To improve spectrum sensing performance, a cooperative spectrum sensing method based on **information** **geometry** and fuzzy c-means clustering algorithm is proposed in this paper. In the process of signal feature extraction, a feature extraction method combining decomposition, recombination, and **information** **geometry** is proposed. First, to improve the spectrum sensing performance when the number of cooperative secondary users is small, the signals collected by the secondary users are split and reorganized, thereby logically increasing the number of cooperative secondary users. Then, in order to visually analyze the signal detection problem, the **information** **geometry** theory is used to map the split and recombine signals onto the manifold, thereby transforming the signal detection problem into a geometric problem. Further, use geometric tools to extract the corresponding statistical characteristics of the signal. Finally, according to the extracted features, the appropriate classifier is trained by the fuzzy c-means clustering algorithm and used for spectrum sensing, thus avoiding complex threshold derivation. In the simulation results and performance analysis section, the experimental results were further analyzed, and the results show that the proposed method can effectively improve the spectrum sensing performance.

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Abstract: A probabilistic description is essential for understanding the dynamics of stochastic systems far from equilibrium, given uncertainty inherent in the systems. To compare different Probability Density Functions (PDFs), it is extremely useful to quantify the difference among different PDFs by assigning an appropriate metric to probability such that the distance increases with the difference between the two PDFs. This metric structure then provides a key link between stochastic systems and **information** **geometry**. For a non-equilibrium process, we define an infinitesimal distance at any time by comparing two PDFs at times infinitesimally apart and sum these distances in time. The total distance along the trajectory of the system quantifies the total number of different states that the system undergoes in time and is called the **information** length. By using this concept, we investigate the **information** **geometry** of non-equilibrium processes involved in disorder-order transitions between the critical and subcritical states in a bistable system. Specifically, we compute time-dependent PDFs, **information** length, the rate of change in **information** length, entropy change and Fisher **information** in disorder-to-order and order-to-disorder transitions and discuss similarities and disparities between the two transitions. In particular, we show that the total **information** length in order-to-disorder transition is much larger than that in disorder-to-order transition and elucidate the link to the drastically different evolution of entropy in both transitions. We also provide the comparison of the results with those in the case of the transition between the subcritical and supercritical states and discuss implications for fitness.

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Since we deal with a multi-parameter statistical problem, we adopt a differential **geometry** viewpoint in the spirit of the theory of **information** **geometry** 2 . This allows us to characterise the manifold of identifiable parameters as the quotient of the parameter space with respect to a group of transformations leaving the output state invariant (see Theorem 1), thus extending our previous results for discrete time quantum Markov chains 31 . An analogous differential geometric construction has been presented in 33,34 for parametrisations of discrete matrix product states,

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Abstract: Stochastic processes are ubiquitous in nature and laboratories, and play a major role across traditional disciplinary boundaries. These stochastic processes are described by different variables and are thus very system-specific. In order to elucidate underlying principles governing different phenomena, it is extremely valuable to utilise a mathematical tool that is not specific to a particular system. We provide such a tool based on **information** **geometry** by quantifying the similarity and disparity between Probability Density Functions (PDFs) by a metric such that the distance between two PDFs increases with the disparity between them. Specifically, we invoke the **information** length L ( t ) to quantify **information** change associated with a time-dependent PDF that depends on time. L ( t ) is uniquely defined as a function of time for a given initial condition. We demonstrate the utility of L ( t ) in understanding **information** change and attractor structure in classical and quantum systems.

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ABSTRACT: the visual and especially tactile quality of a surface is called a texture. Stochastic textures with features spanning many length scales arise in a range of contexts in sciences, from nano size structures like synthetic bone to a ocean wave height distributions and cosmic phenomena like inter-galactic cluster void distributions. The samples here came from the papermaking industry but such a reduction of large frequently noisy spatial data sets is useful in a range of materials and contexts at all scales. Aim is reducing the size of structure without any loss in **information** by using **information** **geometry**. We are giving answer to “how far apart are two distributions?” e.g. Gaussian (m, s2 ): Euclidian distance between two distributions has no `natural’ statistical significance. **Information** **geometry** seeks first the shape of the (multidimensional) surface and once the surface is known the shortest curve between two points representing the distributions is the ‘natural’ metric, **Information** **geometry** provides a natural metric to discriminate among formation textures.

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Abstract—Aiming at the problem of high false alarm rate with respect to adaptive threshold in the ship detection from synthetic aperture radar (SAR) images, a novel strategy increasing robustness when using local adaptive threshold is proposed. In this article, we establish a fusion detection model based on a combination of the **information** **geometry** and surface **geometry**. **Information** **geometry** from a metric viewpoint can increase the contrast between targets and clutter in SAR image. Local surface feature gives a brief application of adaptive threshold method in ship detection from SAR images by means of the constant false-alarm-rate. Experiments indicate that the proposed **geometry**-based approach can eﬀectively detect ship targets from complex background SAR images by using the method of fusion processing.

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We investigated **information** **geometry** associated with order-to-disorder and disorder- to-order transitions in a 0D Ginzburg-Landau model where the formation (disappearance) of an ordered state is modelled by the transition from a unimodal (bimodal) to bimodal (unimodal) PDF of a stochastic variable x. Our 0D model permitted us to perform a detailed statistical analysis. We considered on-critical quenching with a pair of forward and backward processes FP and BP for disorder-to-order (critical to subcritical) and order- to-disorder (subcritical to critical) transitions, respectively by selecting the initial PDF of FP/BP the same as the final equilibrium PDF of BP/FP. A pair of disorder-to-order and order-to-disorder transitions models a burst, for example, in the gene expression consisting of a pair of induction and repression (e.g. see [50]). In such bistable systems, a continuous switching between ordered and disordered states is often observed, the transition occurring in bursts interspersed by a quiescent period (e.g. see [50]). For our cyclic order-disorder transition, an initial condition represents the ‘resting’ state between the two bursts. We thus paid particular attention to the effect of initial conditions on **information** change by comparing on-quenching and off-quenching cases.

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We investigated **information** **geometry** associated with order-to-disorder and disorder-to-order transitions in a 0D Ginzburg–Landau model where the formation (disappearance) of an ordered state is modelled by the transition from a unimodal (bimodal) to bimodal (unimodal) PDF of a stochastic variable x. Our 0D model permitted us to perform a detailed statistical analysis. We considered on-critical quenching with a pair of forward and backward processes FP and BP for disorder-to-order (critical to subcritical) and order-to-disorder (subcritical to critical) transitions, respectively by selecting the initial PDF of FP/BP the same as the final equilibrium PDF of BP/FP. A pair of disorder-to-order and order-to-disorder transitions models a burst, for example, in the gene expression consisting of a pair of induction and repression (e.g., see [50]). In such bistable systems, a continuous switching between ordered and disordered states is often observed, the transition occurring in bursts interspersed by a quiescent period (e.g., see [50]). For our cyclic order-disorder transition, an initial condition represents the “resting” state between the two bursts. We thus paid particular attention to the effect of initial conditions on **information** change by comparing on-quenching and off-quenching cases.

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In particular, we computed how the rate of **information** change and the resulting total **information** length L ∞ depend on the position of an initial Gaussian peak. We found that for all choices of f ( x ) , the unstable fixed points yield comparatively small L ∞ , even though they are farthest away from the final equilibrium points. It is particularly interesting that L ∞ as a function of initial position qualitatively follows f ( x ) , indicating the close connection between the **information** **geometry** and the underlying forcing.

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to be useful in many problems of practical importance as for instance, distributed sparse detection [18], sparse support recovery [48], energy detection [35], MIMO radar processing [31,45], network secrecy [13], Angular Resolution Limit in array processing [27], detection performance for informed communication systems [33], just to name a few. In addition, the Chernoff **information** bound can be tight for a minimal s-divergence over parameter s ∈ ( 0, 1 ) . Generally, this step requires to solve numerically an optimization problem [41] and often leads to a complicated and uninformative expression of the optimal value of s. To circumvent this difficulty, a simplified case of s = 1/2 is often used corresponding to the well-known Bhattacharyya divergence [47] at the price of a less accurate prediction of P e (N) . In **information** **geometry**, parameter s is often called α, and the s-divergence is the

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Abstract: We elucidate the effect of different deterministic nonlinear forces on geometric structure of stochastic processes by investigating the transient relaxation of initial PDFs of a stochastic variable x under forces proportional to − x n (n = 3, 5, 7) and different strength D of δ-correlated stochastic noise. We identify the three main stages consisting of nondiffusive evolution, quasi-linear Gaussian evolution and settling into stationary PDFs. The strength of stochastic noise is shown to play a crucial role in determining these timescales as well as the peak amplitude and width of PDFs. From time-evolution of PDFs, we compute the rate of **information** change for a given initial PDF and uniquely determine the **information** length L( t ) as a function of time that represents the number of different statistical states that a system evolves through in time. We identify a robust geodesic (where the **information** changes at a constant rate) in the initial stage, and map out geometric structure of an attractor as L( t → ∞ ) ∝ µ m , where µ is the position of an initial Gaussian PDF. The scaling exponent m increases with n, and also varies with D (although to a lesser extent). Our results highlight ubiquitous power-laws and multi-scalings of **information** **geometry** due to nonlinear interaction. Keywords: stochastic processes; Fokker-Planck equation; **information** length

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Abstract: We introduce the Symplectic Structure of **Information** **Geometry** based on Souriau’s Lie Group Thermodynamics model, with a covariant definition of Gibbs equilibrium via invariances through co-adjoint action of a group on its moment space, defining physical observables like energy, heat, and moment as pure geometrical objects. Using Geometric (Planck) Temperature of Souriau model and Symplectic cocycle notion, the Fisher metric is identified as a Souriau Geometric Heat Capacity. Souriau model is based on affine representation of Lie Group and Lie algebra that we compare with Koszul works on G/K homogeneous space and bijective correspondence between the set of G-invariant flat connections on G/K and the set of affine representations of the Lie algebra of G. In the framework of Lie Group Thermodynamics, an Euler-Poincaré equation is elaborated with respect to thermodynamic variables, and a new variational principal for thermodynamics is built through an invariant Poincaré-Cartan-Souriau integral. The Souriau-Fisher metric is linked to KKS (Kostant-Kirillov-Souriau) 2-form that associates a canonical homogeneous symplectic manifold to the co-adjoint orbits. We apply this model in the framework of **Information** **Geometry** for the action of an affine Group for exponentiel families, and provide some illustrations of use cases for multivariate Gaussian densities. **Information** **Geometry** is presented in the context of seminal work of Fréchet and his Clairaut-Legendre equation. Souriau model of Statistical Physics is validated as compatible with Balian gauge model of thermodynamics. We recall the precursor work of Casalis on affine group invariance for natural exponential families.

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Abstract: We elucidate the effect of different deterministic nonlinear forces on geometric structure of stochastic processes by investigating the transient relaxation of initial PDFs of a stochastic variable x under forces proportional to − x n (n = 3, 5, 7) and different strength D of δ-correlated stochastic noise. We identify the three main stages consisting of nondiffusive evolution, quasi-linear Gaussian evolution and settling into stationary PDFs. The strength of stochastic noise is shown to play a crucial role in determining these timescales as well as the peak amplitude and width of PDFs. From time-evolution of PDFs, we compute the rate of **information** change for a given initial PDF and uniquely determine the **information** length L ( t ) as a function of time that represents the number of different statistical states that a system evolves through in time. We identify a robust geodesic (where the **information** changes at a constant rate) in the initial stage, and map out geometric structure of an attractor as L ( t → ∞ ) ∝ µ m , where µ is the position of an initial Gaussian PDF. The scaling exponent m increases with n, and also varies with D (although to a lesser extent). Our results highlight ubiquitous power-laws and multi-scalings of **information** **geometry** due to nonlinear interaction. Keywords: stochastic processes; Fokker-Planck equation; **information** length

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We investigated the **information** **geometry** of non-equilibrium processes involved in classical and quantum systems. Specifically, we introduced τ ( t ) as a dynamical time scale quantifying **information** change and calculated L( t ) by measuring the total elapsed time t in units of τ. As a unique representation of the total number of statistically different states that a PDF evolves through in reaching a final PDF, L ∞ was demonstrated to be a novel diagnostic for mapping out an attractor structure. In particular, L ∞ preserves a linear **geometry** of a linear process while manifesting nonlinear **geometry** in a cubic (nonlinear) process; it takes its minimum value at the stable equilibrium point. In the case of a chaotic attractor, L ∞ exhibits a sensitive dependence on initial conditions like a Lyapunov exponent. Thus, L ∞ is a useful diagnostic for mapping out an attractor structure. To illustrate that L can be applied to any data as long as time-dependent PDFs can be constructed from the data, we presented the analysis of different classical music (e.g. see [16]). Finally, the width of PDFs was shown to play a dual role in **information** length in quantum systems. It cannot be over-emphasized that L is path-specific and is a dynamical measure of the metric, capturing the actual statistical change that occurs during time evolution. This path-specificity would be crucial when it is desirable to control certain quantities according to the state of the system (e.g. time-dependent PDFs) at any given time (e.g. see [18]). Due to the generality of our methodology, we envision a large scope for further applications to different phenomena.

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Mathematical models describing physical, biological, and other real-life phe- nomena contain parameters whose values must be estimated from data. Over the past decade, a powerful framework called “sloppiness” has been developed that relies on **Information** **Geometry** [1] to study the uncertainty in this proce- dure [10, 17, 56, 57, 58, 55]. Although the idea of using the Fisher **Information** to quantify uncertainty is not new (see for example [20, 45]), the study of sloppi- ness gives rise to a particular observation about the uncertainty of the procedure and has potential implications beyond parameter estimation. Specifically, sloppi- ness has enabled advances in the field of systems biology, drawing connections to sensitivity [25, 19, 24], experimental design [4, 37, 25], identifiability [47, 55, 13], robustness [17], and reverse engineering [19, 14]. Sethna, Transtrum and co-authors identified sloppiness as a universal property of highly parameterised mathematical models [61, 56, 54, 25]. More recently a non-local version of sloppiness has emerged, called predictive sloppiness [33]. However, the precise interpretation of sloppiness remains a matter of active discussion in the literature [4, 26, 29].

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Mathematical models describing physical, biological, and other real-life phe- nomena contain parameters whose values must be estimated from data. Over the past decade, a powerful framework called “sloppiness” has been developed that relies on **Information** **Geometry** [1] to study the uncertainty in this proce- dure [10, 17, 56, 57, 58, 55]. Although the idea of using the Fisher **Information** to quantify uncertainty is not new (see for example [20, 45]), the study of sloppi- ness gives rise to a particular observation about the uncertainty of the procedure and has potential implications beyond parameter estimation. Specifically, sloppi- ness has enabled advances in the field of systems biology, drawing connections to sensitivity [25, 19, 24], experimental design [4, 37, 25], identifiability [47, 55, 13], robustness [17], and reverse engineering [19, 14]. Sethna, Transtrum and co-authors identified sloppiness as a universal property of highly parameterised mathematical models [61, 56, 54, 25]. More recently a non-local version of sloppiness has emerged, called predictive sloppiness [33]. However, the precise interpretation of sloppiness remains a matter of active discussion in the literature [4, 26, 29].

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component of the ‘Using and applying’ attainment target. Even at the ‘foundation’ level at Key Stage 4 (for the lower attaining 14-16 year olds), students are to be taught to distinguish between practical demonstrations and proofs and to show step- by-step deduction in solving a geometrical problem (DfEE 1999:78). For teachers teaching this, or a similar, curriculum, the challenge is to develop teaching methods which do not turn pupils off or get them solely to learn by rote (as appears to have been the case in the past). This will certainly require new pedagogical approaches which are likely to involve technology like dynamic **geometry**, as well as discursive methods of engagement and methods of assessment which reduce the pressure to rote learn.

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In the 1982 version of this paper, rather than start with a whole sh, four basic square tiles similar to those shown in Figure 17 were used.. The tiles p, q, r, s are shown in an arragem[r]

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Presently, we contemplate a renewed effort to calculate the full **geometry** of the MSSM vac- uum moduli space. The computing power required for applying the previously described algorithm with ∼ 1000 GIOs and ∼ 50 fields is well beyond what is accessible by standard personal computers. The use of supercomputers is envisaged. For this reason, our goal here is significantly more modest, and we only unveil aspects of the **geometry** for the electroweak sector. That is, we study a subsector of the full vacuum moduli space that is given by the additional constraints that the vacuum expectation values of the quark fields vanish:

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We have studied the logarithmic corrections to the entanglement entropy of a minimally coupled scalar field in the subtracted **geometry** black hole background. Our main results are collected in formulas (5.1.19-5.1.24). They all diﬀer from the corresponding results for non-subtracted black holes, indicating that the agreement of subtracted and non-subtracted entropies does not extend beyond the tree level. On the other hand, the subtracted results approach the original ones for the extremal BPS case in the appropriate limit. We noticed that the logarithmic correction term universally changes sign for all cases of subtracted black holes. For the schwarzchild case we found the interpolating solution which for certain choices of the Harrison parameters gives a vanishing logarithmic correction.

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