# MPC Seminar

This is the homepage of the Chapman University **Mathematics, Physics, and Computation Seminars**
(**MPC Seminar**)

*Seminar Organizers:* Roman Buniy and Peter Jipsen

## Spring 2019

The seminar talks are held in **Keck Center for Science and Engineering, KC 171** (Center St. Orange, CA 92866, intersection of Center St. And Sycamore St.), **usually on Wednesday at 4 pm**.
Sometimes there will be a change of time or venue and the announcement will reflect this change.

See [http://www.chapman.edu/about/maps-directions/index.aspx Maps and directions], Keck Center is Building 28 on the Campus Map [https://www.chapman.edu/about/_files/maps-and-directions/current-maps/campus-map.pdf Campus map]

### Thursday, May 16, 2019 at 3 pm, in Keck 370, tea and cookies at 2:40pm (same room)

#### *Speaker:* **Dr. Purbita Jana, The Institute of Mathematical Sciences, Chennai, India**

*Title:* **Intuitionistic topological systems and their connections with Heyting algebra and Gödel algebra**

*Abstract:* (joint work with Antonio Di Nola and Revaz Grigolia) This talk will suggest a new approach of representation of a Heyting algebra as an I-topological system. I-topological systems will be introduced following the notion of topological system introduced by S. Vickers which is a triple (X, |=, A) consisting of a non-empty set X, a frame A and a relation between X and A satisfying logic of finite observations or geometric logic. It is well known that frame is the Lindenbaum algebra of geometric logic where as the Lindenbaum algebra of intuitionistic logic is a Heyting algebra. Hence to define I-topological systems, intuitionistic logic plays a crucial role. Moreover, we will focus on the categorical relationships between the I-topological system, Esakia space and Heyting algebra (and its particular case of Gödel algebra).

### Tuesday, April 16, 2019 at 5 pm, in Keck 370, tea and cookies at 3:40pm (same room)

#### *Speaker:* **Dr. Michele Piazzai, University of Amsterdam, Netherlands**

*Title:* **The Ecology of Ideas: Computational Evidence of Self-Organization**

*Abstract:* Concepts are cognitive structures that people use to categorize objects and organize their knowledge of the world. Common examples include food and furniture, but also rock'n'roll, quantum physics, and blockchain. For individuals, concepts are the main tools of learning and inference; for groups of individuals (i.e., social systems), they are indispensable instruments of coordination and exchange. But how can the idiosyncratic knowledge of individuals in a social system converge on shared concepts, so as to allow coordination? Conventional wisdom in social science suggests that designated mediators, such as music critics, facilitate this convergence by endorsing the learning and usage of particular concepts. By contrast, I will argue that rational decisions embedded in everyday social interaction provide the necessary and sufficient conditions for concepts' convergence. No mediators are needed for the agents to develop a shared conceptual structure: as commonly occurs in complex adaptive systems, design emerges in the absence of centralized designers. Simulation results are presented that illustrate this point.

### Thursday, April 11, 2019 at 4 pm, in Keck 370, tea and cookies at 3:40pm (same room)

#### *Speaker:* **Dr. Michele Piazzai, University of Amsterdam, Netherlands**

*Title:* **The Ecology of Ideas: Empirical Evidence of Evolutionary Pressure**

*Abstract:* Building on the notion that the collective cognition of agents engaged in social interaction constitutes a complex adaptive system, I present a theoretical framework to describe agents' shared conceptual structure as a self-replicating network of objects, ideas, and decisions. Finite cognitive resources at the agents' disposal create a competitive environment where concepts vie to be assigned to objects in categorization decisions, so as to survive in the agents' memory. This struggle for survival gives the socio-cognitive system traits characteristic of an ecology. I identify two sources of ecological interdependence among concepts, including horizontal relations between concepts located at the same level of abstraction and vertical relations between concepts located at different levels. An empirical model is developed to test whether these relations affect the selection of concepts in agents' decisions and the evolution of their shared conceptual structure. To test this model, I analyze one decade's worth of categorization decisions made by users of a crowd-sourced music encyclopedia whose objective is sorting records into genres and sub-genres.

### Thursday, March 28, 2019 at 5 pm, in Keck 171, tea and cookies at 4:40pm (same room)

#### *Speaker:* **Prof. Giovanni Sambin, University of Padova, Italy**

*Title:* **Half way to Grothendieck's aim (embedding of pointwise topology into pointfree topology)**

*Abstract:* For a mathematician, the most interesting motivation for a different foundation is when it offers a new perspective or even a solution to an old problem. Developing topology over a very "weak" foundation offers a precise mathematical expression and proof to the expectation, which is well present since the time of Grothendieck, of a notion of space generalising that of topological space.
The well-known link between pointfree and standard topology (with points) is a categorical adjunction between topological spaces Top and locales Loc. Using a "weak" foundation to develop topology, one is compelled to preserve more information than usual, and thus replace topological spaces with "concrete spaces", and locales with "positive topologies". Contrary to a common expectation, it is precisely this "useless" information what offers a solution to Grothendieck's aim. That is, the above adjunction becomes a categorical embedding of concrete spaces into positive topologies.
This embedding does not look possible without exploiting the presence of the new (intrinsically existential) notion of positivity relation.
Half way because this leaves the problem open of finding a suitable notion of positivity (formal closed subsets) in the framework of arbitrary Grothendieck topologies.

### Wednesday, March 13, 2019 at 4 pm, in Keck 171, tea and cookies at 3:45pm in Keck 370

#### *Speaker:* **Dr. Matt Pusey, University of Oxford**

*Title:* **Anomalous weak values and contextuality: robustness and imaginary**
parts

*Abstract:* I will discuss extensions to my previous work linking anomalous weak
values with contextuality. In particular, I will show that using
transformation noncontextuality obviates the experimentally
problematic requirement that the post-selection is projective. I will
also discuss the status of the imaginary part of weak values. My talk
will be based on joint work with Ravi Kunjwal and Matteo Lostaglio:
https://arxiv.org/abs/1812.06940

### Wednesday, February 27, 2019 at 4 pm, in Keck 171, tea and cookies at 3:45pm in Keck 370

#### *Speaker:* **Prof. Christos Tzounis, California State Polytechnic University, Pomona**

*Title:* **Metric of an Evaporating Black Hole**

*Abstract:* We present an approximate time-dependent metric in ingoing Eddington-Finkelstein coordinates for an evaporating nonrotating black hole as a first-order perturbation of the Schwarzschild metric, using the linearized back reaction from a realistic approximation to the stress-energy tensor for the Hawking radiation in the Unruh quantum state.

### Wednesday, February 13, 2019 at 4 pm, in Keck 171, tea and cookies at 3:45pm in Keck 370

#### *Speaker:* **Prof. Peter Rakitzis, Institute of Electronic Structure and Laser, Heraklion, Greece**

*Title:* **Ultrahigh-Density Spin-Polarized H and D Atoms Observed via Magnetization Quantum Beats**

*Abstract:* We measure nuclear and electron spin-polarized H and D densities of at least $10^{20}$ cm$^{−3}$ and $10^{19}$ cm$^{−3}$, i.e., at pressures of 5 bar and 0.5 bar, respectively, with ∼10-50 ns lifetimes, from the photodissociation of HBr and DI with circularly polarized UV light pulses. We observe the hyperfine quantum beating of the H and D magnetization with a pickup coil, i.e., the respective 0.7 and 3 ns periodic transfer of polarization from the electrons to the nuclei and back. These pulsed densities are ∼7 orders of magnitude higher than that produced by conventional continuous-production methods, and are sufficient for three novel applications: (a) laser-driven ion acceleration of spin-polarized electrons, protons, or deuterons, (b) the preparation of nuclear-spin-polarized molecules, and (c) the demonstration of spin-polarized D-T or D$^{-3}$He laser fusion at large laser facilities such as NIF, for which a reactivity enhancement of ∼50% is expected.

### Tuesday, February 5 -- Saturday, February 9, 2019 in Keck 370

#### **11th Annual Chapman University CECAT Workshop on Pointfree Mathematics**

TUESDAY, February 5th:

1:00 - 2:00pm **M. Andrew Moshier**, *Chapman University*, Weakening relations

THURSDAY, February 7th:

10:30 - 11:30am: **Peter Jipsen**, *Chapman University*, Involutive residuated lattices and relation algebras

11:30 - 12:30pm: **Sara Vannucci**, *University of Salerno*, Semiring and Semimodule Issues in Residuated Lattices

1:00 - 2:15pm: **Rick Ball**, *University of Denver*, Pointfree Integration

2:15 - 3:15pm: **Anna Laura Suarez**, *University of Birmingham*, D-frames and frame coproducts

3:30 - 4.30pm: **Alex Kurz**, *Chapman University*, On the duality theory of weakening relations

FRIDAY, February 8th

10:30 - 11:30am: **Ales Pultr**, *Charles University*, Strong Hausdorff properties of frames

11:30 - 12:30pm: **Diego Valota**, *University of Milan*, Computing Spectra via Dualities in the MTL hierarchy

2:00 - 3:00pm: **Olim Tuyt**, *University of Bern*, Algebraic finite model property of a modal Gödel logic

3:00 - 4:15pm: **Rick Ball**, *University of Denver*, Pointfree Integration, continued

SATURDAY, February 9th

1:00 - 5:00pm: Discussion session

## Fall 2018

The seminar talks are held in Keck Center for Science and Engineering, KC 171 (Center St. Orange, CA 92866, intersection of Center St. And Sycamore St.). Sometimes there will be a change of venue and the announcement will reflect this change.

See [http://www.chapman.edu/discover/maps-directions/index.aspx Maps and directions], Keck Center is Building 28 on the Campus Map [https://www.chapman.edu/about/_files/maps-and-directions/current-maps/campus-map.pdf Campus map]

### Thursday, December 6th 2018 at 4:15pm, in Keck 171, tea and cookies at 3:45pm in Keck 370

#### *Speaker:* ** Dr. Bogdan Suceava, CSUF **

*Title:* ** Strictly Convex Hypersurfaces Satisfying Weingarten-Type Inequalities **

*Abstract: * Linear Weingarten surfaces in three-dimensional ambient space satisfy a relation between mean curvature and Gaussian
curvature: aH^2+bK=c. We investigate whether for hypersurfaces invariant to inversions of dimensions 3, 4, and 5, there are
curvature inequalities similar to the classical Weingarten condition. We also consider the globalization of these pointwise
inequalities. This question is suggested by the investigations of Bang-Yen Chen’s fundamental inequalities, as we reflect
on the geometric interpretations of these relations.Additionally, we plan to discuss other related inequalities, investigated in recent
works written with Mihaela Vajiac, Nicholas Brubaker, and Leonard Giugiuc, respectively.

### Friday, November 30th 2018 at 2:00pm, in Keck 171, tea and cookies at 1:30pm in Keck 370

#### *Speaker:* ** Dr. Apostolos Tzimoulis, Chapman University postdoc **

*Title:* ** Proof theory and algebraic semantics for predicate logics **

*Abstract: * I will start with recasting classical first-order logic in an algebraic and proof-theoretic framework based on Lawvere's theory of hyperdoctrines. Then I will discuss the problem of obtaining general semantics for predicate non-classical logics, provide some examples, and argue that algebraic and proof-theoretic insight can help us understand better and solve this problem.

### Thursday, November 29th 2018 at 4:15pm, in Keck 171, tea and cookies at 3:45pm in Keck 370

#### *Speaker:* ** Dr. Sabine Hossenfelder (Frankfurt Institute for Advanced Studies)**

*Title:* ** Do women get fewer citations than men? **

*Abstract: * I will talk about the results of a citation analysis on
publication data from the arXiv and inspire in which we explored gender
differences. I will further explain how we can use bibliometric analysis
to improve the efficiency of knowledge discovery.

There is also a public talk at 7pm, See Public Talk, Argyros Forum, 7pm

### Conference: Monday to Friday, November 12th to November 16th in Sandhu Conference Center

#### *Speaker:* ** Advances in operator theory with applications to mathematical physics **

*CONFERENCE:* ** Advances in operator theory with applications to mathematical physics **

*Abstract: * For a complete schedule, and a list of abstracts, see: Conference Webpage.

### Monday, November 5th 2018 at 4:15pm, in Keck 171, tea and cookies at 3:45pm in Keck 370

#### *Speaker:* ** Dr. Uwe Kahler, Universidade de Aveiro, **

*Title:* ** Curvature detection using Taylorlets **

*Abstract: * The problem in detection of nodules in medical images consists of two parts: the detection of edges and the detection of curvature. For the detection of edges as elements of the wavefront set of an image shearlets appeared in the last decade as the principal approach based on approximation. But elements of the wavefront set have a problem in the sense that they are singularities with prescribed direction, but not prescribed curvature. To overcome this problem higher order shearlets, so-called Taylorlets were introduced. While we will discuss them in this talk we will also point out and discuss a principal mathematical problem arising in their application: the problem of construction of a Schwartz function with infinitely many generalized vanishing moments. We will show that Meyer’s frequency-based approach does not fit this case and provide a space-based method for its generation.

### Thursday, November 1st 2018 at 4:15pm, in Keck 171, tea and cookies at 3:45pm in Keck 370

#### *Speaker:* ** Dr. Roman Buniy, Chapman University **

*Title:* ** Tripartite entanglement of qudits **

*Abstract: * We provide an in-depth study of tripartite entanglement of qudits. We start with a short review of tripartite entanglement invariants, prove a theorem about the complete list of all allowed values of three (out of
the total of four) such invariants, and give several bounds on the
allowed values of the fourth invariant. After introducing several
operations on entangled states (that allow us to build new states from
old states) and deriving general properties pertaining to their
invariants, we arrive at the decomposition theorem as one of our main
results. The theorem relates the algebraic invariants of any
entanglement class with the invariants of its corresponding components
in each of its direct sum decompositions. This naturally leads to the
definition of reducible and irreducible entanglement classes. We
explicitly compute algebraic invariants for several families of
irreducible classes and show how the decomposition theorem allows
computations of invariants for compounded classes to be carried out
efficiently. This theorem allows us to compute the invariants for the
infinite number of entanglement classes constructed from irreducible
components. We proceed with the complete list of the entanglement
classes for three qutrits with decompositions of each class into
irreducible components, and provide a visual guide to interrelations of
these decompositions. We conclude with numerous examples of building
classes for higher spin qudits.

### Friday, October 26th at 3:00pm, in Keck 171, tea and cookies at 2:30pm in Keck 370

#### *Speaker:* ** Dr. Askery Canabarro, Federal University of Alagoas - BRAZIL **

*Title:* ** Statistical and Machine Learning for physicists, but not just Physics (part 2) **

*Abstract: * Machine Learning has become one of the most exciting areas of modern research and application. In these talks we provide an introduction to the core concepts and tools of machine learning in a way easily understood and intuitive to physicists. The review begins by covering fundamental concepts in ML and modern statistics such as overfitting, regularization, and generalization before moving on to more advanced topics in both supervised and unsupervised learning, for instance: ensemble deep learning, auto ML and so on. We illustrate the ideas with problems we are currently involved in.

Part 2 of a series of 2 talks.

### Thursday, October 25th at 4:15pm, in Keck 171, tea and cookies at 3:45pm in Keck 370

#### *Speaker:* ** Dr. Askery Canabarro, Federal University of Alagoas - BRAZIL **

*Title:* ** Statistical and Machine Learning for physicists (part 1) **

*Abstract: * Machine Learning has become one of the most exciting areas of modern research and application. In these talks we provide an introduction to the core concepts and tools of machine learning in a way easily understood and intuitive to physicists. The review begins by covering fundamental concepts in ML and modern statistics such as overfitting, regularization, and generalization before moving on to more advanced topics in both supervised and unsupervised learning, for instance: ensemble deep learning, auto ML and so on. We illustrate the ideas with problems we are currently involved in.

Part 1 of a series of 2 talks.

### Friday, October 19th at 1:00pm, in Keck 171, tea and cookies at 2:00pm in Keck 370

#### *Speaker:* ** Dr. Erik Linstead, Chapman University **

*Title:* ** A Convoluted Talk **

*Abstract: * Convolutional neural networks (CNNs) represent the current state-of-the-art in machine learning for computer vision. In this talk we will discuss some interesting applications of CNNs to non-traditional domains, as well as explore what happens to CNNs when we ignore computational efficiency to more closely align with neural physiology.

### Thursday, October 18th at 4:15pm, in Keck 171, tea and cookies at 3:45pm in Keck 370

#### *Speaker:* ** Dr. Sandu Popescu, IQS, Chapman University **

*Title:* ** Dynamical quantum non-locality **

*Abstract: * During the 50 years since its discovery, the Aharonov–Bohm effect has had a significant impact on the development of physics. Its arguably deepest implication, however, has been virtually ignored.

Next year will be the 60th anniversary of the discovery of the Aharonov–Bohm (AB) effect, one of the most surprising and quintessential effects in quantum mechanics. Since its discovery in 1959, the AB effect has made a significant impact on the development of physics. It has been generalized in a variety of directions — from a rather straightforward dual effect such as the Aharonov–Casher effect, to the celebrated Berry phase, to non-Abelian gauge theories, to Wilson loops, to anyons. During these past 60 years the impact of the AB effect has been significant indeed.

And during all these past 60 years, what I believe to be by far the deepest implication of the AB effect (discovered by Yakir Aharonov and described in his Tel Aviv University lecture notes and elsewhere) has been virtually ignored. It is an implication that transcends the specific context from which it originates, and goes directly to the very core of quantum physics: the quantum equations of motion are non-local. Without appreciating this fact, it is safe to say that no real understanding of the nature of quantum mechanics is possible.

### Thursday, October 4th at 4:15pm, in Keck 171, tea and cookies at 3:45pm in Keck 370

#### *Speaker:* ** Dr. Justin Dressel, Chapman University **

*Title:* ** Strengthening weak measurements for qubit tomography and multitime correlators **

*Abstract: * We re-examine the measurement strength needed to perform two recent quantum information tasks with qubits: state tomography using weak values, and determining multitime correlators. Traditionally these protocols have required weak measurements that are minimally disturbing, meaning that the coupling between an investigated quantum system and a measurement device has no appreciable influence on the evolution of the system. We show that the weakness of the interaction is not in fact necessary when measuring qubits. For the case of state tomography, we report an experiment performed with neutron matter-waves that extends the notion of generalized eigenvalues for the neutron's path system to allow the exact determination of weak values using both strong and weak interactions. Experimental evidence is given that strong interactions outperform weak ones both for precision and accuracy. For the case of obtaining multitime correlators, we show a method that uses sequential generalized measurements. Specifically, if a correlator can be expressed as an average of nested (anti)commutators of operators that square to the identity, then that correlator can be determined exactly from the average of a measurement sequence of arbitrary strength. We show that both two-point and four-point (out-of-time-ordered) correlators belong to this useful class of qubit correlators.

### Thursday, September 27th at 4:15pm, in Keck 171, tea and cookies at 3:45pm in Keck 370

#### *Speaker:* ** Dr. Matthew Leifer, Chapman University**

*Title:* ** Fine Tunings and the Nature of Quantum Reality **

*Abstract: * Despite many years of research, there is still no universally agreed upon realist interpretation of quantum theory. In this talk, I argue that the main problem is to deal with the fine-tunings implied by no-go theorems about realist approaches to quantum theory, such as Bell’s theorem. We should seek to either eliminate these fine tunings or explain them as emergent. I will give an overview of the various fine-tunings that exist in quantum theory, due to nonlocality, contextuality, lack of time-symmetry, and results on the reality of the quantum state. I will explain how we can quantify each fine tuning, and exploit them in quantum information processing tasks. If time permits, I will outline two approaches to solve the fine-tuning problem based on block universe models with retrocausality and many-worlds.

### Tuesday, September 18th at 4:15pm in Keck 171, tea and cookies at 3:45pm in Keck 370

#### *Speaker:* ** Professor H. Turgay Kaptanoglu, Department of Mathematics, Bilkent University, Ankara **

*Title:* ** Singular Integral Operators With Bergman-Besov Kernels on the ball **

*Abstract: * Although the boundedness of the Bergman-Besov projection operators from Lebesgue classes onto Bergman-Besov spaces has been studied for several decades, the study of the boundedness of the same operators as singular integral operators between different Lebesgue classes are rather new. Some initial work has recently been done by Cheng, Fang, Wang, Yu for the weighted Bergman operator on the unit disc and by Cheng, Hou, Liu for the Drury-Arveson operator. Also Zhao has investigated certain sub-cases of the same problem as Bergman projections. The methods they employ are sporadic and specific to the particular cases they are interested in.

### The 4th SYSMICS Workshop: Friday-Monday, September 14-17, in Sandhu Conference Center D1

#### *Speaker:* ** The 4th SYSMICS Workshop **

*Title:* ** Topic of the workshop: "Duality in Algebra and Logic” **

*Abstract: * Workshop Webpage: http://math.chapman.edu/~jipsen/sysmics/

### Friday, August 31st at 3:30pm, in Beckman 404

#### *Speaker:* ** Dr. Philip Mannheim, Professor of Physics, University of Connecticut **

*Title:* ** The Crisis in Fundamental Physics **

*Abstract: * Cosmology deals with the the astrophysical macroscopic universe on large
scales while fundamental physics deals with the particle physics
microscopic universe on small ones. Recently it has become apparent that
large and small scale physics are actually intertwined leading to an
astro-particle picture of the universe. At the present time this picture
has achieved great success, but at the same time it has led to many open
questions and challenges, challenges which threaten to potentially
undermine the entire picture. These challenges include dark matter, dark
energy, the cosmological constant problem, quantum gravity, the status of
supersymmetry, the multiverse picture, extra space-time dimensions, and the nature of the Higgs boson. In this talk we review these issues and suggest that their resolution would require a paradigm shift in our view of the universe.

### Friday, August 31st at 1:00pm, Keck 171, lunch with the speaker in the Faculty Club at noon

#### *Speaker:* ** Dr. Philip Mannheim, Professor of Physics, University of Connecticut **

*Title:* ** Why physicists are interested in differential geometry **

*Abstract: * Ever since Einstein's development of gravity theory, general relativity and differential geometry have been central components of physics research and of our understanding of the universe. Of special interest is how gravity can interface with the other fundamental forces, the nuclear force, the weak force, and especially the electromagnetic force. In this talk we describe some of the motivation and central achievements for general relativity, and discuss some proposed generalizations of it such as torsion and Weyl geometry that might lead to a purely geometric unification of the fundamental forces.

### Thursday, August 30th at 4:00pm, Keck 171, tea and cookies at 3:30pm in Keck 370

#### *Speaker:* ** Dr. Philip Mannheim, Professor of Physics, University of Connecticut **

*Title:* ** Quantum Conformal Gravity **

*Abstract: * Conformal symmetry is a natural symmetry in physics since it is the full symmetry of the light cone. If all particles are to get their masses by symmetry breaking then conformal symmetry is the symmetry of the unbroken Lagrangian. Like Yang-Mills theories conformal symmetry has a local extension, namely conformal gravity, a pure metric-based candidate alternative to the non-conformal invariant standard Newton-Einstein theory of gravity. With its dimensionless coupling constant quantum conformal gravity is power counting renormalizable. Since its equations of motion are fourth-order derivative equations conformal gravity has long been thought to possess unacceptable ghost states of negative norm that would violate unitarity. However on constructing the quantum Hilbert space Bender and Mannheim found that this not to be the case. Conformal gravity is thus offered as a completely consistent and unitary quantum theory of gravity, one that requires neither the extra dimensions nor the supersymmetry of string theory. As formulated via local conformal invariance there is no intrinsic classical gravity, with gravity instead being intrinsically quantum-mechanical, with the observed classical gravity being output rather than input. The contribution of the graviton loops of conformal gravity enables conformal gravity to solve the cosmological constant problem. Like Yang-Mills the potential of conformal gravity contains both a Newtonian term and a linear potential. Together with a quadratic potential that the theory also contains conformal gravity is able to explain the systematics of galactic rotation curves without any need for galactic dark matter. Since all mass is to be dynamical there cannot be a fundamental double-well Higgs potential in the theory. Instead, the Higgs boson is generated dynamically, with the hierarchy problem then being solved.

### Thursday, August 23rd at 4:00pm, Keck 171, tea and cookies at 3:30pm in Keck 370

#### *Speaker:* ** Dr. Gunduz Caginalp, Professor of Mathematics, University of Pittsburgh**

*Title:* ** Volatility Maxima as a Forecaster of Trading Price Extrema **

*Abstract: * This is joint work with Carey Caginalp. The relationship between price volatility and a market extremum is examined using a fundamental economics model of supply and demand. By examining randomness through a microeconomic setting, we obtain the implications of randomness in the supply and demand, rather than assuming that price has randomness on an empirical basis. Within a very general setting the volatility has a maximum that precedes the extremum of the price. A key issue is that randomness arises from the supply and demand, and the variance in the stochastic differential equation governing the logarithm of price must reflect this. Analogous results are obtained by further assuming that the supply and demand are dependent on the deviation from fundamental value of the asset.