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The article presents a mathematical study constructing the integer divisors in residue classes when r, s, n are integers satisfying 0 ⩽ r < s < n, s ⩾ nα, α > 1/4, and when gcd(r,s) = 1. It presents techniques for factoring an integer, along with refinements and an improved presentation, which can find divisors in residue classes in polynomial time. It suggests that the divisors of n equivalent to r can upper bounded by O((α - 1/4)-3/2).

A complete two-dimensional local field K of mixed characteristic with finite second residue field is considered. The existence of a completely ramified extension L of K such that L is a standard field is assumed. It is proved that the rank of the quotient U(1)KK/TK, where TK is the closure of the torsion subgroup, is equal to the degree of the constant subfield of K over ℚp. I. B. Zhukov constructed a set of generators of this quotient in the case where K is a standard field. In this paper, two natural generalizations of this set are considered, and it is proved that one of them generates the entire group and the other generates its subgroup of finite index. Bibliography: 5 titles. [ABSTRACT FROM AUTHOR]

A sequence $(a_n)_{n=0}^{\infty}$ over ℤ is an LP-sequence if for every prime p and integer n ≥ 0 we have $a_n \equiv \prod_{i=0}^{l-1} a_{n_{i}}$ (mod p), when $n= \sum_{i=0}^{l-1} n_ip^i$ is a base p expansion of n. In this paper, we study sequences $(a_n)_{n=0}^{\infty}$ such that both $(a_n)_{n=0}^{\infty}$, $(a_{dn})_{n=0}^{\infty}$ are LP-sequences for some d ≥ 2. One of those sequences is a counter-example to a conjecture of McIntosh [15]. [ABSTRACT FROM AUTHOR]

Let p be a prime and let f be any cusp form of level l ∈ {2,3,5,7,13} whose weight satisfy a certain congruence modulo (p-1). Then we exhibit explicit linear combinations of the coefficients of f that must be divisible by p. For a normalized Hecke eigenform, this translates (under mild restrictions) into the pth coefficient itself being divisible by a prime ideal above p in the ring generated by the coefficients of f. This provides many instances of the so-called non-ordinary primes. We also discuss linear relations satisfied universally on the space of modular forms of these levels. These results extend recent work of Choie, Kohnen and Ono in the level 1 case. [ABSTRACT FROM AUTHOR]

In this paper, we study various arithmetic properties of d + n/d, where d runs through all the τ(n) positive divisors of n. For example, denoting by ϖ(n) the number of prime values among these sums, we study how often ϖ(n) > 0 and also ϖ(n) = τ(n), and we also evaluate the average value of ϖ(n). We estimate some character sums with d + n/d and study the distribution of quadratic nonresidues and primitive roots among these sums on average over n ≤ x. [ABSTRACT FROM AUTHOR]

In connection with problems of information theory, we study arithmetical progressions constructed at the points of elliptic curves over a finite field. For certain types of such curves, we establish the distribution of the quadratic residues at the x-coordinates of the sequence of points corresponding to progressions if the elliptic curves is defined over a simple field. A description of the set of all progressions on elliptic curves over a finite field is also given. [ABSTRACT FROM AUTHOR]

Proteins are highly flexible and large amplitude deformations of their structure, also called slow dynamics, are often decisive to their function. We present a two-level rendering approach that enables visualization of slow dynamics of large protein assemblies. Our approach is aligned with a hierarchical model of large scale molecules. Instead of constantly updating positions of large amounts of atoms, we update the position and rotation of residues, i.e., higher level building blocks of a protein. Residues are represented by one vertex only indicating its position and additional information defining the rotation. The atoms in the residues are generated on-the-fly on the GPU, exploiting the new graphics hardware geometry shader capabilities. Moreover, we represent the atoms by billboards instead of tessellated spheres. Our representation is then significantly faster and pixel precise. We demonstrate the usefulness of our new approach in the context of our collaborative bioinformatics project. [ABSTRACT FROM AUTHOR]

Abstract: We prove the existence of equiangular tight frames having elements drawn from either or whenever n is either for , or a power of a prime such that . We also find a simple explicit expression for the prime power case by establishing a connection to a -element equiangular tight frame based on quadratic residues. [Copyright &y& Elsevier]

The article focuses on density and polynomial congruencies. The author presents an algebraic problem for determining solutions of a congruence as the modulus varies. He discusses a mathematical analysis of the problem and offers solutions, incorporating Christopher Hooley's theorem on irreducible polynomials. A lemma on a polynomial with version of the remainder theorem also is presented.

The article presents a mathematical problem relating to lattice points. It presents the problem of a photographer who has to take pictures of a high school marching band while standing near them on the field, using the fewest number of photographs such that every person is visible and unobstructed in at least one of the photographs. Several answers to the problem using the Chinese Remainder Theorem as a key component also are presented.

In the first part of this article, we had introduced the notion of quadratic reciprocity and dwelt briefly on its history, which goes back all the way to the work of fermat. Then we discussed the Law of Quadratic Reciprocity ('QRL'), which Gauss named Theorema Aureum. Following this, we gave a not too well known proof of the QRL due to G Rousseau. Now we give two more proofs of the QRL , drawing respectively from ideas in linear algebra and field extensions; they too are not very well known. [ABSTRACT FROM AUTHOR]

We consider a congruence ρ on a semigroup S as a subsemigroup of the direct product S × S. We prove that if ρ has finite derivation type (FDT), then so does S. [ABSTRACT FROM AUTHOR]

In this paper, we describe strong P-congruences and sublattice-structure of the strong P-congruence lattice C P(S) of a P-inversive semigroup S(P). It is proved that the set of all strong P-congruences C P(S) on S(P) is a complete lattice. A close link is discovered between the class of P-inversive semigroups and the well-known class of regular ⋆-semigroups. Further, we introduce concepts of strong normal partition/equivalence, C-trace/kernel and discuss some sublattices of C P(S). It is proved that the set of strong P-congruences, which have C-traces (C-kernels) equal to a given strong normal equivalence of P (C-kernel), is a complete sublattice of C P(S). It is also proved that the sublattices determined by C-trace-equaling relation θ and C-kernel-equaling relation κ, respectively, are complete sublattices of C P(S) and the greatest elements of these sublattices are given. [ABSTRACT FROM AUTHOR]

We describe the structure of 0-simple countably compact topological inverse semigroups and the structure of congruence-free countably compact topological inverse semigroups. [ABSTRACT FROM AUTHOR]

Are there perfect squares which on division by 7 leave remainder 3? Are there perfect squares which on division by 3 leave remainder 7? (A "remainder of 7" on division by 3 is the same as a remainder of 1.) The answers: 'NO' and 'YES', respectively. These facts are stated by number theorists as follows: 3 is a quadratic non-residue modulo 7; 7 is a quadratic residue modulo 3. The notion of quadratic residue is far reaching, and the key theorem here is the Law of Quadratic Reciprocity , first stated by Euler in 1783, but with out proof, and first proved by Gauss, in 1796. The theorem is easy to state but is mysterious, as it reveals a connection between two questions that appear unconnected. Let p, q be distinct odd primes; then the questions are: "Is p a quadratic residue modulo p?" and "Is q a quadratic residue modulo p?" Gauss had a high regard for this result and called it Theorema Aureum, the Golden Theorem. Though it has been proved in many different ways, it retains its mystery. In this two part article we give three proofs of the theorem. The first one, described in this part, is based on group theory. [ABSTRACT FROM AUTHOR]

Mytilus edulis attaches to solid surfaces in turbulent environment by making a byssus, which contains 3,4-dihydroxyphenlalanine (DOPA). The DOPA residues are key components that are believed to be primarily responsible for underwater chemisorption of the adhesion proteins to substrates. This study describes the symthesis and adhesive properties of a pyrene-containing dipeptide, Boc-Lys(pyr)-DOPA. Boc-Lys(pyr)-DOPA forms fibrous-micelles by the self-assembly of hydrophobic pyrenes. The micelle surface constitutes of DOPA residues. Self-assembled Boc-Lys(pyr)-DOPA had good adhesive properties on various substrates. Our results advocate a new type of adhesive material and a may load to new material developments. [ABSTRACT FROM AUTHOR]

Activated Protein C (APC) inactivates factor VIIIa by cleavage at Arg336 and Arg562 within the A1 and A2 subunits, respectively, with reaction at the former site occurring at a rate ∼25-fold faster than the latter. Recombinant factor VIII variants possessing mutations within the P4-P3′ sequences were used to determine the contributions of these residues to the disparate cleavage rates at the two P1 sites. Specific activity values for 336(P4-P3′)562, 336(P4-P2)562, and 336(P1′-P3′)562 mutants, where indicated residues surrounding the Arg336 site were replaced with those surrounding Arg562, were similar to wild type (WT) factor VIII; whereas 562(P4-P3′)336 and 562(P4-P2)336 mutants showed specific activity values <1% the WT value. Inactivation rates for the 336 site mutants were reduced ∼6-11-fold compared with WT factor VIIIa, and approached values attributed to cleavage at Arg562. Cleavage rates at Arg336 were reduced ∼100-fold for 336(P4-P3′)562, and ∼9-16-fold for 336(P4-P2)562 and 336(P1′-P3′)562 mutants. Inhibition kinetics revealed similar affinities of APC for WT factor VIIIa and 336(P4-P3′)562 variant. Alternatively, the 562(P4-P3′)336 variant showed a modest increase in cleavage rate (∼4-fold) at Arg562 compared with WT, whereas these rates were increased by ∼27- and 6-fold for 562(P4-P3′)336 and 562(P4-P2)336, respectively, using the factor VIII procofactor form as substrate. Thus the P4-P3′ residues surrounding Arg336 and Arg562 make significant contributions to proteolysis rates at each site, apparently independent of binding affinity. Efficient cleavage at Arg336 by APC is attributed to favorable P4-P3′ residues at this site, whereas cleavage at Arg562 can be accelerated following replacement with more optimal P4-P3′ residues. [ABSTRACT FROM AUTHOR]

The epithelial Na+ channel (ENaC) is typically formed by three homologous subunits (α, β, and γ) that possess a characteristic large extracellular loop (ECL) containing 16 conserved cysteine (Cys) residues. We investigated the functional role of these Cys residues in Na+ self-inhibition, an allosteric inhibition of ENaC activity by extracellular Na+. All 16 Cys residues within α and γ ECLs and selected β ECL Cys residues were individually mutated to alanine or serine residues. The Na+ self-inhibition response of wild type and mutant channels expressed in Xenopus oocytes was determined by whole cell voltage clamp. Individual mutation of eight α (Cys-1, -4, -5, -6, -7, -10, -13, or -16), one β (Cys-7), and nine γ(Cys-3, -4, -6, -7, -10, -11, -12, -13, or -16) residues significantly reduced the magnitude of Na+ self-inhibition. Na+ self-inhibition was eliminated by simultaneous mutations of either the last three a ECL Cys residues (Cys-14, -15, and -16) or Cys-7 within both α and β ECLs. By analyzing the Na+ self-inhibition responses and the effects of a methanethiosulfonate reagent on channel currents in single and double Cys mutants, we identified five Cys pairs within the αECL (αCys-1/αCys-6, αCys-4/αCys-5, αCys-7/αCys-16, αCys-10/αCys-13, and αCys-11/αCys-12) and one pair within the γECL (γCys-7/γCys-16) that likely form intrasubunit disulfide bonds. We conclude that approximately half of the ECL Cys residues in the α and γ ENaC subunits are required to establish the tertiary structure that ensures a proper Na+ self-inhibition response, likely by formation of multiple intrasubunit disulfide bonds. [ABSTRACT FROM AUTHOR]

Residue number systems are attractive due to their basic arithmetic operations, such as addition and multiplication, are carry-free; this facilitates low-power, high-speed implementations. But converting from residue to conventional notation is generally problematic. In this paper, we propose a high-speed architecture for reverse conversion; this is based on Base Extension and the Chinese Remainder Theorem. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to use the generalized Bernoulli numbers, Gauss sums and the mean value theorems of Dirichlet L-functions to study the asymptotic property of the difference between a quadratic residue and its inverse modulo p (a prime), and to give an interesting hybrid mean value formula. [ABSTRACT FROM AUTHOR]

Let $V$ be a complete discrete valuation ring with residue field $k$ of characteristic $p>0$ and fraction field $K$ of characteristic zero. Let ${\cal S}$ be a formal scheme over $V$ and let $\mathfrak{X}\to {\cal S}$ be a locally projective formal abelian scheme. In this paper we prove that, under suitable natural conditions on the Hasse–Witt matrix of $\mathfrak{X}\otimes_V V/\mathit{pV}$, the kernel of the Frobenius morphism on $\mathfrak{X}_k$ can be canonically lifted to a finite and flat subgroup scheme of $\mathfrak{X}$ over an admissible blow-up of ${\cal S}$, called the ‘canonical subgroup of $\mathfrak{X}$’. This is done by a careful study of torsors under group schemes of order $p$ over $\mathfrak{X}$. We also present a filtration on ${\rm H}^1(\mathfrak{X},\mu_p)$ in the spirit of the Hodge–Tate decomposition. [ABSTRACT FROM AUTHOR]

Conformational fluctuations in proteins have emerged as a potentially important aspect of biological function, although the precise relationship and the implications have yet to be fully explored. Numerous studies have reported that the binding of ligand can influence fluctuations. However, the role of the binding site in mediating these fluctuations is not known. Of particular interest is whether in addition to serving as structural scaffolds for recognition and catalysis, active-site residues may also play a role in modulating the cooperative network. To address this question, we employ an experimentally validated ensemble-based description of proteins to elucidate the extent to which perturbations at different sites can influence the cooperative network in the protein. Applying this method to a database of test proteins, it is found statistically that binding sites are located in regions most able to affect the cooperative network, even for cooperative interactions between residues distant to the binding sites. This indicates that the conformational manifold under native conditions is determined by the network of cooperative interactions within the protein and suggests that proteins have evolved to use these conformational fluctuations in carrying out their functions. Furthermore, because the energetic coupling pattern calculated for each protein is robust and relatively insensitive to sequence, these studies further suggest that binding sites evolved in regions of the protein that are inherently poised to take advantage of the fluctuations in the native structure. [ABSTRACT FROM AUTHOR]

Abstract: If S is a nonempty, finite subset of the positive integers, we address the question of when the elements of S consist of various mixtures of quadratic residues and nonresidues for infinitely many primes. We are concerned in particular with the problem of characterizing those subsets of integers that consist entirely of either (1) quadratic residues or (2) quadratic nonresidues for such a set of primes. We solve problem (1) and we show that problem (2) is equivalent to a purely combinatorial problem concerning families of subsets of a finite set. For sets S of (essentially) small cardinality, we solve problem (2). Related results and some associated enumerative combinatorics are also discussed. [Copyright &y& Elsevier]

Abstract: In this note we shall improve some congruences of G.S. Kazandzidis and D.F. Bailey to higher prime power moduli, by studying the relation between irregular pairs of the form and a refined version of Wolstenholme''s theorem. [Copyright &y& Elsevier]

For two distinct primes p, q, we describe those clones on a set of size pq that contain a given group operation and all constant operations. We show that each such clone is determined by congruences and commutator relations. Thus we obtain that there is only a finite number of such clones on a fixed set. [ABSTRACT FROM AUTHOR]

The article examines the relationship between quadratic residues and the Frobenius coin problem. It presents a chart showing the non-representable Frobenius numbers for 5 and 7 and the squares of these numbers. It defines the terms quadratic residue and the coin problem of Frobenius. It provides an example which describes the case for five and seven-cent coins.

The article discusses an experimental investigation involving the United States Environmental Protection Agency's method to estimate the alpha activity of water samples (EPA Method 900.0), which is termed the gross alpha-particle activity (GAA). The investigation involved looking at the gross alpha-particle activity of 226Ra, the time between sample preparation and analysis, and the residue mass and geometry. The investigation showed that the factors presented can cause the gross alpha-particle activity of a sample to vary by more than a factor of three for replicate samples, and to substantially overestimate the alpha activity of the residue.

Abstract: Efromovich-Pinsker and Stein blockwise-shrinkage estimates are traditionally studied via upper-bound oracle inequalities, which bound the estimate''s risk from above by the oracle''s risk plus a remainder term. These bounds allow one to establish sufficient conditions for attaining the oracle''s risk. To explore necessary conditions, this article develops a lower-bound oracle inequality, which bounds the estimate''s risk from below by the oracle''s risk minus a remainder term. In particular, the lower bound implies that thresholds must vanish for attaining the oracle''s risk. [Copyright &y& Elsevier]

A generalized Chinese remainder theorem (CRT) for multiple integers from residue sets has been studied recently, where the remainders in a residue set are not ordered. In this correspondence, we first propose a majority method and then based on the proposed majority method we present a sharpened dynamic range of multiple integers that can be uniquely determined from their residue sets. [ABSTRACT FROM AUTHOR]

Using the group consisting of the eight Möbius transformations x, – x, 1 / x , − 1 / x , (x − 1) / (x + 1) , (x + 1) / (1 − x) , (x + 1) / (x − 1) , and (1 − x) / (x + 1) , we present an enumerative proof of the classical result for when the element 2 is a quadratic residue in the finite field Fq. [ABSTRACT FROM AUTHOR]

With growing understanding of the differences between solid residues from the fluidized bed combustion of petroleum coke and of coal, the significance of fuel-derived and sorbent-derived components of the ash has become clearer It is well documented that hydration of the ashes is necessary prior to disposal or utilization or as a reactivation method. Initially, hydration of the lime was thought to involve water reacting only with CaO to form Ca(OH)2 but when the free lime content of the ashes is looked at before and after hydration, it is apparent that the process is more complex. Detailed analyses have shown that the free lime can decrease and vary within the same ash in different particle size ranges. The complexity of the reactions is reflected in problems with the assessment of the free lime content of the materials and the effect of hydration on different particle size fractions of the ash. The free lime content of the ash is significantly lower than expected based on the elemental analysis. Bed ash from the circulating fluidized bed combustion boilers owned and operated by the Nelson Industrial Steam Company Ltd. (NISCO) was examined in detail to elucidate the fate of calcium in the ash during hydration, using a range of techniques. The objective of the study is to determine the amount of CaO available for hydration/reactivation and to better understand interactions of Ca and other mineral components of the ash. Analysis results indicate that in NISCO ashes up to about 6% of the analytical CaO may be combined as acid soluble and - insoluble OCCs (other calcium compounds). This implies up to about 10% less free lime than would be inferred from standard chemical analyses. About 1% of the missing CaO can be present as acid insoluble Ca and Mg vanadates, with up to 2% bound in soluble OCCs. The remaining 3-4% is still not accounted for It is clear that even very minor quantities of mineral matter, other than CaO or CaSO4, associated mainly with the coarser size fractions, are important. The amount of bound water in the hydrated ash, other than that combined in portlandite or brucite, can be as large as 3-5%. This cannot be ignored when sample mass change on hydration or heating is used as a measure of the extent of CaO to Ca(OH)2 conversion. [ABSTRACT FROM AUTHOR]

This paper presents a linear and quadratic residual bound for eigenvalues of an indefinite possible singular Hermitian matrix. These bounds are a generalization of results on a semidefinite Hermitian matrix [Z. Drmač and V. Hari, SIAM J. Matrix Anal. Appl., 18 (1997), pp. 21–29]. The bounds here contain an extra factor which depends on the norm of a J-unitary matrix, where J is diagonal matrix with ±1 on its diagonal. [ABSTRACT FROM AUTHOR]

Abstract: In recent years, the conversion of residue numbers to a binary integer has been intensively studied. The Chinese Remainder Theorem (CRT) is a solution to this conversion problem of a number to the Residue Number System with a general moduli set. This paper presents a new division-free conversion approach for the conversion of residue numbers to a binary integer. The algorithm differs from others employing a great number of division instructions by using shift instructions instead. These simple instructions keep the power consumption lower. This algorithm can also be implemented with a lookup table or upon a vector machine. Both make the conversion process efficient. This division-free algorithm employs the concept of Montgomery multiplication algorithm. There are two variations of Montgomery algorithm proposed, which are algorithms MMA and IMA. The algorithm MMA is to transform the input number into the output presentation of Montgomery algorithm. Algorithm IMA is therefore inverse the computation of Montgomery algorithm to obtain the multiplicand. These two algorithms are in the complexity of O(n), where n is ⌈log2 q i ⌉. q i is a modulus. The proposed algorithm for converting the residues to a binary integer therefore runs on O(n ×log m) times on O(m) processors. There are O(log m) iterations of O(n) complexity. Compared with the traditional conversion algorithm, the advantages of this proposed algorithm are not only in employing simpler operations but also in performing fewer iterations. [Copyright &y& Elsevier]

The article presents solutions to several mathematical problems. 420 is the answer to a problem that asked about the total number of five digit numbers with all digit nonzero and no digit that are divisible by 25. 4 dimes is the answer to a problem that asked about the total number of dimes if there are $2.15 in change entirely of dimes and quarters, and the number of quarters is more than that of dimes. 396 is the answer to a problem that asked about the total number of positive integral factors of 605.

The orientation between the backbone residues of proteins is defined based on the local configurations and the corresponding preferences are analyzed by statistics. It is found that all the residue pairs have some specific preferences of orientations. The statistical analysis is mainly concentrated in the orientational distributions for two kinds of groupings of residues based on the hydrophobicity and secondary structural features. The statistics for such two types of groupings shows different orientational preferences. It is found that for the former grouping the orientational preference is rather weak, while for the later a kind of strong orientational preferences. This suggests that the formation of local structures and of secondary structures are highly related to the orientational preferences. [ABSTRACT FROM AUTHOR]

In this, paper, we present several construction methods for low-correlation zone (LCZ) sequence sets. First, we propose a design scheme for binary LCZ sequence sets with parameters (2n+1 - 2, M, L, 2). In this scheme, we can freely set the LCZ length L and the resulting LCZ sequence sets have the size M, which is almost optimal with respect to Tang, Fan, and Matsufuji bound. Second, given a q-ary LCZ sequence set with parameters (N, M, L, ϵ) and even q, we construct another q-ary LCZ sequence set with parameters (2N, 2M, L, 2ϵ) or (2N, 2M, L - 1, 2ϵ). Especially, the new set with parameters (2N, 2M, L, 2) can be optimal in terms of the set size if a q-ary optimal LCZ sequence set with parameters (N, M, L, 1) is used. [ABSTRACT FROM AUTHOR]

Inspired by a paper of Sárközy [4] we study sets of integers and sets of residues with the property that all sums and all products are distinct. [ABSTRACT FROM AUTHOR]

For a finite set A of positive integers, we study the partition function pA(n). This function enumerates the partitions of the positive integer n into parts in A. We give simple proofs of some known and unknown identities and congruences for pA(n). For n in a special residue class, pA(n) is a polynomial in n. We examine these polynomials for linear factors, and the results are applied to a restricted m-ary partition function. We extend the domain of pA and prove a reciprocity formula with supplement. In closing we consider an asymptotic formula for pA(n) and its refinement. [ABSTRACT FROM AUTHOR]

Recent works have used the theory of modular forms to establish linear congruences for the partition function and for traces of singular moduli. We show that this type of phenomenon is completely general, by finding similar congruences for the coefficients of any weakly holomorphic modular form on any congruence subgroup $\Gamma_0 (N)$. In particular, we give congruences for a wide class of partition functions and for traces of CM values of arbitrary modular functions on certain congruence subgroups of prime level. [ABSTRACT FROM AUTHOR]

A novel and flexible approach to the design and implementation of fast Fourier transforms (FFTs) of standard (fixed-radix) or non-standard length is described. By adopting a two-factor formulation, based upon either the prime factor algorithm or the common factor algorithm, where one factor, N1, is an integer power, and the other, N2, a small integer (odd- or even-valued), the (N1×N2)-point 1D discrete Fourier transform (DFT) can be carried out via the ‘row–column’ method with the design of just two generic DFT/FFT modules. Processing the small row-DFTs coefficient-by-coefficient, rather than DFT-by-DFT, it is seen how the usual requirement for matrix transposition of the row-DFT output can be eliminated. This reduces the processing requirement to a number of independent processes in which each process generates a set of small-DFT coefficients prior to computing a fixed-radix FFT of the resulting coefficient set. Hardware efficiency/simplicity is achieved by computing the small-DFT coefficients via a modified form of the Goertzel DFT (GDFT) filter, referred to here as the ‘dual-coefficient’ GDFT filter, which simultaneously computes two DFT coefficients. The resulting decomposition maps onto just floorl½N2floorr+1 processing elements (PEs), via the single-instruction multiple-data computing architecture, in which each PE comprises two types of module: one corresponding to a dual-coefficient GDFT filter bank, the other to the associated dual-data FFTs. This yields flexible-length FFTs that lend themselves naturally to multi-level parallelisation, overcoming the communication problems associated with mapping sequentially optimal FFT algorithms onto multi-processor computing architectures. [ABSTRACT FROM AUTHOR]

Established methods for transmembrane protein segment prediction are often based upon hydrophobicity analysis. Classical wavelet multiscale methods have proved successful in the prediction task. However, they implicitly model protein chain residues as being equally spaced. Our main motivation is to challenge this assumption by developing a new multiscale "lifting" technique that utilizes irregularly spaced residues, where the spacing is derived from resolved three-dimensional (3D) information obtained from similar aligned proteins. For different protein families we calculate asymmetrical dissimilarity matrices of order 20 that estimate the "distance" between residue types. We use our new adaptive lifting technique to regress the Kyte and Doolittle hydrophobicity index upon residues (now irregularly spaced using information from the distance matrices) and use the regression to predict transmembranar segments. We compare the results obtained through our method with the ones obtained through the usage of classical wavelets and show that incorporating 3D resolved structure improves overall prediction (both in terms of the existence of predicted segments compared to experimentally determined ones and the proportion of correctly predicted segments). [ABSTRACT FROM AUTHOR]

The incongruence length difference (ILD) test may produce artificially large significance values with the addition not only of uninformative characters, but also of informative characters not relevant to the groups in conflict. Previously reported problems with the ILD test involved cases of false positives, reporting high incongruence when none is expected. Under certain conditions, the test can suffer with the opposite problem (false negatives), reporting non-significant values in cases of high incongruence. These opposing effects can be combined in a data set, such that a comparison over all partitions appears as congruent, while some of the pair-wise comparisons are reported as significantly incongruent. © The Willi Hennig Society 2006. [ABSTRACT FROM AUTHOR]

A criterion is proposed to compare systematic hypotheses based on multiple sources of information under a diverse set of interpretive assumptions (i.e., sensitivity analysis of Wheeler, 1995 ). This metric, the Meta-Retention Index (MRI), is the retention index (RI) of Farris calculated over the set of conventional homologous qualitative characters (ordered, unordered, Sankoff, etc.) and molecular fragment characters sensu Wheeler (1996, 1999 ). The superiority of this measure to other similar measures (e.g., incongruence length difference test) comes from its independence from partition information. The only values that participate in its calculation are the minimum, maximum and observed cost (= cladogram cost) of each character. The partition (morphology, gene locus) from which the variant may have come is irrelevant. In the special cases where there is only a single data partition, this measure is equivalent to the conventional RI; and in the case where there are single fragment characters per partition (contiguous molecular loci as data sets) the measure is identical to the complement of the Rescaled Incongruence Length Difference (RILD) of Wheeler and Hayashi (1998 ). The MRI can serve as an optimality criterion for deciding among systematic hypotheses based on the same data, but different sets of analysis assumptions (e.g., character weights, indel costs). The MRI may lose discriminatory power in situations where a minority of highly congruent characters is given high weight. This situation can be detected and seems unlikely to occur frequently in real data sets. © The Willi Hennig Society 2006. [ABSTRACT FROM AUTHOR]

For two particular classes of elliptic curves, we establish congruences relating the coefficients of their corresponding modular forms to combinatorial objects. These congruences resemble a supercongruence for the Apéry numbers conjectured by Beukers and proved by Ahlgren and Ono in [1]. We also consider the trace Tr2k(Γ0(N), n) of the Hecke operator Tn acting on the space of cusp forms S2k(Γ0(N)). We show that for (n, N) = 1, these traces interpolate p-adically in the weight aspect. [ABSTRACT FROM AUTHOR]

Abstract: Consider complex subspaces of symmetric 2×2 complex matrices under real congruences (sending M to AMA T with A real and invertible). The orbits for dimension one were classified (in different language) two decades ago. Those for dimension two were recently classified by Coffman. It turns out that these two problems are in fact dual to each other, and Coffman’s classification yields a simpler expression for the orbits in dimension one. [Copyright &y& Elsevier]

In this paper, new robust M-estimation techniques are developed for combating impulsive noise and multiple-access interference in communication systems. Power functions of the proposed robust estimators are obtained by minimizing a non- quadratic residual function derived from the Huber's minimax robust estimation theory. Maximum peaks of these power functions are used for estimation of communication signals as well as direction of arrival. A strong advantage of the proposed robust M-estimation algorithms is a decreased sensitivity of the estimates with respect to an actual unknown distribution of random noises and interferences. Simulation results demonstrate that the proposed robust algorithms, the robust median, and robust Huber estimators, offer significant performance gain over the conventional and minimum-variance distortionless-response estimators, with the best results given by the robust Huber estimator. [ABSTRACT FROM AUTHOR]

Abstract: The paper presents an alternative perspective on the mixture estimation problem. First, observations are counted into a histogram. Secondly, rough and enhanced parameter estimation followed by the separation of observations is done. Finally, the residue is distributed between the components by the Bayes decision rule. The number of components, the mixture component parameters and the component weights are modelled jointly, no initial parameter estimates are required, the approach is numerically stable, the number of components has no influence upon the convergence and the speed of convergence is very high. The alternative perspective is compared to the EM algorithm and verified through several data sets. The presented algorithm showed significant advantages compared to the competitive methods and has already been successfully applied in reliability and fatigue analyses. [Copyright &y& Elsevier]

We let G be a group, and we let k be a natural number. We assume that G is nilpotent of class at most k, and that every (k + 1)-ary congruence preserving function on G is a polynomial function. We show that then every congruence preserving function on G (of any finite arity) is a polynomial function. [ABSTRACT FROM AUTHOR]

Abstract: In this note we give a negative answer to the Problem 20 in the article by Baczyński “Residual implications revisited. Notes on the Smets–Magrez Theorem” [Fuzzy Sets and Systems 145 (2) (2004)]. [Copyright &y& Elsevier]

A matrix A ϵ Mn(ℂ) is called unitoid if it is congruent to a diagonal matrix. Necessary and sufficient conditions are given on the canonical angles of a unitoid matrix so that sufficiently small perturbations remain unitoid. This, in particular, resolves the question of when simultaneous diagonalizability of two Hermitian matrices is retained under perturbation. [ABSTRACT FROM AUTHOR]