... ** The ** asymptpotics ** of ** ** eigenvalues ** ** and ** ** trace ** ** formula ** ** of ** ** operator ** ** associated ** ** with ** ** one ** ** singular ** problem Nigar M Aslanova Institute ** of ** Mathematics ** and ** Mechanics ** of ** NAS ** of ** Azerbaijan, ... ** the ** corresponding ** operator ** by L, so L = L0 + q In this article ** the ** ** asymptotics ** ** of ** ** eigenvalues ** ** and ** ** the ** ** trace ** ** formula ** ** of ** ** operator ** L will be studied In [1] ** the ** regularized traces ** of ** all orders ** of ** ... on ** the ** right hand side ** of ** (1.12) by F (z) Then for large |z| by Rouches’ theorem ** the ** number ** of ** ** the ** zeros ** of ** F (z) inside ** the ** contour equals ** the ** number ** of ** ** the ** zeros sin number ** of ** ** the ** zeros of...

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... ** The ** asymptpotics ** of ** ** eigenvalues ** ** and ** ** trace ** ** formula ** ** of ** ** operator ** ** associated ** ** with ** ** one ** ** singular ** problem Nigar M Aslanova Institute ** of ** Mathematics ** and ** Mechanics ** of ** NAS ** of ** Azerbaijan, ... ** the ** corresponding ** operator ** by L, so L = L0 + q In this article ** the ** ** asymptotics ** ** of ** ** eigenvalues ** ** and ** ** the ** ** trace ** ** formula ** ** of ** ** operator ** L will be studied In [1] ** the ** regularized traces ** of ** all orders ** of ** ... on ** the ** right hand side ** of ** (1.12) by F (z) Then for large |z| by Rouches’ theorem ** the ** number ** of ** ** the ** zeros ** of ** F (z) inside ** the ** contour equals ** the ** number ** of ** ** the ** zeros sin number ** of ** ** the ** zeros of...

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... deﬁning ** the ** Y -valued ** Ito ** process ** and ** establishing ** the ** ** Ito ** formula, in Secs ** and ** we construct ** the ** ** Ito ** integral of L(X, Y )-valued adapted ** random ** functions w.r.t an X-valued symmetric ** Gaussian ** ** random ** ... Mathematik, Bd 58 Leipzig, 1983 10 D H Thang, Vector symmetric ** random ** ** measures ** ** and ** ** random ** integrals, Theor Proba Appl 37 (1992) 526–533 11 D H Thang, On ** Ito ** stochastic integral ** with ** ** respect ** ** to ** ... technique similar ** to ** ** the ** proof of Lemma 3.2 ** and ** ** the ** ** Ito** s method in [6] we can deﬁne ** the ** ** random ** integral f dZ for ** random ** functions f ∈ M Theorem 3.4 Let X, Y be Banach spaces of type Then there exists...

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... reduce ** the ** evaluation ** of ** M(R+ ) to ** the ** evaluation ** of ** M(R+ ), which we have already done in Lemma M(R+ ) and M(R+ ) are related in ** the ** following way ** The ** tiles ** of ** ** the ** upper half R+ ** of ** ** the ** ** hexagon ** ** with ** ... rhombus ** tilings ** ** of ** ** the ** ** hexagon ** and perfect matchings ** of ** ** the ** ** hexagon** s ‘dual’ graph G (see Figure for ** the ** construction ** of ** ** the ** dual graph and Figure **3a ** for ** the ** correspondence between ** tilings ** and matchings) ... be ** a ** planar bipartite graph ** with ** reﬂective ** symmetry,** which splits into ** two ** parts after removal ** of ** ** the ** vertices ** of ** ** the ** ** symmetry ** axis We can clearly assume that ** the ** ** symmetry ** axis is ** the ** x-axis Label...

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... From ** a ** standard k **-packing ** of T , remove an **(a ** − 1)2 -grid ** and ** replace it ** with ** an ** a2** -grid ** covering ** the same area The result is ** a ** ** packing ** of k − **(a ** − 1)2 + ** a2** = k + **2a ** − = n ** equilateral ** triangles ... we get ** a ** ** packing ** of T by n2 ** equilateral ** triangles ** with ** sides of length n Such ** a ** conﬁguration is called an n2 -grid When T is ** a ** ** unit ** ** equilateral ** ** triangle,** the ** packing ** is ** a ** standard n2 **-packing ** ... ** equilateral ** ** triangle ** covered by smaller ** equilateral ** triangles Theorem 3.5 When n is odd ** and ** n ≥ 7, T1 (n) ≤ − n−3 Proof Consider ** a ** ** covering ** of ** a ** ** unit ** ** equilateral ** ** triangle ** T ** with ** ** equilateral ** triangles...

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... B., Tilings ** of ** ** the ** ** sphere ** ** with ** ** right ** ** triangles ** III: ** the ** ** asymptotically ** obtuse case, preprint [8] Dawson, R J MacG., and Doyle, B., Tilings ** of ** ** the ** ** sphere ** ** with ** ** right ** ** triangles ** IV: ** the ** ** asymptotically ** ... classify ** the ** ** right ** ** triangles ** that tile ** the ** ** sphere ** and have vertex signatures ** with ** ** asymptotically ** acute V1 , completing ** the ** classiﬁcation ** of ** ** right ** ** triangles ** that tile ** the ** ** sphere ** ** The ** main result ** The ** ... ** of ** ** the ** ** sphere ** ** with ** ** right ** ** triangles,** Discrete Geometry, Pure Appl Math 253, Marcel Dekker, (2003) 207–214 [6] Dawson, R J MacG., and Doyle, B., Tilings ** of ** ** the ** ** sphere ** ** with ** ** right ** ** triangles ** II: The...

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... vectors ** **** (0, ** ** 2, ** 8), ** **** (0, ** 5, 0), ** **** (1, ** 0, 10), ** **** (1, ** ** 3, ** ** 2), ** **(2, ** 1, 4), and (4, 0, 0) Its area is 1/80 that ** of ** ** the ** ** sphere ** Theorem ** The ** triangle T8 does not tile ** the ** ** sphere ** Proof: As before, we examine ** the ** ... B., Tilings ** of ** ** the ** ** sphere ** ** with ** ** right ** ** triangles ** III: ** the ** asymptotically obtuse families, preprint [7] Dawson, R J MacG., and Doyle, B., Tilings ** of ** ** the ** ** sphere ** ** with ** ** right ** ** triangles ** IV: ** the ** asymptotically ... c) ** of ** ** the ** ** triangles ** Tn Proposition ** The ** vertex vectors ** of ** Tn are precisely ** **** (0, ** ** 2, ** ** n),** ** **** (1, ** ** 3, ** ** 2), ** (4, 0, 0); and, if n is even, ** **** (1, ** 0, 3n−4 ) and **(2, ** 1, n ); and also, if n is even and ≤ 8, (0,...

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... triangle divide that ** of ** ** the ** ** sphere ** Proposition ** The ** only ** right ** ** triangles ** ** with ** V1 = (1, 2, 1) that tile ** the ** ** sphere ** are ** the ** (90◦ , 90◦ , 90◦ ) and (90◦ , 108◦ , 54◦ ) ** triangles ** Proof: ** The ** ** triangles ** mentioned ... (a, b, c), and ** the ** families ** of ** ** triangles ** ** with ** those β sources, as ** asymptotically ** acute, ** asymptotically ** ** right,** or ** asymptotically ** ** obtuse,** depending on ** the ** limiting value ** of ** β(n) (There is some overlap ... Tilings ** of ** ** the ** ** Sphere ** ** with ** Isosceles ** Triangles ** Disc and Comp Geom 30 (2003) 467-487 [3] Dawson, R J MacG., Single-split tilings ** of ** ** the ** ** sphere ** ** with ** ** right ** ** triangles ** in Discrete Geometry: In Honor of...

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... Antipsychotic ** and ** mood stabilizer efficacy ** and ** tolerability in pediatric ** and ** adult patients ** with ** ** bipolar ** I mania: ** A ** comparative ** analysis ** ** of ** acute, randomized, placebo-controlled trials ** Bipolar ** ... University ** of ** Calgary, Faculty ** of ** Medicine Subjects ** and ** Materials Data were available from clients who provided information to the company’s database from January 2001 (when the database was incorporated ... designs: on-off control ** of ** their tantrums ** and ** rages was demonstrated ** with ** this same formula [30] Further, in ** a ** database ** analysis ** ** of ** ** a ** large sample ** of ** 358 adults ** with ** ** bipolar ** ** disorder,** more than half...

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... strongly ** to ** Then lim Tn ** A ** tr = n→∞ ** A ** ** TRACE ** ** FORMULA ** ** AND ** ** APPLICATION ** Lemma 2.5 ([22, Corollary 3.8]) Let ** A,** B ∈ L(H) have the property that both AB ** and ** BA lie in the space of ** trace ** class operators, ... development Algebra i Analiz, 4(5):1–44, 1992 **A ** ** TRACE ** ** FORMULA ** ** AND ** ** APPLICATION ** 11 [4] J-F Bony, V Bruneau, ** and ** G Raikov Resonances ** and ** spectral shift function near the Landau levels Ann Inst Fourier ... Robert ** and ** X.P Wang Time-delay ** and ** spectral density for ** Stark ** Hamiltonians II Asymptotics of ** trace ** formulae Chinese Ann Math Ser B, 12(3):358–383, 1991 ** A ** Chinese summary appears in Chinese Ann Math...

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