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[MATH] [MATH] . Due to Equation ( 16 [EQUATION] [MATH] . According to the inductive step, [EQUATION] [MATH] [MATH] Hence we have a cycle, which is a contradiction to satisfying . This concludes the proof of this lemma. |
Corollary 4 For any [MATH] [MATH] Corollary follows by invoking Lemma with [MATH] and relying on Null Player Lemma 6 Let [MATH] . If [MATH] satisfies [MATH] and EF , it holds for every player index [MATH] that |
[EQUATION] Proof. We prove the claim by induction over [MATH] and [MATH] , player index and number of players respectively. Base cases: |
1. The assertion holds for [MATH] and [MATH] due to Lemma 2. The assertion holds for [MATH] and [MATH] Due to Corollary Assume the assertion holds for [MATH] and [MATH] , for every [MATH] . Next, we show it holds for [MATH] , for [MATH] , while dealing with [MATH] afterwards. |
Case 1: Assume by contradiction that [MATH] EF suggests that the sum of probabilities remains [MATH] , and due to the inductive step we have |
[MATH] for [MATH] . Hence [EQUATION] Denote by [MATH] a player such that [EQUATION] and let [MATH] for [MATH] that satisfy [EQUATION] |
Consider the following game [EQUATION] where [MATH] [MATH] [MATH] for [MATH] and [MATH] for [MATH] . Note that all players but [MATH] and [MATH] are non-strategic, or alternatively every strategy they select has the same satisfaction level w.r.t. users [MATH] . We have the following cycle: |
[MATH] [MATH] . Due to Equation ( 18 ), we have [EQUATION] [MATH] . According to the inductive step, [EQUATION] [MATH] [MATH] Hence we have a cycle, which is a contradiction to satisfying |
Case 2: [MATH] . Due to EF [EQUATION] Denote by [MATH] a player such that [EQUATION] and let [MATH] for [MATH] that satisfy [EQUATION] |
Consider the following game: [EQUATION] where again [MATH] [MATH] [MATH] for [MATH] and [MATH] for [MATH] . We have the following cycle: |
[MATH] [MATH] . Due to Equation ( 20 [EQUATION] [MATH] . According to the inductive step, [EQUATION] [MATH] [MATH] Hence we obtained a cycle, which is a contradiction to satisfying . The only missing ingredient is the case of [MATH] Due to Lemma it holds that [MATH] ; thus, due to Symmetry we have |
[EQUATION] Using this technique we obtain [MATH] for every player [MATH] . Finally, due to EF [EQUATION] Since the Shapley mediator satisfies the assertion, we have |
[EQUATION] as well. This concludes the proof of the lemma. Proof of Theorem The proof of Theorem Follows directly from Lemma A.7 Proof of Theorem |
Proof. Recall that under the Shapley mediator it holds that [EQUATION] The following analysis holds for every [MATH] as defined in the model. For ease of notation, we handle the functions [MATH] and [MATH] as set functions. Namely, for [MATH] let |
[EQUATION] and [EQUATION] In addition, recall that a strategy profile [MATH] is often referred to as the set of items selected by the players. Next, we lower-bound the payoff of a player as a function of the social welfare. |
Lemma 7 For every strategy profile [MATH] it holds that [EQUATION] Proof. For a user [MATH] , exactly one of the following holds: |
If [MATH] , then [MATH] holds, with equality when player [MATH] offers the least satisfying item to [MATH] If [MATH] , player [MATH] gets the full difference between the terms: [MATH] , as well as at least [MATH] times [MATH] |
These cases are disjoint; thus, [EQUATION] Notice that if [MATH] it holds that [MATH] , and if [MATH] then [MATH] As a result, [EQUATION] |
which concludes the proof of this lemma. One more necessary definition is the following: Definition 2 (Submodular function) We say that [MATH] is submodular if for any [MATH] such that [MATH] and every [MATH] it holds that - |
[EQUATION] Note that by definition of [MATH] it is a monotonically increasing set function. In addition, Lemma 8 [MATH] is submodular. |
Proof. For arbitrary [MATH] and [MATH] , we need to show that [EQUATION] If [MATH] , then by monotonicity the right-hand side of Equation ( 23 ) equals zero while the left-hand side is non-negative. Alternatively, if [MATH] then [MATH] . Moreover, [MATH] ; hence, |
[EQUATION] By summing Equation ( 23 ) over all users, we get: Corollary 5 The social welfare function [MATH] is submodular. We are now ready to prove the theorem: denote the optimal solution as [MATH] and an arbitrary PNE profile [MATH] . Since [MATH] is in equilibrium it follows that [MATH] . Therefore - |
[EQUATION] Due to Lemma we have: [EQUATION] Summing Equation ( 25 ) over all players: [EQUATION] Observe that due to submodularity, for every [MATH] it holds that |
[EQUATION] Thus: [EQUATION] By substituting Equation ( 27 ) into Equation ( 26 ) we get: [EQUATION] Finally, by Equations ( 24 ) and ( 28 ) we have [MATH] , and |
[EQUATION] After upper-bounding the [MATH] , our objective is to show a game instance which achieves this bound. Consider a symmetric [MATH] -player game with [MATH] users, and items [MATH] . In addition: |
[EQUATION] The optimal social welfare is obtained when each player selects a unique item (e.g. player [MATH] selects [MATH] ). In that case, the Shapley mediator will display an item to every user with probability [MATH] ; hence the social welfare is [MATH] Observe that the strategy profile [MATH] is in equilibrium: co... |
[EQUATION] For [MATH] we get [MATH] ; therefore [MATH] is an equilibrium profile. Overall, [EQUATION] This concludes the proof of Theorem |
A.8 Proof of Proposition Proof. Consider a game with one user and one player with one strategy [MATH] , such that [MATH] for some [MATH] . It holds that [MATH] , while [MATH] . Therefore, [MATH] , which can be arbitrarily large. |
A.9 Proof of Proposition Proof. Let [MATH] be arbitrarily small, and consider the satisfaction matrix [EQUATION] Let [MATH] and [MATH] . Under TOP , the only PNE is [MATH] with [MATH] , but [MATH] ; therefore, [MATH] , which can be arbitrarily large. |
A.10 Proof of Lemma Proof. Fix an arbitrary user [MATH] and a strategy profile [MATH] , and denote [EQUATION] Observe that [EQUATION] |
The minimum value obtained by the function [MATH] in the segment [MATH] is [MATH] . Notice that Equation ( 29 ) holds for all users concurrently; hence |
[EQUATION] Ultimately, [EQUATION] Appendix B On user utility As mentioned in Section , numerical calculations show that the upper bound on [MATH] is far from tight. We present here the methods employed to obtain the tighter (numerical) bound. |
Recall that under the Shapley mediator, [EQUATION] Due to linearity, user utility cannot be less than [MATH] times the minimum utility one user can get. Hence, we focus on a game with one user. In addition, despite [MATH] being a function of the strategy profile, it is more convenient to present it as a function of the... |
[EQUATION] Note that [EQUATION] Hence [MATH] can be presented as [EQUATION] Therefore, [EQUATION] Thus, by taking the partial derivatives to zero, one can find the argmin point (more precisely, the argmin vector) of [MATH] . We solved this system of linear equations numerically for various values of [MATH] . The result... |
Appendix C Personalized offers The model defined in Section enables each player [MATH] to choose a single item out of [MATH] . In this section, we extend our model to a more general case, where players may select several items and offer each user the one item which satisfies him the most. |
For reader convenience, we repeat the part of the model being reconsidered: The set of items (e.g. possible ad formats/messages to select from) available to player [MATH] is denoted by [MATH] , which we assume to be finite. A strategy of player [MATH] is an item from [MATH] |
Consider the case where each player [MATH] is limited to choose up to [MATH] items from [MATH] , where [MATH] is fixed. Formally, the strategy space of each player [MATH] is [MATH] , and we keep on using [MATH] to represent her strategy. In addition, users are now targeted personally – for each user [MATH] , player [MA... |
We again define the characteristic function of the cooperative game as [MATH] . The coalition payoff is [MATH] ’s highest satisfaction with items offered by the coalition members. Hence, the Shapley value of each player remains the same, and the proof of Theorem holds as is. In addition, Theorem and Lemma did not make ... |
As for Theorem , a few modifications are required. The strategy of selecting a set [MATH] is modeled as choosing all resources associated with intervals that are subsets of [MATH] , where [MATH] . Namely, |
[EQUATION] Thus, there is an induced one-to-one function from the power set of items to the power set of resources, [MATH] Mapping between items and resources, we define the set of possible strategies of player [MATH] [EQUATION] Using these modifications, the proof of Theorem given in Subsection A.3 now holds. |
# Source: arxiv 1806.01255 # Title: Chemical Oscillation in Ultracold Chemistry # Sections: all # Downloaded: 2026-03-02T08:56:12.198727+00:00 |
Chemical Oscillation in Ultracold Chemistry Abstract We demonstrate the occurrence of oscillatory reactions in the ultra-cold chemistry of atom-molecular Bose-Einstein condensate. Nonlinear oscillations in the mean-field dynamics occur for a specific range of elliptic modulus, giving rise to both in- and out-phase modu... |
Introduction After the experimental realization of atomic Bose–Einstein condensate (BEC) at nanokelvin temperatures, a major research effort over the past few years has been to extend the techniques of atom cooling and trapping |
to molecular systems for realizing molecular Bose-Einstein condensates. Complex spectral structure of molecules has made it difficult to cool them to the ultracold regime by direct laser-cooling techniques that has successfully worked for atoms. Although significant progress has been made for capturing molecules in dif... |
, such techniques have not been successful in preparing dense samples of molecules in specific quantum states. An alternative pathway has now been followed for realizing molecular condensates through the conversion of pre-cooled atomic BECs |
. This approach successfully exploits the existence of scattering resonances for connecting ultracold atoms to transient resonant states. For example, a two-photon stimulated Raman transition in a [MATH] BEC has been used to produce [MATH] molecules in a single rotational-vibrational state |
. Ultracold molecules have also been formed through photoassociation (PA) . An immediate advantage of atom-molecule co-trapping is that it offers longer order of interaction time compared to the molecular crossed beam methods. This can facilitate the study of cold chemistry for especially slow processes. The prospect o... |
, although the coherence properties of these systems have not been probed in great detail. Experimentally PA can be effectively used to produce coherent coupling between atomic and molecular BECs for investigating the atom-molecular interaction within the life time of trapped molecules. |
Chemical reaction at ultracold temperature can be surprisingly efficient, aided by nonlinear scattering effects. At such temperature, disordered movement of high momenta particles is absent. Hence, conventional description |
of collision dependent reaction, based on Maxwell-Boltzman statistics gets replaced with the framework of quantum statistics, where reactants are characterized by their de-Broglie wavelengths. The first phenomenological model in this direction was proposed by Heinzen et al. |
, where a mean-field ansatz was used to describe the coherent formation of diatomic molecules in BEC. In this approach, reactants are represented by their corresponding fields, while the density operator replaces reactant concentration. The mean field description for coupled atom-molecule BEC |
(AMBEC) has been developed, which includes pair correlations, quantum fluctuations and thermal effects. Dynamics of AMBEC is described by a modified coupled non-linear Schr [MATH] dinger equation (NLSE). The difference from pure two-species BEC |
, described by a coupled NLSE, arises due to inter-conversion, which induces quadratic non-linearity in addition to the cubic non-linearity arising from s-wave scattering. Simultaneous appearance of cubic and quadratic non-linearity in AMBEC provides novel cross-phase modulation |
, affecting the conversion process in the presence of PA. Fig schematically depicts the reaction pathways between atoms and molecules at ultraccold temperature, that can circumvent the conventional chemical barriers. In the mean field approach, suitable ground state solutions can asymptotically connect two different co... |
The present work is devoted to the study of oscillatory chemical reactions in the atom-molecular system. The role of various non-linear interactions and that of PA will be probed in detail. Oscillatory kinetics refers to spontaneous progression of reaction in both forward and backward directions, appearing to violate t... |
and do not last forever, dying away slowly as the mixture settles into an unchanging state. In the present case, nonlinear oscillations are found to set in for restricted values of elliptic modulus, giving rise to both in- and out-phase modulations in the atom-molecule population density. PA is found to play the key ro... |
, differing significantly from the two-BEC case, with one class of soliton necessarily accompanied by a background. Interestingly, nonlinear excitations in the form of cnoidal waves, similar to the ones in atomic BEC, for both the repulsive and attractive domains, are found as exact solutions, wherein the heavy molecul... |
Model Two atoms can be combined together to form a molecule through the absorption of a photon from an applied optical field during an atomic collision. In recent years, Feshbach resonances |
have come into prominence in the study of ultracold atomic gases, wherein positions of resonances can be adjusted using applied magnetic fields. It is possible to control the interactions between atoms and molecules appropriately by tuning resonances to near-zero collision energy. This interaction close to the absolute... |
. Since the energy produced in this exoergic process is very low, the reaction products remain in the trap. We are considering only the elementary reactions that proceed without forming any identifiable intermediate species. The most elementary second-order reaction, diatomic molecule formation , is represented by: |
[MATH] In this case, possible product formation leads to different possibilities for the quantum statistics: [MATH] [MATH] and [MATH] , where [MATH] stands for bosonic and [MATH] for fermionic counterpart. Interestingly, these conversions correspond to well-known field theroretical models, the Lee-Van Hove model of mes... |
. We consider a chemical system of the first type, where bosonic enhancement of the chemical dynamics is the strongest. Table illustrates different orders of chemical reaction in a general scenario. Zeroth-order reaction physically represents an exchange of species with reservoir, while the First-order reaction models ... |
[EQUATION] which can be generalized to include the effect of multiple concurrent reactions, particle loss and dissipation. For the rest of the paper, we neglect these effects and assume the reaction rate to be much larger than the ground state energies, i.e., [MATH] , where [MATH] and [MATH] label the corresponding gro... |
[EQUATION] Here [MATH] [MATH] and [MATH] measure the strength of atom-atom, molecule-molecule and atom-molecule interactions respectively. [MATH] and [MATH] stand for atomic and molecular trapping potentials, with [MATH] being the strength of PA. The parameter [MATH] is the energy mismatch in converting the atoms to mo... |
are kept unchanged for notational convenience. Equations of motion for the atomic and molecular mean fields, are given by [EQUATION] |
Evidently, for nonzero [MATH] , the overall particle number is conserved, [EQUATION] Here [MATH] ’s are taken as, [MATH] for [MATH] ., for which the continuity equation can be written as, |
[EQUATION] This condition is invariant under scale transformation and Galilean boost . Under scaling, density and phase change as, [MATH] [MATH] , while for the boost by an amount [MATH] , the changes are [MATH] and [MATH] . In the following, we first consider the static configurations, which yield the asymptotic equil... |
Trapping Configuration Confining traps are usually approximated by harmonic potentials. Trap frequency in general, can be time-dependent. Depending on the sign of trap frequency [MATH] , oscillator potential can either be confining or expulsive. Interestingly, in the mean field approach, the wave packet dynamics in pre... |
[EQUATION] Here [MATH] represents the amplitude of the pulse and [MATH] is the similarity variable [EQUATION] where [MATH] and [MATH] are the dimensionless width and center of the self-similar wave. The quadratically chirped phase is given by |
[EQUATION] where the parameters [MATH] [MATH] and [MATH] are to be determined. They are related to the phase-front curvature, the frequency shift, and the phase offset, respectively. In the general case of time-dependent harmonic trapping, condensate profile gets appropriately modulated in time, along with the conditio... |
[EQUATION] The first one, involves the motion related to chirping, which can be expressed as a Schr [MATH] dinger eigenvalue problem via the Cole-Hopf transformation |
. Taking advantage of this connection, it can be shown that, corresponding to each solvable quantum-mechanical system, one can identify a soliton configuration. The fact that the Schr [MATH] dinger equation can be exactly solved for a variety of potential, gives us freedom to control the dynamics of the BEC in a number... |
Ground state configuration The ground state is governed by the values of densities and phases that minimize the energy per unit volume. Energy density for the case of constant density and phase, assuming same chemical potential for both the condensate components, is given by |
[EQUATION] Here, [MATH] is the phase difference between the condensate components, [MATH] , leading to a phase correlation in the presence of PA, different from the density-density correlations arising from inter-species interaction. As mentioned earlier, this term can arise from the two-photon (Raman) process or a dir... |
[EQUATION] where [MATH] is the density difference. For convenience, this state equation can be written as a cubic polynomial in [MATH] |
[EQUATION] where, [MATH] and [MATH] . For this configuration, we keep the mismatch part to zero. Generally, cubic equations of state are much used in thermodynamical system, arises due to the addition of a co-volume parameter [MATH] and an attractive pressure term inversely proportional to [MATH] (molar volume) to the ... |
As will be evident later, the condensate configuration will no longer be stable upto this strength of inter-conversion. Here, critical strength does not mark any appearance of phase transition. It is about quantifying a particular ratio of [MATH] . The solid-red and dashed-red curves are explicated in the presence of a... |
Stability We now examine the effect of small variation of the condensate density due to the presence of phonons, which can potentially destabilize the condensation. In case of a single species condensate, stability depends on the sign of interaction. For repulsive interaction, condensate is found to be stable, whereas ... |
. To examine stability, we insert a weak perturbation of the form [MATH] of frequency [MATH] and wave vector [MATH] around the steady state solution |
[EQUATION] where [MATH] are the real densities to start with and [MATH] satisfies [MATH] . Linearizng in [MATH] , one obtains the dispersion relation. The corresponding gain/loss spectrum can be put into a matrix form [MATH] , with the diagonal elements [MATH] [MATH] and the off-diagonal elements [MATH] [MATH] and [MAT... |
Oscillatory Excitations We now investigate the dynamics of AMBEC, concentrating on the possibility of chemical oscillations. Oscillatory excitations manifest in several chemical reactions, most well-known being the Belousov-Zhabotinsky reaction |
, where the products exhibit periodic changes either in space or in time and give rise to remarkable spatio-temporal pattern . In the present case, both linear and quadratic cnoidal waves are found as exact solutions. The linear case is analogous to the one in atomic BEC, whereas the quadratic one is novel to the atom-... |
[EQUATION] here, [MATH] with [MATH] being the velocity. The cnoidal wave excitations exist only in the presence of a fast varying plane wave component: [MATH] . A lengthy calculation leads to the amplitudes of periodic pair, |
[EQUATION] along with [MATH] and the wave vector, [MATH] It is seen that the nature of atom-atom interaction decides the sign of energy mismatch, [MATH] , with [MATH] and [MATH] . We consider [MATH] |
and obtain explicit solutions for general values of couplings. Accessible density parameters are ensured for [MATH] . PA plays crucial role in determining the front velocity |
and can be used to control how quickly the reactants are used up. This is in sharp contrast to the prediction of usual chemical kinematics, where rates do not depend on the number of participating particles and tend to zero at low temperature. The parameter controlling PA, leads to two different physical situations in ... |
appears as an exact solution. Densities of atomic and molecular BECs are explicated in Fig.( ). Both the densities are characterized by two-frequency modulations. |
Cnoidal chemical waves in a background Unlike the quadratic oscillatory excitations for both the components, the AMBEC system also exhibits periodic atomic density waves in a constant molecular background, |
[EQUATION] with [MATH] . Amplitudes of the atomic and molecular densities are found to be of the form, [MATH] [MATH] with [MATH] [MATH] and [MATH] . Physical solutions are assured for [MATH] , implying both intra and inter-species interactions must be attractive. Interesting to note that, this class of solution exhibit... |
Homo-density Gapped Solitons In addition to the cnoidal waves, bright localized solitons for both the atomic and molecular components, are found as exact solutions: |
[EQUATION] The mean field equations yield two distinct configurations [EQUATION] with [MATH] . The consistency conditions allow only two discrete values for the parameter [MATH] [MATH] . Above solutions exist only for [MATH] , i.e., when the self-interactions (atom-atom or molecule-molecule) are repulsive and the cross... |
Conclusion To summarize, we have investigated the reaction kinetics associated with distinct set of collective chemical waves with different density distributions. Photoassociation is found to dominantly regulate the rate of reactions and product formation. This can be used to produce selectively dense mixture of atoms... |
# Source: arxiv 1806.01309 # Title: Homoclinic chaos and its organization in a nonlinear optics model # Sections: all # Downloaded: 2026-03-02T08:51:05.353416+00:00 |
Homoclinic chaos and its organization in a nonlinear optics model Abstract Abstract We developed a powerful computational approach to elaborate on onset mechanisms of deterministic chaos due to complex homoclinic bifurcations in diverse systems. Its core is the reduction of phase space dynamics to symbolic binary repre... |
chaos, bifurcation, homoclinic, symbolic, toolkit pacs: 05.45.Ac,05.45.-a, 02.30.Oz New directions in science are launched by new tools much more often than by new concepts. F.Dyson Freeman ( 1998 Break-through discovery of deterministic chaos in [infra-red gas] lasers in nonlinear optics was pioneered and established ... |
In this letter we demonstrate how our newly developed toolkit, called “Deterministic Chaos Prospector (DCP)” lets one quickly and fully disclose and elaborate on the origins of complex chaotic dynamics in a 6D model of a resonant 3-level optically-pumped laser (OPL) Moloney et al. 1987 ); Forysiak et al. 1991 . In addi... |
The 3-level optically pumped laser model Moloney et al. 1987 ); Forysiak et al. 1991 is given by [EQUATION] with bifurcation parameters [MATH] [MATH] , and [MATH] , being the Rabi flopping quantities representing the electric field amplitudes at pump and emission frequencies, and the cavity loss parameter, resp.; [MATH... |
Both structural and dynamical instability in laser model (1) is due to an abundance of homoclinic bifurcations ( [MATH] ) of the saddle [MATH] , whose 1D unstable separatrix [MATH] (and [MATH] ) densely fills out the two spatially-symmetric wings of the butterfly-shaped strange attractor (Fig. 1a,e) Afraimovich et al. ... |
[EQUATION] As such, the periodic sequence [MATH] . or [MATH] , corresponds to [MATH] converging to the equilibrium state [MATH] or a periodic orbit emerging from though AH-birfurcation, while the sequence [MATH] or [MATH] corresponds to [MATH] converging to [MATH] and so forth. Wherever small parameter variations do no... |
An aperiodic binary sequence is associated with chaotic dynamics that is characterized by the sensitive dependence on small parameter variations that change [MATH] -progressions and corresponding symbolic sequences (Fig. 1c). Changes occurs at homoclinic bifurcations when [MATH] comes back to saddle [MATH] The primary ... |
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