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  1. Device is reset. Handwriting recognition In initial versions (Newton OS 1., x ,) the handwriting recognition gave e x tremely mi x ed results for users and was
  2. Is assumed that the shape of the function (and even its norm \int | f ( x )|^2, x , ) depend on the character used to denote its argument. If the Greek letter is
  3. x . \for all y. (So SY \to x y) # (( \phi (0) \land \for all x . \, ( \phi (, x ,) \to \phi (So) )) \to \for all x . \phi ( x ) for any \Mather_\, formula
  4. Form of the Heaviside step function used in signal processing, defined as:: u (, x ,) = \begin 0,& x < 0 \\ \franc, & x = 0 \\ 1,& x > 0,\end where the value of
  5. Function returns a number's sign without respect to its value. Therefore, x ,SGN ( x )abs ( x ). The signup function is a form of the Heaviside step function
  6. Series of" primitive notions ", either a precise notion of what we mean by, x , x \, ( or, for that matter," to be equal" ) has to be well established first
  7. Equivalent: * d satisfies the ultrametric inequality d ( x , y ) \LEQ \ma x (d (, x , z),d (y, z )) for all x , y,z in F. * v (a) \LE 1 \Right arrow v (1+a)
  8. and are the elements of F, then the polynomial ( x − a1) ( x − a2) ··· (, x ,− an) + 1 has no zero in F. By contrast, the fundamental theorem of algebra
  9. That is substitutable for x \, in \phi\, the formula \fi x _t \to \e x ists, x ,\, \phi is universally valid. Non-logical a x ioms Non-logical a x ioms are
  10. Of radiogenic. Primordial has an abundance of only 31.5 ppm (= 9340 ppm, x ,0.337 %),comparable to that of neon (18.18 ppm). The Martian atmosphere
  11. The" only" norm on R1,in the sense that, for every norm | | · | | on R1,| |, x ,| | = | | 1 | | · | x |. The comple x absolute value is a special case of the
  12. Derivative of the real absolute value function is the signup function, sgn (, x ,), which is defined as: \SGN ( x ) = \franc, for x ≠ 0. The absolute value
  13. The following definition. For any comple x number: z = x + in, \,where, x ,and y are real numbers, the absolute value or modulus of z is denoted | z | and
  14. A x ioms: # \for all x . \not (So = 0) # \for all x . \for all y. (So SY \to, x ,y) # (( \phi (0) \land \for all x . \, ( \phi ( x ) \to \phi (So) )) \to
  15. For e x ample, the open interval (0,1) does not have the least element: if, x ,is in (0,1),then so is x /2,and x /2 is always strictly smaller than x . So
  16. Sign function is constant at all points. Therefore, the second derivative of |, x ,| with respect to x is zero everywhere e x cept zero, where it is undefined. The
  17. Is the signup function, sgn ( x ),which is defined as: \SGN ( x ) = \franc, for, x , ≠ 0. The absolute value function is not differentiable at x 0. For applications
  18. Closed, because if a1,a2,…, an are the elements of F, then the polynomial (, x ,− a1) ( x − a2) ··· ( x − an) + 1 has no zero in F. By contrast, the
  19. Zero is conventional. So for all nonzero points on the real number line, : u (, x ,) = \franc. \, The absolute value function has no concavity at any point, the
  20. Is trivially true for any field. If F is algebraically closed and p (, x ,) is an irreducible polynomial of FX, then it has some root a and therefore p (
  21. Processing, defined as:: u ( x ) = \begin 0,& x < 0 \\ \franc, & x = 0 \\ 1,&, x ,> 0,\end where the value of the Heaviside function at zero is conventional. So
  22. A number's sign without respect to its value. Therefore, x SGN ( x )abs (, x ,). The signup function is a form of the Heaviside step function used in signal
  23. If a1,a2,…, an are the elements of F, then the polynomial ( x − a1) (, x ,− a2) ··· ( x − an) + 1 has no zero in F. By contrast, the fundamental
  24. That has to be smooth at x 0. One of such appro x imations is given by:: |, x ,| \appro x \franc \math rm (km) where k>0. The appro x imation improves as k
  25. In the sense that, for every norm | | · | | on R1,| | x | | = | | 1 | | · |, x ,|. The comple x absolute value is a special case of the norm in an inner product
  26. Value function is the signup function, sgn ( x ),which is defined as: \SGN (, x ,) = \franc, for x ≠ 0. The absolute value function is not differentiable at x 0.
  27. At all points. Therefore, the second derivative of | x | with respect to, x ,is zero everywhere e x cept zero, where it is undefined. The absolute value
  28. Ultrametric inequality d ( x , y ) \LEQ \ma x (d ( x , z ), d (y, z )) for all, x , y, z in F. * v (a) \LE 1 \Right arrow v (1+a) \LE 1 \te x t a \in F. An
  29. In addition, If: z x + i y r (\cos \phi + i \sin \phi) \, and: \overlie =, x ,- in is the comple x conjugate of z, then it is easily seen that: \begin | z | &
  30. Included in the basic retail package, a problem that was later solved with 2., x ,Newton devices - these were bundled with a serial cable and the appropriate
  31. Is universally valid. This means that, for any variable symbol x \, the formula, x , x \, can be regarded as an a x iom. Also, in this e x ample, for this not to fall
  32. Of the real absolute value given in (2) – (10) above. In addition, If: z, x ,+ i y r (\cos \phi + i \sin \phi) \, and: \overlie = x - in is the comple x
  33. X is equal to its absolute value considered as a comple x number since:: |, x ,+ i0 | \sort \sort = | x |. Similar to the geometric interpretation of the
  34. Can be seen as motivating the following definition. For any comple x number: z =, x ,+ in, \,where x and y are real numbers, the absolute value or modulus of z is
  35. Defined as: | z | = \sort. It follows that the absolute value of a real number, x ,is equal to its absolute value considered as a comple x number since:: | x + i0
  36. Value considered as a comple x number since:: | x + i0 | \sort \sort = |, x ,|. Similar to the geometric interpretation of the absolute value for real
  37. And a term t\, \! That is substitutable for x \, in \phi\, the formula \for all, x ,\, \phi \to \fi x _t is universally valid. Where the symbol \fi x _t stands for
  38. Step function used in signal processing, defined as:: u ( x ) = \begin 0,&, x ,< 0 \\ \franc, & x = 0 \\ 1,& x > 0,\end where the value of the Heaviside
  39. Used in signal processing, defined as:: u ( x ) = \begin 0,& x < 0 \\ \franc, &, x , = 0 \\ 1,& x > 0,\end where the value of the Heaviside function at zero is
  40. The Latin alphabet, such as Hawaiian, and Italian, which uses the letters j, k, x , y and w only in foreign words. It is unknown whether the earliest alphabets
  41. Is an irreducible polynomial of FX, then it has some root a and therefore p (, x ,) is a multiple of x − a. Since p ( x ) is irreducible, this means that p ( x )
  42. At x 0. Sometimes, a smooth appro x imation is required that has to be smooth at, x ,0. One of such appro x imations is given by:: | x | \appro x \franc \math rm (km)
  43. 0) \land \for all x . \, ( \phi ( x ) \to \phi (So) )) \to \for all x . \phi (, x ,) for any \Mather_\, formula \phi\ with one free variable. The standard
  44. Smooth appro x imation The absolute value function does not have a derivative at, x ,0. Sometimes, a smooth appro x imation is required that has to be smooth at x 0.
  45. And the following are equivalent: * d satisfies the ultrametric inequality d (, x , y) \LEQ \ma x (d ( x , z ), d (y, z )) for all x , y,z in F. * v (a) \LE 1
  46. Separate single letters) would follow the letters d, e,g, l,n, r,t, x ,and z respectively. Nor is, in a dictionary of English, the le x ical section
  47. Let \Mather\, be a first-order language. For each variable x \, the formula, x ,= x \, is universally valid. This means that, for any variable symbol x \, the
  48. Function is continuous everywhere. It is differentiable everywhere e x cept for, x ,= 0. It is monotonically decreasing on the interval (−∞,0 and monotonically
  49. x ) = \franc, for x ≠ 0. The absolute value function is not differentiable at, x ,0. For applications in which a well-defined derivative may be needed, however
  50. Be able to claim P (t)\, Again, we are claiming that the formula \for all, x ,\phi \to \fi x _t is valid, that is, we must be able to give a" proof" of

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