The Splitting Condition for The Weighted -Complex

AbstractWe generalise the classical Bombieri–Vinogradov theorem for short intervals to a non-abelian setting. This leads to variants of the prime number theorem for short intervals where the primes lie in arithmetic progressions that are “twisted” by a splitting condition in a Galois extension of number fields. Using this result in conjunction with the recent work of Maynard, we prove that rational primes with a given splitting condition in a Galois extensionL/$\mathbb{Q}$exhibit bounded gaps in short intervals. We explore several arithmetic applications related to questions of Serre regarding the non-vanishing Fourier coefficients of cuspidal modular forms. One such application is that for a given modularL-functionL(s, f), the fundamental discriminantsdfor which thed-quadratic twist ofL(s, f) has a non-vanishing central critical value exhibit bounded gaps in short intervals.

MULTIBUMP SOLUTIONS OF NONLINEAR PERIODIC SCHRÖDINGER EQUATIONS IN A DEGENERATE SETTING

Communications in Contemporary Mathematics ◽

10.1142/s0219199705001763 ◽

2005 ◽

Vol 07 (03) ◽

pp. 269-298 ◽

Cited By ~ 21

Author(s):

NILS ACKERMANN ◽

TOBIAS WETH

Keyword(s):

Schrodinger Equations ◽

Schrödinger Equations ◽

New Approach ◽

Nodal Domains ◽

Splitting Condition

We prove the existence of infinitely many geometrically distinct two bump solutions of periodic superlinear Schrödinger equations of the type -Δu + V(x)u = f(x,u), where x ∈ ℝN and lim |x| → ∞u(x) = 0. The solutions we construct change sign and have exactly two nodal domains. The usual multibump constructions for these equations rely on strong non-degeneracy assumptions. We present a new approach that only requires a weak splitting condition. In the second part of the paper we exhibit classes of potentials V for which this splitting condition holds.

Betti splitting from a topological point of view

Journal of Algebra and Its Applications ◽

10.1142/s0219498820501169 ◽

2019 ◽

Vol 19 (06) ◽

pp. 2050116

Author(s):

Davide Bolognini ◽

Ulderico Fugacci

Keyword(s):

Monomial Ideal ◽

Betti Numbers ◽

Simplicial Complexes ◽

Point Of View ◽

Topological Spaces ◽

Topological Properties ◽

Topological Approach ◽

Graded Betti Numbers ◽

Alexander Duality ◽

Splitting Condition

A Betti splitting [Formula: see text] of a monomial ideal [Formula: see text] ensures the recovery of the graded Betti numbers of [Formula: see text] starting from those of [Formula: see text] and [Formula: see text]. In this paper, we introduce an analogous notion for simplicial complexes, using Alexander duality, proving that it is equivalent to a recursive splitting condition on links of some vertices. We provide results ensuring the existence of a Betti splitting for a simplicial complex [Formula: see text], relating it to topological properties of [Formula: see text]. Among other things, we prove that orientability for a manifold without boundary is equivalent to the admission of a Betti splitting induced by the removal of a single facet. Taking advantage of our topological approach, we provide the first example of a monomial ideal which admits Betti splittings in all characteristics but with characteristic-dependent resolution. Moreover, we introduce new numerical descriptors for simplicial complexes and topological spaces, useful to deal with questions concerning the existence of Betti splitting.

Mechanics and Acoustic Emission Fractal Characteristics of Surrounding Rock of Tantalum–Niobium Mine Under Splitting Condition

Natural Resources Research ◽

10.1007/s11053-021-09967-6 ◽

2021 ◽

Author(s):

Kang Zhao ◽

Yufeng Song ◽

Yajing Yan ◽

Shuijie Gu ◽

Junqiang Wang ◽

...

Keyword(s):

Acoustic Emission ◽

Surrounding Rock ◽

Fractal Characteristics ◽

Splitting Condition

A splitting condition using block theory.

10.1307/mmj/1029001713 ◽

1976 ◽

Vol 23 (3) ◽

pp. 203-206 ◽

Cited By ~ 1

Author(s):

Stephen M. Gagola

Keyword(s):

Block Theory ◽

Splitting Condition

Beam-Splitting Condition in a Broadside Symmetric Leaky-Wave Antenna of Finite Length

IEEE Antennas and Wireless Propagation Letters ◽

10.1109/lawp.2008.2002906 ◽

2008 ◽

Vol 7 ◽

pp. 609-612 ◽

Cited By ~ 18

Author(s):

A. Sutinjo ◽

M. Okoniewski ◽

R.H. Johnston

Keyword(s):

Finite Length ◽

Beam Splitting ◽

Leaky Wave ◽

Leaky Wave Antenna ◽

Wave Antenna ◽

Splitting Condition

Ray Optics Analysis Explanation of Beam-Splitting Condition in Fabry-Pérot Antennas

IEEE Access ◽

10.1109/access.2018.2839201 ◽

2018 ◽

Vol 6 ◽

pp. 32360-32366

Author(s):

Etienne Perret ◽

Raul Guzman Quiros

Keyword(s):

Beam Splitting ◽

Ray Optics ◽

Fabry Perot ◽

Splitting Condition

On the Ledrappier–Young formula for self-affine measures

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004115000419 ◽

2015 ◽

Vol 159 (3) ◽

pp. 405-432 ◽

Cited By ~ 13

Author(s):

BALÁZS BÁRÁNY

Keyword(s):

Iterated Function System ◽

Sufficient Conditions ◽

Lyapunov Dimension ◽

Dominated Splitting ◽

Triangular Matrices ◽

Iterated Function ◽

Strong Separation ◽

Strong Separation Condition ◽

Lower Triangular Matrices ◽

Splitting Condition

AbstractLedrappier and Young introduced a relation between entropy, Lyapunov exponents and dimension for invariant measures of diffeomorphisms on compact manifolds. In this paper, we show that a self-affine measure on the plane satisfies the Ledrappier–Young formula if the corresponding iterated function system (IFS) satisfies the strong separation condition and the linear parts satisfy the dominated splitting condition. We give sufficient conditions, inspired by Ledrappier and by Falconer and Kempton, that the dimensions of such a self-affine measure is equal to the Lyapunov dimension. We show some applications, namely, we give another proof for Hueter–Lalley's theorem and we consider self-affine measures and sets generated by lower triangular matrices.