An agent at time t and space p is therefore more specific. BUMP reads: implies then the bipolar interaction. The logical form of bipolar quantum entanglement qualifies BDL as a causal logic for equilibrium-based bipolar deduction. These measures lead to the unification of energy and information as well as system level equilibrium and harmony. This is discussed in the next section.

The concepts of bipolar elementary energy and information laid a basis for modeling bipolar quantum superposition of multiple bipolar quantum agents BQAs cf. Figure 1 shows that two or more bipolar variables can be integrated into a multidimensional bipolar quantum superposition. The unification is realized in the background independent BQG cf. Part I. Figure 2 shows an equilibrium-based unification of matter and antimatter atoms as a bipolar quantum cellular automaton BQCA — a multidimensional bipolar dynamic equilibrium emerged from bipolar interaction and bipolar quantum superposition.

It is shown that the BQCA also leads to the unification of wave-particle duality [15,16]. Thus, not only can BQG serve as a geometry of light, it can also serve as the geometry of Nature. Figure 2. In this section we show that the BQCA model for matter-antimatter unification presented in last section can be scaled to physical and biological system models with conservational, regenerating, degenerating or repression and activation properties. These properties make it possible to describe different biological systems as BQAs at the molecular, cell and organism levels.

Without a formal geometrical and logical basis for quantum mechanics, after seven decades since its inception, quantum biology [20] is still a research area in its infancy. Although the Yin and Yang of Nature have been recognized as fundamental and ubiquitous bipolar coexistence in biology [21] and genomics [22], no formal geometrical and logical model had been available for reasoning on the Yin and Yang for thousands of years until recently.

The above dilemma can be best illustrated with two mysteries. The maximum efficiency eventually fades away and they may have to change color from green to yellow in the fall. In quantum biology research, however, we still do not have a quantum mechanical model for biological growth and aging. It is still a mystery how exactly the incoming photons can contribute to the inner working of photosynthesis. Scientists have found that such navigation ability is based on the magnetic field of the Earth [cf.

Therefore, the BQCA in Figure 2 c can be scaled to molecule, cell and system levels through further superposition and entanglement Figure 3. At the quantum level, x,y can be the elementary bipolar energy of an electron-positron pair within an atom. At the atomic level, x,y can be the total bipolar energy of all the bipolar pairs within an atom. At the molecule, cell or organism levels, x,y can be the total bipolar energy of all the bipolar pairs of a lower level. With this unification, two bipolar variables x,y and u,v can be interactive through bipolar quantum superposition or entanglement.

Thus, internal and external bipolar interaction can be posited as the source of causality for biological and mental functionalities with formal bipolar equilibrium-based definability. Theorem 1 shows that BQG presents a unified geometrical and analytical basis for quantum mechanics, quantum biology, and quantum fractality with nonlinear dynamic normal or abnormal bipolar fractality. Figure 3 shows that the background independent nature of BQG makes it possible to host bipolar fractals for qualitative and quantitative analysis anywhere and anytime.

While geometric shapes or patterns have been a focus in the fractal geometry of Nature [7], the background independent nature of BQG bridges a gap from the fractals of Nature to the quantum nature of fractals and vice versa. It has the potential for bipolar quantum swarm intelligence at the system, cell, molecular, atom, and quantum levels as well, which may find applications in life sciences such as cancer research [25,26]. Figure 3. Equilibrium-based bipolar scalability and fractality of BQCA: a Bipolar scalability and fractality; b BQG as a background independent bipolar fractal geometry.

With complete background independence, BQG makes bipolar quantum cellular automaton-based fractal emergence, branching, regulation, and communication possible through bipolar quantum superposition and entanglement. A BQA as a dynamic equilibrium is to the Yin and the Yang of bipolar relativity as gravity is to space and time of general relativity but with fundamentally different syntax, semantics, and basic postulates. While space and time are not direct opposites and cannot form a bipolar dynamic equilibrium, bipolar interaction and complementarity is posited to cause the emergence of BQAs as well as spacetime through bipolar quantum superposition and entanglement in a bipolar dynamic equilibrium process [1,2].

Since the Yin and the Yang are two reciprocal and interdependent opposites of a dynamic equilibrium that are completely background independent and ubiquitous, BQG is fundamentally different from Euclidian, Hilbert, and spacetime geometries. The new geometry is quadrant-irrelevant and shape-free because bipolar identity, interaction, superposition, separation, and entanglement can be accounted for in the geometry without quadrants.

With the shape-free and quadrant-irrelevant properties, BQG can support bipolar fractality anywhere in any amount of bipolar energy or information for investigating into the quantum nature of shaped fractals in micro and meso scales. Shapes and quadrants can, however, be added where observer is involved and background dependent information is needed.

Thus, BQG can subsume other geometries and can be used to reason on space and time as well. This is illustrated as follows:. The ubiquitous Yin Yang 1 regulator protein acts as both a repressor and an activator in gene expression regulation [22]. Thus, the functionality of the regulator can be characterized as a bipolar variable yin, yang. Activation regulation may lead to biological growth.

When the absolute total energy of each row and each column of the organizational matrix M t is greater than 1. Repression Yin is the opposite of activation Yang of the ubiquitous Yin Yang 1 regulator protein [22]. Such regulation may lead to biological degeneration or aging. When the absolute total energy of each row and column of the organizational matrix M t is less than 1.

For instance, a cell may grow and divide into two cells. This can be mathematically characterized with Eq. After a cellular division, each bipolar vector can be regulated by its own bipolar regulatory matrix M i as a local regulation center for further energy conservation, growth, cell division or degeneration. All the local regulation centers can be regulated by a global regulation center M G. In normal growth , all E i are regulated by the global regulation center M G. Otherwise, some E i could be out of control with abnormal growth.

Bipolar cell division leads to the concept of bipolar fractal branching. An equilibrium-based principle of normal and abnormal bipolar fractal branching is derived as follows. Bipolar fractal branching principle: In a normal bipolar fractal branching , a BQCA observes the bipolar dynamic equilibrium condition, where the absolute energy of the BQCA equals to the total of all its branches or elements at any time t regardless of local bipolar balance or imbalance.

Otherwise, there must be abnormal bipolar fractal branching with unregulated growth in a non-equilibrium state. Theorem 2: A necessary but insufficient condition of normal bipolar fractal branching is branching toward the eternal equilibrium state 0,0 of BQG.

## The Second Condition for Equilibrium

Proof: If the condition is violated, the energy of a branch would be greater than that of the BQCA itself and the branching would be abnormal. Therefore, the condition is necessary. The condition is insufficient because it does not guarantee the total energy of all branches and elements to be equal to that of the BQCA itself at any time.

Figure 5 a shows a sketch of normal bipolar fractal branching of a single branch where bole is greater than branch; Figure 5 b shows a sketch of abnormal case where branch is greater than bole. Evidently, normal or abnormal bipolar fractal branching can be quantized, plotted, and analyzed within the equilibrium-based and background independent BQG but that is impossible without bipolarity in other geometries.

Figure 5. Single normal and abnormal bipolar fractal branching in BQG. While truth has been deemed the essence of being since Aristotle, without bipolarity truth-based logic and linear algebra are incapable of bipolar causal interaction, self-organization, and dynamic regulation for energy-information conservation, degeneration, regeneration and oscillation due to bipolar cancelation. For instance,. Of course, we can attempt to use a positive regulation matrix.

But a positive matrix does not show bipolar interaction and balancing cause-effect relation toward a bipolar dynamic equilibrium, non-equilibrium or oscillating state [2,16]. It provides a holistic, unitary, and analytical framework of quantum mechanics and quantum biology for the complex interaction and regulation of quantum agents with an equilibrium-based scalable quantum automata theory [2,15,16,18]. While quantum mechanics heavily relies on probability and do not lend itself as an analytical system, the analytical nature of the bipolar equilibrium-based approach provides a geometrical and logical basis toward a computational paradigm of quantum agents, quantum biology and quantum intelligence [15].

In light of the above, BQG and BDL has been proposed as an equilibrium-based geometrical and logical foundation for quantum physics and biophysics with an equilibrium-based interpretation of quantum superposition. Although it is questionable whether the equilibrium-based geometrical and logical system is what Einstein sought for physics in the last century, BQG has been proven completely background independent cf.

It has been show that the two together leads to an analytical paradigm of quantum mechanics and quantum biology. Furthermore, BDL does satisfy the simplicity criterion set forth by Einstein and has passed a major falsifiability test with a logical exposition ofthe longstanding puzzle of Dirac 3-polarizer experiment. While background independent geometry has been advocated by Lee Smolinin the quest for quantum gravity [8], no formal logical system has been reported for completely background independent geometrical reasoning with logically definable causality besides BDL [2].

A distinguishing factor lies in YinYang bipolar complementarity. No matter time is real or unreal, fundamentally different from the Yin and the Yang of Nature, space and time are not bipolar interactive and cannot form bipolar dynamic equilibrium, symmetry or quantum superposition for quantum gravity. This could be the reason why other approaches to quantum gravity so far stopped short in finding a unique logical foundation as a general theoretical basis for physics. It is contended that the equilibrium-based approach has opened an Eastern road toward quantum gravity [2,16] and will lead to a quantum reincarnation of philosophy [17].

It may well be that, at its most basic level, there is no randomness in nature, no fundamentally statistical aspect to the laws of evolution. Everything, up to the most minute detail, is controlled by invariable laws. Every significant event in our universe takes place for a reason, it was caused by the action of physical law, not just by chance.

We hope to inspire more physicists to do so, to consider seriously the possibility that quantum mechanics as we know it is not a fundamental, mysterious, impenetrable feature of our physical world, but rather an instrument to statistically describe a world where the physical laws, at their most basic roots, are not quantum mechanical at all.

Not only has BQCA logically unified matter and antimatter atoms [16], it also has led to a cellular model for quantum biology with scalable bipolar quantum fractality. BQCA is expected to serve as a mathematical basis for swarm quantum intelligence as well. Some questions remain, however, including:. While all three questions are gigantic research topics. The third one is a most critical one for quantum computing. The BQCA interpretation of quantum mechanics provides an equilibrium-based analytical basis for quantum measurement, decoherence and collapse.

Without bipolarity, there seems to be no way to call upon anything external factor to disrupt the autonomous evolution of a quantum entanglement in the existing decoherence and collapse theories in quantum mechanics [cf. Notably, limitations of fractal geometry have been identified. In particular, experimental result shows that the many orders of magnitude of the fractal power law described in the seminal book by Mandelbrot [7] are not supported in an unequivocal way by data collected [10].

While this limitation is observed in macro scales, bipolar quantum geometry provides a platform for investigating into the quantum nature of fractals. It is interesting to ask: If the universe is a bipolar quantum fractal in a dynamic equilibrium, does the order of magnitude of a quantum fractal power law have to have an upper limit? Since fractals in Nature exhibit statistical similarity with a heavy role of chance, how fractals can have deterministic property is another question. A possible answer is that certain statistical similarity at macro scales of Nature can be caused by bipolar quantum superposition of the different types of particles in different combinations at the meso scales.

## Isaac Newton and the equilibrium theory | Everything about solar energy

The unlimited number of combinations may lead to the nondeterministic property at a higher level. But at the quantum level, some not all fundamental properties might be deterministic such as bipolar equilibrium and non-equilibrium conditions. Philosophically, truth has been asserted as the essence of being since Aristotle.

Is truth the essence of being and Nature? Could BDL be the logic of physics? Could the essence of being be bipolar dynamic equilibrium a bipolar quantum superposition or entanglement? To answer these questions, we need to recall some historical event. However, other mathematicians were braver. They took a leap into the unknown and decided that negative numbers could be used during calculations, as long as they had disappeared upon reaching the solution. The trailblazers were the Chinese who by BC were able to solve simultaneous equations involving negative numbers.

The Ancient Greeks rejected negative numbers as absurd, by AD, the Indians had written the rules for the multiplication of negative numbers and years later, Arabic mathematicians realized the importance of negative debt.

Why were they such an abstract concept? And how did they finally get accepted? It is pointed out [2] that the debate on truth, bipolarity and isomorphism can be deemed a continuation of the debate on negative numbers extended to the logical arena. They are linked by the prices of goods and factor services.

The interdependence of markets is concealed by the partial equilibrium approach. Markets consist of buyers and sellers. Thus an economic system consists of millions of economic decision-making units who are motivated by self-interest. Each one pursues his own goal and strives for his own equilibrium independently of the others. In traditional economic theory the goal of a decision-making agent, consumer or producer, is maximisation of something.

The firm maximizes profit, subject to the technological constraint of the production function. A worker determines his supply of labour by maximising satisfaction derived from work-leisure opportunities, subject to a given wage rate. A general equilibrium is defined as a state in which all markets and all decision-making units are in simultaneous equilibrium. A general equilibrium exists if each market is cleared at a positive price, with each consumer maximising satisfaction and each firm maximising profit.

The interdependence between individuals and markets requires that equilibrium for all product and factor markets as well as for all participants in each market must be determined simultaneously in order to secure a consistent set of prices. General equilibrium emerges from the solution of a simultaneous equation model, of millions of equations in millions of unknowns. The unknowns are the prices of all factors and all commodities and the quantities purchased and sold of factors and commodities by each consumer and each producer.

In principle a simultaneous-equation system has a solution if the number of independent equations is equal to the number of unknowns in the system. This approach has been followed by the founder of general equilibrium analysis Leon Walras. The most ambitious general equilibrium model was developed by the French economist Leon Walras In his Elements of Pure Economics Walras argued that all prices and quantities in all markets are determined simultaneously through their interaction with one another. For example, each consumer has a double role: he buys commodities and sells services of factors to firms.

Thus for each consumer we have a set of equations consisting of two subsets: one describing his demands of the different commodities, and the other his supplies of factor inputs. Similarly, the behaviour of each firm is presented by a set of equations with two subsets one for the quantities of commodities that it produces, and the other for the demand for factor inputs for each commodity produced. In a general equilibrium system of the Walrasian type there are as many markets as there are commodities and factors of production. In a commodity market the number of demand functions is equal to the number of consumers, and the number of the supply functions is equal to the number of firms which produce the commodity.

In each factor market the number of demand functions is equal to the number of firms multiplied by the number of commodities they produce. The number of supply functions is equal to the number of consumers who own ex hypothesis the factors of production. A necessary but not sufficient condition for the existence of a general equilibrium is that there must be in the system as many independent equations as the number of unknowns.

Thus the first task in establishing the existence of a general equilibrium is to describe the economy by means of a system of equations, defining how many equations are required to complete and solve the system. For example, assume that an economy consists of two consumers, A and B, who own two factors of production, K and L These factors are used by two firms to produce two commodities, X and Y. It is assumed that each firm produces one commodity, and each consumer buys some quantity of both.

It is also assumed that both consumers own some quantity of both factors but the distribution of ownership of factors is exogenously determined.

Since the number of equations is equal to the number of unknowns, one should think that a general equilibrium solution exists. In this model the absolute level of prices cannot be determined. With this device prices are determined only as ratios: each price is given relative to the price of the numeraire. If we assign unity to the price of the numeraire, we attain equality of the number of simultaneous equations and unknown variables the number of unknowns is reduced to 17 in our example. However, the absolute prices are still not determined: they are simply expressed in terms of the numeraire.

This indeterminacy can be eliminated by the introduction explicitly in the model of a money market, in which money is not only the numeraire, but also the medium of exchange and store of wealth. Even if there is equality of independent equations and unknowns, there is no guarantee that a general equilibrium solution exists. The proof of the existence of a general equilibrium solution is difficult.

Leon Walras was never able to prove the existence of a general equilibrium.

### When Price is Higher than Equilibrium

In Arrow and Debreu provided a proof of the existence of a general equilibrium in perfectly competitive markets, in which there are no indivisibilities and no increasing returns to scale. Furthermore, in Arrow and Hahn proved the existence of a general equilibrium for an economy with limited increasing returns and monopolistic competition, without indivisibilities. Apart from the existence problem, two other problems are associated with an equilibrium: the problem of its stability and the problem of its uniqueness.

These problems can best be illustrated with the partial-equilibrium example of a demand-supply model. Assume that a commodity is sold in a perfectly competitive market, so that from the utility-maximising behaviour of individual consumers there is a market demand function, and from the profit-maximising behaviour of firms there is a market supply function. An equilibrium exists when at a certain positive price the quantity demanded is equal to the quantity supplied. At such a price there is neither excess demand nor excess supply. The latter is often called negative excess demand.

Thus an equilibrium price can be defined as the price at which the excess demand is zero the market is cleared and there is no excess demand. The equilibrium is stable if the demand function cuts the supply function from above. In this case an excess demand drives price up, while an excess supply excess negative demand drives the price down figure The equilibrium is unstable if the demand function cuts the supply function from below. In this case an excess demand drives the price down, and an excess supply drives the price up figure In figure It is obvious that at P e 1 there is a stable equilibrium, while at P e 2 the equilibrium is unstable.

Finally in figure In fact the three basic questions related to the existence, stability and uniqueness of an equilibrium can be expressed in terms of the excess demand function. To see this we redraw below figures For each of these cases we have derived the relevant excess demand function by subtracting Q s from Q D at all prices. From the redrawn diagrams in conjunction with the corresponding ones There are as many equilibria as the number of times that the excess demand curve E P intersects the vertical price-axis figure The equilibrium is stable if the slope of the excess demand curve is negative at the point of its intersection with the price-axis figure The equilibrium is unstable if the slope of the excess demand curve is positive at the point of its intersection with the price-axis figure If the excess demand function does not intersect the vertical axis at any one price, an equilibrium does not exist figure The above analysis of the existence, stability and uniqueness in terms of excess demand functions can be extended to general equilibrium analysis.

Now we use graphical analysis to show the general equilibrium of a simple economy in which there are two factors of production, two commodities each produced by a firm and two consumers. This is known as the 2 x 2 x 2 general equilibrium model. Furthermore we will be concerned with the static properties of general equilibrium and not with the dynamic process of reaching the state of such an equilibrium, the latter having been sketched in the preceding section.

These factors are homogeneous and perfectly divisible. Only two commodities are produced, X and Y. Technology is given. Each production function exhibits constant returns to scale. There are two consumers in the economy, A and B, whose preferences are represented by ordinal indifference curves, which are convex to the origin, exhibiting diminishing marginal rate of substitution between the two commodities. Finally, it is assumed that the consumers are sovereign, in the sense that their choice is not influenced by advertising or other activities of the firms.