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The separability of quantum systems, particularly in mixed states, remains a fundamental challenge in quantum mechanics. The Werner density matrix, a widely studied representation of bipartite mixed quantum systems, has provided valuable insights into entanglement, nonlocality, and separability. In this work, we propose an extension to the Werner density matrix by introducing a classical bias while preserving the independence of the two particles. Using the Peres-Horodecki separability criterion, we demonstrate that this biased Werner density matrix enlarges the entangled state region, suggesting that the classical bias enhances the formation of quantum entanglement. Furthermore, we show that the entanglement enhancement persists when classical dependence is introduced, indicating that bias is the key factor driving this effect. Our results suggest that careful mixing of classical and quantum components can render a state separable, even when both components exhibit correlations and entanglement., Comment: Submitted for a publication

Among living organisms, there are species that change their patterns on their body surface during their growth process and those that maintain their patterns. Theoretically, it has been shown that large-scale species do not form distinct patterns. However, exceptionally, even large-scale species like giraffes form and maintain patterns, and previous studies have shown that the growth plays a crucial role in pattern formation and transition. Here we show how the growth of the domain contributes to Turing bifurcation based on the reaction-diffusion system by applying the Gray-Scott model to the reaction terms, both analytically and numerically, focusing on the phenomenon of pattern formation and maintenance in large species like giraffes, where melanocytes are widely distributed. After analytically identifying the Turing bifurcation related to the growth rate, we numerically verify the pattern formation and maintenance in response to the finite-amplitude perturbations of the blue state specific to the Gray-Scott model near the bifurcation. Furthermore, among pairs of the parameters that form Turing patterns in a reaction-diffusion system on a fixed domain, we determine a pair of the parameters that maximizes the growth rate for the Turing bifurcation in a reaction-diffusion system on a time-dependently growing domain. Specifically, we conduct a numerical analysis to pursue the pair of the parameters in the Turing space that can be the most robust in maintaining the patterns formed on the fixed domain, even as the domain grows. This study may contribute to specifically reaffirming the importance of growth rate in pattern formation and understanding patterns that are easy to maintain even during growth.

This study examines how patterns on mammal body surfaces change as they transition from juveniles to adults and with seasonal variations. Our previous research suggests that patterns formed in infancy may fade due to the growing domain effects, typically linked to the body's surface expanding as it grows, but this transition is influenced by various factors. Here, we focus on how heat, which can change with body growth, affects this process. Generally, smaller organisms lose heat more easily due to their higher surface area-to-volume ratio, while larger organisms retain heat more effectively. In fact, thermoregulation during infancy is a crucial factor directly influencing survival. We propose a theoretical model that incorporates both growing domain and metabolic rate effects to explain the mechanisms behind these pattern transitions. The model suggests that Turing patterns formed during juvenile stages disperse in adulthood due to domain growth and changes in metabolic rates affecting reaction rates. Numerical analysis shows that when both growing domain and metabolic rate effects are considered, patterns disperse more rapidly than when only growing domain effects are accounted for. Furthermore, we discuss potential correlations among pattern transitions, growth, and thermoregulation mechanisms, emphasizing the role of metabolic rates in maintaining body temperature. Our findings shed light on the intricate relationship between growth, thermoregulation, and pattern transitions in mammals, possibly useful for further research in this field.

In this study, we introduce and explore a delay differential equation that lends itself to explicit solutions in the Fourier-transformed space. Through the careful alignment of the initial function, we can construct a highly accurate solution to the equation. These findings open new avenues for understanding delay systems, demonstrating the efficacy of Fourier transform techniques in capturing transient oscillatory dynamics., Comment: submitted for publication, 9pages 3 figures. Ver2: added a paragraph at the end of Sec.2 and aknowledgements; Ver 3: Updated texts and figures, and added references

We present a new approach to examine transient dynamics in a class of non-autonomous delay differential equations. Exact solutions for these equations are obtained using the Lambert W function alongside an appropriately chosen initial function. These solutions provide a reliable approximation for transient dynamics when the initial functions are not markedly distinct. We explore a non-autonomous equation that exhibits a distinctive phenomenon of reviving dynamics as an illustrative example. The derived exact solutions effectively encapsulate the qualitative characteristics of this reviving dynamics over various delay values., Comment: submitted for a publication: Ver2; extensively revised with a new figure and added references; Further major revision with modified main equation and figures (Published version)

The patterns observed on the body surface of living organisms have traditionally been attributed solely to ecological strategies. However, this study investigates a fascinating phenomenon in Pelodiscus sinensis, where patterns formed on the plastron during embryonic and juvenile stages, which are not externally visible, ultimately disappear in adulthood. This exploration suggests the existence of mechanisms beyond ecological purposes in the formation of body surface patterns in living organisms. While numerous studies have examined pattern formation mechanisms, limited research has focused on the dispersion of preexisting patterns. This study aims to investigate the actual dispersion of patterns on the plastron of P.sinensis.Our research explores the role of osteoblasts expressing the enzyme cyp26b1, retinoic acid, and melanoblasts/melanophores on the ventral part of the plastron. We propose a hypothesis based on a reaction-diffusion system with a time-dependent growing spatial domain. This mathematical framework suggests the occurrence of the dispersion phenomenon. Specifically, we focus on the dilution term within the system under the growing-domain condition. In the context of black-pattern formation, we propose that variations in retinoic acid concentration, indirectly influenced by osteoblasts expressing the enzyme cyp26b1 during the embryonic/juvenile stage, contribute to the observed patterns. This hypothesis is grounded in the concept of prioritized osteogenesis and ossification during the embryonic/juvenile phase. This study expands our understanding of the species' survival strategy, highlighting the significance of bone biology and retinoic acid regulation. The findings have broader implications beyond P.sinensis, contributing to our knowledge of pattern formation and bone development in other organisms., Comment: a) Short version submitted for a publication, b) Supplementary Videos in

We discuss a new type of delay differential equation that exhibits resonating transient oscillations. The power spectrum peak of the dynamical trajectory reaches its maximum height when the delay is suitably tuned. Furthermore, our analysis of the resonant conditions for this equation has revealed a new connection between the solutions of the transcendental trigonometric equation and the Lambert W function. These results offer fresh insights into the nonlinear dynamics induced by delayed feedback., Comment: submitted for a publication; ver2. added proof to make a conjecture into a theorem, updating references and minor corrections; ver3. added Appendix and corrections of typos

Recently, we have studied a delay differential equation which has a coefficient that is a linear function of time. The equation has shown the oscillatory transient dynamics appear and disappear as the delay is increased between zero to asymptotically large delay. We here propose and study another equation that shows similar transient oscillations. It has an extra exponential gaussian factor on the delayed feedback term. It is shown that this equation is analytically tractable with the use of the Lambert $W$ function. This equation is also studied numerically to confirm some of the properties inferred from the analytical solution. We also have found that the amplitude of transient oscillation changes and goes through a maximum as we increase the value of the delay. In this sense, the proposed equation is one of the simplest dynamical equations that brings out a resonant behavior without any external oscillating inputs., Comment: 18 pages, 7 figures, version 2: Some typos and errors fixed; version 3: An author and reference added, and errors fixed; version 4: extensively revised final published version; Version 5: Corrected some technical errors and updated figures and tables accordingly from the published version

We search a simplest and minimal way to determine whether a given quantum system is entangled or separable. For this end, we propose binary correlation measurements in which restricted knowledge of only zero or non-zero correlations is available. We consider the concrete investigation on a pure state for two particles, each particle having two basis states (the 2x2 system). We show that, even with this limited information from the binary correlation measurements, we can still reach the known minimum of three measurements for entanglement detection. We next consider the comparable problem applied to the mixed density matrix. The mixed quantum case appears to require more detailed information, which we illustrate by studying the concrete example of the Werner density matrix., Comment: 11 pages, under submisssion. On version 2 there are the following revisions:some error in a proof was fixed; Reference updated and reflecting comments from readers, emphasis of the paper, and the title was changed. The technical contents, however, are the same as the version 1. The version 3 updated references and the discussion accordingly

We consider a quantum entangled state for two particles, each particle having two basis states, which includes an entangled pair of spin 1/2 particles. We show that, for any quantum entangled state vectors of such systems, one can always find a pair of observable operators X, Y with zero-correlations = . At the same time, if we consider the analogous classical system of a "classically entangled" (statistically non-independent) pair of random variables taking two values, one can never have zero correlations (zero covariance, E[XY] - E[X]E[Y] = 0). We provide a general proof to illustrate the different nature of entanglements in classical and quantum theories., Comment: 11 pages, Discussion on the quantum pigeonhole effect and references added in version 2; Extension to mixed states is added in the appendix in version 3 (The appendix is not included in the published version in PTEP)

It is well known that when a pair of random variables is statistically independent, it has no-correlation (zero covariance, $E[XY] - E[X]E[Y] = 0$), and that the converse is not true. However, if both of these random variables take only two values, no-correlation entails statistical independence. We provide here a general proof. We subsequently examine whether this equivalence property carries over to quantum mechanical systems. A counter-example is explicitly constructed to show that it does not. This observation provides yet another simple theorem separating classical and quantum theories., Comment: 7 pages, 2 figures, minor revision, To appear in Progress of Theoretical and Experimental Physics

We present here a new model and algorithm which performs an efficient Natural gradient descent for Multilayer Perceptrons. Natural gradient descent was originally proposed from a point of view of information geometry, and it performs the steepest descent updates on manifolds in a Riemannian space. In particular, we extend an approach taken by the "Whitened neural networks" model. We make the whitening process not only in feed-forward direction as in the original model, but also in the back-propagation phase. Its efficacy is shown by an application of this "Bidirectional whitened neural networks" model to a handwritten character recognition data (MNIST data)., Comment: 16pages

We present here a system with collection of random walks relaying a signal in one dimension in the presence of delays. We are interested in the time for a signal to travel from one end (start) to the other end (finish) of the lined group of random walkers. The delay is introduced at the point when the signal is transferred from each walker to the next one. It is found that there is an optimal number of walkers for the signal to travel fastest when delays are present. We discuss implications of this model and associated behaviors to physical and biological systems., Comment: 6 pages 3 Figures, To be presented and to appear in the proceedings of 9th European Nonlinear Dynamics Conference, 25-30 June 2017, Budapest, Hungary

We present here a new extended model of the gambler's ruin problem by incorporating delays in receiving of rewards and paying of penalties. When there is a difference between two delays, an exact analysis of the ruin probability is difficult. We derive an approximate scheme to find an effective shift in the initial assets of the gambler. Through comparison against computer simulations, this approximation is shown to work for small differences between the two delays., Comment: submitted for a publication

We propose a new approach for solving combinatorial optimization problem by utilizing the mechanism of chases and escapes, which has a long history in mathematics. In addition to the well-used steepest descent and neighboring search, we perform a chase and escape game on the "landscape" of the cost function. We have created a concrete algorithm for the Traveling Salesman Problem. Our preliminary test indicates a possibility that this new fusion of chases and escapes problem into combinatorial optimization search is fruitful., Comment: 3 pages, 4 figures. To appear in the Proceedings of the International Symposium on Artificial Life and Robotics (AROB20th), Beppu, Oita Japan, January 21-23, 2015

We present here two examples of stochastic modelings of social phenomena. The first topic is pedestrian counter flow. Two groups of model pedestrians move in opposite directions and create congestions. It will be shown that this congestion becomes worst where individuals are given certain stochastic freedom to avoid another in front compared to the case that they are bound to more strict rules. The second example model tournaments. We present here a rather unexpected feature of tournaments that the probability to reach the top position is higher than that of finishing up at lower positions for not only the number one ranked player, but also for a range of top players. This "inversion characteristics" are shown to be observed with simple mathematical model tournaments as well as in the real tournaments., Comment: submitted for a publication

We present here a simple equation explicitly incorporating non-locality, which reproduces quantized energy levels of the bound states for the square well potentials. Introduction of this equation is motivated by studies of differential equation with time delayed feedback, which can be viewed as describing temporal non-locality., Comment: 4 pages, 1 figure, submitted for a publication

We are studying the effect of converting caught targets into new chasers in the context of the recently proposed `group chase and escape' problem. Numerical simulations have shown that this conversion can substantially reduce the lifetimes of the targets when a large number of them are initially present. At the same time, it also leads to a non-monotonic dependence on the initial number of targets, resulting in the existence of a maximum lifetime. As a counter effect for this conversion, we further introduce self-multiplying abilities to the targets. We found that the longest lifetime exists when suitable combination of these two effects is created., Comment: submitted for publication

We present here a new conjecture for the nature of the Mersenne prime numbers by connecting it with the Collatz-Kakutani problem. By introducing a natural path length on the basis of the Collatz-Kakutani tree, we conjecture that this path length of a Mersenne prime from the root of the Collatz-Kakutani tree is approximately proportional to the index of the Mersenne prime. We also discuss difference of behaviors between Mersenne numbers and Mersenne primes., Comment: 10 pages, 4 figures, submitted for a publication

We describe here a new concept of one group chasing another, called "group chase and escape", by presenting a simple model. We will show that even a simple model can demonstrate rather rich and complex behavior. In particular, there are cases in which an optimal number of chasers exists for a given number of escapees (or targets) to minimize the cost of catching all targets. We have also found an indication of self-organized spatial structures formed by both groups., Comment: 13 pages, 12 figures, accepted and to appear in New Journal of Physics

We present simple classical dynamical models to illustrate the idea of introducing a stochasticity with non-locality into the time variable. For stochasticity in time, these models include noise in the time variable but not in the "space" variable, which is opposite to the normal description of stochastic dynamics. Similarly with respect to non-locality, we discuss delayed and predictive dynamics which involve two points separated on the time axis. With certain combinations of fluctuations and non-locality in time, we observe a ``resonance'' effect. This is an effect similar to stochastic resonance, which has been discussed within the normal context of stochastic dynamics, but with different mechanisms. We discuss how these models may be developed to fit a broader context of generalized dynamical systems where fluctuations and non-locality are present in both space and time., Comment: 12 pages, 5 figures, Accepted and to appear in Physica A. (reference corrected for ver. 2)

We study a system whose dynamics are governed by predictions of its future states. A general formalism and concrete examples are presented. We find that the dynamical characteristics depend on how to shape the predictions as well as on how far ahead in time to make them. We also report that noise can induce oscillatory behavior, which we call "predictive stochastic resonance"., Comment: Presented and To appear in the Proc. of 9th Granada Seminar, (Granada, Spain, September 11-15, 2006)

We present a simple dynamical model to address the question of introducing a stochastic nature in a time variable. This model includes noise in the time variable but not in the "space" variable, which is opposite to the normal description of stochastic dynamics. The notable feature is that these models can induce a "resonance" with varying noise strength in the time variable. Thus, they provide a different mechanism for stochastic resonance, which has been discussed within the normal context of stochastic dynamics., Comment: Presented and to apper in the AIP Conf. Proc. of the 8th Int. Symp. on Frontier of Fundamental Physics (FFP8, Oct. 17-19, 2006, Madrid, Spain)