MatheAss 9.0 - News

 

MatheAss  is also revised from time to time between updates, mostly based on user hints. In order to benefit from subsequent additions, users registered for version 9.0 can install the current version without their license file being overwritten. Whether you are registered for version 9.0 is displayed in the Help/Info menu item.
Users registered for older versions can install MatheAss 9.0 as a test version without overwriting the old version.

What's new in Matheass 9.0?

Settings of 2D graphics (from February 2023)

The graphics can be moved by dragging them with the left mouse button and can be zoomed centrally with the mouse wheel.
You can zoom separately in x and y direction by dragging with both mouse buttons.

The other functions of the previous context menu have been replaced by the buttons
Aspect 1:1,  Center and  Settings on the right side.

Where possible, an area is chosen in the initial drawing where all the essential points are visible.
You can return to this setting by double-clicking in the drawing.

The following program parts have been added:

Algebra

Prime tuples
In an interval [a,b], all prime twins (p,p+2), prime cousins (p,p+4), sexy primes (p,p+6) and prime triplets are determined. 
Prime twins between 1 and 200

(3|5) (5|7) (11|13) (17|19) (29|31) (41|43) (59|61)
(71|73) (101|103) (107|109) (137|139) (149|151)
(179|181) (191|193) (197|199)

15 pairs of prime twins
Prime triplets between 1 and 100

(3|5|7) (5|7|11) [7|11|13] (11|13|17) [13|17|19]   
(17|19|23) [37|41|43] (41|43|47) [67|71|73] 

9 triplet prime triplets
4 of the form (p|p+2|p+6) and 4 of the form [p|p+4|p+6]
Calculating percentages
The base value G, the percentage value W, the percentage p or p%, the growth factor q and the final value E are calculated if two independent values are entered.
Given:
¯¯¯¯¯¯  
           basic value G = 150 
        percentage p% = 2.5% = 0.025 = 1/40 

Results:
¯¯¯¯¯¯¯
 percentage value W = 3.75 
         growth factor q = 102,5% = 1,025 = 41/40 
             final value E = 153.75
Given:
¯¯¯¯¯¯
 percentage value W = -120 
         growth factor q = 95% = 0,95 = 19/20

Results:
¯¯¯¯¯¯¯
          basic value G = 2400 
        percentage p% = -5% = -0.05 = -1/20 
             final value E = 2280
Calculation with big integers (since april 2021)
Calculation with two big integers  a  and  b  with a maximum of 10,000 digits.

Geometry

Special straight lines in a triangle
The program determines the equations of the perpendiculars, the bisectors of the sides, the bisectors of the angles and the heights of a triangle. In addition, the centers and radii of the circumference, the inscribed circle and the three excircles.
Given:
¯¯¯¯¯¯
   Vertices:    A(1|0)   B(5|1)   C(3|6)
 
Results:
¯¯¯¯¯¯¯
       Sides:   a :  5·x + 2·y = 27
                    b :  3·x - y = 3
                    c :  x - 4·y = 1
 
    Incircle:   Mi(3,119|1,962)          r i = 1,390  
 
 Excircles:   Ma(7,626|6,136)        ra = 4,346
                   Mb(-4,356|5,784)       rb = 6,910
                   Mc(3,248|-2,427)       rc = 2,900
Arbitrary Polygons   (since november 2022)
The sides and angles of the polygon are now also calculated and it is checked whether the polygon is convex, concave or self-intersecting.
In addition, convex polygons are checked whether they have an incircle and/or a circumcircle.
Vertices:                              Area  A = 16
 A(1|2)                            
 B(4,5|0,5)                   Perimeter  p = 15,54498
 C(6|4)
 D(4,5|5,5)                   Centroid of vertices: 
 E(1|4)                         CV(3,4|3,2)
                                      
                                    Centroid of area: 
                                    CA(3,46875|3,07813)

Sides:                           Angles:
 |AB| = 3,8078866          ∡BAE = 113,19859°
 |BC| = 3,8078866          ∡CBA = 90°
 |CD| = 2,1213203          ∡DCB = 111,80141°
 |DE| = 3,8078866          ∡EDC = 111,80141°
 |EA| = 2                         ∡AED = 113,19859°

Cyclic polygon
Circumcircle:  M(3,5|3)  r=2,6925824

Cyclic polygon:

Mappings of Polygons
Displacement, straight line mirroring, point mirroring, rotation, centric stretching and shear can be applied to an n-gon.
The input has been made clearer and the construction lines can be drawn in the diagram. 
Counter image
    A(1|1), B(5|1), C(5|5), D(3|7), E(1|5), 

1. Translation: dx=2, dy=1  ☑
    A(3|2), B(7|2), C(7|6), D(5|8), E(3|6), 

2. Rotation: Z(2|-1), α=-60° ☑
    A(5,0981|-0,36603), B(7,0981|-3,8301),    
    C(10,562|-1,8301), D(11,294|0,90192), 
    E(8,5622|1,634),
Tangent lines to circles   (since february 2021)
The following tangents will be calculated:
  • The tangent to a circle k at a point B.
  • The tangents to a circle k through a point P outside the circle
  • The tangents to a circle k parallel to a straight line g
  • The tangents to two circles  k1  and  k2 
Given:
¯¯¯¯¯
  k1 : M(5|8) ,   r =5
  k2 : M(-1|2) ,   r =3

Outer tangents
¯¯¯¯¯¯¯¯¯¯¯¯
  t1: -4,2923·x + 7,04104·y = -6,36427
  t2: -7,04104·x + 4,29230·y = 40,3643

Inner tangents
¯¯¯¯¯¯¯¯¯¯¯¯
  t3: 1,21895·x + 2,55228·y = 12,3709
  t4: -2,55228·x - 1,21895·y = -8,3709
Distances_on_the_Sphere   (since december 2021)

The distance between two points on a sphere is calculated.

GPS decimal
¯¯¯¯¯¯¯¯¯¯¯
  Berlin : 52.523403, 13.4114
New York : 40.714268, -74.005974

GPS dms
¯¯¯¯¯¯¯
  Berlin : 52° 31' 24.2508" N, 13° 24' 41.0400" E
New York : 40° 42' 51.3648" N, 74°  0' 21.5064" W
  .
  .
  .
  
Distance
¯¯¯¯¯¯¯¯¯¯
   d = r · α [rad] = 6385,112

Analysis

Sequences and Series   (since may 2021)
The program determines the first n terms of a sequence  (ai)  and the associated series (sum of the sequence terms) if the first terms of the sequence and a recourse formula  ai=ƒ(a0, a1, ... , ai-1)  or an explicit function  ai = ƒ(i)  are given.
The sequence of odd numbers e.g. can be defined explicitly by  ai = 2·i + 1  or  recursively by  ai = ai-1 + 2  with  a0=1 . 
Folge
¯¯¯¯¯
( a[ i ] ) = (1; 3; 5; 7; 9; 11; 13; 15; 17; 19)

Reihe
¯¯¯¯¯
( Σ a[ i ] ) = (1; 4; 9; 16; 25; 36; 49; 64; 81; 100)
Factoring Polynomials
The program calculates the rational zeros and the linear factorization of a polynomial.
p(x) = x5 - 9·x4 - 82/9·x3 + 82·x2 + x - 9
       = (1/9)·(9·x5 - 81·x4 - 82·x3 + 738·x2 + 9·x - 81)
       = (1/9)·(3·x - 1)·(3·x + 1)·(x - 9)·(x - 3)·(x + 3)

Rational Zeros: 1/3, -1/3, 9, 3, -3
Transforming Polynomials
A polynomial  p(x)  can be shifted or stretched in the x-direction and y-direction.
ƒ(x) =  - 1/4·x4 + 2·x3 - 16·x + 21

Shifted by  dx = -2 ,  dy = 0

ƒ(x + 2) =  - 1/4·x4 + 6·x2 + 1
GCD and LCM of polynomials   (since february 2021)
The greatest common divisor (GCD) and the least common multiple (LCM) of two polynomials p1(x) and p2(x) are determined..
p1(x) =  4·x6 - 2·x5 - 6·x4- 18·x3 - 2·x2 + 24·x + 8
p2(x) =  10·x4- 14·x3 - 22·x2 + 14·x + 12

GCD(p1,p2) =  x2 - x - 2
LCM(p1,p2) =  40·x8 - 36·x7 - 76·x6 - 144·x5 + 88·x4+ 356·x3 - 4·x2 - 176·x - 48
Calculus of Polynomial Functions
The program carries out the curve discussion for polynomial function. This means that the derivatives and the antiderivative are determined, the function is examined for rational zeros, for extremes, for inflection points and for symmetry. 
Function :
¯¯¯¯¯¯¯¯
ƒ(x) = 3·x4 - 82/3·x2 + 3
       = 1/3·(9·x4 - 82·x2 + 9)
       = 1/3·(3·x - 1)·(3·x + 1)·(x - 3)·(x + 3)

Derivations :
¯¯¯¯¯¯¯¯¯¯
ƒ'(x)  = 12·x3 - 164/3·x
ƒ"(x)  = 36·x2 - 164/3
ƒ'"(x) = 72·x

Antiderivative:
¯¯¯¯¯¯¯¯¯¯¯¯
ƒ(x) = 3/5·x5 - 82/9·x3 + 3·x + c

…
Calculus of Rational Functions
The program carries out the curve discussion for a rational function. That is, the derivatives, the definition gaps and the continuous continuation are determined. The function is examined for zeros, extrema, points of inflection: and the behavior for | x | → ∞. 
Function :
¯¯¯¯¯¯¯¯
            3·x3 + x2 - 4         (x - 1)·(3·x2 + 4·x + 4)  
ƒ(x) = —————— = ———————————
               4·x2 - 16                4·(x - 2)·(x + 2)       

Definition gaps :
¯¯¯¯¯¯¯¯¯¯¯¯¯
x = 2  Pol mit Vorzeichenwechsel
x =-2  Pol mit Vorzeichenwechsel

Derivations :
¯¯¯¯¯¯¯¯¯¯
               3·(x4 - 12·x2)             3·(x2·(x2 - 12))   
ƒ'(x) = ———————— = —————————
            4·(x4 - 8·x2 + 16)       4·(x - 2)2·(x + 2)2 

                     6·(x3 + 12·x)                6·(x·(x2 + 12))  
ƒ"(x) = ——————————— = ————————
             x6 - 12·x4 + 48·x2 - 64        (x - 2)3·(x + 2)3 
…
Integral Calculus   (since February 2021 with arc lengths)
  ƒ1(x) = cosh(x)
  ƒ2(x) = x^2+1

  Limits of integration [a;b]  from  -2 to 2

  Oriented content :  A1 = -2,07961
  Absolute content :  A2 = 2,07961

  Arc lengths      :  L1[a;b] = 7,254   L2[a,b] = 9,294

Stochastics

Statistics
In the statistics section, the histogram was supplemented by a box plot.
Logistic Regression
The program determines a curve fit for a series of measurements to the logistic function  
with the parameters    a1 = ƒ(0)·S ,  a2 = ƒ(0) ,  a3 = S - ƒ(0) ,  und  a4 = -k·S  and the saturation limit S .
Data from:  "hopfenwachstum.csv"

Saturation limit:  6
      Dark figure:  1

                            4,0189                 
ƒ(x) = ————————————————
            0,66981 + 5,3302 · e^(-0,35622·t)


Inflection point W(5,8226/3)

Maximum growth rate ƒ'(xw) = 0,53433

8 Values 
Coeff.of determin.   = 0,99383916
Correlation coeff.    = 0,99691482
Standard deviation = 0,16172584
Data series from Johns Hopkins University (JHU) on the corona pandemic are attached as CSV files.

Linear Algebra

Linear Optimization   (since February 2022)
The program determines the optimal solution for a two-variable objective function with linear inequalities as boundary conditions.
Objective function:   
  ƒ(x,y) = 140·x + 80·y → Maximum

Constraints:
  x ≥ 0
  y ≥ 0
  x ≤ 600
  y ≤ 700
  x + y ≤ 750
  3·x + y ≤ 1200

Maximum
  x = 225   y = 525
  ƒ(x,y) = 73500

Registration

How much costs MatheAss 9.0?

  29 € for the private license

  79 € for the school license

360 € for the extended school license, with which the serial number may be passed on to the pupils.

How much is the update?

  10 € for owners of a private license

  30 € for owners of a school license

  90 € for owners of an extended school license

How can I pay?

Here by PayPal :

MatheAss 9.0


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