Mathematical study of h-index sequences

This paper studies mathematical properties of h-index sequences as developed by Liang [Liang, L. (2006). h-Index sequence and h-index matrix: Constructions and applications. Scientometrics,69(1), 153–159]. For practical reasons, Liming studies such sequences where the time goes backwards while it is more logical to use the time going forward (real career periods). Both type of h-index sequences are studied here and their interrelations are revealed. We show cases where these sequences are convex, linear and concave. We also show that, when one of the sequences is convex then the other one is concave, showing that the reverse-time sequence, in general, cannot be used to derive similar properties of the (difficult to obtain) forward time sequence. We show that both sequences are the same if and only if the author produces the same number of papers per year. If the author produces an increasing number of papers per year, then Liang’s h-sequences are above the “normal” ones. All these results are also valid for g- and R-sequences. The results are confirmed by the h-, g- and R-sequences (forward and reverse time) of the author.     

On some n-normed linear space valued difference sequences

The main aim of this article is to extend the notion of strongly Cesàro summable and strongly lacunary summable real sequences to n-normed linear space valued (n-nls valued) difference sequences. Consequently we introduce the spaces |σ₁|(X,?) and N_θ(X,?), respectively, where X is an n-normed space and ? is a difference operator. We investigate these spaces for completeness as well as for the relationship between these spaces.     

DACs: Bringing direct access to variable-length codes

We present a new variable-length encoding scheme for sequences of integers, Directly Addressable Codes (DACs), which enables direct access to any element of the encoded sequence without the need of any sampling method. Our proposal is a kind of implicit data structure that introduces synchronism in the encoded sequence without using asymptotically any extra space. We show some experiments demonstrating that the technique is not only simple, but also competitive in time and space with existing solutions in several applications, such as the representation of LCP arrays or high-order entropy-compressed sequences.     

Periodic solutions for a kind of Rayleigh equation with two deviating arguments

In this paper, we use the coincidence degree theory to establish new results on the existence of T-periodic solutions for the Rayleigh equation with two deviating arguments of the form

x^″+f(x^′(t))+g₁(t,x(t-τ₁(t)))+g₂(t,x(t-τ₂(t)))=p(t).     

Impact of CCL2 and Its Receptor CCR2 Gene Polymorphism in North Indian Population: A Comparative Study in Different Ethnic Groups Worldwide

Chemokine are small, inducible pro-inflammatory cytokines involved in many biological processes, such as migration of leukocytes, atherosclerosis, angiogenesis, tumor growth, and metastasis. Chemokine are also known to influence tumor cell’s activity. Specifically, tumor cells express chemokine receptors in a non random manner suggesting a role of chemokine in metastatic destination of tumor cells. The present study was conducted to determine distribution of (Chemokine receptor 2) CCR2 V64I, Chemokine ligand 2 CCL2 I/D, and CCL2 2518 A>G gene polymorphisms in North Indian population and compare with different populations globally. Polymerase chain reaction (PCR)-based analysis was conducted in 200 normal healthy individuals of similar ethnicity. Allelic frequencies in wild type (GG) of CCR2 V64I G>A were 63 % G; CCL2 I/D 42 % II; CCL2 2518 A>G 40.5 % A. The minor variant allele frequency in our population was as follows: 19.5 % for CCR2 V64I, 35.5 % for CCL2 I/D, 35.3 % for CCL2 2518 A>G. We further compared frequency distribution for these genes with various published studies in different ethnicity. Our results suggested that frequency in chemokine genes exhibit distinctive pattern in India that could be attributed to ethnicity variation. This could assist in high-risk screening of human exposed to environmental carcinogens and cancer predisposition in different ethnic groups. Thus, they signify an impact of ethnicity and provide a basis for future epidemiological and clinical studies.     

Kernel recursive generalized mixed norm algorithm

This work studies the problem of kernel adaptive filtering (KAF) for nonlinear signal processing under non-Gaussian noise environments. A new KAF algorithm, called kernel recursive generalized mixed norm (KRGMN), is derived by minimizing the generalized mixed norm (GMN) cost instead of the well-known mean square error (MSE). A single error norm such as l_p error norm can be used as a cost function in KAF to deal with non-Gaussian noises but it may exhibit slow convergence speed and poor misadjustments in some situations. To improve the convergence performance, the GMN cost is formed as a convex mixture of l_p and l_q norms to increase the convergence rate and substantially reduce the steady-state errors. The proposed KRGMN algorithm can solve efficiently the problems such as nonlinear channel equalization and system identification in non-Gaussian noises. Simulation results confirm the desirable performance of the new algorithm.     

Alternative convergence criteria for iterative methods of solving nonlinear equations

Let χ_m+1=T(χ_m) or even χ_m+1=T(χ_m,χ_m?1, …, χ_m?q), m=1,2,3 … be an iteration method for solving the nonlinear problem F(χ)=0, where F(χ) and its derivatives possess all of the properties required by T(χ_m). Then if it can be established that for the problem at hand ∥F(χ_m+1)∥?β_m∥F(χ_m)∥, ? m > M₀ (M₀<∞) and 0?β_m<1 , definitions are established and theorems proven concerning convergence, uniqueness and bounds on the error after ‘m’ successive iterations of a new approach to convergence properties T(χ_m). These charateristics are referred to as “alternate” (local, global) convergence properties and none of the proofs given are restricted to any specific type of method such as, e.g. contraction mapping types. Application of results obtained are illustrated using Newton's method as well as the general concept of Newton-like methods.     

Information extraction with automatic knowledge expansion

POSIE (POSTECH Information Extraction System) is an information extraction system which uses multiple learning strategies, i.e., SmL, user-oriented learning, and separate-context learning, in a question answering framework. POSIE replaces laborious annotation with automatic instance extraction by the SmL from structured Web documents, and places the user at the end of the user-oriented learning cycle. Information extraction as question answering simplifies the extraction procedures for a set of slots. We introduce the techniques verified on the question answering framework, such as domain knowledge and instance rules, into an information extraction problem. To incrementally improve extraction performance, a sequence of the user-oriented learning and the separate-context learning produces context rules and generalizes them in both the learning and extraction phases. Experiments on the “continuing education” domain initially show that the F1-measure becomes 0.477 and recall 0.748 with no user training. However, as the size of the training documents grows, the F1-measure reaches beyond 0.75 with recall 0.772. We also obtain F-measure of about 0.9 for five out of seven slots on “job offering” domain.     

Polyphosphonium-based ion bipolar junction transistors

Advancements in the field of electronics during the past few decades have inspired the use of transistors in a diversity of research fields, including biology and medicine. However, signals in living organisms are not only carried by electrons but also through fluxes of ions and biomolecules. Thus, in order to implement the transistor functionality to control biological signals, devices that can modulate currents of ions and biomolecules, i.e., ionic transistors and diodes, are needed. One successful approach for modulation of ionic currents is to use oppositely charged ion-selective membranes to form so called ion bipolar junction transistors (IBJTs). Unfortunately, overall IBJT device performance has been hindered due to the typical low mobility of ions, large geometries of the ion bipolar junction materials, and the possibility of electric field enhanced (EFE) water dissociation in the junction. Here, we introduce a novel polyphosphonium-based anion-selective material into npn-type IBJTs. The new material does not show EFE water dissociation and therefore allows for a reduction of junction length down to 2 μm, which significantly improves the switching performance of the ion transistor to 2 s. The presented improvement in speed as well the simplified design will be useful for future development of advanced iontronic circuits employing IBJTs, for example, addressable drug-delivery devices.There has been a recent interest in developing diodes^1–4 and transistors^4–8 that conduct and modulate ion currents. Such non-linear iontronic components are, for example, interesting as they allow further control of ions in, for instance, electrophoretic drug delivery devices. A range of microfabricated diodes,^9–11 transistors,^12,13 and circuits^9,14 has been constructed using ion-selective membranes. These membranes contain fixed charges of either polarity, compensated by mobile ions of opposite charge (counter-ions). When immersed in an electrolyte, counter-ions can move through the membrane, while ions with the same charge as the fixed charges (co-ions) are repelled. This renders the membrane selective for the counter-ion and can therefore be considered as p- or n-type ion conductors. By combining two membranes of opposite polarity, a bipolar membrane (BM) configuration is obtained¹⁵ (Figure 1(a)). The BM junction can be biased by an ion current in the reverse and forward directions, respectively.^16,17 This modulates the ion concentration inside the BM, and thus the ionic conductivity, which then results in an current rectification.^2,18 In the three-terminal ion bipolar junction transistor¹² (IBJT), an ion-selective base (B) is connected to oppositely selective emitter (E) and collector (C), forming two BM configurations (EB and BC) (Figure 1(b)). pnp- and npn-IBJTs have been constructed¹⁴ from photolithography patterned poly(styrene sulfonate) (PSS, p-selective) and quaternized poly(vinylbenzyl chloride) (n-selective) as emitter, collector, and base. In these devices, a neutral poly(ethylene glycol) (PEG) electrolyte is typically inserted into the junction to separate the base from the emitter and collector,¹² in order to avoid¹⁹ electric field enhanced (EFE) water dissociation¹⁶ (Figure 1(a)). EFE water dissociation is typically observed in BMs²⁰ and produces water ions inside the BM under reverse bias, which prevents proper IBJT operation. In PEG-IBJTs, the current between the emitter and collector (I_C) is thus modulated by controlling the ion concentration inside the PEG-junction.²¹ Ions are injected or extracted into the junction depending on the bias of the base (V_EB). In a npn-IBJT, a positive bias is typically applied between emitter and collector (V_EC), thus allowing anions to migrate from the emitter to the collector. In the cut-off mode (Figure 1(c)), a negative bias V_EB is applied, resulting in reverse bias of both EB and BC. Cations in the junction will migrate into the base, while anions will primarily migrate into the collector, due to the higher collector bias. This base current (I_B) will extract ions from the junction, which decreases the ionic conductivity in the junction resulting in a low I_C. Eventually, the resistive characteristics for ion charge transport, between the emitter and collector, will be entirely dominated by the junction. This gives that most of the applied V_EC is consumed across the junction with only a minimal voltage potential drop across the emitter and base terminals.Open in a separate windowFIG. 1.(a) The modes of operation for a BM; forward bias (high conduction and ion accumulation), reverse bias (low conduction and ion depletion), and EFE water dissociation (high conduction, formation of ions). (b) Illustrations of an npn-IBJT, with anion-selective emitter (E) and collector (C) forming a junction with a cation-selective base (B). (c) In cut-off mode, the base and collector extract ions from the junction, prohibiting co-ion migration through the base. (d) In active mode, the forward biased EB injects ions into the base, thus allowing anions from the emitter to migrate as co-ions through the base into the collector.In the active-mode of the npn-IBJT (Figure 1(d)), the V_EB bias at the base is reversed (i.e., now positive). This causes injection of cations, from the base, and anions, from the emitter, into the junction. As the ion concentration increases, anions from the emitter can start to drift across the junction to the collector, thus a high I_C is obtained. The high concentration of ions inside the junction is reflected in a low resistive value for ion transport. This now causes the voltage to drop over the emitter and collector terminals, thus lowering the EB forward bias and the injection of ions from the base. At the collector-junction interface, the extraction of anions produces an ion depletion zone and a corresponding voltage drop. Thus, in the active-mode, the applied V_EC is primarily consumed across the emitter and collector terminals and also at the collector-junction interface.The switching speed of an IBJT should be strongly correlated to the distance separating the emitter and collector,¹⁴ as this length determines the volume that needs to be filled or emptied with ions causing modulation of ions in the junction. To achieve a fast-switching IBJT, the junction volume, i.e., the collector-emitter separation, should be as small as possible. However, EFE water dissociation must be avoided since this process ruin the IBJT operation. EFE water dissociation is, in part, driven by the appearance of a large potential drop across a small distance, as occurring at the interface of a BM under reverse bias, producing a high electric field that accelerates the forward reaction rate of water auto-dissociation.¹⁶ Miniaturization of the collector-emitter distance is therefore problematic, as the separation inside the EB and BC BMs evidently also mush shrink, resulting in higher reverse bias electric fields across the BMs and thus promoting EFE water dissociation. The problem of EFE water dissociation in an IBJT primarily manifests itself in the cut-off mode, as water ions are generated in the reversed biased EB and BC BMs. These ions produce an elevated cut-off I_C, and hence deteriorate the IBJTs on–off performance. Here, we report an IBJT, in which the EFE water dissociation is avoided by the use of a novel polyphosphonium-based anion-selective material, which previously has been shown to prevent EFE water dissociation in BM diodes.¹¹ This allows the collector and emitter to directly contact the base without an intermediate PEG-layer. Without the need for a PEG-separator inside the BMs, the collector-emitter distance is reduced to only 2 μm.Polyphosphonium-based npn-IBJTs were produced following the same manufacturing protocol as was reported for polyphosphonium-based ion diodes.¹¹ Conjugated polymer electrodes and cation-selective base was patterned from ∼200 nm thick poly(3,4-ethylenedioxythiophene):polystyrene sulfonate film on polyethylene terephthalate-sheets using photolithography and dry-etching. The base was rendered electronically insulating by chemical overoxidation via exposure to sodium hypochlorite through a mask. A 2 μm thick SU8-layer was patterned on-top of this configuration, with an opening defining the actual junction. 1 μm thick polyphosphonium-based anion-selective emitter and collector were deposited and patterned using photolithography and dry-etching, to overlap with the base at the opening of the SU8. Finally, a second 10 μm thick layer of SU8 was used to seal the junction. The membranes were hydrated by incubation in dH2O for 24 h before any measurements were carried out. Aqueous 0.1M NaCl electrolytes were used during the measurement. All electrical measurements were performed using a Keithley 2602 source meter.The switching characteristics of the npn-IBJT were obtained by applying V_EC of 10 V and alternating V_EB at ±3 V for various duration of time, see Figure ​Figure2.2. A periodic 5 s switching with 8 Hz measurement rate was used to record the dynamics of the turn-on/off characteristics of the device. When V_EB switches from −3 to +3 V, there is a quick increase in the I_B, as ions from the base and emitter migrate into the emitter/base junction. After a delay of ∼0.25 s, I_C starts to increase due to the increased ion concentration in the emitter/base junction and the subsequent diffusion of anions into the base. As the I_C increases, the I_B decreases as the voltage drop between the emitter and base decreases, and after ∼2 s I_C reaches 90% of the steady state on-current level. For longer on-switching times, the I_B and I_C stay stable over 30 s, after which a small increase is observed. This current-drift in both I_B and I_C is likely due to the contribution of co-ion migration. As cations from the base migrate into the emitter as co-ions, the conductivity in the emitter increases, leading to an increased I_C value. This increases the ion concentration at the base, which gives less selective ion injection and thus more cation injection from the base, i.e., a higher I_B.Open in a separate windowFIG. 2.Emitter-collector current response as the IBJT is switched between cut-off (V_EB = −3 V) and active mode (V_EB = 3 V) for V_EC = 10 V, at 5 s and 120 s periods.As V_EB is switched back to −3 V, there is a sharp negative peak in I_E as ions are extracted from the junction, which occur mainly through the base (cations) and collector (anions) terminals. As the ion concentration in the base drops, I_C decreases. The transistor turns off to 10% of the value of the steady state on-current within ∼2 s, regardless of the duration of the on-state. The constant turn-off time indicates that ions are not accumulating to a significant extent inside the junction during the on-steady state but are instead constantly transported out of the junction. When all co-ions have been extracted from the junction, the Donnan exclusion prevents subsequent injection of anions into the base, and I_C is therefore low. The on/off ratio of I_C reaches above 100.A transfer curve was obtained by scanning V_EB between −3 and +3 V while keeping V_EC at 10 V (Figure 3(a)). As expected, both I_C and I_B remain low for negative V_EB. In this range, both EB and BC are biased in reverse direction. As V_EB turns positive, the EB configuration is switched into forward bias and ions are injected into the junction. This leads to a linear increase in I_C vs. V_EB. For the reverse scan, a minor hysteresis is observed for both the I_C and I_B scans, again probably due to the contribution of co-ion migration due to long time operation of the device.Open in a separate windowFIG. 3.Transfer and output curves. (a) The transfer curve is low for negative V_EB and increases linearly for positive V_EB with approximately zero threshold. (b) The output curves show I_C saturating with respect of V_EC for positive V_EB.The transistor output characteristics were obtained by scanning V_EC at different V_EB values (Figure 3(b)). The saturation regime, i.e., the bias mode was both EB and BC are in forward bias, was avoided as this has negative impact on the stability of the device. As reported for previous IBJT devices, the output characteristics show a clear saturation behaviour of I_C across the entire range of V_EC. Further, the I_C increases linearly with V_EB. The increase of both I_C and I_B when operating for extended periods of time in the active mode is again attributed to the addition and inclusion of co-ions in the junction. The current gain (I_C/I_B) at V_EC = 10 V decreases with V_EB and reaches 43.9, 17.9, and 10.7 for V_EB = 1 V, 2 V, and 3 V, respectively. For higher base bias voltages, the ion concentration increases in the junction and thus the injection selectivity decreases.In comparison with previously reported IBJTs,^12,14,21 the lack of a neutral electrolyte layer in the junction has an overall positive effect on the device characteristics. Main performance improvements are found in a decrease in the turn-on time from 9 s (for npn-IBJT²¹) to 2 s, for devices with comparable junction widths and heights. The main contribution to the improved switching speed is likely the decreased length between the emitter and collector. Interestingly, simulations have shown that an extended space charge region (ESCR), for a PEG-IBJT in cut-off mode, can extend several micrometers away from the collector.²² Thus, a PEG-IBJT with an emitter-collector separation of single micrometers should show an increased cut-off current due to the ESCR overlapping in the junction. However, by omitting the PEG in the junction, the ESCR is reduced due to screening from the fixed charges in the BM layers. This enables the IBJT, reported here, to operate with retained low cut-off currents. On-off ratios and ion current gains are approximately equal to previous IBJTs,^12,14,21 at above 100 and 10, respectively. The on–off ratio and ion current gain are more dependent on the selectivity of the membranes and the charge of the junction.Further, the need to separate the layers in a PEG-IBJT puts high demands on the patterning resolution and alignment accuracy to reduce the separation between emitter/collector and base. As polyphosphonium allows the IBJT to be built without separation of layers, miniaturization of the junction is relatively easier to obtain. The switching speed can potentially be further improved by retaining the base material between the emitter and collector (see Figure 1(b)), thus allowing for a more direct pathway for I_C. This design would, however, require a much more accurate layer alignment or that the base patterned on top of the emitter and collector layers. In general, such modifications of device geometry are simpler to accomplish with the non-EFE water dissociating polyphosphonium as fewer active layers are used, suggesting a further use of polyphosphonium to improve switching speed and miniaturization of IBJTs. Such further advancement in IBJT performance would be welcomed, for example, in the continued work towards complex ionic circuits¹⁴ to regulate signalling in bioelectronics and in drug delivery applications, in which generation of dynamic and complex gradients, at high spatial resolution, is of generic interest.     

Montgomery multiplication over rings

Montgomery multiplication of two elements a and b of a finite field F_q is defined as abr^-1 where r is a fixed field element in . In this paper we define Montgomery multiplication of elements a(x) and b(x) in a polynomial ring modulo the ideal generated by a reducible polynomial f(x). We then show that Montgomery multiplication over a field represented by a polynomial ring modulo an irreducible pentanomial can be performed more efficiently in terms of time delay by embedding the field in a quotient of a polynomial ring modulo a reducible trinomial. The trinomial has a degree that is slightly higher than that of the pentanomial, thereby increasing the number of gates in the multiplier by a small amount.     

Binary matrices with specified properties

The binary matrices A which are circulant with one or more of the following properties: (1) A is symmetric, i.e. A = A^T, (2) A is orthogonal, i.e. AA^T = I (mod 2), (3) A has low multiplicative order, i.e. A^m = I, occur often in communication, control and network theory problems. In this paper we construct a number of such matrices. The results are based on the theory of power-residues modulo an odd prime p, and the fact that the algebra of all p × p circulant matrices is isomorphic to the algebra of polynomials modulo (x^p?1).     

Convergent genomic signatures of high-altitude adaptation among domestic mammals

Abundant and diverse domestic mammals living on the Tibetan Plateau provide useful materials for investigating adaptive evolution and genetic convergence. Here, we used 327 genomes from horses, sheep, goats, cattle, pigs and dogs living at both high and low altitudes, including 73 genomes generated for this study, to disentangle the genetic mechanisms underlying local adaptation of domestic mammals. Although molecular convergence is comparatively rare at the DNA sequence level, we found convergent signature of positive selection at the gene level, particularly the EPAS1 gene in these Tibetan domestic mammals. We also reported a potential function in response to hypoxia for the gene C10orf67, which underwent positive selection in three of the domestic mammals. Our data provide an insight into adaptive evolution of high-altitude domestic mammals, and should facilitate the search for additional novel genes involved in the hypoxia response pathway.     

Some properties of submatrices in a solution to the matrix equation AXB=C with applications

Suppose that AXB=C is a consistent matrix equation and partition its solution X into a 2×2 block form. In this article we give some formulas for the maximal and minimal ranks of the submatrices in a solution X to AXB=C. From these formulas, we derive necessary and sufficient conditions for the submatrices to be zero and nonsingular, respectively. As applications, we give a group of formulas for the maximal and minimal ranks of submatrices in generalized inverses of matrices and their properties.     

GIMC architecture for linear systems with a single I/O delay

A new feedback controller architecture is presented for linear systems with a single I/O delay in the generalized internal model control (GIMC) framework. According to the doubly coprime factorization of these systems, traditional GIMC strategy is extended to linear systems with a single I/O delay. The distinguished feature of the control system architecture is that high tracking performance and good external disturbance rejection could be done separately by a nominal Smith predictor part and a finite dimensional conditional controller. First, a nominal Smith predictor part could be designed to deal with command tracking performance. Second, according to the left coprime factorization of the nominal controller, a finite dimensional conditional controller could be considered for external disturbance rejection, when the controlled plant should be assumed to be a square one. Finally, a simple design example is illustrated the effectiveness of the presented method.Finally, a simple design example is illustrated the effectiveness of the presented method.     

Strongly and uniformly convergent Green's function expansions

The convergence of Green's function expansions used in the exact analytical treatment of problems involving boundaries of different shapes is a property crucial in obtaining their solution. Existing expansions in most cases suffer from two serious setbacks: they do not converge uniformly in their region of validity, exhibiting a slow and conditional convergence near the source (singular) point and, even worse, they change expression when the field point moves past the source point. For such reasons they are unsuited for the solution of singular integral equations, in which values of the Green's function G at the source point do appear inside the integral. These inadequacies are met head-on by extracting the singular behavior in a closed-form term. Additional simple terms are also extracted to improve the convergence of the expansion of the remaining, non-singular part of G. The so-obtained new eigenfunction expansions for G converge uniformly over the whole region of their validity and very strongly (exponentially) near the source point. They are particularly suited for the solution of singular integral equations by the Carleman-Vekua method, otherwise known as the method of regularization by solving the dominant equation. These new expansions can be further subjected to a Watson transformation yielding a third expansion exhibiting strong convergence in regions where the convergence of the preceding series weakens, and vise versa. All these considerations are illustrated in this paper by means of a two-dimensional harmonic Green's function of a line source inside a rectangular shield, which is useful in a variety of shieldedline configurations. Extensions to different dielectric sublayers, to wave (Helmholtz) Green's functions, etc., are also discussed.     

Building user profiles based on sequences for content and collaborative filtering

Modeling user profiles is a necessary step for most information filtering systems – such as recommender systems – to provide personalized recommendations. However, most of them work with users or items as vectors, by applying different types of mathematical operations between them and neglecting sequential or content-based information. Hence, in this paper we study how to propose an adaptive mechanism to obtain user sequences using different sources of information, allowing the generation of hybrid recommendations as a seamless, transparent technique from the system viewpoint. As a proof of concept, we develop the Longest Common Subsequence (LCS) algorithm as a similarity metric to compare the user sequences, where, in the process of adapting this algorithm to recommendation, we include different parameters to control the efficiency by reducing the information used in the algorithm (preference filter), to decide when a neighbor is considered useful enough to be included in the process (confidence filter), to identify whether two interactions are equivalent (δ-matching threshold), and to normalize the length of the LCS in a bounded interval (normalization functions). These parameters can be extended to work with any type of sequential algorithm.We evaluate our approach with several state-of-the-art recommendation algorithms using different evaluation metrics measuring the accuracy, diversity, and novelty of the recommendations, and analyze the impact of the proposed parameters. We have found that our approach offers a competitive performance, outperforming content, collaborative, and hybrid baselines, and producing positive results when either content- or rating-based information is exploited.     

The scattering of X-rays

Present Status of the Problem.—The scattering of X-rays is one of the outstanding problems of electromagnetic radiation which has not been solved satisfactorily. All theories (based on classical electrodynamics) presented thus far do not explain either the diminution in the scattering coefficient, or the observed asymmetry in the scattering, or both. Among such theories we may mention:J. J. Thompson's Theory.—Assuming that the scattering is done by a point electron, and making use of certain additional hypotheses, Thomson showed that the scattering coefficient of any substance is given by

$\text{σ=}\text{8πNpe}{}^4\text{3m}{}^2\text{c}{}^4$

and that the intensity of the scattered radiation is given by

$\text{I}{}_{\mathrm{\theta }}\text{=I}\text{e}{}^4\text{(I+cos}{}^2\text{θ)}\text{2r}{}^2\text{m}{}^2\text{c}{}^4$

where N is the number of atoms per c.c., p the number of electrons per atom, e the electronic charge, m the electronic mass, c the velocity of light, I_θ the intensity of the scattered radiation at an angle θ between the incident beam and the radius vector joining the centre of the electron and the point P distant r from the electron, and I is the intensity of the incident beam. This theory explains neither the asymmetry nor the decrease in the coefficient of scattering.Schott's Theory.—Among other things, the assumption is here made that the atom consists of coaxal rings of electron. The electrons in each ring are spaced at equal intervals and revolve with a uniform angular velocity, which, however, may be different for different rings. This theory fails to explain the observed diminution in the scattering coefficient.Debye's Theory.—In its essentials, Debye's theory has the same merits and demerits as that of Schott. Debye assumes that all the electrons in an atom are arranged in a single ring, and that they are spaced at equal intervals. This theory (and also Schott's theory) explains the asymmetry and the “excess scattering,” but is altogether unable to explain the diminution in the scattering coefficient.Modification of the Classical Theory.—The present paper presents a discussion of the possibility of modifying the classical theory (that of J. J. Thomson) so as to account for the decrease in the scattering coefficient as well as the dissymmetry. By assuming that the electron is made up of a number of parts—for simplicity, of two parts—it has been found possible to account for the diminution in the scattering coefficient without, at the same time, explaining the observed asymmetry. To accomplish both objects is what was aimed at in the combination of the present work with that of Debye. In this research the goal has not been perfection between predicted and observed results, but rather to discuss some possible modifications of the classical theory and their consequences.     

On the density of stabilizable plants in the class of unstabilizable plants: The real symmetric disk algebra case

Let A_R denote the set of functions from the disk algebra having real Fourier coefficients. Generalizing a result of Quadrat we show that every unstabilizable multi-input multi-output plant is as close as we want to a stabilizable multi-input multi-output plant in the product topology.