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This note outlines some connections between projective geometry and some designs used in clinical trials in the health sciences. The connections are not immediately obvious but they widen the scope for enrichment work at both the senior high school level and for capstone subjects at the undergraduate level. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the results due to Luzón and Morón, and present a characterization of the generalized Riordan arrays. Using this characterization, we establish a recurrence for the generalized Sheffer sequences, and study some special types of polynomial sequences, including the generalized Lucas u -sequence and v -sequence, and the generalized Humbert polynomial sequence. In particular, by appealing to the factorizations of Riordan arrays, the umbral composition of polynomial sequences and the Hadamard product of series, we establish various relations among these polynomial sequences. Thus, we give a unified approach to some well-known polynomial sequences appeared in combinatorics and the theory of special functions. [ABSTRACT FROM AUTHOR]

We characterize all connected graphs with exactly three distinct normalized Laplacian eigenvalues among which one is equal to 1, and determine all connected bipartite graphs with at least one vertex of degree 1 having exactly four distinct normalized Laplacian eigenvalues. In addition, we find all unicyclic graphs with three or four distinct normalized Laplacian eigenvalues. [ABSTRACT FROM AUTHOR]

We characterize all connected graphs with exactly three distinct normalized Laplacian eigenvalues among which one is equal to 1, and determine all connected bipartite graphs with at least one vertex of degree 1 having exactly four distinct normalized Laplacian eigenvalues. In addition, we find all unicyclic graphs with three or four distinct normalized Laplacian eigenvalues. [ABSTRACT FROM AUTHOR]

We show that, for Hankel matrices, total nonnegativity (resp. total positivity) of order r is preserved by sum, Hadamard product, and Hadamard power with real exponent t ≥ r − 2 . We give examples to show that our results are sharp relative to matrix size and structure (general, symmetric or Hankel). Some of these examples also resolve the Hadamard critical-exponent problem for totally positive and totally nonnegative matrices. [ABSTRACT FROM AUTHOR]

Necessary and Sufficient conditions for an equidistant code to be a 2-TA code are obtained. An explicit construction method is proposed to obtain linear MDS [ p + 1 , 2 , p ] codes over the finite field F p , where p is a prime. These codes can be used as 2-TA codes for p > 2 . In particular, for p = 3 , it is observed that the linear [4, 2, 3] MDS code contradicts a result of Jin and Blaum (2007). The correct version of this result and its proof is given. Existence of some infinite families of equidistant 2-TA codes is shown by using Jacobsthal and Hadamard matrices. Some of these codes are also observed to be good equidistant code (Sinha et al., 2008). [ABSTRACT FROM AUTHOR]

In 1978 Gutman introduced the energy of a graph as the sum of the absolute values of graph eigenvalues, and ever since then graph energy has been intensively studied. Since graph energy is the trace norm of the adjacency matrix, matrix norms provide a natural background for its study. Thus, this paper surveys research on matrix norms that aims to expand and advance the study of graph energy. The focus is exclusively on the Ky Fan and the Schatten norms, both generalizing and enriching the trace norm. As it turns out, the study of extremal properties of these norms leads to numerous analytic problems with deep roots in combinatorics. The survey brings to the fore the exceptional role of Hadamard matrices, conference matrices, and conference graphs in matrix norms. In addition, a vast new matrix class is studied, a relaxation of symmetric Hadamard matrices. The survey presents solutions to just a fraction of a larger body of similar problems bonding analysis to combinatorics. Thus, open problems and questions are raised to outline topics for further investigation. [ABSTRACT FROM AUTHOR]

A characterization of [ABSTRACT FROM AUTHOR]

We study the combined matrix of a nonsingular H-matrix. These matrices can belong to two different H-matrices classes: the most common, invertible class, and one particular class named mixed class. Different results regarding diagonal dominance of the inverse matrix and the combined matrix of a nonsingular H-matrix belonging to the referred classes are obtained. We conclude that the combined matrix of a nonsingular H-matrix is always diagonally dominant and then it is an H-matrix. In particular, the combined matrix in the invertible class remains in the same class. [ABSTRACT FROM AUTHOR]

In this paper, we introduce modular symmetric designs and use them to study the existence of Hadamard matrices modulo 5. We prove that there exist 5-modular Hadamard matrices of order n if and only if [ABSTRACT FROM AUTHOR]

Abstract: We give new constructions for DCEs in which all attributes have the same number of levels. These constructions use several combinatorial structures, such as orthogonal arrays, balanced incomplete block designs and Hadamard matrices. If we assume that only the main effects of the attributes are to be used to explain the results and that all attribute level combinations are equally attractive, we show that the constructed DCEs are D-optimal. [Copyright &y& Elsevier]

Abstract: The use of covariates in block designs is necessary when the experimental errors cannot be controlled using only the qualitative factors. The choice of values of the covariates for a given set-up attaining minimum variance for estimation of the regression parameters has attracted attention in recent times. In this paper, optimum covariate designs (OCD) have been considered for the set-up of the balanced treatment incomplete block (BTIB) designs, which form an important class of test-control designs. It is seen that the OCDs depend much on the methods of construction of the basic BTIB designs. The series of BTIB designs considered in this paper are mainly those as described by Bechhofer and Tamhane (1981) and Das et al. (2005). Different combinatorial arrangements and tools such as Hadamard matrices and different kinds of products of matrices viz Khatri-Rao product and Kronecker product have been conveniently used to construct OCDs with as many covariates as possible. [Copyright &y& Elsevier]

Two Hadamard matrices are considered equivalent if one is obtained from the other by a sequence of operations involving row or column permutations or negations. We complete the classification of Hadamard matrices of order 32. It turns out that there are exactly 13,710,027 such matrices up to equivalence. [ABSTRACT FROM AUTHOR]

Abstract: A complete classification of quaternary complex Hadamard matrices of orders 10, 12 and 14 is given, and a new parametrization scheme for obtaining new examples of affine parametric families of complex Hadamard matrices is provided. On the one hand, it is proven that all 10×10 and 12×12 quaternary complex Hadamard matrices belong to some parametric family, but on the other hand, it is shown by exhibiting an isolated 14×14 matrix that there cannot be a general method for introducing parameters into these types of matrices. [Copyright &y& Elsevier]

In this paper, we provide the Euler-Maclaurin expansions for (offset) trapezoidal rule approximations of the divergent finite-range integrals ∫ab f(x) dx, where f ∈ C∞(a, b) but can have arbitrary algebraic singularities at one or both endpoints. We assume that f(x) has asymptotic expansions of the general forms (These equations cannot be represented into ASCII text) where K,L, and cs, ds, s = 0, 1, . . . , are some constants, ∣K∣+∣L∣ ≠ 0, and γs and δs are distinct, arbitrary and, in general, complex, and different from -1, and satisfy (These equations cannot be represented into ASCII text) Hence the integral ∫ab f(x) dx exists in the sense of Hadamard finite part. The results we obtain in this work extend some of the results in [A. Sidi, Numer. Math. 98 (2004), pp. 371-387] that pertain to the cases in which K = L = 0. They are expressed in very simple terms based only on the asymptotic expansions of f(x) as x → a+ and x → b-. With h = (b - a)/n, where n is a positive integer, one of these results reads (These equations cannot be represented into ASCII text) where I[f] is the Hadamard finite part of ∫ab f(x) dx, C is Euler's constant and ζ(z) is the Riemann Zeta function. We illustrate the results with an example. [ABSTRACT FROM AUTHOR]

In this paper we extend some estimates of the right-hand side of a Hermite–Hadamard type inequality for functions whose derivatives’ absolute values are P-convex. Applications to the trapezoidal formula and special means are introduced. [ABSTRACT FROM AUTHOR]

Abstract: We give a new characterization of skew Hadamard matrices of order n in terms of spectral data for tournaments of order . [Copyright &y& Elsevier]

This note provides a correction and some additions to a 1999 article by Luigia Berardi and Fulvio Zuanni on blocking 3-sets in designs. © 2012 Wiley Periodicals, Inc. J. Combin. Designs 20: 328-331, 2012 [ABSTRACT FROM AUTHOR]

In this paper, we construct a new, previously unknown four-parameter family of complex Hadamard matrices of order 6, the entries of which are described by algebraic functions of roots of various sextic polynomials. We conjecture that the new, generic family G6(4) together with Karlsson's degenerate family K6(3) and Tao's spectral matrix S(0)6 form an exhaustive list of complex Hadamard matrices of order 6. Such a complete characterization might finally lead to the solution of the problem of mutually unbiased bases in ℂ6. [ABSTRACT FROM PUBLISHER]

Abstract: It has been observed by Assmus and Key as a result of the complete classification of Hadamard matrices of order 24, that the extremality of the binary code of a Hadamard matrix of order 24 is equivalent to the extremality of the ternary code of . In this note, we present two proofs of this fact, neither of which depends on the classification. One is a consequence of a more general result on the minimum weight of the dual of the code of a Hadamard matrix. The other relates the lattices obtained from the binary code and the ternary code. Both proofs are presented in greater generality to include higher orders. In particular, the latter method is also used to show the equivalence of (i) the extremality of the ternary code, (ii) the extremality of the -code, and (iii) the extremality of a lattice obtained from a Hadamard matrix of order 48. [Copyright &y& Elsevier]

Abstract: Optimal fractions of resolution V design matrices proposed by Rechtschaffner in 1967 are developed and applied as supersaturated designs in screening experiments. Rechtschaffner matrices allow evaluation of all main factors and two-factor interactions, which in many real-world studies are of practical significance. However, the number of experimental runs increases rapidly with the number of factors in the matrices, which are therefore impractical for more than 5–6 factors. On the contrary, saturated fractions based on Hadamard matrices, which are commonly applied in screening studies, cannot evaluate the interaction effects. Here, a procedure for selecting the optimum fractions of Rechtschaffner matrices is presented and provides supersaturated matrices that are well adapted to a variety of problems, thus allowing the development of screening studies with a relatively small number of experiments. The procedures developed to derive the size-reduced matrices and to evaluate the active factors are discussed and compared in terms of efficiency and reliability, by means of simulation studies and application to a real problem. These fractions are the first supersaturated design matrices capable of estimating interaction effects. Additionally, one important advantage of these supersaturated matrices is that they enable development of follow-up procedures in cases of inconclusive results, by enlarging the matrix and eventually resolving the full Rechtschaffner matrix of departure when it is necessary to evaluate the active factors and their interactions. [Copyright &y& Elsevier]

Boltzmann models from statistical physics, combined with methods from analytic combinatorics, give rise to efficient and easy-to-write algorithms for the random generation of combinatorial objects. This paper proposes to extend Boltzmann generators to a new field of applications by uniformly sampling a Hadamard product. Under an abstract real-arithmetic computation model, our algorithm achieves approximate-size sampling in expected time ${\cal O}$(n${\sqrt n}$) or ${\cal O}$(nσ) depending on the objects considered, with σ the standard deviation of smallest order for the component object sizes. This makes it possible to generate random objects of large size on a standard computer. The analysis heavily relies on a variant of the so-called birthday paradox, which can be modelled as an occupancy urn problem. [ABSTRACT FROM AUTHOR]

Davis, Dillon, and Jedwab all showed the existence of difference sets in groups $${C_{2^{r+2}}\times C_{2^{r}}}$$ . Turyn's bound had previously shown that abelian 2-groups with higher exponents could not admit difference sets. We give a new construction technique that utilizes character values, rational idempotents, and tiling structures to produce Hadamard difference sets in the group $${C_{2^{r+2}}\times C_{2^{r}}}$$ to replicate the result. [ABSTRACT FROM AUTHOR]

Abstract: In the recent years, the trace norm of graphs has been extensively studied under the name of graph energy. The trace norm is just one of the Ky Fan k-norms, given by the sum of the k largest singular values, which are studied more generally in the present paper. Several relations to chromatic number, spectral radius, spread, and to other fundamental parameters are outlined. Some results are extended to more general matrices. [Copyright &y& Elsevier]

Abstract: Let be a Hadamard -design. Suppose that the prime divides , but that does not divide . A result of Klemm implies that every residual design of has -rank at least . Also, every derived design of has -rank at least if . We show that when is a skew Hadamard design, the -ranks of the residual and derived designs are at least even if divides or . We construct infinitely many examples where the -rank is exactly . [Copyright &y& Elsevier]

We propose a new method with minimum experimental run size using the properties of Hadamard matrices through which some φp-optimal exact designs including A-, D- and E-optimal designs are constructed for a linear log contrast model in mixture experiments. [ABSTRACT FROM PUBLISHER]

(M, S)-Optimality criterion is used for classification of nonregular fractional factorial designs. Some properties of trace (Cd) and trace (Cd2) are used and studied in selecting and projecting nonregular designs. The proposed (M, S) criterion is easier to compute and it is also independent of the choice of orthonormal contrasts. The criterion is applied to study the projective properties of nonregular designs when only 24 runs are used for 23 factors. It can also be applied to two-level as well as multi-level fractional factorial designs. (M, S) criterion applied to examine designs constructed from 60 Hadamard matrices of order 24 and obtain lists of designs that attain the maximum or high Estimation Capacity (EC) for various dimensions. [ABSTRACT FROM AUTHOR]

We design the first efficient algorithms and prove new combinatorial bounds for list decoding tensor products of codes and interleaved codes. We show that for every code, the ratio of its list decoding radius (LDR) to its minimum distance stays unchanged under the tensor product operation (rather than squaring, as one might expect). This gives the first efficient list decoders and new combinatorial bounds for some natural codes including multivariate polynomials where the degree in each variable is bounded. We show that for every code, its LDR remains unchanged under m-wise interleaving for an integer m. This generalizes a recent result of Dinur et al. [in Proceedings of the 40th ACM Symposium on Theory of Computing (STOC'08), 2008, pp. 275-284], who proved such a result for interleaved Hadamard codes (equivalently, linear transformations). Using the notion of generalized Hamming weights, we give better list size bounds for both the tensoring and interleaving of binary linear codes. By analyzing the weight distribution of these codes, we reduce the task of bounding the list size to one of bounding the number of close-by low-rank codewords. For decoding linear transformations, using rank reduction together with other ideas, we obtain list size bounds that are tight over small fields. Our results give better bounds on the LDR than what is obtained from the Johnson bound, and yield rather general families of codes decodable beyond the Johnson radius. [ABSTRACT FROM AUTHOR]

Let H = (q1, …. , qr) be a finite set of coprime integers and let n1, n2, … denote the multiplicative semigroup generated by H and arranged in increasing order. The distribution of such sequences has been studied intensively in number theory, and they have remarkable probabilistic and ergodic properties. In particular, the asymptotic properties of the sequence {nkx} are similar to those of independent, identically distributed random variables; here {·} denotes fractional part. In this paper we prove that under mild assumptions on the periodic function f, the sequence f(nkx) obeys the central limit theorem and the law of the iterated logarithm after any permutation of its terms. Note that the permutational invariance of the CLT and LIL generally fails for lacunary sequences f(mkx) even if (mk) has Hadamard gaps. Our proof depends on recent deep results of Amoroso and Viada on Diophantine equations. We will also show that {nkx} satisfies a strong independence property ("interlaced mixing"), enabling one to determine the precise asymptotic behavior of permuted sums SN(σ) = ΣN k=1 f(nσ(k)x). [ABSTRACT FROM AUTHOR]

In this paper we show that every matrix in the class of Sylvester Hadamard matrices of order 2 under H-equivalence can have full row and column sign spectrum, meaning that tabulating the numbers of sign interchanges along any row (or column) gives all integers 0,1,...,2 − 1 in some order. The construction and properties of Yates Hadamard matrices are presented and is established their equivalence with the Sylvester Hadamard matrices of the same order. Finally, is proved that every normalized Hadamard matrix has full column or row sign spectrum if and only if is H-equivalent to a Sylvester Hadamard matrix. This provides us with an efficient criterion identifying the equivalence of Sylvester Hadamard matrices. [ABSTRACT FROM AUTHOR]

Based on the Hadamard product of known frequency domain preamble sequence and its conjugate shift sequence, a new sequence with much better auto-correlation property was constructed. Combined with the arithmetic of cross-correlation, an improved integer frequency offset (IF()) estimation algorithm was proposed. Through Monte-Carlo simulations, the distribution of the expectations and variances of the metric function of newly developed algorithm was analyzed over AWGN channels. The proposed algorithm has been applied in IEEE 802.16d system and simulations were carried out. Simulation results show that the newly proposed method can overcome the serious disturbance introduced by large timing offset. When the SNR is greater than 10 dB, the detection probability of IFO can be accurately obtained without increasing the complexity of calculation. [ABSTRACT FROM AUTHOR]

Abstract: The existence of difference matrices over small cyclic groups is investigated in this computer-aided work. The maximum values of the parameters for which difference matrices exist as well as the number of inequivalent difference matrices in each case is determined up to the computational limit. Several new difference matrices have been found in this manner. The maximum number of rows is 9 for an r ×15 difference matrix over , 8 for an r ×15 difference matrix over , and 6 for an r ×12 difference matrix over ; the number of inequivalent matrices with these parameters is 5, 2, and 7, respectively. [ABSTRACT FROM AUTHOR]

By using the method of differential subordinations, we derive certain sufficient conditions for strongly starlike functions associated with an integral operator. All these results presented here are sharp. [ABSTRACT FROM AUTHOR]

Abstract: A three-parameter family of complex Hadamard matrices of order 6 is presented. It significantly extends the set of closed form complex Hadamard matrices of this order, and in particular contains all previously described one- and two-parameter families as subfamilies. [ABSTRACT FROM AUTHOR]

The problem considered is that of finding D-optimal design for the estimation of covariate parameters and the treatment and block contrasts in a block design set up in the presence of non stochastic controllable covariates, when N = 2(mod 4), N being the total number of observations. It is clear that when N ≠ 0 (mod 4), it is not possible to find designs attaining minimum variance for the estimated covariate parameters. Conditions for D-optimum designs for the estimation of covariate parameters were established when each of the covariates belongs to the interval [-1, 1]. Some constructions of D-optimal design have been provided for symmetric balanced incomplete block design (SBIBD) with parameters b = v, r = k = v - 1, λ =v - 2 when k = 2 (mod 4) and b is an odd integer. [ABSTRACT FROM AUTHOR]

Algebraic theory of coding is one of modern fields of applications of algebra. This theory uses matrix algebra intensively. This paper is devoted to an application of Kronecker's matrix forms of presentations of the genetic code for algebraic analysis of a basic scheme of degeneracy of the genetic code. Similar matrix forms are utilized in the theory of signal processing and encoding. The Kronecker family of the genetic matrices is investigated, which is based on the genetic matrix [C A; U G], where C, A, U, G are the letters of the genetic alphabet. This matrix in the third Kronecker power is the (8*8)-matrix, which contains all 64 genetic triplets in a strict order with a natural binary numeration of the triplets by numbers from 0 to 63. Peculiarities of the basic scheme of degeneracy of the genetic code are reflected in the symmetrical black-and-white mosaic of this genetic (8*8)-matrix. This mosaic matrix is connected algorithmically with Hadamard matrices unexpectedly, which are famous in the theory of signal processing and encoding, spectral analysis, quantum mechanics and quantum computers. Furthermore, many kinds of cyclic permutations of genetic elements lead to reconstruction of initial Hadamard matrices into new Hadamard matrices unexpectedly. This demonstrates that matrix algebra is one of promising instruments and of adequate languages in bioinformatics and algebraic biology. [ABSTRACT FROM AUTHOR]

From each q × q unitary matrix Φ, we construct a family of quantum codes [image omitted], t ≥ 1, for q-state systems which encode (2t + 1)2 q-states into one q-state. We show that such codes are capable of correcting the errors of weight up to t if and only if Φ is a complex Hadamard matrix. [ABSTRACT FROM AUTHOR]

We construct harmonic diffeomorphisms from the complex plane C onto any Hadamard surface M whose curvature is bounded above by a negative constant. For that, we prove a Jenkins-Serrin type theorem for minimal graphs in M x IR over domains of M bounded by ideal geodesic polygons and show the existence of a sequence of minimal graphs over polygonal domains con- verging to an entire minimal graph in Ml x IR with the conformal structure of C. [ABSTRACT FROM AUTHOR]

We propose a group-theoretical approach to the generalized oscillator algebra Aκ? recently investigated in J. Phys. A: Math. Theor. 2010, 43, 115303. The case κ? ≤ 0 corresponds to the noncompact group SU(1,1) (as for the harmonic oscillator and the Pöschl-Teller systems) while the case ? < 0 is described by the compact group SU(2) (as for the Morse system). We construct the phase operators and the corresponding temporally stable phase eigenstates for Aκ in this group-theoretical context. The SU(2) case is exploited for deriving families of mutually unbiased bases used in quantum information. Along this vein, we examine some characteristics of a quadratic discrete Fourier transform in connection with generalized quadratic Gauss sums and generalized Hadamard matrices. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with Hadamard matrices with a certain noncyclic property. First we show that when the first column of a Sylvester Hadamard matrix of order 2m, m≥ 2, a positive integer, is removed, the number of shift distinct row vectors in the matrix is given by 2m - m. Then, for m ≥ 4, we construct an infinite family of Hadamard matrices with a property that when the first column of the Hadamard matrix is removed, all the row vectors of the matrix are shift distinct. These Hadamard matrices are useful in constructing low correlation zone sequences. [ABSTRACT FROM AUTHOR]

Abstract: Using results on Hadamard difference sets, we construct regular graphical Hadamard matrices of negative type of order for every positive integer . If , such a Hadamard matrix is equivalent to a strongly regular graph with parameters . Strongly regular graphs with these parameters have been called max energy graphs, because they have maximal energy (as defined by Gutman) among all graphs on vertices. For odd the strongly regular graphs seem to be new. [Copyright &y& Elsevier]

Abstract: Let denote the set of base sequences , with and of length and and of length . The base sequence conjecture (BSC) asserts that exist (i.e., are non-empty) for all . This is known to be true for and when is a Golay number. We show that it is also true for and . It is worth pointing out that BSC is stronger than the famous Hadamard matrix conjecture. In order to demonstrate the abundance of base sequences, we have previously attached to a graph and computed the for . We now extend these computations and determine the for . We also propose a conjecture describing these graphs in general. [Copyright &y& Elsevier]

An analytical method for getting new complex Hadamard matrices by using mutually unbiased bases and a nonlinear doubling formula is provided. The method is illustrated with the n=4 case that leads to a rich family of eight-dimensional Hadamard matrices that depend on five arbitrary parameters whose modulus is equal to unity. [ABSTRACT FROM AUTHOR]

It is well known that there exists a transversal design TDλ[ k; u] admitting a class regular automorphism group U if and only if there exists a generalized Hadamard matrix GH( u, λ) over U. Note that in this case the resulting transversal design is symmetric by Jungnickel’s result. In this article we define a modified generalized Hadamard matrix and show that transversal designs which are not necessarily symmetric can be constructed from these under a modified condition similar to class regularity even if it admits no class regular automorphism group. [ABSTRACT FROM AUTHOR]

Abstract: A popular measure to assess 2-level supersaturated designs with even number of runs is the criterion. In this paper, we consider 2-level supersaturated designs with odd number of runs which have minimum . We give a more explicit lower bound on than . Conditions of supersaturated designs which attain the lower bounds are given. supersaturated designs attaining the lower bounds are listed for and 7. Hadamard matrices and finite fields are used for constructing supersaturated designs. The lower bound is improved when the number of factors is large, and designs attaining the improved bounds are constructed by using the complements of designs with small number of factors. We also give a method to construct supersaturated designs with odd number of runs from supersaturated designs with even number of runs by deleting a run. [Copyright &y& Elsevier]