On the Square Root of a Bell Matrix

AbstractIn the context of Riordan arrays, the problem of determining the square root of a Bell matrix $$R={\mathcal {R}}(f(t)/t,\ f(t))$$ R = R ( f ( t ) / t , f ( t ) ) defined by a formal power series $$f(t)=\sum _{k \ge 0}f_kt^k$$ f ( t ) = ∑ k ≥ 0 f k t k with $$f(0)=f_0=0$$ f ( 0 ) = f 0 = 0 is presented. It is proved that if $$f^\prime (0)=1$$ f ′ ( 0 ) = 1 and $$f^{\prime \prime }(0)\ne 0$$ f ″ ( 0 ) ≠ 0 then there exists another Bell matrix $$H={\mathcal {R}}(h(t)/t,\ h(t))$$ H = R ( h ( t ) / t , h ( t ) ) such that $$H*H=R;$$ H ∗ H = R ; in particular, function h(t) is univocally determined by a symbolic computational method which in many situations allows to find the function in closed form. Moreover, it is shown that function h(t) is related to the solution of Schröder's equation. We also compute a Riordan involution related to this kind of matrices

The relevant prefixes of coloured Motzkin walks: An average case analysis

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Perancangan Sistem Informasi Administrasi Surat Berbasis Desktop pada Kantor Notaris Hoiril Masuli, Sh, M.Kn

Pada Kantor Notaris Hoiril Masuli, Sh, M.Kn dalam pengelolaan penerimaan berkas masih menggunakan cara manual dan belum terkomputerisasi. Hal ini meyulitkan pegawai dalam pengelolaan data klien, seperti halnya dalam penyimpanan data-data yang masih disimpan dalam bentuk berkas sehingga menyulitkan pegawai dalam mencari data berkas masuk dari klien ataupun berkas yang sudah selesai dibuat serta berkas yang sudah diambil klien. Untuk mengatasi masalah-masalah yang dihadapi, untuk mempercepat pekerjaan dan memudahkan pengelolahaan data perlu adanya sistem yang terkomputerisasi dalam pengelolaan berkas klien. Pada penelitian ini metode yang digunakan adalah berorientasi objek, model penelitian yang digunakan adalah Waterfall serta tools yang digunakan adalah UML (Unified Model Language)

On equalities involving integrals of the logarithm of the Riemann ζ-function and equivalent to the Riemann hypothesis

By using the generalized Littlewood theorem about a contour integral involving the logarithm of an analytic function, we show how an infinite number of integral equalities involving integrals of the logarithm of the Riemann ζ-function and equivalent to the Riemann hypothesis can be established and present some of them as an example. It is shown that all earlier known equalities of this type, viz., the Wang equality, Volchkov equality, Balazard-Saias-Yor equality, and an equality established by one of the authors, are certain special cases of our general approac

On equalities involving integrals of the logarithm of the Riemann ζ-function and equivalent to the Riemann hypothesis

By using the generalized Littlewood theorem about a contour integral involving the logarithm of an analytic function, we show how an infinite number of integral equalities involving integrals of the logarithm of the Riemann ζ-function and equivalent to the Riemann hypothesis can be established and present some of them as an example. It is shown that all earlier known equalities of this type, viz., the Wang equality, Volchkov equality, Balazard–Saias–Yor equality, and an equality established by one of the authors, are certain special cases of our general approach.Показано як за допомогою узагальненої теореми Лiттлвуда про контурний iнтеграл, що мiстить логарифм аналiтичної функцiї, можна отримати нескiнченну кiлькiсть iнтегральних рiвностей, що мiстять iнтеграли вiд логарифма ζ-функцiї Рiмана i є еквiвалентними гiпотезi Рiмана, i наведено кiлька таких рiвностей у якостi прикладу. Показано, що деякi вiдомi рiвностi такого типу, а саме, рiвностi Ванга, Волчкова, Балазарда – Сайаса – Йора та рiвнiсть, що встановлена одним iз авторiв, є частинними випадками нaшого загального пiдход

A Fast Quasi-Conformal Mapping Preconditioner for Electromagnetic Integral Equations

Boundary Element Methods (BEMs) are efficient strategies to numerically solve electromagnetic radiation and scattering problems. Unfortunately, however, classical BEM formulations suffer from ill-conditioning when the frequency is low, or the discretization density is high. In the past, several remedies have been presented for these ill-conditioning problems including preconditioners based on Calderón identities, hierarchical bases, and current decompositions. While effective, these strategies however require ad-hoc procedures including mesh-refinements, new basis function definitions, and adapted fast methods that, if not implemented properly, can become computationally cumbersome

The Tennis Ball Problem

AbstractMallows and Shapiro, (J. Integer Sequences2 (1999)) have recently considered what they dubbed the problem of balls on the lawn. Our object is to explore a natural generalization, the s-tennis ball problem, which reduces to that considered by Mallows and Shapiro in the case s=2. We show how this generalization is connected with s-ary trees, and employ the notion of generating trees to obtain a solution expressed in terms of generating functions

High-pressure behavior of intermediate scapolite : compressibility, structure deformation and phase transition

Scapolites are common volatile-bearing minerals in metamorphic rocks. In this study, the high-pressure behavior of an intermediate member of the scapolite solid solution series (Me47), chemical formula (Na1.86Ca1.86K0.23Fe0.01)(Al4.36Si7.64)O24[Cl0.48(CO3)0.48(SO4)0.01], has been investigated up to 17.79 GPa, by means of in situ single-crystal synchrotron X-ray diffraction. The isothermal elastic behavior of the studied scapolite has been described by a III-order Birch\u2013Murnaghan equation of state, which provided the following refined parameters: V0 = 1110.6(7) \uc53, KV0 = 70(2) GPa (\u3b2V0 = 0.0143(4) GPa 121) and KV\u2032 = 4.8(7). The refined bulk modulus is intermediate between those previously reported for Me17 and Me68 scapolite samples, confirming that the bulk compressibility among the solid solution increases with the Na content. A discussion on the P-induced structure deformation mechanisms of tetragonal scapolite at the atomic scale is provided, along with the implications of the reported results for the modeling of scapolite stability. In addition, a single-crystal to single-crystal phase transition, which is displacive in character, has been observed toward a triclinic polymorph at 9.87 GPa. The high-pressure triclinic polymorph was found to be stable up to the highest pressure investigated

Hopping on the Bethe lattice: Exact results for densities of states and dynamical mean-field theory

We derive an operator identity which relates tight-binding Hamiltonians with arbitrary hopping on the Bethe lattice to the Hamiltonian with nearest-neighbor hopping. This provides an exact expression for the density of states (DOS) of a non-interacting quantum-mechanical particle for any hopping. We present analytic results for the DOS corresponding to hopping between nearest and next-nearest neighbors, and also for exponentially decreasing hopping amplitudes. Conversely it is possible to construct a hopping Hamiltonian on the Bethe lattice for any given DOS. These methods are based only on the so-called distance regularity of the infinite Bethe lattice, and not on the absence of loops. Results are also obtained for the triangular Husimi cactus, a recursive lattice with loops. Furthermore we derive the exact self-consistency equations arising in the context of dynamical mean-field theory, which serve as a starting point for studies of Hubbard-type models with frustration.Comment: 14 pages, 9 figures; introduction expanded, references added; published versio

Pargasite at high pressure and temperature

The P-T phase stability field, the thermoelastic behavior and the P-induced deformation mechanisms at the atomic scale of pargasite crystals, from the "phlogopite peridotite unit" of the Finero mafic-ultramafic complex (Ivrea-Verbano Formation, Italy), have been investigated by a series of in situ experiments: (a) at high pressure (up to 20.1 GPa), by single-crystal synchrotron X-ray diffraction with a diamond anvil cell, (b) at high temperature (up to 823 K), by powder synchrotron X-ray diffraction using a hot air blower device, and (c) at simultaneous HP-HT conditions, by single-crystal synchrotron X-ray diffraction with a resistive-heated diamond anvil cell (Pmax = 16.5 GPa, Tmax = 1200 K). No phase transition has been observed within the P-T range investigated. At ambient T, the refined compressional parameters, calculated by fitting a second-order Birch-Murnaghan Equation of State (BM-EoS), are: V0 = 915.2(8) \uc53 and KP0,T0 = 95(2) GPa (\u3b2P0,T0 = 0.0121(2) GPa-1) for the unit-cell volume; a0 = 9.909(4) \uc5 and K(a)P0,T0 = 76(2) GPa for the a-axis; b0 = 18.066(7) \uc5 and K(b)P0,T0 = 111(2) GPa for the b-axis; c0 = 5.299(5) \uc5 and K(c)P0,T0 = 122(12) GPa for the c-axis [K(c)P0,T0 ~ K(b)P0,T0 > K(a)P0,T0]. The high-pressure structure refinements (at ambient T) show a moderate contraction of the TO4 double chain and a decrease of its bending in response to the hydrostatic compression, along with a pronounced compressibility of the A- and M(4)-polyhedra [KP0,T0(A) = 38(2) GPa, KP0,T0(M4) = 79(5) GPa] if compared to the M(1)-, M(2)-, M(3)-octahedra [KP0,T0(M1,2,3) 64 120 GPa] and to the rigid tetrahedra [KP0,T0(T1,T2) ~ 300 GPa]. The thermal behavior, at ambient pressure up to 823 K, was modelled with Berman's formalism, which gives: V0 = 909.1(2) \uc53, \u3b10 = 2.7(2)*10-5 K-1 and \u3b11 = 1.4(6)*10-9 K-2 [with \u3b10(a) = 0.47(6)*10-5 K-1, \u3b10(b) = 1.07(4)*10-5 K-1, and \u3b10(c) = 0.97(7)*10-5 K-1]. The petrological implications for the experimental findings of this study are discussed