# Existsa

### Group Activity 12A: Rings

An element a 2Risaunit provided there **existsa** 1 2Rsuchthataa 1 is a unity. (1) LetR=f0; 2; 4; 6; 8 gwithadditionand multiplication modulo 10 bearing.

www.math.iit.edu

### ALLEE EFFECTS INA DISCRETE-TIME HOST-PARASITOID MODEL WITH ...

It is proved that there **existsa** population threshold below which the population becomes extinct. When the population is initiated above the threshold, the two age classes will either be synchronizing or 2000 Mathematics Subject Classification.

aimsciences.org

### 3Lecture3

Definition 14 For any real sequence , wesaythat∈ ¯ Risa subsequential limit of if there **existsa** subsequence → For example, the sequence =(−1) has two subsequential limits, −1 and 1.

www.econ.brown.edu

### Integer values of polynomials

Besides the obvious solutiong=f we may want to impose some conditions on the polynomialg; for example, iff (S) Z we wonder if there **existsa** polynomial with integer coecients which parametrizes the setf (S).

www.dm.unipi.it

### MathS-101. Worksheet 13. Limits and Continuity in Rn

Restating this result in terms of the Euclidean metricd, afunctionf: X! Y is continuous if and only if for eachx 2 Xandeach > 0, there **existsa** > 0 such that d (x,y) <=) d (f (x) ,f (y)) <.

math.harvard.edu

### Homework 2 -Solutions

Sincew∈L 2 and|w|≥p, by the pumping lemma, there **existsa** decomposition w=xyzsuchthat|y|>0, |xy|≤pandxy i z∈L 2 foreachi≥0. Fori=0 this implies that xz=0 r 1 0 pisinLwherer<q.

www.cs.iit.edu

### Math 221: Final Exam

Show that if Ris Cohen-Macaulay, then there **existsa** (nonzero) nitely generated R-module Mwhose length and projective dimension are both nite. (4) Let Rbealocal Noetherianring which contains the eld Qofrational numbers, let Gbea nite group actingonR, and let R G Rbethe xedpointsforthe action of G. Show that ...

www.math.harvard.edu

### CHARACTERIZING ASYMPTOTIC ST ABILITY WITHDULAC FUNCTIONS

R n beananalytic vector eld and suppose that the system_ x=F (x) has LAS atp: Is it true that there **existsa** C 1 Dulacfunction B for system (1) dene din W s (p) ?

www.math.grin.edu

### *-normal spaces and decompositions of normality

199 (iii) Functionally *-normal[6]if for every pair of disjoint closed sets A and Boneofwhich is*-closed there **existsa** continuous function f: X! [0,1]such thatf (A) =0 and f (B) =1; (iv) Weakly functionally*-normal (wf*-normal) [6]if for every pair of disjoint*-closed sets A and Bthereexistsa continuous ...

agt.webs.upv.es

### IMO Shortlist 2008

Suppose that there **existsa** circleωtangenttotheray BA beyond A and to the ray BC beyond C , which is also tangent to the lines AD and CD . Prove that the common external tangents of ω 1 andω 2 intersectonω. 1.1.2 Shortlisted Problems 1.

www.imomath.com

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