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# Editorial backlog

1. Arabov M.K., Muhamadiev E.M., Nurov I.D., Sobirov H.I. SIGNS OF EXISTENCE OF LIMITING CYCLES IN THE SECOND ORDER DIFFERENTIAL EQUATIONS
Status:accepted
Abstract.
This article takes consideration into the discovery of the limiting cycles in neighborhood of stationary point of the non - smooth second order differential equation. New conditions are found for coefficient of the equation which ensure the existence of a limiting cycle. On the basis of taken results was made sector division in a plane. Created a package of programs for drawing phase portraits in appropriate sectors.
Date of submission: 19 Aprel 2016 г.
2. Klyachin A.A. On continuity and differentiability of the maximum values of functions
Status: reviewing
Abstract.
This article discusses the functions which are the maximum values of continuous functions on compact subsets of families. These functions are used, for example, in the study of the geometric structure of the equilibrium of different surfaces -- minimal surfaces, surfaces of constant mean curvature, etc. In this paper we find conditions under which such functions are continuous and differentiable.
Date of submission: 17 May 2016 г.
3. Oreshina M.N. Spectral decomposition of a normal operator in a real Hilbert space
Status: reviewing
Abstract.
Unbounded normal operators acting in a real Hilbert space are considered. In this article we carry over the classic results of the spectral theory to the case of such operators. The questions connected with the complexification and decomplexification of normal operators are discussed. We present two real variants of the spectral decomposition and the real functional calculus theorems for unbounded normal operators acting in a real Hilbert space.
Date of submission: 22 May 2016 г.
4. Hasanov Yu.H., Shakiriv I.A. BILATERAL EVALUATION OF THE NORM OF THE FOURIER OPERATOR
Status: reviewing
Abstract.
The lower and the upper uniform estimates of the Lebesgue constants of the classical Fourier operator are not final. A new and more simple integral representation received for it, then on the basis of which the problem of its top assessment is completely solved and the known lower assessment is improved.
Date of submission: 14 July 2016 г.
5. Salo T.M., Skaskiv O.B. The minimum modulus of gap power series and h-measure of exceptional sets
Status: reviewing
Abstract.
For an entire function of the form $f(z)=\sum_{k=0}^{+\infty}f_kz^{n_k}$,  where (nk) is a strictly increasing sequence of non negative integers, we establish conditions when the relations $M_f(r)=(1+o(1)) m_f(r),\quad M_f(r)=(1+o(1))\mu_f(r)$ is true as r→+∞ outside some set E such that $\text{\rm h-meas }(E)=\int_{E}\frac{dh(r)}{r}<+\infty$ where h(r) is positive continuous function increasing to +∞ on [0,+∞) with non-decreasing derivative, $M_f(r)=\max\{|f(z)|\colon |z|=r\},\m_f(r)=\min\{|f(z)|\colon |z|=r\},\ \mu_f(r)=\max\{|f_k|r^{n_k}\colon k\geq 0\}$ the maximum modulus, the minimum modulus and the maximum term of f, respectively.
Date of submission: 22 July 2016 г.
6. Mitrokhin S.I. Об исследовании дифференциального оператора с суммируемым потенциалом с разрывной весовой функцией
Status: reviewing
Abstract.
В работе предлагается новый подход к исследованию дифференциальных операторов с разрывной весовой функцией. Изучены спектральные свойства дифференциального оператора, заданного на конечном отрезке, с разделенными граничными условиями, с суммируемым потенциалом, с условиями <<сопряжения>> в точке разрыва весовой функции. При больших значениях спектрального параметра получена асимптотика фундаментальной системы решений соответствующего дифференциального уравнения, с помощью которой выведено уравнение на собственные значения изучаемого дифференциального оператора. Изучена индикаторная диаграмма и найдена асимптотика собственных значений исследуемого оператора.
Date of submission: 25 Avgust 2016 г.
7. Khushtova F.G. The first boundary value problem in the half-strip for a differential equation with Bessel operator and partial derivative of Riemann-Liouville
Status: reviewing
Abstract.
We study the first boundary value problem in the half-strip for a differential equation with Bessel operator and partial derivative of Riemann-Liouville. The representation of the solution in the case of the zero side condition found in terms of the integral transform with the Wright's function at the kernel. Uniqueness of the solution is proved in the class of functions that satisfy the analogue of Tikhonov's condition.
Date of submission: 16 September 2016 г.
8. Khrystiyanyn A.Y., Lukivska D.V. Quasi-elliptic functions
Status: reviewing
Abstract.
We investigate quasi-elliptic functions (i. e. certain generalization of elliptic functions). For this class of functions analogues of ℘, ζ and σσ Weierstrass functions are constructed and relation between quasi-elliptic and p-loxodromic functions is obtained.
Date of submission: 27 September 2016 г.
9. Krivosheyeva O.A. Invariant subspaces with spectrum of zero density
Status: reviewing
Abstract.
In the paper it is shown that every analytic solution of the homogeneous convolution equation with the characteristic function of a minimal exponential type in the domain of its existence is represented as the series of exponential polynomials which converges uniformly on compact subsets of this domain.
Date of submission: 31 October 2016 г.
10. Baizaev S., Rakhimova M.A. О некоторых функциональных уравнениях в пространствах Шварца и их приложениях
Status: reviewing
Abstract.
В статье изучается вопросы нетривиальной разрешимости функциональных уравнений вида         $(B+r^{2}E)u(r,\theta)=0$
где B− постоянная комплексная матрица порядка n, E− единичная матрица порядка nn, (r,θ)− полярные координаты в пространствах Шварца. Получены многообразия всех решений из указанных пространств и даны приложения результатов к задачам нахождения решений полиномиального роста ряда классов эллиптических систем и переопределенных систем.
Date of submission: 12 December 2016 г.
11. Garifullin R.N., Yamilov R.I. On the integrability of a discrete analogue of the Kaup–Kupershmidt equation
Status: accepted в т.0 №0
Abstract.
We study a new example of equation obtained as a result of a recent generalized symmetry classification of differential-difference equations defined on five points of one-dimensional lattice. We have established that in the continuous limit this new equation goes into the well-known Kaup–Kupershmidt equation. We have also proved its integrability by constructing an L−A pair and conservation laws. Moreover, we present a possibly new scheme for deriving conservation laws from L−A pairs.
Date of submission: 12 December 2016 г.
12. Ishkina Sh.Kh. Combinatorial bounds of overfitting for threshold classifiers
Status: reviewing
Abstract.
Tightening generalization bounds is a fundamental objective of statistical learning theory. However, accurate and computationally efficient bounds are still unknown even for very simple cases. In~this paper, we consider one-dimensional threshold decision rules. We~use the framework of combinatorial theory of overfitting, which is based on a single probabilistic assumption that all partitions of a~set of objects into an observed training sample and a~hidden test sample can occur with equal probability. We~propose a~polynomial algorithm for computing both probability of overfitting and complete cross-validation.
Date of submission: 21 December 2016 г.
13. Galakhov E.I., Salieva O.A. Условия отсутствия решений некоторых неравенств и систем с функциональными параметрами и сингулярными коэффициентами на границе
Status: accepted в т.0 №0
Abstract.
We obtain sufficient conditions for nonexistence of positive solutions of some nonlinear elliptic inequalities and systems that contain the p(x)p(x)-Laplace operators with variable power exponents and coefficients possessing singularity on the boundary.
Date of submission: 28 December 2016 г.
14. Belaidi B., Saidani M. On The Growth of Solutions of Some Higher Order Linear Differential Equations With Meromorphic Coefficients
Status: reviewing
Abstract.
In this paper, we study the growth of meromorphic solutions of the differential equation ....
Date of submission: 06 January 2017 г.
15. Khan N.U., Usman T. CERTAIN GENERATING FUNCTIONS OF HERMITE-BERNOULLI-LEGENDRE POLYNOMIALS
Status: reviewing
Abstract.
In this paper, we introduce a new class of generating functions for Hermite- Bernoulli-Legendre polynomials and investigate certain implicit summation formulas by using different analytical means and applying generating function. We also introduce bi- lateral series associated with the newly-introduced generating function by appropriately specializing a number of known or new partly unilateral and partly bilateral generating functions.
Date of submission: 27 January 2017 г.
16. Shukla I. Simultaneous Quadruple Series Equations Involving Konhauser Biorthogonal Polynomials
Status: reviewing
Abstract.
Spencer and Fano [11] used the biorthogonal polynomials (for the case of k = 2) in carrying out calculations involving penetration of gamma rays through matter. In the present paper an exact solution of simultaneous quadruple series equations involving Konhauser – biorthogonal polynomials of first kind of different indices is obtained by multiplying factor technique due to Noble [13]. This technique has been modified by Thakare [12] to solve dual series equations involving orthogonal polynomials which led to disprove a possible conjecture of Askey [6] that dual series equations involving Jacobi polynomials of different indices cannot be solved. In this paper the solution of simultaneous quadruple series equations involving generalized Laguerre polynomials also have been discussed in a particular case.
Date of submission: 28 January 2017 г.
17. Salimov R.B. The study of the behavior of the singular integral with the Hilbert kernel near a point of the new weak continuity of the density
Status: reviewing
Abstract.
We study the behavior of a singular integral with the Hilbert kernel near a fixed point, where the density vanishes as a negative exponent of the logarithm module of the distance from the fixed point to a variable one at the new conditions.
Date of submission: 08 February 2017 г.
18. Murtazina S.A., Fazlytdinov M.F., Shevtsova T.V., Yumagulov M.G. Operator methods for computing Lyapunov quantities in the problems on the local bifurcations of dynamical systems
Status: reviewing
Abstract.
This work proposes new formulas to calculate Lyapunov quantities in problems of major scripts of local bifurcations of dynamical systems. Considered a dynamical system, described by differential equations, and point mappings. The proposed formulas are obtained from the general operational method of the study of local bifurcations and do not require transition to normal forms and use theorems about the central diversity.
Date of submission: 06 Mart 2017 г.
19. Kononova A.A. On measures generating orthogonal polynomials with similar asymptotic behavior of the ratio near infinity
Status: reviewing
Abstract.
We consider perturbations of orthogonality measure of the system of polynomials that do not change (in some sense) the asymptotical behavior of the ratio of corresponding orthogonal polynomials. The support of the measure consists of the finite number of Jordan curves and may also contain a finite number of mass-points out of the polynomial convex hull of the support of the absolute continuous part of the measure. The problem is a generalization of the problem of compactness of the perturbation of Jacobi operator generated by the perturbation of its spectral measure. A condition, necessary (or necessary and sufficient under some additional restriction) for the stability of the asymptotical behavior of the corresponding orthogonal polynomials is found.
Date of submission: 09 Mart 2017 г.
20. Halder S., Sahoo P. UNIQUENESS OF MEROMORPHIC FUNCTIONS CONCERNING DIFFERENTIAL POLYNOMIALS SHARING A SET
Status: reviewing
Abstract.
In this paper, we investigate the uniqueness of meromorphic functions whose certain nonlinear differential polynomials share a set of values with finite weight and obtain some results that generalize and improve the recent results due to H.Y. Xu [J. Computational Analysis and Applications, 16(2014), 942-954].
Date of submission: 27 Mart 2017 г.
21. Nuraliev F.A., Hayotov A.R., Shadimetov Kh.M. Optimal quadrature formulas with derivatives in Sobolev space
Status: reviewing
Abstract.
In the present paper the problem of construction of optimal quadrature formulas in the sense of Sard in the space L(m)2(0,1)L2(m)(0,1) is considered. Here the quadrature sum consists of values of the integrand at nodes and values of the first and the third derivatives of the integrand at the end points of the integration interval. The coefficients of optimal quadrature formulas are found and the norm of the optimal error functional is calculated for arbitrary natural number N≥m−3N≥m−3 and for any m≥4m≥4 using S.L. Sobolev method which is based on the discrete analogue of the differential operator d2m/dx2md2m/dx2m. In particular, for m=4, 5m=4, 5 optimality of the classical Euler-Maclaurin quadrature formula is obtained. Starting from m=6m=6 new optimal quadrature formulas are obtained. At the end of this work some numerical results are presented.
Date of submission: 30 Mart 2017 г.
22. Krivosheyev A.S., Kuzhaev A.F. On a Leontiev-Levin theorem
Status: reviewing
Abstract.
In this work we investigate relations between different densities of a positive sequence and variables related to them. The results are used to generalise classical proposition on completeness of exponential monomials systems with positive exponents in a convex domain when these exponents haven't density. That proposition was obtained by A.Leontiev and B.Levin independently. This generalisation is given for sufficiently wide class of convex domains, for example, with vertical or horizontal symmetry axes.
Date of submission: 30 Mart 2017 г.
23. Kalyakin L.A. Adiabatic approximation for a model of resonance capture
Status: accepted в т.9 №2
Abstract.
Две модельные задачи о захвате в резонанс анализируются методом усреднения, который приводит к адиабатическому приближению в главном члене асимптотики. Основной целью является приближенное описание области захвата в резонанс. Эта область зависит от дополнительного параметра, входящего в уравнения. Демонстрируется непригодность адиабатического приближения, когда область захвата становится узкой. В этом случае требуется значительное изменение метода усреднения.
Date of submission: 02 Aprel 2017 г.
24. Rathod A. Nevanlinna’s Five-values Theorems for Algebroid Functions
Status: reviewing
Abstract.
By using the second main theorem of the algebroid function, we inves- tigate the problem on two algebroid functions partially sharing five or more values and that improve and generalize the previous results given by Xuan and Gao.
Date of submission: 06 Aprel 2017 г.
25. Khashimov A.R. ЭНЕРГЕТИЧЕСКИЕ ОЦЕНКИ ДЛЯ РЕШЕНИЙ КРАЕВЫХ ЗАДАЧ УРАВНЕНИЯ ТРЕТЬЕГО ПОРЯДКА С КРАТНЫМИ ХАРАКТЕРИСТИКАМИ
Status: reviewing
Abstract.
В статье рассмотрена первая краевая задача для уравнения третьего порядка с кратными характеристиками. Для обобщенного решения уравнения установлена энергетические оценки типа аналога принципа Сен-Венана. С помощью этой оценки выявлены наибольшее класс единственности решений краевых задач в зависимости от геометрических характеристик области.
Date of submission: 07 Aprel 2017 г.
26. Shabat A.B., Эфендиев M.H. О приложениях формулы Фаа-ди-Бруно
Status: reviewing
Abstract.
Two new modifications of the classical Faa-di-Bruno formula are constructed and applications of the obtained formulas in the theory of integrability of nonlinear partial differential equations
Date of submission: 08 Aprel 2017 г.
27. Stepanova I.V. Symmetries of heat and mass transfer equations in viscous fluids
Status: reviewing
Abstract.
The paper is devoted to description of results of application of classical Lie-Ovsyannikov theory to study of heat and mass transfer equations in viscous liquids. Group properties of equations of convective and molecular heat and mass transfer are under consideration. The author analyzed 124 papers and monographs be concerning to mentioned problem.
Date of submission: 10 Aprel 2017 г.
28. Singh D.K., Singh P. Wright Function Associated with Fractional Calculus
Status: reviewing
Abstract.
With the marvelous light of fractional calculus, the study of this paper is based on Wright function and Raizada polynomial.
Date of submission: 17 Aprel 2017 г.
29. Zhukova N.I. The influence of stratification on groups of conformal transformations of pseudo-Riemannian orbifolds
Status: reviewing
Abstract.
Groups of conformal transformations of nn-dimensional pseudo-Riemannian orbifolds $({\mathcal N},g)$ are investigated for n≥3. It is shown that a conformal pseudo-Riemannian geometry is induced on each stratum of that orbifold. For k∈{0,1}∪{3,...,n−1} exact estimates of dimensions of the conformal transformation groups of nn-dimensional pseudo-Rieman\-ni\-an orbifolds admitting k-dimensional strata with essential conformal transformation groups are obtained.
Date of submission: 08 May 2017 г.
30. Kachalov V.I. Pseudoholomorphic functions and their application
Status: reviewing
Abstract.
An analysis of asymptotic methods for solving singularly perturbed problems shows that the solutions obtained by means of these solutions depend in two ways on the small parameter: regularly and singularly. This dependence is particularly clearly demonstrated by the method of regularization of S.A.\,Lomov. Moreover, the regularized solu\-tions of singularly perturbed equations can converge in the usual sense. In this connection, it became necessary to study a special class of functions, pseudoholomorphic functions. This is a very important part of the analy\-sis, it is intended to substantiate the main points of the so-called analytic theory of singular perturbations. On the other hand, the relevance of the theory in question is also dictated by the fact that pseudoholomorphic functions, unlike holomorphic functions, are determined when the condi\-tions of the implicit function theorem are violated.
Date of submission: 16 May 2017 г.
31. Trynin A.Yu. Uniform convergence of sync-approximations on the functional class
Status: reviewing
Abstract.
We obtain a uniform convergence inside the interval $(0, \ pi)$ of the values of the Lagrange-Sturm-Liouville operators for functions from the class. The class is defined by means of one-way moduli of continuity and change
Date of submission: 18 May 2017 г.
32. Bobodzhanov A.A., Safonov V.F. Regularized asymptotics of solutions of integro-differential partial differential equations with rapidly varying kernels
Status: reviewing
Abstract.
The method of Lomov regularization is generalized to partial differential equations with integral operators, the kernel of which contains a rapidly varying exponential factor. The case when the upper limit of the integral operator coincides with the differentiation variable is investigated. For such problems, an algorithm for constructing regularized asymptotics develops. In contrast to the work of Imanaliev M.I., where for analogous problems with slowly varying nuclei only the limit transition is investigated when the small parameter tends to zero, an asymptotic solution of any order (with respect to the parameter) is constructed here.
Date of submission: 19 May 2017 г.
33. Kulaev R.Ch., Shabat A.B. Some properties for Jost functions of a Schr\"odinger equation with potential, which is a distribution
Status: reviewing
Abstract.
Работа посвящена задаче кардинального расширения пространства потенциалов в обратной задаче рассеяния для линейного уравнения Шредингера на числовой прямой. Рассматривается оператор Шредингера с потенциалом из пространства обобщенных функций. Это расширение включает в себя не только потенциалы типа δ-функции, но и экзотику типа функции Кантора. На этом пути устанавливаются условия существования и единственности решений Йоста, изучаются их свойства.
Date of submission: 23 May 2017 г.
34. Fedotov A.I. Hermite-Fejer polynomials as an approximate solution of the singular integro-differential equations
Status: reviewing
Abstract.
An approximate method for solving singular integro-differen-tial equations in periodic case is justified. An approximate solution is sought in a form of Hermite-Fejer polynomials. The convergence of the method is proved and the errors are estimated.
Date of submission: 24 May 2017 г.
35. Konechnaya N.N., Mirzoev K.A. Asymptotics of solutions of a class of linear differential equations
Status: accepted в т.0 №0
Abstract.
In this paper the main term of asymptotics of some fundamen- tal system of solutions of a class of linear differential equations of arbitrary order τy=λyτy=λy at infinity is found, where λλ is a fixed complex number. In this case, the conditions imposed on the coefficients of the differential equation are not related to their smoothness, but only provide a certain power-law growth of the coefficients at infinity. The results obtained are applied to the spectral analysis of the corresponding singular differential operators, including in the case when the expression ττ is a product of two differential expressions.
Date of submission: 26 May 2017 г.
36. Kachalov V.I. On the holomorphic regularization of strongly nonlinear singularly perturbed problems
Status: reviewing
Abstract.
The method of holomorphic regularization, which is a logical extension of the Lomov method, allows one to construct solutions of nonlinear singularly perturbed initial problems in the form of series converging in the usual sense in powers of a small parameter. The method itself is based on a generalization of the Poincare decomposition theorem: in the regular case, solutions depend holomorphically on a small parameter, in the singular case the first integrals inherit this dependence.
Date of submission: 29 May 2017 г.
37. Zikkos E. . A Taylor-Dirichlet series with no singularities on its abscissa of convergence
Status: reviewing
Аннотация.
In this paper it is proved that given any non-negative real number dd, there exists a Taylor-Dirichlet series of the form $\sum_{n=1}^{\infty} \left(\sum_{k=0}^{\mu_n-1}c_{n,k} z^k\right) e^{\lambda_n z},\quad c_{n,k}\in \mathbb{C}$
with no singularities on its abscissa of convergence, such that its associated multiplicity-sequence $\Lambda=\{\lambda_n,\mu_n\}_{n=1}^{\infty}$ has the following properties: (1) the terms of Λ are positive real numbers and uniformly separated, $(\inf_{n\in\mathbb{N}}(\lambda_{n+1}-\lambda_n)>0)$ (2) Λ has density equal to d $\left(\lim_{t\to\infty}\frac{\sum_{\lambda_n\le t}\mu_n}{t}=d<\infty\right)$  (3) the multiplicities of the terms of Λ are unbounded,  $(\mu_n\not=O(1))$  he proof is based on the fact that for this sequence Λ its Krivosheev characteristic $S_{\Lambda}$. We remark that when $\mu_n=1$ for all  $n\in\mathbb{N}$ the result is false by a well known theorem of Polya.
38. Vinnitskii B.V., Sharan V.L., Sheparovych I.B. About some interpolation problem in the class of functions of exponential type in a half-plane
Status: reviewing
Abstract.
The conditions of solvability of the interpolation problem $f(\lambda_{k} )=d_{k}$ are found in the class of functions of exponential type. This results are applied to research of some problem of the function's splitting.
Date of submission: 01 June 2017 г.
39. Klimentov S.B. About Isomorphism of Some Integro-differential Operators
Status: accepted в т.0 №0
Abstract.
In this work representations of <> for solutions of the linear general elliptic system of the first order in the unit circle are considered. The isomorphism of corresponding operators is established in Banach spaces $C^k_\alpha(\overline D)$ and $W^k_p(\overline D)$ k≥1, 0<α<1, p>2.. These results develop and supplement B.V. Boyarsky's works where representa\-tions of <> where obtained. Also this work supplements author’s results on representations of <> with more difficult operators.
Date of submission: 02 June 2017 г.
40. Braichev G.G. Two-sided estimates of the relative growth of functions and their derivatives
Status: reviewing
Abstract.
В работе дано расширенное изложение доклада автора, подготовленного для международной математической конференции по теории функций, посвященной 100-летию чл.-корр. АН СССР А.\,Ф.~Леонтьева. Установлены равномерные двусторонние оценки относительного роста производных двух функций на основе информации об относительном росте самих функций. Рассмотрены примеры применения полученных результатов к исследованию поведения целых функций.
Date of submission: 03 June 2017 г.
41. Eliseev A.G., Ratnikova T.A., Shaposhnikova D.A. Singularly perturbed systems of Volterra integral equations of the second kind
Status: reviewing
Abstract.
In this paper we study the initialization problem for singularly perturbed systems of Volterra integral equations of the second kind. A general approach to the construction of the regularized asymptotics of singularly perturbed integro-differential equations is described in the papers of S.A. Lomov and his progeny. In this paper, we study the problem of the passage to the limit and the estimate of the remainder in the initialization problem for the original singularly perturbed system of Volterra integral equations. Calculations are given, the class of right-hand sides is determined by the example of one singularly perturbed Volterra integral equation of the second kind.
Date of submission: 07 June 2017 г.
42. Shaygardanov Yu.Z. Asymptotics of the solution of an elliptic boundary value problem in neighborhood of the line of the outer touch characteristics of the limit equation
Status: reviewing
Abstract.
The article deals with a boundary value problem for elliptic equations of second order with small parameter ε>0ε>0 in a bounded region $Q\subset \Bbb R^3$ with smooth boundary $\Gamma: \, \varepsilon Au-\partial /\partial x_3u=f(x), \, u|_{\Gamma}=0$.Under the assumption that the set of points external touch characteristics of the limit equation of the boundary is a smooth closed curve the asymptotic expansion of the solution in a neighborhood of this curve was received.
Date of submission: 09 June 2017 г.
43. Rahman S. ON PSEUDO-SLANT SUBMANIFOLDS OF NEARLY LORENTZIAN PARA-SASAKIAN MANIFOLDS
Status: reviewing
Abstract.
The object of the present paper is to study pseudo slant submanifolds of nearly Lorentzian para-Sasakian manifolds. The necessary and sufficient conditions on a totally umbilical proper-slant submanifolds are worked out and obtain some interesting results regarding such submanifolds. The integrability condition of the distribution of pseudo -slant submanifolds of nearly Lorentzian para-Sasakian manifolds are also discussed.
Date of submission: 10 June 2017 г.
44. Avkhadiev F.G. Estimates of Hardy-Rellich's constants for polyharmonic operators and their generalizations
Status: reviewing
Abstract.
We prove lower estimates for functionals defined as maximal constants in Hardy and Rellich type inequalities for polyharmonic opera-tors and their generalizations on domains of the Euclidean space. In proofs we essentially use an integral identity by Ladyzhenskaya and its generalization to the case of polyharmonic operators.
Date of submission: 14 June 2017 г.
45. Poluboyarova N.M. On the instability of extremals of the potential energy functional
Status: reviewing
Abstract.
In this paper study has been done on problem of stability and instability of the potential energy functional. By stability we mean the sign-definiteness of the second variation. The expression for the second variation of the functional is calculated. With the capacitive method it is obtain to make the feature of instability extremals. Proved that stability parabolic extremals are planes. We have written the equation of extremals and the second variation of the functional for n-dimensional surfaces of revolution.
Date of submission: 18 June 2017 г.
46. Sukhov A.B. Levi-flat world: a survey of local theory
Status: reviewing
Abstract.
This expository paper concerns local properties of Levi-flat real analytic manifolds with singularities.
Date of submission: 19 June 2017 г.
47. Novokshenov V.Yu. Discrete integrable equations and special functions
Status: reviewing
Abstract.
A generic scheme based on the Riemann-Hilbert (RH) problem theory is proposed for description of classical special functions satisfying difference equations. Those functions comprise gamma- and zeta functions, as well as orthogonal polynomials with corresponding recurrence relations. It ia shown that all difference equations are the compatibility conditions of certain Lax pair coming from the RH problem. On the other hand, the solutions of RH problem provide analitic properties of the special functions as well as find new functions pretending for the name of special ones. Those are solutions of discrete Painlev\'e equations.
Date of submission: 21 June 2017 г.
48. Imomov A.A., Meyliyev A.H., Тухтаев E.E. Об уточнении одной предельной теоремы из теории критических ветвящихся процессов с дискретным временем
Status: reviewing
Abstract.
We consider critical Galton-Watson branching process. The generating function of distribution of one particle has the infinite second moment, but its tail is regularly varying in sense of Karamata. The speed rate of convergence of distribution of process on nonzero trajectories to the limit law is established.
Date of submission: 22 June 2017 г.
49. Berdellima A. ON A CONJECTURE OF KHABIBULLIN ABOUT A PAIR OF INTEGRAL INEQUALITIES
Status: reviewing
Abstract.
It is known that in general Khabibullin’s conjecture is not true. Sharipov [8] constructed a counterexample when n=2 and α=2. In this paper we develop a method of how to construct a counterexample for the more general case n>2 and α>1/2.
Date of submission: 24 June 2017 г.
50. Gaisin A.M., Gaisina G.A. Estimates of the growth and decrease of functions in the theorem of Macintyre--Efgrafov's type
Status: reviewing
Abstract.
The method of estimating the sum of the Dirichlet series on the system of segments [α,α+δ](α>0 is fixed, δ>0 --- any) based on inequalities for extremal functions from some non-quasianalytic Carleman’s class is given in this paper. An estimate of the rate of tends to zero function in an example of the Macintyre--Evgrafov’s type is proved.
Date of submission: 27 June 2017 г.
51. Isaev K.P., Trunov K.V., Youlmukhametov R.S. Representation of functions in locally convex subspaces of $A^\infty (D)$ by series of exponentials
Status: reviewing
Abstract.
Let $D$ be a bounded convex domain on the complex plane. We consider some locally convex subspaces in the space of analytic functions in $D$ that are infinitely differentiable in $\overline D$. It is proved that such functions are representable by series of exponentials converging in the topology of these subspaces.
Date of submission: 27 June 2017 г.
52. Garayev M. ., Guediri H. ., Sadrawi H. . New Characterizations of Bloch spaces, Bers-type and Zygmund-type spaces and Related Questions
Статус: на рецензии
Аннотация.
We give in terms of Berezin symbols new characterizations of\ the Bloch spaces $\mathcal{B}$ and $\mathcal{B}_{0}$ , Bers-type and the Zygmund-type spaces of analytic functions on the unit disc $\mathbb{D}$ of the complex plane $\mathbb{C}$. Moreover, we discuss some properties of Toeplitz operators on the Bergman space $L_{a}^{2}(\mathbb{D}).$. A new characterization of\ some function space with variable exponents is also given.
53. Kopachevsky N.D., Tsvetkov D.O. Малые движения идеальной стратифицированной жидкости со свободной поверхностью, полностью покрытой крошеным льдом
Status: reviewing
Abstract.
Let a rigid immovable vessel be partially filled with an ideal incompressible stratified fluid. We assume that in an equilibrium state the density of a fluid is a function of the vertical variable $x_3$, i.e., $\rho_0=\rho_0(x_3)$. In this case the gravitational field with constant acceleration $\rho_0=\rho_0(x_3)$ acts on the fluid, here $g>0$ and $\vec e_3$ is unit vector of the vertical axis $Ox_3$, which is directed opposite to $\vec g$. Let Ω be the domain filled with a fluid in equilibrium state, $S$ be rigid wall of the vessel adherent to the fluid, Γ be a free surface completely covered with a crumbled ice. The initial boundary value problem is reduced to the Cauchy problem

$$\begin{equation*} \begin{split} &\mathcal A \frac{d^2x}{dt^2} + \mathcal C x = f(t), \quad x(0)=x^0, \quad x^{'}(0)=x^1, \\ &0<< \mathcal A= \mathcal A^{*} \in \mathcal L(\mathcal H), \quad 0 \leq \mathcal C = \mathcal C^{*} \in \mathcal L(\mathcal H). \end{split} \end{equation*}$$

in some Hilbert space $\mathcal H$. The theorem on strong solvability of initial boundary value problem is proved.
Date of submission: 29 June 2017 г.

54. Baskakov A.G., Uskova N.B. Linear differential operator with an involution as generator of group of operators
Status: reviewing
Abstract.
We consider mixed problem for first order differential equation with involution. By the method of similar operators, he differential operator, which defined by this differential equation, is transformed in orthogonal direct sum of operators. By the main theorem we construct group of operators and we describe the weak solutions of this problem. We use this theorem for Fourier method.
Date of submission: 29 June 2017 г.
55. Ivanova О.А., Melikhov S.N., Melikhov Yu.N. On the commutants of differetiation and translation operators in weighted spaces of entire functions
Status: reviewing
Abstract.
We describe continuous linear operators which commute with the system of partial di?eretiation operators and translation operators in a countable inductive limits of weighted Frechet spaces of entire functions of several complex varibles.
Date of submission: 30 June 2017 г.
56. Shamoyan F.A.. АНАЛИТИЧЕСКИЕ ФУНКЦИИ С ГЛАДКИМ МОДУЛЕМ ГРАНИЧНЫХ ЗНАЧЕНИЙ
Status: reviewing
Abstract.
Пусть $f$ --- аналитическая функция в единичном круге $D$, непрерывная вплоть до его границы $\Gamma, f(z) \neq 0, z \in D$. Предположим $f$ имеет на $Γ$ модуль непрерывности $\omega(|f|,\delta)$. В статье устанавливается оценка $\omega(f,\delta) \leq A\omega(|f|, \sqrt{\delta})$, где $A$ --- некоторое неотрицательное число. Также устанавливается точность данной оценки.                                       Date of submission: 10 July 2017 г.
57. Andriyan S.M., Kroyan A.K., Khachatryan K.A. On Solvability of a Class of Nonlinear Integral Equations in $p$ -adic String Theory
Status: reviewing
Abstract.
In this paper a class of nonlinear integral equations, which has direct application in the $p$ -adic string theory, is studied. The existence of a nontrivial continuous odd and bounded solution on the whole axis is proved. With some additional conditions, the uniqueness of the constructed solution in the certain class of continuous functions is established as well.
Date of submission: 15 July 2017 г.
58. Musin I.Kh. On a Hilbert space of entire functions
Status: reviewing
Abstract.
A weighted Hilbert space of entire functions of nn variables is considered in the paper. The weight function is a convex function on  ${\mathbb C}^n$ depending on modules of variables and growing at infinity faster than a∥z∥ for each a>0. The strong dual of this space is described in terms of the Laplace transform of functionals.
Date of submission: 17 July 2017 г.
59. RamReddy T., Shalini D., Vamshee Krishna D. THIRD ORDER HANKEL DETERMINANT FOR STAR LIKE FUNCTIONS OF ORDER α
Status: reviewing
Abstract.
The objective of this paper is to obtain best possible upper bound to the third Hankel determinant for the class of starlike functions of order αα (0≤α<1), using Toeplitz determinants.
Date of submission: 17 July 2017 г.
60. Haslinger F. Pauli operators and the $\overline\partial$-Neumann problem
Status: reviewing
Abstract.
We apply methods from complex analysis, in particular the \ovprt\ovprt-Neumann operator, to investigate spectral properties of Pauli operators.
Date of submission: 17 July 2017 г.
61. Muravnik A.B. On Qualitative Properties of Solutions of Quasilinear Parabolic Equations Admitting Degenerations at Infinity
Status: reviewing
Abstract.
We consider the Cauchy problem для for quasilinear parabolic equations of the kind $\rho(x)u_t=\Delta u + g(u)|\nabla u|^2$, where the positive coefficient ρ admits a degeneration at infinity, while the coefficient ggeither is a continuous function or admits power singularities such that the power does not exceed one. The long-time behavior of (classical) solutions of the specified problem is investigated.
Date of submission: 21 July 2017 г.
62. Zhapsarbayeva L.K., Kanguzhin B.E., Konyrkulzhayeva M.N. Selfadjoint restriction of the maximal operator on the graph
Status: reviewing
Abstract.
The paper is devoted to the linear differential operators defined on graphs. In this paper Lagrange formula for second-order differential operator on a graph with Kirchhoff conditions at its internal vertices is derived. The conditions for a selfadjointness of the differential operator on the graph are established.
Date of submission: 24 July 2017 г.
63. Ehrgashev T.G. Third Double-Layer Potential for a Generalized Bi-Axially Symmetric Helmholtz Equation
Status: reviewing
Abstract.
The double-layer potential plays an important role in solving boundary value problems for elliptic equations, and in the study of which for a certain equation, the properties of the fundamental solutions of the given equation are used. All the fundamental solutions of the generalized bi-axially symmetric Helmholtz equation were known, and only for the first one was constructed the theory of potential. Here, in this paper, we aim at constructing theory of double-layer potentials corresponding to the third fundamental solution. By using some properties of one of Appell's hypergeometric functions in two variables, we prove limiting theorems and derive integral equations concerning a denseness of double-layer potentials.
Date of submission: 31 July 2017 г.
64. Bandura A.I., Skaskiv O.B. Exhaustion by balls and entire functions of bounded LL-index in joint variables
Status: reviewing
Abstract.
We prove criteria of boundedness of $\mathbf{L}$-index in joint variables which describe local behavior of partial derivatives on sphere in $\mathbb{C}^n$. Some obtained results are new even for entire functions of bounded index in joint variables, i. e. $\mathbf{L}(z)\equiv 1, because we used an exhaustion of$\mathbb{C}^n$by balls instead an exhaustion of$\mathbb{C}^n$by polydiscs. Date of submission: 08 Avgust 2017 г. 65. Baskakov A.G., Dikarev E.E. Spectral Theory of Functions in Research of Partial Differential Operators Status: reviewing Abstract. Spectral properties of differential operators with constant coefficients defined on subspaces of space of bounded continuous functions are studied. Necessary and sufficient conditions of invertibility are obtained under condition of regularity at the infinity (ellipticity type conditions) of polynomial which describes such operators. Spectrum, images and kernels are described. Conditions of compactness of resolvent of differential operators are obtained. Main results are obtained by methods of harmonic analysis and spectral theory of Banach modules. Date of submission: 10 Avgust 2017 г. 66. Davletov D.B., Davletov O.B., Kozhevnikov D.V., Sadykova R.R. Asymptotics of the eigenvalue of the Steklov type boundary value problem for the Laplace operator in a half-strip with a small hole. Status: reviewing Abstract. The paper considers the problem of the Steklov type for the Laplace operator in a half-strip containing a small cavity. On lateral boundaries and on the boundary of a small hole conditions are set Dirichlet, and on the basis of the half-strip -- the Steklov spectral condition. An asymptotic expansion of the eigenvalue to within a small parameter ("Hole diameter"), converging to a simple eigenvalue the limiting problem (without a small cavity) is constructed and rigorously proved. Date of submission: 11 Avgust 2017 г. 67. Saks R.S. Операторы градиент дивергенции и ротор в пространстве$\mathbf{L}_{2}(G)$. Status: reviewing Abstract. Автор изучает структуру пространства$\mathbf{L}_{2}(G)$вектор-функций, квадратично интегрируемых по области$G$трехмерного пространства, и роль операторов градиента дивергенции и ротора в построении базисов в его подпространствах${\mathcal{{A}}}$и${\mathcal{{B}}}$. Доказана самосопряженность продолжения$\mathcal{N}_d$оператора$\nabla\mathrm{div}$в подпространство$\mathcal{A} _ {\gamma}\subset {\mathcal{{A}}}$и базисность системы его собственных функций. Выписаны явные формулы решения спектральной задачи в шаре и условия разложимости вектор-функции в ряд Фурье по собственным функциям градиента дивергенции. Изучена разрешимость краевой задачи:$\nabla\mathrm{div}\,\mathbf{u}+\lambda\,\mathbf{u}=\mathbf{f}$в$G$,$(\mathbf {n}\cdot\mathbf {u})|_{\Gamma}=g$в пространствах Соболева$\mathbf{H}^{s}(G)$порядка s≥0 и в подпространствах${\mathcal{{A}}}$и${\mathcal{{B}}}$. В случае шара эта задача решена полностью: выписаны явные формулы (ряды) для операторов$\mathcal{N}_d$и$\mathcal{N}_d^{-1}$и шкала пространств$\mathcal{A}^p_ {\gamma}\subset \mathbf{H}^{2(p-1)}$, которые они отображают. Попутно изложены аналогичные результаты для оператора ротор и его симметричного расширения$S$в$\mathcal{B}\$. Эта работа есть продолжение исследований автора .,\cite{saUMJ13}-- \cite{twim15} .
Date of submission: 18 Avgust 2017 г.
68. Rubinshtein A.I. On the Bary-Stechkin Theorem
Status: reviewing
Abstract.
We concider the problem on the modulus of continuity for the analogue conjugate functions in the case of functions in the case of functions defined on the diadic group. It is shown that for this case no analogue a Bary--Stechkin theorem.
Date of submission: 18 Avgust 2017 г.


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 << Previous Issue Bulletin of the Korean Mathematical Society (Vol. 27, No. 2) Next Issue >>
 Bull. Korean Math. Soc. 1990 Vol. 27, No. 2, 99—206
 On the consecutive primes Yeu-Hua Too Bull. Korean Math. Soc. 1990 Vol. 27, No. 2, 99—101
 On matrix normed spaces Dong-Yun Shin, Sa-Ge Lee, Chang Ho Byun, and Sang Og Kim Bull. Korean Math. Soc. 1990 Vol. 27, No. 2, 103—112
 A note on $S$-sets in a fixed group hyung Soo Song Bull. Korean Math. Soc. 1990 Vol. 27, No. 2, 113—120
 Properties of the generalized evaluation subgroup of a topological pair Moo Ha Woo Bull. Korean Math. Soc. 1990 Vol. 27, No. 2, 121—126
 An example of partially ordered Shardovsky space Jong Sook Bae and Nak So Sung Bull. Korean Math. Soc. 1990 Vol. 27, No. 2, 127—131
 On the contact conformal curvature tensor Jang Chun Jeong, Jae Don Lee, Ge Hwan Oh, and Jin Suk Pak Bull. Korean Math. Soc. 1990 Vol. 27, No. 2, 133—142
 Hausdorff $\phi$-strong uniqueness Sung Ho Park Bull. Korean Math. Soc. 1990 Vol. 27, No. 2, 143—150
 Necessary and sufficient conditions for the existence of solutions to operators equations Sehie Park Bull. Korean Math. Soc. 1990 Vol. 27, No. 2, 151—155
 Linear abstract Cauchy problem associated with an exponentially bounded $C$-semigroup in a Banach space* Ki Sik Ha, Jai Heui Kim, and Jong Kyu Kim Bull. Korean Math. Soc. 1990 Vol. 27, No. 2, 157—164
 An accelerating monotone iterative technique for nonlinear boundary value problems S. K. Chung Bull. Korean Math. Soc. 1990 Vol. 27, No. 2, 165—172
 Dual operator algebras and $C_0$-operators Il Bong Jung and Yong Chan Kim Bull. Korean Math. Soc. 1990 Vol. 27, No. 2, 173—182
 A characterization of Dirichlet sets Hung Hwan Lee Bull. Korean Math. Soc. 1990 Vol. 27, No. 2, 183—188
 Ultraproducts of locally convex spaces Si Ho Kang Bull. Korean Math. Soc. 1990 Vol. 27, No. 2, 189—196
 The continuity of derivations and module homomorphisms Kil-Woung Jun, Young-Whan Lee, and Dal-Won Park Bull. Korean Math. Soc. 1990 Vol. 27, No. 2, 197—206

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Simple Pendulum

A simple pendulum is a small mass suspended from a string that oscillates under the force of gravity. Here, the gravitational force provides the restoring torque required for oscillation. In a simple pendulum, the string or connecting rod is assumed to be either massless or small compared to the suspended mass that it can be neglected.

We will use the concept that total energy is constant for a body that is executing Simple Harmonic Motion to derive the expression for the period. Also we will neglect any air resistance and assume that the pendulum can oscillate forever after being disturbed.

Consider a pendulum with mass, m suspended from a length l from a pivot as shown below.

The force of gravity on the bob of mass m is mg. When the pendulum is at an angle θ to the vertical as shown, the gravitational force can be split into two components parallel and perpendicular to the string’s direction.

We can see that the restoring torque is $$mg~sin~θ$$ .

If the pendulum is travelling at a speed $$v$$ and has an acceleration $$a$$ ,

The Kinetic energy is

$$KE$$ = $$\frac{1}{2} mv^2$$

Let us consider the lowest point of the pendulum to have zero potential energy. When the pendulum travels upward, the potential energy increases. It can be seen from the diagram that the bob moves vertically a distance $$l~-~lcosϑ$$ as the bob swings by an angle $$θ$$ .

Thus the potential energy is:

$$PE$$ = $$mg(l~-~l ~cos~θ)$$

The total energy is a constant. Thus

$$Total~ Energy$$ = $$E$$ = $$KE + PE$$

$$E$$ = $$\frac{1}{2}~ mv^2 + mg(l~-~l ~cos~θ)$$

Let us differentiate with respect to time (the left hand side is zero since differential of a constant is zero)

$$\frac{dE}{dt}$$ = $$0$$ = $$\frac{d}{dt} ~\left(\frac{1}{2}~ mv^2 + mg(l~-~l ~cos~θ)\right)$$

$$0$$ = $$\frac{d}{dt}~ (\frac{1}{2}~ mv^2) + \frac{d}{dt} (mgl – mgl~ cos~θ)$$

Thus we have,

$$0$$ = $$mv. {dv}{dt} + 0~ -~ mgl(-sin~θ).\frac{dθ}{dt}$$

$$τ$$ = $$\frac{dθ}{dt}$$

And

$$v$$ = $$lω$$

(Linear velocity is the angular velocity (τ) times radius for motion along a circular arc)

$$0$$ = $$ml^2 τ.{dτ}{dt} + mgl sin~θ.τ$$

$$0$$ = $$ml ~\frac{d^2~ θ}{dt^2} + mg sin~θ$$

For small oscillations i.e. $$θ$$ being small, we can approximate

$$sin~θ ≈ θ$$

$$-mg~θ$$ = $$ml ~\frac{d^2 θ}{dt^2}$$

$$\frac{d^2 θ}{dt^2}$$ = $$-\frac{g}{l}~ θ$$

This is the equation of Simple Harmonic Motion whose general form is:

$$\frac{d^2 x}{dt^2}$$ = $$-ω^2 x$$

Where ω is the circular frequency for an oscillation time period T,

$$ω$$ = $$\frac{2π}{T}$$

Thus the angular frequency for a simple pendulum by comparing the two differential equations is:

$$ω$$ = $$\frac{2π}{T}$$ = $$√{\frac{g}{l}}$$

In other words,

$$Time~ period$$ = $$T$$ = $$2π√{\frac{l}{g}}$$


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# Question #2015759: op. man.

Question: Determine the maximum possible flow through the following network from source to sink, given the arc flow capacities. Note that some of the branches are undirected (the flow can go both ways), while others are directed.

Solution: The solution consists of 48 words (1 page)
Deliverables: Word Document

0


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# Total derivative: elementary question

1. Feb 3, 2010

### mnb96

Hello,
Given a function $$f(x_1,\ldots,x_n)$$, the total derivative for $x_1$ is:

$$\frac{df}{dx_1}=\frac{\partial f}{\partial x_1}+\frac{\partial f}{\partial x_2}\frac{dx_2}{dx_1}+\ldots+\frac{\partial f}{\partial x_n}\frac{dx_n}{dx_1}$$

Now, if the $x_i$ are just variables, can we say that:

$$\frac{dx_i}{dx_j}=\delta_{ij}$$

?...If so, then the total derivative would equal the partial derivative:

$$\frac{df}{dx_1}=\frac{\partial f}{\partial x_1}$$

is that correct?

2. Feb 3, 2010

### Tac-Tics

Correct. Total derivatives are only used when the parameters of a function are not independent. This is really a bastardization of the term function, since all formal parameters are independent by definition. But we let the physicists do it anyway, because they're so swell.

It makes more sense, I think, if you parametrize all the variables over one parameter (say t):

$$f(x_1(t), x_2(t), \dots, x_n(t))$$

And then take the regular Calc I derivative with respect to t

$$\frac{d}{dt} f(x_1(t), x_2(t), ..., x_n(t)) = \frac{\partial f}{\partial x_1}\frac{dx_1}{dt} + \frac{\partial f}{\partial x_2}\frac{dx_2}{dt} + \dots + \frac{\partial f}{\partial x_n}\frac{dx_n}{dt}$$

3. Feb 3, 2010

### mnb96

Thanks a lot!
I have just got something which is still bugging my mind.

$$x=f(u,v)$$

and

$$v=g(x)$$

we want to compute the total derivative $$\frac{df}{du}$$.
This would yield:

$$\frac{\partial f}{\partial u}+\frac{\partial f}{\partial v}\frac{dv}{du}$$

Now, what about $$\frac{dv}{du}$$. Is it supposed to be zero???

4. Feb 3, 2010

### Tac-Tics

It occurs to me that I made a slight mistake in my explanation above. You can see on the RHS of my last equation, I'm missing a $$\frac{\partial f}{\partial x}$$ term, so it's not exactly equal to the total derivative except in the case where x1(t) = t.

You can see how tricky these problems can get!

In your second example, I'm not sure the best way to interpret it. Do you realize you're using the variable x twice?

Anyway, the moral in the story is these problems are often written with a particular physical set up in mind, and the details are only summarized by the equations used. Numbers get mistaken for FUNCTIONS of real numbers, and it is impossible to attack the problem in an algebraic way. You have to reason about the problem and use techniques like the total derivative which are specially designed to minimize the negative effects of the notation used in calculus.

5. Feb 3, 2010

### mnb96

I can't see any mistake in your explanation!
Why is that?
Also wikipedia http://en.wikipedia.org/wiki/Total_derivative reports the same example you mentioned: it is at the and of the section "Differentiation with indirect dependencies"

I agree with you that these things are tricky!

I suspect the tricky example I made arises when you deal with curvilinear coordinates.
Some time ago I had a problem which I thought I understood, but I am afraid I still don't.

I hope you don't mind if I ask you to take a look at it. But in case you have no time, thanks anyway!

6. Feb 3, 2010

### Ben Niehoff

This is correct. The assumption here is that the x_i might have hidden dependencies between them. This sort of situation occurs in, e.g. thermodynamics.

This is not true at all. If you use partial derivatives,

$$\frac{\partial x_i}{\partial x_j} = \delta_{ij}$$

then it is true. But if the x_i have hidden dependencies between them, then the total derivative will not be trivial.

Not at all, that makes no sense. I hope the above explanation helps you see why.

7. Feb 3, 2010

### mnb96

I think there is a fundamental gap in my knowledge.
Perhaps I don't fully understand the meaning of $dx_2/dx_1$ when they are two variables.

Let's make a very simple example:

$$f(x,y)=x+y$$

the total derivative is:

$$\frac{df}{dx}=\frac{\partial f}{\partial x}+\frac{\partial f}{\partial y}\frac{dy}{dx} = 1 + \frac{dy}{dx}$$

Now, what do we replace $$\frac{dy}{dx}$$ with???
This notation assumes that y is a function of x, which is not true!

Last edited: Feb 3, 2010


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Sharedwww / ribet / index.htmlOpen in CoCalc
Author: William A. Stein
Antwerp Modular Forms Conference Photo 1972

# Antwerp Modular Forms 1972 Conference Photo

(scanned by William Stein)

 Larger Numbered Scan Small Scan (no numbers) Medium Size Scan (1.1MB) 600DPI Scan (2.8MB)

Scroll below the photo for number key and some interesting remarks about the conference.

(Thanks to Ken Ribet for sending me the photo and Ribet and Patterson for identifying many people in it.)

## Name Key

 If you think you know the identity of somebody not identified below, please send me an email ([email protected]). 1 may be Roger Cook, later a Professor at Sheffield, England, and now retired I believe Alain Robert Frans Oort ? M.Van Tieghem (Leuven) Georges Poitou Kenku (Nigeria) ? Balwant Singh ?? ? David Tingley Jean-Marc Fontaine Zuckerman ? Oliver Atkin M. Van Tilborgh (techn asst W.Kuyk) Armand Borel B. Ditters (Nijmegen, Netherlands) Nina Vleugels (secretary W. Kuyk) M. Hazewinkel (Amsterdam) Jutta Kneser Martin Kneser John Loxton J.H.H. Chalk of Toronto ?? Miles Reid Hendrik W. Lenstra H.P.F. Swinnerton-Dyer Heinz Helling (Bielefeld, Germany) Horst G\"unter Zimmer (Saarbr\"ucken, Germany) G. Schreier (Louvain) Jean Cougnard ? R.W. van der Waall (Amsterdam) Jean-Pierre Serre ? R. Parimala ?? Charlie Toll (now at NSA) ? Marc Best (Technical University of Twente, The Netherlands) J. Den Boer (Nijmegen) ? Hostess Rita Willem Kuyk (one of the editors of the LNM volumes) Labute Oskar Herrman (Heidelberg, Germany) Lucile Begueri (Mme Poitou) L. Vanhamme (Brussel) Steenbrink (Nijmegen) Mme Ap\'ery Roger Ap\'ery ? ? Ken Ribet David Masser Mike Razar Milne St\"ohr (Berlin, Germany) H. Kisilevsky Albrecht Fr\"ohlich A.E. Brouwer (Amsterdam) Goro Shimura ? Max Deuring ? ? Meinhard Peters (M\"unster, Germany) J. Bogo (Assistant to Kuyk) F. van Oystaeyen (Assistant to Kuyk) Eli Donkar Kuang-Yen Shih ? ? Robert Rankin Samuel J. Patterson J\"urgen Elstrodt Alan Beardon Erik Lippa (Ph.D. from University of Michigan, 1971) ? ? Enrico Bombieri?? Cees de Vreugd D.C. van Leijenhorst (now at the Catholic University of Nijmegen, The Netherlands) Nick Katz Pierre Deligne Notes: There is an unidentified woman with no number to Serre's left between 34 and 37 horizontally and between 35 and 54 vertically. She might be Michele Lynch. According to Elstrod: "According to my list of names (written down from memory immediately after the conference), the "unidentified woman to the left of Serre between 34 and 37 horizontally" is Yvette Amice (albeit with a question-mark)). Near the top I mysteriously put an extra 84 on a person whose face is down. This is not Deligne, who is in the lower right.

## Interesting E-mail From Participants

 Ken Ribet: These are the famous volumes "Modular Forms in 1 Variable I-IV". Some of the numbers are 349, 350, 476; these are volumes II, III, IV. I'm talking about Lecture Notes in Math. There are later volumes (601 and 627) from a conference in Bonn in 1976." The food and wine were great. My only regret is that I didn't really start learning about wine until after the conference. I have no idea what we were drinking, but I remember that it was great. A lot of the tables had the idea that you could get a second bottle of wine as soon as possible; they (we) did this by draining the first bottle as soon as we sat down. I don't know whether there was a finite supply of wine for the whole group at each meal -- but this must have been the case. I really got drunk the last night. They served suckling pig and had Alsatian wine with it. Samual J. Patterson: "There are people who were there that I do not recognise on the photo (e.g. Serre, Eichler, Cassels, Kenku, Petersson, Birch, Bombieri, Alan Baker, Curt Meyer, Leutbecher, Oliver Atkin -in a totally random order). Perhaps one or another will come to me. The conference was financed very generously by NATO. This was in the middle of the Vietnam war and there was a lot of discomfort and discussion about this fact. One of the uses to which the NATO money was put was there there were about three attractive hostesses in bright yellow dresses. One of these is Nr. 42; I have no idea what her name was. Also the food was excellent - thanks in part to NATO. Some were graduate students who did not continue with mathematics - generally there were a large number of graduate students like myself there - also financed by NATO, to the disgust of the likes of Grothendieck, Langlands and Godement. However it was my first conference and nothing has really lived up to it since then. I profited enormously from it." Bill Casselman: Other people missing in the picture are Andy Ogg, Jacques Velu and Robert Langlands. I think there were more than 3 of the hostesses in yellow dresses, who were called "the daffodils" by many. Eli N. Donkar: From: "Eli N. Donkar" To: [email protected] Date: 04/16/04 11:49 am  William, My wife stumbled across this photo on the web because I was (correctly) identified in it. I was a graduate student at Princeton at the time, studying with Shimura. After graduate school I continued in number theory briefly at Johns Hopkins, but ended up switching careers doing financial forecasting for the Social Security Administration for the last 27 years. I have not had time to inspect the picture carefully, but I should be able to come up with a few names of my contemporaries who attended. The one correction I did notice that you should make is for number 70 (sitting on my left). His full name is Kuang-Yen Shih. Shih was my classmate at Princeton and Ph.D. brother (i.e. also a student of Shimura). Boy did it bring back memories. One comment to add to Patterson's memories is that not only was the food out of this world, but the wine was superb (I think Serre and the French contingent had a hand in the selections) and it was very free flowing. A virtual nirvana for a poor graduate student of the time. Although the food and wine were great, the mathematics was even better and we certainly spent many more hours in the day talking and doing mathematics than we spent eating and drinking wine. From: Jurgen Elstrodt My former student Olav Richter (Dept. of Math. University of Denton, Texas) drew my attention to the Antwerp Modular Forms 1972 Conference Photo in the Internet. I took part in the conference and I am No. 75 on the Photo (probably S.J. Patterson identified me). My photo of the participants is slightly different from the one in the internet, hence some people can better be identified on my photo. In addition to the names on your list, I can identify the following people: 7. Kenku (Nigeria) 18. Ditters (Netherlands) 19. Hostess Braaksma (?) 20. M. Hazewinkel (Amsterdam) 28 Heinz Helling (Bielefeld, Germany) 29. Horst G�nter Zimmer (Saarbr�cken, Germany) 30 may be G�nter K�hler (W�rzburg), but could also be Wulf-Dieter Geyer (Erlangen). I cannot remember to have seen K�hler on the conference. 42. Hostess Rita 45. Oskar Herrman (Heidelberg, Germany) 46. I could not identify this woman when I wrote down my list. According to my list of names (written down from memory immediately after the conferece), the "unidentified woman to the left of Serre between 34 and 37 horizontally" is Yvette Amice (albeit with a question-mark). 57. St�hr (Berlin, Germany) 66. Meinhard Peters (M�nster, Germany) 67. + 68. Assistants to Kuyk  Several participants are not on the photo, e.g. Eichler, Cassels, Petersson, Bombieri, Langlands, Curt Meyer, Leutbecher (Munich), Bauermeister (G�ttingen), Lange, Cohn. I still have a list of the participants who lived in the students dormitory (with the names of the participants and their rooms). I hope my list will help you to fill some gaps. Best wishes J�rgen Elstrodt Subject: 1972 Summer School on Modular Functions From: Willem Kuijk To: [email protected] Date: 05/26/04 09:01 am  Dear Dr Stein, Thank you for bringing back to my memory all that concerns and concerned the 1972 Antwerp Summer School, that was a most perhaps the most important professional event in my life. It may interest you to know that, before I did a final offer for such a conference to Serre at ICM Nice, I tried out to have several conferences with Serre as star speaker. From that time on we have had a friendly colleagial relationship. Prior to 1967/68 at the IAS where I assisted him writing a "bouquin" as he called it ("abelian l-adic representations and elliptic curves") he was at my colloquium at Mc Gill U. and I ventured a few years later in Nice to propose that a big conference on Number Theory should be conducive to changing math. by the turn of the century. To my surprise and pleasure he agreed and he consulted all the leading French mathematicians (Cartier, Poitou, etc but I am sure not Grothendieck) as to what should be a central topic intermediate between all the then extant math fields, so that chances for interdisciplinary results including math. applications (sic) would in due time be most likely. Out came the Modular Function bid, and Serre left me totally free to organize it in my way, which implied I conceived of a friendly environment with relaxation and work and good food and wine. In vino veritas. The work involved cost me more than three years of intensive work including the proceedings. We agreed to not thank Nato too explicitly and friendly in the Proceedings, because we agreed that funding such enterprises should not be a task for a military defense organization. (Nato stopped funding the affair two weeks before the conference began because of protests in the Dutch and French press etc.etc.). I am thankful to you to have made efforts to bring out elements that resulted from the summer school. For so many years now I bear within me a treasure of knowledge regarding the school, but nobody asked me about it; clearly, because I did not belong to the coryphees of the subject. After the school I did what I should do on account of my nature: study physiology, brain physiology and heart ph., foundations of math, educational math. and what have you that I was interested in. I also did exams in philosophy, wrote a 1977 book on "Complementarity in Mathematics", thereby coining the term CIMath. I had only 10 PH.D. students in math among which Jan Denef in Leuven. It was H.W.Lenstra (also a student that I shared with Oort) who wrote in 1994: (forgive me my pride even if L's pronouncement were not true: "without K no 'Antwerp', and without "Antwerp" no solution to the Fermat problem". He referred to the fact that I was called by the Be government when in Princeton to set up a math. dpt at the new university here. I accepted on the condition that they would support and fund a conference in my style on a central math. topic , etc... And I got what I wanted. Seven years ago upon retiring I got a massive aneurysmal brain bleeding and, after two most splendid near death experiences (in which members of the SuSchool and McGill staff + students figured prominently) during a week of coma, I got up and (limpingly though) running again. Ever since I study 6 hours a day, but cannot very easily go to places. Here is the c.v. I wrote. it is somewhere on the net where my name is Kuijk-Gruenbauer to honour my mother and and the memory of her family. Willem Kuijk-Gr�nbauer (former name Willem Kuyk) studied Pure Mathematics and Theoretical Physics in Amsterdam and gained a D.Sc. degree with a doctoral thesis in Number Theory and Galois Theory, only with the wish to return to working in Applied Mathematics and the foundations of Physics later on. When employed by the Mathematical Centre in Amsterdam, Canadian job propositions guided him by way of a postdoctorate fellowship into an Associate professorship at McGill University in Montreal, that lasted eight years and ended with the decision to accept an offer by the Belgian Government to set up a department of Mathematics at the new Antwerp University. This decision jibed with his earlier proposal, made during a visiting membership at the IAS (Princeton) to J-P. Serre of the Coll�ge de France, to bring all the world's leading Number theorists and their Ph.D. students together into one Summer School, and then see what happens in the way of a rejuvenation of the number theoretic branches of Mathematics. This three week School by the name 1972 Antwerp Summer School on Modular Functions in one variable (with four Springer LNM proceedings) marked the beginning of the Mathematics department at Antwerp University, and, more importantly, it has been a stimulus to many old schools in Number Theory as well as to the establishment of new ones; to the point where it has been conducive to the solution of a series of long-standing conjectures such as de Fermat's Last Theorem, the Weil conjectures etc. Early retirement from the pure Mathematics area finally freed Kuyk's way to take up his interest in regulation systems in Biology (body heat and heart control) and the foundation of Physics (wave equations and questions regarding physical dimensions and the quantum theory of motion). Thanks for listening, Willem From: Rob van der Waall To: [email protected] Date: 06/02/04 07:08 am  Dear William Stein, Yes, I visited the congress on Modular Forms at Antwerp in 1972, and I am identified correctly on the photograph. I can add two persons wanted: 39: Marc Best (an expert in algebraic coding theory; among other things, he did classify all the perfect codes over any alphabeth not only prime powers (up to one small isolated case,later done by Hu if I am not mistaKen); affiliated with the Technical University of Twente, The Netherlands; died by his own hand about 16 years ago); 82: D.C. van Leijenhorst (now at the Catholic University of Nijmegen,The Netherlands) 38 is not Bill Casselmann, as far as I can remember; William, I do not remember your face, unfortunately, About the conference: it was a phantastic experience to me, and to all attendencies I suppose. Best regards, Robert van der Waall ( now at The University of Amsterdam).


text:
# Initialising and Accessing Elements in Tuple

## Introduction

Tuples are basically a data type in python. These tuples are an ordered collection of elements of different data types. Furthermore, we represent them by writing the elements inside the parenthesis separated by commas. We can also define tuples as lists that we cannot change. Therefore, we can call them immutable tuples. Moreover, we access elements by using the index starting from zero. We can create a tuple in various ways. Let us learn about initialising and accessing elements in tuple.

## Initialising and Accessing Elements in Tuple

It is not necessary to initialise an empty tuple. Furthermore, we can directly insert elements into the tuple. Moreover, we can access elements in a tuple just like we do in lists and strings that is, by using the indexing method. Let us learn about initialising and accessing elements in tuple in detail through various examples.

## Initialising the Tuples

We can initialise a tuple in many ways. We can either initialise a tuple empty or with the elements. There are basically two ways to initialise empty tuples. We can do so either by writing just empty parenthesis () or by using the tuple() constructor. They are as follows:

### Example 1: Initialising empty tuple using empty parenthesis()

>>>emptytup1 = ()

>>>emptytup1

()

### Example 2: Initialising empty tuple using tuple() constructor

>>>emptytup2 = tuple()

>>>emptytup2

()

Besides, we can also create tuples directly with the elements inside them. Furthermore, it is not necessary to initialise a tuple before inserting the elements. There are several ways to initialise or create a tuple. Let us have a look at them.

## Initialising Tuples with Elements

We can create or initialise a tuple in various ways by using different types of elements. Since a tuple can contain all elements of the same data type as well as of mixed data types as well. Therefore, we have multiple ways of creating tuples. Let us look at few examples of creating tuples in python.

Example 1: Tuple with integers as elements

>>>tup = (22, 33, 5, 23)

>>>tup

(22, 33, 5, 23)

As you can see here, we have only integer type of elements in the tuple.

Example 2: Tuple with mixed data type

>>>tup2 = ('hi', 11, 45.7)

>>>tup2

('hi', 11, 45.7)

As you can see here, we have 'string', 'integer', and 'float' data type elements all in the same tuple tup2.

Besides, we can use the tuple() constructor also to create a tuple. We can do this as follows:

>>>tup = tuple ((22, 45, 23, 78, 6.89))

>>>tup

(22, 45, 23, 78, 6.89)

## Accessing Elements in a Tuple

We can access elements in a tuple in the same way as we do in lists and strings. Hence, we can access elements simply by indexing and slicing. Furthermore, the indexing is simple as in lists, starting from the index zero. Let us look at a few examples of accessing the elements in a tuple.

>>>tup1 = (10, 3, 4, 22, 1)

# for accessing the first element of the tuple

>>>tup1[0]

10

# accessing the third element of the tuple

>>>tup1[2]

4

>>>tup1[10]

# gives error as the index is only up to 4

IndexError: tuple index out of range

>>>tup1[1+3]

# the expression inside the square brackets results in an integer index 4. Hence, we get the element at the 4th index.

1

Therefore, in this way, we can perform the function of initialising and accessing elements in tuple. In further topics, we will study in more detail indexing and slicing. That will further help us to understand the concept of initialising and accessing elements in tuple.

Q1. State true or false:

We cannot initialise an empty tuple.

A1. False, we can initialise the tuple as empty i.e., with no element inside.

Q2. How do we perform accessing of elements in a tuple?

a) slicing

b) indexing

c) initializing

d) both a and b

A2. d) both a and b

Q3. Which error is displayed if we try to give an index that is not available in a tuple?

A3. If the index is not present python gives the IndexError.

IndexError: tuple index out of range

Q4. >>> tup [4-1]

We can access an element by giving an index as an expression as shown above. State true or false.

A4. True, we can give an expression inside square brackets if it results in an integer that is valid for the tuple index.

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text:
IGV custom genome/Biostarhandbook Chapter 16
0
0
Entering edit mode
4.4 years ago
Ricky ▴ 50

Dear all,

being new to NGS data analysis and I am currently trying to work myself through the Biostarhandbook. Although it appears to be a rather simple task I am struggeling with generating a custom genome for ebola in IGV. I followed the instructions (code) given in chapter 16 of the handbook to generate the annotation (gff) file from the ebola genbank file. However, although IGV displays the genome sequence it does not show the annotation although the file is provided during the create .genome process. Also manually loading the gff file into the genome does not change anything. My fasta genome file looks like so:
cat 1976.fa | head >AF086833 CGGACACACAAAAAGAAAGAAGAATTTTTAGGATCTTTTGTGTGCGAATAACTATGAGGAAGATTAATAA TTTTCCTCTCATTGAAATTTATATCGGAATTTAAATTGAAATTGTTACTGTAATCACACCTGGTTTGTTT CAGAGCCACATCACAAAGATAGAGAACAACCTAGGTCTCCGAAGGGAGCAAGGGCATCAGTGTGCTCAGT TGAAAATCCCTTGTCAACACCTAGGTCTTATCACATCACAAGTTCCACCTCAGACTCTGCAGGGTGATCC AACAACCTTAATAGAAACATTATTGTTAAAGGACAGCATTAGTTCACAGTCAAACAAGCAAGATTGAGAA TTAACCTTGGTTTTGAACTTGAACACTTAGGGGATTGAAGATTCAACAACCCTAAAGCTTGGGGTAAAAC ATTGGAAATAGTTAAAAGACAAATTGCTCGGAATCACAAAATTCCGAGTATGGATTCTCGTCCTCAGAAA ATCTGGATGGCGCCGAGTCTCACTGAATCTGACATGGATTACCACAAGATCTTGACAGCAGGTCTGTCCG TTCAACAGGGGATTGTTCGGCAAAGAGTCATCCCAGTGTATCAAGTAAACAATCTTGAAGAAATTTGCC  and my gff file like so: ##gff-version 3 AF086833 EMBL gene 56 3026 . + . ID=AF086833.3;gene=NP AF086833 EMBL gene 3032 4407 . + . ID=AF086833.9;gene=VP35 AF086833 EMBL gene 4390 5894 . + . ID=AF086833.13;gene=VP40 AF086833 EMBL gene 5900 8305 . + . ID=AF086833.20;gene=GP AF086833 EMBL gene 8288 9740 . + . ID=AF086833.34;gene=VP30 AF086833 EMBL gene 9885 11518 . + . ID=AF086833.41;gene=VP24;note=putative AF086833 EMBL gene 11501 18282 . + . ID=AF086833.47;gene=L  Could anybody point me towards a solution? Cheers Ricky IGV RNA-Seq genome • 2.7k views ADD COMMENT 0 Entering edit mode Did you try "Open File" or "Import Regions"? ADD REPLY 0 Entering edit mode According to the instructions I added the gff file in Create Genome/Optional/Gene file when I also set the fasta file for the genome. I also tried to add the gff file manually after opening the genome in IGV using "Open File" and according to your suggestion now used "Import Regions", no success though. ADD REPLY 0 Entering edit mode There is not much to see in the genome unless you load an alignment file in. Have you done that after creating the "custom" genome? Remember to zoom in significantly before you will start seeing features on the GTF track/read in the alignment window. ADD REPLY 0 Entering edit mode The genome is rather small (ca 19 kb) an relatively gene rich so annotation should be easily visible, zooming doesn't help at all. I can load a bam file and alignment of reads is displayed nicely, so that works, but still no genes. ADD REPLY 2 Entering edit mode It is possible that IGV has started checking fasta sequence identifiers strictly since these directions were written. There is a mismatch between the chromosome name in the 1976.fa file and the 1976-genes.gff file. Try the following: 1. In IGV select a different genome. Go into Genomes --> Manage Genome List. Delete the custom Ebola genome you made. 2. Open 1976.fa file in an editor and remove the version number from the accession number. Change >AF086833.2 to >AF086833. Save the file. If you have a previous 1976.fa.fai file in the directory where you saved the genome delete that so IGV will be forced to recreate the index. 3. Follow the directions to make a new genome using this edited sequence file and the GFF file. I just confirmed that this works. ADD REPLY 0 Entering edit mode Thanks a lot for the suggestion, changed the file to  cat 1976-genes.gff
##gff-version 3
AF086833    EMBL    gene    56  3026    .   +   .   ID=AF086833;gene=NP
AF086833    EMBL    gene    3032    4407    .   +   .   ID=AF086833;gene=VP35
AF086833    EMBL    gene    4390    5894    .   +   .   ID=AF086833;gene=VP40
AF086833    EMBL    gene    5900    8305    .   +   .   ID=AF086833;gene=GP
AF086833    EMBL    gene    8288    9740    .   +   .   ID=AF086833;gene=VP30
AF086833    EMBL    gene    9885    11518   .   +   .   ID=AF086833;gene=VP24;note=putative
AF086833    EMBL    gene    11501   18282   .   +   .   ID=AF086833;gene=L

Solved the problem somehow partially, now IGV displays at least the last entry of the list (gene L) in its correct position. Strangely enough the other ones are not shown, although I cannot see any difference in formatting now between the lines.

1
Entering edit mode

I asked you to edit the sequence file header (fasta) not the GFF file. Use the original GFF file as is. Can you try it again?

0
Entering edit mode

Sorry for that, was a bit confused. The header of my genome sequence fasta file is:

\$ cat 1976.fa | head
>AF086833
CGGACACACAAAAAGAAAGAAGAATTTTTAGGATCTTTTGTGTGCGAATAACTATGAGGAAGATTAATAA

so no version number in AF086833 and I changed back the gff file to:

##gff-version 3
AF086833    EMBL    gene    56  3026    .   +   .   ID=AF086833.3;gene=NP
AF086833    EMBL    gene    3032    4407    .   +   .   ID=AF086833.9;gene=VP35
AF086833    EMBL    gene    4390    5894    .   +   .   ID=AF086833.13;gene=VP40
AF086833    EMBL    gene    5900    8305    .   +   .   ID=AF086833.20;gene=GP
AF086833    EMBL    gene    8288    9740    .   +   .   ID=AF086833.34;gene=VP30
AF086833    EMBL    gene    9885    11518   .   +   .   ID=AF086833.41;gene=VP24;note=putative
AF086833    EMBL    gene    11501   18282   .   +   .   ID=AF086833.47;gene=L

Did the genome again in IGV and indeed, now it displays the genes. However, overlapping genes are represented as a single bar. Is there any possibility I can change that to overlapping bars arranged on top of each other?

I meanwhile found out: change visualization from collapsed to expanded

0
Entering edit mode

I think that is the way IGV handles annotation files. You could split the genes into multiple GFF files and then load them to see if that sort of does what you want.


text:
Why does a constructible set in a Noetherian topological space contain an open subset dense in its closure?

In a Noetherian topological space, a constructible set is a finite union of locally closed sets. This is a conclusion on constructible sets:

Every constructible set contains a dense open subset of its closure.

Now neglect the Noetherian condition, and give $\mathbb{R}$ the usual topology. $\{0\}$ is closed, thus "constructible". The only open set contained in $\{0 \}$ is the empty set. It is certain that the empty set is not dense in the closure of $\{0 \}$ which is equal to $\{0 \}$.

So, the Noetherian condition is necessary.

But the result does not seem clear to me. Would you please give me a proof or a reference?

Many thanks.

-

The claim is that if $A$ is constructible, then there is a set $D \subseteq A$ such that $\mathrm{cl}_X D \supseteq A$ and $D$ is relatively open in $\mathrm{cl}_X A$; $D$ need not be open in the whose space $X$. In your example $A = \mathrm{cl}_\mathbb{R} A =$ $\{0\}$, and we may take $D=A$: certainly $D$ is dense in $A$, and $D = \mathrm{cl}_\mathbb{R} A$ is also an open subset of $\mathrm{cl}_\mathbb{R} A$.
@ShinyaSakai: Yes, there is. In the reals with the usual topology let $A=\mathbb{Q}\cap [0,1]$; then $\mathrm{cl}A=[0,1]$, so no non-empty subset of $A$ is relatively open in $\mathrm{cl}A$. – Brian M. Scott Jul 31 '11 at 2:04


text:
# Quark Matter 2012

12-18 August 2012
US/Eastern timezone

## Di-electron differential cross section in Au+Au collisions at different beam energies at STAR

15 Aug 2012, 09:10
20m
Diplomat ()

### Diplomat

Oral Presentation Electroweak probes

### Speaker

Dr Huang Bingchu (Brookhaven National Lab)

### Description

Di-leptons serve as clean and bulk penetrating probes to study the properties of the strongly interacting hot and dense medium created in heavy ion collisions. They are produced in all stages of the heavy-ion collisions and are not affected by strong interactions, hence can probe the entire evolution of the collision. Di-lepton production in the low mass range ($M_{ll}<1.1$ GeV/$c^{2}$) allows the study of vector meson in-medium properties, an observable possibly connected to chiral symmetry restoration. In the intermediate mass region (\$1.1

### Primary author

Dr Huang Bingchu (Brookhaven National Lab)

 Slides