Mon Jan 19 21:57:36 MST 1998 euler% math Mathematica 3.0 for Solaris Copyright 1988-96 Wolfram Research, Inc. -- Terminal graphics initialized -- In[1]:= In[2]:= In[3]:= (* ----------[ M a t h e m a t i c a ]---------- *) 0. Second In[4]:= (* ---------- Initialization ---------- *) 0. Second In[5]:= (* ---------- Inequalities ---------- *) 0. Second In[6]:= (* => True *) 0. Second In[7]:= E^Pi > Pi^E 0.02 Second Out[7]= True In[8]:= (* => [True, False] *) 0. Second In[9]:= x/: Im[x] = 0 0. Second Out[9]= 0 In[10]:= {x^4 - x + 1 > 0, x^4 - x + 1 > 1} 0. Second 4 4 Out[10]= {1 - x + x > 0, 1 - x + x > 1} In[11]:= Remove[x] 0. Second In[12]:= (* => True *) 0. Second In[13]:= (*assume[Abs[x] < 1]*) 0. Second In[14]:= -1 < x && x < 1 0. Second Out[14]= -1 < x && x < 1 In[15]:= (* x > y > 0 and k, n > 0 => k x^n > k y^n *) 0. Second In[16]:= (*assume[x > y, y > 0]*) 0. Second In[17]:= 2*x^2 > 2*y^2 0. Second 2 2 Out[17]= 2 x > 2 y In[18]:= (*assume[k > 0]*) 0. Second In[19]:= k*x^2 > k*y^2 0. Second 2 2 Out[19]= k x > k y In[20]:= (*assume[n > 0]*) 0. Second In[21]:= k*x^n > k*y^n 0. Second n n Out[21]= k x > k y In[22]:= (* x > 1 and y >= x - 1 => y > 0 *) 0. Second In[23]:= (*assume[x > 1, y >= x - 1]*) 0. Second In[24]:= y > 0 0. Second Out[24]= y > 0 In[25]:= (* x >= y, y >= z, z >= x => x = y = z *) 0. Second In[26]:= (*assume[x >= y, y >= z, z >= x]*) 0. Second In[27]:= {x == y, x == z, y == z} 0. Second Out[27]= {x == y, x == z, y == z} In[28]:= (* x < -1 or x > 3 *) 0. Second In[29]:= << Algebra`InequalitySolve` 0.2 Second In[30]:= InequalitySolve[Abs[x - 1] > 2, x] 0.18 Second Out[30]= x < -1 || x > 3 In[31]:= (* x < 1 or 2 < x < 3 or 4 < x < 5 *) 0. Second In[32]:= InequalitySolve[Expand[(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)] < 0,\ > x] 0.1 Second Out[32]= x < 1 || (2 < x) < 3 || (4 < x) < 5 In[33]:= (* x < 3 or x >= 5 *) 0. Second In[34]:= InequalitySolve[6/(x - 3) <= 3, x] 0.12 Second Out[34]= x < 3 || x >= 5 In[35]:= (* => 0 <= x < 4 *) 0. Second In[36]:= InequalitySolve[Sqrt[x] < 2, x] InequalitySolve::npi: A nonpolynomial equation or inequality encountered. The solution set may be incorrect. 0.07 Second Out[36]= x < 4 In[37]:= (* => x is real *) 0. Second In[38]:= InequalitySolve[Sin[x] < 2, x] 0.09 Second Out[38]= True In[39]:= (* => x != pi/2 + n 2 pi *) 0. Second In[40]:= InequalitySolve[Sin[x] < 1, x] 0.08 Second Pi Pi Out[40]= x < -- || x > -- 2 2 In[41]:= (* The next two examples come from Abdubrahim Muhammad Farhat,\ > _Stability Analysis of Finite Difference Schemes_, Ph.D. dissertation, University of New Mexico, Albuquerque, New Mexico, December 1993 => 0 <= A <= 1/2 *) 0. Second In[42]:= InequalitySolve[Abs[2*A*(Cos[t] - 1) + 1] <= 1, A] InequalitySolve::nineq: Abs[1 + 2 A (-1 + Cos[t])] <= 1 is not a formula constructed with univariate polynomial equations and inequalities in A. 0.16 Second Out[42]= InequalitySolve[Abs[1 + 2 A (-1 + Cos[t])] <= 1, A] In[43]:= (* => 125 A^4 + 24 A^2 - 48 < 0 or |A| < 2/5 sqrt([8 sqrt(6) -\ > 3]/5) *) 0. Second In[44]:= InequalitySolve[A^2*(Cos[t] - 4)^2*Sin[t]^2 < 9, A] InequalitySolve::nineq: 2 2 2 A (-4 + Cos[t]) Sin[t] < 9 is not a formula constructed with univariate polynomial equations and inequalities in A. 0.22 Second 2 2 2 Out[44]= InequalitySolve[A (-4 + Cos[t]) Sin[t] < 9, A] In[45]:= (* => |x| < y *) 0. Second In[46]:= InequalitySolve[{x + y > 0, x - y < 0}, {x, y}] LogicalExpand::elist: List encountered during logical expansion of {x + y > 0, x - y < 0}. InequalitySolve::nlin: {x + y > 0, x - y < 0} is not a formula constructed with linear equations and inequalities in {x, y}. 0.02 Second Out[46]= InequalitySolve[{x + y > 0, x - y < 0}, {x, y}] In[47]:= (* ---------- Quit ---------- *) 0. Second In[48]:= Quit[] real 10.35 user 2.04 sys 0.43