BROKEN LINES SIGNALS AND SMARANDACHE STEPPED FUNCTIONS

BROKEN LINES SIGNALS AND SMARANDACHE STEPPED FUNCTIONS Mircea Eugen Şelariu

1.

INTRODUCTION

The evolutive supermathematical functions (FSM ─ Ev) are discussed in the papers [1], [2], [3], [4], [5], [6], [7], [8]. They are combinations of the four types of FSM (centric functions (FC), excentric functions (FE), elevated functions (FEl) and exotic functions (FEx), taken by two, called centricoexcentric functions, elevatoexotic functions, centricoelevated functions, and so forth).

Aside from the centric(super)matematical functions (FC), that are only of centric α variable, all the others can be of excentric θ variable. In this way, between the three types of excentric functions (FE, FE, FEx) there are combinations between those of variable θ and those of variable α, as further proceeded.

From the website http://www.scritub.com/tehnica-mecanica/SEMNALE-ELECTRICE93336.php (Fig. 1) we present the main signals used in technics.

Fig. 1 Sinusoidal centric signals  and linear signals in straight line segments  and  A number of Smarandache stepped functions are presented in the paper [13], so named in honor of the distinguished Professor Dr. Math. Florentin Smarandache, from the University of New Mexico, Gallup campus (USA). They were developed by means of excentric circular supermathematical functions (FSM–CE). Families of such Smarandache stepped functions, represented using FSM–CE excentric amplitude of excentric θ variable, are shown in Figure 2.

y1 = aex[θ, S(1,0 )],

Plot[Evaluate[Table[{đ?‘Ą − ArcSin[đ?‘ Sin[đ?‘Ą]]}, {đ?‘ , −1, +1}], {đ?‘Ą, −Pi, 2Pi}]] ďƒ cu pasul 0,2

y2 = aex[θ, S(-1,0 )]

6

10 4

5 2

2

2

4

6

5

5

10

2

5

y1 = aexm[θ, S(1,0 )], 30 20 10

15

10

5

5

10

15

10 20 30

Fig. 2 Smarandache stepped functions represented using FSM–CE aexθ ď ° and aexmθ ď Ź and in 3D

ď ą They have the equation: (1) aexθ = θ – arcsin[s.sin(θ ─ Îľ)] = θ – bexθ, Figure 1ď ° or the modified one: (2) aexmθ = C.θ – arcsin[s.sin(n.θ ─ Îľ)], Figure 1ď ą with C ďƒŽ [0,25; 2] and n = 2.

Plot[Evaluate[Table[{(đ?‘Ą − ArcSin[−0.1đ?‘ Sin[đ?‘Ą]]) − Cos[3đ?‘Ą]â „Sqrt[1 − (−0.1đ?‘ Sin[3đ?‘Ą])2 ]}, {đ?‘ , −10,10}], {đ?‘Ą, −Pi, 2Pi}]]

Plot[{(đ?‘Ą − ArcSin[Sin[đ?‘Ą]]) − Cos[3đ?‘Ą]â „Sqrt[1 − (Sin[3đ?‘Ą])2 ] , đ?‘Ą − ArcSin[−Sin[đ?‘Ą]] − Cos[3đ?‘Ą]â „Sqrt[1 − (−Sin[3đ?‘Ą])2 ]}, {đ?‘Ą, −Pi, 2Pi}]

6

6

4

4

2

2

2

2

4

2

6

2

2

2

4

4

Plot[{(đ?‘Ą − ArcSin[−Sin[đ?‘Ą]â „Sqrt[2 + 2Cos[đ?‘Ą]]]) − Cos[3đ?‘Ą]/ Sqrt[1 − (Sin[3đ?‘Ą])^2], (đ?‘Ą − ArcSin[Sin[đ?‘Ą]â „Sqrt[2 − 2Cos[đ?‘Ą]]]) −Cos[3đ?‘Ą]â „Sqrt[1 − (Sin[3đ?‘Ą])^2]}, {đ?‘Ą, −Pi, 2Pi}]

4

6

Plot[{(đ?‘Ą − ArcSin[−Sin[đ?‘Ą]â „Sqrt[2 + 2Cos[đ?‘Ą]]]) − Cos[3đ?‘Ą]/ Sqrt[1 − (Sin[3đ?‘Ą])^2], (đ?‘Ą − ArcSin[Sin[đ?‘Ą]â „Sqrt[2 − 2Cos[đ?‘Ą]]]) −Cos[3đ?‘Ą]â „Sqrt[1 − (Sin[3đ?‘Ą])^2]}, {đ?‘Ą, −Pi, 2Pi}]

8

6 6

4 4

2

2

2

2

4

6

2

2

4

6

2

2 4

4

Fig. 3 Fascicle of graphs of FSM–CE Smarandache aexmθ with cascade aspect

Some graphs of the fascicle of Smarandache stepped functions have a “waterfallâ€? or cascade aspect, such as those shown in Figure 3 ď ´ and because some graphics are in blue, of which were extracted, for s = Âą 1, Smarandache stepped functions; some steps are visibly harder to escalate, but not impossible.

2. BROKEN LINEAR FUNCTIONS OR SAW TOOTH LINEAR SIGNALS Broken linear functions or symmetric saw tooth linear signals can be represented using FSM–CE beta excentric of excentric variable θ : (3) bexθ = arcsin[s.sin(θ ─ Îľ)], (Fig. 2 ď ´) and asymmetric saw tooth linear signals using FSM–CE beta excentric of centric variable Îą : đ?’”.đ?‘ đ?‘–đ?‘›(đ?œ˝ ─ đ?œş), (4) BexÎą = arcsin , (Fig. 2 ď ľ) 2 √1+đ?‘ −2đ?’”đ?‘?đ?‘œđ?‘ (đ?œ˝ ─ đ?œş)

It can be observed from Figure 2 that for extreme values of numerical linear excentricity s, i.e. for s = ± 1, the graphs of the functions are in broken straight lines, respectively they are symmetrical triangular signals for bexθ and asymmetrical triangular signals for Bexα.

In the next paragraph (§ 3) it is shown how these signals can be converted into step signals, respectively in FSM–CE Smarandache stepped functions using modified FSM–CE bexθ (bexmθ).

bexθ

2

2

Bexα

3 2

3 2

2

3 2

2

2

3 2

2

2

2

Fig. 4 The graphs FSM–CE bexθ  and Bexα 

3. TRANSFORMATION OF LINEAR BROKEN FUNCTIONS IN EVOLUTIVE SMARANDACHE STEPPED FUNCTIONS We will proceed as follows: From the family of FSM–CE bexθ Âą BexÎą, shown, for example, in Figure 5, for s ďƒŽ [-1, +1] with the step 0,2 ď ° we will select those of s = Âą 1 ď Ź, next by transformation FSM–CE bexθ, videlicet: s.sin(đ?›‰ ─ đ?›†) (5) bexθ + BexÎą = arcsin[s.sin(θ ─ Îľ)] + arcsin ďƒ 2 √1+đ?‘ −2đ?‘ cos(đ?›‰ ─ đ?›†)

(6)

bexmθ + Bexmι = C1.θ + bexθ ¹ C2ι + Bexι = C.θ + arcsin[s.sin(θ ─ ξ)] ¹ arcsin

s.sin(� ─ �)

,

√1+đ?‘ 2 −2đ?‘ cos(đ?›‰ ─ đ?›†)

(C = C1 + C2) next, in footnote ď ą, the Smarandache stepped functions graphs are shown (see Fig. 5).

Plot[Evaluate[Table[{ArcSin[0.2đ?‘ Sin[đ?‘Ą]] + ArcSin[0.2đ?‘ Sin[đ?‘Ą]/ Sqrt[1 + (0.2đ?‘ )^2 − 0.4đ?‘ Cos[đ?‘Ą]]]}, {đ?‘ , −5,5}], {đ?‘Ą, 0,2Pi}]]

Plot[Evaluate[Table[{ArcSin[0.2đ?‘ Sin[đ?‘Ą]] − ArcSin[0.2đ?‘ Sin[đ?‘Ą]/ Sqrt[1 + (0.2đ?‘ )^2 − 0.4đ?‘ Cos[đ?‘Ą]]]}, {đ?‘ , −5,5}], {đ?‘Ą, 0,2Pi} ]] 1.5

2

1.0

1 0.5

1

2

3

4

5

6

1

2

1

2

3

4

5

6

0.5

1 1.0

2

1.5

1.5

2

1.0 1

0.5 1

2

3

4

5

6

3

4

5

6

0.5

1

1.0

2

1.5

Plot[Evaluate[Table[{1.5đ?‘Ą + ArcSin[0.1đ?‘ Sin[đ?‘Ą]] + ArcSin[0.1đ?‘ Sin[đ?‘Ą] / Sqrt[1 + (0.1đ?‘ )^2 − 0.2đ?‘ Cos[đ?‘Ą]]]}, {đ?‘ , 10,10}], {đ?‘Ą, −2Pi, 2Pi} ]

Plot[Evaluate[Table[{0.5đ?‘Ą + đ??›đ??žđ??ąđ?›‰ − đ?? đ??žđ??ąđ?›‚}, {đ?‘ , 1,1}, {đ?‘Ą, −2Pi, 2Pi}]

4

5

6

4

2

2

2

4

6

6

4

2

2

4

6

2

5 4

Fig.5 The graphs FSM–CE 1,5.θ + bexθ + BexÎą ď ´ and 0,5.θ + bexθ - BexÎą ď ľ In this way, the resulting function, the stepped Smarandache function, is a FSM–CE excentric evolutive function of both variables. They can be added and / or subtracted, as in Figure 5, or multiplied and / or divided. Or the functions can be of 2θ, 3θ, ‌ or nθ (see Fig. 6).

Plot[Evaluate[Table[{ArcSin[0.2đ?‘ Sin[3đ?‘Ą]] + ArcSin[0.2đ?‘ Sin[đ?‘Ą] /Sqrt[1 + (0.2đ?‘ )^2 − 0.4đ?‘ Cos[đ?‘Ą]]]}, {đ?‘ , −5,5}], {đ?‘Ą, 0,2Pi} ]]

Plot[Evaluate[Table[{ArcSin[0.2đ?‘ Sin[đ?‘Ą]] + ArcSin[0.2đ?‘ Sin[2đ?‘Ą] /Sqrt[1 + (0.2đ?‘ )^2 − 0.4đ?‘ Cos[2đ?‘Ą]]]}, {đ?‘ , −5,5}], {đ?‘Ą, 0,2Pi} ]]

3

3

2

2

1

1 1

2

3

4

5

6

1

1

2

2

3

3

3

3

2

2

1

1

1

2

3

4

5

6

1

2

3

4

1

2

3

4

6

5

5

6

4

6

1

1

2

2

3

3

y = 2,4đ?‘Ą − bexθ + BexÎą

y = 2 t + bexθ + Bexι

15

10 5

10 6

4

2

5

2 5 10

1

2

3

4

5

6

Fig.6 The graphs FSM–CE 1,5.θ + bex3θ + BexÎą ď ´ and 0,5.θ + bexθ – Bex2Îą ď ľ (θ ≥ Îą ≥ t)

Plot[Evaluate[Table[{ArcSin[0.2đ?‘ Sin[2đ?‘Ą]/ Sqrt[1 + (0.2đ?‘ )^2 − 0.4đ?‘ Cos[2đ?‘Ą]]] −ArcSin[0.2đ?‘ Sin[2đ?‘Ą]]}, {đ?‘ , −5,5}], {đ?‘Ą, 0,2Pi}]]

Plot[Evaluate[Table[{ArcSin[0.2đ?‘ Sin[đ?‘Ą]] + ArcSin[0.2đ?‘ Sin[2đ?‘Ą] /Sqrt[1 + (0.2đ?‘ )^2 − 0.4đ?‘ Cos[2đ?‘Ą]]]}, {đ?‘ , −5,5}], {đ?‘Ą, 0,2Pi} ]]

1.5

3

1.0

2

0.5

1 1

2

3

4

5

1

6

0.5

1

1.0

2

1.5

3

2

3

4

5

6

1.5

3

1.0

2

0.5

1

1

2

3

4

5

6

6

4

2

2

0.5

1

1.0

2

1.5

4

6

3

Plot[{3đ?‘Ą + ArcSin[Sin[2đ?‘Ą]â „Sqrt[2 − 2Cos[2đ?‘Ą]]] − ArcSin[Sin[2đ?‘Ą]],3đ?‘Ą + ArcSin[−Sin[2đ?‘Ą]â „Sqrt[2 + 2Cos[2đ?‘Ą]]] − ArcSin[Sin[2đ?‘Ą]]}, {đ?‘Ą, −Pi, 2Pi}]

Plot[{2đ?‘Ą + ArcSin[−Sin[đ?‘Ą]] + ArcSin[−Sin[2đ?‘Ą]/ Sqrt[2 + 2Cos[2đ?‘Ą]]],2đ?‘Ą + ArcSin[Sin[đ?‘Ą]] + ArcSin[Sin[2đ?‘Ą]â „Sqrt[2 − 2Cos[2đ?‘Ą]]]}, {đ?‘Ą, −2Pi, 2Pi} ]

15

10

10

5

5 6

2

2

4

4

2

6

2

4

6

5

5 10

10

Fig. 7 The graphs of evolutive stepped Smarandache FSM–CE 3.θ + bex2θ + Bex2Îą ď ´ and 2.θ + bexθ – Bex2Îą ď ľ (θ ≥ Îą ≥ t) Plot[{ArcSin[−0.8 Sin[đ?‘Ą]â „Sqrt[1 + (0.8)2 + 1.6Cos[đ?‘Ą]]] − ArcSin[−0.8Sin[đ?‘Ą]]}, {đ?‘Ą, −2Pi, 2Pi}]

6

4

Plot[{ArcSin[0.8 Sin[đ?‘Ą]â „Sqrt[1 + (0.8)2 − 1.6Cos[đ?‘Ą]]] − ArcSin[0.8Sin[đ?‘Ą]]}, {đ?‘Ą, −2Pi, 2Pi}]

0.4

0.4

0.2

0.2

2

2 0.2 0.4

4

6

6

4

2

2

4

6

0.2 0.4

Fig. 8 The graphs of some evolutive FSM–CE found in science It's very interesting how, by simply adding the term nθ, the graphs transformation is so profound. Actually, for n = 2, distributing 2t in θ and ι to the two beta excentric (bexθ and Bexι) functions, we reach the excentric amplitude (aexθ and Aexι) functions, as result from relation (1).

In Figure 8, out of topic, the graphs of some functions are presented, which the author met in the scientific literature, and to which, in due course, did not know the equation, which is not currently the case.

From Figure 9 ď Ź ď ľ it results a new possibility to represent exactly the graphs of rectangular signals, in addition to their representation by derivative excentric FSM–CE of excentric dexθ variable, and by the excentric quadrilobic functions FSM–QE siqθ and coqθ of equations: đ???đ??žđ??ąđ?›‰ = 1 + đ??œđ??¨đ??Şđ?›‰ =

(7) {

đ??Źđ??˘đ??Şđ?›‰ =

đ?‘ .cos(đ?œƒâˆ’đ?œ€) √1−đ?‘ 2 đ?‘ đ?‘–đ?‘›2 (đ?œƒâˆ’đ?œ€) đ?‘?đ?‘œđ?‘ đ?œƒ

√1−đ?‘ 2 đ?‘ đ?‘–đ?‘›2 (đ?œƒâˆ’đ?œ€) đ?‘ đ?‘–đ?‘›đ?œƒ √1−đ?‘ 2 đ?‘?đ?‘œđ?‘ 2 (đ?œƒâˆ’đ?œ€)

Plot[Evaluate[Table[{ArcSin[0.2đ?‘ Sin[3đ?‘Ą]] + ArcSin[0.2đ?‘ Sin[đ?‘Ą] /Sqrt[1 + (0.2đ?‘ )^2 − 0.4đ?‘ Cos[đ?‘Ą]]]}, {đ?‘ , −5,5}], {đ?‘Ą, 0,2Pi}]]

Plot[Evaluate[Table[{ArcSin[0.2đ?‘ Sin[2đ?‘Ą]â „Sqrt[1 + (0.2đ?‘ )^2 − 0.4đ?‘ Cos[2đ?‘Ą]]] /Abs[ArcSin[0.2đ?‘ Sin[đ?‘Ą]]]}, {đ?‘ , −5,5}], {đ?‘Ą, −2Pi, 4Pi}]]

3 4

2 2

1

1

2

3

4

5

6

5

1

2

2

4

5

10

5

10

3 3

1.5

2

1.0

1

0.5 1

2

3

4

5

6

5

1

0.5

2

1.0

3

1.5 Plot[Evaluate[Table[{2.4đ?‘Ą − ArcSin[0.2đ?‘ Sin[3đ?‘Ą − ArcSin[0.2đ?‘ Sin[đ?‘Ą]/Sqrt[1 + (0.2đ?‘ )^2 − 0.4đ?‘ Cos[đ?‘Ą]]]} , {đ?‘ , 5,5}], {đ?‘Ą, 0,2Pi}]

Plot[{đ?‘Ą + Cos[đ?‘Ą]â „Sqrt[1 − (Sin[đ?‘Ą])2 ] + ArcSin[−Sin[2đ?‘Ą]â „Sqrt[2 + 2Cos[2đ?‘Ą]]] /Abs[ArcSin[−Sin[đ?‘Ą]]], đ?‘Ą + Cos[đ?‘Ą]â „Sqrt[1 − (Sin[đ?‘Ą])^2] − ArcSin[−Sin[2đ?‘Ą]â „Sqrt[2 + Cos[2đ?‘Ą]]]/Abs[ArcSin[−Sin[đ?‘Ą]]]}, {đ?‘Ą, −2Pi, 4Pi}]

15

15 10

10 5

5 5

1

2

3

4

5

6

5

10

5

Fig. 9 The graphs FSM–CE 2,4.θ + bex3θ + BexÎą ď ´ and C.θ + Bex2Îą / bexθ ď ľ (θ ≥ Îą ≥ t)

4. SMARANDACHE STEPPED FUNCTIONS ACHIEVED BY EXCENTRIC QUADRILOBIC FUNCTIONS OF NUMERICAL LINEAR EXCENTRICITY s= ¹ 1 Excentric quadrilobic functions FSM–QE siqθ and coqθ have the equations from the relations (7) and the graphs from Figure 10. It can be observed without difficulty that for s = ¹ 1 perfect rectangular functions / signals are obtained.

Plot[Evaluate[Table[{Cos[đ?‘Ą]â „Sqrt[1 − (0.2đ?‘ Sin[đ?‘Ą])2 ]}, {đ?‘ , −5, 5}], {đ?‘Ą, 0,2Pi} ]]

Plot[Evaluate[Table[{Sin[đ?‘Ą]â „Sqrt[1 − (0.2đ?‘ Cos[đ?‘Ą])2 ]}, {đ?‘ , −5, 5}], {đ?‘Ą, 0,2Pi} ]]

1.0

1.0

0.5

0.5

1

2

3

4

5

1

6

0.5

0.5

1.0

1.0

2

3

4

5

6

s=Âą1 1.0

1.0

0.5

0.5

1

2

3

4

5

1

6

0.5

0.5

1.0

1.0

2

3

4

5

6

Fig. 10 The graphs FSM– QE coqθ ď ´ and siqθ ď ľ

coqθ + siqθ

6

4

coqθ ─ siqθ

2

2

1

1

2

2

4

6

6

4

2

2

1

1

2

2

4

6

θ + coqθ + siqθ

θ – coq2θ + siqθ 10

10

5 5

5 5

5

5

10

10

5 5

Plot[{Cos[đ?‘Ą]â „Sqrt[1 − (Sin[đ?‘Ą])2 ] + Cos[2đ?‘Ą]â „Sqrt[1 − (Sin[2đ?‘Ą])2 ] − Sin[đ?‘Ą]â „Sqrt[1 − (Cos[đ?‘Ą])2 ]}, {đ?‘Ą, −2Pi, 2Pi}]

Plot[{Cos[đ?‘Ą]â „Sqrt[1 − (Sin[đ?‘Ą])2 ] + Cos[2đ?‘Ą]â „Sqrt[1 − (Sin[2đ?‘Ą])2 ] + 2 Sin[đ?‘Ą]â „Sqrt[1 − (Cos[đ?‘Ą])2]}, {đ?‘Ą, −2Pi, 2Pi}]

3

4

2

2 1

6

4

2

2

4

6

6

4

2

2

4

6

1

2 2

4

3

(Sin[đ?‘Ą])2]

Plot[{−(Cos[đ?‘Ą]â „Sqrt[1 − (Sin[đ?‘Ą])2 ] + Cos[2đ?‘Ą]â „Sqrt[1 − (Sin[2đ?‘Ą])2 ] + 2 Sin[đ?‘Ą]â „Sqrt[1 − (Cos[đ?‘Ą])2])}, {đ?‘Ą, −2Pi, 2Pi}

Plot[{đ?‘Ą + 2 Cos[đ?‘Ą]â „Sqrt[1 − − Cos[2đ?‘Ą]/ Sqrt[1 − (Sin[2đ?‘Ą])^2] + Sin[đ?‘Ą]/ Sqrt[1 − (Cos[đ?‘Ą])^2]}, {đ?‘Ą, −2Pi, 2Pi}]

4 6 4

2

2 6

4

2

2

4

6

6

4

2

2

4

6

2

2

4 6

4

Plot[{(2 Cos[0.5đ?‘Ą]â „Sqrt[1 − (Sin[0.5đ?‘Ą])^2] + Cos[2đ?‘Ą]/ Sqrt[1 − (Sin[2đ?‘Ą])^2] + 2 Sin[3đ?‘Ą]â „Sqrt[1 − (Cos[3đ?‘Ą])^2])} , {đ?‘Ą, −2Pi, 2Pi}]

6

4

Plot[{(2 Cos[đ?‘Ą]â „Sqrt[1 − (Sin[đ?‘Ą])^2] + Cos[2đ?‘Ą]/ Sqrt[1 − (Sin[2đ?‘Ą])^2] + 2 Sin[3đ?‘Ą]â „Sqrt[1 − (Cos[3đ?‘Ą])^2])} , {đ?‘Ą, −2Pi, 2Pi}]

4

4

2

2

2

2

4

6

6

4

2

2

2

2

4

4

4

Fig. 11 The graphs FSM– QE coqmθ ď ´ siqnθ ď ľ and their combinations for s = 1

6

BIBLIOGRAPHY 1

Mircea E. Şelariu

2

Mircea E. Şelariu

3

Mircea E. Şelariu

4

Mircea E. Şelariu

5

Mircea E. Şelariu

6

Mircea E. Şelariu

7

Mircea E. Şelariu

8

Mircea E. Şelariu

9 10

Mircea E. Şelariu Mircea E. Şelariu

11

Mircea E. Şelariu

12

Mircea E. Şelariu

13

Mircea E. Şelariu

14

Mircea E. Şelariu, Smarandache Florentin, Niţu Marian Mircea E. Şelariu,

15

NEMĂRGINIREA ŞI MĂREŢIA SUPERMATEMATICII. NOTA Internet I : FUNCŢII SUPERMATEMATICE EFECTIVE (FSEf) NEMĂRGINIREA ŞI MĂREŢIA SUPERMATEMATICII. NOTA Internet II : FUNCŢII SUPERMATEMATICE EVOLUTIVE NEMĂRGINIREA ŞI MĂREŢIA SUPERMATEMATICII. Internet NOTA III : FUNCŢII SM REPREZENTÂND SEMNALE LINIARE FRÂNTE NEMĂRGINIREA ŞI MĂREŢIA SUPERMATEMATICII. Internet NOTA IV : FUNCŢII SUPERMATEMATICE EXCENTRICOELEVATE NEMĂRGINIREA ŞI MĂREŢIA SUPERMATEMATICII. Internet NOTA V : FUNCŢII SUPERMATEMATICE EVOLUTIVE EXCENTRICE DE VARIABILE EXCENTRICOCENTRICE NEMĂRGINIREA ŞI MĂREŢIA SUPERMATEMATICII. Internet NOTA VI : FUNCŢII SUPERMATEMATICE EVOLUTIVE ELEVATOEXCENTRICE DE θ şi, respectiv, α NEMĂRGINIREA ŞI MĂREŢIA SUPERMATEMATICII. Internet NOTA VII : FUNCŢII SUPERMATEMATICE EVOLUTIVE ELIPTICE CENTRICOEXCENTRICE DE PERIOADĂ 4K(k) NEMĂRGINIREA ŞI MĂREŢIA SUPERMATEMATICII Internet Nota VIII:FUNCŢII SUPERMATEMATICE EVOLUTIVE ELIPTICE CENTRICOEXCENTRICE DE PERIOADĂ 2π Ed. POLITEHNICA, Timişoara, 2007 SUPERMATEMATICA. FUNDAMENTE Ed. POLITEHNICA, Timişoara, 2012 SUPERMATEMATICA. FUNDAMENTE VOL.I ŞI VOL. II EDIŢIA A 2-A Ed, Matrix Rom, Buc, 2015 SUPERMATEMATICA. FUNDAMENTE VOL.I ŞI VOL. II EDIŢIA A 3-A Com.VII Conf. Internaţ. de Ing. Manag. şi SUPERMATEMATICA Tehn. TEHNO’95, Timişoara Vol. 9 Matematică Aplicată, pag. 41…64 Scientia Magna Vol. 3 (2007), SMARANDACHE STEPPED No. 1, 81-92 FUNCTIONS INTERNATIONAL JOURNAL OF CARDINAL FUNCTIONS AND GEOMETRY INTEGRAL FUNCTIONS Vol. 1 (2012), No. 1, 27 - 40 TECHNO-ART OF SELARIU SUPERMATHEMATICS FUNCTIONS VOL I ŞI VOL. II

American Research Press, 2007