BROKEN LINES SIGNALS AND SMARANDACHE STEPPED FUNCTIONS

Transcription

1 BROKEN LINES SIGNALS AND SMARANDACHE STEPPED FUNCTIONS Mircea Eugen Şelariu. INTRODUCTION The evolutive supermathematical functions (FSM Ev) are discussed in the papers [], [], [], [], [], [], [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 (Fig. ) we present the main signals used in technics. Fig. Sinusoidal centric signals and linear signals in straight line segments and A number of Smarandache stepped functions are presented in the paper [], 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 Plot[Evaluate[Table[{t ArcSin[sSin[t]]}, {s,, +}], {t, Pi, Pi}]] cu pasul 0, y = aex[θ, S(,0 )], y = aex[θ, S(-,0 )] 0 0 y = aex m [θ, S(,0 )], Fig. Smarandache stepped functions represented using FSM CE aexθ and aexmθ and in D They have the equation: () aexθ = θ arcsin[s.sin(θ ε)] = θ bexθ, Figure or the modified one: () aex m θ = C.θ arcsin[s.sin(n.θ ε)], Figure with C [0,; ] and n =.

3 Plot[Evaluate[Table[{(t ArcSin[ 0.sSin[t]]) Cos[t] Sqrt[ ( 0.sSin[t]) ]}, {s, 0,0}], {t, Pi, Pi}]] Plot[{(t ArcSin[Sin[t]]) Cos[t] Sqrt[ (Sin[t]) ], t ArcSin[ Sin[t]] Cos[t] Sqrt[ ( Sin[t]) ]}, {t, Pi, Pi}] Plot[{(t ArcSin[ Sin[t] Sqrt[ + Cos[t]] ]) Cos[t]/ Sqrt[ (Sin[t])^], (t ArcSin[Sin[t] Sqrt[ Cos[t]] ]) Cos[t] Sqrt[ (Sin[t])^]}, {t, Pi, Pi}] 8 Plot[{(t ArcSin[ Sin[t] Sqrt[ + Cos[t]] ]) Cos[t]/ Sqrt[ (Sin[t])^], (t ArcSin[Sin[t] Sqrt[ Cos[t]] ]) Cos[t] Sqrt[ (Sin[t])^]}, {t, Pi, Pi}] Fig. Fascicle of graphs of FSM CE Smarandache aexmθ with cascade aspect Some graphs of the fascicle of Smarandache stepped functions have a waterfall or cascade aspect, such as those shown in Figure and because some graphics are in blue, of which were extracted, for s = ±, Smarandache stepped functions; some steps are visibly harder to escalate, but not impossible.. 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 : () bexθ = arcsin[s.sin(θ ε)], (Fig. ) and asymmetric saw tooth linear signals using FSM CE beta excentric of centric variable α : () Bexα = arcsin s.sin(θ ε), +s scos(θ ε), (Fig. )

4 It can be observed from Figure that for extreme values of numerical linear excentricity s, i.e. for s = ±, 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 ( ) it is shown how these signals can be converted into step signals, respectively in FSM CE Smarandache stepped functions using modified FSM CE bexθ (bex m θ). bexθ Bexα Fig. The graphs FSM CE bexθ and Bexα

5 . TRANSFORMATION OF LINEAR BROKEN FUNCTIONS IN EVOLUTIVE SMARANDACHE STEPPED FUNCTIONS We will proceed as follows: From the family of FSM CE bexθ ± Bexα, shown, for example, in Figure, for s [-, +] with the step 0, we will select those of s = ±, next by transformation FSM CE bexθ, videlicet: s.sin(θ ε) () bexθ + Bexα = arcsin[s.sin(θ ε)] + arcsin +s s cos (θ ε) () bex m θ + Bex m α = C.θ + bexθ ± C α + Bexα = C.θ + arcsin[s.sin(θ ε)] ± arcsin s.sin(θ ε), +s s cos (θ ε) (C = C + C ) next, in footnote, the Smarandache stepped functions graphs are shown (see Fig. ). Plot[Evaluate[Table[{ArcSin[0.sSin[t]] + ArcSin[0.sSin[t]/ Sqrt[ + (0.s)^ 0.sCos[t]]]}, {s,,}], {t, 0,Pi}]] Plot[Evaluate[Table[{ArcSin[0.sSin[t]] ArcSin[0.sSin[t]/ Sqrt[ + (0.s)^ 0.sCos[t]]]}, {s,,}], {t, 0,Pi} ]] Plot[Evaluate[Table[{.t + ArcSin[0.sSin[t]] + ArcSin[0.sSin[t] / Sqrt[ + (0.s)^ 0.sCos[t]]]}, {s, 0,0}], {t, Pi, Pi} ].0. Plot[Evaluate[Table[{0.t + bexθ Bexα}, {s,,}, {t, Pi, Pi}] Fig. The graphs FSM CE,.θ + bexθ + Bexα and 0,.θ + bexθ - Bexα 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, or multiplied and / or divided. Or the functions can be of θ, θ, or nθ (see Fig. ).

6 Plot[Evaluate[Table[{ArcSin[0.sSin[t]] + ArcSin[0.sSin[t] /Sqrt[ + (0.s)^ 0.sCos[t]]]}, {s,,}], {t, 0,Pi} ]] Plot[Evaluate[Table[{ArcSin[0.sSin[t]] + ArcSin[0.sSin[t] /Sqrt[ + (0.s)^ 0.sCos[t]]]}, {s,,}], {t, 0,Pi} ]] y =,t bexθ + Bexα y = t + bexθ + Bexα 0 0 Fig. The graphs FSM CE,.θ + bexθ + Bexα and 0,.θ + bexθ Bexα (θ α t) Plot[Evaluate[Table[{ArcSin[0.sSin[t]/ Sqrt[ + (0.s)^ 0.sCos[t]]] ArcSin[0.sSin[t]]}, {s,,}], {t, 0,Pi}]] Plot[Evaluate[Table[{ArcSin[0.sSin[t]] + ArcSin[0.sSin[t] /Sqrt[ + (0.s)^ 0.sCos[t]]]}, {s,,}], {t, 0,Pi} ]] 0..0.

7 Plot[{t + ArcSin[Sin[t] Sqrt[ Cos[t]] ] ArcSin[Sin[t]],t + ArcSin[ Sin[t] Sqrt[ + Cos[t]]] ArcSin[Sin[t]]}, {t, Pi, Pi}] Plot[{t + ArcSin[ Sin[t]] + ArcSin[ Sin[t]/ Sqrt[ + Cos[t]]],t + ArcSin[Sin[t]] + ArcSin[Sin[t] Sqrt[ Cos[t]] ]}, {t, Pi, Pi} ] Fig. 7 The graphs of evolutive stepped Smarandache FSM CE.θ + bexθ + Bexα and.θ + bexθ Bexα (θ α t) 0 Plot[{ArcSin[ 0.8 Sin[t] Sqrt[ + (0.8) +.Cos[t]] ] ArcSin[ 0.8Sin[t]]}, {t, Pi, Pi}] Plot[{ArcSin[0.8 Sin[t] Sqrt[ + (0.8).Cos[t]] ] ArcSin[0.8Sin[t]]}, {t, Pi, Pi}] 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 =, distributing t in θ and α to the two beta excentric (bexθ and Bexα) functions, we reach the excentric amplitude (aexθ and Aexα) functions, as result from relation (). 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 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: (7) { dexθ = + coqθ = siqθ = s.cos (θ ε) s sin (θ ε) cosθ s sin (θ ε) sinθ s cos (θ ε)

8 Plot[Evaluate[Table[{ArcSin[0.sSin[t]] + ArcSin[0.sSin[t] /Sqrt[ + (0.s)^ 0.sCos[t]]]}, {s,,}], {t, 0,Pi}]] Plot[Evaluate[Table[{ArcSin[0.s Sin[t] Sqrt[ + (0.s)^ 0.sCos[t]] ] /Abs[ArcSin[0.sSin[t]]]}, {s,,}], {t, Pi, Pi}]] Plot[Evaluate[Table[{.t ArcSin[0.sSin[t ArcSin[0.sSin[t]/Sqrt[ + (0.s)^ 0.sCos[t]]]}, {s,,}], {t, 0,Pi}]. Plot[{t + Cos[t] Sqrt[ (Sin[t]) ] + ArcSin[ Sin[t] Sqrt[ + Cos[t]] ] /Abs[ArcSin[ Sin[t]]], t + Cos[t] Sqrt[ (Sin[t])^] ArcSin[ Sin[t] Sqrt[ + Cos[t]] ]/Abs[ArcSin[ Sin[t]]]}, {t, Pi, Pi}] 0 0 Fig. 9 The graphs FSM CE,.θ + bexθ + Bexα and C.θ + Bexα / bexθ (θ α t). SMARANDACHE STEPPED FUNCTIONS ACHIEVED BY EXCENTRIC QUADRILOBIC FUNCTIONS OF NUMERICAL LINEAR EXCENTRICITY s= ± Excentric quadrilobic functions FSM QE siqθ and coqθ have the equations from the relations (7) and the graphs from Figure 0. It can be observed without difficulty that for s = ± perfect rectangular functions / signals are obtained.

9 Plot[Evaluate[Table[{Cos[t] Sqrt[ (0.sSin[t]) ]}, {s,, }], {t, 0,Pi} ]].0.0 Plot[Evaluate[Table[{Sin[t] Sqrt[ (0.sCos[t]) ]}, {s,, }], {t, 0,Pi} ]] s = ± Fig. 0 The graphs FSM QE coqθ and siqθ coqθ + siqθ coqθ siqθ

10 0 θ + coqθ + siqθ θ coqθ + siqθ 0 0 Plot[{Cos[t] Sqrt[ (Sin[t]) ] + Cos[t] Sqrt[ (Sin[t]) ] Sin[t] Sqrt[ (Cos[t]) ]}, {t, Pi, Pi}] 0 Plot[{Cos[t] Sqrt[ (Sin[t]) ] + Cos[t] Sqrt[ (Sin[t]) ] + Sin[t] Sqrt[ (Cos[t]) ]}, {t, Pi, Pi}] Plot[{t + Cos[t] Sqrt[ (Sin[t]) ] Cos[t]/ Sqrt[ (Sin[t])^] + Sin[t]/ Sqrt[ (Cos[t])^]}, {t, Pi, Pi}] Plot[{( Cos[0.t] Sqrt[ (Sin[0.t])^] + Cos[t]/ Sqrt[ (Sin[t])^] + Sin[t] Sqrt[ (Cos[t])^])}, {t, Pi, Pi}] Plot[{ (Cos[t] Sqrt[ (Sin[t]) ] + Cos[t] Sqrt[ (Sin[t]) ] + Sin[t] Sqrt[ (Cos[t]) ])}, {t, Pi, Pi} Plot[{( Cos[t] Sqrt[ (Sin[t])^] + Cos[t]/ Sqrt[ (Sin[t])^] + Sin[t] Sqrt[ (Cos[t])^])}, {t, Pi, Pi}] Fig. The graphs FSM QE coqmθ siqnθ and their combinations for s =

11 BIBLIOGRAPHY Mircea E. Şelariu NEMĂRGINIREA ŞI MĂREŢIA SUPERMATEMATICII. NOTA Internet I : FUNCŢII SUPERMATEMATICE EFECTIVE (FSEf) Mircea E. Şelariu NEMĂRGINIREA ŞI MĂREŢIA SUPERMATEMATICII. NOTA Internet II : FUNCŢII SUPERMATEMATICE EVOLUTIVE Mircea E. Şelariu NEMĂRGINIREA ŞI MĂREŢIA SUPERMATEMATICII. Internet NOTA III : FUNCŢII SM REPREZENTÂND SEMNALE LINIARE FRÂNTE Mircea E. Şelariu NEMĂRGINIREA ŞI MĂREŢIA SUPERMATEMATICII. Internet NOTA IV : FUNCŢII SUPERMATEMATICE EXCENTRICOELEVATE Mircea E. Şelariu NEMĂRGINIREA ŞI MĂREŢIA SUPERMATEMATICII. Internet NOTA V : FUNCŢII SUPERMATEMATICE EVOLUTIVE EXCENTRICE DE VARIABILE EXCENTRICOCENTRICE Mircea E. Şelariu NEMĂRGINIREA ŞI MĂREŢIA SUPERMATEMATICII. Internet NOTA VI : FUNCŢII SUPERMATEMATICE EVOLUTIVE ELEVATOEXCENTRICE DE θ şi, respectiv, α 7 Mircea E. Şelariu NEMĂRGINIREA ŞI MĂREŢIA SUPERMATEMATICII. Internet NOTA VII : FUNCŢII SUPERMATEMATICE EVOLUTIVE ELIPTICE CENTRICOEXCENTRICE DE PERIOADĂ K(k) 8 Mircea E. Şelariu NEMĂRGINIREA ŞI MĂREŢIA SUPERMATEMATICII Internet Nota VIII:FUNCŢII SUPERMATEMATICE EVOLUTIVE ELIPTICE CENTRICOEXCENTRICE DE PERIOADĂ π 9 Mircea E. Şelariu SUPERMATEMATICA. FUNDAMENTE Ed. POLITEHNICA, Timişoara, Mircea E. Şelariu SUPERMATEMATICA. FUNDAMENTE Ed. POLITEHNICA, Timişoara, 0 VOL.I ŞI VOL. II EDIŢIA A -A Mircea E. Şelariu SUPERMATEMATICA. FUNDAMENTE Ed, Matrix Rom, Buc, 0 VOL.I ŞI VOL. II EDIŢIA A -A Mircea E. Şelariu SUPERMATEMATICA Com.VII Conf. Internaţ. de Ing. Manag. şi Tehn. TEHNO 9, Timişoara Vol. 9 Matematică Aplicată, pag. Mircea E. Şelariu SMARANDACHE STEPPED FUNCTIONS Scientia Magna Vol. (007), No., 8-9 Mircea E. Şelariu, Smarandache Florentin, CARDINAL FUNCTIONS AND INTEGRAL FUNCTIONS INTERNATIONAL JOURNAL OF GEOMETRY Vol. (0), No., 7-0 Niţu Marian Mircea E. Şelariu, TECHNO-ART OF SELARIU SUPERMATHEMATICS FUNCTIONS VOL I ŞI VOL. II American Research Press, 007