text:
Subjects -> MATHEMATICS (Total: 1118 journals)     - APPLIED MATHEMATICS (92 journals)    - GEOMETRY AND TOPOLOGY (23 journals)    - MATHEMATICS (819 journals)    - MATHEMATICS (GENERAL) (45 journals)    - NUMERICAL ANALYSIS (26 journals)    - PROBABILITIES AND MATH STATISTICS (113 journals) MATHEMATICS (819 journals)            First | 1 2 3 4 5
 Showing 401 - 538 of 538 Journals sorted alphabetically Journal of Geological Research       (Followers: 1) Journal of Geovisualization and Spatial Analysis Journal of Global Optimization       (Followers: 7) Journal of Global Research in Mathematical Archives       (Followers: 1) Journal of Homotopy and Related Structures Journal of Honai Math Journal of Humanistic Mathematics       (Followers: 1) Journal of Hyperbolic Differential Equations Journal of Indian Council of Philosophical Research Journal of Industrial Mathematics       (Followers: 2) Journal of Inequalities and Applications Journal of Infrared, Millimeter and Terahertz Waves       (Followers: 3) Journal of Integrable Systems       (Followers: 1) Journal of K-Theory Journal of Knot Theory and Its Ramifications       (Followers: 2) Journal of Liquid Chromatography & Related Technologies       (Followers: 7) Journal of Logical and Algebraic Methods in Programming       (Followers: 1) Journal of Manufacturing Systems       (Followers: 4) Journal of Mathematical Analysis and Applications       (Followers: 5) Journal of mathematical and computational science       (Followers: 7) Journal of Mathematical and Fundamental Sciences Journal of Mathematical Behavior       (Followers: 4) Journal of Mathematical Chemistry       (Followers: 5) Journal of Mathematical Extension       (Followers: 3) Journal of Mathematical Finance       (Followers: 9) Journal of Mathematical Imaging and Vision       (Followers: 7) Journal of Mathematical Logic       (Followers: 3) Journal of Mathematical Modelling and Algorithms       (Followers: 1) Journal of Mathematical Neuroscience       (Followers: 10) Journal of Mathematical Sciences       (Followers: 1) Journal of Mathematical Sciences and Applications       (Followers: 4) Journal of Mathematical Sociology       (Followers: 3) Journal of Mathematics Journal of Mathematics and Statistics       (Followers: 8) Journal of Mathematics and Statistics Studies       (Followers: 3) Journal of Mathematics and the Arts       (Followers: 2) Journal of Mathematics Education at Teachers College       (Followers: 4) Journal of Mathematics in Industry Journal of Mathematics Research       (Followers: 6) Journal of Metallurgy       (Followers: 8) Journal of Modern Mathematics Frontier Journal of Multidisciplinary Modeling and Optimization       (Followers: 1) Journal of Multivariate Analysis       (Followers: 13) Journal of Natural Sciences and Mathematics Research Journal of Nonlinear Analysis and Optimization : Theory & Applications       (Followers: 4) Journal of Nonlinear Mathematical Physics       (Followers: 3) Journal of Nonlinear Science       (Followers: 1) Journal of Numerical Cognition       (Followers: 1) Journal of Optimization       (Followers: 5) Journal of Peridynamics and Nonlocal Modeling Journal of Problem Solving       (Followers: 2) Journal of Progressive Research in Mathematics       (Followers: 5) Journal of Pseudo-Differential Operators and Applications Journal of Pure and Applied Algebra       (Followers: 4) Journal of Quantitative Linguistics       (Followers: 6) Journal of Scientific Computing       (Followers: 19) Journal of Scientific Research Journal of Symbolic Computation       (Followers: 2) Journal of the Australian Mathematical Society Journal of the Egyptian Mathematical Society Journal of the European Mathematical Society       (Followers: 2) Journal of the Indian Mathematical Society       (Followers: 1) Journal of the Institute of Mathematics of Jussieu Journal of the London Mathematical Society       (Followers: 3) Journal of the Nigerian Mathematical Society       (Followers: 1) Journal of Theoretical and Applied Physics       (Followers: 9) Journal of Topology and Analysis Journal of Transport and Supply Chain Management       (Followers: 16) Journal of Turbulence       (Followers: 10) Journal of Uncertainty Analysis and Applications       (Followers: 1) Journal of Universal Mathematics Journal of Urban Regeneration & Renewal       (Followers: 11) JRAMathEdu : Journal of Research and Advances in Mathematics Education       (Followers: 7) JUMLAHKU : Jurnal Matematika Ilmiah STKIP Muhammadiyah Kuningan       (Followers: 2) JURING (Journal for Research in Mathematics Learning)       (Followers: 1) Jurnal Ilmiah AdMathEdu Jurnal Matematika       (Followers: 1) Jurnal Matematika Integratif Jurnal Matematika, Sains, Dan Teknologi Jurnal Natural Jurnal Pendidikan Matematika Raflesia Jurnal Penelitian Pembelajaran Matematika Sekolah Jurnal Penelitian Sains (JPS) Jurnal Riset Pendidikan Matematika Jurnal Sains Matematika dan Statistika Jurnal Tadris Matematika Jurnal Teknologi dan Sistem Komputer Kontinu : Jurnal Penelitian Didaktik Matematika       (Followers: 2) Kreano, Jurnal Matematika Kreatif-Inovatif       (Followers: 6) Le Matematiche Lettera Matematica Lietuvos Matematikos Rinkinys       (Followers: 3) Limits : Journal of Mathematics and Its Applications       (Followers: 1) Linear Algebra and its Applications       (Followers: 24) Linear and Multilinear Algebra       (Followers: 8) Lithuanian Mathematical Journal LMS Journal of Computation and Mathematics Lobachevskii Journal of Mathematics Logic and Analysis       (Followers: 1) Logic Journal of the IGPL       (Followers: 1) Logica Universalis manuscripta mathematica MaPan : Jurnal Matematika dan Pembelajaran Marine Genomics       (Followers: 2) Matemáticas, Educación y Sociedad Matematicheskie Zametki Mathematica Eterna Mathematica Scandinavica       (Followers: 1) Mathematical Analysis and its Contemporary Applications Mathematical and Computational Forestry & Natural-Resource Sciences Mathematical Communications Mathematical Computation       (Followers: 1) Mathematical Geosciences       (Followers: 4) Mathematical Journal of Interdisciplinary Sciences       (Followers: 1) Mathematical Medicine and Biology: A Journal of the IMA       (Followers: 2) Mathematical Methods in the Applied Sciences       (Followers: 5) Mathematical Methods of Statistics       (Followers: 4) Mathematical Modelling and Analysis       (Followers: 1) Mathematical Modelling of Natural Phenomena       (Followers: 2) Mathematical Models and Methods in Applied Sciences       (Followers: 2) Mathematical Models in Engineering Mathematical Notes Mathematical Proceedings of the Cambridge Philosophical Society       (Followers: 2) Mathematical Programming Computation       (Followers: 3) Mathematical Sciences Mathematical Social Sciences       (Followers: 1) Mathematical Theory and Modeling       (Followers: 13) Mathematical Thinking and Learning       (Followers: 4) Mathematics and Statistics       (Followers: 5) Mathematics Education Forum Chitwan       (Followers: 2) Mathematics Education Journal       (Followers: 3) Mathematics Education Research Journal       (Followers: 20) Mathematics in Science and Engineering Mathematics of Control, Signals, and Systems (MCSS)       (Followers: 5) Mathématiques et sciences humaines       (Followers: 6) Mathematische Annalen       (Followers: 1) Mathematische Nachrichten       (Followers: 1) Mathematische Semesterberichte Mathematische Zeitschrift       (Followers: 1) MathLAB Journal       (Followers: 4) MATICS       (Followers: 2) Matrix Science Mathematic       (Followers: 1) Measurement Science Review       (Followers: 3) Mediterranean Journal of Mathematics Memetic Computing Mendel : Soft Computing Journal Metaheuristics Metals and Materials International Metascience       (Followers: 1) Metrology and Instruments / Метрологія та прилади       (Followers: 1) Milan Journal of Mathematics MLQ- Mathematical Logic Quarterly       (Followers: 1) Modelling in Civil and Environmental Engineering       (Followers: 5) MONA : Matematik- og Naturfagsdidaktik       (Followers: 4) Monatshefte fur Mathematik Moroccan Journal of Pure and Applied Analysis       (Followers: 4) Moscow University Mathematics Bulletin MSOR Connections       (Followers: 1) Multiscale Modeling and Simulation       (Followers: 4) MUST : Journal of Mathematics Education, Science and Technology       (Followers: 4) Nagoya Mathematical Journal Nano Research       (Followers: 4) Nanotechnologies in Russia       (Followers: 1) Natural Resource Modeling       (Followers: 1) New Mathematics and Natural Computation       (Followers: 1) Nonlinear Analysis : Modelling and Control       (Followers: 1) Nonlinear Analysis : Theory, Methods & Applications       (Followers: 1) Nonlinear Analysis: Hybrid Systems Nonlinear Analysis: Real World Applications       (Followers: 2) Nonlinear Differential Equations and Applications NoDEA Nonlinear Oscillations       (Followers: 1) North Carolina Journal of Mathematics and Statistics North-Holland Mathematical Library       (Followers: 1) North-Holland Mathematics Studies North-Holland Series in Applied Mathematics and Mechanics       (Followers: 1) Note di Matematica NTM Zeitschrift für Geschichte der Wissenschaften, Technik und Medizin       (Followers: 4) Numeracy : Advancing Education in Quantitative Literacy Numerical Analysis and Applications       (Followers: 1) Numerical Functional Analysis and Optimization       (Followers: 2) Numerical Linear Algebra with Applications       (Followers: 8) Numerical Mathematics : Theory, Methods and Applications Numerische Mathematik Open Journal of Discrete Mathematics       (Followers: 6) Open Journal of Modelling and Simulation       (Followers: 2) Operations Research Letters       (Followers: 11) Optimization Letters       (Followers: 2) Optimization Methods and Software       (Followers: 8) Opuscula Mathematica       (Followers: 109) Order       (Followers: 1) ORiON P-Adic Numbers, Ultrametric Analysis, and Applications PAMM : Proceedings in Applied Mathematics and Mechanics       (Followers: 1) Parallel Processing Letters       (Followers: 3) Peking Mathematical Journal Periodica Mathematica Hungarica       (Followers: 1) Perspectivas da Educação Matemática Petroleum Science       (Followers: 1) Philosophia Mathematica       (Followers: 1) Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences       (Followers: 10)
Similar Journals
 Journal of Pseudo-Differential Operators and ApplicationsJournal Prestige (SJR): 0.664 Citation Impact (citeScore): 1Number of Followers: 0      Hybrid journal (It can contain Open Access articles) ISSN (Print) 1662-9981 - ISSN (Online) 1662-999X Published by Springer-Verlag  [2658 journals]
• Nonnegative solutions and multiple solutions for a class of weighted
Hamiltonian subelliptic systems

Abstract: Abstract In this paper, we investigate existence results for the following system \begin{aligned} \left\{ \begin{aligned}&-\Delta _X u= v ^{p-1}v{\omega _1(x)}, \\&-\Delta _X v= u ^{q-1}u{\omega _2(x)}, x\in \Omega \subset {\mathbb {R}}^N \end{aligned} \right. \end{aligned} with homogeneous Dirichlet boundary conditions, where $$X=\{X_{1},X_{2},\ldots ,X_{m}\}$$ is a system of smooth vector fields defined on a neighborhood of $${\overline{\Omega }}$$ that satisfies the Hörmander’s condition, $$\triangle _{X}= -\sum _{i=1}^{m}X_{i}^{*} X_i$$ on $$\Omega$$ . Furthermore, $$\partial \Omega$$ is non-characteristic with respect to X and the weights $$\omega _i(x)$$ satisfy certain growth conditions involving negative powers of the C–C metric to isolated singular points. We construct a proper functional setting and use a version of linking theorem to examine the existence of nontrivial solutions where the growth of nonlinearities (p, q) are below certain critical hyperbola related to the generalized Métivier index of X. This work extends the results in Djairo G. de Figueiredo et al. (Annali di Matematica 187:531–545, 2008) to the class of subelliptic finitely degenerate operators and improves the recent multiple solutions in C.T. Anh et al. (Adv Nonlinear Anal 8(1): 661–678, 2017) to the case of superlinear system. Furthermore, the existence of nonnegative solutions and ground-state solutions is proved.
PubDate: 2021-10-07

• $$L^p$$ L p properties of non-Archimedean fractional differentiation
operators

Abstract: Abstract Let $$D^{\alpha }$$ be the Vladimirov–Taibleson fractional differentiation operator acting on complex-valued functions on a non-Archimedean local field. The identity $$D^{\alpha }D^{-\alpha }f=f$$ was known only for the case where f has a compact support. Following a result by Samko about the fractional Laplacian of real analysis, we extend the above identity in terms of $$L^p$$ -convergence of truncated integrals. Differences between real and non-Archimedean cases are discussed.
PubDate: 2021-10-05

• Different approach to the decomposition theory of $$HM^p_{q,{\varDelta _{\nu }}}$$ H M q , Δ ν p Hardy–Morrey spaces

Abstract: Abstract B-maximal functions in $$HM^p_{q,{\varDelta _{\nu }}}$$ Hardy–Morrey spaces, related to a Laplace–Bessel differential operator, are studied. We give B-maximal characterization of $$HM^p_{q,{\varDelta _{\nu }}}$$ Hardy–Morrey spaces and study the atomic decomposition theory which has the same cancellation properties of the $$H^{p}_{\varDelta _{\nu }}({\mathbb {R}}^{n}_{+})$$ Hardy spaces.
PubDate: 2021-10-04

• Amrein–Berthier and Logvinenko–Sereda uncertainty principles for the
Hankel–Stockwell transform

Abstract: Abstract The aim of this paper is to prove two new uncertainty principles for the Hankel–Stockwell transform namely Amrein–Berthier and Logvinenko–Sereda theorems.
PubDate: 2021-10-04

• Equivalence of K-functionals and modulus of smoothness for Laguerre type
operator

Abstract: Abstract The aim of this article is to establish the equivalence between a K-functional and a modulus of smoothness generated by $$D{\alpha }$$ operator on $${\mathbb {K}}$$ in the metrics of $$L^{2}_{\alpha }({\mathbb {K}})$$ with a certain weight.
PubDate: 2021-09-27

• Two classes of bilinear fractional integral operators and their
commutators on generalized fractional Morrey spaces

Abstract: Abstract In this paper, the authors prove that the following form of bilinear fractional integral operator \begin{aligned} B_{\alpha }(f,g)(x):=\int _{\mathbb {R}^{n}}\frac{f(x-y)g(x+y)}{ y ^{n-\alpha }}\mathrm {d}y \end{aligned} is bounded from the product of generalized fractional Morrey spaces $${\mathcal {L}}^{p_{1},\eta _{1},\varphi }(\mathbb {R}^{n})\times {\mathcal {L}}^{p_{2},\eta _{2},\varphi }(\mathbb {R}^{n})$$ to space $${\mathcal {L}}^{q,\eta -\alpha ,\varphi }(\mathbb {R}^{n})$$ , and also bounded from the product space $${\mathcal {L}}^{p_{1},\varphi }$$ $$(\mathbb {R}^{n})\times {\mathcal {L}}^{p_{2},\varphi }(\mathbb {R}^{n})$$ to $${\mathcal {L}}^{q,\varphi ^{\frac{q}{p}}}(\mathbb {R}^{n})$$ . Moreover, the boundedness of commutator $$[b_{1},b_{2},{\mathcal {I}}_{\alpha }]$$ generated by $$b_{1}, b_{2}\in \mathrm {BMO}(\mathbb {R}^{n})$$ and bilinear fractional integral operator $${\mathcal {I}}_{\alpha }$$ defined as in (1.1) is bounded from the product space $${\mathcal {L}}^{p_{1},\eta _{1},\varphi }(\mathbb {R}^{n})\times {\mathcal {L}}^{p_{2},\eta _{2},\varphi }(\mathbb {R}^{n})$$ to space $${\mathcal {L}}^{q,\eta -\alpha ,\varphi }(\mathbb {R}^{n})$$ ; as a corollary, the boundedness of the commutator $$[b_{1},b_{2},{\mathcal {I}}_{\alpha }]$$ on generalized Morrey space is also obtained.
PubDate: 2021-09-26

• Group classification and constructing conservation laws for

Abstract: Abstract Finding the symmetries and conservation laws of the nonlinear fractional differential equations are topics, which have many applications in various fields of science and engineering. In this manuscript, firstly, by using the Lie symmetry analysis, we obtain the Lie point symmetries of the time-fractional Sawada–Kotera–Parker–Dye equation. After that, by using the infinitesimal generators, we present the corresponding reduced fractional differential equations and fractional conservation laws.
PubDate: 2021-09-18

• Elliptic boundary value problems associated with isometric group actions

Abstract: Abstract Given a manifold with boundary endowed with an action of a discrete group on it, we consider the algebra of operators generated by elements in the Boutet de Monvel algebra of pseudodifferential boundary value problems and shift operators acting on functions on the manifold and its boundary. Provided that the group is of polynomial growth and its action is isometric, we construct a Chern character for elliptic elements in this algebra with values in a de Rham type cohomology of the fixed point manifolds for the group action and obtain an index formula in terms of this Chern character. Our index formula contains as special cases the index formula by Fedosov for boundary value problems in the Boutet de Monvel algebra and the index formula by Nazaikinskii, Savin and Sternin for operators on a closed manifold associated with an isometric group action.
PubDate: 2021-09-15

• The convergence constants and non linear approximations of fusion frames

Abstract: Abstract In this paper, we study the unconditional constant and nonlinear N-term approximation of fusion frames in Hilbert spaces. We show that the unconditional constant and greedy constant are bounded by a constant which is associated with the fusion frame bounds. We prove that the unconditional constant of cross fusion frame expansions satisfies the similar properties as long as the cross g-frame expansions stay uniformly bounded away from zero. Finally, we show that fusion frames satisfy the quasi greedy and almost greedy conditions. Moreover, we prove that fusion Riesz bases satisfy the greedy condition.
PubDate: 2021-08-29

• Nonlinear commutators on Morrey–Banach spaces

Abstract: Abstract We establish the boundedness of the nonlinear commutator introduced by Rochberg and Weiss (Ann Math 118:315–347, 1983) on Morrey–Banach spaces. In particular, we have the boundedness results on the Morrey–Lorentz spaces and the Morrey spaces with variable exponents.
PubDate: 2021-08-20

• Uncertainty principles for the offset linear canonical transform

Abstract: Abstract In this paper, we extend some different uncertainty principles (UPs), including Miyachi’s UP and Morgan’s UP for the offset linear canonical transform (OLCT) which provides a more general framework for a number of well-known linear integral transforms such as Fourier transform, fractional Fourier transform, linear canonical transform.
PubDate: 2021-08-16

• An interplay between parameter (p, q)-Boas transform and linear
canonical transform

Abstract: Abstract In this paper, we define and study parameter (p, q)-Boas transform of a signal in linear canonical transform domain. Also, we define the complex signals associated with parameter (p, q)-Boas transform and linear canonical transform and prove the generalized Boas transform product theorem (an analog of Bedrosian’s theorem) in the linear canonical transform domain. Further, using a linear combination of a signal and its parameter (p, q)-Boas transform, we define a $$\eta$$ -linear canonical Boas transform and study some of its properties. Finally, we propose some $$\eta$$ -linear canonical complex signals and give analysis of their linear canonical transform spectrum and eigenfunction property.
PubDate: 2021-08-10

• Gabor multipliers associated with the Bessel–Kingman hypergroup

Abstract: Abstract We consider the Bessel–Kingman hypergroup $$\big ([0,+\infty [,*\big )$$ and we denote by $$d\mu _\alpha ,\ \alpha \geqslant -\frac{1}{2},\$$ the measure defined on $$[0,+\infty [$$ by $$d\mu _\alpha (x)=\frac{x^{2+1}}{2^\alpha \ \Gamma (\alpha +1)}\ dx.\$$ We define the Gabor multiplier $$\mathscr {G}_{u,v}(\sigma )$$ associated with two square integrable functions u, v on $$[0,+\infty [$$ with respect to the measure $$d\mu _\alpha$$ called window functions and $$\sigma$$ is a measurable function on $$[0,+\infty [$$ called a signal. We prove that $$\mathscr {G}_{u,v}(\sigma )$$ is a bounded linear operator on $$L^2(d\mu _\alpha )$$ and it is compact. Next, we define the Schatten von-Neumann class $$S^p,\ p\in \ [1,+\infty ],\$$ and we show that the Gabor multiplier $$\mathscr {G}_{u,v}(\sigma )$$ belongs to the class $$S^p.\$$ We give also a formula of trace when $$\sigma \in \ L^1(d\mu _\alpha )$$ . Also, we define the Landau–Pollak–Slebian operator and we give its connection with the Gabor multiplier. Last, we study the boundedness and compactness of the Gabor multipliers for more window functions $$u,v\in \ L^p(d\mu _\alpha ),\ p\in \ [1,+\infty ]$$ .
PubDate: 2021-07-27

• Pseudo-fractional differential equations and generalized g-Laplace
transform

Abstract: Abstract In this article, we introduce a generalized g-Laplace transform and discuss some essential results of integral transform theory, in particular, involving a $$\psi$$ -Hilfer pseudo-fractional derivative and function convolution. In this sense, we investigated the existence and uniqueness of known solutions for a pseudo-fractional differential equation.
PubDate: 2021-07-24

• Semilinear wave equation on compact Lie groups

Abstract: Abstract In this note, we study the semilinear wave equation with power nonlinearity $$u ^p$$ on compact Lie groups. First, we prove a local in time existence result in the energy space via Fourier analysis on compact Lie groups. Then, we prove a blow-up result for the semilinear Cauchy problem for any $$p>1$$ , under suitable sign assumptions for the initial data. Furthermore, sharp lifespan estimates for local (in time) solutions are derived.
PubDate: 2021-07-14

• Fundamental solutions and decay rates for evolution problems on the torus
$$\mathbb {T}^n$$ T n

Abstract: Abstract In this paper we study large-time behavior evolution problems on the n-dimensional torus $$\mathbb {T}^n$$ , $$n \ge 1$$ . Here we analyze the solutions to these problems, studying their regularity and obtaining estimates of them. The main tools we use is the toroidal Fourier transform, together with Fourier series and a version of the Hardy-Littlewood inequality, applied to our case of the n-dimensional torus $$\mathbb {T}^n$$ . We use this inequality to find an estimate of solutions to evolution problems.
PubDate: 2021-07-14

• Time–frequency analysis associated with the k-Hankel Gabor transform on
$${\mathbb {R}}^{d}$$ R d

Abstract: Abstract In this paper, we investigate the k-Hankel Gabor transform on $${\mathbb {R}}^{d}$$ in some problems of time–frequency analysis. Firstly, we present the main theorems of Harmonic analysis as Plancherel’s, Lieb’s and inversion formulas for this transform. Next, we formulate some novel uncertainty principles including the Heisenberg and logarithmic uncertainty principles, Benedick–Amrein–Berthier’s uncertainty principle, local uncertainty principles and Shapiro’s uncertainty principle. In sequel, we introduce the localization operators associated with the k-Hankel Gabor transform on $${\mathbb {R}}^{d}$$ and we develop corresponding theory. In particular we study their trace class properties and we prove that they are in the Schatten–von Neumann.
PubDate: 2021-07-01

• $$\psi$$ ψ -Mittag–Leffler pseudo-fractional operators

Abstract: Abstract In this paper we present new types of pseudo-fractional integral and derivatives on a semiring $$([a,b],\oplus ,\odot )$$ . These operators are called the $$\psi$$ -Riemann–Liouville–Mittag–Leffler pseudo-fractional integrals and derivatives and $$\psi$$ -Caputo–Mittag–Leffler pseudo-fractional derivatives. Some properties, like the semigroup law, composition relations between pseudo-fractional differentiation and pseudo-fractional integration operators, composition relations between pseudo-fractional derivatives, among others, are discussed. Finally, as an application, we proved two versions of the fundamental theorem of fractional calculus.
PubDate: 2021-06-29

• Mixed radial-angular integrability for rough maximal operators

Abstract: Abstract In this paper, we study the mixed radial-angular integrability for maximal operators related to a class of rough singular integrals along “polynomial curves". Under assuming that the kernel satisfies certain rather weak size condition on sphere, we obtain the boundedness of such maximal operators on the mixed radial-angular Lebesgue spaces. Meanwhile, on the corresponding vector-valued versions are also established.
PubDate: 2021-06-16

• FIOs with cusp singularities and open umbrellas

Abstract: Abstract We study the composition calculus of Fourier integral operators F, with Morin singularities. This composition calculus falls outside of the classical transverse or clean intersection calculus of FIOs and it is motivated by the inverse problems studied in Felea and Greenleaf (Math Res Lett 17(5):867–886, 2010) and Felea and Nolan (J Fourier Anal Appl 21(4):799–821, 2015). In this case, the normal operator $$F^*F$$ is an operator with the wave front set in $$\Delta \cup \Lambda$$ where $$\Lambda$$ is a singular canonical relation called an open umbrella. We show that the operator $$F^*F$$ has the same order on $$\Delta \setminus \Lambda$$ and on $$\Lambda \setminus \Delta$$ . This result is a generalization of the results obtained in Felea and Greenleaf (2010) and Felea and Nolan (2015) where the canonical relations have the fold/cusp and cusp/cusp singularities.
PubDate: 2021-06-15

JournalTOCs
School of Mathematical and Computer Sciences
Heriot-Watt University
Edinburgh, EH14 4AS, UK
Email: journaltocs@hw.ac.uk
Tel: +00 44 (0)131 4513762


text:
# How do you solve -4x > 12?

Apr 30, 2018

$x < - 3$

#### Explanation:

it is given

$- 4 x > 12$

So

$\implies 4 x < - 12 \text{ }$ [$\div$ by a negative $\to$ flip the sign]

$x < \frac{- 12}{4} \implies x < - 3$

so the solution of this question is $x < - 3$.

Apr 30, 2018

$x < - 3$

#### Explanation:

Given: $- 4 x > 12$.

Divide by $4$ on both sides.

$\frac{- 4 x}{4} > \frac{12}{4}$

$- x > 3$

Switch signs. That means that the inequality sign flips as well.

$\implies x < - 3$

format: plain


text:
01-11-2020, 06:08 AM (This post was last modified: 01-11-2020 06:15 AM by Thomas Okken.)
Post: #26
 Thomas Okken Senior Member Posts: 1,304 Joined: Feb 2014
(01-11-2020 05:37 AM)tcab Wrote:  I managed to compile Free42 on the Raspberry Pi. As Thomas said, its dead easy:
Code:
sudo apt install libgtk-3-dev git clone https://github.com/thomasokken/free42.git cd free42/gtk make ./free42bin

Not sure whether I should have run the build-gtk script instead?, but the resulting binary free42bin using the steps above seems to have worked OK (though I did get a few compile warnings flying past...)

The difference is that the build-gtk script checks if /usr/include/alsa/asoundlib.h exists, and adds AUDIO_ALSA=1 to the 'make' command line if it does, and it strips the binaries to make them smaller.

Regarding the warnings, those are probably related to GTK. In the later GTK 3 versions, a few APIs were deprecated, but I'm sticking with them in order to be able to continue building in Ubuntu 12.04, which has a rather old version of GTK 3 (3.4.2). I don't know if it's possible to suppress those warnings, but they are harmless at any rate.

(01-11-2020 05:37 AM)tcab Wrote:  Print42 was unable to automatically detect the location of the Free42 printout text file (Print42 tries to guess via inspecting the Free42 state file, which it thinks is in /home/pi/.local/share/Free42/state but the Free42 state file is not there - I'll have to find out where the proper location is and tweak that in the next release of Print42).

The location of the directory containing the state file is hard-coded in gtk/shell_main.cc, line 408, and the state, print-out, and keymap file names, which include the Free42 directory name, are hard coded starting at line 421, six lines in total. Search for /.free42/ to make sure you find all occurrences.

I guess it's a bit ugly but I didn't anticipate people wanting to store those files somewhere other than \$HOME/.free42

(01-11-2020 05:37 AM)tcab Wrote:  Here is a photo of my setup.

Cool!

(01-09-2020 06:08 PM)Bill (Smithville NJ) Wrote:  Enjoy your new little computer.

I'm liking it so far! It has already helped me fix a bug that I hadn't been able to reproduce in my other Linux environments. Although that bug is caused by bad behavior in GTK or in the display driver, so that is a mixed blessing, I guess.

I have had dropped WiFi connections a couple of times, although I'm not 100% sure the Pi is at fault there. It could also be a problem in my router (Netgear WNDR4300 running a not-very-recent version of DD-WRT) or in the MacBook Air I'm using to VNC into the Pi. I haven't noticed WiFi flakiness before, but I don't usually VNC between machines at home for long periods of time, either, so that will require some investigation.
 « Next Oldest | Next Newest »


text:
# Solving 59x2+82x+65 using the Quadratic Formula

For your equation of the form "ax2 + bx + c = 0," enter the values for a, b, and c:

 a x2 + b x + c = 0
Reset

You entered:
59x2+82x+65=0.

There are no solutions in the real number domain.
There are two complex solutions: x = -0.69491525423729 + 0.78663060243202i, and x = -0.69491525423729 - 0.78663060243202i,
where i is the imaginary unit.

## Here's how we found that solution:

You entered the following equation:
(1)           59x2+82x+65=0.

For any quadratic equation ax2 + bx + c = 0, one can solve for x using the following equation, which is known as the quadratic formula:
(2)

In the form above, you specified values for the variables a, b, and c. Plugging those values into Eqn. 1, we get:
(3)           $$x=-82\pm\frac{\sqrt{82^2-4*59*65}}{2*59}$$

which simplifies to:
(4)           $$x=-82\pm\frac{\sqrt{6724-15340}}{118}$$

Now, note that b2-4ac is a negative number. Specifically in our case, 6724 - 15340 = -8616.
(5)           $$x=-82\pm\frac{\sqrt{-8616}}{118}$$

This means that our solution will require finding the square root of a negative number. There is no real number solution for this, so our solution will be a complex number (that is, it will involve the imaginary number i, defined as the square root of -1.).
Let's calculate the square root:
(6)           $$x=-82\pm\frac{92.822411086978i}{118}$$

This equation further simplifies to:
(7)           $$x=-\frac{-82}{118}\pm0.78663060243202i$$

Solving for x, we find two solutions which are both complex numbers:
x = -0.69491525423729 + 0.78663060243202i
and
x = -0.69491525423729 - 0.78663060243202i

Both of these solutions are complex numbers.
These are the two solutions that will satisfy the equation 59x2+82x+65=0.

### Notes

A quadratic equation is an equation that has the form:
ax2 + bx + c = 0,
where x is an unknown. A, b, and c are constants. A and b are referred to as coefficients. Furthermore, it should be noted that a cannot be equal to 0.

Calculating a solution to a quadratic equation may seem daunting. Fortunately, you have this handy-dandy quadratic equation solver. Acutally, quadratic equations can be readily solved using the quadratic formula, which is the same technique used by this quadratic equation solver. Try it, and it will explain each of the steps to you. This is the quadratic formula:

Since there are always 2 solutions to a square root (one negative, one positive), solving the quadratic equation results in 2 values for x. The two solutions for x (which may be positive or negative, real or complex) are called roots. Under extraordinary circumstances, the two roots may equal each other, producing one solution for x.

Why do we need to be able to solve quadratic equations? Quadratic equations are needed to find answers to many real-world problems. For example, to compute whether a braking car can stop fast enough to avoid hitting something would require the use of s quadratic equation.

As mentioned above, in the equation ax2+bx+c=0, a cannot be zero. If a were 0, then ax2 = 0x2 = 0 for any value of x, so our equation becomes 0 + bx + c = 0, which is the same as bx + c = 0, which is no longer a quadratic equation. In fact, bx + c = 0 is a linear equation, which is much simpler to solve than a quadratic equation.

We hope you find this quadratic equation calculator useful. We encourage you to plug in different values for a, b, and c. But, if you just want to use it to calculate the answers to your quadratic equations, that's cool too. Thank you for your interest in Quadratic-Equation-Calculator.com.


text:
# Could not load file or assembly '...' or one of its dependencies

I have an assembly MyEventAssembly.dll that subscribes to various events in the tridion event system. It references another assembly MySharedAssembly.dll. MySharedAssembly.dll references Tridion.Common.dll, Tridion.ContentManager.dll, Tridion.ContentManager.Common.dll, Tridion.ContenetManage.Publisher.dll and Tridion.Logging.dll.

I stop IIS, stop all Tridion services and end the dllhst3g.exe. I copy MyEventAssembly.dll, MySharedAssembly.dll into C:\Program Files (x86)\Tridion\bin. I then start IIS and the Tridion Content Manager Service Host service. I then access the content manager through my web browser where I am presented with the following message

Could not load file or assembly 'MySharedAssembly, Version=1.0.0.0, Culture=neutral, PublicKeyToken=null' or one of its dependencies. The system cannot find the file specified.

Unable to Initialize TDSE object. Initialization of the extension 'C:\Program Files (x86)\Tridion\bin\MyEventAssembly.dll' failed.

System.Reflection.RuntimeAssembly.GetExportedTypes(RuntimeAssembly,ObjectHandleOnStack)
System.Reflection.RuntimeAssembly.GetExportedTypes()
Tridion.ContentManager.Extensibility.TcmExtension.InitializeExtensions()
Tridion.ContentManager.Session.InitializeSession()
Tridion.ContentManager.Session..ctor(String,String,UserContext)
UtilitiesTDS.GetUserContext
TDSE.Initialize

I also get the same error in the windows event log on the content manager server.

If I access the content manager a second time through the web browser I receive no error message and the UI loads as expected.

I have tried putting the dependent assemblies (MySharedAssembly.dll, Tridion.Common.dll, Tridion.ContentManager.dll, Tridion.ContentManager.Common.dll, Tridion.ContenetManage.Publisher.dll and Tridion.Logging.dll) into various bin directories in the Tridion folder such as C:\Program Files (x86)\Tridion\bin and C:\Program Files (x86)\Tridion\webservices\bin but with no success.

Where should I put the referenced assemblies? Is there any way I can solve it without registering the assemblies in the GAC?

• Put your shared assembly in GAC Mar 26 '15 at 13:45
• @user978511 Is there a solution that doesn't involve putting the assemblies in the GAC? Mar 26 '15 at 13:50
• AFAIK there's no other solution Mar 26 '15 at 14:11
• You could use a tool like ILMerge to create a single assembly. Then you can reference that in Tridion.ContentManager.config. Mar 26 '15 at 14:12
• I see two Answers in the form of Comments... Mar 27 '15 at 8:27

After reading the following blog post

http://www.curlette.com/?p=1403

It appears that Tridion likes to have a single dll. I've now given up with the idea that I can make my proposed solution work. Instead I've used ILMerge as suggested by Peter Kjaer to combine all my assemblies into one.

2. Create a new console project and add ILMerge.exe as a reference.
3. Add the following code to the console project and run it to create the merged assembly

static void Main(string[] args)
{
var ilmerge = new ILMerging.ILMerge();
ilmerge.SetInputAssemblies(new string[] {
@"C:\path\to\MyEventAssembly.dll",
@"C:\path\to\MySharedAssembly.dll"
});
// if you get any errors when merging you might have to uncomment this
//ilmerge.SetTargetPlatform("v4", @"C:\Windows\Microsoft.NET\Framework\v4.0.30319");
ilmerge.OutputFile = @"C:\dev\MyEventAssembly.ILMerged.dll";
ilmerge.Merge();
}

Using the MyEventAssembly.ILMerged.dll from the event system now works as expected with no errors.

I'll update this with the equivalent post-build script once I've written it.


text:
Working Paper Article Version 1 This version is not peer-reviewed

# Remarks on Nonsingular Models as Hayward and Magnetized Black Hole with Rational Nonlinear Electrodynamics

Version 1 : Received: 13 November 2020 / Approved: 16 November 2020 / Online: 16 November 2020 (16:04:52 CET)

How to cite: Il'ich Kruglov, S. Remarks on Nonsingular Models as Hayward and Magnetized Black Hole with Rational Nonlinear Electrodynamics. Preprints 2020, 2020110434 Il'ich Kruglov, S. Remarks on Nonsingular Models as Hayward and Magnetized Black Hole with Rational Nonlinear Electrodynamics. Preprints 2020, 2020110434

## Abstract

A Hayward black hole and magnetically charged black hole based on rational nonlinear electrodynamics with the Lagrangian ${\cal L} = -{\cal F}/(1+2\beta{\cal F})$ (${\cal F}$ is a field invariant) are considered. It was shown that the metric function in both models possesses a de Sitter core without singularities as $r\rightarrow 0$. The behavior of the Hawking temperature and the heat capacity in these models are similar. The phase transitions take place when the Hawking temperature has the maximum, and black holes are thermodynamically stable at some event horizon radii when the heat capacity is positive. We show that the source of gravity in the Hayward model is questionable.

## Keywords

Magnetically charged black holes; Hawking temperature;; heat capacity; phase transitions

## Subject

PHYSICAL SCIENCES, Acoustics

Views 0


text:
MathSciNet bibliographic data MR916928 30D05 (58F08 58F11) Makienko, P. M. Iterations of analytic functions in ${\bf C}\sp *$${\bf C}\sp *$. (Russian) Dokl. Akad. Nauk SSSR 297 (1987), no. 1, 35--37; translation in Soviet Math. Dokl. 36 (1988), no. 3, 418–420 Links to the journal or article are not yet available

For users without a MathSciNet license , Relay Station allows linking from MR numbers in online mathematical literature directly to electronic journals and original articles. Subscribers receive the added value of full MathSciNet reviews.


text:
User:
• Matrices
• Algebra
• Geometry
• Funciones
• Trigonometry
• Coordinate geometry
• Combinatorics
 Suma y resta Producto por escalar Producto Inversa
 Monomials Polynomials Special products Equations Quadratic equations Radical expressions Systems of equations Sequences and series Inner product Exponential equations Matrices Determinants Inverse of a matrix Logarithmic equations Systems of 3 variables equations
 2-D Shapes Areas Pythagorean Theorem Distances
 Graphs Definition of slope Positive or negative slope Determine slope of a line Ecuación de una recta Equation of a line (from graph) Quadratic function Posición relativa de dos rectas Asymptotes Limits Distancias Continuity and discontinuities
 Sine Cosine Tangent Cosecant Secant Cotangent Trigonometric identities Law of cosines Law of sines
 Ecuación de una recta Posición relativa de dos rectas Distancias Angles in space Inner product

Areas
Surface area of Cylinders

A cylinder is a 3-dimensional figure with two circular bases and a curved face.

The surface area of a cylinder the sum of the areas of its two bases and its curved face.
To conceptualize surface area of cylinders, we can imagine that the lateral area of a cylinder can be "unrolled" into a rectangle with one side equaling the circumference of the circle and the other side equal to the height of the cylinder (unless it is oblique).

It is pretty easy to calculate their areas and then add them up.

SAcylinder=

 Find the surface area of the cylinder:

SAcylinder=

r=5
h=15
SAcylinder=


text:
## Cryptology ePrint Archive: Report 2014/690

Efficient Interval Check in the Presence of Malicious Adversaries

Genqiang Wu and Yeping He and Yi Lu and Liping Ding

Abstract: We consider the following problem: Assuming that Alice and Bob have an integer interval $[a, e]$ and an integer $b$ respectively, for a commitment $c$ to $b$, Alice and Bob jointly check whether $b$ is within $[a, e]$ without revealing their inputs, where either party may behave maliciously. A special case of the problem is the secure integer comparison in the malicious model. This problem mainly arises from location-based access control systems where one party needs to assure to the other party that its location is within some definite area.

Our main result is a constant-round protocol that exhibit the square of $\log e$ communication and the square of $\log e$ exponentiations with simulation-based security. At the heart of the construction is perfect $k$-ary index and corresponding zero-knowledge proof techniques. We consider a more general case of the problem where the interval is substituted by a union of intervals.

Category / Keywords: cryptographic protocols / private interval check, secure integer comparison, malicious model, zero-knowledge proof, $k$-ary tree index, location-based access control


text:
# BOLCHEM: model applications

## Simulation of an ozone episode over Italy

In this study ( download the extended abstract here), we focused on an ozone episode that occurred over Italy during the first week of July 1999, when a west moving wave coming from northern Africa crossed the Italian peninsula with temperatures reaching values up to 20°C at 850 hPa level, creating ideal meteorological conditions for the developing of photochemical pollution at lower atmospheric levels, especially over the highly industrialized and urbanized region of Po valley.

alt=/>
alt=/>
alt=/>

The BOLCHEM was run with an horizontal resolution of 20 km. The lower level was approximately at 20m above the surface. Two different simulations were accomplished for the period 1-7 July 1999. using the chemical mechanism SAPRC90 and CB-IV, respectively.

The simulated ozone concentrations (units in $LaTeX code: \mu g/m^3$) using the two chemical schemes at the lowest model level oare shown in the figures at the left side (left SAPRC90, right CB-IV). Figures refer to 2$LaTeX code: ^{nd}$ July at 15 UTC (40 hour simulation). The ozone fields show a very similar spatial distribution. Differences in the maximum concentrations can be appreciated: in general SAPRC-90 simulation shows higher values than CB-IV one.

The time series (figures below) of ozone concentration at three rural and one sub-urban (Torino Lingotto) stations are showed at figures below. The diurnal cycle is generally well reproduced by the two chemical mechanisms and the concentrations computed with the two schemes are similar.

alt=/>
alt=/>
alt=/>
alt=/>

## Model simulations of the ozone concentrations over Europe during the summer 2003 heat wave

This work ( Villani et al., 2005) presents the effect of different meteorological drivers and chemical mechanisms on the resulting chemical concentrations computed with BOLCHEMand CHIMERE(Vautard et al. ,2001). In particular, ozone concentrations were simulated with the two models for some days of the heat wave event of the summer 2003 over Europe, from 30$LaTeX code: ^{th}$ July to 3$LaTeX code: ^{rd}$ August. These five days are representative of the beginning of a particularly intense episode of photochemical pollution over Northern Europe.

The simulations were performed using the same emissions, initial and boundary conditions for the chemical species. The comparison aimed at investigating the O3 concentrations reproduced with:

• CHIMERE driven by different meteorological fields, which are modelled by meteorological models MM5 and BOLAM;
• CHIMERE and BOLCHEM, characterised by two different chemical mechanisms (MELCHIOR2 in CHIMERE, and SAPRC90 in BOLCHEM), but driven by the same meteorological fields (BOLAM).

Figures show the ozone concentration simulated using the different modelling sistems on 3$LaTeX code: ^{rd}$ August 2003 at 15UTC.
.

alt=/>
$LaTeX code: O_3 (\mu g/m^3)$ simulated with BOLCHEM
alt=/>

alt=/>
$LaTeX code: O_3 (\mu g/m^3)$ simulated with CHIMERE-BOLAM
alt= />
$LaTeX code: O_3 (\mu g/m^3)$ simulated with CHIMERE-MM5

References

R. Vautard, M. Beekmann, J. Roux, and D. Gomber. 2001. "Validation of a hybrid forecasting system for the ozone concentrations over the Paris area". Atmos. Environ., 35(14), 2449-2461.

M. G. Villani, P. D'Aulerio, M. D'Isidoro, A. Buzzi, F. Fierli, A. Maurizi, M. Mircea and F. Tampieri. "Model Simulations of the Ozone Concentrations over Europe During the Summer 2003 Heat Wave".First ACCENT Symposium, Urbino, Sept. 12th-16th 2005. Download extended abstract

## Saharan dust

In order to simulate the transport of Saharan dust over Italy, a dust emission scheme has been coupled to BOLCHEM. The dust forecast is a mandatory request for the air quality models since the mineral dust particles i) affect the ozone formation by uptaking gases as HNO3, H2O2, HO2, etc.; ii) alter the atmospheric radiation budget and consequently affect the convective activity and the latent heat fluxes in the areas where water evaporates from surface and iii) contribute substantially to the total aerosol mass usually employed in developing the EU regulations. The dust emission scheme presently implemented in BOLCHEM considers the distribution of preferential dust source areas, soil types and surface conditions like vegetation cover and soil moisture. The flux of dust at surface is calculated as a function of friction velocity that, in turn, depends on winds. Figures show the simulated dust field, integrated over all levels, and the MODIS image (right) at 16 july 2003, 12.35 UTC, respectively.

alt=/>
Simulated Saharan dust transport.
alt=/>
MODIS image (Source: NASA.)

## Point source release tests

work in progress... ...


text:
# Toronto Math Forum

## MAT334-2018F => MAT334--Tests => Final Exam => Topic started by: Victor Ivrii on December 18, 2018, 06:14:31 AM

Title: FE-P3
Post by: Victor Ivrii on December 18, 2018, 06:14:31 AM
Find all singular points, classify them, and find residues at these points of
$$f(z)= \tan (z) + z\cot^2(z);$$
infinity included.
Title: Re: FE-P3
Post by: hz12 on December 18, 2018, 11:13:44 AM
f(z) = $\frac{\mathrm{sin}\mathrm{}(z)}{\mathrm{cos}\mathrm{}(z)}+z\frac{\mathrm{cos}\mathrm{}\wedge 2(z)}{\mathrm{sin}\mathrm{}\wedge 2(z)}$
=$\frac{{{\mathrm{sin}}^{\mathrm{3}}\left(z\right) +\ }{\mathrm{zcos}}^{\mathrm{3}}\left(z\right)\ \ \ \ =g}{{\mathrm{cos} \left(z\right)\ }{sin}^2\left(z\right)\ \ \ \ =h}$

Cos(z)sin$\mathrm{\wedge}$2(z) = 0

cos(z) = 0 or sin$\mathrm{\wedge}$2(z) = 0

so z =k$\pi$ or $z=\frac{\pi }{2}+k\pi$

1, when z = k$\pi$

g = ${sin}^3(z)+z{cos}^3\left(z\right)\neq 0$                            h= ${\mathrm{cos} \left(z\right)\ }{sin}^2\left(z\right)=0$
$h^=-{sin}^3\left(z\right)+2{cos}^2\left(z\right){\mathrm{sin} \left(z\right)\ }=0$                    ${\ h}^{''}\neq 0$

So pole of order = 2

2, when z =$\frac{\pi }{2}+k\pi$

g = ${sin}^3(z)+z{cos}^3\left(z\right)\neq 0$                            h= ${\mathrm{cos} \left(z\right)\ }{sin}^2\left(z\right)=0$
$h^=-{sin}^3\left(z\right)+2{cos}^2\left(z\right){\mathrm{sin} \left(z\right)\ }\ \neq 0$

So pole of order = 1
Title: Re: FE-P3
Post by: Ziqi Zhang on December 18, 2018, 11:17:50 AM
I think when z=0, it should have pole of order 1. Because at z=0, the numerator is not 0 when taking derivative once and denominator is not 0 when taking derivative twice.
Title: Re: FE-P3
Post by: Ziqi Zhang on December 18, 2018, 11:28:38 AM
Residue at z=0: 0+cos2(0)=1

Residue at z=kπ, but k≠0: 0-2(kπ)cos(kπ)sin(kπ)+cos2(kπ)=(-1)n

Residue at z=0.5π+kπ: -1+0=-1
Title: Re: FE-P3
Post by: Zixuan Miao on December 18, 2018, 11:37:21 AM
for infinity case, it is a non-isolated singularity
Title: Re: FE-P3
Post by: Siying Li on December 18, 2018, 12:45:02 PM
I think the residue at 0 should be 0..
Here's my answer: (in calculation I ignore k for convenience)

${\rm f}\left({\rm z}\right){\rm =}{\tan \left({\rm z}\right)\ }{\rm +z}{{\cot }^{{\rm 2}} \left(z\right)=\frac{{\sin \left(z\right)\ }}{{\cos \left(z\right)\ }}+\frac{z{{\cos }^2 \left(z\right)\ }}{{{\sin }^2 \left(z\right)\ }}\ }=\frac{{{\sin }^3 (z)\ }+z{{\cos }^3 (z)\ }}{{\cos \left(z\right)\ }{{\sin }^2 \left(z\right)\ }}$
If ${\cos \left(z\right)\ }=0$, then ${\rm z=}\frac{\pi}{2} + k\pi$

For numerator, ${{\sin }^3 (\frac{\pi}{2})\ }+\frac{\pi}{2}{{\cos }^3 \left(\frac{\pi}{2}\right)\ne 0\ }$, the numerator has zero order of zeros

For denominator, ${\cos \left(\frac{\pi}{2}\right)\ }{{\sin }^2 \left(\frac{\pi}{2}\right)=0\ ,-{{\sin }^3 \left(\frac{\pi}{2}\right)\ }+\frac{\pi}{2}{{\cos }^2 \left(\frac{\pi}{2}\right)\ }{\sin \left(\frac{\pi}{2}\right)\ }\ne 0\ }$, the denominator has zeros of order 1

Then ${\rm f}\left({\rm z}\right)$ has simple pole ${\rm z=}\frac{\pi}{2}+k\pi$

If ${{\sin }^{{\rm 2}} \left(z\right)\ }=0$, then ${\rm z=0}$ or ${\rm z=}k\pi$

When ${\rm z=0}$,

For numerator, ${{\sin }^3 (0)\ }+0{{\cos }^3 \left(0\right)=0\ },3{{\sin }^2 \left(0\right)+{{\cos }^3 \left(0\right)\ }+3{{\cos }^2 (0)\ }{\rm sin}?(0)\ }\ \ne 0\$, the numerator has zero order of 1

For denominator, ${\cos \left(0\right)\ }{{\sin }^2 \left(0\right)=0\ ,-{{\sin }^3 \left(0\right)\ }+2{{\cos }^2 \left(0\right)\ }{\sin \left(0\right)\ }=0\ },\ -3{{\sin }^2 \left(0\right){\cos \left(0\right)\ }-4{\cos \left(0\right)\ }{{\sin }^{{\rm 2}} \left(0\right)\ }+2cos^3\left(0\right)\ne 0\ }$, the denominator has zeros of order 2

Then ${\rm f}\left({\rm z}\right)$ has simple pole ${\rm z=0}$

When ${\rm z=}k\pi$,

For numerator, ${{\sin }^3 (\pi )\ }+0{{\cos }^3 \left(\pi \right)\ne 0\ }$ the numerator has zero order of 0

For denominator, ${\cos \left(\pi \right)\ }{{\sin }^2 \left(\pi \right)=0\ ,-{{\sin }^3 \left(\pi \right)\ }+2{{\cos }^2 \left(\pi \right)\ }{\sin \left(\pi \right)\ }=0\ },\ -3{{\sin }^2 \left(\pi \right){\cos \left(\pi \right)\ }-4{\cos \left(\pi \right)\ }{{\sin }^{{\rm 2}} \left(\pi \right)\ }+2cos^3\left(\pi \right)\ne 0\ }$, the denominator has zeros of order 2

Then ${\rm f}\left({\rm z}\right)$ has pole ${\rm z=}k\pi$ with order 2

${\rm Res}\left({\rm f}\left({\rm z}\right),\frac{\pi}{2}+k\pi\right)=\frac{{{\sin }^3 (\frac{\pi }{{\rm 2}})\ }+\frac{\pi}{2}{{\cos }^3 (\frac{\pi}{2})\ }}{-{{\sin }^3 \left(\frac{\pi }{{\rm 2}}\right)\ }+2{{\cos }^2 \left(\frac{\pi }{{\rm 2}}\right)\ }{\sin \left(\frac{\pi }{{\rm 2}}\right)\ }}=\frac{1}{-1}=-1$
${\rm Res}\left({\rm f}\left({\rm z}\right),k\pi\right)=\frac{3{{\sin }^2 \left(\pi\right)+{{\cos }^3 \left(\pi\right)\ }+3{{\cos }^2 (\pi)\ }{\rm sin}(\pi)\ }}{1!}=\frac{-1}{1}=-1$
Since ${\rm f}\left({\rm z}\right)=\frac{{{\sin }^3 (z)\ }+z{{\cos }^3 (z)\ }}{{\cos \left(z\right)\ }{{\sin }^2 \left(z\right)\ }}=\frac{{\left(\int^{\infty }_0{{\left(-1\right)}^n\frac{z^{2n+1}}{\left(2n+1\right)!}}\right)}^3+\int^{\infty }_0{{\left(-1\right)}^n\frac{z^{2n+1}}{\left(2n\right)!}}}{\int^{\infty }_0{{\left(-1\right)}^n\frac{z^{2n}}{\left(2n\right)!}}\ {\left(\int^{\infty }_0{{\left(-1\right)}^n\frac{z^{2n+1}}{\left(2n+1\right)!}}\right)}^2}$ has no term of ${\left({\rm z-0}\right)}^{{\rm -}{\rm 1}}$, then
${\rm Res}\left({\rm f}\left({\rm z}\right),0\right)=0$
Title: Re: FE-P3
Post by: Yifei Wang on December 18, 2018, 12:55:26 PM
let $w = \frac{1}{z}$ then $Z = \frac{1}{w}$
${Z\to\infty}$ ${w\to0}$

$f(z) = f(\frac{1}{w}) = \frac{sin{\frac{1}{w}}}{cos\frac{1}{w}}+\frac{1}{w}\frac{cos^2{\frac{1}{w}}}{sin^2{\frac{1}{w}}}$

at ${w\to0}$, we have a non-isolated singularity
Title: FE-P3 official
Post by: Victor Ivrii on December 20, 2018, 05:04:59 AM
$\newcommand{\Res}{\operatorname{Res}}$
(a) Singular points are of $g(z)=\tan(z)$ and $h(z)=z\cot ^2(z)$, that is $z_n= (n+\frac{1}{2})\pi$ and $w_n = \pi n$.

(b) $z_n$ are simple poles and $\Res (f,z_n)= \Res (\tan(z), z_n)= \frac{\sin(z)}{(\cos (z))'}\bigr|_{z=z_n} =-1$.

$w_0$ is a simple pole
$$\Res (f,w_0)= \Res (z\cot^2(z), w_0)= \Res (z\cot (z) \times \cot(z), 0)= \Res (\cot(z),0)= 1$$
because $\lim _{z\to 0} z\cot(z)=1$.

$w_n$ with $n\ne 0$ are double poles and
\begin{align*}
\Res (f,w_n)=&\Res (z\cot^2 (z), w_n)= \Res ((\pi n +w)\cot^2 (w) , 0) =\\
&\pi n\Res (\cot^2 (w) , 0) +\Res (w\cot^2(w),0) =1
\end{align*}
because $\Res (\cot^2 (w) , 0)=0$ (since $\cot^2(w)$ is an even function) and $\Res (w\cot^2(w),0) =1$ we already calculated.

$\infty$ is a not isolated singularity and therefore residue here is not defined.


text:
RESEARCH GROUP

MATHEMATICAL STRUCTURES OF THE UNIVERSE

#### Selected publications

 Author:Title:Abstract:Journal:Full Record: ANDORAND NOT Author:Title:Abstract:Journal:Full Record: ANDORAND NOT Author:Title:Abstract:Journal:Full Record: from to   type ALL researchconferencereviewsbookspopular res per page (134 res)
 61. M. Heller, Z. Odrzygó¼d¼, L. Pysiak, W. SasinAnatomy of Malicious SingularitiesJournal of Mathematical Physics, vol. 48, pp. 092504-092511 (2007). Abstract:Abstract 62. Michael Heller, Leszek Pysiak and Wies³aw SasinConceptual Unification of Gravity and QuantaInternational Journal of Theoretical Physics, vol. 46, pp. 2494-2512 (2007). [abstract] [preprint] [journal] [download] Abstract:We present a model unifying general relativity and quantum mechanics. The model is based on the (noncommutative) algebra A on the groupoid Γ=E×G where E is the total space of the frame bundle over spacetime, and G the Lorentz group. The differential geometry, based on derivations of A , is constructed. The eigenvalue equation for the Einstein operator plays the role of the generalized Einstein’s equation. The algebra A , when suitably represented in a bundle of Hilbert spaces, is a von Neumann algebra ℳ of random operators representing the quantum sector of the model. The Tomita–Takesaki theorem allows us to define the dynamics of random operators which depends on the state φ. The same state defines the noncommutative probability measure (in the sense of Voiculescu’s free probability theory). Moreover, the state φ satisfies the Kubo–Martin–Schwinger (KMS) condition, and can be interpreted as describing a generalized equilibrium state. By suitably averaging elements of the algebra A , one recovers the standard geometry of spacetime. We show that any act of measurement, performed at a given spacetime point, makes the model to collapse to the standard quantum mechanics (on the group G). As an example we compute the noncommutative version of the closed Friedman world model. Generalized eigenvalues of the Einstein operator produce the correct components of the energy-momentum tensor. Dynamics of random operators does not “feel” singularities. 63. Michael Heller, Zdzis³aw Odrzygó¼d¼, Leszek Pysiak, and Wies³aw SasinAnatomy of malicious singularities J. Math. Phys., vol. 48, p. 092504 (2007). [abstract] [preprint] [journal] Abstract:As well known, the b boundaries of the closed Friedman world model and of Schwarzschild solution consist of a single point. We study this phenomenon in a broader context of differential and structured spaces. We show that it is an equivalence relation ρ, defined on the Cauchy completed total space math of the frame bundle over a given space-time, that is responsible for this pathology. A singularity is called malicious if the equivalence class [p0] related to the singularity remains in close contact with all other equivalence classes, i.e., if p0 ∊ cl[p] for every p ∊ E. We formulate conditions for which such a situation occurs. The differential structure of any space-time with malicious singularities consists only of constant functions which means that, from the topological point of view, everything collapses to a single point. It was noncommutative geometry that was especially devised to deal with such situations. A noncommutative algebra on math, which turns out to be a von Neumann algebra of random operators, allows us to study probabilistic properties (in a generalized sense) of malicious singularities. Our main result is that, in the noncommutative regime, even the strongest singularities are probabilistically irrelevant. 64. Leszek M. Soko³owskiGeneral relativity, gravitational energy and spin-two fieldInt. J. Geom. Meth. Mod. Phys. , vol. 4, pp. 1-23 (2007). [abstract] [preprint] Abstract:(Lectures given at the 42 Karpacz Winter School of Theoretical Physics, L±dek Zdrój, Poland, 6-11 February 2006, "Current Mathematical Topics in Gravitation and Cosmology"): In my lectures I will deal with three seemingly unrelated problems: i) to what extent is general relativity exceptional among metric gravity theories? ii) is it possible to define gravitational energy density applying field-theory approach to gravity? and iii) can a consistent theory of a gravitationally interacting spin-two field be developed at all? The connecting link to them is the concept of a fundamental spin-2 field. A linear spin-2 field encounters insurmountable inconsistencies when coupled to gravity. After discussing the inconsistencies of any coupling of the linear spin-2 field to gravity, I exhibit the origin of the fact that a gauge invariant field has the variational metric stress tensor which is gauge dependent. I give a general theorem explaining under what conditions a symmetry of a field Lagrangian becomes also the symmetry of the stress tensor. It is a conclusion of the theorem that any attempt to define gravitational energy density in the framework of a field theory of gravity must fail. Finally I make a very brief introduction to basic concepts of how a certain kind of a necessarily nonlinear spin-2 field arises in a natural way from vacuum higher derivative gravity theories. This specific spin-2 field consistently interacts gravitationally. 65. Andrzej WoszczynaMechanika zaburzeñ skalarnych w radiacyjnym wszech¶wieciePrace Komisji Astrofizyki PAU, vol. 11, p. 117 (2007). [pdf.pl] Abstract:Abstract 66. Leszek M. Soko³owski, Andrzej StaruszkiewiczOn the issue of gravitonsClass. Quant. Grav. , vol. 23, pp. 5907-5918 (2006). [abstract] [preprint] [journal] Abstract:We investigate the problem of whether one can anticipate any features of the graviton without a detailed knowledge of a full quantum gravity. Assuming that in linearized gravity the graviton is in a sense similar to the photon, we derive a curious large number coincidence between the number of gravitons emitted by a solar planet during its orbital period and the number of its nucleons. In Einstein's GR the analogy between the graviton and the photon is ill founded. A generic relationship between quanta of a quantum field and plane waves of the corresponding classical field is broken in the case of GR. The graviton cannot be classically approximated by a generic pp wave nor by the exact plane wave. Most important, the ADM energy is a zero frequency characteristic of any asymptotically flat spacetime and this means that any general relationship between energy and frequency is a priori impossible. In particular the formula $E=\hbar \omega$ does not hold. The graviton must have features different from those of the photon and these cannot be predicted from classical general relativity.
print all >>


text:
# Simple question to help understand the exponential map

Just trying to understand the exponential map in Lie groups. How do I answer this question:

Show that the Lie algebra of a Lie subgroup $H \subset G$ can be identified with $$\mathfrak{h}=\{X\in \mathfrak{g}:\exp(tX)\in H\text{ for all sufficiently small }t\in \mathbb{R}\}$$

?

I understand that there is a correspondence between Lie subalgebras $\mathfrak{h}\subset \mathfrak{g}$ and connected Lie subgroups $H \in G$, but what's the relationship with $\exp$ here?

Let $X\in \mathfrak{g}$, $X$ defines on $G$ a vector field invariant by left multiplication $\hat X(g)=dL_g.X$ where $L_g(u)=gu$. $exp(X)$ is the value of the flow $\phi_X$ of $\hat X$ at $1$. $dexp_0(\hat X)=X$, thus if $exp(tX)\in H$, ${d\over{dt}}exp(tX)=X\in T_IH=\mathfrak{h}$.
• I still don't quite fully understand. If $X \in \mathfrak{g}$, then isn't $X$ itself a left invariant vector field? Why do we need $\hat{X}$? Also why is $L_g(u)=gu.exp(X)$ instead of just $gu$? Sorry if this is trivial, my understanding of this topic is pretty poor at the moment. May 11, 2018 at 19:02
• For me, $X$ is an element of the tangent space at the identity $T_IG$ and $\hat X$, the left-invariant vector associated to $X$: $\hat X(I)=X$. May 11, 2018 at 20:14