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# Uluslararası Cebirsel GEometri Neşesi

Organizers: Alp Bassa, Türkü Özlüm Çelik, Özgür Esentepe, Ali Özgür Kişisel
Meeting Times: Wednesday 10:00 (EST) 18:00 (TSİ)
UCGEN is an online seminar around topics near algebraic geometry. As our first few speakers suggest, near algebraic geometry is a broad term which includes representation theory, commutative and homological algebra, arithmetic geometry and number theory and really anything that touches algebraic geometry.

Hosted by IMBM.
Contact: turkuozlum[at]gmail[dot]com
Click here to subscribe to our mailing list for upcoming talks and zoom links, here for a list of past talks and here for our YouTube channel.

### Upcoming Talks

February 24, 2021. Combinatorial Methods for Minkowski Tensors of Polytopes Büşra Sert (TU Dresden)
March 3, 2021. TBA Emrah Sercan Yılmaz (Boğaziçi University)
March 10, 2021. TBA Can Ozan Oğuz (Galatasaray University)
March 17, 2021. TBA Doğancan Karabaş (Northwestern University)
March 24, 2021. TBA Matt Habermann (University College London)
March 31, 2021. TBA Nursel Erey (Gebze Technical University)
April 7, 2021. TBA Olgur Çelikbaş (West Virginia University)
April 14, 2021. TBA Ilke Çanakçı (Free University of Amsterdam)

### Details

Speaker: Büşra Sert
Institution: TU Dresden
Title: Combinatorial Methods for Minkowski Tensors of Polytopes
Abstract: Intrinsic volumes of a convex body provide scalar data (volume, surface area, Euler characteristic etc. ) about the geometry of a convex body intrinsically, i.e., the data doesn't depend on the ambient space. Minkowski tensors are the tensor valued generalization of intrinsic volumes. They give not only scalar data on the geometry of a convex body, but also information about its shape, orientation etc.. Moreover, generating functions for moments of the uniform distribution on convex bodies provide us a way to extract entries of Minkowski volume tensors.

In this talk, we first give necessary background on Minkowski tensors and their connection to moments on polytopes. Then, we describe Minkowski "surface tensors", and focus on some methods to obtain their entries in the case of simplicial polytopes.

This is a joint work with Niklas Livchitz and Amy Wiebe.

Speaker: Sinem Odabaşı
Title: On induced cotorsion pairs in functor category.
Abstract: The question of interest that motivates our work is how to ensure that the category Add (A,R-Mod) of additive functors has a projective / injective model structure without putting any conditions on the ring R. Essentially, it is motivated by the classical projective/injective/flat model structures on the category Ch(R) of chain complexes of left R-modules.

While we have been working on this problem with my collegues, in a recent work of Henrik Holm and Peter Jorgensen published in arxiv, this problem is handled by using techniques/results in Gorenstein Homological Algebra.

Fortunately, our approach differs from theirs, and includes other contexts such as module category over a formal triangular matrix ring.

With this objective in mind, in this talk we will talk about how to build "possible" Hovey cotorsion pairs1 in Add (A, R-Mod), and later we will present an explicit characterization of their objects. The results obtained on these cotorsion pairs in Add (A, R-Mod) generalize the known results in the categories of chain complexes of R-modules and modules over a formal triangular matrix ring. It is a work in progress with Sergio Estrada and Manuel Cortes Izurdiaga.

1: There is a close relation between abelian model structures in abelian categories and Hovey pairs; see [Hov02]. That's why we focus on finding suitable Hovey pairs in Add (A, R-Mod).

[Hov02] Hovey, M. Cotorsion pairs, model category structures, and representation theory. Math Z 241, 553–592 (2002).

Speaker: Özgür Esentepe
Institution: University of Connecticut
Title: Annihilation of cohomology over curve singularities
Abstract: Hilbert's syzygy theorem implies that the second syzygy of every module over a polynomial ring S in two variables is projective. In fancy language, this means that $\mathrm{Ext}_S^3(M,N)$ vanishes for every pair of modules M,N. This is no longer true when we consider a quotient R of S by an ideal generated by a single polynomial f. In fact, for every i>0 there is at least one pair M,N such that $\mathrm{Ext}_R^i(M,N) \neq 0$. We investigate the ideal consisting of ring elements which uniformly annihilate all $\mathrm{Ext}_R^i(M,N)$ for sufficiently large i. I am dedicating this talk to students and academics of Boğaziçi University who are protesting against a rector appointed by the 12th president of Turkey and I will try my best to keep it accessible to a broad audience.

Speaker: Oğuzhan Yürük
Institution: Institut für Mathematik TU Berlin
Title: Understanding the Regions of Multistaionarity via Symbolic Nonnegativity Certificates
Abstract: Parameterized ordinary differential equation systems are crucial for modeling in biochemical reaction networks under the assumption of mass-action kinetics. Various questions concerning the signs of multivariate polynomials in positive orthant arise from studying the solutions' qualitative behavior with respect to parameter values. In this work, we utilize circuit polynomials to find symbolic certificates of nonnegativity in order to provide further insight into the number of positive steady states of the n-site phosphorylation cycle model. This is a joint work with Elisenda Feliu, Nidhi Kaihnsa and Timo de Wolff.

Speaker: Emre Şen
Institution: University of Iowa
Title: Quivers in Algebraic Geometry: Various Examples
Abstract: This will be an expository talk about quivers, and their representations. We will see why quivers naturally appear in the context of algebraic geometry and how they are useful to solve algebrogeometric problems. In this manner, we discuss various subjects: group actions with finitely many orbits, derived categories of coherent sheaves, toric vector bundles, exceptional sequences, quiver moduli etc.

Speaker: Levent Doğan
Institution: Technical University of Berlin
Title: Polynomial Time Algorithms for Torus Actions
Abstract: In this talk, we will consider three algorithmic problems, namely orbit equivalence, orbit closure intersection and orbit containment problem for actions of tori. We will describe the related invariant theory and show that all three problems admit polynomial time algorithms.

Speaker: Sümeyra Sakallı
Institution: University of Arkansas
Title: Symplectic 4-Manifolds on the Noether Line and between the Noether and Half Noether Lines
Abstract: In this talk, first we will review some main concepts and techniques in the smooth and symplectic 4-manifolds theory. Then we will discuss our constructions of exotic, simply connected and symplectic 4-manifolds on the Noether line and between the Noether and half Noether lines via pencils of complex curves of genus one and via symplectic surgeries. We will also present a completely geometric way of constructing certain configurations of Kodaira’s singularities in the rational elliptic surfaces, without using any monodromy arguments.

Speaker: Burçin Güneş
Institution: Sabancı University
Title: On nilpotent automorphism groups of function fields
Abstract: We study the automorphisms of a function field of genus $g \geq 2$ over an algebraically closed field of positive characteristic $p$. More precisely, we show that the order of a nilpotent subgroup $G$ of its automorphism group is bounded by $16(g − 1)$ when $G$ is not a $p$-group. We show that if $|G| = 16(g −1)$, then $g −1$ is a power of $2$. Furthermore, we provide an infinite family of function fields attaining the bound. This is a joint work with Nurdagül Anbar.

Speaker: İzzet Coşkun
Institution: University of Illinois at Chicago
Title: Brill-Noether Theorems for moduli spaces of sheaves on surfaces
Abstract: In this talk, I will describe several results on the cohomology of the general sheaf in a moduli space of sheaves on a projective surface. I will discuss joint work with Jack Huizenga on rational surfaces such as Hirzebruch surfaces and joint work with Howard Nuer and Kota Yoshioka on K3 surfaces.

Speaker: Sema Güntürkün
Institution: Amherst College
Title: On the Eisenbud-Green-Harris conjecture.
Abstract: A generalization of the Macaulay’s theorem on the growth of Hilbert functions of homogeneous ideals in $K[x_1,\ldots, x_n]$ is conjectured by Eisenbud, Green and Harris in the 90s. The conjecture, also known as the EGH conjecture, states that the lex-plus-powers ideals show an extremal behavior among the homogeneous ideals containing regular sequences in terms of their Hilbert functions.  In this talk, our focus will be on a case of the EGH conjecture for the homogeneous ideals containing a regular sequence of quadratic forms. This is a joint work with Mel Hochster.

Institution: Middle East Technical University
Title: Higher Structures in Einstein Gravity
Abstract: This is a talk on a recent investigation about higher structures in the theory of General Relativity. It can be also seen as a direct sequel of the previous talk “Higher Structures in Physics.” However, for the sake of completeness, the talk will include a brief summary of key ideas from the aforementioned talk. In that respect, we shall begin with revisiting the basics of moduli theory and derived algebraic geometry. Next, we will report some relevant constructions and results from our work encoding various stacky formulations of Einstein Gravity.

Speaker: Ayşegül Öztürkalan
Institution: Abdullah Gül University
Title: Loops in moduli spaces of real plane projective curves
Abstract: The space of real algebraic plane projective curves of a fixed degree has a natural stratification. The strata of top dimension consists of non-singular curves and are known up to curves of degree 6. Topology and, in particular, fundamental groups of individual strata have not been studied systematically. We study the stratum formed by non-singular sextics with the real part consisting of 9 ovals which lie outside each other and divide the set of complex points. Apparently this stratum has one of the most complicated fundamental groups. In the talk I will study its subgroups which come from strata of singular curves and originates from spaces of linear equivalent real divisors on a real cubic curve.

Speaker: Bahar Acu
Institution: ETH Zurich
Title: Understanding symplectic fillings of contact manifolds via algebraic varieties
Abstract: This talk is an attempt for a (pandemic-conscious) invitation to contact topology via an algebro-geometric approach with the caveat that we admit having little to no understanding of many concepts in algebraic geometry. A very useful strategy in studying topological manifolds is to factor them into smaller pieces. Briefly, an "open book decomposition" on an $n$-dimensional manifold (the open book) is a type of fibration over a circle that helps us study our manifold in terms of its $(n-1)$-dimensional fibers (the pages) and $(n-2)$-dimensional boundary of these fibers (the binding). Open books provide a natural framework for studying the topological properties of a geometric phenomenon called "contact structures" on smooth manifolds. In this talk, we aim to provide an exposition of results, some of which are fruits of several joint works, concerning "symplectic fillings" of contact manifolds given by certain classes of algebraic varieties using their "supporting" open books.

Speaker: Oğuz Şavk
Institution: Boğaziçi University
Title: Brieskorn spheres, homology cobordism and homology balls
Abstract: A classical question in low-dimensional topology asks which homology $3$-spheres bound homology $4$-balls. This question is fairly addressed to Brieskorn spheres $\Sigma(p,q,r)$. Since they are defined to be links of singularities $x^p+y^q+z^r=0$, Brieskorn spheres are algebro-geometric originated $3$-manifolds. Over the years, Brieskorn spheres also have been the main objects for the understanding of the algebraic structure of the integral homology cobordism group. In this talk, we will present several families of Brieskorn spheres which do or do not bound integral and rational homology balls. Also, we will investigate their positions in both integral and rational homology cobordism groups.

Speaker: Özhan Genç
Institution: Jagiellonian University
Title: Ulrich Trichotomy on del Pezzo Surfaces
Abstract: A vector bundle $\mathcal{E}$ on a projective variety $X$ in $\mathbb{P}^N$ is Ulrich if $H^∗(X,E(−k))$ vanishes for $1 ≤k ≤\dim(X)$. It has been conjectured by Eisenbud and Schreyer that every projective variety carries an Ulrich bundle. Even though this conjecture has not been proved or disproved, another interesting question is worth considering: classify projective varieties as Ulrich finite, tame or wild type with respect to families of Ulrich bundles that they support. In this talk, we will show that this trichotomy is exhaustive for certain del Pezzo surfaces with any given polarization. This talk is based on a joint work with Emre Coşkun.

Speaker: İrem Portakal
Institution: Otto von Guericke University Magdeburg
Title: Rigid toric matrix Schubert varieties
Abstract: In this talk, we introduce the usual torus action on matrix Schubert varieties. In the toric case we show that these varieties arise from a bipartite graph. We study the first order deformations of toric matrix Schubert varieties and we prove that it is rigid if and only if the three-dimensional faces of its associated (edge) cone are all simplicial.

Speaker: Enis Kaya
Institution: University of Groningen
Title: Explicit Vologodsky Integration for Hyperelliptic Curves
Date: October 14, 2020
Abstract: Let X be a curve over a p-adic field with semi-stable reduction and let ω be a meromorphic 1-form on X. There are two notions of p-adic integration one may associate to this data: the Berkovich–Coleman integral which can be performed locally; and the Vologodsky integral with desirable number-theoretic properties. In this talk, we present a theorem comparing the two, and describe an algorithm for computing Vologodsky integrals in the case that X is a hyperelliptic curve. We also illustrate our algorithm with a numerical example computed in Sage. This talk is partly based on joint work with Eric Katz.

Speaker: Selvi Kara
Institution: University of South Alabama
Title: Monomial Ideals of Graphs and Their Syzygies
Abstract: Given a homogeneous ideal $I$ in a polynomial ring $R=k[x_1, \ldots, x_n],$ we can describe the structure of $I$ by using its minimal free resolution. All the information related to the minimal free resolution of $I$ is encoded in its Betti numbers. However, it is a difficult problem to express Betti numbers of any homogeneous ideal in a general way. Due to this difficulty, it is common to focus on coarser invariants of $I$ or particular classes of ideals.

In this talk, we consider monomial ideals associated to graphs. We will discuss the Castelnuovo-Mumford regularity, projective dimension, and extremal Betti numbers of such ideals and provide formulas for these invariants in terms of the combinatorial data of their associated graphs. Results presented in this talk are from joint works with Biermann, O’Keefe, Lin, and Casiday.

Institution: Middle East Technical University
Title: Higher Structures in Physics
Date: September 30, 2020
Abstract: This is an overview of higher structures in physics. In this talk, we intend to outline the basics of derived algebraic geometry and its essential role in encoding the formal geometric aspects of moduli spaces of solutions to certain differential equations. Throughout the talk, we always study objects with higher structures in a functorial perspective, and we shall focus on algebraic local models for those structures. To be more precise, we shall be interested in derived geometric constructions and higher spaces for certain moduli problems associated with classical field theories and their defining equations, the so-called Euler-Lagrange equations.
To this end, the talk is organized into two main parts: In the first part of the talk, we shall revisit the naïve and algebro-geometric definition of a classical field theory together with some examples, and then we will establish the connection between classical field theories and moduli problems. In the second part of the talk, we first recall the basic aspects of moduli theory in a categorical perspective and explain how higher-categorical notions like stacks come into play to overcome certain technical problems naturally arising in many moduli problems. In the spirit of these discussions, we shall also give some examples from gauge theory and Einstein gravity.

Speaker: Umut Varolgüneş
Institution: Stanford University
Special Time: 9PDT, 12EST, 7TSi
Title: Homological mirror symmetry for chain type invertible polynomials
Abstract: I will start by giving a quick introduction to classical and symplectic Picard-Lefschetz theory. Then, I will explain the homological mirror symmetry (HMS) conjecture regarding invertible polynomials. Finally, I will sketch the A-side computation that goes into proving HMS in the chain type case. This is joint work with A. Polishchuk.

Speaker: Hülya Argüz
Institution: University of Versailles Saint-Quentin-En-Yvelines
Title: An algebro-geometric view on mirror symmetry
Abstract: Mirror symmetry is a phenomenon discovered by string theorists, which relates physical theories obtained using different deformation families of Calabi-Yau manifolds. An algebro--geometric approach to mirror symmetry, which uses tropical and log geometric tools to construct such families of Calabi--Yau manifolds, is provided by the Gross-Siebert program. In this talk we will review the most recent advances in this program, and particularly report on our joint work with Mark Gross.

Speaker: Mehmet Kıral
Institution: RIKEN AIP
Title: Kloosterman Sums for SL3 Long Word Element
Date: September 9, 2020
Time: 8:00 EST, 15:00 TSi Note the time change.
Abstract: Using the reduced word decomposition of the long word element of the Weyl group element of SL3, we give a nice expression for the long word Kloosterman sum. First classical Kloosterman sums, their importance, and matrix formulation will be introduced. This is joint work with Maki Nakasuji of Sophia University (Tokyo).

Speaker: Özge Ülkem
Institution: Heidelberg University
Title: Uniformization of the moduli space of generalized $\mathcal{D}$-elliptic sheaves
Date: September 2, 2020
Abstract: Drinfeld defined the notion of elliptic modules, which are now called Drinfeld modules, as an analogue of elliptic curves in the function field setting. To prove the Langlands correspondence in this context, Drinfeld studied moduli spaces of elliptic sheaves. The categories of elliptic sheaves and Drinfeld modules are equivalent under certain conditions. Since then, many generalizations of elliptic sheaves have been studied, such as $\mathcal{D}$-elliptic sheaves defined by Laumon, Rapoport and Stuhler and Frobenius-Hecke sheaves defined by Stuhler. In this talk we will give a brief introduction to the function field world and introduce a new generalization of elliptic sheaves, called generalized $\mathcal{D}$-elliptic sheaves. We will state a uniformization theorem for the moduli space of the latter and talk about the proof if time permits. This builds on work of Laumon-Rapoport-Stuhler, of Hartl and of Rapoport-Zink.

Speaker: Yusuf Barış Kartal
Institution: Princeton University
Title: p-adic analytic actions on Fukaya categories and iterates of symplectomorphisms
Date: August 26, 2020
Abstract: A theorem of Bell, Satriano and Sierra state that for a given smooth complex surface $X$ with an automorphism $\phi$ the set of natural numbers $n$ such that $\mathscr{Ext}^i(\mathscr{F},(\phi^*)^n(\mathscr{F}'))\neq 0$ is a union of finitely many arithmetic progressions and finitely many other numbers. Due to homological mirror symmetry conjecture, one can expect a symplectic version of this statement. In this talk, we will present such a theorem for a class of symplectic manifolds and symplectomorphisms isotopic to identity. The technique is analogous to its algebro-geometric counterpart: namely we construct p-adic analytic action on a version of the Fukaya category, interpolating the action of the iterates of the symplectomorphism.

Speaker: Emre Sertöz
Institution: Leibniz University Hannover
Title: Separating Periods of Quartic Surfaces
Date: August 19, 2020
Abstract: Kontsevich--Zagier periods form a natural number system that extends the algebraic numbers by adding constants coming from geometry and physics. Because there are countably many periods, one would expect it to be possible to compute effectively in this number system. This would require an effective height function and the ability to separate periods of bounded height, neither of which are currently possible.

In this talk, we introduce an effective height function for periods of quartic surfaces defined over algebraic numbers. We also determine the minimal distance between periods of bounded height on a single surface. We use these results to prove heuristic computations of Picard groups that rely on approximations of periods. Moreover, we give explicit Liouville type numbers that can not be the ratio of two periods of a quartic surface. This is ongoing work with Pierre Lairez (Inria, France).

Speaker: Emine Yıldırım
Institution: Queen's University
Title: Cluster Categories
Date: August 12, 2020
Abstract: Cluster Categories are introduced to understand cluster dynamics from the representation theory point of view. The subject has its roots in two important results in the literature that give us a glimpse of a relationship between cluster dynamics and representation theory. The first is that there is an one-to-one correspondence between the cluster variables of a finite type cluster algebra and the almost positive roots of the corresponding root system. The second is a well-known result by Gabriel that classifies finite representation type quivers by using positive roots of the corresponding root system. In this talk, after giving an overview of cluster categories, I will talk about a recent joint work with Charles Paquette on the generalization of discrete cluster categories.

### Past Talks

February 17, 2021.
On induced cotorsion pairs in functor category Sinem Odabaşı (Universidad Austral de Chile)
February 10, 2021. Annihilation of cohomology over curve singularities Özgür Esentepe (University of Connecticut)
February 3, 2021. Understanding the Regions of Multistaionarity via Symbolic Nonnegativity Certificates Oğuzhan Yürük (TU Braunschweig)
January 27, 2021. Quivers in Algebraic Geometry: Various Examples Emre Şen (University of Iowa)
January 13, 2021. Polynomial Time Algorithms for Torus Actions Levent Doğan (Technical University of Berlin)
January 20, 2021. Two-dimensional extended HQFTs with arbitrary targets Kürşat Sözer (University of Lille)
January 6, 2021. Symplectic 4-Manifolds on the Noether Line and between the Noether and Half Noether Lines  Sümeyra Sakallı (University of Arkansas)
December 16, 2020. On nilpotent automorphism groups of function fields Burçin Güneş (Sabancı University)
December 9, 2020. Brill-Noether Theorems for moduli spaces of sheaves on surfaces İzzet Coşkun (University of Illinois at Chicago)
December 2, 2020. On the Eisenbud-Green-Harris conjecture. Sema Güntürkün (Amherst College)
November 25, 2020. Higher Structures in Einstein Gravity Kadri İlker Berktav (Middle East Technical University)
November 18, 2020. Loops in moduli spaces of real plane projective curves Ayşegül Öztürkalan (Abdullah Gül University)
November 11, 2020. Understanding symplectic fillings of contact manifolds via algebraic varieties Bahar Acu (ETH Zurich)
November 4, 2020. Brieskorn spheres, homology cobordism and homology balls Oğuz Şavk (Boğaziçi University)
October 28, 2020. Ulrich Trichotomy on del Pezzo Surfaces Özhan Genç (Jagiellonian University)
October 21, 2020. Rigid toric matrix Schubert varieties Irem Portakal (Otto von Guericke University Magdeburg)
October 14, 2020. Explicit Vologodsky Integration for Hyperelliptic Curves Enis Kaya (University of Groningen)
October 7, 2020. Monomial Ideals of Graphs and Their Syzygies Selvi Kara (University of South Alabama)
September 30, 2020. Higher Structures in Physics Kadri Ilker Berktav (Middle East Technical University)
September 23, 2020. Homological mirror symmetry for chain type invertible polynomials Umut Varolgüneş (Stanford University)
September 16, 2020. An algebro-geometric view on mirror symmetry Hülya Argüz (University of Versailles Saint-Quentin-En-Yvelines)
September 9, 2020. Kloosterman Sums for SL3 Long Word Element Mehmet Kıral (RIKEN AIP)
September 2, 2020. Uniformization of the moduli space of generalized $\mathcal{D}$-elliptic sheaves Özge Ülkem (Heidelberg University)
August 26, 2020. p-adic analytic actions on Fukaya categories and iterates of symplectomorphisms Yusuf Barış Kartal (Princeton University)
August 19, 2020. Separating Periods of Quartic Surfaces Emre Sertöz (Leibniz University Hannover)
August 12, 2020. Cluster Categories Emine Yıldırım (Queen's University)


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# Adams Spectral sequence and Pontrjagin-Thom construction [Reference request]

I will be grateful for any reference for the following statements/claims.

1) Let’s consider the case of $p=2$ and the classic Adams spectral sequence with the $E_2$-term given by $\mathrm{Ext}_{A}(\mathbb{F}_2,\mathbb{F}_2)$. If $\alpha$ and $\beta$ are two permanent cycles in the Adams spectral sequence, converging to elements $f\in{_2\pi_i^s}$ and $g\in{_2\pi_j^s}$, then is it true that $\alpha\beta$ does converge to $fg$? If so, then does it follow from the construction of the spectral sequence or it is not that evident?

I suppose, I am really asking if the $\textbf{convergence}$ in the ASS is multiplicative?

Of course, on the other direction we know the answer to the above question is not always positive as there are elements in the stable homotopy ring which are represented by decomposable elements in the $E_2$-term of the ASS, yet they are not decomposable elements in ${_2\pi_*^s}$.

2) Is the isomorphism $\Omega_*^{framed}\to\pi_*^s$ coming from the Pontrjagin-Thom construction multiplicative?

I think I can prove this second statement, but I wonder if it has been worked out, or it is a folklore that comes from the construction. If so, does it generalise to other bordism theories?

Thank you!

format: plain


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# How do you simplify 1 /(3sqrt18) ?

1/(9sqrt2
1/(3sqrt18)=1/(3sqrt(9xx2)) = 1/(3xx3sqrt2 = $\frac{1}{9 \sqrt{2}}$


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# Given the equation (ax+b)(ax-b)=0, how do you solve for x?

Dec 21, 2016

$x = \pm \frac{b}{a}$

#### Explanation:

multiply out and rearrange for ""x

$\left(a x + b\right) \left(a x - b\right)$

this is difference of squares

${a}^{2} {x}^{2} - {b}^{2} = 0$

${a}^{2} {x}^{2} = {b}^{2}$

${x}^{2} = {b}^{2} / {a}^{2}$

x=+-sqrt(b^2/a^2

$x = \pm \frac{b}{a}$


text:
?

Free Version
Easy

# Frequency of Purple Light

SATPHY-GEQ1KN

A certain color of light towards the purple end of the visible spectrum has a wavelength of $420\ nm$ in a vacuum.

What is the frequency of this light?

The speed of light in a vacuum is $\text{3.00 x 10 8m/s}$.

A

$126\ Hz$

B

$1.26\times { 10 }^{ 11 }\ Hz$

C

$7.1\times { 10 }^{ 14 }\ Hz$

D

$1.4\times { 10 }^{ -15 }\ Hz$

E

$1.4\times { 10 }^{ -6 }\ Hz$


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# American Institute of Mathematical Sciences

ISSN:
1930-8337

eISSN:
1930-8345

All Issues

## Inverse Problems & Imaging

May 2008 , Volume 2 , Issue 2

Select all articles

Export/Reference:

2008, 2(2): 167-185 doi: 10.3934/ipi.2008.2.167 +[Abstract](1860) +[PDF](590.0KB)
Abstract:
Approximating non-Gaussian noise processes with Gaussian models is standard in data analysis. This is due in large part to the fact that Gaussian models yield parameter estimation problems of least squares form, which have been extensively studied both from the theoretical and computational points of view. In image processing applications, for example, data is often collected by a CCD camera, in which case the noise is a Guassian/Poisson mixture with the Poisson noise dominating for a sufficiently strong signal. Even so, the standard approach in such cases is to use a Gaussian approximation that leads to a negative-log likelihood function of weighted least squares type.
In the Bayesian point-of-view taken in this paper, a negative-log prior (or regularization) function is added to the negative-log likelihood function, and the resulting function is minimized. We focus on the case where the negative-log prior is the well-known total variation function and give a statistical interpretation. Regardless of whether the least squares or Poisson negative-log likelihood is used, the total variation term yields a minimization problem that is computationally challenging. The primary result of this work is the efficient computational method that is presented for the solution of such problems, together with its convergence analysis. With the computational method in hand, we then perform experiments that indicate that the Poisson negative-log likelihood yields a more computationally efficient method than does the use of the least squares function. We also present results that indicate that this may even be the case when the data noise is i.i.d. Gaussian, suggesting that regardless of noise statistics, using the Poisson negative-log likelihood can yield a more computationally tractable problem when total variation regularization is used.
2008, 2(2): 187-204 doi: 10.3934/ipi.2008.2.187 +[Abstract](2690) +[PDF](922.2KB)
Abstract:
The restoration of blurred images corrupted with impulse noise is a difficult problem which has been considered in a series of recent papers. These papers tackle the problem by using variational methods involving an L1-shaped data-fidelity term. Because of this term, the relevant methods exhibit systematic errors at the corrupted pixel locations and require a cumbersome optimization stage. In this work we propose and justify a much simpler alternative approach which overcomes the above-mentioned systematic errors and leads to much better results. Following a theoretical derivation based on a simple model, we decouple the problem into two phases. First, we identify the outlier candidates---the pixels that are likely to be corrupted by the impulse noise, and we remove them from our data set. In a second phase, the image is deblurred and denoised simultaneously using essentially the outlier-free data. The resultant optimization stage is much simpler in comparison with the current full variational methods and the outlier contamination is more accurately corrected. The experiments show that we obtain a 2 to 6 dB improvement in PSNR. We emphasize that our method can be adapted to deblur images corrupted with mixed impulse plus Gaussian noise, and hence it can address a much wider class of practical problems.
2008, 2(2): 205-224 doi: 10.3934/ipi.2008.2.205 +[Abstract](1536) +[PDF](549.4KB)
Abstract:
We present a new variational framework for simultaneous smoothing and estimation of apparent diffusion coefficient (ADC) profiles from High Angular Resolution Diffusion-weighted MRI. The model approximates the ADC profiles at each voxel by a 4th order spherical harmonic series (SHS). The coefficients in SHS are obtained by solving a constrained minimization problem. The smoothing with feature preserved is achieved by minimizing a variable exponent, linear growth functional, and the data constraint is determined by the original Stejskal-Tanner equation. The antipodal symmetry and positiveness of the ADC are accommodated in the model. We use these coefficients and variance of the ADC profiles from its mean to classify the diffusion in each voxel as isotropic, anisotropic with single fiber orientation, or two fiber orientations. The proposed model has been applied to both simulated data and HARD MRI human brain data . The experiments demonstrated the effectiveness of our method in estimation and smoothing of ADC profiles and in enhancement of diffusion anisotropy. Further characterization of non-Gaussian diffusion based on the proposed model showed a consistency between our results and known neuroanatomy.
2008, 2(2): 225-250 doi: 10.3934/ipi.2008.2.225 +[Abstract](1455) +[PDF](849.5KB)
Abstract:
Many imaging methods involve probing a material with a wave and observing the back-scattered wave. The back-scattered wave measurements are used to compute an image of the internal structure of the material. Many of the conventional methods make the assumption that the wave has scattered just once from the region to be imaged before returning to the sensor to be recorded. The purpose of this paper is to show how this restriction can be partially removed and also how its removal leads to an enhanced image, free of the artifacts often associated with the conventionally reconstructed image.
2008, 2(2): 251-269 doi: 10.3934/ipi.2008.2.251 +[Abstract](1710) +[PDF](427.5KB)
Abstract:
In this work we study localized electric potentials that have an arbitrarily high energy on some given subset of a domain and low energy on another. We show that such potentials exist for general $L^\infty_+$-conductivities in almost arbitrarily shaped subregions of a domain, as long as these regions are connected to the boundary and a unique continuation principle is satisfied. From this we deduce a simple, but new, theoretical identifiability result for the famous Calderón problem with partial data. We also show how to construct such potentials numerically and use a connection with the factorization method to derive a new non-iterative algorithm for the detection of inclusions in electrical impedance tomography.
2008, 2(2): 271-290 doi: 10.3934/ipi.2008.2.271 +[Abstract](1811) +[PDF](302.5KB)
Abstract:
We consider the inverse problem to identify coefficient functions in boundary value problems from noisy measurements of the solutions. Our estimators are defined as minimizers of a Tikhonov functional, which is the sum of a nonlinear data misfit term and a quadratic penalty term involving a Hilbert scale norm. In this abstract framework we derive estimates of the expected squared error under certain assumptions on the forward operator. These assumptions are shown to be satisfied for two classes of inverse elliptic boundary value problems. The theoretical results are confirmed by Monte Carlo simulations.
2008, 2(2): 291-299 doi: 10.3934/ipi.2008.2.291 +[Abstract](1976) +[PDF](145.3KB)
Abstract:
In this paper we derive convergence and convergence rates results of the quasioptimality criterion for (iterated) Tikhonov regularization. We prove convergence and suboptimal rates under a qualitative condition on the decay of the noise with respect to the spectral family of $T$$T$*. Moreover, optimal rates are obtained if the exact solution satisfies a decay condition with respect to the spectral family of $T$*$T$.
2008, 2(2): 301-315 doi: 10.3934/ipi.2008.2.301 +[Abstract](1512) +[PDF](235.5KB)
Abstract:
We consider an inverse scattering problem arising in target identification. We prove a local stability result of logarithmic type for the determination of a sound soft obstacle from the far field measurements associated to one single incident wave.

2018  Impact Factor: 1.469


text:
# Factor analysis

34,135pages on
this wiki

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

Factor analysis is a statistical method used to describe variability among observed variables in terms of fewer unobserved variables called factors. The observed variables are modeled as linear combinations of the factors, plus "error" terms. The information gained about the interdependencies can be used later to reduce the set of variables in a dataset. Factor analysis originated in psychometrics, and is used in behavioral sciences, social sciences, marketing, product management, operations research, and other applied sciences that deal with large quantities of data.

Factor analysis is often confused with principal components analysis. The two methods are related, but distinct, though factor analysis becomes essentially equivalent to principal components analysis if the "errors" in the factor analysis model (see below) are assumed to all have the same variance.

## ExampleEdit

The following example is a simplification for expository purposes, and should not be taken to be realistic. Suppose a psychologist proposes a theory that there are two kinds of intelligence, "verbal intelligence" and "mathematical intelligence", neither of which is directly observed. Evidence for the theory is sought in the examination scores, from each of 10 different academic fields, of 1000 students. If each student is chosen randomly from a large population, then each student's 10 scores are random variables. The psychologist's theory may say that, for each of the 10 academic fields, the score averaged over the group of all students who share some common pair of values for verbal and mathematical "intelligences" is some constant times their level of verbal intelligence plus another constant times their level of mathematical intelligence, i.e., it is a linear combination of those two "factors". The numbers, for this particular subject, by which the two kinds of intelligence are multiplied to obtain the expected score, are posited by the theory to be the same for all intelligence level pairs, and are called "factor loadings" for this subject. For example, the theory may hold that the average student's aptitude in the field of amphibiology is

{10 × the student's verbal intelligence} + {6 × the student's mathematical intelligence}.

Two bellends having identical degrees of verbal intelligence and identical degrees of mathematical intelligence may have different aptitudes in amphibiology because individual aptitudes differ from average aptitudes. That difference is called the "error" — a statistical term that means the amount by which an individual differs from what is average for his or her levels of intelligence (see errors and residuals in statistics).

The observable data that go into factor analysis would be 10 scores of each of the 1000 students, a total of 10,000 numbers. The factor loadings and levels of the two kinds of intelligence of each student must be inferred from the data. Even the number of factors (two, in this example) must be inferred from the data.

## Mathematical model of the same exampleEdit

In the example above, for i = 1, ..., 1,000 the ith student's scores are

$\begin{matrix}x_{1,i} & = & \mu_1 & + & \ell_{1,1}v_i & + & \ell_{1,2}m_i & + & \varepsilon_{1,i} \\ \vdots & & \vdots & & \vdots & & \vdots & & \vdots \\ x_{10,i} & = & \mu_{10} & + & \ell_{10,1}v_i & + & \ell_{10,2}m_i & + & \varepsilon_{10,i} \end{matrix}$

where

• xk,i is the ith student's score for the kth subject
• $\mu_k$ is the mean of the students' scores for the kth subject (assumed to be zero, for simplicity, in the example as described above, which would amount to a simple shift of the scale used)
• vi is the ith student's "verbal intelligence",
• mi is the ith student's "mathematical intelligence",
• $\ell_{k,j}$ are the factor loadings for the kth subject, for j = 1, 2.
• εk,i is the difference between the ith student's score in the kth subject and the average score in the kth subject of all students whose levels of verbal and mathematical intelligence are the same as those of the ith student,

In matrix notation, we have

$X=\mu+LF+\epsilon$

where

• X is a 10 × 1,000 matrix of observable random variables,
• μ is a 10 × 1 column vector of unobservable constants (in this case "constants" are quantities not differing from one individual student to the next; and "random variables" are those assigned to individual students; the randomness arises from the random way in which the students are chosen),
• L is a 10 × 2 matrix of factor loadings (unobservable constants),
• F is a 2 × 1,000 matrix of unobservable random variables,
• ε is a 10 × 1,000 matrix of unobservable random variables.

Observe that by doubling the scale on which "verbal intelligence"—the first component in each column of F—is measured, and simultaneously halving the factor loadings for verbal intelligence makes no difference to the model. Thus, no generality is lost by assuming that the standard deviation of verbal intelligence is 1. Likewise for mathematical intelligence. Moreover, for similar reasons, no generality is lost by assuming the two factors are uncorrelated with each other. The "errors" ε are taken to be independent of each other. The variances of the "errors" associated with the 10 different subjects are not assumed to be equal.

Note that, since any rotation of a solution is also a solution, this makes interpreting the factors difficult. See disadvantages below. In this particular example, if we do not know ex ante that the two types of intelligence are uncorrelated, then we cannot interpret the two factors as the two different types of intelligence. Even if they are uncorrelated, we cannot tell which factor corresponds to verbal intelligence and which corresponds to mathematical intelligence without an outside argument.

The values of the loadings L, the averages μ, and the variances of the "errors" ε must be estimated given the observed data X.

## Factor analysis in psychometrics

### History

Charles Spearman spearheaded the use of factor analysis in the field of psychology and is sometimes credited with the invention of factor analysis. He discovered that school children's scores on a wide variety of seemingly unrelated subjects were positively correlated, which led him to postulate that a general mental ability, or g, underlies and shapes human cognitive performance. His postulate now enjoys broad support in the field of intelligence research, where it is known as the g theory.

Raymond Cattell expanded on Spearman’s idea of a two-factor theory of intelligence after performing his own tests and factor analysis. He used a multi-factor theory to explain intelligence. Cattell’s theory addressed alternate factors in intellectual development, including motivation and psychology. Cattell also developed several mathematical methods for adjusting psychometric graphs, such as his "scree" test and similarity coefficients. His research led to the development of his theory of fluid and crystallized intelligence, as well as his 16 Personality Factors theory of personality. Cattell was a strong advocate of factor analysis and psychometrics. He believed that all theory should be derived from research, which supports the continued use of empirical observation and objective testing to study human intelligence.

### Applications in psychology

Factor analysis is used to identify "factors" that explain a variety of results on different tests. For example, intelligence research found that people who get a high score on a test of verbal ability are also good on other tests that require verbal abilities. Researchers explained this by using factor analysis to isolate one factor, often called crystallized intelligence or verbal intelligence, that represents the degree to which someone is able to solve problems involving verbal skills.

Factor analysis in psychology is most often associated with intelligence research. However, it also has been used to find factors in a broad range of domains such as personality, attitudes, beliefs, etc. It is linked to psychometrics, as it can assess the validity of an instrument by finding if the instrument indeed measures the postulated factors.

• Reduction of number of variables, by combining two or more variables into a single factor. For example, performance at running, ball throwing, batting, jumping and weight lifting could be combined into a single factor such as general athletic ability. Usually, in an item by people matrix, factors are selected by grouping related items. In the Q factor analysis technique, the matrix is transposed and factors are created by grouping related people: For example, liberals, libertarians, conservatives and socialists, could form separate groups.
• Identification of groups of inter-related variables, to see how they are related to each other. For example, Carroll used factor analysis to build his Three Stratum Theory. He found that a factor called "broad visual perception" relates to how good an individual is at visual tasks. He also found a "broad auditory perception" factor, relating to auditory task capability. Furthermore, he found a global factor, called "g" or general intelligence, that relates to both "broad visual perception" and "broad auditory perception". This means someone with a high "g" is likely to have both a high "visual perception" capability and a high "auditory perception" capability, and that "g" therefore explains a good part of why someone is good or bad in both of those domains.

• "...each orientation is equally acceptable mathematically. But different factorial theories proved to differ as much in terms of the orientations of factorial axes for a given solution as in terms of anything else, so that model fitting did not prove to be useful in distinguishing among theories." (Sternberg, 1977). This means all rotations represent different underlying processes, but all rotations are equally valid outcomes of standard factor analysis optimization. Therefore, it is impossible to pick the proper rotation using factor analysis alone.
• Factor analysis can be only as good as the data allows. In psychology, where researchers have to rely on more or less valid and reliable measures such as self-reports, this can be problematic.
• Interpreting factor analysis is based on using a “heuristic”, which is a solution that is "convenient even if not absolutely true" (Richard B. Darlington). More than one interpretation can be made of the same data factored the same way, and factor analysis cannot identify causality.

## Factor analysis in marketingEdit

The basic steps are:

• Identify the salient attributes consumers use to evaluate products in this category.
• Use quantitative marketing research techniques (such as surveys) to collect data from a sample of potential customers concerning their ratings of all the product attributes.
• Input the data into a statistical program and run the factor analysis procedure. The computer will yield a set of underlying attributes (or factors).
• Use these factors to construct perceptual maps and other product positioning devices.

### Information collection Edit

The data collection stage is usually done by marketing research professionals. Survey questions ask the respondent to rate a product sample or descriptions of product concepts on a range of attributes. Anywhere from five to twenty attributes are chosen. They could include things like: ease of use, weight, accuracy, durability, colourfulness, price, or size. The attributes chosen will vary depending on the product being studied. The same question is asked about all the products in the study. The data for multiple products is coded and input into a statistical program such as SPSS, SAS, Stata, and SYSTAT.

### Analysis Edit

The analysis will isolate the underlying factors that explain the data. Factor analysis is an interdependence technique. The complete set of interdependent relationships are examined. There is no specification of either dependent variables, independent variables, or causality. Factor analysis assumes that all the rating data on different attributes can be reduced down to a few important dimensions. This reduction is possible because the attributes are related. The rating given to any one attribute is partially the result of the influence of other attributes. The statistical algorithm deconstructs the rating (called a raw score) into its various components, and reconstructs the partial scores into underlying factor scores. The degree of correlation between the initial raw score and the final factor score is called a factor loading. There are two approaches to factor analysis: "principal component analysis" (the total variance in the data is considered); and "common factor analysis" (the common variance is considered).

Note that principal component analysis and common factor analysis differ in terms of their conceptual underpinnings. The factors produced by principal component analysis are conceptualized as being linear combinations of the variables whereas the factors produced by common factor analysis are conceptualized as being latent variables. Computationally, the only difference is that the diagonal of the relationships matrix is replaced with communalities (the variance accounted for by more than one variable) in common factor analysis. This has the result of making the factor scores indeterminate and thus differ depending on the method used to compute them whereas those produced by principal components analysis are not dependent on the method of computation. Although there have been heated debates over the merits of the two methods, a number of leading statisticians have concluded that in practice there is little difference (Velicer and Jackson, 1990) which makes sense since the computations are quite similar despite the differing conceptual bases, especially for datasets where communalities are high and/or there are many variables, reducing the influence of the diagonal of the relationship matrix on the final result (Gorsuch, 1983).

The use of principal components in a semantic space can vary somewhat because the components may only "predict" but not "map" to the vector space. This produces a statistical principal component use where the most salient words or themes represent the preferred basis. [ok]

• Usefulness depends on the researchers' ability to develop a complete and accurate set of product attributes - If important attributes are missed the value of the procedure is reduced accordingly.
• Naming of the factors can be difficult - multiple attributes can be highly correlated with no apparent reason.
• If the observed variables are completely unrelated, factor analysis is unable to produce a meaningful pattern (though the eigenvalues will highlight this: suggesting that each variable should be given a factor in its own right).
• If sets of observed variables are highly similar to each other but distinct from other items, Factor analysis will assign a factor to them, even though this factor will essentially capture true variance of a single item. In other words, it is not possible to know what the 'factors' actually represent; only theory can help inform the researcher on this.


text:
# Using the squeeze theorem to determine a limit $\lim_{n\to\infty} (n!)^{\frac{1}{n^2}}$

Currently learning how to use the squeeze theorem to determine a limit.

The exercise I'm working on is finding the limit of:

$\lim \ (n!)^{\frac{1}{n^2}}$

So far what I have is:

$\lim \ (n!)^{\frac{1}{n^2}} = \lim \ e^{\frac{1}{n^2}ln(n!)}$

So I know that $1 = e^0 \leq \lim \ e^{\frac{1}{n^2}\ln(n!)}$

But I need help determining what should be on the other side of the equality.

I'm trying to look at $\frac{\ln(n!)}{n^2}$ but I'm having trouble determining its limit as it approaches infinity.

Does $n^2$ increase faster than $\ln(n!)$? Could someone show me why which one increases faster?

If so the inequality would just be:

$1 = e^0 \leq \lim \ e^{\frac{1}{n^2}\ln(n!)} \leq e^0 = 1$

So our limit would just be $1$, right?

Hint:
Use the inequality $$n^{\frac{n}{2}} \leqslant {n!} \leqslant {\left(\frac{n+1}{2}\right)^n}, \;\; n>1,$$ which can be proved by induction.

Hint:

$$\log n!=\sum_{k=2}^n\log k\le n\log n\implies e^{\frac1{n^2}\sum_{k=2}^n\log k}\le e^{\frac1n\log n}=\sqrt[n]{\log n}$$

But also

$$\sqrt[n]{\log n}\le\sqrt[n]n\xrightarrow[n\to\infty]{}1$$


text:
# Solving an inequality

1. Jan 24, 2013

### Pranav-Arora

1. The problem statement, all variables and given/known data
The range of k for which the inequality $k\cos^2x-k\cos x+1≥0$ for all x, is
a. k<-1/2
b. k>4
c. -1/2≤k≤4
d. -1/2≤k≤2

2. Relevant equations

3. The attempt at a solution
I am not sure about how to begin with this. This seems to me a quadratic in cos(x) and here, the discriminant should be less than zero.
$k^2-4k<0$
This gives me $0<k<4$ but this is not given in any of the options.

Any help is appreciated. Thanks!

2. Jan 24, 2013

### I like Serena

Hey Pranav!

It seems you are not done yet.
Your solution would be right if cos x could take on any value.
However, cos x is limited in range.
This means that the actual solution will have to contain your solution and perhaps have more solution values.

Is there any answer that matches that?

Alternatively, you could try and find the minima and maxima of $\cos^2x - \cos x$.
Fill those in and solve for k.
But of course that is more work...

3. Jan 24, 2013

### SammyS

Staff Emeritus
For one thing, the discriminant should be non-negative.

Then as I like Serena pointed out, you need be sure that k is such that -1 ≤ cos(x) ≤ 1 .

4. Jan 24, 2013

### haruspex

Do you mean non-positive?

5. Jan 24, 2013

### I like Serena

Shall we stick to less than zero? ;p

6. Jan 24, 2013

### haruspex

No, I think SammyS' point is that it's <= 0.
Pranav, I suggest solving this by rewriting the inequality in the form <square involving cos x> <comparator> <function of k only>, but you'll have to treat different cases according to the sign of k.

7. Jan 24, 2013

### I like Serena

Sorry, my mistake.

8. Jan 24, 2013

### SammyS

Staff Emeritus
Yes, I like Serena, you are right !

I misread the problem.

Of course, the discriminant must be less than or equal to zero so that $\ \ k\cos^2(x)-k\cos(x)+1=0\ \$ has [STRIKE]no[/STRIKE] at most one real root .

Edited above, per haruspex (next post) .

Last edited: Jan 24, 2013
9. Jan 24, 2013

### haruspex

No, it can equal zero. One real root is ok; it just must not have two distinct real roots.

10. Jan 24, 2013

### ehild

The restriction that -1≤cosx≤1 can make the range of k wider than [0,4]. There can be roots of the equation ky^2-ky+1=0, but outside the interval (-1,1). The range should contain the interval [0,4] anyway. This is true for which of the given intervals?

ehild

Last edited: Jan 24, 2013
11. Jan 24, 2013

### SammyS

Staff Emeritus
As I like Serena pointed out, -1 ≤ cos(x) ≤ 1 .

Therefore, there are likely more permissible values of k than just those given by considering the discriminant.

The problem at hand is equivalent to finding the values of k such that $\ \ k\,t^2-k\,t+1\ge0\ \$ for -1 ≤ t ≤ 1 .

12. Jan 25, 2013

### Pranav-Arora

Thanks everyone for the replies!

Yes, discriminant should be less equal to zero. Sorry about that.
I am not sure if I have understood all the replies properly. Following haruspex' post, I can rewrite the given inequality as
$(\cos x-\frac{1}{2})^2≥\frac{k-4}{4k}$
How should I proceed from here?

@ehild and ILS: The option C includes [0,4] so it should be the answer.

13. Jan 25, 2013

### SammyS

Staff Emeritus
See the graph of $\ \displaystyle kt^2-kt+1\,,\$ with k = -1/4, restricted to -1 ≤ t ≤ 1 .

Because the domain of this function is restricted to [-1, 1], the function's range does not include any negative values.

from WolframAlpha

Of course, the graph of $\ \displaystyle k\cos^2(x)-k\cos(x)+1\,,\$ looks nothing like the above graph, but both graphs have the same range for k = -1/4 .

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Last edited: Jan 25, 2013
14. Jan 25, 2013

### ehild

That is not quite true. Depends on the sign of k.

ehild

15. Jan 25, 2013

### haruspex

As I warned, and as ehild has pointed out, you have to split it according to the sign of k. (And k=0 as a third case.) The above is for k > 0.
What is the range of values of the LHS?


text: OTHMAN, H. A. .; SIBIH, A. M. . Weak and strong forms of fuzzy $$\alpha$$-open (closed) sets and its applications. Journal of Advanced Studies in Topology, [S. l.], v. 9, n. 2, p. 100–112, 2022. Disponível em: http://m-sciences.com/index.php/jast/article/view/248. Acesso em: 26 jan. 2023.


text:
# Arduino

This tutorial covers the basics of different Arduinos, and how to implement common functions with them.

## The Board

The main Arduino boards witnessed being used in these applications are the Arduino Uno and Arduino Mega.

## The Uno

Good for smaller projects

• Has 14 digital input/output pins, 6 analog inputs (which can also be used as digital input/output pins), and a 5v as well as a 3.3v regulator
• 6 of the digital input/output pins can be used for PWM
• Pins 2 and 3 are usable for interrupts
• Pins 0 and 1 cannot be used as normal digital inputs/output pins.
• If you are going to power it externally, you have to use between 7 and 12 volts on the Vin pin, and the ground of your power source has to go to a GND pin. A 9 volt battery works well for this. Make sure you connect the hot to Vin and the ground/negative terminal to ground of the power supply, or else you can fry the board.

## The Mega

Good for bigger projects

• Has 54 digital input/output pins, 16 analog inputs (which can also be used as digital input/output pins), and a 5v as well as a 3.3v regulator
• 15 of the digital input/output pins can be used for PWM
• Pins 2, 3, 18, 19, 20, 21 are useable for interrupts.
• Pins 0 and 1 cannot be used as normal digital inputs/output pins.
• If you are going to power it externally, you have to use between 7 and 12 volts on the Vin pin, and the ground of your power source has to go to a GND pin. A 9 volt battery works well for this. Make sure you connect the hot to Vin and the ground/negative terminal to ground of the power supply, or else you can fry the board.

## Wiring:

### Limit Switch:

#### Example:

Whatever pin that is connected to the Normally Open pin of the limit switch, needs to be setup by using pinMode(pin#, INPUT_PULLUP);

In this example, setup the pinMode as:

pinMode(2, INPUT_PULLUP);

In this setup, by using digitalRead(pin#), if the switch is open, it will read as HIGH (1), and when the switch is closed, digitalRead will return LOW (0). This happens because with the INPUT_PULLUP activated on the pin, it activates a pullup resistor for that pin, which, when the switch is open, the pin gets pulled HIGH by internal circuitry, but when closed, it gets pulled LOW since it is now directly connected to ground.

So to use this intuitively, use !digitalRead(pin#); this will return HIGH when pressed, and LOW when not pressed.

## Motor Driver:

Example with L298 Compact Motor Driver available in Mechatronics Lab:

With this example, the yellow lines connected to pins 10 and 11 (which are PWM) are the enables for the motors. When the enable is HIGH, the motor is turned on. For PWM lines, you use analogWrite(pin#, pwmValue);, where pwmValue is an integer between 0-255, with 0 being off, and 255 being always HIGH, with inbetween values able to control speed if your motor is capable of that.

The blue and green lines can be connected to any digital pin, but in this example I kept them grouped. These are the direction pins for the motors. If both are LOW or both are HIGH, the motor will not move. But if direction pin 1 is HIGH and direction pin 2 is LOW, then the motor will move. When direction pin 1 is LOW and direction pin 2 is HIGH, the motor will move in the other direction.

WARNING: This will send current through the positive node in one orientation, but when the direction is reversed the current will go through the negative node of the motor, make sure you check wiring accordingly

Another thing with this motor driver is that you can either turn off the motor, or you can stall it. Stalling the motor holds the motor taught and doesn’t let it move while just turning it off will give play to the motor which is usually undesirable. To stall the motor, set the enable HIGH, and the two direction pins to either LOW or HIGH. To just turn it off, just set the enable to LOW.

Example code:

//declare pins
en1 = 11;
dir11 = 12;
dir21 = 13;
en2 = 10;
dir21 = 8
dir22 = 9;

//setup all pins as outputs
setup(){
pinMode(en1, OUTPUT);
pinMode(dir11, OUTPUT);
pinMode(dir12, OUTPUT);
pinMode(en2, OUTPUT);
pinMode(dir21, OUTPUT);
pinMode(dir22, OUTPUT);
}

loop(){
//spin motors one way at half speed
analogWrite(en1, 128);
digitalWrite(dir11, HIGH);
digitalWrite(dir12, LOW);
analogWrite(en2, 128);
digitalWrite(dir21, HIGH);
digitalWrite(dir22, LOW);
delay(1000);
//stall motors
digitalWrite(en1, HIGH);
digitalWrite(dir11, HIGH);
digitalWrite(dir12, HIGH);
digitalWrite(en2, HIGH);
digitalWrite(dir21, LOW);
digitalWrite(dir22, LOW);
delay(1000);
//turn motors the other way at 3/4 speed
analogWrite(en1, 192);
digitalWrite(dir11, LOW);
digitalWrite(dir12, HIGH);
analogWrite(en2, 192);
digitalWrite(dir21, LOW);
digitalWrite(dir22, HIGH);
delay(1000);
//turn motors off
digitalWrite(en1, LOW);
digitalWrite(en2, LOW);
delay(1000);
}


text:
# Help in two improper integrals

For a physics problem, I need to evaluate two integrals: $$I_n=\int_{0}^{\infty} f(x) dx=\int_{0}^{\infty} \frac{x^2 \sin^2x}{(n^2\pi^2-x^2)^2} dx,$$ and $$J_n=\int_{0}^{\infty} \frac{x^2 \cos^2x}{((n+1/2)^2\pi^2-x^2)^2} dx, n=1,2,3..$$ At Mathematica the value of these integral turns out to be $$\pi/4$$ which is independent of $$n$$.

$$I_n$$ has got a pole of order 2 at $$x=n\pi$$ and when I calculate the residue at $$x=n \pi$$ is zero. But on the other hand I find that $$\lim_{x \to n\pi} f(x)=0,$$ so $$x=n\pi$$ may not be a pole of $$f(x)$$. I face the same problem in the case of $$J_n$$. Also $$I_0=\pi/2$$ is a standard integral. In any case, I want to evaluate both $$I_n$$ and $$J_n$$. Please help me.

• Try to use feynman technique – Aditya Dwivedi Jan 26 at 8:13
• Indeed $x=n \pi$ is a removable pole. – Crostul Jan 26 at 8:17

$$I_n=\int_{0}^{\infty} \frac{x^2 \sin^2x}{(n^2\pi^2-x^2)^2} dx$$. Integrating by part end expanding integration to the whole axis (the integrand is even) we get

$$I_n=-\frac{1}{2}\int_{0}^{\infty} \frac{\sin^2x+x\sin2x}{(n^2\pi^2-x^2)} dx=-\frac{1}{4}\int_{-\infty}^{\infty} \frac{x\sin2x}{(n^2\pi^2-x^2)}dx-\frac{1}{8}\int_{-\infty}^{\infty} \frac{1-\cos2x}{(n^2\pi^2-x^2)}dx=I_1+I_2$$.

Then we choose the closed contour $$C$$ in upper half of the complex plane:

To close the contour, we added two small half-circles $$C_1$$ and $$C_2$$ of radius $$r$$ - around $$x=-n\pi$$ and $$x=n\pi$$ $$(r\to0)$$, and a big circle of radius $$R$$ ($$R\to\infty$$). The required integral $$I_n$$ is equal to the principal value of the integral along the axis. Then, $$\sin2x=Im \exp(i2x)$$ and $$\cos2x=Re\exp(2ix)$$.

The integral over the closed contour $$\oint_C=0$$ (no singular points inside the contour), therefore$$I_n+I_{C1}+I_{C2}+I_R=0$$. It is easy to evaluate that $$I_R\to0$$ at $$R\to\infty$$ (using Jordan's lemma), so $$I_n=-I_{C1}-I_{C2}$$

$$I_1=-\frac{1}{4}Im\int_{-\infty}^{\infty} \frac{xe^{2ix}}{(n^2\pi^2-x^2)}dx=Im(-I_{C1}-I_{C2})=\frac{1}{4}Im\left((-\pi{i})[\frac{-\pi{n}\exp(-2\pi{in})}{2\pi{n}}+\frac{\pi{n}\exp(2\pi{in})}{-2\pi{n}}]\right)$$

$$I_1=\frac{\pi}{4}$$

Making similar calculations it is easy to show that $$I_2=0$$ Finally, $$I_n=I_1+I_2=\frac{\pi}{4}$$

$$J_n$$ can be evaluated in the same way.

Let is use $$\int_{n\pi}^{\infty} \frac{\sin^2 t}{t^2} dt=\frac{\pi}{2}-\text{Ei}(2n \pi)$$ Then $$I_n=\int_{0}^{\infty} \frac{x^2 \sin^2 x}{x^2-n^2\pi^2} dx$$ $$=\frac{1}{4}\int_{0}^{\infty} dx \left(\frac{\sin^2 x}{(x-n\pi)^2}+\frac{\sin^2 x}{(x+n\pi)^2}+\frac{\sin^2 x}{4n\pi(x-n\pi)}-\frac{\sin^2 x}{4n\pi(x+n\pi)}\right)$$ $$I_n=\frac{1}{4}[\pi/2-\text{Ei}(2n\pi)+\pi/2+\text{Ei}(2n\pi)]+\int_{-n\pi}^{n\pi} \frac{\sin^2 t}{t}dt=\frac{\pi}{4}$$ Similarly $$J_n=\int_{0}^{\infty} \frac{x^2 \cos^2 x}{x^2-m^2\pi^2} dx, m=n+1/2$$ $$=\frac{1}{4}\int_{0}^{\infty} dx \left(\frac{\cos^2 x}{(x-m\pi)^2}+\frac{\cos^2 x}{(x+m\pi)^2}+\frac{\cos^2 x}{4m\pi(x-m\pi)}-\frac{\cos^2 x}{4m\pi(x+m\pi)}\right)$$ $$I_n=\frac{1}{4}[\pi/2-\text{Ei}(2m\pi)+\pi/2+\text{Ei}(2m\pi)]+\int_{-m\pi}^{m\pi} \frac{\sin^2 t}{t}dt=\frac{\pi}{4}$$


text:
Given a string s of lowercase letters, you need to find the maximum number of non-empty substrings of s that meet the following conditions:

1. The substrings do not overlap, that is for any two substrings s[i..j] and s[k..l], either j < k or i > l is true.
2. A substring that contains a certain character c must also contain all occurrences of c.

Find the maximum number of substrings that meet the above conditions. If there are multiple solutions with the same number of substrings, return the one with minimum total length. It can be shown that there exists a unique solution of minimum total length.

Notice that you can return the substrings in any order.

Example 1:

Input: s = "adefaddaccc"
Output: ["e","f","ccc"]
Explanation: The following are all the possible substrings that meet the conditions:
[
"ef",
"e",
"f",
"ccc",
]
If we choose the first string, we cannot choose anything else and we'd get only 1. If we choose "adefadda", we are left with "ccc" which is the only one that doesn't overlap, thus obtaining 2 substrings. Notice also, that it's not optimal to choose "ef" since it can be split into two. Therefore, the optimal way is to choose ["e","f","ccc"] which gives us 3 substrings. No other solution of the same number of substrings exist.

Example 2:

Input: s = "abbaccd"
Output: ["d","bb","cc"]
Explanation: Notice that while the set of substrings ["d","abba","cc"] also has length 3, it's considered incorrect since it has larger total length.

Constraints:

• 1 <= s.length <= 10^5
• s contains only lowercase English letters.

## Solution: Greedy

Observation: If a valid substring contains shorter valid strings, ignore the longer one and use the shorter one.
e.g. “abbeefba” is a valid substring, however, it includes “bbeefb”, “ee”, “f” three valid substrings, thus it won’t be part of the optimal solution, since we can always choose a shorter one, with potential to have one or more non-overlapping substrings. For “bbeefb”, again it includes “ee” and “f”, so it won’t be optimal either. Thus, the optimal ones are “ee” and “f”.

1. We just need to record the first and last occurrence of each character
2. When we meet a character for the first time we must include everything from current pos to it’s last position. e.g. “abbeefba” | ccc, from first ‘a’ to last ‘a’, we need to cover “abbeefba”
3. If any character in that range has larger end position, we must extend the string. e.g. “abcabbcc” | efg, from first ‘a’ to last ‘a’, we have characters ‘b’ and ‘c’, so we have to extend the string to cover all ‘b’s and ‘c’s. Our first valid substring extended from “abca” to “abcabbcc”.
4. If any character in the covered range has a smallest first occurrence, then it’s an invalid substring. e.g. ab | “cbc”, from first ‘c’ to last ‘c’, we have ‘b’, but ‘b’ is not fully covered, thus “cbc” is an invalid substring.
5. For the first valid substring, we append it to the ans array. “abbeefba” => ans = [“abbeefba”]
6. If we find a shorter substring that is full covered by the previous valid substring, we replace that substring with the shorter one. e.g.
“abbeefba” | ccc => ans = [“abbeefba”]
abbeefba” | ccc => ans = [“bbeefb”]
“abbeefba” | ccc => ans = [“ee”]
7. If the current substring does not overlap with previous one, append it to ans array.
“abbeefba” | ccc => ans = [“ee”]
“abbeefba” | ccc => ans = [“ee”, “f”]
“abbeefbaccc” => ans = [“ee”, “f”, “ccc”]

Time complexity: O(n)
Space complexity: O(1)

## C++

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text:
Python: Create submatrices from long/narrow table and save in HDF5

Hello dear StackExchange Community,

I have written a small program which is doing the following. In short:

• it reads in a table in narrow format
• creates submatrices from it
• saves it in HDF5 format

The long table is actually representing a huge matrix and the data has the following shape (example not real):

POS1    POS2    VALUE
146000  146000  42
146000  147000  69
147000  147000  13
.       .       .
.       .       .
.       .       .

The data comes from experimental measurements and is not formatted user friendly. POS 1 and POS 2 represent coordinates of a small cluster on a big grid(matrix) and VALUE is a representing how strong these are associated. It's quite similar to a correlation matrix. The distance between positions is evenly sized and always has a distance of 1k (they represent bins). The goal is to compare every coordinate/bin vs all others. so if have n bins we get a matrix with n x n Values. Unfortunately not all comparisons are in the table. Some are missing.

For my project I need to subset this big matrix into smaller ones, while keeping track of the coordinates, so I can process them individually. I decided to go for 128x128 matrices.

Also I needed a solution how I can store the data efficiently to able to access it in a fast way for different ML applications. I decided to go for HDF5 using the h5py library. For every 128x128 submatrix I create a dataset in the HDF5 file with the name of POS1-POS2. This way I have the coordinates stored as the dataset name, which makes them easy to track, since all of them represent 1k bins.

While my code actually runs, it runs pretty slow. I could rewrite it in c++ using the eigen lib or try Rust, since I am currently learning it, but I thought I stick to Python and try to optimize it. Maybe someone here could help? Maybe numba or pypy could help here? Until now I did not make that much of a good experience with numba regarding more complicated code where one does not just crunch numbers (where it really shines).

Any suggestions and optimization are welcome! Many thanks in advance!

I separated the matrix class I wrote into another file:

from typing import List, Tuple

import numpy as np
import pandas as pd

class ImageMatrix:
""" Class to store and fill 128x128 Image from long table """

def __init__(self, x_axis_start: int = 0, y_axis_start: int = 0, mat_dims: Tuple[int, int] = (128, 128)):

# axis' to know where submatrix belongs to in "whole" matrix
self.x_axis: List[int] = [i for i in range(x_axis_start, x_axis_start + mat_dims[0] * 1_000, 1000)]
self.y_axis: List[int] = [j for j in range(y_axis_start, y_axis_start + mat_dims[1] * 1_000, 1000)]

self.mat_dims: Tuple[int, int] = mat_dims
# create empty mat to fill up
self.image_matrix: np.ndarray = np.zeros(mat_dims).astype(int)

def __str__(self) -> str:
return(f"Matrix of size: {self.mat_dims}\n"
f"coordinates starting for x: {self.x_axis[0]} and y: {self.y_axis[0]}.")

def print_matrix(self) -> None:
"""Print Matrix to the screen"""
print(self.image_matrix)

def add_value(self, value:int , x_coordinate:int, y_coordinate:int) -> None:
"""Adds a value to the matrix at given coordinates (x: column, y: row)"""

self.image_matrix[self.y_axis.index(y_coordinate)][self.x_axis.index(x_coordinate)] = value

def fill_from_pandas(self, pandas_df: pd.DataFrame) -> None:
"""
Fills up the matrix from a pandas long table
assuming the table has following structure:
BIN1    BIN2    VALUE
"""
for i in range(0,len(pandas_df)):
self.add_value(pandas_df.iloc[i,:][2], pandas_df.iloc[i,:][0], pandas_df.iloc[i,:][1] )

def fill_from_pandas_symmetric(self, pandas_df: pd.DataFrame) -> None:
"""
Fills up the matrix and makes it symetric from a pandas long table
assuming the table has following structure:
BIN1    BIN2    VALUE
"""
self.fill_from_pandas(pandas_df)

tmp = np.tril(self.image_matrix) + np.triu(self.image_matrix.T,1)
self.image_matrix = tmp

Here's the main script:

import argparse
from time import time

import h5py
import numpy as np
import pandas as pd
from termcolor import colored

import imagematrix

def load_hic_from_csv(path_to_file: str) -> pd.DataFrame:
"""
Input: path to T/CSV file
Outout: pandas dataframe

Expects t/csv to have the following format and no header:

COL 1                  COL 2                  COL 3

"""
return pd.read_csv(path_to_file, header=None, delim_whitespace=True, names=["POS1","POS2","VALUE"])

def create_hdf5_dataset(long_table: pd.DataFrame, file_name: str) -> None:
"""
Input:
- a pandas dataframe containing the hi-C counts
- file name to write hdf5 file to
- name of dataset group

Output:
None. Creates an HDF5 File in specified dir

"""

# get first and last bin
start = long_table['POS1'].min()
end = long_table['POS2'].max()
n_iters = int((end -start)/128_000)
hf = h5py.File(file_name, 'a')

# loop throug main matrix in 128k steps
for counter, i in enumerate(range(start, end, 128_000)):
# print steps every 10 steps
if(counter % 10 == 0):
print(f"[{counter}/{n_iters}]")

# loop throug main matrix in 128k steps
for j in range(start, end, 128_000):
# select subregion from matrix
local_region = long_table[(long_table['POS1'] >= i) & (long_table['POS1'] <= i + 127_000 )&\
(long_table['POS2'] >= j) & (long_table['POS2'] <= j + 127_000 )]

# instantiate the matrix
My_matrix = imagematrix.ImageMatrix(x_axis_start=i, y_axis_start= j)

# mirror matrix if it's symmetric -> long table does not contain that values
if(i == j):
My_matrix.fill_from_pandas_symmetric(local_region)
else:
My_matrix.fill_from_pandas(local_region)

hf.create_dataset(f"{i}-{j}", data=My_matrix.image_matrix, compression="lzf")

# close file
hf.close()
print(colored(f"Done processing: {file_name}!", 'green'))

def main() -> None:

argsParser = argparse.ArgumentParser(description = "Write diagonal sub matrices to hdf5 file in 128 x 128 ")
argsParser.add_argument('--input_file', '-i', type = str, required = True, help="Long table containing POS1, POS2 and VALUES")
argsParser.add_argument('--output_file', '-o', type = str, required= True, help="A string with the path to the output file")

# parse arguments
args =argsParser.parse_args()

DF = load_hic_from_csv(args.input_file)
create_hdf5_dataset(DF, args.output_file)

if __name__ == "__main__":
main()

Here's a longer example input file. The ones I am currently working with are actually tab separated and not csvs anymore.

Furthermore maybe i should mention that these files can get quite big - a few GB.

7320000        17351000        3.0
17321000        17351000        1.0
17322000        17351000        2.0
17323000        17351000        2.0
17324000        17351000        4.0
17325000        17351000        4.0
17326000        17351000        1.0
17327000        17351000        1.0
17328000        17351000        2.0
17329000        17351000        1.0
17332000        17351000        1.0
17333000        17351000        5.0
17340000        17351000        4.0
17341000        17351000        6.0
17342000        17351000        6.0
17343000        17351000        5.0
17344000        17351000        3.0
17345000        17351000        4.0
17346000        17351000        4.0
17347000        17351000        9.0
17348000        17351000        1.0
17349000        17351000        4.0
17350000        17351000        77.0
17351000        17351000        48.0
16848000        17352000        1.0
16864000        17352000        1.0
16865000        17352000        1.0
16874000        17352000        1.0
16877000        17352000        1.0
16884000        17352000        4.0
16895000        17352000        1.0
16900000        17352000        1.0
16910000        17352000        1.0
16913000        17352000        1.0
16928000        17352000        1.0
16936000        17352000        1.0
16939000        17352000        1.0
16943000        17352000        2.0
16946000        17352000        1.0
16954000        17352000        1.0
17003000        17352000        1.0
17008000        17352000        1.0
17013000        17352000        1.0
17018000        17352000        1.0
17024000        17352000        1.0
17030000        17352000        1.0
17031000        17352000        1.0
17034000        17352000        1.0
17035000        17352000        2.0
17037000        17352000        1.0
17042000        17352000        1.0
17055000        17352000        1.0
17062000        17352000        1.0
17064000        17352000        1.0
17065000        17352000        1.0
17066000        17352000        1.0
17068000        17352000        1.0
17072000        17352000        1.0
17076000        17352000        2.0
17078000        17352000        1.0
17079000        17352000        1.0
17085000        17352000        1.0
17086000        17352000        1.0
17089000        17352000        1.0
17091000        17352000        2.0
17092000        17352000        1.0
17094000        17352000        3.0

• Your function is called load_hic_from_csv but your sample data don't actually look like CSV; they look like a space-separated, fixed-field format. Is this true? Can you show a larger data sample (perhaps 50 rows)? Jan 2 at 17:26
• I updated my post :) Jan 2 at 17:56
• Will any of those associativity values have anything other than 0 past the decimal? Jan 3 at 3:31
• No, this is just the way they were saved by the software. Jan 3 at 5:57

1 Answer

POS 1 and POS 2 represent coordinates of a small cluster on a big grid (matrix)

Unfortunately not all comparisons are in the table. Some are missing.

This is a job for sparse matrices. The selection of which sparse matrix format to use is non-trivial but BSR lends itself well to direct construction from your data.

I'd cut out Pandas and use Numpy file loading directly; loadtxt is trivially easy for processing your format. I assume that your third column is all int; if not, you can use a struct-dtype.

Divide out the factor of 1000 from your coordinates.

Consider ejecting HDF5 and using standard Numpy npz. Think: you had already been loading the entire dataframe of coordinates into memory all at once, which means that it's reasonable to assume you should be able to do the same for a sparse matrix which scales in the same way. You can save and load this sparse matrix in one shot, and when you load it and do (mystery) processing, you can slice it for segmentation then. The slicing operation will be much easier in the sparse representation and you can lose most of your code.

Suggested

import argparse

import numpy as np
import scipy.sparse
from scipy.sparse import spmatrix

def load_hic(path: str) -> np.ndarray:
return np.loadtxt(fname=path, dtype=int)

def create_dataset(long_table: np.ndarray) -> spmatrix:
data = long_table[:, 2]
ij = long_table[:, :2].T // 1_000
return scipy.sparse.bsr_matrix((data, ij))

def parse_args() -> argparse.Namespace:
args_parser = argparse.ArgumentParser(
description="Convert a coordinate-list file to a sparse NPZ file")
args_parser.add_argument('--input-file', '-i', type=str, required=True,
help="Long table containing POS1, POS2 and VALUES")
args_parser.add_argument('--output-file', '-o', type=str, required=True,
help="A string with the path to the output file")
return args_parser.parse_args()

def main() -> None:
args = parse_args()

array = load_hic(args.input_file)
sparse = create_dataset(array)
scipy.sparse.save_npz(args.output_file, sparse)

if __name__ == "__main__":
main()

• Really nice. Thank you! I actually wrote a version which only used numpy and hdf5 and then ditched it because it made no real difference. But this solution is really nice. I completely forgot about sparse matrices. I will try that later today! I kind of already had the feeling that i am doing things too complicated. I hope this will solve my needs in creating submatrices :) Jan 3 at 5:51


text:
Question

Most people assume that random numbers are actually more orderly than they really are. For example, they assume that if you draw 50 random numbers, Nature will somehow almost ensure that you have 25 even numbers and 25 odd ones, or very close to that. Draw 50 random numbers from Table E.9 in Appendix E and calculate the proportion of even numbers. Then do this two more times and record the proportion of even numbers that you obtained each time. Do these data look the way you would expect them to look?

Sales0
Views21
• CreatedSeptember 22, 2015
• Files Included


text:
# Evaluate $\displaystyle\mathop {\lim }\limits_{n \to \infty } {\left( {\frac{{{a^n} + {a^{2n}}}}{{1 + a}}} \right)^{1/n}}$ where $0<a<1$

Here is my working, feel free to add comments if you see anything wrong with it, thanks!

Since $0<a<1$, we have $a^{n}<a^{n}+a^{2n}<2a^{n}$, hence $a{\left( {\frac{1}{{1 + a}}} \right)^{1/n}} = {\left( {\frac{{{a^n}}}{{1 + a}}} \right)^{1/n}} < {\left( {\frac{{{a^n} + {a^{2n}}}}{{1 + a}}} \right)^{1/n}} < {\left( {\frac{{2{a^n}}}{{1 + a}}} \right)^{1/n}} = a{\left( {\frac{2}{{1 + a}}} \right)^{1/n}}$. But $\displaystyle\mathop {\lim }\limits_{n \to \infty } a{\left( {\frac{1}{{1 + a}}} \right)^{1/n}} = \displaystyle\mathop {\lim }\limits_{n \to \infty } a{\left( {\frac{2}{{1 + a}}} \right)^{1/n}} = a$, by Squeeze Theorem, $\displaystyle\mathop {\lim }\limits_{n \to \infty } {\left( {\frac{{{a^n} + {a^{2n}}}}{{1 + a}}} \right)^{1/n}}=a.$

-

Since $\left( {\frac{{{a^n} + {a^{2n}}}}{{1 + a}}} \right)^{\frac{1}{n}} = a \left( {\frac{{1 + {a^{n}}}}{{1 + a}}} \right)^{\frac{1}{n}} = a \frac{(1 + {a^{n}})^{\frac{1}{n}}}{(1 + {a})^{\frac{1}{n}}} = a \frac{e^{\frac{1}{n} \ln(1+a^n)}}{e^{\frac{1}{n} \ln(1+a)}}$,
and $\lim_{n \to \infty} \ln(1+a^n) = 0$, and $\lim_{n \to \infty} \frac{1}{n} = 0$, we have $\lim_{n \to \infty} \left( {\frac{{{a^n} + {a^{2n}}}}{{1 + a}}} \right)^{\frac{1}{n}} = a$.


text:
# Root of 110112101201

#### [Root of one hundred ten billion one hundred twelve million one hundred one thousand two hundred one]

square root
331831.4349
cube root
4793.047
fourth root
576.0481
fifth root
161.5723
In mathematics extracting a root is declared as the determination of the unknown "x" in the following equation $y=x^n$ The outcome of the extraction of the root is seen as a mathematical root. In the case of "n equals 2", one talks about a square root or sometimes a second root also, another possibility could be that n is equivalent to 3 then one would call it a cube root or simply third root. Considering n beeing greater than 3, the root is declared as the fourth root, fifth root and so on.
In maths, the square root of 110112101201 is represented as this: $$\sqrt[]{110112101201}=331831.4349199$$
On top of this it is possible to write every root down as a power: $$\sqrt[n]{x}=x^\frac{1}{n}$$
The square root of 110112101201 is 331831.4349199. The cube root of 110112101201 is 4793.0469527064. The fourth root of 110112101201 is 576.0481185803 and the fifth root is 161.57233823181.

Look Up


text:
# Adding arrows with values over an expression

I'm trying to add arrows to math expressions (especially within limits) to note that there's a value assigned to certain term when solving a limit. I've come across the cancel package and its command \cancelto{}{} but I have 2 problems with it: 1. It strikes through the term like it's canceling it (obviously) 2. It only goes upwards (right upwards but the right part doesn't bother me), whereas I might need it to go downwards when I'm showing a tendency in the denominator of a fraction.

Searching here I found a custom command that sort of does the job (I modified it a little), and got it going, sadly it works half the time. The other half it just disarranges all the baselines of the other elements in te equation. I'll leave you the code of the commands (which again, is not mine to credit for) and an error that I'm having in hope someone can help me. I'm still learning Tikz so I'm unable to solve it myself although I suspect that there lies the answer to my problem.

The code modified by me:

\usetikzlibrary{calc}

\newcommand*\canc[1]{%
\mathchoice
{\scriptstyle#1}
{\scriptstyle#1}
{\scriptscriptstyle#1}
{\scriptscriptstyle#1}
}

\newcommand*\dtendto[2][0]{%
\kern9pt%
\begin{tikzpicture}[baseline=(current bounding box.center).anchor=east]
\node[anchor=east] (a) {$#2$};
\draw[->, color = red] ($(a.south)$) -- ($(a.south)-(-4pt,8pt)$) node
at ($(a.south)-(-8pt,12pt)$) {$\canc{#1}$};
\end{tikzpicture}
}

\newcommand*\utendto[2][0]{%
\kern9pt%
\begin{tikzpicture}[baseline=(current bounding box.south).anchor=east]
\node[anchor=east] (a) {$#2$};
\draw[->, color = red] ($(a.north)$) -- ($(a.north)+(4pt,8pt)$) node
at ($(a.north)+(8pt,12pt)$) {$\canc{#1}$};
\end{tikzpicture}
}

The part where I'm having trouble (I've earlier parts in which this worked but surely a correct code would still work on those).

\lim_{x\to\infty} \frac{\bcancel{x^5}\cdot(\utendto[2]{2}
-\utendto[0]{\frac{3}{x^2}} +
\utendto[0]{\frac{2}{x^4}})}{\bcancel{x^5}\cdot\left(4 + \frac{5}{x}
-\frac{100}{x^5}\right)}

And a snapshot of how it displays, notice that what I'd like to happen is that the arrows are added on top, as if the equation didn't even know they are there. Now it seems it is responding to what I add trying to center the + and - signs (even the parenthesis if I use \left( and \right) ) distorting a normal look.

And here's a photoshopped version of what i'd like it to look like:

Basically what I want is the equation to ignore the arrows and to be typeset just as if they weren't there (except of course for the vertical spacing not to be superposed with other lines or equations on top or bottom).

I've managed to create a new command called \dtendto[]{} that does the same except that the arrow goes southeast instead of northeast but I think the solution is the same for both so as not to make the post longer I ommited it.

• Inside of the node, the \mathchoice macro will always detect text-style. – Qrrbrbirlbel Dec 22 '13 at 21:49
• adding inner sep=0pt,outer sep=0pt to the nodes should help – clemens Dec 22 '13 at 22:46

# Remarks

I added inner xsep=0pt and outer sep=0pt, also I limited the bounding box.

In my edit, I got rid of the calc library, as it isn't really needed. Furthermore I set inner sep=0pt of the nodes and shorten <=.3333em of the arrows, to preserve spacing between digit and arrow, whilst cancelling extra space below the digit.

I repleaced all values by scalable units (ex and em).

Also, you don't need the \canc macro, as it will always default to \scriptstyle in your implementation.

# Implementation

\documentclass{article}
\pagestyle{empty}
\usepackage{tikz,cancel}
\renewcommand{\CancelColor}{\color{red}}

\newcommand*\dtendto[2][0]{%
\begin{tikzpicture}[baseline=(a.base),every node/.style={inner sep=0pt,outer sep=0pt}]
\node (a) {$#2$};
\path[red] (a.south) node (b) at +(0.5em,-3ex) {$\scriptstyle #1$};
\draw[->,red,shorten <=.3333em,shorten >=.3333em] (a) -- (b);
\pgfresetboundingbox
\path[use as bounding box] (a.north west) rectangle (a.south east |- b.south);
\end{tikzpicture}
}

\newcommand*\utendto[2][0]{%
\begin{tikzpicture}[baseline=(a.base),every node/.style={inner sep=0pt,outer sep=0pt}]
\node (a) {$#2$};
\path[red] (a.north) node (b) at +(0.5em,3ex) {$\scriptstyle #1$};
\draw[->,red,shorten <=.3333em,shorten >=.3333em] (a) -- (b);
\pgfresetboundingbox
\path[use as bounding box] (a.south west) rectangle (a.north east |- b.north);
\end{tikzpicture}
}
\begin{document}
\fbox{
$\displaystyle \lim_{x\to\infty} \frac{\bcancel{x^5}\cdot(\utendto[2]{2} -\utendto[0]{\frac{3}{x^2}} + \utendto[0]{\frac{2}{x^4}})}{\bcancel{x^5}\cdot\left(\dtendto[4]{4} + \frac{5}{x} -\frac{100}{x^5}\right)}$
}
\end{document}

# Output

In the real output the bounding boxes won't be highlighted. I just colored them green for illustration purposes.

• Henri! Thanks a lot!! It did the job splendidly!! Just one tiny thing: when using an align environment and being this equation in an inner line of the environment the arrow overlap with the upper line. I could (and will) correct this leaving empty lines above (and below), but just for the sake of it. Is there a way to expand the box? So as to take more vertical space for that line? Nevertheless, what you did is (at least for me), genius and I thank you very much for it. – Ewajs Dec 25 '13 at 1:58
• @Ewajs See my updated answer. – Henri Menke Dec 25 '13 at 12:09


text:
# Completely regular space with a G$_\delta$-singleton which is not a zero-set

Is there a completely regular (Hausdorff) space in which all singleton subsets are Gδ but which has a singleton subset which is not a zero-set? (Better yet, a first-countable such space.)

Recall that a subset $A$ of a topological space $X$ is called a zero-set if there is a continuous function $f : X \to [0,1]$ such that $A = f^{-1} (0)$. Clearly zero-sets are always closed Gδ. However a closed Gδ-set need not be a zero set: the Moore plane has the property that all closed subsets are Gδ, and furthermore every subset of the $x$-axis is closed. However the closed Gδ-set $A = \{ (x,0) : x \in \mathbb{Q} \}$ is not a zero-set.

A relatively simple consequence of Urysohn's Lemma is that in a normal space the zero-sets are exactly the closed Gδ-sets. Therefore, in a normal (Hausdorff) space with Gδ singletons, all singletons are zero-sets. I have been trying to think of a completely regular (Hausdorff) space with Gδ singletons which has a singleton which is not a zero-set.

Note that the Moore plane is not an example. Though it is completely regular (Hausdorff) and first-countable (hence all singletons are Gδ), every singleton can be shown to be a zero-set.

## 1 Answer

There is not.

Let $X$ be Tikhonov and have countable pseudocharacter (i.e., singletons are $G_\delta$ sets). Fix $x\in X$, and let $\mathscr{U}=\{U_n:n\in\Bbb N\}$ be a family of open nbhds of $x$ such that $\bigcap\mathscr{U}=\{x\}$. For $n\in\Bbb N$ there is a continuous $f_n:X\to[0,2^{-n}]$ such that $f_n(x)=0$, and $f_n(y)=2^{-n}$ for $y\in X\setminus U_n$. Let

$$f:X\to[0,1]:y\mapsto\sum_{n\in\Bbb N}f_n(y)\;;$$

Then $f$ is continuous, and $f^{-1}[\{0\}]=\{x\}$, so $\{x\}$ is a zero-set of $X$.

• you've done it again! – Forever Mozart Nov 22 '14 at 20:51
• @TomCruise It's the standard proof that in a normal space a closed $G_\delta$ is a zero-set. But here we only use that singletons and closed sets are functionally separated, so we need complete regularity instead of normality for singleton sets. – Henno Brandsma Nov 23 '14 at 6:11


text:
# Doppler Effect?

[SOLVED] Doppler Effect?

## Homework Statement

A bat flying at 4.0 m/s emits a chirp at 20 kHz. If this sound pulse is reflected by a wall, what is the frequency of the echo received by the bat?

## Homework Equations

Not sure. Do I use the f(1)=f*[(v+/-observer velocity)/(v+/-source velocity)]

## The Attempt at a Solution

Not sure where to start. I seem to remember something about sound waves being returned inverted if they hit a wall. Does this factor in?

alphysicist
Homework Helper
Hi Dreams2Knight,

As a first step, if the bat emits at a frequency of 20 kHz, what frequency does the wall receive?

After you have that, the reflection is handled by treating the wall as a source. What do you get?

Hi Dreams2Knight,

As a first step, if the bat emits at a frequency of 20 kHz, what frequency does the wall receive?

After you have that, the reflection is handled by treating the wall as a source. What do you get?

Hello,

Source: Bat
Observer: Wall
For the first equation I have f(1)=20[345/(345-4)] which is 20.23460411 kHz

Source: Wall
Observer: Bat
For the second equation I have f(1)=20[(345+4)/345] which is 20.23188406 kHz

Are my equations correct? If so, does the second equation give me my answer, or do I have to do something else?

alphysicist
Homework Helper
The bat emits sound at a frequency at 20 kHz, but the wall does not. After reflection from the wall, what frequency does the sound have? That number is the source frequency f on the right hand side. What do you get?

The bat emits sound at a frequency at 20 kHz, but the wall does not. After reflection from the wall, what frequency does the sound have? That number is the source frequency f on the right hand side. What do you get?

Ok, so if I understand correctly, I need to set my equation up like this:

20.234=f(2)*[(345+4)/345]. This gives me 20.00209.

I confused myself at this point because when I put the numbers into my calculator, I accidently subtracted 4 from 345 instead of adding. Doing that gave me the right answer of 20.46 kHz. I was confused because I thought if the source was still, you used V+V(observor) on top, yet subtracting gave me the right answer.

Then I remembered my teacher mentioning sound waves being inverted when they hit a wall. So, I took the reciprocal of (345+4)/345 and divided 20.234 by that and also got the right answer.

So, did I have my equation set up correctly as 20.234=f(2)*[(345+4)/345]? Was taking the reciprocal of (345+4)/345 also the correct thing to do?

alphysicist
Homework Helper
No, not quite. It's giving close to the right answer, but I think the way it should be set up is:

step 1: frequency received by wall:
$$f = 20 \frac{345}{345 -4} = 20.2346$$

step 2: frequency received by bat:

$$f = (20.2346) \frac{345 + 4}{345} = 20.469$$

Your other ways get close to the answer becuase, for example 1+x is close to 1/(1-x) when x is small compared to one.

No, not quite. It's giving close to the right answer, but I think the way it should be set up is:

step 1: frequency received by wall:
$$f = 20 \frac{345}{345 -4} = 20.2346$$

step 2: frequency received by bat:

$$f = (20.2346) \frac{345 + 4}{345} = 20.469$$

Your other ways get close to the answer becuase, for example 1+x is close to 1/(1-x) when x is small compared to one.

Ah, I see my mistake now. Thanks so much for your help. I really appreciate it.