Form F1 Research Articles

ON THE DOMAIN OF THE CONVERGENCE OF TAYLOR-DIRECHLET SERIES WITH COMPLEX EXPONENTS

The article deals with Dirichlet series of the form F(z) = Σ+∞ k=0 akezλk , as well as Taylor-Dirichlet series of the form F1(z) =+∞Σk=0 zmkakezλk , where (λk) is some sequence of pairwise different complex numbers, and mk is a sequence of non-negative integers, ak ∈ C. The main statements of the work are Statements 1–3 and relate to the description and relationship between the domains of convergence, absolute convergence, and the domains of existence of the maximum term of the above series F and F1.

Characterization of Collagen from Three Genetic Lines (Gray, Red and F1) of Oreochromis niloticus (Tilapia) Skin in Young and Old Adults.

From tilapia (Oreochromis niloticus) farming, the by-products have been identified as a source of collagen that could be used for the development of dermocosmetics or pharmaceutical products. However, the characteristics of collagen related to a specific strain or culture must be well defined prior to its application. Collagen was extracted from the skin of three strains of tilapia: red YY males (YY: two Y-type sex chromosomes), XX gray females, and the F1: offspring of crossing red YY males with XX gray females; at different ages in the adult phase, using acetic acid and pepsin enzyme. The characteristics of acid-soluble collagen (ASC) and pepsin-soluble collagen (PSC) were shown by SDS-PAGE band profiles to be similar to bovine collagen type I (SIGMA), the PSC of gray tilapia being more fragile to temperature changes, consistent with the results of fractional viscosity. The characteristics of the F1 progeny were prioritized for being a commercially productive and sustainable source for the extraction of collagen, and the ASC form, being the one with the greatest stability and advantage over PSC, of importance to our investigations, leads to a controlled digestion as in the case of peptide induction, and also in the development of natural products in the pharmaceutical and/or dermocosmetic industry. Evaluations of the triple helix structure by FT-IR, X-ray diffraction and UV–visible spectroscopy give similar results between the strains: red, gray, and F1, and between ages in the adult form F1 (15, 24, and 36 months of age). Consequently, the skin of tilapia in adult form is recommended sustainably for up to 24 months of age where the collagen is obtained with the use of acetic acid without enzymatic treatment.

Characterization of heavy metals in textile sludge with hydrothermal carbonization treatment

Presence of heavy metals in sludge can severely limit land application due to their bioavailability. The current work studied distribution and risk as well as leaching toxicity of heavy metals in textile sludge treated with hydrothermal carbonization (HTC) at different conditions. Treatment temperature and time can significantly affect characteristics of heavy metals in sludge. For the treatment at 220 °C and 3 h, the content of Cu, Cr, Mn, and Zn existed in form F1 + F2 (water soluble and bound to carbonate and Fe-Mn oxides) dropped by 4.7, 7.1, 8.8, and 7.3%, while the content of Cu, Cr, and Mn in form F4 (bound to quartz, feldspars, etc) increased by 12.9, 19.1, and 10.6%, respectively. This effectively lowered the bioavailability and leaching rate of heavy metals in sludge. Addition of weak alkaline Al(OH)3 could efficiently force the transformation of F1 to F4, possibly because of the increased pH value of sludge. HTC processing might be an effective way of fixing heavy metals in textile sludge.

Solution of Functional Equations Related to Elliptic Functions

Functional equations of the form f(x + y)g(x − y) = Σ =1 α j (x)β j (y) as well as of the form f1(x + z)f2(y + z)f3(x + y − z) = Σ =1 φ j (x, y)ψ j (z) are solved for unknown entire functions f, g,α j , β j : ℂ → ℂ and f1, f2, f3, ψ j : ℂ → ℂ, φ j : ℂ2 → ℂ in the cases of n = 3 and m = 4.

Star-Configurations in ℙn and the Weak-Lefschetz Property

We find that some union of two star-configurations in ℙ2 has generic Hilbert function. Applying the result, we prove that some Artinian quotients of a coordinate ring of a star-configuration in ℙn satisfy the weak-Lefschetz property. More precisely, let 𝕏 and 𝕐 be star-configurations in ℙ2 of type (2, s) and (2, s + 1) defined by forms F1,…, Fs, and G1,…, Gs, L, respectively, with deg(Fi) = deg(Gi) ≤2 for i = 1,…, s and s ≥ 3. If L is a general linear form in R = 𝕜[x0, x1, x2], then R/(I𝕏 + I𝕐) has the weak-Lefschetz property with a Lefschetz element L, which extends the result of [21].

Third-Generation Capsule Endoscopy Outperforms Second-Generation Based on the Detectability of Esophageal Varices.

Background and Aim. The third-generation capsule endoscopy (SB3) was shown to have better image resolution than that of SB2. The aim of this study was to compare SB2 and SB3 regarding detectability of esophageal varices (EVs). Methods. Seventy-six consecutive liver cirrhosis patients (42 men; mean age: 67 years) received SB3, and 99 (58 men; mean age, 67 years old) received SB2. All patients underwent esophagogastroduodenoscopy within 1 month prior to capsule endoscopy as gold standard for diagnosis. The diagnosis using SB3 and SB2 for EVs was evaluated regarding form (F0–F3), location (Ls, Lm, and Li), and the red color (RC) sign of EVs. Results. SB2 and SB3 did not significantly differ on overall diagnostic rates for EV. Sensitivity, specificity, positive predictive value, and negative predictive value of SB2/SB3 for EV diagnosis were, respectively, 65%/81%, 100%/100%, 100%/100%, and 70%/62%. However, the diagnostic rates for EV form F1 were 81% using SB3 and 52% using SB2 (P = 0.009). Further, the diagnostic rates for Ls/Lm varices were 79% using SB3 and 81% using SB2, and, for Li, varices were 84% using SB3 and 52% using SB2 (P = 0.02). Conclusion. SB3 significantly improved the detectability of EVs compared with SB2.

Isomorphic pairs of quadratic forms

Let k be a field of characteristic distinct from 2, V a finite dimensional vector space over k. We call two pairs of quadratic k-forms (f1,g1), (f2,g2) on V isomorphic if there exists an isomorphism s:V→V such that f2=f1∘s, g2=g1∘s. We prove that if f1+tg1≃f2+tg2 over k(t) and either the form f1+tg1 is anisotropic, or det(f1+tg1) is a squarefree polynomial of degree at least dimV−1, then the pairs (f1,g1) and (f2,g2) are isomorphic.

Syzygies of differentials of forms

Given a standard graded polynomial ring R=k[x1,…,xn] over a field k of characteristic zero and a graded k-subalgebra A=k[f1,…,fm]⊂R, one relates the module ΩA/k of Kähler k-differentials of A to the transposed Jacobian module D⊂∑i=1nRdxi of the forms f1,…,fm by means of a Leibniz mapΩA/k→D whose kernel is the torsion of ΩA/k. Letting D denote the R-submodule generated by the (image of the) syzygy module of ΩA/k and Z the syzygy module of D, there is a natural inclusion D⊂Z coming from the chain rule for composite derivatives. The main goal is to give means to test when this inclusion is an equality – in which case one says that the forms f1,…,fm are polarizable. One surveys some classes of subalgebras that are generated by polarizable forms. The problem has some curious connections with constructs of commutative algebra, such as the Jacobian ideal, the conormal module and its torsion, homological dimension in R and syzygies, complete intersections and Koszul algebras. Some of these connections trigger questions which have interest in their own.

The Structure of Low-Temperature Tridymite in Silica Refractories

The paper presents results of investigations aimed at determining the structure of low-temperature tridymite in refractory silica materials. In the initial phase of works, the literature and Internet databases containing structural data of crystalline compounds were reviewed in search of a model of tridymite structure which best fits the one forming in silica materials. The results of reviewing the models of polymorphous tridymite forms which are available in the world literature have been verified by Rietveld method, using a sample of silica material that contained tridymite without cristobalite and residual quartz. On the basis of analyses it was found that tridymite in refractory silica materials occurs in a form of two low-temperature polymorphous forms C1 and F1, instead of one form – Cc, as it was assumed before. The developed model has been used in a quantitative analysis (Rietvield method) of phase composition of several silica refractories.

Abelian functional equations, planar web geometry and polylogarithms

We study some abelian functional equations (Afe). They are equations in the F i ’s of the form F 1(U 1) + ... + F N (U N ) = 0 where the U i ’s are real rational functions in two variables. First we prove that the local measurable solutions are actually analytic and we characterize their components as solutions of linear differential equations constructed from the U i ’s. Then we propose two methods for solving Afe. Next we apply these methods to the explicit resolution of generalized versions of classical (inhomogeneous) Afe satisfied by low order polylogarithms. Interpreted in the framework of web geometry, these results give us new nonlinearizable maximal rank planar webs. Then we observe that there is a relation between these webs and certain configurations of points in $$\mathbb{C}\mathbb{P}^{2} ,$$ which leads us to define the notion of web associated to a configuration: these webs seem of high rank and could provide numerous new exceptional webs. Finally, we use the preceding results to show that, under weak regularity assumptions, the trilogarithm is the only function which satisfies the Spence–Kummer equation.

The metalla-Pinner reaction between Pt(iv)-bound nitriles and alkylated oxamic and oximic forms of hydroxamic acids

The nitrile ligands in the platinum(IV) complexes trans-[PtCl4(RCN)2] (R=Me, Et, CH2Ph) and cis/trans-[PtCl4(MeCN)(Me2SO)] are involved in a metalla-Pinner reaction with N-methylbenzohydroxamic acid (N-alkylated form of hydroxamic acid, hydroxamic form; F1), PhC(=O)N(Me)OH, to achieve the imino species [PtCl4[NH=C(R)ON(Me)C(=O)Ph]2 (1-3) and [PtCl4[NH=C(Me)ON(Me)C(=O)Ph](Me2SO)] (7), respectively. Treatment of trans-[PtCl4(RCN)2] (R=Me, Et) and cis/trans-[PtCl4(MeCN)(Me2SO)] with the O-alkylated form of a hydroxamic acid (hydroximic form), i.e. methyl 2,4,6-trimethylbenzohydroximate, 2,4,6-(Me3C6H2)C(OMe)=NOH (F2A), allows the isolation of [PtCl4[NH=C(R)ON=C(OMe)(2,4,6-Me3C6H2)]2] (5, 6) and [PtCl4[NH=C(Me)ON=C(OMe)(2,4,6-Me3C6H2)](Me2SO)] (8), correspondingly. In accord with the latter reaction, the coupling of nitriles in trans-[PtCl4(EtCN)2] with methyl benzohydroximate, PhC(OMe)=NOH (F2B), gives [PtCl4[NH=C(Et)ON=C(OMe)Ph]2] (4). The addition proceeds faster with the hydroximic F2, rather than with the hydroxamic form F1. The complexes 1-8 were characterized by C, H, N elemental analyses, FAB+ mass-spectrometry, IR, 1H and 13C[1H] NMR spectroscopies. The X-ray structure determinations have been performed for both hydroxamic and hydroximic complexes, i.e. 2 and 6, indicating that the imino ligands are mutually trans and they are in the E-configuration.

On Simultaneous Diagonal Inequalities

in integers x1, . . . , xs has been considered by a number of authors over the last quartercentury, starting with the work of Cook [9] and Pitman [13] on the case t = 2. More recently, Brudern and Cook [8] have shown that the above system is soluble provided that s is sufficiently large in terms of k and t and that the forms F1, . . . , Ft satisfy certain additional conditions. What has not yet been considered is the possibility of allowing the forms F1, . . . , Ft to have different degrees. However, with the recent work of Wooley [18], [20] on the corresponding problem for equations, the study of such systems has become a feasible prospect. In this paper we take a first step in that direction by studying the analogue of the system considered in [18] and [20]. Let λ1, . . . , λs and μ1, . . . , μs be real numbers such that for each i either λi or μi is nonzero. We define the forms F (x) = λ1x 3 1 + · · ·+ λsxs G(x) = μ1x 2 1 + · · ·+ μsxs

The force-free magnetosphere around an oblique rotator

The force-free magnetosphere around an oblique rotator

On the Representation of Numbers by the Direct Sums of Some Binary Quadratic Forms

The systems of bases are constructed for the spaces of cusp forms Sk(€0(3);u) (ki6), Sk(€0(7);u) (ki3) and Sk(€0(11);u) (k i 3). Formulas are obtained for the number of representation of a positive integer by the sum of k binary quadratic forms of the kind x 2 +x1x2+x 2 (6 i k i 17), of the kind x 2 +x1x2+2x 2 (3 i k i 11) and of the kind x 2 + x1x2 + 3x 2 (3 i k i 7). Let Fk denote a direct sum of k binary quadratic forms F1 with the same negative discriminant Aq (q is prime e 3 (mod 4)). The question of the rep- resentation of natural numbers by the direct sum of some binary quadratic forms was for the Þrst time considered by Petersson (1). In particular, he constructed the basis for the space of cusp forms Sk(€0(3);u) for an arbi- trary integer k i 6 using the Dedekind e-function and obtained formulas for the number of representation r(n;Fk) of the positive integer n by Fk when F1 = x 2 + x1x2 + x 2 and 2 i k i 6. Lomadze (2, 3) obtained formulas for r(n;Fk) when F1 = x 2 + x1x2 + x 2 (2 i k i 17) and F1 = x 2 + x1x2 + 2x 2 (2 i k i 11) and showed that they obey a fairly deÞnite law. To this end, he constructed the bases of the spaces Sk(€0(3);u) and Sk(€(7);u) apart for each k using a generalized multiple theta series. Merzlyakov (4) con- structed the basis of the space Sk(€0(3);u) for an arbitrary integer k i 6 and obtained formulas for r(n;Fk) when F1 = x 2 +x1x2 +x 2 (2 i k i 11). But these formulas do not obey any law. In the present paper, using the generalized multiple theta series, the systems of bases for the space of cusp forms Sk(€0(3);u) with an integer k i 6 and for the spaces Sk(€0(7);u) and Sk(€0(11);u) with an integer k i 3 are constructed. After that, these bases are used to obtain the formulas for r(n;Fk) when F1 = x 2 + x1x2 + x 2 (6 i k i 17), F1 = x 2 + x1x2 + 2x 2

On the Number of Representations of Positive Integers by a Direct Sum of Binary Quadratic Forms with Discriminant −23

Explicit exact formulas are obtained for the number of representations of positive integers by some direct sums of quadratic forms F1 = x 2 + x1x2 + 6x 2 and ˆ1 = 2x 2 + x1x2 + 3x 2 .

Efficient numerical method for finding the initial response of quantum processes to changes in the potential

The time-dependent Schrodinger equation is solved for an electron with potentials of the form f1(r) + f2(r, t)U(t), where U(t) is the unit step function. We use a product formulation, solving for F(r, t) where the wave function Ψ(r, t) = F(r, t)Φ(r)e-iEt/h in which Φ(r)e-iEt/h is the solution for t < 0. A simple implementation of the product formulation that does not use absorbing boundary conditions and is explicit, without using split operators, is applied in two examples. The first example pertains to resonances in tunneling with square barriers when the barrier height varies sinusoidally with time. The initial response of quantum tunneling to oscillations in the barrier height shows a buildup for oscillations at the resonance frequency and an off-resonance response that diminishes to approach the steady-state solution after several cycles. The second example is the initial response of a hydrogen atom to an intense applied electric field. In both examples the response is delayed, and for tunneling particles the delay is approximately equal to the semiclassical value for the duration of barrier traversal, defined as the time for traversing the inverted barrier. © 1996 John Wiley & Sons, Inc.

Algebraically invariant extensions of 𝜎-finite measures on Euclidean space

Let G G be a group of algebraic transformations of R n {{\mathbf {R}}^n} , i,e., the group of functions generated by bijections of R n {{\mathbf {R}}^n} of the form ( f 1 , … , f n ) ({f_1}, \ldots ,{f_n}) where each f i {f_i} is a rational function with coefficients in R {\mathbf {R}} in n n -variables. For a function γ : G → ( 0 , ∞ ) \gamma :G \to (0,\infty ) we say that a measure μ \mu on R n {{\mathbf {R}}^n} is γ \gamma -invariant when μ ( g [ A ] ) = γ ( g ) ⋅ μ ( A ) \mu (g[A]) = \gamma (g)\cdot \mu (A) for every g ∈ G g \in G and every μ \mu -measurable set A A . We will examine the question: "Does there exist a proper γ \gamma -invariant extension of μ ? \mu ? We prove that if μ \mu is σ \sigma -finite then such an extension exists whenever G G contains an uncountable subset of rational functions H ⊂ ( R ( X 1 , … , X n ) ) n H \subset {({\mathbf {R}}({X_1}, \ldots ,{X_n}))^n} such that μ ( { x : h 1 ( x ) = h 2 ( x ) } ) = 0 \mu (\{ x:{h_1}(x) = {h_2}(x)\} ) = 0 for all h 1 , h 2 ∈ H , h 1 ≠ h 2 {h_1},{h_2} \in H,{h_1} \ne {h_2} . In particular if G G is any uncountable subgroup of affine transformations of R n , γ ( g ) {{\bf {R}}^n},\gamma (g{\text {)}} is the absolute value of the Jacobian of g ∈ G g \in G and μ \mu is a γ \gamma -invariant extension of the n n -dimensional Lebesgue measure then μ \mu has a proper γ \gamma -invariant extension. The conclusion remains true for any σ \sigma -finite measure if G G is a transitive group of isometries of R n {{\mathbf {R}}^n} . An easy strengthening of this last corollary gives also an answer to a problem of Harazisvili.

Fundamental Frequencies of U-Tubes in Tube Bundles

The natural frequencies of U-tubes on multiple supports have been studied as a complement to the author’s previous work on the vibration of straight heat-exchanger tubes (reference [1]). A rapid estimation procedure for fundamental frequencies of tubes on symmetrical support spacings has been developed by expressing the frequency in the form f1≥12π•a1•1Ls2•EIμ where the square root contains the tube properties of Young’s modulus, cross-sectional second moment, and linear density; Ls is a characteristic length of support spacing; and a1 is a dimensionless number which is a strong function of the support geometry (as expressed by the ratio of bend radius to span lengths) and weak function of Poisson’s ratio and of tube axial moment of inertia. a1 has been plotted as a function of the ratio of the bend radius to the straight-span length, for usual values of Poisson’s ratio and small axial moment of inertia. The underlying assumptions for the use of such plots are examined and the theoretical basis for the statement of a lower bound is given, in order to show where the use of this method is applicable.

Dynamical scaling and the growth of diffusion-limited aggregates

The validity of dynamical scaling for the growth of aggregates in a bath of particles of concentration c0 is investigated. A length scale ξ∼c0−1/(d−D) is introduced which governs the crossover from a fractal of dimension D to compact aggregate behavior. For static scaling a variable x=R/ξ is introduced where R is the radius of gyration of the aggregate. The cluster size N is found to vary as N(c0,R)=(ξ/a)DxDg(x), where a is the particle size and the static structure function is well represented by g(x)=(A+Bx)d−D. To consider dynamical scaling the asymptotic dynamics are first examined. For R≪ξ the radius of gyration and cluster size are respectively found to grow with time as R(c0,t)∼(c0t)1/(2+D−d) and N(c0,t)∼(c0t)D/(2+D−d). While for R≫ξ one finds R(c0,t)∼c01/(d−D)t and N(c0,t)∼c01+d/(d−D)td. If a scaled time variable τ=Dft/ξ2 is introduced where Df is a microscopic diffusion constant, then the previous asymptotic results can be incorporated into dynamical scaling forms for R(c0,t) and N(c0,t). These are R(c0,t)/ξ=x=τ1/(2+D−d)f1(τ) where f1(τ) is well represented by the form f1(τ)=[(A1+B1τ)]1−1/(D+2−d); while N(c0,t)=(ξ/a)DτD/(2+D−d) f2(τ) where f2(τ) can be written [A2+B2τ]d−D/(2+D−d). Using these results all concentration c0 and time t regimes can be investigated. Dynamical scaling is found to hold within the errors inherent in computer simulations due to depletion effects of finite lattices.

Female Heliothis subflexa (Lepidoptera: Noctuidae) Sex Pheromone: Chemistry and Congeneric Comparisons 1

Chemical, chromatographic, and mass spectral analyses of the heptane extracts of excised female Heliothis subflexa ovipositors showed that the females produce 11 compounds that are analgous to known Heliothis spp. sex pheromones. Seven of these compounds have been previously identified from female H. virescens . In addition to these seven, female H. subflexa produce the (Z)-isomers of 7-, 9-, and 11-hexadecen-1-ol acetate and 1-hexadecanol acetate. The ratio of (Z)-11-hexadecenal to (2)-9- hexadecenal found in H. subflexa was significantly different from ratios found in other female Heliothis spp. Chromatographic analyses of the heptane extracts of the ovipositors of F1 females resulting from the H. subflexa (♀) × H. virescens (♂) cross and female progeny form F1 hybrid females backcrossed 34 times to H. virescens showed that they both produce a blend of pheromonal compounds that is qualitatively and quantitatively similar to compounds produced by H. virescens females. Field and laboratory behavioral studies indicated that the C16 acetates and the ratio of (Z)-11-hexadecenal: (Z)-9- hexadecenal secreted by female H. subflexa play significant roles in the maintentance of the specificity of the female H. subflexa pheromonal signal.