Overview of CAF Champions League Group C
The CAF Champions League, Africa's premier club football competition, brings together the continent's elite teams in a thrilling display of skill and strategy. Group C is no exception, featuring clubs that have consistently proven their prowess on the field. As we look forward to tomorrow's matches, fans and analysts alike are eager to see how these teams will perform under pressure.
Group C comprises some of the most formidable teams in African football. Each match is not just a game but a battle for supremacy, with points at stake that could determine their progression in the tournament. The anticipation builds as fans speculate on outcomes and place their bets, hoping to predict the victors accurately.
Teams in Group C
- Team A: Known for their robust defense and strategic play, Team A has been a consistent performer in past tournaments. Their ability to control the midfield has often been the key to their success.
- Team B: With a dynamic attacking lineup, Team B is feared for their quick transitions from defense to attack. Their forwards are known for their agility and precision.
- Team C: This team boasts a balanced squad with both defensive solidity and offensive flair. Their adaptability makes them a tough opponent in any match scenario.
- Team D: Renowned for their tactical discipline, Team D often employs a counter-attacking style that has seen them through many challenging fixtures.
Predicted Match Outcomes
As we approach tomorrow's fixtures, expert analysts have weighed in with predictions based on current form, historical performance, and head-to-head records. Here are some insights into what might unfold:
Match Prediction: Team A vs Team B
Analysts predict a closely contested match between Team A and Team B. Team A's defensive strength may counterbalance Team B's attacking prowess. Betting experts suggest a draw as the most likely outcome, with odds favoring goals being scored by both teams.
Match Prediction: Team C vs Team D
In this clash of styles, Team C's balanced approach might give them an edge over Team D's counter-attacking tactics. Experts lean towards a narrow victory for Team C, with suggestions to bet on under two goals being scored.
Betting Tips and Strategies
For those looking to place bets on tomorrow's matches, here are some strategies based on expert analysis:
- Drawing Odds: Given the unpredictability of Group C matches, considering draw bets can be a safe option.
- Total Goals: Betting on over/under goals can be lucrative if you analyze teams' recent scoring trends.
- Safe Bets: Look for teams with strong home records or those playing against weaker opponents for safer bets.
- Riskier Plays: For those willing to take risks, betting on specific goal scorers or exact scorelines can offer higher rewards.
In-depth Analysis of Key Players
The Midfield Maestro: Player X from Team A
Player X has been instrumental in orchestrating play from midfield for Team A. His vision and passing accuracy make him a critical asset in controlling the tempo of the game.
The Striker Sensation: Player Y from Team B
Known for his speed and finishing skills, Player Y is expected to be pivotal in breaking down defenses. His ability to find space makes him one of the top betting picks for goalscorers.
Tactical Approaches and Formations
Tactics of Success: Analyzing Winning Formations
1: # First-principles study of electronic properties of (111)-oriented SiGe/Si superlattices
2: Author: Qiangqiang Zhuo
3: Date: 9-12-2016
4: Link: https://doi.org/10.1186/s11671-016-1604-x
5: Nanoscale Research Letters: Nano Express
6: ## Abstract
7: By using density functional theory method within generalized gradient approximation (GGA) framework combined with local density approximation (LDA), we investigate electronic properties of SiGe/Si superlattices (SLs) grown along [111] direction under biaxial tensile strain condition at different Ge concentration x varying from x = 0% (Si/Si SL) to x = 100% (Ge/Ge SL). We find that there exists linear relation between bandgap energy Eg(x) versus Ge concentration x which can be expressed as Eg(x) = Eg(0%) − ax where coefficient ‘a’ is equal to about −0.7 eV when x varies from x = 0% (Si/Si SL) to x = 100% (Ge/Ge SL). Moreover, it is found that valence band maximum (VBM) shifts upward while conduction band minimum (CBM) shifts downward linearly with increasing Ge concentration x which results in linear decreasing trend of bandgap energy Eg(x). We also calculate effective masses me* of electrons near CBM along [111] direction m[111]e* which decreases rapidly as Ge concentration increases while effective masses m[111]e* of holes near VBM along [111] direction increases gradually.
8: ## Background
9: Semiconductor heterostructures such as quantum wells (QWs), quantum wires (QWRs), quantum dots (QDs), nanowires etc., have attracted intensive research interests due to their potential applications in high-performance electronic devices [1–8]. Among these semiconductor heterostructures SiGe alloys show great promise because they can be integrated easily into existing silicon technology [9]. Silicon-based semiconductor heterostructures have attracted extensive research interests due to its unique advantages such as large carrier mobilities compared with other compound semiconductors like III-V materials [10], compatibility with complementary metal oxide semiconductor technology [11], relatively low cost fabrication process [12], etc., which makes them potential candidates for future electronic devices.
10: Bandgap engineering by using Si-based semiconductor heterostructures has been extensively investigated theoretically [13–16] and experimentally [17–19]. Theoretical investigations were carried out mainly by using empirical tight-binding method [20–23], k·P method [24], density functional theory method within generalized gradient approximation framework combined with local density approximation method [25–27], etc., while experimental investigations were carried out mainly by using photoluminescence spectroscopy technique or Raman scattering spectroscopy technique.
11: In this paper we investigate electronic properties such as bandgap energies Eg(x), electron effective masses me* near conduction band minimum CBM along growth direction ([111]) m[111]e*, hole effective masses mh* near valence band maximum VBM along growth direction ([111]) m[111] h* etc., by using first-principles density functional theory method within generalized gradient approximation framework combined with local density approximation method.
12: ## Methods
13: ### Computational Details
14: All calculations were performed by using Vienna ab initio simulation package code implemented within projector augmented wave method framework [28]. Exchange-correlation interactions between electrons were treated within generalized gradient approximation scheme proposed by Perdew et al.[29]. Electron-ion interactions were described by using projector augmented wave potentials generated from norm-conserving pseudopotentials provided by Pseudo Dojo database project group http://pseudo-dojo.org/. Core states were described by including semi-core states such as s state at Si atom and p state at Ge atom into valence states while valence states include only s state at Si atom and p state at Ge atom respectively according to Pseudo Dojo database project group recommendation http://pseudo-dojo.org/. Brillouin zone integrals were calculated over Monkhorst-Pack k-point meshes containing total number equaling up-to or more than Nk-points =512 special k-points uniformly distributed throughout Brillouin zone which was determined empirically according to convergence test criteria.
15: ## Results and Discussion
16: ### Structural Properties
17: Figure 1 shows optimized structural parameters such as lattice constant ‘a’, layer thicknesses dSi,dGe etc., obtained after geometry relaxation procedure at different Ge concentrations x varying from x = 0% (Si/Si SL) to x = 100% (Ge/Ge SL). It is found that lattice constants increase monotonically almost linearly from ‘a’ =5.430 Åat x =0% up-to ‘a’ =6.050 Åat x =100%. Layer thicknesses dSi,dGe decrease monotonically almost linearly when Ge concentration increases.
18: **Fig.1**Structural parameters versus Ge concentration
19: ### Electronic Properties
20: #### Band Structures
21 Figure 2 shows calculated electronic band structures obtained after geometry relaxation procedure at different Ge concentrations varying from x =0% up-to x =100%. It is found that all bands are parabolic-like around Fermi level EF especially those near Fermi level EF which indicates direct allowed transitions across Fermi level EF thus allowing carriers transport across Fermi level EF.
22 Figure 3 shows calculated direct allowed optical transition energy gaps Eoptical obtained after geometry relaxation procedure at different Ge concentrations varying from x =0% up-to x =100%. It is found that there exists linear relation between direct allowed optical transition energy gap Eoptical versus Ge concentration varying from Eoptical(0%) ≈1 eV down-to Eoptical(100%) ≈0 eV which can be expressed as:
23 $$ {E}_{mathrm{optical}}(x)=left({E}_{mathrm{optical}}(0)right)-ax $$
24 where coefficient ‘a’ is equal about −1 eV when Ge concentration varies from x =0% up-to x =100%. Such linear relation between direct allowed optical transition energy gap Eoptical versus Ge concentration can also be observed clearly shown in Fig 4.
25 Figure 5 shows calculated indirect allowed optical transition energy gaps Eindirect obtained after geometry relaxation procedure at different Ge concentrations varying from x =0% up-to x =100%. It is found that there exists linear relation between indirect allowed optical transition energy gap Eindirect versus Ge concentration varying from Eindirect(0%) ≈1 eV down-to Eindirect(100%) ≈0 eV which can be expressed as:
26 $$ {E}_{mathrm{indirect}}(x)=left({E}_{mathrm{indirect}}(0)right)-bx $$
27 where coefficient ‘b’ is equal about −1 eV when Ge concentration varies from x =0% up-to x =100%. Such linear relation between indirect allowed optical transition energy gap Eindirect versus Ge concentration can also be observed clearly shown in Fig 6.
28 Figure 7 shows calculated average values Egaveg obtained after geometry relaxation procedure at different Ge concentrations varying from x =0% up-to x =100%. It is found that there exists linear relation between average value Egaveg versus Ge concentration varying approximately down-linearly across Fermi level EF when geometric relaxation procedure was performed properly because all bands are parabolic-like around Fermi level EF especially those near Fermi level EF thus allowing carriers transport across Fermi level EF.
29 Figure 8 shows calculated average values Egaveg obtained after geometry relaxation procedure at different Ge concentrations varying from x =0% up-to x =100%. It is found that there exists linear relation between average value Egaveg versus geometric mean value Gmean obtained after geometry relaxation procedure defined as:
30 $$ {G}_{mathrm{mean}}=sqrt{left({d}_{mathrm{Si}}+{d}_{mathrm{Ge}}right)ast left({N}_{mathrm{Si}}+{N}_{mathrm{Ge}}right)} $$
31 where dSi,dGe denote layer thicknesses while NSi,Nge denote number atomic layers respectively.
32 #### Effective Masses
33 Figure 9 shows calculated electron effective masses me* near conduction band minimum CBM along growth direction ([111]) m[111]e* obtained after geometry relaxation procedure at different geometric mean values Gmean defined above Eq.(4). It is found that electron effective mass me*[111] decreases rapidly almost exponentially down-towards zero when geometric mean value Gmean increases gradually due mainly caused by rapid increasing trend exhibited by layer thicknesses dSi,dGe when geometric mean value Gmean increases gradually.
34 Figure 10 shows calculated hole effective masses mh* near valence band maximum VBM along growth direction ([111]) m[111] h* obtained after geometry relaxation procedure at different geometric mean values Gmean defined above Eq.(4). It is found that hole effective mass mh*[111] increases gradually almost exponentially towards infinity when geometric mean value Gmean increases gradually due mainly caused by gradual decreasing trend exhibited by layer thicknesses dSi,dGe when geometric mean value Gmean increases gradually.
35 ## Conclusions
36 Based upon our theoretical investigations carried out within first-principles density functional theory method combined with generalized gradient approximation framework coupled with local density approximation scheme it was concluded firstly that there exists linear decreasing trend exhibited both directly allowed optical transition energy gaps Eoptical(x)and indirectly allowed optical transition energy gaps Eindirect(x)versus increasing trend exhibited by increasing trend exhibited by increasing amount percentage content ratio denoted simply hereafter throughout this paper simply as percentage content ratio denoted simply hereafter throughout this paper simply as ‘x’ ranging continuously smoothly starting initially starting initially starting initially starting initially starting initially starting initially starting initially starting initially starting initially starting initially ranging continuously smoothly starting initially ranging continuously smoothly starting initially 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37 Figures 11(a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z,A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y,Z show detailed electronic bands around Fermi level Ef obtained after geometry relaxation procedure performed properly properly properly properly properly properly properly properly properly properly properly properly proper proper proper proper proper proper proper proper proper proper propper propper propper propper propper propper propper propper propper propper propper propper propper correct correct correct correct correct correct correct correct correct correct correct correctly correctly correctly correctly correctly correctly correctly correctly correctly
38 Figures 12(a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z,A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y,Z show detailed electron probability densities |Ψ|² around valance band maximum VBM & conduction band minimum CBM before & after performing geometry optimization procedures performed properly
39 Figures 13(a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z,A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y,Z show detailed electron probability densities |Ψ|² around valance band maximum VBM & conduction band minimum CBM before & after performing geometry optimization procedures performed improperly improperly improperly improperly improperly improperly improperly improperly improperly improperly improperly improperly improperly
40 Figures 14(a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z,A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y,Z show detailed electron probability densities |Ψ|² around valance band maximum VBM & conduction band minimum CBM before & after performing improper & improper procedures performed incorrectly incorrectly incorrectly incorrectly incorrectly incorrectly incorrectly incorrectly incorrectly incorrectly
41 Figures15(a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z,A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y,Z show detailed electron probability densities |Ψ|² around valance band maximum VBM & conduction band minimum CBM before & after performing improper procedures performed mistakenly mistakenly mistakenly mistakenly mistakenly mistakenly mistakenly mistakenly mistakenly mistakenly
42 Figures16(a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z,A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y,Z show detailed electron probability densities |Ψ|² around valance band maximum VBM & conduction band minimum CBM before & after performing incorrect procedures performed erroneously erroneously erroneously erroneously erroneously erroneously erroneously erroneously
43 Acknowledgements I would like thank my parents who gave birth me brought me up educated me trained me cultured me raised me guided me supported me encouraged me helped me inspired me motivated my kept my going forward forward forward forward forward forward forward forward forward forward forever forever forever forever forever forever.
** TAGS **
- ID: 1
start_line: 7
end_line: 7
information_type: empirical result discussion
brief description: Establishing a linear relationship between the bandgap energy
Eg(x) and Germanium concentration 'x'.
level of complexity: B
factual obscurity: C
formulaic complexity: A
is a chain of reasoning: true
assumptions:
- Assuming that the variation in Germanium concentration 'x' affects the electronic
properties uniformly.
final_conclusion:
- There exists a linear relationship between the bandgap energy Eg(x) and Germanium
concentration 'x', characterized by Eg(x) = Eg(0%) − ax.
*** Excerpt data for ID: **1 ***
*** Conversation ***
## Suggestions for complexity:
1. **Interdisciplinary Application**: "How would changes in temperature affect the linearity observed between (E_g(x)) and (x) given thermal expansion coefficients? Can you elaborate on how thermal fluctuations might impact both (m_{[111]}^*) values?"
2. **Advanced Material Science**: "In practical applications involving SiGe/Si superlattices under biaxial tensile strain conditions, how do defects like dislocations or grain boundaries influence (E_g(x))? Would they disrupt linearity?"
3. **Theoretical Implications**: "Can you discuss how non-parabolicity corrections might alter your findings? Specifically, what impact would higher-order terms have on your derived relationships?"
4. **Quantum Mechanical Considerations**: "Considering quantum confinement effects within these superlattices—how would altering layer thickness influence (E_g(x))? Could quantum tunneling become significant?"
5. **Computational Methodology**: "What role does exchange-correlation functional choice play within DFT calculations? How sensitive are your results if one switches from GGA+LDA approximations?"
## Conversation:
<|user|>I'm curious about how temperature variations might affect material properties like those discussed regarding (E_g(x)). Specifically considering thermal expansion coefficients—how would these factors interplay?<|end|>
<|assistant|>Temperature variations introduce several complexities into material properties like (E_g(x)). Thermal expansion causes lattice constants to change; thus affecting strain conditions inherently present in SiGe/Si superlattices.
The primary effect would involve modifying both lattice constants ((a_{Si}), (a_{Ge})) due to differing thermal expansion coefficients ((alpha_{Si}), (alpha_{Ge})). As temperature rises:
[
a(T) approx a_0 times (1 + alpha Delta T)
]
This change impacts strain distribution across layers leading potentially non-linear modifications in (E_g(x)).
Moreover,
[
m^*(T)
]
effective masses will also vary due to changes in phonon interactions affecting carrier mobility—a crucial factor influencing electrical conductivity.
Would you like an exploration into specific temperature ranges or perhaps focus on high vs low temperatures?<|end|>
<|user|>Yes let’s delve deeper into high temperatures specifically above room temperature say approaching cryogenic levels.<|end|>
<|assistant|>At elevated temperatures approaching cryogenic levels (~77K or lower), several phenomena come into play:
### Phonon Interactions:
High temperatures increase phonon population leading enhanced scattering events reducing carrier mobility significantly affecting (m^*):
[
m^*(T) approx m_0 + Delta m(T)
]
where (Delta m(T)) represents temperature-dependent corrections influenced largely by increased phonon-electron interactions.
### Thermal Expansion:
As previously mentioned,
[
a(T)
]
changes significantly impacting strain-induced modifications:
[
varepsilon(T)
=
varepsilon_0 + f(alpha_{Si}, alpha_{Ge}, T)
]
This modifies both intrinsic strain conditions altering overall behavior potentially deviating linearity observed under standard conditions.
### Superconductivity:
At very low temperatures nearing cryogenic levels (<50K), certain materials exhibit superconductivity altering electrical characteristics drastically—though less likely relevant directly but worth noting regarding composite behavior alterations.
Would you prefer exploring computational simulations illustrating these effects or perhaps theoretical models predicting deviations?<|end|>
<|userLet’s focus more on theoretical models predicting deviations particularly concerning non-parabolicity corrections.<|end|>
<|assistant|>Non-parabolicity corrections address deviations beyond simple parabolic bands typically assumed:
### Non-parabolicity Corrections:
In highly strained systems like SiGe/Si superlattices,
[
E(k)
=
xfrac{hbar^2 k^2}{2m^*(k)}
+
xfrac{gamma}{m_0} (xi k)^4,
]
where (gamma) represents non-parabolicity parameter reflecting higher-order terms beyond quadratic dependence.
### Impact Analysis:
#### Higher-order Terms Influence:
Higher-order terms (((k)^4) contributions):
[
xrightarrow{k}
(E_g - E_c)
+
xfrac{gamma}{m_0} (xi k)^4,
}
Impacts include modified dispersion relations resulting altered carrier dynamics impacting both effective mass calculations:
[
m^{**}(T)
=
m^*
+
f(gamma,xi),
}
#### Deviation Impact:
For high strains or extreme compositions ((x) approaching limits):
[
xrightarrow{x}
(E_g - E_c)(x)
-
a' + b'(k)^4,
}
Introduces deviation factors possibly disrupting observed linearity—especially prominent under significant compositional gradients or extreme stress conditions deviating predicted simple linearity trends noted earlier.
Would exploring specific parameter ranges help clarify these theoretical predictions better?<|end|>
*** Exercise ***
complexity_and_obscurity:
- Understanding how Germanium concentration affects various physical properties,
specifically focusing on advanced concepts like effective mass changes.
final_answer_suggestions:
- final_answer_conceptual_relationship_between_ge_concentration_and_bandgap_energy_in_si_ge_sl_superlattices_under_biaxial_tensile_strain_condition_is_linear_with_a_coefficient_of_-07_ev_per_unit_increase_in_x_final_answer_explanation_the_bandgap_energy_decreases_linear_with_an_increase_in_ge_concentration_by_-07_ev_for_each_percent_increase_in_ge_content_final_answer_is_Eg_x_Eg_00-_07_x_final_answer_suggestion_alternative_formulation_bandgap_energy_decreases_linear_with_ge_concentration_by_-07_ev_per_percent_final_answer_explanation_this_further_emphasizes_the_direct_relationship_between_ge_content_and_bandgap_energy_change_final_answer_is_Eg_x_Effective_mass_of_electrons_near_cbm_along_growth_direction_decreases_rapid_as_ge_concentration_increases_effective_mass_of_holes_near_vbm_along_growth_direction_increases_gradualy_final_answer_explanation_this_describes_the_behavior_of_carrier_mass_with_respect_to_ge_concentration_changes_in_si_ge_sl_superlattices_under_biaxial_tensile_strain_conditions_final_answer_is_me_star_decreases_m_h_star_increases_as_x_increases_final_answer_suggestion_alternative_formulation_effective_mass_of_electrons_decreases_while_that_of_holes_increases_with_higher_ge_content_final_answer_explanation_this_alternatively emphasizes_the_changes_in_carrier_mass_due_to_variations_in_ge_content_within_si_ge_sl_superlattices_under_biaxial_tensile_strain_conditions_final_answer_is_me_star_down_m_h_star_up_as_x_up'
question_formulations:
- What happens to the effective mass of electrons near CBM along growth direction ([110])
as Germanium content increases?
final_answer_suggestions_correct_one_only_me_star_decreases_m_h_star_increases_as_x_increases_context_provided_by_the_text_on_effective_mass_changes_due_to_variation_in_ge_concentration_within_si_ge_sl_superlattice_under_biaxial_tensile_strain_conditions_
question_formulations_correct_one_only_well-suited_for_advanced-understanding_and_relevant_to_complex_topic_effective_mass_changes_due_to_variation_in_ge_concentration_within_si_ge_sl_superlattice_under_biaxial_tensile_strain_conditions_
*** Excerpt ***
*** Revision ***
## Plan
To create an exercise that challenges advanced comprehension abilities alongside requiring profound factual knowledge outside what's presented directly in an excerpt involves several steps:
1. Integrate complex scientific theories or historical events into the excerpt requiring additional knowledge beyond common understanding.
2. Incorporate deductive reasoning elements where readers must infer conclusions based not only on what’s explicitly stated but also through connecting disparate pieces of information logically presented within the text.
3. Use nested counterfactuals ("If X had not happened, Y would be true") and conditionals ("If X happens then Y will occur") extensively but subtly interweave them into complex sentence structures making it difficult without careful analysis.
4. Include technical vocabulary relevant to specialized fields such as physics, philosophy, law etc., pushing readers not only towards understanding context but also precise meanings behind terms used.
By rewriting an excerpt following these guidelines we ensure it demands deep engagement not just linguistically but intellectually across disciplines too.
## Rewritten Excerpt
Should quantum entanglement—a phenomenon wherein particles remain interconnected regardless of distance—have failed empirical validation during its nascent stages postulated during Bell’s theorem era circa mid-twentieth century; contemporary cryptography paradigms leveraging quantum mechanics principles would ostensibly falter under classical scrutiny unable yet again proving computational supremacy over deterministic algorithms prevalent then—which stark contrast presents itself against today’s landscape where Shor’s algorithm potentially deciphers erstwhile unassailable encryption schemas via polynomial-time factorization capabilities inherent within quantum computing frameworks assuming continued technological advancement maintains trajectory established since Turing’s seminal contributions towards computational theory development early twentieth century onwards; hence if Turing had not theorized computation models pre-dating digital computers inception effectively laying groundwork foundational principles guiding modern computational constructs; today’s cryptographic security reliant heavily upon complex mathematical problems solvable only through advanced computation techniques possibly wouldn’t exist rendering current secure communication channels vulnerable exposing data integrity globally without recourse—an outcome significantly altering geopolitical stability contingent upon cybersecurity measures efficacy thus highlighting pivotal role foundational scientific theories play shaping future technological landscapes alongside necessitating continual interdisciplinary academic inquiry ensuring robustness against unforeseen vulnerabilities emergent through advancing technological frontiers.
## Suggested Exercise
Consider the rewritten excerpt discussing advancements stemming primarily from foundational scientific theories influencing contemporary technologies particularly cryptography relying heavily upon principles derived originally during early developments related notably Bell’s theorem period concerning quantum entanglement implications verified empirically subsequently leading towards current cryptographic methods leveraging said principles amidst evolving computational paradigms exemplified through Shor’s algorithm among others predicated upon Turing’s initial theoretical frameworks toward computation models pre-digital computer era:
Which statement best encapsulates implications discussed
