scholarly journals On dimensions supporting a rational projective plane

2019 ◽  
Vol 11 (03) ◽  
pp. 535-555 ◽  
Author(s):  
Lee Kennard ◽  
Zhixu Su
Keyword(s):  
Projective Plane ◽  
Open Problem ◽  
Sufficient Conditions ◽  
Quadratic Residue ◽  
Bernoulli Numbers ◽  
Existence Results ◽  
Necessary And Sufficient ◽  
Simply Connected ◽  
Existence Question ◽  
Smooth Closed Manifold

A rational projective plane ([Formula: see text]) is a simply connected, smooth, closed manifold [Formula: see text] such that [Formula: see text]. An open problem is to classify the dimensions at which such a manifold exists. The Barge–Sullivan rational surgery realization theorem provides necessary and sufficient conditions that include the Hattori–Stong integrality conditions on the Pontryagin numbers. In this paper, we simplify these conditions and combine them with the signature equation to give a single quadratic residue equation that determines whether a given dimension supports a [Formula: see text]. We then confirm the existence of a [Formula: see text] in two new dimensions and prove several non-existence results using factorization of the numerators of the divided Bernoulli numbers. We also resolve the existence question in the Spin case, and we discuss existence results for the more general class of rational projective spaces.

scholarly journals The pseudoidentity problem and reducibility for completely regular semigroups

2001 ◽  
Vol 63 (3) ◽  
pp. 407-433 ◽  
Author(s):  
Jorge Almedia ◽  
Peter G. Trotter
Keyword(s):  
Word Problem ◽  
Finite Groups ◽  
Open Problem ◽  
Sufficient Conditions ◽  
90Th Birthday ◽  
Regular Semigroups ◽  
Completely Regular ◽  
Strengthened Version ◽  
Completely Regular Semigroups ◽  
Necessary And Sufficient

Dedicated to George Szekeres on the occasion of his 90th birthdayNecessary and sufficient conditions for equality over the pseudovariety CR of all finite completely regular semigroups are obtained. They are inspired by the solution of the word problem for free completely regular semigroups and clarify the role played by groups in the structure of such semigroups. A strengthened version of Ash's inevitability theorem (κ-reducibility of the pseudovariety G of all finite groups) is proposed as an open problem and it is shown that, if this stronger version holds, then CR is also κ-reducible and, therefore, hyperdecidable.

An Inverse Eigenvalue Problem for Spring-Mass Systems with Partial Mass Connected to the Ground

2013 ◽  
Vol 353-356 ◽  
pp. 3308-3311
Author(s):  
Xia Tian ◽  
Chuan Xiao Li
Keyword(s):  
Total Mass ◽  
Sufficient Conditions ◽  
Inverse Eigenvalue Problem ◽  
Spring Stiffness ◽  
Positive Mass ◽  
Mass System ◽  
Inverse Eigenvalue ◽  
Physical Elements ◽  
Necessary And Sufficient ◽  
Simply Connected

Consider the simply connected spring-mass system with partial mass connected to the ground. The inverse mode problem of constructing the physical elements of the system from two eigenpairs, the grounding spring stiffness and total mass of the system is considered. The necessary and sufficient conditions for constructing a physical realizable system with positive mass and stiffness elements are established. If these conditions are satisfied, the grounding spring-mass system may be constructed uniquely. The numerical methods and examples are given finally.

scholarly journals A few remarks on the values of the Bernoulli polynomials at rational arguments and some relations with ζ(2k + 1)

2021 ◽  
Vol 27 (4) ◽  
pp. 180-186
Author(s):  
André Pierro de Camargo ◽  
◽  
Giulliano Cogui de Oliveira Teruya ◽  
Keyword(s):  
Fourier Expansion ◽  
Sufficient Conditions ◽  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Necessary And Sufficient Conditions ◽  
Rational Multiple ◽  
Necessary And Sufficient ◽  
Rational Arguments

A problem posed by Lehmer in 1938 asks for simple closed formulae for the values of the even Bernoulli polynomials at rational arguments in terms of the Bernoulli numbers. We discuss this problem based on the Fourier expansion of the Bernoulli polynomials. We also give some necessary and sufficient conditions for ζ(2k + 1) be a rational multiple of π2k+1.

An Inverse Mode Problem for the Rod on Elastic Foundation

2013 ◽  
Vol 353-356 ◽  
pp. 3198-3201
Author(s):  
Xia Tian ◽  
Chuan Xiao Li
Keyword(s):  
Numerical Methods ◽  
Elastic Foundation ◽  
Discrete Model ◽  
Sufficient Conditions ◽  
Spring Stiffness ◽  
Positive Mass ◽  
Mass System ◽  
Physical Elements ◽  
Necessary And Sufficient ◽  
Simply Connected

Consider the rod on elastic foundation. Its discrete model is the simply connected spring-mass system with partial mass connected to the ground. The inverse mode problem of constructing the physical elements of the system from two eigenpairs, the spring stiffness of the system is considered. The necessary and sufficient conditions for constructing a physical realizable system with positive mass and stiffness elements are established. If these conditions are satisfied, the rod on the elastic foundation may be constructed uniquely. The numerical methods and examples are given finally.

An Inverse Mode Problems for Simply Connected Central Symmetric Spring-Mass Systems

2012 ◽  
Vol 166-169 ◽  
pp. 3399-3402
Author(s):  
Xia Tian ◽  
Chuan Xiao Li
Keyword(s):  
Numerical Methods ◽  
Total Mass ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Positive Mass ◽  
Mass System ◽  
Physical Elements ◽  
Necessary And Sufficient ◽  
Simply Connected

Given the odd /even degree eigenpair and the even degree eigenpair of a simply connected central symmetric spring-mass system respectively. The inverse mode problem of constructing the physical elements of the system from two eigenpairs and the total mass of the system is considered. The necessary and sufficient conditions for constructing of a physical realizable system with positive mass and stiffness elements are established. If these conditions are satisfied, the simply connected central symmetric spring-mass system may be constructed uniquely. The numerical methods and examples are given finally.

scholarly journals Some Conditions on Trans-Sasakian Manifolds to Be Homothetic to Sasakian Manifolds

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1887
Author(s):  
Sharief Deshmukh ◽  
Amira Ishan ◽  
Olga Belova ◽  
Suha B. Al-Shaikh
Keyword(s):  
Sufficient Conditions ◽  
Sasakian Manifold ◽  
Necessary And Sufficient Conditions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Sasakian Manifolds ◽  
3 Dimensional ◽  
Necessary And Sufficient ◽  
Simply Connected

In this paper, we study 3-dimensional compact and connected trans-Sasakian manifolds and find necessary and sufficient conditions under which these manifolds are homothetic to Sasakian manifolds. First, four results in this paper deal with finding necessary and sufficient conditions on a compact and connected trans-Sasakian manifold to be homothetic to a compact and connected Sasakian manifold, and the fifth result deals with finding necessary and sufficient condition on a connected trans-Sasakian manifold to be homothetic to a connected Sasakian manifold. Finally, we find necessary and sufficient conditions on a compact and simply connected trans-Sasakian manifold to be homothetic to a compact and simply connected Einstein Sasakian manifold.

The moments of the number of exits from a simply connected region

1998 ◽  
Vol 30 (1) ◽  
pp. 167-180 ◽  
Author(s):  
Robert Illsley
Keyword(s):  
Integral Formula ◽  
Sufficient Conditions ◽  
Moment Matrix ◽  
Expected Number ◽  
Factorial Moments ◽  
Gaussian Case ◽  
Simply Connected Region ◽  
Necessary And Sufficient ◽  
Simply Connected ◽  
Stationary Vector

We generalise the work of Cramér and Leadbetter, Ylvisaker and Ito on the level crossings of a stationary Gaussian process to multivariate processes. Necessary and sufficient conditions for the existence of the expected number of crossings E(C) of the boundary of a region of ℝp by a stationary vector stochastic process are obtained, along with an explicit formula for E(C) in the Gaussian case. A rigorous proof of Belyaev's integral formula for the factorial moments of the number of exits from a region of ℝp is given for a class of processes which includes Gaussian processes having a finite second order spectral moment matrix. Applications to χ2 processes are briefly considered.

scholarly journals ON PRIME-LIKE RADICALS

2010 ◽  
Vol 82 (1) ◽  
pp. 113-119 ◽  
Author(s):  
S. TUMURBAT ◽  
H. FRANCE-JACKSON
Keyword(s):  
Polynomial Ring ◽  
Prime Ring ◽  
Open Problem ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Prime Radical ◽  
Polynomial Rings ◽  
Necessary And Sufficient

AbstractA radical γ is prime-like if, for every prime ring A, the polynomial ring A[x] is γ-semisimple. In this paper, we study properties of prime-like radicals. In particular, we give necessary and sufficient conditions for a radical γ containing the prime radical β to be prime-like. This allows us to easily find distinct special radicals that coincide on simple rings and on polynomial rings, which answers a question put by Ferrero. It also allows us to reformulate a long-standing open problem of Gardner in terms of prime-like radicals.

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