On the analytic solution of certain functional and difference equations

1983 ◽  
Vol 389 (1796) ◽  
pp. 1-13 ◽  
Keyword(s):  
Functional Equation ◽  
Analytic Solution ◽  
Sufficient Conditions ◽  
Irreducible Polynomial ◽  
Constant Solution ◽  
Simple Change ◽  
Change Of Variable ◽  
The Difference ◽  
Necessary And Sufficient ◽  
Complex Coefficients

Let P(u, v) be an irreducible polynomial with complex coefficients and let q ≥ 2 be an integer. We establish the necessary and sufficient conditions under which the functional equation P(f(z), f(z q )) = 0, (F) has a non-constant analytic solution that is either regular in a neighbour­hood of the point z = 0 or has a pole at this point (theorem 1). By a simple change of variable, the difference equation P ( F(Z) , F ( Z + 1)) = 0, (D) can be proved under the same restrictions to have a non-constant solution of the form F(Z) = Σ ∞ j=I f j e -jq z , which is regular in the strip Re Z ≥ X 0 , |Im Z | < π/2 ln q , if X 0 is sufficiently large (theorem 2).

A Direct Rotatability Criterion for Spherical Four-Bar Linkages

1995 ◽  
Vol 117 (4) ◽  
pp. 597-600 ◽  
Author(s):  
K. C. Gupta ◽  
R. Ma
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Absolute Value ◽  
The Difference ◽  
Necessary And Sufficient ◽  
Four Bar Linkage

The necessary and sufficient conditions for the full input rotatability in a spherical four-bar linkage are proved. The direct criterion is: for all twist angles α in the range [0, π], the excess (deficit) of the sum of the frame and input twist angles over (from) π should, in absolute value, be greater than that for the coupler and follower twist angles; the difference between the follower and input twist angles, in absolute value, should be greater than that for the coupler and follower twist angles. Application of the direct criterion to full rotatability of other links are discussed and some variations in the form of the criterion are developed.

A Direct Rotatability Criterion for Spherical Four-Bar Linkages

1994 ◽  
Author(s):  
R. Ma ◽  
K. C. Gupta
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Absolute Value ◽  
The Difference ◽  
Necessary And Sufficient ◽  
Four Bar Linkage

Abstract The necessary and sufficient conditions for the full input rotatability in a spherical four bar linkage are proved. The direct criterion is: for all twist angles α in the range [0, π], the excess (deficit) of the sum of the frame and input twist angles over (from) π should, in absolute value, be greater than that for the coupler and follower twist angles; the difference between the follower and input twist angles, in absolute value, should be greater than that for the coupler and follower twist angles. Application of the direct criterion to full rotatability of other links are discussed and some variations in the form of the criterion are developed.

scholarly journals Two-Vertex Generators of Jacobians of Graphs

10.37236/7302 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
David Brandfonbrener ◽  
Pat Devlin ◽  
Netanel Friedenberg ◽  
Yuxuan Ke ◽  
Steffen Marcus ◽  
...  
Keyword(s):  
Empirical Evidence ◽  
Random Graphs ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
The Difference ◽  
Necessary And Sufficient

We give necessary and sufficient conditions under which the Jacobian of a graph is generated by a divisor that is the difference of two vertices. This answers a question posed by Becker and Glass and allows us to prove various other propositions about the order of divisors that are the difference of two vertices. We conclude with some conjectures about these divisors on random graphs and support them with empirical evidence.

scholarly journals Existence of a Period-Two Solution in Linearizable Difference Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
E. J. Janowski ◽  
M. R. S. Kulenović
Keyword(s):  
Difference Equation ◽  
Linear Equation ◽  
Difference Equations ◽  
Initial Conditions ◽  
Sufficient Conditions ◽  
Minimal Period ◽  
Real Numbers ◽  
Existence And Nonexistence ◽  
The Difference ◽  
Necessary And Sufficient

Consider the difference equationxn+1=f(xn,…,xn−k),n=0,1,…,wherek∈{1,2,…}and the initial conditions are real numbers. We investigate the existence and nonexistence of the minimal period-two solution of this equation when it can be rewritten as the nonautonomous linear equationxn+l=∑i=1−lkgixn−i,n=0,1,…,wherel,k∈{1,2,…}and the functionsgi:ℝk+l→ℝ. We give some necessary and sufficient conditions for the equation to have a minimal period-two solution whenl=1.

ON THE COHEN–MACAULAYNESS OF SOME GRADED RINGS

2008 ◽  
Vol 07 (01) ◽  
pp. 109-128
Author(s):  
D. P. PATIL ◽  
G. TAMONE
Keyword(s):  
Numerical Semigroup ◽  
Sufficient Conditions ◽  
Local Rings ◽  
Graded Rings ◽  
Semigroup Rings ◽  
Associated Graded Ring ◽  
Blowing Up ◽  
The Difference ◽  
Necessary And Sufficient ◽  
Numerical Semigroup Rings

Let (R,𝔪) be a 1-dimensional Cohen–Macaulay local ring of multiplicity e and embedding dimension ν ≥ 2. Let B denote the blowing-up of R along 𝔪 and let I be the conductor of R in B. Let x ∈ 𝔪 be a superficial element in 𝔪 of degree 1 and [Formula: see text], [Formula: see text]. We assume that the length [Formula: see text]. This class of local rings contains the class of 1-dimensional Gorenstein local rings (see 1.5). In Sec. 1, we prove that (see 1.6) if the associated graded ring G = gr 𝔪(R) is Cohen–Macaulay, then I ⊆ 𝔪s + xR, where s is the degree of the h-polynomial h R of R. In Sec. 2, we give necessary and sufficient conditions (see Corollaries 2.4, 2.5, 2.9 and Theorem 2.11) for the Cohen–Macaulayness of G. These conditions are numerical conditions on the h-polynomial h R, particularly on its coefficients and the degree in comparison with the difference e - ν. In Sec. 3, we give some conditions (see Propositions 3.2, 3.3 and Corollary 3.4) for the Gorensteinness of G. In Sec. 4, we give a characterization (see Proposition 4.3) of numerical semigroup rings which satisfy the condition [Formula: see text].

scholarly journals Application of a Theorem in Stochastic Models of Elections

2010 ◽  
Vol 2010 ◽  
pp. 1-30 ◽  
Author(s):  
Norman Schofield ◽  
Christopher Claassen ◽  
Ugur Ozdemir ◽  
Alexei Zakharov
Keyword(s):  
Formal Model ◽  
Sufficient Conditions ◽  
The United States ◽  
Character Trait ◽  
Necessary Condition ◽  
Presidential Candidates ◽  
Sufficient Conditions For Convergence ◽  
The Difference ◽  
Centripetal Effect ◽  
Necessary And Sufficient

Previous empirical research has developed stochastic electoral models for Israel, Turkey, and other polities. The work suggests thatconvergence to an electoral center(often predicted by electoral models) is a nongeneric phenomenon. In an attempt to explain nonconvergence, a formal model based onintrinsic valenceis presented. This theory showed that there are necessary and sufficient conditions for convergence. The necessary condition is that a convergence coefficientcis bounded above by the dimensionwof the policy space, while a sufficient condition is that the coefficient is bounded above by 1. This coefficient is defined in terms of the difference in exogenous valences, the “spatial coefficient”, and the electoral variance. The theoretical model is then applied to empirical analyses of elections in the United States and Britain. These empirical models include sociodemographic valence and electoral perceptions of character trait. It is shown that the model implies convergence to positions close to the electoral origin. To explain party divergence, the model is then extended to incorporate activist valences. This extension gives a first-orderbalance conditionthat allows the party to calculate the optimal marginal condition to maximize vote share. We argue that the equilibrium positions of presidential candidates in US elections and by party leaders in British elections are principally due to the influence of activists, rather than the centripetal effect of the electorate.

Vectors

1996 ◽  
Author(s):  
Gerhard Oertel
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Unit Vector ◽  
Cross Product ◽  
Nonzero Vector ◽  
Algebraic Representation ◽  
The Difference ◽  
Necessary And Sufficient ◽  
Three Dimensional Space

The reader, even if familiar with vectors, will find it useful to work through this chapter because it introduces notation that will be used throughout this book. We will take vectors to be entities that possess magnitude, orientation, and sense in three-dimensional space. Graphically, we will represent them as arrows with the sense from tail to head, magnitude proportional to the length, and orientation indicated by the angles they form with a given set of reference directions. Two different kinds of symbol will be used to designate vectors algebraically, boldface letters (and the boldface number zero for a vector of zero magnitude), and subscripted letters to be introduced later. The first problems deal with simple vector geometry and its algebraic representation. Multiplying a vector by a scalar affects only its magnitude (length) without changing its direction. Problem 1. State the necessary and sufficient conditions for the three vectors A, B, and C to form a triangle. (Problems 1–9, 12–14, 19–23, and 25 from Sokolnikoff & Redheffer, 1958.) Problem 2. Given the sum S = A + B and the difference D = A – B, find A and B in terms of S and D (a) graphically and (b) algebraically. Problem 3. (a) State the unit vector a with the same direction as a nonzero vector A. (b) Let two nonzero vectors A and B issue from the same point, forming an angle between them; using the result of (a), find a vector that bisects this angle. Problem 4. Using vector methods, show that a line from one of the vertices of a parallelogram to the midpoint of one of the nonadjacent sides trisects one of the diagonals. Two vectors are said to form with each other two distinct products: a scalar, the dot product, and a vector, the cross product.

scholarly journals Necessary and Sufficient Conditions for Positive Solutions of Second-Order Nonlinear Dynamic Equations on Time Scales

2011 ◽  
Vol 2011 ◽  
pp. 1-11
Author(s):  
Zhi-Qiang Zhu
Keyword(s):  
Time Scales ◽  
Difference Equations ◽  
Nonlinear Dynamic ◽  
Sufficient Conditions ◽  
Nonoscillatory Solutions ◽  
Nonlinear Dynamic Equation ◽  
Nonlinear Dynamic Equations ◽  
Necessary And Sufficient Criteria ◽  
The Difference ◽  
Necessary And Sufficient

This paper is concerned with the existence of nonoscillatory solutions for the nonlinear dynamic equation on time scales. By making use of the generalized Riccati transformation technique, we establish some necessary and sufficient criteria to guarantee the existence. The last examples show that our results can be applied on the differential equations, the difference equations, and the -difference equations.

scholarly journals Convex sequences of higher order

Filomat ◽  
2018 ◽  
Vol 32 (13) ◽  
pp. 4655-4663
Author(s):  
Daniel Sofonea ◽  
Ioan Ţincu ◽  
Ana Acu
Keyword(s):  
Real Number ◽  
Sufficient Conditions ◽  
Higher Order ◽  
Necessary And Sufficient Conditions ◽  
Difference Operators ◽  
Real Sequence ◽  
Convex Sequence ◽  
Different Types ◽  
The Difference ◽  
Necessary And Sufficient

In this paper we study the class of convex sequences of higher order defined using the difference operators and investigate their properties. The notion of the convex sequence of order r ? N will be extended for r a real number. Some necessary and sufficient conditions such that a real sequence belongs to the class of convex sequences of higher order r ? R are introduced. Using different types of means we will investigate the convexity of higher order for real sequences.

scholarly journals On Decomposition of Stochastic Finite-State Systems

1973 ◽  
Vol 2 (8) ◽  
Author(s):  
Mogens Nielsen
Keyword(s):  
Information Flow ◽  
Care Transitions ◽  
Sufficient Conditions ◽  
Stochastic Automata ◽  
Finite State ◽  
Classical Models ◽  
Decomposition Models ◽  
New Type ◽  
The Difference ◽  
Necessary And Sufficient

<p>Various decomposition models for stochastic finite-state systems (stochastic automata with or without output) are discussed. A new type of information flow - the next state from an interconnection-component instead of the present state - is introduced in well known loop-free and feed-back decomposition models. Results on these modified decomposition models are stated (like necessary and sufficient conditions for a system to be decomposed) are stated, and the difference between these and the corresponding results on the classical models are discussed with respect to f. ex. don't care transitions and synthesis.</p>

Sign in / Sign up

Export Citation Format

Share Document