From In-Out Duality to the Foundation of Social Quantum Mechanics

This paper introduces a framework of in-out duality, merging insights from quantum mechanics with social sciences to illuminate the complex interplay between internal potentialities and external manifestations. It articulates foundational, mathematical axioms (Entanglement, Homogeneity, Emergence, and Measurement) that underpin the dynamics of systems, emphasizing the interconnectedness and emergent behaviors resulting from internal and external interactions. By exploring quantum concepts like coherence, entanglement, and superposition, the paper proposes an interdisciplinary approach termed Quantum Social Mechanics. This approach challenges classical paradigms, advocating for a reevaluation of conventional notions through the lens of quantum principles. The paper argues that understanding the universe’s complexities requires a synthesis of motion states and potential states, suggesting a paradigm shift towards integrating quantum mechanics into the philosophical foundation of social theory. Through this comprehensive framework, the paper aims to foster a deeper understanding of the universe’s interconnected nature and the dynamic processes that govern the emergence of complex systems and behaviors.

Duality theorem for smash coproduct over quantum groupoids

The duality theorem of generalized weak smash coproducts of weak module coalgebras and comodule coalgebras over quantum groupoids is studied.Let H be a weak Hopf algebra,C a left weak H-comodule coalgebra and D a left weak H-module coalgebra.First,a weak generalized smash coproduct C×lH D over quantum groupoids is defined and the module and comodule structures on it are constructed.The weak generalized right smash coproduct C×rL D is similar.Then some isomorph-isms between them are obtained.Secondly,by introducing some concepts of a weak convolution invertible element,a weak co-inner coaction and a strongly relative co-inner coaction,a sufficient condition for C×rH D to be isomorphic to Cv D is obtained,where v∈WC（C,H）and the coaction of H on D is right strongly relative co-inner.Finally,the duality theorem for a generalized smash coproduct over quantum groupoids,（C×lH H）×lH H≌Cv（H×lH H）,is obtained.

APPROXIMATE DUALITY OF g-FRAMES IN HILBERT SPACES

In this article, we introduce and characterize approximate duality for g-frames. We get some important properties and applications of approximate duals. We also obtain some new results in approximate duality of frames, and generalize some of the known results in approximate duality of frames to g-frames. We also get some results for fusion frames, and perturbation of approximately dual g-frames. We show that approximate duals are stable under small perturbations and they are useful for erasures and reconstruction.

Duality Computing in Quantum Computers

In this letter,we propose a duality computing mode,which resembles particle-wave duality property whena quantum system such as a quantum computer passes through a double-slit.In this mode,computing operations arenot necessarily unitary.The duality mode provides a natural link between classical computing and quantum computing.In addition,the duality mode provides a new tool for quantum algorithm design.

Elementary analysis of interferometers for wave particle duality test and the prospect of going beyond the complementarity principle

A distinct method to show a quantum object behaving both as wave and as particle is proposed and described in some detail. We make a systematic analysis using the elementary methodology of quantum mechanics upon Young＇s two-slit interferometer and the Mach-Zehnder two-arm interferometer with the focus placed on how to measure the interference pattern （wave nature） and the which-way information （particle nature） of quantum objects. We design several schemes to simultaneously acquire the which-way information for an individual quantum object and the high-contrast interference pattern for an ensemble of these quantum objects by placing two sets of measurement instruments that are well separated in space and whose perturbation of each other is negligibly small within the interferometer at the same time. Yet, improper arrangement and cooperation of these two sets of measurement instruments in the interferometer would lead to failure of simultaneous observation of wave and particle behaviors. The internal freedoms of quantum objects could be harnessed to probe both the which-way information and the interference pattern for the center-of-mass motion. That quantum objects can behave beyond the wave-particle duality and the complementarity principle would stimulate new conceptual examination and exploration of quantum theory at a deeper level.

General Quantum Interference Principle and Duality Computer

In this article, we propose a general principle of quantum interference for quantum system, and based on this we propose a new type of computing machine, the duality computer, that may outperform in principle both classical computer and the quantum computer. According to the general principle of quantum interference, the very essence of quantum interference is the interference of the sub-waves of the quantum system itself A quantum system considered here can be any quantum system： a single microscopic particle, a composite quantum system such as an atom or a molecule, or a loose collection of a few quantum objects such as two independent photons. In the duality computer, the wave of the duality computer is split into several sub-waves and they pass through different routes, where different computing gate operations are performed. These sub-waves are then re-combined to interfere to give the computational results. The quantum computer, however, has only used the particle nature of quantum object. In a duality computer, it may be possible to find a marked item from an unsorted database using only a single query, and all NP-complete problems may have polynomial algorithms. Two proof-of-the-principle designs of the duality computer are presented： the giant molecule scheme and the nonlinear quantum optics scheme. We also propose thought experiment to check the related fundamental issues, the measurement efficiency of a partial wave function.

DUALITY OF WEIGHTED MULTIPARAMETER TRIEBEL-LIZORKIN SPACES

In this paper, we reintroduce the weighted multi-parameter Triebel-Lizorkin spaces Fp^a,q（w;R^n1×R^n2） based on the Frazier and Jawerth＇ method in [11]. This space was firstly introduced in [18]. Then we establish its dual space and get that （Fp＇q）＊ = CMOp^-a,q＇ for 0 ~p≤ 1.

KUHN-TUCKER CONDITION AND WOLFE DUALITY OF PREINVEX SET-VALUED OPTIMIZATION

The optimality Kuhn-Tucker condition and the wolfe duality for the preinvex set-valued optimization are investigated. Firstly, the concepts of alpha-order G-invex set and the alpha-order S-preinvex set-valued function were introduced, from which the properties of the corresponding contingent cone and the alpha-order contingent derivative were studied. Finally, the optimality Kuhn-Tucker condition and the Wolfe duality theorem for the alpha-order S-preinvex set-valued optimization were presented with the help of the alpha-order contingent derivative.

Prime Factorization in the Duality Computer

We give algorithms to factorize large integers in the duality computer. We provide three duality algorithms for factorization based on a naive factorization method, the Shor algorithm in quantum computing, and the Fermat＇s method in classical computing. All these algorithms may be polynomial in the input size.

The Wave-Particle Duality—Does the Concept of Particle Make Sense in Quantum Mechanics? Should We Ask the Second Quantization?

The quantum object is in general considered as displaying both wave and particle nature. By particle is understood an item localized in a very small volume of the space, and which cannot be simultaneously in two disjoint regions of the space. By wave, to the contrary, is understood a distributed item, occupying in some cases two or more disjoint regions of the space. The quantum formalism did not explain until today the so-called “collapse” of the wave-function, i.e. the shrinking of the wave-function to one small region of the space, when a macroscopic object is encountered. This seems to happen in “which-way” experiments. A very appealing explanation for this behavior is the idea of a particle, localized in some limited part of the wave-function. The present article challenges the concept of particle. It proves in the base of a variant of the Tan, Walls and Collett experiment, that this concept leads to a situation in which the particle has to be simultaneously in two places distant from one another—situation that contradicts the very definition of a particle. Another argument is based on a modified version of the Afshar experiment, showing that the concept of particle is problematic. The concept of particle makes additional difficulties when the wave-function passes through fields. An unexpected possibility to solve these difficulties seems to arise from the cavity quantum electrodynamics studies done recently by S. Savasta and his collaborators. It involves virtual particles. One of these studies is briefly described here. Though, experimental results are needed, so that it is too soon to conclude whether it speaks in favor, or against the concept of particle.

Duality theorem for weak L-R smash products

This paper gives a duality theorem for weak L-R smash products, which extends the duality theorem for weak smash products given by Nikshych.

Wave–particle duality in a Raman atom interferometer

We theoretically investigate the wave–particle duality based on a Raman atom interferometer, via the interaction between the atom and Raman laser, which is similar to the optical Mach–Zehnder interferometer. The wave and which-way information are stored in the atomic internal states. For the φ- π- π /2 type of atom interferometer, we find that the visibility(V) and predictability(P) still satisfy the duality relation, P2+ V2≤ 1.

Sufficiency and Duality for Nondif ferentiable Multiobjective Fractional Programming Problems with (Φ,ρ,α)-V-Invexity

A new concept of(Φ,ρ,α)-V-invexity for differentiable vector-valued functions is introduced,which is a generalization of differentiable scalar-valued(Φ,ρ)-invexity.Based upon the(Φ,ρ,α)-V-invex functions,sufficient optimality conditions and MondWeir type dual theorems are derived for a class of nondifferentiable multiobjective fractional programming problems in which every component of the objective function and each constraint function contain a term involving the support function of a compact convex set.

Long Non-coding RNA ANRIL in Gene Regulation and Its Duality in Atherosclerosis

The antisense transcript long non-coding RNA（lnc RNA）（antisense non-coding RNA in the INK4 locus, ANRIL） is an antisense of the cyclin-dependent kinase inhibitor 2 B（CDKN2B） gene on chromosome 9 p21 that contains an overlapping 299-bp region and shares a bidirectional promoter with alternate open reading frame（ARF）. In the context of gene regulation, ANRIL is responsible for directly recruiting polycomb group（Pc G） proteins, including polycomb repressive complex-1（PRC-1） and polycomb repressive complex-2（PRC-2）, to modify the epigenetic chromatin state and subsequently inhibit gene expression in cis-regulation. On the other hand, previous reports have indicated that ANRIL is capable of binding to a specific site or sequence, including the Alu element, E2 F transcription factor 1（E2F1）, and CCCTC-binding factor（CTCF）, to achieve trans-regulation functions. In addition to its function in cell proliferation, adhesion and apoptosis, ANRIL is very closely associated with atherosclerosis-related diseases. The different transcripts and the SNPs that are related to atherosclerotic vascular diseases（ASVD-SNPs） are inextricably linked to the development and progression of atherosclerosis. Linear transcripts have been shown to be a risk factor for atherosclerosis, whereas circular transcripts are protective against atherosclerosis. Furthermore, ANRIL also acts as a component of the inflammatory pathway involved in the regulation of inflammation, which is considered to be one of the causes of atherosclerosis. Collectively, ANRIL plays an important role in the formation of atherosclerosis, and the artificial modification of ANRIL transcripts should be considered following the development of this disease.

OPTIMALITY CONDITIONS AND DUALITY RESULTS FOR NONSMOOTH VECTOR OPTIMIZATION PROBLEMS WITH THE MULTIPLE INTERVAL-VALUED OBJECTIVE FUNCTION

In this paper, both Fritz John and Karush-Kuhn-Tucker necessary optimality conditions are established for a （weakly） LU-efficient solution in the considered nonsmooth multiobjective programming problem with the multiple interval-objective function. Further, the sufficient optimality conditions for a （weakly） LU-efficient solution and several duality results in Mond-Weir sense are proved under assumptions that the functions constituting the considered nondifferentiable multiobjective programming problem with the multiple interval- objective function are convex.

SYMPLECTIC DUALITY SYSTEM ON PLANE MAGNETOELECTROELASTIC SOLIDS

By means of the generalized variable principle of magnetoelectroelastic solids, the plane magnetoelectroelastic solids problem was derived to Hamiltonian system. In symplectic geometry space, which consists of original variables, displacements, electric potential and magnetic potential, and their duality variables, lengthways stress, electric displacement and magnetic induction, the effective methods of separation of variables and symplectic eigenfunction expansion were applied to solve the problem. Then all the eigen-solutions and the eigen-solutions in Jordan form on eigenvalue zero can be given, and their specific physical significations were shown clearly. At last, the special solutions were presented with uniform loader, constant electric displacement and constant magnetic induction on two sides of the rectangle domain.

The Duality on Vector Optimization Problems

Duality framework on vector optimization problems in a locally convex topological vector space are established by using scalarization with a cone-strongly increasing function.The dualities for the scalar convex composed optimization problems and for general vector optimization problems are studied.A general approach for studying duality in vector optimization problems is presented.

Optimality conditions and duality for a class of nondifferentiable multiobjective generalized fractional programming problems

This paper studies a class of multiobjective generalized fractional programming problems, where the numerators of objective functions are the sum of differentiable function and convex function, while the denominators are the difference of differentiable function and convex function. Under the assumption of Calmness Constraint Qualification the Kuhn-Tucker type necessary conditions for efficient solution are given, and the Kuhn-Tucker type sufficient conditions for efficient solution are presented under the assumptions of （F, α, ρ, d）-V-convexity. Subsequently, the optimality conditions for two kinds of duality models are formulated and duality theorems are proved.

ON SUFFICIENCY AND DUALITY OF SOLUTIONS FOR QUASI B_s-INVEX SEMI-INFINITE PROGRAMMING

A class of functions called quasi B s invex and pseudo B s invex functions are introduced by using the concept of symmetric gradient. The examples of quasi B s invex and pseudo B s invex functions are given. The sufficient optimality conditions and Mond Weir type duality results are obtained for a nondifferentiable nonlinear semi infinite programming problem involving quasi B s invex and pseudo B s invex functions.

Optimality Conditions and Duality for Nonsmooth Multiobjective Programms with Generalized Essential Convexity

In this paper, we introduce generalized essentially pseudoconvex function and generalized essentially quasiconvex function, and give sufficient optimality conditions of the nonsmooth generalized convex multi-objective programming and its saddle point theorem about cone efficient solution. We set up Mond-Weir type duality and Craven type duality for nonsmooth multiobjective programming with generalized essentially convex functions, and prove them.