scholarly journals An explicit result for primes between cubes

2016 ◽  
Vol 55 (2) ◽  
pp. 177-197 ◽  
Author(s):  
Adrian W. Dudek
Keyword(s):  
Explicit Result

scholarly journals Cohomological localization of $$ \mathcal{N} $$ = 2 gauge theories with matter

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Guido Festuccia ◽  
Anastasios Gorantis ◽  
Antonio Pittelli ◽  
Konstantina Polydorou ◽  
Lorenzo Ruggeri
Keyword(s):  
Vector Field ◽  
Extended Supersymmetry ◽  
General Framework ◽  
Gauge Theories ◽  
Killing Vector ◽  
Killing Vector Field ◽  
Explicit Result ◽  
Yang Mills ◽  
Self Duality ◽  
Witten Theory

Abstract We construct a large class of gauge theories with extended supersymmetry on four-dimensional manifolds with a Killing vector field and isolated fixed points. We extend previous results limited to super Yang-Mills theory to general $$ \mathcal{N} $$ N = 2 gauge theories including hypermultiplets. We present a general framework encompassing equivariant Donaldson-Witten theory and Pestun’s theory on S4 as two particular cases. This is achieved by expressing fields in cohomological variables, whose features are dictated by supersymmetry and require a generalized notion of self-duality for two-forms and of chirality for spinors. Finally, we implement localization techniques to compute the exact partition function of the cohomological theories we built up and write the explicit result for manifolds with diverse topologies.

scholarly journals ON THE QUANTUM THEORY OF MASSLESS SPIN-3/2 FIELD IN MINKOWSKI SPACETIME

2006 ◽  
Vol 03 (07) ◽  
pp. 1303-1312 ◽  
Author(s):  
WEIGANG QIU ◽  
FEI SUN ◽  
HONGBAO ZHANG
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Quantum Theory ◽  
Casimir Effect ◽  
Minkowski Spacetime ◽  
Quantum Field ◽  
Explicit Result ◽  
Massless Spin ◽  
The Many ◽  
The One

From the modern viewpoint and by the geometric method, this paper provides a concise foundation for the quantum theory of massless spin-3/2 field in Minkowski spacetime, which includes both the one-particle's quantum mechanics and the many-particle's quantum field theory. The explicit result presented here is useful for the investigation of spin-3/2 field in various circumstances such as supergravity, twistor programme, Casimir effect, and quantum inequality.

First passage to a general threshold for a process corresponding to sampling at Poisson times

1971 ◽  
Vol 8 (03) ◽  
pp. 573-588 ◽  
Author(s):  
Barry Belkin
Keyword(s):  
Markov Processes ◽  
Classical Case ◽  
First Passage ◽  
Explicit Result ◽  
Threshold Functions ◽  
Linear Threshold Functions ◽  
Linear Threshold ◽  
Special Cases ◽  
Weiner Process ◽  
Semi Markov Processes

The problem of computing the distribution of the time of first passage to a constant threshold for special classes of stochastic processes has been the subject of considerable study. For example, Baxter and Donsker (1957) have considered the problem for processes with stationary, independent increments, Darling and Siegert (1953) for continuous Markov processes, Mehr and McFadden (1965) for Gauss-Markov processes, and Stone (1969) for semi-Markov processes. The results, however, generally express the first passage distribution in terms of transforms which can be inverted only in a relatively few special cases, such as in the classical case of the Weiner process and for certain stable and compound Poisson processes. For linear threshold functions and processes with non-negative interchangeable increments the first passage problem has been studied by Takács (1957) (an explicit result was obtained by Pyke (1959) in the special case of a Poisson process). Again in the case of a linear threshold, an explicit form for the first passage distribution was found by Slepian (1961) for the Weiner process. For the Ornstein-Uhlenbeck process and certain U-shaped thresholds the problem has recently been studied by Daniels (1969).

An explicit André–Oort type result for

2015 ◽  
Vol 159 (1) ◽  
pp. 153-163 ◽  
Author(s):  
ROLAND PAULIN
Keyword(s):  
Field Theory ◽  
Class Field Theory ◽  
Roots Of Unity ◽  
Type Result ◽  
Explicit Result ◽  
Class Field ◽  
Singular Moduli ◽  
Special Points

AbstractUsing class field theory we prove an explicit result of André–Oort type for $\mathbb{P}^1(\mathbb{C}) \times \mathbb{G}_m(\mathbb{C})$. In this variation the special points of $\mathbb{P}^1(\mathbb{C})$ are the singular moduli, while the special points of $\mathbb{G}_m(\mathbb{C})$ are defined to be the roots of unity.

scholarly journals On Merton’s Problem for Life Insurers

2004 ◽  
Vol 34 (1) ◽  
pp. 5-25 ◽  
Author(s):  
Mogens Steffensen
Keyword(s):  
Durable Goods ◽  
Optimal Investment ◽  
Pension Insurance ◽  
Explicit Result ◽  
Life Insurers ◽  
Utility Optimization ◽  
Special Cases ◽  
Finite State ◽  
Merton's Problem ◽  
Optimal Consumption And Investment

This paper deals with optimal investment and redistribution of the free reserves connected to life and pension insurance contracts in form of dividends and bonus. Formulated appropriately this problem can be viewed as a modification of Merton’s problem of optimal consumption and investment with a very particular form of consumption and utility hereof. Both are linked to a finite state Markov chain. We distinguish between utility optimization of dividends, where a semi-explicit result is obtained, and utility optimization of bonus payments. The latter connects to the financial notion of durable goods and allows for an explicit solution only in very special cases.

scholarly journals Elliptic curve variants of the least quadratic nonresidue problem and Linnik’s theorem

2017 ◽  
Vol 14 (01) ◽  
pp. 255-288
Author(s):  
Evan Chen ◽  
Peter S. Park ◽  
Ashvin A. Swaminathan
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Riemann Hypothesis ◽  
Complex Multiplication ◽  
Symmetric Power ◽  
Explicit Result ◽  
Generalized Riemann Hypothesis ◽  
Quadratic Nonresidue ◽  
Trace Of Frobenius

Let [Formula: see text] and [Formula: see text] be [Formula: see text]-nonisogenous, semistable elliptic curves over [Formula: see text], having respective conductors [Formula: see text] and [Formula: see text] and both without complex multiplication. For each prime [Formula: see text], denote by [Formula: see text] the trace of Frobenius. Assuming the Generalized Riemann Hypothesis (GRH) for the convolved symmetric power [Formula: see text]-functions [Formula: see text] where [Formula: see text], we prove an explicit result that can be stated succinctly as follows: there exists a prime [Formula: see text] such that [Formula: see text] and [Formula: see text] This improves and makes explicit a result of Bucur and Kedlaya. Now, if [Formula: see text] is a subinterval with Sato–Tate measure [Formula: see text] and if the symmetric power [Formula: see text]-functions [Formula: see text] are functorial and satisfy GRH for all [Formula: see text], we employ similar techniques to prove an explicit result that can be stated succinctly as follows: there exists a prime [Formula: see text] such that [Formula: see text] and [Formula: see text]

Uniform bounds for the number of solutions to Yn = f(X)

1986 ◽  
Vol 100 (2) ◽  
pp. 237-248 ◽  
Author(s):  
J.-H. Evertse ◽  
J. H. Silverman
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Solutions To Equations ◽  
Number Of Solutions ◽  
Explicit Result ◽  
Uniform Bounds ◽  
The Past ◽  
Image Position ◽  
Integral Solutions

Let K be an algebraic number field and f(X) ∈ K[X]. The Diophantine problem of describing the solutions to equations of the formhas attracted considerable interest over the past 60 years. Siegel [12], [13] was the first to show that under suitable non-degeneracy conditions, the equation (+) has only finitely many integral solutions in K. LeVeque[7] proved the following, more explicit, result. Letwhere a ∈ K* and αl,…,αk are distinct and algebraic over K. Then (+) has only finitely many integral solutions unless (nl,…,nk) is a permutation of one of the n-tuples

A practical form of Lagrange–Hamilton theory for ideal fluids and plasmas

2003 ◽  
Vol 69 (3) ◽  
pp. 211-252 ◽  
Author(s):  
JONAS LARSSON
Keyword(s):  
Differential Geometry ◽  
Classical Mechanics ◽  
Vector Analysis ◽  
Lie Derivative ◽  
Explicit Result ◽  
Lie Derivatives ◽  
Vector Calculus ◽  
Starting Point ◽  
Ideal Fluids ◽  
Standard Vector

Lagrangian and Hamiltonian formalisms for ideal fluids and plasmas have, during the last few decades, developed much in theory and applications. The recent formulation of ideal fluid/plasma dynamics in terms of the Euler–Poincaré equations makes a self-contained, but mathematically elementary, form of Lagrange–Hamilton theory possible. The starting point is Hamilton's principle. The main goal is to present Lagrange–Hamilton theory in a way that simplifies its applications within usual fluid and plasma theory so that we can use standard vector analysis and standard Eulerian fluid variables. The formalisms of differential geometry, Lie group theory and dual spaces are avoided and so is the use of Lagrangian fluid variables or Clebsch potentials. In the formal ‘axiomatic’ setting of Lagrange–Hamilton theory the concepts of Lie algebra and Hilbert space are used, but only in an elementary way. The formalism is manifestly non-canonical, but the analogy with usual classical mechanics is striking. The Lie derivative is a most convenient tool when the abstract Lagrange–Hamilton formalism is applied to concrete fluid/plasma models. This directional/dynamical derivative is usually defined within differential geometry. However, following the goals of this paper, we choose to define Lie derivatives within standard vector analysis instead (in terms of the directional field and the div, grad and curl operators). Basic identities for the Lie derivatives, necessary for using them effectively in vector calculus and Lagrange–Hamilton theory, are included. Various dynamical invariants, valid for classes of fluid and plasma models (including both compressible and incompressible ideal magnetohydrodynamics), are given simple and straightforward derivations thanks to the Lie derivative calculus. We also consider non-canonical Poisson brackets and derive, in particular, an explicit result for incompressible and inhomogeneous flows.

scholarly journals Refined floor diagrams from higher genera and lambda classes

2021 ◽  
Vol 27 (3) ◽  
Author(s):  
Pierrick Bousseau
Keyword(s):  
Explicit Result ◽  
Change Of Variables ◽  
Hirzebruch Surfaces ◽  
Witten Theory ◽  
Generating Series ◽  
Higher Genus ◽  
Degeneration Formula ◽  
Correspondence Theorem ◽  
Floor Diagrams

AbstractWe show that, after the change of variables $$q=e^{iu}$$ q = e iu , refined floor diagrams for $${\mathbb {P}}^2$$ P 2 and Hirzebruch surfaces compute generating series of higher genus relative Gromov–Witten invariants with insertion of a lambda class. The proof uses an inductive application of the degeneration formula in relative Gromov–Witten theory and an explicit result in relative Gromov–Witten theory of $${\mathbb {P}}^1$$ P 1 . Combining this result with the similar looking refined tropical correspondence theorem for log Gromov–Witten invariants, we obtain a non-trivial relation between relative and log Gromov–Witten invariants for $${\mathbb {P}}^2$$ P 2 and Hirzebruch surfaces. We also prove that the Block–Göttsche invariants of $${\mathbb {F}}_0$$ F 0 and $${\mathbb {F}}_2$$ F 2 are related by the Abramovich–Bertram formula.

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