Search

We have argued in previous notes (1) that the “wave” nature of quantum mechanics follows from the Lorentz invariant for a constant p: A= -Et+px which holds for both a particle with rest mass and a photon. In A, t and x are separated indicating independence which violates the Newtonian view x(t). “A” leads to dA/dx partial = p and dA/dt= -E. Creating eigenvalue equations yields: -id/dx exp(ipx) = p exp(ipx) and id/dt exp(-iEt) = E exp(-iEt). Thus, x and t may be separated. Given that exp(ipx) exists in all of space (because there is no time), it represents a somewhat spatially invariant probability. In the case of a photon, Maxwell showed one may write wave equations for the electric El and magnetic B fields using his electromagnetic equations with no charge or currents.. He showed the wave travels at the speed of light in a vacuum and thus characterized light as an electromagnetic wave. This is not the same statement as exp(ipx) which is a probability wave. One may note that Maxwell’s equations are Lorentz invariant, so the ideas of the first paragraph should hold. In this note, however, we consider the Poynting equation: d/dt partial (.5( eoElEl+1/uoBB)) + 1/uo grad dot El x B = function(charge and current). 1/uo ElxB is considered to be a momentum density. We note Maxwell’s equations contain d/dt partial and d/dx partial, so time and space are considered independent (unlike x(t)), and act on El and B. The presence of charge and current densities breaks spatial invariance i.e. creates local regions. We argue in this note that d/dx and d/dt are generators of translations in space and time. We further argue that Maxwell’s theoretical creation of a photon follows from introducing spatial invariance into the problem, i.e. removing all charge and current densities. Spatial invariance in Newtonian mechanics, however, is associated with a constant i.e. a constant speed and momentum and equal probability to be at any x. Given that d/dx and d/dt appear, one has a function of space and time which seems to be completely at odds with spatial invariance and the idea of a constant momentum. We have argued in (2), however, that constant motion, even though represented by v=constant, is associated with the force that created the momentum, and this manifests itself in a spatial density which is probabilistic. Thus the presence of d/dt and d/dx in Maxwell’s electromagnetic equations, even when charge and current densities are removed, implies the same thing. Thus we argue that Maxwell’s electromagnetic equations are probabilistic. We investigate these ideas in this note. We further argue that it is spatial invariance, together with the presence of d/dx partial which yields the periodic form exp(ipx). Thus it is not a coincidence that wave equations appear for El and B. &nbsp

It is known that if a photon hits a surface with a different index of refraction (at an angle within a certain range), it has a specific probability “a” to reflect and 1-a to refract. In this note we ask: Where does this statistical nature come from? In previous notes, we have considered Fermat’s principle applied separately to explain two dimensional reflection and refraction. We have also used the idea of a difference of squares and energy conservation in one dimensional reflection/refraction to find values of the probabilities. The question we ask is: Why are there probabilities in the first place? Is there some general idea at work despite the complicated details of the photon interacting with the atoms of the new medium.We argue the general notion is due to the idea of spatial invariance which is linked to the translation operator d/dx which in turn in linked to a p impulse occurring at a given x .Thus the variables p and x are linked through the notion of an impulse existing at some x. The idea that d/dx may operate on a function of constant p and x to yield p, implies the function of p,x should have physical meaning. This is a problem as classically a particle moving with constant p should not be associated with any function of x except a constant. (All x points are the “same”.) Thus we argue that a “probability” in x is established with a constantly moving particle (or even photon). The eigenfunction of d/dx (or rather -id/dx) which exhibits some kind of spatial invariance is exp(ipx). This contains two different x probability distributions cos(px) and sin(px). We noted that d/dx is associated with p due to an impulse hit at say an x point. Thus various exp(ipx)s with different weights should appear together in a physical problem with impulses such as reflection/refraction. The weights of the various exp(ip1 x), exp(ip2 x) etc become linked to probabilities associated with the state 1, 2 etc. Thus the origin of these probabilities (i.e. their specific values) follow from the probabilistic form exp(ipx) which follows from d/dx. (In (1) we have argued that the x distributions in exp(ipx) may be further linked to the force which “created” the p in the first place, with different p values having different distributions cos(px), sin(px)).

It was argued in (1) that Fermat’s least time principle for light (in a two dimensional (x,y) reflection or refraction scenario) minimizes time by actually varying it in terms of a spatially invariant variable i.e. x (i.e. taking d/dx) with the flat face of the mirror/medium along the x axis at y=0. This is equivalent to conserving momentum in the x direction. Thus d/dx is associated with (px) which may be associated with the Lorentz invariant -Et+p dot r. In other words, d/dx emerges as a momentum operator (up to a factor) and d/dt as an energy operator, yet both are generators in time and space linked with invariance of constant values i.e. conservation. Furthermore an eigenvalue equation: -id/dx exp(ipx) = p exp(ipx) and id/dt exp(-iEt) = E exp(-iEt) automatically (by the nature of eigenvalue) leads to orthogonal eigenfunctions linked to conservation of energy and momentum. In special relativity rest mass mo represents a constant in time which may exist at any point x. In fact, it is often thought of as a point, for example a center of mass point. The actual object with mo, however, may be large with moving parts all within a “container”. What then does x,t in -Et+px mean for constant E,p? It seems that x,t are center-of-mass coordinates for the overall system. We argued in previous notes that 1/p is proportional to a wavelength so for p→0, mo may be at any x point with the same probability. Within the system with moving parts, however, there is motion and an x scale for each moving piece. For a single particle bound system, there is a single energy at each x which is like mo and a potential V(x) and KE(x). In (2) we argued that given the free particle relationship -EE+pp= -momo (c=1) one may retain a constant E value by replacing E with E-V(x) and pp with (and not ). How do the probabilities appear? They are linked to -Et+px with frequency hbar/E and wavelength hbar/p and are specifically manifested by exp(-iEt) and exp(ipx). For and we use spatial probabilities i.e. W(x) = Sum over p a(p)exp(ipx). Each exp(ipx) is linked to the operator -id/dx and invariance, but pp is linked with the operator -d/dx d/dx and one does not use an eigenfunction of this operator. Thus the notion of conservation of momentum is retained in the exp(ipx)s and their linear combination W(x) yields a function such that an average kinetic energy exists at each x point i.e KE(x)=KEclassical = -1/2m d/dx dW/dx / W. This is combined with a V(x) to ensure constant E at each x point i.e. create a kind of mo piece linked with id/dt. We argue that E being constant at each x is a statistical equilibrium type of result. Each exp(ipx) is associated with an average motion of x=p/m t and given that exp(ipx) and exp(-ipx) are included, there is also the notion of which is positive and negative in different regions and integrates to 0. prms(x) from becomes the classical p(x). Thus bound state quantum mechanics seems to be based on a free particle treatment which follows from the special relativistic Lorentz invariants -EE+pp=-momo and -Et+px = constant, directly linked to conservation ideas. Conservation of momentum of an internal bound particle is maintained through exp(ipx), while by itself is not linked with conservation as there is no eigenstate, but is combined with a V(x) to create a constant En linked with the entire system (and P=0) which then becomes associated with invariance in time and the generator id/dt. In this note, we try to argue that the notion of W*(x)W(x) (where W(x)=wavefunction= Sum over p a(p)exp(ipx)) is spatial density and P(p)=a*(p)a(p) based on spatial invariance and conservation of momentum for a single particle bound state even though there seems to be no spatial invariance or notion of conservation of momentum in W(x) which is simply a solution of the time-independent Schrodinger equation. One may move from a W(x) scenario (i.e. Schrodinger equation) to a W*(x)W(x) picture, by trying to “pick out” invariant (p only terms) using the orthogonality of the exp(ipx)s. Thus W*(x)W(x) (spatial density) emerges naturally in this approach.

The Maxwell-Boltzmann (MB) probability factor exp(-.5mvv/T) where T is temperature may be derived through the time reversal reaction balance of elastic two body collisions i.e. e1+e2=e3+e4 and f(e1)f(e2)=f(e3)f(e4). The latter relation may be assumed a priori. Taking ln of it and equating to the former multiplied by -1/T yields the MB relation. In this note we argue that this calculation may be thought of in terms of information. Even though f(e1)f(e2) is assumed a priori and ln taken to convert it into a sum, in information science ln(f(ei)) is called information. In physics, one may think of it in terms of a change relative to the value itself e.g. d/dT ln(f(e1)) = df(e1)/dT / f(e1). Two body elastic collisions represent the interactions in the system, thus: E=e1+e2= ei+ej. In other words the only information before and after an interaction is a given E. We argue that equilibrium tries to minimize information so f(e1) and f(e2) together representing the collision e1+e2 should not represent any new information. Thus one expects G(e1+e2) to represent this probability such that G splits into two identical parts. This may occur through addition or multiplication. Multiplication allows for f(e1) to drop to zero as e1 → infinite. Thus we argue that an information type argument may lead to G(e1+e2) = f(e1)f(e2) which in turn leads to the MB factor. In the case of a potential we consider the idea of spatial invariance which already appears in Fermat’s principle and is linked with conservation of momentum in the direction of an invariant variable. For no potential one clearly has spatial invariance of .5mvv. With a potential, one has spatial invariance of E1= .5mv(x)v(x) +V(x). Assuming invariance of this variable together with the loss of information from f(e1)f(e2) one may suggest f(E) = exp(-(.5mvv+V(x))/T) ((A)). This begs the question: How does d/dx and conservation which appear in Fermat’s principle due to spatial invariance appear here? We argue that d/dx, the generator of translations, is linked to spatial invariance. We again use the idea of equilibrium trying to minimize information. We use density d(x), but write it as information i.e. ln(d(x)) and argue that a change in this value should equal a change in a scalar given piece of information in the system namely V(x)/T1 where T1 is a constant to create proper units. This may alternatively be written with a d/dx as in Fermat’s principle to create a conservation law i.e. d/dx ln(d(x)) = -d/dx V(x)/T1. This is equivalent to a pressure difference balanced by a force times density. Thus informational ideas may yield Netwonian mechanical balance. Information considerations lead to d(x) = exp(-V(x)/T1) so d(x) vanishes for large x for say an oscillator potential. Using information minimization for a single point x i.e. f(v) = exp(-.5mvv/T) one may say that having T1=T yields spatial invariance for the total energy over x as suggested in ((A)). Thus we argue that ideas of minimization of information together with invariance may be key to obtaining the Maxwell-Boltzmann factor and may also account for the Newtonian pressure -force*density “conservation”.

A quantum particle is often described as exhibiting dual particle-wave nature. A particle seems to be an object which obeys Newton’s laws/special relativity with a wave also being a phenomenon existing in Newtonian mechanics (special relativity). In other words the particle and wave are governed by x-t external variables which have infinite resolution. It was originally thought that x and t in addition to being external were absolute, but special relativity changed this idea. We have suggested in previous notes that a free quantum particle with energy E and momentum p creates its own space time i.e time intervals are proportional to 1/E and space to 1/p. This means no external clock is needed and different internal rulers and clocks exist for different E,p. It also suggests finite time and space intervals. In fact, an absolute is introduced namely A=0 = -Et+px with t and x being independent. This suggests that t→b/E and x→b/p. A is the free particle Lagrangian (both relativistic and nonrelativistic) with x and t being considered independent. Thus we suggest that the external x ruler and t clock of Newtonian mechanics represent an average of the internal picture and are also described by A, but with x/t=v and not x and t being treated as independent. In (1) we showed that having discrete x spacing with a running time (i.e. Netwon’s time variable) and insisting on spatial invariance leads to dA/dx partial = p. Thus instead of considering particle wave duality we suggest an underlying wave structure which creates its own spacetime intervals (1/E, 1/p), but may be mapped in certain situations to the external running x and t of Newton’s theory. For example, consider an electron (a quantum particle) with a fixed p moving over a distance much greater than its wavelength. It may be mapped to Newton’s external running x,t values with x=vt, but this is an average mapping into continuous x and t. In reality the situation is more complicated because there may be a Bohm-Aharonov effect occurring. Thus the underlying wave picture is in fact physically important and linked to interactions. Thus a self-ruler is not simply a measuring device - interactions may occur unlike in the case of the external ruler. Given that p defines a kind of spatial invariance approximated by a periodic function exp(ipx) (eigenfunction of -id/dx) and that Newton’s running time is not needed in a quantum bound state equilibrium, one may wonder what one should do in the case of a particle bound in a V(x) potential. In Newtonian mechanics V(x) suggests non-spatial invariance and v(x) velocity changes with x as well in contrast with exp(ipx) which is periodically invariant. In fact Newtonian mechanics (and Lagrangian/Hamiltonian as well) focus on d/dt i.e x(t). We have argued above that time intervals 1/E are independent of 1/p spatial ones. It may be noticed, however, that different p values lead to different wavelengths and so even though there is a kind of spatial invariance for each, when compared to each other this invariance is broken and superpositions of exp(ipx) may create local x probability densities. At the same they create spatial momentum densities because -d/dx (d/dx) 1/2m yields a kind of average kinetic energy when normalized. This seems to describe the x-p uncertainty relationship. In order to create spatial resolution (other than the built-in one in the wavelength) one needs to superpose different momenta, each of which are spatially invariant according to their periodic scheme exp(ipx). Thus one may create an average 1/2m and obtain a vrms. This may be described in terms of Newtonian external running x and t variables which have infinite resolution, but this vrms does not map to a physical state i.e to exp(i (mvrms) x) localized at x. Rather the various exp(ipx)’s which create vrms are the physical entities as may be seen by knocking a particle out of a bound state. The physical exp(ipx) momenta states have spatial invariance and so cannot be associated with a specific position in space. Only their superposition creates the notion of specific position in space, but then there is a distribution of p. This seems to be the basis for the x-p uncertainty principle. Thus we suggest that the particle of particle-wave duality is created through a mapping of a physical wave scenario to a continuous (infinite resolution) x-t scheme which in reality is an abstraction because each p,E free particle state has it sown internal clock-ruler i.e. 1/E and 1/p time and space units. Given that 1/E and 1/p have finite values the idea of dt and dx tending to 0 only makes sense as an abstraction which only holds in certain situations.

The nonrelativistic classical action of a free particle is A=.5mt vv is equivalent to time multiplied by a constant. v=dx/dt =constant for free motion and infinite x resolution is assumed. One may write v=x/t so that A=.5mxx/t.. Assuming x and t are independent which they are not in Newton’s scenario yields dA/dx partial = p where p is momentum. A=.5mt vv with v=constant, however, leads to dA/dx partial = 0, thus there is a dilemma. To resolve this dilemma we assume a finite unknown resolution to x, but let t have infinite resolution.Thus we write v = Dx / delta t. where Dx is the unknown x resolution distance as the smallest x interval distance allowed. In other words Dx does not tend to 0 so v is really an average speed within this resolution scheme. One may take a number line and mark off Dx points from the origin, but the origin may be shifted. In other words there is x invariance in this scenario which suggests a constant in this direction related to this invariance. The question is: What is this constant? One might consider either velocity or momentum (which is nonrelativsitically m (rest mass) multiplied by v. One, however, wants a theory which is the same in a lab frame as in a frame moving with a constant velocity and (p,E) and (x,t) are 4-vectors in special relativity, while v is not part of a 4-vector. Thus we consider a p as being linked to an invariance in the x direction for nonelativistic motion. Alternatively one may note that Fermat’s principle associates d/dx i.e. spatial invariance in the x direction with constant momentum in this direction. Fermat’s principle, however, applies to light so perhaps there is a similarity between light and a free particle with rest mass. We also suggest that spatial invariance is linked with probability i.e. the probability to be at one x versus another. Both Lagrangian and Hamiltonian theory deal largely with time derivatives. One may note that dv/dx=0 for a free particle, but we are interested in a function different from v, one that yields the constant of motion associated with invariance. Weseek a function A such that dA/dx partial =p where dx is smaller than Dx i.e. may tend to zero. The classical action written with v=Dx/Dt written as x/t for simplicity with Dx and Dt being independent within the resolution ranges. In Newton’s scenario the resolution ranges tend to zero so one never has x,t independence, but with fixed resolution one does. Thus dA/dx partial = p describes the x invariance. In particular holding t constant (say t=0) yields A=px such that the invariance in x is described by nb/p where b is a constant and n=1, 2,3 … This suggests a resolution scheme for the invariant x i.e. Dx = b/p. Then v= Dx/ delta t. dA/dt partial = E which is a constant and so this also suggests an invariance in time. For x=0, A=-Et suggesting a periodicity in t of b/E. This is independent of b/p except for the constant b which ensures that A=0. We focus in this note, however, on resolution b/p in the x direction and its link to b/p length units which define Dx. Thus by forcing the notion of a finite resolution in the x direction, unlike the Newtonian scenario, one is forced to think of p (momentum) as not simply being mv (nonrelativistic), but as related to an invariance in the x direction. The function defining this invariance, namely A the classical action, also serves to define Dx, the x resolution suggesting a periodic scheme in space similar to that found in light i.e the periodicity in the electric and magnetic fields cos(-Et+px). Thus even though A = .5m Dx Dx/ delta t, and Dx/delta t = v, one may vary Dx within the resolution range by a tiny amount dx which does tend to zero and leave delta t unchanged. This demonstrates the invariance in the x direction of shifting Dx points which is associated with the conserved quantity momentum as shown by dA/dx partial = p. These same results hold for a relativistic free particle with A= -Et + px, but E = mocc/ sqrt(1-vv/cc) and p = mv/ sqrt(1-vv/cc).

Physically in a two ray problem, with one ray being an incident ray, one has the freedom to choose the incident angle which means that starting at a point x,y the angle of incidence dictates the point at which the ray hits the medium or mirror. Such a two ray approach may apply to reflection from a stationary or moving mirror (constant speed in the negative y direction) or refraction. Fermat’s principle, however, requires extremizing total time traveled which means taking a derivative. In general (1) time is written in terms of distance and so the derivative pertains to a “free” spatial variable which is usually the x variable because constraints exist in the y direction. For example a stationary mirror or medium surface exists at a specific y point and this cannot be varied. For a moving mirror the motion in the y direction represents a constraint. Thus it seems that in such problems Fermat’s principle relies on a free spatial variable which suggests an invariance in the problem and hence a conservation law in the direction of this free variable as discussed in Part I. This conservation law pertains to a hypothetical velocity which is along the x axis, but represents a hypotenuse, with c/n1 (n1=index of refraction) or relative velocity being a projection of 90-A and 90-B where A and B are angles of incidence and refraction/reflection. Thus the hypothetical velocity is not the actual physical x velocity motion which is not conserved. Fermat’s principle does not mention momentum, but in this note we ask whether it leads to conservation of momentum in the direction of spatial invariance i.e. the x direction? To make the link we suggest that time reversal symmetry is also required, but that this seems to be linked to spatial invariance in one direction. One may use Fermat’s principle to solve reflection from a stationary mirror and find that A=B (angle of incidence=angle of reflection). It is possible to have px,py and b px, b py represent the initial and final momenta and still have trigonometric relationships. Time reversal invariance, however, suggests that running a movie forwards and backwards are the same so b must be 1. From this problem one may see that a momentum component may change if there is a discontinuity with the change in the direction normal to this discontinuity. Thus py changes to -py, but px remains constant. This is also linked to time reversal symmetry we argue as a discontinuity (step function) may be characterized by one value say n1 before the discontinuity and n2 after. Thus running the movie backwards and forwards represents two different problems. In the x direction, however, there may be no discontinuity and so for this component the movie played forwards and backwards is the same. Thus we conclude that momentum conservation exists in the direction of spatial invariance in Fermat’s principle.Spatial invariance must exist otherwise one may not take a derivative to time to extremize total time, but this very spatial invariance seems to also suggest time reversal symmetry in the same direction. In a previous note (2) we argued that given that momentum is of the form 1/wavelength and that wavelength represents resolution, there is a kind of entropy represented by this resolution. Fermat’s principle may show that A and B (incident and reflected/refracted angles) differ. This we suggest means that a time reversed picture represents a different physical picture. We have argued above that A and B differing means conservation of momentum in the x direction, but no conservation of the magnitude of momentum. Thus we argue that a lack of time reversal symmetry with a physical problem suggests different resolutions (i.e. changes in this wavelength based entropy) although the change may be either towards greater or lesser resolution.

In part I of this note, we argued that exp(ipx) written as a 2-vector is solution of a 2x2 rotation matrix with cos(p dx) like terms operating on (a(px),b(px)) to yield d/dx operating on this same vector i.e. the translation operator. We did not give a reason why this should be the case. In this note, we focus on developing bound state quantum mechanics as a statistical theory in parallel with classical statistical theory. As argued in previous notes, the free particle “probability” becomes the central object because there is no acceleration in the statistical picture. It must possess an invariance related to all x points because none is preferential. Ideas of periodicity and invariance, linked to a statistical picture, seem to arise and lead to the complex periodic/spatially invariant form exp(ipx). A number of the ideas in this note have been touched upon in previous notes, but here we try to develop the ideas strictly by comparing with the classical statistical scenario and stressing spatial invariance.

We show that a Parrondo paradox can emerge in two-state quantum walks without resorting to experimentally intricate high-dimensional coins. To achieve such goal we employ a time-dependent coin operator without breaking the translation spatial invariance of the system.

Recently, a backdoor data poisoning attack was proposed, which adds mislabeled examples to the training set, with an embedded backdoor pattern, aiming to have the classifier learn to classify to a target class whenever the backdoor pattern is present in a test sample. We address post-training detection of innocuous perceptible backdoors in DNN image classifiers, wherein the defender does not have access to the poisoned training set. This problem is challenging because without the poisoned training set, we have no hint about the actual backdoor pattern used during training. We identify two properties of perceptible backdoor patterns - spatial invariance and robustness - based upon which we propose a novel detector using the maximum achievable misclassification fraction (MAMF) statistic. We detect whether the trained DNN has been backdoor-attacked and infer the source and target classes. Our detector outperforms other existing detectors experimentally.

Existing techniques to encode spatial invariance within deep convolutional neural networks (CNNs) apply the same warping field to all the feature channels. This does not account for the fact that the individual feature channels can represent different semantic parts, which can undergo different spatial transformations w.r.t. a canonical configuration. To overcome this limitation, we introduce a learnable module, the volumetric transformer network (VTN), that predicts channel-wise warping fields so as to reconfigure intermediate CNN features spatially and channel-wisely. We design our VTN as an encoder-decoder network, with modules dedicated to letting the information flow across the feature channels, to account for the dependencies between the semantic parts. We further propose a loss function defined between the warped features of pairs of instances, which improves the localization ability of VTN. Our experiments show that VTN consistently boosts the features' representation power and consequently the networks' accuracy on fine-grained image recognition and instance-level image retrieval., ECCV 2020

Based on a dual-size shift invariance sparse linear array, this paper presents a novel algorithm for the localization of mixed far-field and near-field sources. First, by constructing a cumulant matrix with only direction-of-arrival (DOA) information, the proposed algorithm decouples the DOA estimation from the range estimation. The cumulant-domain quarter-wavelength invariance yields unambiguous estimates of DOAs, which are then used as coarse references to disambiguate the phase ambiguities in fine estimates induced from the larger spatial invariance. Then, based on the estimated DOAs, another cumulant matrix is derived and decoupled to generate unambiguous and cyclically ambiguous estimates of range parameter. According to the coarse range estimation, the types of sources can be identified and the unambiguous fine range estimates of NF sources are obtained after disambiguation. Compared with some existing algorithms, the proposed algorithm enjoys extended array aperture and higher estimation accuracy. Simulation results are given to validate the performance of the proposed algorithm.

In this paper, tractable stability and performance conditions are presented for systems consisting of an infinite number of spatially invariant, i.e., identical subsystems that are described by (non)linear differential equations and interconnected (partly) through packet-based communication networks. These networks transmit packets asynchronously and independently of each other and are equipped with scheduling protocols that determine which actuator, sensor, or controller node is allowed access to the network. The overall system is modeled as an infinite interconnection of spatially invariant hybrid subsystems. To underline the relevance of this framework, it is shown how two well-known and natural system configurations can be captured in this hybrid modeling framework. Moreover, for the resulting overall infinite-dimensional hybrid system, a proper solution concept is introduced, which is necessary as many standard concepts do not apply as Zeno behavior is inevitable for the systems under study. Based on the proposed hybrid modeling framework, conditions leading to a maximally allowable transmission interval (MATI) for all of the individual communication networks are derived such that uniform global asymptotic stability (UGAS) or $\mathcal {L}_{p}$ -stability of the overall system is guaranteed. Interestingly, by exploiting the interconnection structure, the conditions guaranteeing UGAS or $\mathcal {L}_{p}$ -stability can be stated locally in the sense that they only involve the (local) dynamics of one subsystem in the interconnection and local conditions on the scheduling protocol. Finally, it is shown that in the linear case the derived conditions can even be stated in terms of “local” LMIs, making them amenable for computational verification.

Convolutional neural networks (CNNs or ConvNets) are a popular group of neural networks that belong to a wider family of methods known as deep learning. The secret for their success lies in their carefully designed architecture capable of considering the local and global characteristics of the input data. Initially, CNNs have been designed to process image data efficiently, and for this, they were developed with properties such as local connectivity, spatial invariance, and hierarchical features. With these properties, CNNs have propelled breakthroughs across several research areas and have recently been applied in psychiatry and neurology to investigate brain disorders. In this chapter, we will present a theoretical introduction to CNNs. We will then illustrate their use in brain disorders by reviewing exemplary applications to neuroimaging and electroencephalogram data.

Convolutional neural networks in deep learning models have dominated the recent image recognition works. But the lack of capacity to maintain spatial invariance makes identification of micronucleus cells as a classic task in digital pathology still a challenge task. In this paper, a novel convolutional neural network for feature maps spatial transformation (FSTCNN) is proposed, which incorporates a Spatial Transformer Network. Our model allows the spatial manipulation of data within the network, provides the ability of active spatial transformation for neural network without any extra supervision. We compared the results of inserting STN into different convolutional layers and found that such a network can transform the input image more steadily, correct the image to one certain position, make it fill the whole screen to create a better environment for image recognition. The results show a distinct advantage over other convolutional neural networks for medical image recognition.

The concentrations of polycyclic aromatic hydrocarbons (PAHs) have been determined in the gaseous phase and in various particulate matter (PM) size fractions at different locations in and outside of Stockholm, Sweden, representative of street level, urban and rural background. The focus has been on the seldom determined but highly carcinogenic dibenzopyrene isomers (DBPs) dibenzo[a,l]pyrene, dibenzo[a,e]pyrene, dibenzo[a,i]pyrene and dibenzo[a,h]pyrene. PAHs with 3 rings were found to be mainly associated with the vapor phase (>90%) whereas PAHs with 5–6 rings were mostly associated with particulate matter (>92%) and the 4-ringed PAHs partitioned between the two phases. PAH abundance was determined to be in the order street level > urban background > rural background with the PM10 street level 2010 mean of benzo[a]pyrene (B[a]P) reaching 0.24 ng/m3, well below the EU annual limit value of 1 ng/m3. In addition, higher PAH concentrations were found in the sub-micron particle fraction (PM1) as compared to the super-micron fraction (PM1-10) with the abundance in PM1 varying between 57 and 86% of the total PAHs. The B[a]P equivalent concentrations derived for DB[a,l]P and total DBPs exceeded 1–2 and 2–4 times, respectively, that of B[a]P at the four sampling sites; therefore underestimation of the cancer risk posed by PAHs in air could be made if the DBPs were not considered in risk assessment using the toxic equivalency approach, whilst the high correlation (p < 0.001) found in the relative concentrations supports the use of B[a]P as a marker substance for assessment of the carcinogenic risk associated to PAHs. However, the big difference in concentration ratios of B[a]P and the DBPs between the present study and some literature data calls for further research to evaluate the temporal and spatial invariance of the B[a]P/DBP ratios.

We obtain an invariance principle for two-dimensional Brownian sheet where the underlying random field need not be independent or stationary. The invariance principle is applied to obtain and analyze a frequency-domain test for spatial unit root against local-to-unity alternatives.

Convolutional neural networks(CNNs) has become one of the primary algorithms for various computer vision tasks. Handwritten character recognition is a typical example of such task that has also attracted attention. CNN architectures such as LeNet and AlexNet have become very prominent over the last two decades however the spatial invariance of the different kernels has been a prominent issue till now. With the introduction of capsule networks, kernels can work together in consensus with one another with the help of dynamic routing, that combines individual opinions of multiple groups of kernels called capsules to employ equivariance among kernels. In the current work, we have implemented capsule network on handwritten Indic digits and character datasets to show its superiority over networks like LeNet. Furthermore, it has also been shown that they can boost the performance of other networks like LeNet and AlexNet., Comment: Accepted in IEEE Applied Signal Processing Conference 2018(ASPCON 2018 ) held on December 7-9, 2018 at Jadavpur University, Kolkata, India

The ability to detect and track objects in the visual world is a crucial skill for any intelligent agent, as it is a necessary precursor to any object-level reasoning process. Moreover, it is important that agents learn to track objects without supervision (i.e. without access to annotated training videos) since this will allow agents to begin operating in new environments with minimal human assistance. The task of learning to discover and track objects in videos, which we call \textit{unsupervised object tracking}, has grown in prominence in recent years; however, most architectures that address it still struggle to deal with large scenes containing many objects. In the current work, we propose an architecture that scales well to the large-scene, many-object setting by employing spatially invariant computations (convolutions and spatial attention) and representations (a spatially local object specification scheme). In a series of experiments, we demonstrate a number of attractive features of our architecture; most notably, that it outperforms competing methods at tracking objects in cluttered scenes with many objects, and that it can generalize well to videos that are larger and/or contain more objects than videos encountered during training., Comment: Accepted at AAAI 2020. Code: https://github.com/e2crawfo/silot. Visualizations: https://sites.google.com/view/silot

Convolutional Neural Networks (CNNs) have brought us the exceptionally significant improvement in the performance of the variety of visual tasks, such as object classification, semantic segmentation or linear regression. However, these powerful neural models suffer from the lack of spatial invariance. In this paper, we introduce the end-to-end system that is able to learn such invariance including in-plane and out-of-plane rotations. We performed extensive experiments on variations of widely known MNIST dataset, which consist of images subjected to deformations. Our comparative results show that we can successfully improve the classification score by implementing so-called Spatial Transformer module.

Designed three visual psychological experiments to explore human ability on transformed texture discrimination and classical texture classification, and make some comparisons between human beings and computer algorithms. Some interesting results are obtained from these visual psychological experiments: (1) the invariance performance of sevral computer algorithms is appropriate to human beings; (2) texture is almost rotation invariant for human beings although single “b” and single “p” have significant different comprehension; (3) texture appearance depends on visual resolution. The biggest invariant scale of spatial scaling is roughly 2.3 times for human visual system at some fixed observation conditions; (4) some computer algorithms (such as MBF and Gabor) have achieved better performance than human beings under some restrictive conditions for fair comparison.

The low-energy dynamics of a generic self-gravitating medium can be studied by using effective field theory (EFT) in terms of four derivatively coupled scalar fields. Imposing SO(3) internal spatial invariance, the theory describes fluids, superfluids, solids and supersolids. Dynamical and thermodynamical properties of the medium are dictated by internal symmetries of the effective theory. From the analysis of cosmological perturbations it emerges that in the scalar sector, be- sides the gravitational potential, there is a non-adiabatic mode corresponding to the perturbations of the entropy per particle σ . Perfect fluids and solids are adiabatic with σ constant in time, while for superfluids and supersolids σ has non-trivial dynamics. Tensor perturbations are massive for solids and supersolids. Such an effective approach can be used to give a very general modelling of the dark sector based on symmetries.

In the previous chapter it has been demonstrated that a periodic linear corrugation patterned on the surface of a 1DPC can extract the BSW-CE. Due to the spatial invariance of the surface structure, such configuration does not require a specific localization of the emitter.

International audience; In this paper, we introduce a sparse direction-of-arrival (DOA) estimation algorithm for sensor arrays presenting multiple scales of spatial invariance. We exploit the Khatri-Rao structure of the over-complete steering vector dictionary, corresponding to this array geometry, in order to devise a computationally efficient sparse estimation approach. This approach is based on an iterative refinement and pruning strategy of the dictionary. We show, in numerical simulations, that our approach outperforms the state-of-the-art approach based on a Candecomp/Parafac (CP) decomposition, proposed by Miron et al. in 2015.

The locally deepest regions of major sulci, the sulcal pits, are thought to be the first cortical folds to develop and are closely related to functional areas. We examined the spatial distribution of sulcal pits across the entire cortical region, and assessed the hemispheric asymmetry in their frequency and distribution in a large group of normal adult brains. We automatically extracted sulcal pits from magnetic resonance imaging data using surface-based methods and constructed a group map from 148 subjects. The spatial distribution of the sulcal pits was relatively invariant between individuals, showing high frequency and density in specific focal areas. The left and right sulcal pits were spatially covariant in the regions of the earliest developed sulci. The sulcal pits with great spatial invariance appear to be useful as stable anatomical landmarks. We showed the most significant asymmetry in the frequency and spatial variance of sulcal pits in the superior temporal sulcus, which might be related to the lateralization of language function to the left hemisphere, developing more consistently and strongly than for the right. Our analyses support previous empirical and theoretical studies, and provide additional insights concerning the anatomical and functional development of the brain.

Within this paper fractal transformations are presented as a powerful new shape recognition technique. The motivation behind using fractal transformations is to develop a high speed shape recognition technique which will be scale invariant. A review is given of the most popular existing shape recognition techniques. There then follows a full mathematical analysis of the new technique together with a proof of the authors Fractal Invariance Theorem, the new theorem at the centre of the recognition technique. Through the mathematical analysis it becomes apparent that the fractal recognition technique possesses the remarkable property that it is able to distinguish between similar objects. Details are then given of the practical implementation of the technique together with an algorithm for making the technique rotationally invariant. The technique is then applied to a selection of real world objects and a comparison made with the popular moment invariants technique. This shows that the fractal technique is faster than the technique of moment invariants, and also requires less initial information to be effective. Finally conclusions are drawn and further work detailed.

Compressive sensing (CS) for urban operations and through-the-wall radar imaging has been shown to be successful in fast data acquisition and moving target localizations. However, the research work in this area thus far has assumed prior effective wall removal, allowing proper detection of indoor targets. In this paper, we show that wall removal techniques, operating with full data volume and applying backprojection imaging methods, can be equally effective under reduced data volume and within the sparse signal reconstruction framework. Specifically, we demonstrate that the spatial filtering and the singular value decomposition based approaches, which, respectively, exploit the spatial invariance and the strength of the EM wall return, for suppression of the wall reflections, can be employed using few measurements, thus allowing CS to be applied to data with higher target-to-wallclutter ratio.

Two aspects concerning the production of shapes in handwriting are distinguished in the present paper. The first aspect is indicated by thespatial variability of letters measured across replications. Spatial variability is assumed to reflect the basic spatial noise observed in everyday writing. The second aspect deals with the geometric characteristics of letter shapes, measured by the ratio of the vertical over the horizontal letter size (Y/X-ratio). The main question is whether the geometric characteristics and basic spatial noise originate from a common source. More specifically, we are interested in whether Y/X-ratio as well as spatial variability will alter across changed circumstances, or whether Y/X-ratio will alter without a change in spatial variability. Subjects wrote the simple letter sequencelelele in conditions with and without vision under three scaling conditions requirements (small, normal and large letter sizes). The main results were that geometric aspects of letters altered (Y/X-ratio) under no vision and under the scaling requirement to write in a small format. In contrast, shapes were produced with unchanged spatial variability in all conditions of vision and scaling requirements. The results suggest that alterations of geometric aspects of letters across changed circumstances do not necessarily involve an increase in spatial variability.

Ocean impulse response functions computed with the time‐domain parabolic equation model are shown to be consistent with response functions measured in a bottom‐limited ocean located off the east coast of the United States. The geoacoustics used in the model are developed from Deep Sea Drilling Project sites near the experimental area. Measured impulse responses are obtained by correlating the measured source signature of a 25‐ to 150‐Hz linear, frequency‐modulated signal with the signals received on a 15‐element vertical line array. Correlation coefficients computed as a function of depth are used as a measure of comparison between measured and modeled responses. The capability of the time‐domain parabolic equation to model wide‐angle (68°–75°) multipaths and sub‐bottom layers with high impedance contrasts is shown. Correlation loss between response functions measured from different transmissions at the same range is shown to be attributable to the degree to which spatial invariance of the ocean can be assumed.

This paper develops linear quadratic robust control theory for a class of spatially invariant distributed control systems that appear in areas of economics such as New Economic Geography, management of ecological systems, optimal harvesting of spatially mobile species, and the like. Since this class of problems has an infinite dimensional state and control space it would appear analytically intractable. We show that by Fourier transforming the problem, the solution decomposes into a countable number of finite state space robust control problems each of which can be solved by standard methods. We use this convenient property to characterize 'hot spots' which are points in the transformed space that correspond to 'breakdown' points in conventional finite dimensional robust control, or where instabilities appear or where the value function loses concavity. We apply our methods to a spatial extension of a well known optimal fishing model.

SUMMARY The properties of real roads along the lateral axis cannot not be adequately explained with the classic model of the isotropic road. With a realistic assumption of spatial invariance (homogeneity), a more general surface model has been formulated and compared with the experimental data. A good match of measurements and “theory” has been achieved. A simple parametric approximation is developed for the coherence function on the basis of the extended unevenness model. In addition to the undulation exponent of the road surface and the track width, the curve is characterized by three other key figures. The adaptation to the experimental data results in comparatively small yet justifiable parameter intervals. Due to the further development of filter methods for the synthesis of random irregularities, a model family is now available which facilitates a satisfactory balance between model accuracy and numerical complexity for most applications. The main difficulties would appear to lie in the formulation o...

In this paper, we consider a class of spatially distributed systems which have a special property known as spatial invariance. It is well-known that for such problems, the problem of designing decentralized controllers is hard. In this paper, we generalize some previously known results and show that for a certain class of problems, the control problem has a convex reformulation. We employ the notion of partially-ordered sets and the associated notion of incidence algebras to introduce a class of systems called poset causal systems. We show that poset causal systems are a fairly large class of systems that properly include some other classes of systems studied in the literature (namely cone-causal and funnel causal systems). Finally we show that the set of poset-causal controllers for poset-causal plants are amenable to a convex parameterization.

The bioelectric impedance technique is a non-invasive method that provides the analysis of blood volume changes in the arteries. This is made possible by an interpretation of the impedance signal variations. In this paper, time and spatial variations of such impedance signals are studied on recordings made on limbs of 15 healthy subjects at rest. For that purpose, the scalogram of each signal has been computed and quantitative measures based on energies were determined. The results show that the signals are statistically time invariant on three anatomical segments of the limbs: pelvis, thigh and calf. p Value varies between 0.20 and 0.52 for the absolute energies computed on scalograms of signals recorded at 5 min intervals. Moreover, the analysis made on the two legs of each subject shows that the signals are spatial invariant on the three anatomical segments. p Value varies between 0.0785 and 1.000 for the absolute energies computed on the scalograms of signals recorded simultaneously on the two legs. These conclusions will therefore help the clinicians in studying the temporal variations of physiological parameters on limbs with the impedance technique. Moreover, the results on the spatial invariance make possible the comparisons of these parameters with those given by other acquisition techniques.