Artin stack over an arbitrary field.

Varieties of Algebras with Superinvolution of Almost Polynomial Growth

Abstract: Let A be an associative algebra with superinvolution ∗ over a field of characteristic zero and let $c_{n}^{\ast }(A)$ be its sequence of corresponding ∗-codimensions. In case A is finite dimensional, we prove that such sequence is polynomially bounded if and only if the variety generated by A does not contain three explicitly described algebras with superinvolution. As a consequence we find out that no intermediate growth of the ∗-codimensions between polynomial and exponential is allowed.

Local Theta correspondence of Tempered Representations and Langlands parameters

Abstract: In this paper, we give an explicit determination of the theta lifting for symplectic-orthogonal and unitary dual pairs over a nonarchimedean field $F$ of characteristic $0$. We determine when theta lifts of tempered representations are nonzero, and determine the theta lifts in terms of the local Langlands correspondence.

Reductive group actions

Abstract: In this paper, we study rationality properties of reductive group actions which are defined over an arbitrary field of characteristic zero. Thereby, we unify Luna's theory of spherical systems and Borel-Tits' theory of reductive groups. In particular, we define for any reductive group action a generalized Tits index whose main constituents are a root system and a generalization of the anisotropic kernel. The index controls to a large extent the behavior at infinity (i.e., embeddings). For k-spherical varieties (i.e., where a minimal parabolic has an open orbit) we obtain explicit (wonderful) completions of the set of rational points. For local fields this means honest compactifications generalizing the maximal Satake compactification of a symmetric space. Our main tool is a k-version of the local structure theorem.

Rigid geometry of curves and their Jacobians

Abstract: This book presents some of the most important aspects of rigid geometry, namely its applications to the study of smooth algebraic curves, of their Jacobians, and of abelian varieties - all of them defined over a complete non-archimedean valued field. The text starts with a survey of the foundation of rigid geometry, and then focuses on a detailed treatment of the applications. In the case of curves with split rational reduction there is a complete analogue to the fascinating theory of Riemann surfaces. In the case of proper smooth group varieties the uniformization and the construction of abelian varieties are treated in detail. Rigid geometry was established by John Tate and was enriched by a formal algebraic approach launched by Michel Raynaud. It has proved as a means to illustrate the geometric ideas behind the abstract methods of formal algebraic geometry as used by Mumford and Faltings. This book should be of great use to students wishing to enter this field, as well as those already working in it.

On the uniqueness of weak solution for the 2-D Ericksen--Leslie system

Abstract: In this paper, we prove the uniqueness of weak solutions to the two dimensional full Ericksen-Leslie system with the Leslie stress and general Ericksen stress under the physical constrains on the Leslie coefficients. This question remains unknown even in the case when the Leslie stress is vanishing. The main technique used in the proof is Littlewood-Paley analysis performed in a very delicate way. Different from the earlier result in [28], we introduce a new metric and explore the algebraic structure of the molecular field.

Galois groups and cohomological functors

Abstract: Let $q=p^s$ be a prime power, $F$ a field containing a root of unity of order $q$, and $G_F$ its absolute Galois group. We determine a new canonical quotient $\mathrm{Gal}(F_{(3)}/F)$ of $G_F$ which encodes the full mod-$q$ cohomology ring $H^*(G_F,\mathbb{Z}/q)$ and is minimal with respect to this property. We prove some fundamental structure theorems related to these quotients. In particular, it is shown that when $q=p$ is an odd prime, $F_{(3)}$ is the compositum of all Galois extensions $E$ of $F$ such that $\mathrm{Gal}(E/F)$ is isomorphic to $\{1\}$, $\mathbb{Z}/p$ or to the nonabelian group $H_{p^3}$ of order $p^3$ and exponent $p$.