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# Graduate/Postdoc Topology and Singularities Seminar

## Fall 2021 / Spring 2022

This year the seminar will continue to be on Zoom, hosted by Laurentiu Maxim and Botong Wang. The meeting information is below. We meet on Mondays, at 10am. Seminar announcements will be sent to the "singularities" mailing list. To subscribe, please send an email at: join-singularities@lists.wisc.edu (or, if you get an error message, just email maxim@math.wisc.edu). We plan to have all talks recorded.

The seminar is targeted at junior mathematicians with an interest in the topological study of singularities. We ask the speakers to take this fact into consideration when preparing their talks.

Topic: Topology and Singularities Seminar

Meeting ID: 997 5395 0856 Passcode: singular

date speaker title
Sept 10-11

TIBAR60 Conference

Sept 20 Botong Wang (UW-Madison) Perverse sheaves on varieties with large fundamental group and the Singer-Hopf conjecture
Sept 27 Rodolfo Aguilar Aguilar (Sofia, Bulgaria) Arrangements, fundamental groups and homology planes
October 4 Yongqiang Liu (USTC, China) $L^2$-type invariants and cohomology jump loci for smooth complex quasi-projective varieties
Dec 6 Jacob Matherne (Bonn) TBA

## Abstracts

### Botong Wang

The Singer-Hopf conjecture predicts that for a compact aspherical manifold $X$ of real dimension 2$n$, the Euler characteristic of $X$ has sign $(-1)^n$. In a joint work with Yongqiang Liu and Laurentiu Maxim, we made a stronger conjecture in the complex algebraic setting that if a smooth complex projective variety is aspherical, then all perverse sheaves on the variety has nonnegative Euler characteristics. We confirm this conjecture in a special case, when the fundamental group admits a faithful, semisimple, cohomologically rigid representation. Our approach is to construct a complex variation of Hodge structure on the variety and relate the nonnegativity of Euler characteristics with certain curvature conditions on the period domain. Joint work with Donu Arapura.

### Rodolfo Aguilar Aguilar

We will show some examples of homology planes: smooth, affine complex surfaces with trivial reduced integral homology, of log-general type which have an infinite fundamental group. These could be the first examples where such infinitude is shown, they arise as partial compactifications of the complement of an arrangement of lines in the complex projective plane.

### Irma Pallarés Torres

The Brasselet-Schürmann-Yokura conjecture for rational homology manifolds

Abstract: The Brasselet-Schürmann-Yokura conjecture is a conjecture of characteristic classes of singular spaces formulated by J. P. Brasselet, J. Schürmann, and S. Yokura. The conjecture predicts the equality between the Hodge L-class and the Goresky-MacPherson L-class for compact complex algebraic varieties that are rational homology manifolds. In this talk, I will explain two proofs of the conjecture. The first, via classical Hodge theory for projective varieties. This is a joint work with J. Fernández de Bobadilla. The second, using the theory of mixed Hodge modules for general compact algebraic varieties. This is a joint work with J. Fernández de Bobadilla and M. Saito.

### Manuel González Villa

On a quadratic form associated with the nilpotent part of the monodromy of a curve

Abstract: Joint work with Lilia Alanís-López, Enrique Artal Bartolo, Christian Bonatti, Xavier Gómez-Mont, and Pablo Portilla Cuadrado. We study the nilpotent part N of certain pseudo-periodic automorphisms of surfaces appearing in singularity theory. We associate a quadratic form Q defined on the first (relative to the boundary) homology group of the Milnor fiber F of any germ analytic curve on a normal surface. Using the twist formula and techniques from mapping class group theory, we prove that the form Q obtained after killing ker N is definite positive, and that its restriction to the absolute homology group of F is even whenever the Nielsen-Thurston graph of the monodromy automorphism is a tree. The form Q is computable in terms of the Nielsen-Thurston or the dual graph of the semistable reduction, as illustrated with several examples. Numerical invariants associated to Q are able to distinguish plane curve singularities with different topological types but the same spectral pairs or Seifert form. Finally, we discuss a generic linear germ defined on a superisolated surface with not smooth ambient space.

### Feng Hao

Holomorphic 1-forms, smoothness of morphisms to abelian varieties and some linearity properties

Abstract: In this talk I will discuss some results on how the existence of nowhere vanishing holomorphic 1-forms on a smooth complex projective variety X affect the singularities of morphisms from X to (simple) abelian varieties. Also, I will discuss some results on the linearity of the set of holomorphic 1-forms admitting zeros, which is related to the study of singularities of Albanese map. This is a joint work with Yajnaseni Dutta and Yongqiang Liu.

### Alex Suciu

Ab-exact extensions and Milnor fibrations of arrangements

Abstract: We study the lower central series, the Alexander invariants, and the cohomology jump loci of groups arising as split extensions with trivial monodromy in first homology with appropriate coefficients. We use these techniques to gain further understanding of the Milnor fibration of the complement of a hyperplane arrangement.

### Rares Rasdeaconu

Moduli spaces of stable, rank 1, torsion free sheaves on real curves

Abstract: The counting of rational curves representing primitive homology classes on complex or real K3 surfaces is governed by the Yau-Zaslow formula and its real analog, respectively. A natural approach to extend such formulae to the non-primitive case requires the computation of the Euler characteristic of moduli spaces of stable, rank one sheaves on curves which are possibly reducible and non-reduced. The recent developments in this direction will be presented (joint work with V. Kharlamov).

Positive factorizations of monodromies on links of isolated complex surface singularities

Abstract: We present a generalization of a classical result concerning smooth germs of surfaces, by showing that monodromies on links of isolated complex surface singularities associated with reduced holomorphic map germs are a product of right-handed Dehn twists. We explore some consequences of this theorem in the realm of mapping class groups of surfaces. If time permits, we will talk about the higher-dimensional counterpart of the result and how it possibly provides an obstruction to the smoothability of isolated singularities.

Minimal Exponents and a conjecture of Teissier

Abstract: The minimal exponent, defined by M. Saito, is an invariant of hypersurface singularities which is a refinement of the log canonical threshold. One expects singularities to worsen when intersecting with a smooth hypersurface, and Teissier conjectured in the 80's an inequality describing how much the minimal exponent can decrease in this process. I will describe joint work with Mircea Mustaţă (building on work he has done with Eva Elduque) in which we prove this conjecture.

### Daniel Bath

Towards a Logarithmic Comparison Theorem for Quasi-Free Divisors

Abstract: One way to compute the cohomology of the complement of a hypersurface {f=0} is to compute the cohomology of the de Rham complex of meromorphic forms with poles of arbitrary order along f. Sitting inside this complex is the logarithmic de Rham complex, consisting of certain forms with poles of order at most one. It is a long standing question to find necessary and sufficient conditions on the hypersurface ensuring that the natural inclusion of complexes is a quasi-isomorphism, that is, that f satisfies the Logarithmic Comparison Theorem. The modern approach utilizes D-module techniques and the "best" sufficient conditions require assuming that (among other things) the singular locus of f is, morally, as large as possible. To relax this assumption, we introduce a new variant of the Logarithmic Comparison Theorem where: logarithmic derivations are replaced with a submodule closed under Lie brackets; the logarithmic de Rham complex is replaced with a new de Rham complex. I will discuss our proposed strategy for proving this Quasi-Free Logarithmic Comparison Theorem which employs curious Bernstein--Sato polynomial-type constructions. Work in progress; joint with Luis Narvaez-Macarro and Francisco Castro-Jimenez.

### Dominik Wrazidlo

Intersection spaces and Poincaré completion

Abstract: In this survey talk, I will present a two-step method that assigns rational Poincaré duality spaces to certain compact oriented pseudomanifolds by modifying only a neighborhood of the singular set. The first step of the method consists of the intersection space construction, which was invented by Banagl, and was later extended by Agustín and Fernández de Bobadilla to a larger class of pseudomanifolds of arbitrary stratification depth. I will explain work in progress about the study of the rational homotopy type of intersection spaces. The second step of the method is a joint work with T. Essig, in which we extend intersection spaces to rational Poincaré duality spaces in certain cases by constructing a fundamental class. We achieve such a "Poincaré completion" by attaching a finite number of cells to the intersection space, which generalizes work of Klimczak. The focus of this talk lies on a survey of main ideas, examples, and possible future directions of research.

### Jörg Schürmann

(Equivariant) characteristic classes of singular toric varieties

Abstract: In this survey talk, we discuss for singular toric varieties different (equivariant) characteristic classes like Chern, Todd, L and Hirzebruch classes, with applications to (weighted) lattice points counting and Euler-MacLaurin type formulae for lattice polytopes. This is about joint work with Laurentiu Maxim.

### Jose Ignacio Cogolludo

Homological invariants of even Artin kernels

Abstract: Even Artin groups form a special family generalizing right-angle Artin groups that are associated with even-labeled graphs. Some of their properties as groups can be described in terms of their defining graph. In this talk we will discuss different properties associated with the homology of cocyclic subgroups and other finiteness conditions. This is a joint work in progress with C. Martínez-Pérez and R. Blasco-García.

Alexander modules and Mellin transform

Abstract: I will talk about the study of Alexander modules of algebraic varieties using Gabber and Loeser's Mellin transform. The main strength of this approach is that it allows the application of the full machinery of the theory of perverse sheaves, and even mixed Hodge modules. We obtain new results about the structure of Alexander modules, especially about their torsion part and, in the multivariable case, their artinian submodules. It also yields a mixed Hodge structure on the maximal artinian submodules of the Alexander modules. This is based on joint work with Eva Elduque, Christian Geske, Laurentiu Maxim and Botong Wang.

### Owen Barrett

The derived category of the abelian category of constructible sheaves

Abstract: Nori proved in 2002 that given a complex algebraic variety $$X$$, the bounded derived category of the abelian category of constructible sheaves on $$X$$ is equivalent to the usual triangulated category $$D(X)$$ of bounded constructible complexes on $$X$$. He moreover showed that given any constructible sheaf $$\mathcal F$$ on $$\mathbb{A}^n$$, there is an injection $$\mathcal F\hookrightarrow\mathcal G$$ with $$\mathcal G$$ constructible and $$H^i(\mathbb{A}^n,\mathcal G)=0$$ for $$i>0$$.

In this talk, I’ll discuss how to extend Nori’s theorem to the case of a variety over an algebraically closed field of positive characteristic, with Betti constructible sheaves replaced by $$\ell$$-adic sheaves. This is the case $$p=0$$ of the general problem which asks whether the bounded derived category of $$p$$-perverse sheaves is equivalent to $$D(X)$$, resolved affirmatively for the middle perversity by Beilinson.

### Ruijie Yang

Decomposition theorem for semisimple local systems

Abstract: In this talk, I would like to present a new and geometric proof of Sabbah's Decomposition Theorem, which asserts that any semisimple local system remains semisimple after taking the direct image under proper algebraic maps.

Sabbah's proof is D-module-theoretic in nature and relies on his theory of polarizable twistor D-modules, which generalizes Saito's theory of polarizable Hodge modules. Instead, we combine the topological approach developed by De Cataldo-Migliorini and Simpson's theory of mixed twistor structures to give the new proof. Along the way, we obtain new results about the cohomology of semisimple local systems. Joint work with Chuanhao Wei.

## Fall 2018

The Seminar meets at 10.30 to 11:30 on Fridays in Van Vleck 901.

date speaker title
Oct 5 Fenglin Li "Hasse principle and u-invariant"
Oct 26 Fenglin Li "Hasse principle and u-invariant (II)"
Nov 2 José Rodríguez "Maximum likelihood degree"

### Abstracts

#### Nov 2: José Rodríguez

Maximum likelihood degree

In statistics, point estimation uses sample data to calculate the "best estimate" of an unknown population parameter. For example, the sample average can be used to estimate the population mean. While there are many different point estimators, some of the most common ones are the maximum likelihood estimator (MLE), method of moments, and generalized method of moments (GMM).

In algebraic statistics statistical models are studied through the lens of algebra, geometry, and combinatorics. From model selection to inference, this interdisciplinary field has seen applications in a wide range of statistical procedures. In this talk, I will review maximum likelihood estimation and the maximum likelihood degree (ML degree) for discrete models. In particular, I will discuss how the ML degree gives a measure of algebraic complexity of the point estimate for MLE and how we can compute it using tools from topology and geometry. If time permits I will also discuss how we can use maximum likelihood degrees to study singularities.

## Fall 2017

The Seminar meets at 3:30 to 4:30 pm on Wednesdays in Van Vleck 901.

date speaker title
Oct 4 Eva Elduque "Twisted Alexander Modules of Complex Essential Hyperplane Arrangement Complements (I)"
Oct 11 Eva Elduque "Twisted Alexander Modules of Complex Essential Hyperplane Arrangement Complements (II)"
Oct 18 Sebastian Baader "Dehn twist length in mapping class groups"
Oct 25 Cancelled
Nov 1 Christian Geske "Algebraic Intersection Spaces (I)"
Nov 8 Christian Geske "Algebraic Intersection Spaces (II)"
Nov 15 Laurentiu Maxim "Stratified Morse Theory: an overview (I)"
Nov 22 Thanksgiving break
Nov 29 Laurentiu Maxim "Stratified Morse Theory: an overview (II)"
December 6 Alexandra Kjuchukova "Singular branched covers of four-manifolds and applications"
December 13 TBD "TBA"

## Spring 2017

Fridays at 11:00 VV901

The Seminar meets on Fridays at 11:00 pm in Van Vleck 901, and is coordinated by Alexandra Kjuchukova, Manuel Gonzalez Villa and Botong Wang.

date speaker title
Jan 27 Christian Geske "Intersection Spaces and Equivariant Moore Approximation I"
Feb 3 Christian Geske "Intersection Spaces and Equivariant Moore Approximation II"
Feb 10 Sashka "The Wirtinger Number of a knot equals its bridge number I"
Feb 17 Sashka "The Wirtinger Number of a knot equals its bridge number II"
Feb 24 Christian Geske "Intersection Spaces and Equivariant Moore Approximation III"
Mar 3 Manuel Gonzalez Villa "Multiplier ideals of irreducible plane curve singularities"

## Fall 2016

Wednesdays at 14:30 VV901

The Seminar meets on Wednesdays at 14:30 pm in Van Vleck 901 (except on October 26th when we will meet in Van Vleck 903), and is coordinated by Alexandra Kjuchukova, Manuel Gonzalez Villa and Botong Wang.

date speaker title
Sept. 14 (W) Laurentiu Maxim "Alexander-type invariants of hypersurface complements"
Sept. 21 (W) Botong Wang "Cohomology jump loci"
Sept. 28 (W) Alexandra Kjuchukova "On the Bridge Number vs Meridional Rank Conjecture"
Oct 5 (W) Manuel Gonzalez Villa "Introduction to Newton polyhedra"
Oct 12 (W) Manuel Gonzalez Villa "More on Newton polyhedra"
Oct 26 (W) Christian Geske "Intersection Spaces"
Nov 2 (W) Christian Geske "Intersection Spaces Continued"
Nov 9 (W) CANCELLED
Nov 16 (W) Eva Elduque "Braids and the fundamental group of plane curve complements"
Nov 30 (W) Laurentiu Maxim "Novikov homology of hypersurface complements"
Dec 7 (W) CANCELLED
Dec 14 (W) Eva Elduque Specialty Exam: "Twisted Alexander invariants of plane curve complements"

## Spring 2016

Mondays at 3:20 B139VV

The old Graduate Singularities Seminar will meet as a Graduate/Postdoc Topology and Singularities Seminar in Fall 2015 and Spring 2016.

The seminar meets on Mondays at 3:20 pm in Van Vleck B139. During Spring 2016 we will cover first chapters the book Singularities in Topology by Alex Dimca (Universitext, Springer Verlag, 2004). If you would like to participate giving one of the talks, please contact Eva Elduque or Christian Geske.

date speaker title
Feb. 8 (M) Christian Geske Section 1.1 and 1.2: Category of complexes and Homotopical category
Feb. 15 (M) Eva Elduque Sections 1.3 and 1.4: Derived category and derived functors
Feb. 22 (M) Botong Wang Sections 2.1 and 2.2: Generalities on Sheaves and Derived tensor products
Feb. 29 (M) Christian Geske Hypercohomology and Holomorphic Differential Forms on Analytic Varieties
Mar. 7 (M) Eva Elduque Section 2.3: Direct and inverse image
Mar. 14 (M) Cancelled
Mar. 28 (M) Cancelled
Apr. 4 (M) Cancelled
Apr. 11 (M) Christian Geske Section 2.3 cont.
Apr. 18 (M) Cancelled
Apr. 25 (M) Cancelled
May. 2 (M) Cancelled

## Abstracts

(From the back cover of Dimca's book) Constructible and perverse sheaves are the algebraic counterpart of the decomposition of a singular space into smooth manifolds, a great geometrical idea due to R. Thom and H. Whitney. These sheaves, generalizing the local systems that are so ubiquitous in mathematics, have powerful applications to the topology of such singular spaces (mainly algebraic and analytic complex varieties).

This introduction to the subject can be regarded as a textbook on Modern Algebraic Topology, which treats the cohomology of spaces with sheaf coefficients (as opposed to the classical constant coefficient cohomology).

The first five chapters introduce derived categories, direct and inverse images of sheaf complexes, Verdier duality, constructible and perverse sheaves, vanishing and characteristic cycles. They also discuss relations to D-modules and intersection cohomology. The final chapters apply this powerful tool to the study of the topology of singularities, of polynomial functions and of hyperplane arrangements.

Some fundamental results, for which excellent sources exist, are not proved but just stated and illustrated by examples and corollaries. In this way, the reader is guided rather quickly from the A-B-C of the theory to current research questions, supported in this by a wealth of examples and exercises.

## Fall 2015

Thursdays 4pm in B139VV

date speaker title
Sept. 24 (Th) KaiHo (Tommy) Wong Twisted Alexander Invariant for Knots and Plane Curves
Oct. 1 (Th) Alexandra (Sashka) Kjuchukova Linking numbers and branched covers I
Oct. 8 (Th) Alexandra (Sashka) Kjuchukova Linking numbers and branched covers II
Oct. 15 (Th) Manuel Gonzalez Villa On poles of zeta functions and monodromy conjecture I
Oct. 22 (Th) Yun Su (Suky) Pretalk Higher-order degrees of hypersurface complements., Survey on Alexander polynomial for plane curves.
Oct. 29 (Th) Yun Su (Suky) Aftertalk Higher-order degrees of hypersurface complements.
Nov. 5 (Th) Manuel Gonzalez Villa On poles of zeta functions and monodromy conjecture II
Nov. 12 (Th) Manuel Gonzalez Villa On poles of zeta functions and monodromy conjecture III
Nov. 19 (Th) Eva Elduque Stiefel-Whitney classes
Dec. 3 (Th) Eva Elduque Grass-mania!
Dec. 10 (Th) KaiHo (Tommy) Wong Pretalk Milnor Fiber of Complex Hyperplane Arrangements

## Abstracts

### Th, Sep 24: Tommy

Twisted Alexander Invariant of Knots and Plane Curves.

I will introduced three invariants of knots and plane curves, fundamental group, Alexander polynomial, and twisted Alexander polynomial. Some basic examples will be used to illustrate how Alexander polynomial or twisted Alexander polynomial can be computed from the fundamental group. If time permits, I will survey some known facts about twisted Alexander invariant of plane curves.

### Th, Oct 1 and 8: Sashka

Linking numbers and branched coverings I and II

Let K be a knot in S^3, and let M be a non-cyclic branched cover of S^3 with branching set K. The linking numbers between the branch curves in M, when defined, are an invariant of K which can be traced back to Reidemeister and was used by Ken Perko in the 60s to distinguish 25 new knot types not detected by their Alexander Polynomials. In addition to this classical result, recent work in the study of branched covers of four-manifolds with singular branching sets leads us to consider the linking of other curves in M besides the branch curves.

In these two talks, I will outline Perko's original method for computing linking in a branched cover, and I will give a brief overview of its classical applications. Then, I'll describe a suitable generalization of his method, and explain its relevance to a couple of open questions in the classification of branched covers between four-manifolds.

### Th, Oct 15, Nov 5 and Nov 12: Manuel

On poles of zeta functions and monodromy conjecture I and II

Brief introduction to topological and motivic zeta functions and their relations. Statement of the monodromy conjecture. Characterization and properties of poles of the in the case of plane curves. Open problems in the case of quasi-ordinary singularities.

### Th, Nov 19: Eva

Stiefel-Whitney classes

Not all elements in the Z_2 cohomology ring of the base space of a real vector bundle are created equal. We will define the Stiefel-Whitney classes and give evidence of why they are the cool kids of the cohomology dance. For example, they will tell us information about when a manifold is the boundary of another one or when we can’t embed a given projective space into R^n.

### Th, Dec 3: Eva

Grass-mania!

In this talk, we will talk about the grassmannians, both the finite and infinite dimensional ones. We will define their canonical vector bundles, which turn out to be universal in some sense, and give them a CW structure to compute their cohomology ring. As an application, we will prove the uniqueness of the Stiefel-Whitney classes defined in the last talk.

This talk is for the most part self contained, so it doesn't matter if you missed the previous one.

### Th, Dec 10: Tommy

A line is one of the simplest geometric objects, but a whole bunch of them could provide us open problems!

I will talk about some past results on line arrangements, that are whole bunches of lines. I will speak a little bit on why line arrangements or plane arrangements stand out from other hypersurfaces in the study of topological singularity theory.

## Spring 2014

We continue with Professor Alex Suciu's work.

## Fall 2014

We follow Professor Alex Suciu's work this semester.

But we will not meet at a regular basis.

## Spring 2014

We meet on Tuesdays 3:30-4:25pm in room B211.

date speaker title
Feb. 25 (Tue) Yongqiang Liu Monodromy Decomposition I
Mar. 4 (Tue) Yongqiang Liu Monodromy Decomposition II
Mar. 25 (Tue) KaiHo Wong Conjecture of lower bounds of Alexander polynomial
Apr. 8 (Tue) Yongqiang Liu Nearby Cycles and Alexander Modules

## Fall 2013

We are learning Hodge Theory this semester and will be following three books:

1. Voisin, Hodge Theory and Complex Algebraic Geometry I & II

2. Peters, Steenbrink, Mixed Hodge Structures

We meet weekly on Wednesdays from 12 at noon to 1pm in room 901.

date speaker title
Sep. 18 (Wed) KaiHo Wong Discussions on book material
Sep. 25 (Wed) Yongqiang Liu Milnor Fibration at infinity of polynomial map
Oct. 9 (Wed) KaiHo Wong Discussions on book material
Oct. 16 (Wed) Yongqiang Liu Polynomial singularities
Nov. 13 (Wed) KaiHo Wong Discussions on book material

## Spring 2013

date speaker title
Feb. 6 (Wed) Jeff Poskin Toric Varieties III
Feb.13 (Wed) Yongqiang Liu Intersection Alexander Module
Feb.20 (Wed) Yun Su (Suky) How do singularities change shape and view of objects?
Feb.27 (Wed) KaiHo Wong Fundamental groups of plane curves complements
Mar.20 (Wed) Jörg Schürmann (University of Münster, Germany) Characteristic classes of singular toric varieties
Apr. 3 (Wed) KaiHo Wong Fundamental groups of plane curves complements II
Apr.10 (Wed) Yongqiang Liu Milnor fiber of local function germ
Apr.17 (Wed) 2:45pm-3:45pm (Note the different time) KaiHo Wong Formula of Alexander polynomials of plane curves

## Abstracts

### Wed, 2/27: Tommy

Fundamental groups of plane curves complements

I will sketch the proof of the Zariski-Van Kampen thereon and say some general results about the fundamental groups of plane curves complements. In particular, we will investigate, under what conditions, these groups are abelian. Some simple examples will be provided. And if time permits, some classical examples of Zariski and Oka will be computed.

## Fall 2012

date speaker title
Sept. 18 (Tue) KaiHo Wong Organization and Milnor fibration and Milnor Fiber
Sept. 25 (Tue) KaiHo Wong Algebraic links and exotic spheres
Oct. 4 (Thu) Yun Su (Suky) Alexander polynomial of complex algebraic curve (Note the different day but same time and location)
Oct. 11 (Thu) Yongqiang Liu Sheaves and Hypercohomology
Oct. 18 (Thu) Jeff Poskin Toric Varieties II
Nov. 1 (Thu) Yongqiang Liu Mixed Hodge Structure
Nov. 15 (Thu) KaiHo Wong Euler characteristics of hypersurfaces with isolated singularities
Nov. 29 (Thu) Markus Banagl, University of Heidelberg High-Dimensional Topological Field Theory, Automata Theory, and Exotic spheres

## Abstracts

### Thu, 10/4: Suky

Alexander polynomial of complex algebraic curve

I will extend the definition of Alexander polynomial in knot theory to an complex algebraic curve. From the definition, it is clear that Alexander polynomial is an topological invariant for curves. I will explain how the topology of a curve control its Alexander polynomial, in terms of the factors. Calculations of some examples will be provided.


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Lower semicontinuity of the indicator function: stochastic processes

Let $X$ be a Markov process given on a metric space $\mathcal X$ by a transition semigroup $P_t$ acting on $\mathbb B(\mathcal X)$ - the set of all bounded and Borel measurable functions. Such a function is said to be $\mathcal C$-lower semicontinuous (l.s.c.) if $$\mathsf P_x\{\liminf\limits_{t\downarrow 0}f(X_t)\geq f(x)\}=1$$ for any $x\in \mathcal X$. I wonder under which conditions on $P_t$ a function $1_A(x)$ is l.s.c. for any open $A$?

Not to be confused with a usual definition of a l.s.c. function which is not based on the processes.

As I understand it means that starting in an open set, with probability one the process stays there for some positive time. If I am not wrong, that holds for any process with cadlag paths since there exists $\lim\limits_{t\downarrow 0}\,\,X_t = x$ so if $x\in A$ - open, then $\lim\limits_{t\downarrow 0}1_A(X_t) = 1$.

-
Yes cadlag (or, more generally right-continuous) paths gives a positive answer. You do need to choose a good version of the process otherwise $\liminf_{t\downarrow 0}f(X_s)$ will not be measurable. Assuming joint measurability, separability of the metric space and the strong Markov property, then your condition will be true if and only if the process is right-continuous. – George Lowther Oct 11 '11 at 1:47
@GeorgeLowther: Thank you very much. Would you put this comment as an answer if you have time? I will accept it. – Ilya Oct 11 '11 at 7:02
Sorry to make such a basic question, but how does $P_x$ relates to $P_t$. For me, $P_x$ is a probability on $\mathcal{X}^{\mathbb{R}}$... in fact, there is a probability $P$ on $\mathcal{X}^{\mathbb{R}}$, and $P_x = P(\cdot | X_0 = x)$. How does $P_t$ determine $P_x$? – André Caldas Oct 11 '11 at 14:55
@AndréCaldas: sorry for confuse you with symbols: in fact I used different fonts for the probability measure and the semigroup. $$\mathsf P_x = \mathsf P\{\cdot|X_0 = x\}$$ and $$P_tf(x) = \mathsf E_x[f(X_t)] :=\int\limits_{\mathcal X}f(y)\mathsf P_x\{X_t\in dy\}$$ – Ilya Oct 11 '11 at 15:01
Is this the same as saying that $$P_t(A) = \int_{\mathcal{X}} P_x(X_t \in A) dx?$$ – André Caldas Oct 11 '11 at 16:44

Notice that $$\{\liminf_{t \downarrow 0} f(X_t) \geq f(x)\} = \bigcap_{n \in \mathbb{N}} \left\{\liminf_{t \downarrow 0} f(X_t) > f(x) - \frac{1}{n}\right\}.$$ So, since $\left\{\liminf_{t \downarrow 0} f(X_t) > f(x) - \frac{1}{n}\right\}$ decreases when $n \rightarrow \infty$, what you want is that $$P_x\left\{\liminf_{t \downarrow 0} f(X_t) > f(x) - \frac{1}{n}\right\} = 1$$ for every $n \in \mathbb{N}$. That is, for every $\varepsilon > 0$, $$P_x\left\{\liminf_{t \downarrow 0} f(X_t) > f(x) - \varepsilon\right\} = 1$$
So, the condition is equivalent to $$P_x\left(\bigcap_{t > 0} \bigcap_{0 < s < t} \{f(X_s) > f(x) - \varepsilon\}\right) = 1$$ for every $\varepsilon > 0$.
Now, assuming $f = 1_A$, we have that if $x \not \in A$, then the above equation is always true. For $x \in A$, the condition becomes $$P_x\left(\bigcap_{t > 0} \bigcap_{0 < s < t} \{f(X_s) = 1\}\right) = 1.$$ But this is the same as $$P_x\left(\bigcap_{t > 0} \bigcap_{0 < s < t} \{X_s \in A\}\right) = 1.$$ Since the sets $\bigcap_{0 < s < t} \{X_s \in A\}$ increase with $t$, we can take a sequence $t_j \downarrow 0$, and conclude that the condition becomes $$P_x\left(\bigcap_{0 < s < t} \{X_s \in A\}\right) \uparrow 1,$$ when $t \downarrow 0$.
I don't know how to pass from this to $P_t$, since I am not familiar with Markov processes. But I guess that you might be able to conclude what @George Lowther said in his comment: the process is right-continuous.

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# Programming notes¶

This is a notebook that was created in class on February 11, 2020. The goal was to review some basic programming in Python to be sure your background was strong enough to complete Homework 2.

## Tuples:¶

In [1]:
(1, 2, 3)

## Lists:¶

In [2]:
[1, 2, 3]

## For ... in ...:¶

In [3]:
for x in ["cat", "dog", 53]:
print(2*x)
y = "x is {}".format(x)
print(y)

## Range:¶

Ranges represent a list of consecutive values.

In [4]:
range(5)

You can convert a range to a list.

In [5]:
list(range(5))

Some information about range can be discovered by evaluating range?.

In [6]:
range?

list(range(3, 10))
In [7]:
list(range(1, 11, 2))

Out[7]:
[1, 3, 5, 7, 9]
In [8]:
list(range(11, 0, -2))

Out[8]:
[11, 9, 7, 5, 3, 1]
In [9]:
for i in range(11, 0, -2):
print(i)

11
9
7
5
3
1

# Summing the elements in a tuple, list or range:¶

In [10]:
sum(range(11, 0, -2))

Out[10]:
36

## While and break¶

Here we compute the sums $$\sum_{i=1}^N i$$ for increasing values of $N$ until the first time the sum exceeds 1000.

In [11]:
i = 1
total = 0
while True:
total = total + i
print("total = {} when i = {}".format(total, i))
if total > 1000:
break
i = i + 1

total = 1 when i = 1
total = 3 when i = 2
total = 6 when i = 3
total = 10 when i = 4
total = 15 when i = 5
total = 21 when i = 6
total = 28 when i = 7
total = 36 when i = 8
total = 45 when i = 9
total = 55 when i = 10
total = 66 when i = 11
total = 78 when i = 12
total = 91 when i = 13
total = 105 when i = 14
total = 120 when i = 15
total = 136 when i = 16
total = 153 when i = 17
total = 171 when i = 18
total = 190 when i = 19
total = 210 when i = 20
total = 231 when i = 21
total = 253 when i = 22
total = 276 when i = 23
total = 300 when i = 24
total = 325 when i = 25
total = 351 when i = 26
total = 378 when i = 27
total = 406 when i = 28
total = 435 when i = 29
total = 465 when i = 30
total = 496 when i = 31
total = 528 when i = 32
total = 561 when i = 33
total = 595 when i = 34
total = 630 when i = 35
total = 666 when i = 36
total = 703 when i = 37
total = 741 when i = 38
total = 780 when i = 39
total = 820 when i = 40
total = 861 when i = 41
total = 903 when i = 42
total = 946 when i = 43
total = 990 when i = 44
total = 1035 when i = 45

## Random numbers:¶

In [12]:
import random

In [13]:
random.randint?

Print a random sequence of twenty zeros or ones.

In [14]:
for i in range(20):
print(random.randint(0, 1))

0
1
0
0
1
1
0
1
1
0
0
0
1
0
0
1
1
0
1
1

Print a random sequence of twenty coin flips:

In [15]:
for i in range(20):
x = random.randint(0, 1)
if x == 0:
if x == 1:
coin = "Tails"
print(coin)

Heads
Tails
Tails
Tails
Tails
Tails
Tails

In [16]:
def flip(n):
"""Flip a coin n times, and print the results."""
for i in range(n):
x = random.randint(0, 1)
if x == 0:
if x == 1:
coin = "Tails"
print(coin)

In [17]:
flip(5)

Heads
Tails
Tails

In [18]:
flip?

## Ternary¶

In [19]:
def ternary(n):
# Below we peform sanity checks on the input.
assert type(n) == int, 'n must be an integer'
assert n>0, 'n must be a positive integer.'

rep = '' # This will store the ternary representation.
while n != 0:
trit = n % 3
rep = str(trit) + rep # add the new trit on the left.
n = n // 3 # Update the value of n.
return rep

In [20]:
ternary(10)

Out[20]:
'101'

## Working with strings:¶

In [21]:
s = "My dog's name is Sugar."

Iterating through the characters in a string:

In [22]:
for c in s:
print(c, end="-")

M-y- -d-o-g-'-s- -n-a-m-e- -i-s- -S-u-g-a-r-.-

The length of a string:

In [23]:
len(s)

Out[23]:
23

The first character:

In [24]:
s[0]

Out[24]:
'M'

The second:

In [25]:
s[1]

Out[25]:
'y'

The 23rd character:

In [26]:
s[22]

Out[26]:
'.'

There is no 24th character in s.

In [27]:
s[23]

---------------------------------------------------------------------------
IndexError                                Traceback (most recent call last)
<ipython-input-27-82823ef8b99e> in <module>()
----> 1 s[23]

IndexError: string index out of range

Another way to iterate through the characters in a string:

In [28]:
for i in range(len(s)):
c = s[i]
print(c, end="-")

M-y- -d-o-g-'-s- -n-a-m-e- -i-s- -S-u-g-a-r-.-

Iterating through backwards:

In [29]:
for i in range(len(s)-1, -1, -1):
c = s[i]
print(c, end="-")

.-r-a-g-u-S- -s-i- -e-m-a-n- -s-'-g-o-d- -y-M-

## Converting a string to a number:¶

In [30]:
int("0")

Out[30]:
0
In [31]:
int("23")

Out[31]:
23

## Substrings:¶

Recall what s is:

In [32]:
s

Out[32]:
"My dog's name is Sugar."

The capital S appears at index 17, and the r appears at index 21:

In [33]:
print(s[17])
print(s[21])

S
r

To get the substring Sugar you can type:

In [34]:
s[17:22]

Out[34]:
'Sugar'

This is called a "slice." The meaning of numbers is similar to the range function. We are extracting the substring beginning in index 17 up index 22 (excluded).

You can also use a step size of 2 if you wanted:

In [35]:
s[17:22:2]

Out[35]:
'Sgr'

## Interval Arithmetic¶

In [36]:
from mpmath import iv

Use 3 bits of precision:

In [37]:
iv.prec = 3

This will find an interval with endpoints using 3 bits of precision that contains 1.3.

In [38]:
iv.mpf("1.3")

Out[38]:
mpi('1.25', '1.5')

The value None is returned if the truth of a comparison can not be decided from the intervals.

In [39]:
print(iv.mpf("1.3") < iv.mpf("1.4"))

None

Normally you would get true or false:

In [40]:
print(iv.mpf("1.3") < iv.mpf("2.1"))

True

Assume $x \neq y$. If you want to decide if $x < y$, you can repeat a comparisson at higher and higher precision until you get a definitive answer.

In [41]:
x = "1.3"
y = "1.4"
prec = 1
while True:
iv.prec = prec
value = (iv.mpf(x) < iv.mpf(y))
print("At {} of bits of precision, the comparisson returned {}.".format(prec, value))
if value == True or value == False:
# The comparison worked
break
prec = prec + 1
print("The comparison is {}.".format(value))

At 1 of bits of precision, the comparisson returned None.
At 2 of bits of precision, the comparisson returned None.
At 3 of bits of precision, the comparisson returned None.
At 4 of bits of precision, the comparisson returned None.
At 5 of bits of precision, the comparisson returned True.
The comparison is True.

In [42]:
x = iv.pi
x

Out[42]:
mpi('3.125', '3.25')

x.delta gives the width of the interval (as an interval).

In [43]:
x.delta

Out[43]:
mpi('0.125', '0.125')

Print out the interval to two significant digits:

In [44]:
iv.nstr(x,2)

Out[44]:
'[3.1, 3.3]'

To three, and store the value in x.

In [45]:
x = iv.nstr(x,3)
x

Out[45]:
'[3.13, 3.25]'

We can use a substring to pull out the first number.

In [46]:
x[1:5]

Out[46]:
'3.13'

Here is a fancier way of extracting the first number from the string. If s is a string, you can type s.split? to find out what split does. Play around with it!

In [47]:
x.split("[")[1].split(",")[0]

Out[47]:
'3.13'


text:
Dr. Cho’s Website
Course Materials

# Python exercises

Institute for Environmental and Spatial Analysis...University of North Georgia

## 1   Arithmetic calculator

Keywords: keyboard input, looping, slicing, branching

simple_calculator.py

Write a simple calculator that supports four basic arithmetic operations (+, -, *, and /) for two floating-point numbers.

1. Print “? ” and read an expression
2. If the expression is quit, stop
3. Split the expression into three parts: left operand, operator, and right operand
4. Branch based on the operator
5. Calculate the expression and print its result
6. Go to step 1

## 2   Dice rolling

Keywords: keyboard input, probability, random()

dice_rolling.py

Python provides a pseudo-random number generator in the random module. We’ll use the random() function from this module to write a dice rolling simulator. This function takes no arguments and returns a pseudo-random float between $[0,1)$. That is, $0\le\text{random()}<1$. The workflow of this program should be as follows:

1. Print “Roll? ” and read a string input
2. If the input is quit, stop
3. Take a random number from random()
4. Print one of 1–6 based on the random number from step 3
5. Go to step 1

Do NOT use randint(). Only use these three functions: random(), print(), and input().

## 3   HTTP client

Keywords: http, http.client, open

http_client.py

Write a simple HTTP client that downloads https://prd-tnm.s3.amazonaws.com/StagedProducts/Elevation/13/ArcGrid/USGS_NED_13_n34w085_ArcGrid.zip and saves it as USGS_NED_13_n34w085_ArcGrid.zip.

## 4   Recursive factorial

factorial.py

Write a factorial function that recursively calls itself to calculate and return the factorial of an integer. \begin{align*} n!&=n\times(n-1)!\\ 0!&=1 \end{align*}

## 5   Circle equation

Keywords: math

circle.py

The following is an equation of a circle with radius $r$ centered at $(0,0)$ in an $x$-$y$ plane: $x^2+y^2=r^2.$

1. Express this equation in radius $r$ and angle $\theta$.
2. Write a function that returns $y$ when $r$ and $x$ are passed to it as arguments.
3. Write a function that returns $r$ and $\theta$ when $x$ and $y$ are given.

## 6   Draw circles

Keywords: turtle, tkinter

turtle_circle.py

Draw a circle using trigonometry and the turtle module.

tkinter_circle.py

Draw a circle using trigonometry and the tkinter module.

## 7   Read and write GeoTIFF files

Keywords: matplotlib.pyplot, PIL.Image

1. Read elevation.tif in nc_spm_08_grass7_exercise.zip and plot it.
2. Save the GeoTIFF file as elevation_new.tif and compare the original and new files.
3. Calculate its extent using TIFF tags

## 8   Moving average

Keywords: numpy, matplotlib.pyplot, PIL.Image

moving_average.py

1. Read elevation.tif in nc_spm_08_grass7_exercise.zip as a numpy array
2. Calculate the moving average of nine neighbor cells
3. Create movavg.tif with the moving average result

## 9   Read and write shapefiles

Keywords: geopandas, shapely

shapefile_write.py Create new shapefiles polys.shp, lines.shp, and points.shp with some geometries.

## 10   Vector analysis

Keywords: geopandas, shapely

vector_analysis.py

1. Create 500-meter buffers around schools_wake.shp in nc_spm_08_grass7_exercise.zip
2. Clip roadsmajor.shp in nc_spm_08_grass7_exercise.zip to the buffers
3. Select the roadsmajor features that intersect with the school buffers

## 11   EarthExplorer client

Keywords: json

earthexplorer_client.py

1. Send a JSON request using the POST method for retreiving an API key
2. Parse the JSON response to extract the API key

## 12   Conway’s game of life

Keywords: matplotlib.animation.FuncAnimation

game_of_life.py

Write Conway’s game of life.

## 13   Moving ball

Keywords: tkinter

moving_ball.py

1. Draw a blue ball
2. Make the ball move in a diagonal direction
3. Allow it to bounce off of the walls
4. Control the ball using the arrow keys

## 14   Quiz: Average elevation

Keywords: matplotlib.pyplot, PIL.Image, geopandas, shapely

FirstLastname_average_elevation.py

In this analysis, we want to calculate the average elevation of a certain census tract.

1. Read elevation.tif and census_wake2000.shp in nc_spm_08_grass7_exercise.zip
2. Plot elevation.tif and overlay census_wake2000.shp
3. Calculate the average elevation of the census tract 25
4. Print its result using the format f'Tract ID:\t{tract_id}\nCount:\t\t{count}\nTotal:\t\t{total}\nAverage:\t{total/count}'


text:
# Bayesian inference of parameter governing Markov transition matrix

A 3-state Markov chain $X = \{x_i : i \in \{1, \cdots, N\}\}$ is observed, and its transition matrix $P$ is assumed to be of the form

$$\begin{pmatrix} (1-a)^2 & 2a(1-a) & a^2 \\ b(1-a) & (1-a)(1-b) + ab & a(1-b) \\ b^2 & 2b(1-b) & (1-b)^2 \end{pmatrix}$$

with $a$ and $b$ assigned a uniform prior distribution between 0 and 1.

Let $c_{ij}$ denote the number of transitions from state $i$ to $j$ in the observed chain $X$. I'm interested in sampling from the posterior distribution $$p(a,b|X) \propto p(X|a, b)p(a,b) = \prod_{i,j} {P_{ij}}^{c_{ij}}$$

This can be further developed as $$(1-a)^{2c_{11}}\ (2a(1-a))^{c_{12}}\ a^{2c_{13}}\ (b(1-a))^{c_{21}}\ ((1-a)(1-b) + ab)^{c_{22}}\ (a(1-b))^{c_{23}}\ b^{2c_{31}}\ (2b(1-b))^{c_{32}}\ (1-b)^{2c_{33}}$$ but I don't know if it can be factored into a much nicer form. I'm wondering if there is any way to sample from this distribution without resorting to simulation methods such as MCMC.

EDIT: improved question

• The first step would be to construct the likelihood for an observed chain. Given a fixed p and q, what is the probability of an observed chain? – aleshing Jul 14 '18 at 5:28
• Given the transition matrix $P$, one way of expressing the likelihood would be $\prod_{i=1}^{N-1} P_{X_i, X_{i+1}}$. – Gonzalo Benegas Jul 14 '18 at 5:42
• Ok now write that out in terms of $p$ and $q$ (hint: using indicator functions or counts of specific transitions observed is the best way to go) – aleshing Jul 14 '18 at 14:11
• Another hint: suppose $c_{22}=0$, and simplify the likelihood in this case. Using your independent uniform priors you should be able to find the posterior in closed form. – aleshing Jul 16 '18 at 21:15
• In that case it seems $a$ and $b$ would have beta posterior distributions. But if the assumption does not hold? Do you suggest anything else? – Gonzalo Benegas Jul 16 '18 at 22:11


text:
# Two Semantics for a Language of Reactive Objects

1 MEIJE - Concurrency, Synchronization and Real-time Programming
CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : We are studying semantics of a small object-based language, with the following main characteristics: parallelism, dynamicity, high order parameters, notion of a global instant, and reactivity. We give formal semantics using two related formalisms , namely $\pi$-calculus and the so-called «Chemical Abstract Machine» (\Name{CHAM}). These formalisms are both powerful enough to express dynamicity and high order parameters, but they give distinct insights into the language. In $\pi$-calculus, emphasis is put on communications through channels; on the other hand, in \Name{CHAM}, emphasis is put on more abstract and general interactions between program parts. We finally prove adequacy of the $\pi$-calculus semantics w.r.t. the \Name{CHA- M} semantics.
Keywords :
Type de document :
Rapport
RR-2511, INRIA. 1995
Domaine :

https://hal.inria.fr/inria-00074167
Contributeur : Rapport de Recherche Inria <>
Soumis le : mercredi 24 mai 2006 - 14:39:29
Dernière modification le : samedi 27 janvier 2018 - 01:31:28
Document(s) archivé(s) le : dimanche 4 avril 2010 - 21:19:30

### Identifiants

• HAL Id : inria-00074167, version 1

### Citation

Frédéric Boussinot, Cosimo Laneve. Two Semantics for a Language of Reactive Objects. RR-2511, INRIA. 1995. 〈inria-00074167〉

### Métriques

Consultations de la notice

## 108

Téléchargements de fichiers


text:
Page 1 of 1

### Determining the cathode/anode

Posted: Fri Mar 06, 2020 9:48 pm
How would you determine the cathode or anode of a cell by only knowing the E value of the half reactions (assuming the you don't know which electrode is being oxidized or which is being reduced)?

### Re: Determining the cathode/anode

Posted: Fri Mar 06, 2020 11:36 pm
the anode is the one with the most negative potential

### Re: Determining the cathode/anode

Posted: Sat Mar 07, 2020 12:31 am
You can use the equation Ecell = Eright - Eleft to deduce the cathode and anode. Eright would be the cathode and Eleft would be the anode.

### Re: Determining the cathode/anode

Posted: Sat Mar 07, 2020 1:09 am
The total Ecell should always be positive (unless the cell was powered by some non-spontaneous energy source). So if you have the standard E values for the half reactions (both in reduction form) you should flip the one that makes the total Ecell the most positive. The flipped half reaction is the oxidation reaction and it would occur at the anode.

### Re: Determining the cathode/anode

Posted: Sat Mar 07, 2020 10:37 am
Since the half reactions are usually given to us as reductions, the standard potential of the cathode should be greater than the anode.
This is because $E^{\circ}$ is the standard potential of the half reaction. The greater it is, the more likely it is that the reaction will occur. We are always given reduction half reactions, and we know that reduction occurs at the cathode. Therefore, the reaction with the highest potential occurs at the cathode.
If the half reactions are oxidation reactions, then E represents the oxidation potential and the reaction with the greater number occurs at the anode.

### Re: Determining the cathode/anode

Posted: Sat Mar 07, 2020 10:58 am
The more negative cell potential has the most powerful reducing power and is therefore the anode.

### Re: Determining the cathode/anode

Posted: Sat Mar 07, 2020 11:19 pm
Samuel G Rivera - Discussion 4I wrote:How would you determine the cathode or anode of a cell by only knowing the E value of the half reactions (assuming the you don't know which electrode is being oxidized or which is being reduced)?

To determine which is the anode, and which is the cathode, you look at the standard reaction potentials and whichever is most positive, that is the cathode; the other is the anode. In a concentration cell, the larger concentration is the cathode.

### Re: Determining the cathode/anode

Posted: Sun Mar 08, 2020 12:36 am
Use Ecell = Ecathode - Eanode... in a galvanic cell, make sure that Ecell is positive, so the cathode needs to be more positive and the anode more negative.

### Re: Determining the cathode/anode

Posted: Sun Mar 08, 2020 12:57 am
The most negatively reduced standard cell potential occurs at the cathode and the most positive standard cell potential occurs at the anode.

### Re: Determining the cathode/anode

Posted: Sun Mar 08, 2020 6:32 pm
the anode is the more negative cell potential and the cathode is the more positive one


text:
시간 제한 메모리 제한 제출 정답 맞은 사람 정답 비율
1 초 128 MB 4 2 2 50.000%

## 문제

In spite of the scarcity of snowfall in Madrid, interest in winter sports is growing in the city, especially with regard to skiing. Many people spend several weekends or even full weeks improving their skills in the mountains.

In this problem we deal with only one of the multiple alpine skiing disciplines: slalom. A course is constructed by laying out a series of gates, which are formed by two poles. The skier must pass between the two poles forming each gate. The winner is the skier who takes the least time to complete the course while not missing any of the gates.

You have recently started to learn to ski, but you have already set yourself the goal of taking part in the Winter Olympic Games of 2018, for which Madrid will presumably present a candidature. As part of the theoretical training, you need to write a program that calculates, given a starting point and a series of gates, the minimum-length path starting from the point given and passing through each gate until you reach the last one, which is the finish line. You may assume that the gates are horizontal and are ordered from highest to lowest, so that you need to pass through them in order. You consider yourself an accomplished skier, so you can make any series of turns, no matter how difficult, and your only concern is minimizing the total length of the path.

## 입력

The first line of each case gives the number of gates n (1 ≤ n ≤ 1 000). The next line contains two floating point numbers, the Cartesian coordinates x and y of the starting position, in that order. Next come n lines with three floating point numbers each, y x1 x2, meaning that the next gate is a horizontal line from (x1, y) to (x2, y). You can safely assume that x1 < x2. The values of y are strictly decreasing and are always smaller than that of the starting position. The last gate represents the finish line. All coordinates are between −500 000 and 500 000, inclusive. A value of 0 for n means the end of the input. A blank line follows each case.

## 출력

For each test case, output a line with the minimum distance needed to reach the finish line. Your answer should be accurate to within an absolute or relative error of 10−7.

## 예제 입력

2
0 2
1 1 2
0 0.5 3

3
0 4
3 1 2
2 -1 0
1 1 2

0

## 예제 출력

2.41421356237
4.24264068712


text:
# Imagining the imaginary

Algebra Level 4

A student has a lesson on Complex Numbers and makes the following conclusions:

1. $$\mathbb{R} \subset \mathbb{I}$$
2. $$\mathbb{I} \subset \mathbb{C}$$
3. $$\sqrt[-1]{-i} = i$$
4. $$\sqrt[]{\bar{z}z} = |z|$$
5. $$z = r\cos(\theta)+ir\sin(\theta)$$

Which of the above is/are false?

Notes:

• $$i = \sqrt {-1}$$ is the imaginary unit
• $$\mathbb{R}$$ is the set of all real numbers
• $$\mathbb{I}$$ is the set of all imaginary numbers
• $$\mathbb{C}$$ is the set of all complex numbers
• $$z=a+ ib; a,b \in \mathbb R$$ is the general form of a complex number
• $$\bar{z} = a-ib$$ is the conjugate of $$z$$
• $$|z| = \sqrt{a^2+b^2}$$ is the modulus of $$z$$
×


text:
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## Q. 12.EX.5

COMPUTING COMBINED COST OF ORDERING AND HOLDING. Sharp, Inc. (from Examples 3 and 4) wants to determine the combined annual ordering and holding costs.
APPROACH $\blacktriangleright$ Apply Equation (12-5), using the data in Example 3.

$TC = \frac{D}{Q}S + \frac{Q}{2}H$                 (12-5)

## Verified Solution

SOLUTION $\blacktriangleright$
$TC = \frac{D}{Q}S + \frac{Q}{2}H = \frac{1,000}{200}(\ 10) + \frac{200}{2}(\ .50)$ = (5)($10) + (100)($ .50) = $50 +$50 = $100 INSIGHT $\blacktriangleright$ These are the annual setup and holding costs. The$100 total does not include the actual cost of goods. Notice that in the EOQ model, holding costs always equal setup (order) costs.

LEARNING EXERCISE $\blacktriangleright$ Find the total annual cost if D = 1,200 units in Example 3. [Answer: \$109.54.]
RELATED PROBLEMS $\blacktriangleright$ 12.11, 12.14, 12.15, 12.16 (12.33b,c; 12.35e; 12.36a,b are available in MyOMLab)


text:
# How to implement dependent type theory I

I am spending a semester at the Institute for Advanced Study where we have a special year on Univalent foundations. We are doing all sorts of things, among others experimenting with type theories. We have got some real experts here who know type theory and Coq inside out, and much more, and they’re doing crazy things to Coq (I will report on them when they are done). In the meanwhile I have been thinking how one might implement dependent type theories with undecidable type checking. This is a tricky subject and I am certainly not the first one to think about it. Anyhow, if I want to experiment with type theories, I need a small prototype first. Today I will present a very minimal one, and build on it in future posts.

Make a guess, how many lines of code does it take to implement a dependent type theory with universes, dependent products, a parser, lexer, pretty-printer, and a toplevel which uses line-editing when available?

# Eff 3.0

Matija and I are pleased to announce a new major release of the eff programming language.

In the last year or so eff has matured considerably:

• It now looks and feels like OCaml, so you won’t have to learn yet another syntax.
• It has static typing with parametric polymorphism and type inference.
• Eff now clearly separates three basic concepts: effect types, effect instances, and handlers.
• How eff works is explained in our paper on Programming with Algebraic Effects and Handlers.
• We moved the source code to GitHub, so go ahead and fork it!

# Programming with Algebraic Effects and Handlers

With Matija Pretnar.

Abstract: Eff is a programming language based on the algebraic approach to computational effects, in which effects are viewed as algebraic operations and effect handlers as homomorphisms from free algebras. Eff supports first-class effects and handlers through which we may easily define new computational effects, seamlessly combine existing ones, and handle them in novel ways. We give a denotational semantics of eff and discuss a prototype implementation based on it. Through examples we demonstrate how the standard effects are treated in eff, and how eff supports programming techniques that use various forms of delimited continuations, such as backtracking, breadth-first search, selection functionals, cooperative multi-threading, and others.

ArXiv version: arXiv:1203.1539v1 [cs.PL]

While making a comment on Stackoverflow I noticed something: suppose we have a term in the $\lambda$-calculus in which no abstracted variable is used more than once. For example, $\lambda a b c . (a b) (\lambda d. d c)$ is such a term, but $\lambda f . f (\lambda x . x x)$ is not because $x$ is used twice. If I am not mistaken, all such terms can be typed. For example:

# fun a b c -> (a b) (fun d -> d c) ;;
- : ('a -> (('b -> 'c) -> 'c) -> 'd) -> 'a -> 'b -> 'd = <fun>

# fun a b c d e e' f g h i j k l m n o o' o'' o''' p q r r' s t u u' v w x y z ->
q u i c k b r o w n f o' x j u' m p s o'' v e r' t h e' l a z y d o''' g;;
- : 'a -> 'b -> 'c -> 'd -> 'e -> 'f -> 'g -> 'h -> 'i -> 'j ->
'k -> 'l -> 'm -> 'n -> 'o -> 'p -> 'q -> 'r -> 's -> 't ->
('u -> 'j -> 'c -> 'l -> 'b -> 'v -> 'p -> 'w -> 'o -> 'g ->
'q -> 'x -> 'k -> 'y -> 'n -> 't -> 'z -> 'r -> 'a1 -> 'e ->
'b1 -> 'c1 -> 'i -> 'f -> 'm -> 'a -> 'd1 -> 'e1 -> 'd -> 's
-> 'h -> 'f1) -> 'v -> 'b1 -> 'z -> 'c1 -> 'u -> 'y -> 'a1
-> 'w -> 'x -> 'e1 -> 'd1 -> 'f1 = <fun>

What is the easiest way to see that this really is the case?

A related question is this (I am sure people have thought about it): how big can a type of a typeable $\lambda$-term be? For example, the Ackermann function can be typed as follows, although the type prevents it from doing the right thing in a typed setting:

# let one = fun f x -> f x ;;
val one : ('a -> 'b) -> 'a -> 'b =
# let suc = fun n f x -> n f (f x) ;;
val suc : (('a -> 'b) -> 'b -> 'c) -> ('a -> 'b) -> 'a -> 'c =
# let ack = fun m -> m (fun f n -> n f (f one)) suc ;;
val ack :
((((('a -> 'b) -> 'a -> 'b) -> 'c) ->
(((('a -> 'b) -> 'a -> 'b) -> 'c) -> 'c -> 'd) -> 'd) ->
((('e -> 'f) -> 'f -> 'g) -> ('e -> 'f) -> 'e -> 'g) -> 'h) -> 'h = <fun>

That’s one mean type there! Can it be “explained”? Hmm, why does ack compute the Ackermann function in the untyped $\lambda$-calculus?

# Running a classical proof with choice in Agda

As a preparation for my part of a joint tutorial Programs from proofs at MFPS 27 at the end of this month with Ulrich Berger, Monika Seisenberger, and Paulo Oliva, I’ve developed in Agda some things we’ve been doing together.

Using

for giving a proof term for classical countable choice, we prove the classical infinite pigeonhole principle in Agda: every infinite boolean sequence has a constant infinite subsequence, where the existential quantification is classical (double negated).

As a corollary, we get the finite pigeonhole principle, using Friedman’s trick to make the existential quantifiers intuitionistic.

This we can run, and it runs fast enough. The point is to illustrate in Agda how we can get witnesses from classical proofs that use countable choice. The finite pigeonhole principle has a simple constructive proof, of course, and hence this is really for illustration only.

The main Agda files are

These are Agda files converted to html so that you can navigate them by clicking at words to go to their definitions. A zip file with all Agda files is available. Not much more information is available here.

The three little modules that implement the Berardi-Bezem-Coquand, Berger-Oliva and Escardo-Oliva functionals disable the termination checker, but no other module does. The type of these functionals in Agda is the J-shift principle, which generalizes the double-negation shift.

# Bob Harper has a blog

Bob Harper of CMU, has recently started a blog, called Existential Type, about programming languages. He is a leading expert in Programming Languages. I remember being deeply inspired the first time I heard him talk. I was an incoming graduate student at CMU and he presented what the programming languages people at CMU did. His posts are fun to read, unreserved and very educational. Highly recommended!

# Delimited continuations in eff

[UPDATE 2012-03-08: since this post was written eff has changed considerably. For updated information, please visit the eff page.]

Let’s keep the blog rolling! Here are delimited continuations in eff, and a bunch of questions I do not know the answers to.

# How eff handles built-in effects

[UPDATE 2012-03-08: since this post was written eff has changed considerably. For updated information, please visit the eff page.]

From some of the responses we have been getting it looks like people think that the io effect in eff is like unsafePerformIO in Haskell, namely that it causes an effect but pretends to be pure. This is not the case. Let me explain how eff handles built-in effects.

# Programming with effects II: Introducing eff

[UPDATE 2012-03-08: since this post was written eff has changed considerably. For updated information, please visit the eff page.]

This is a second post about the programming language eff. We covered the theory behind it in a previous post. Now we turn to the programming language itself.

Please bear in mind that eff is an academic experiment. It is not meant to take over the world. Yet. We just wanted to show that the theoretical ideas about the algebraic nature of computational effects can be put into practice. Eff has many superficial similarities with Haskell. This is no surprise because there is a precise connection between algebras and monads. The main advantage of eff over Haskell is supposed to be the ease with which computational effects can be combined.

# Programming with effects I: Theory

[UPDATE 2012-03-08: since this post was written eff has changed considerably. For updated information, please visit the eff page.]

I just returned from Paris where I was visiting the INRIA ?r² team. It was a great visit, everyone was very hospitable, the food was great, and the weather was nice. I spoke at their seminar where I presented a new programming language eff which is based on the idea that computational effects are algebras. The language has been designed and implemented jointly by Matija Pretnar and myself. Eff is far from being finished, but I think it is ready to be shown to the world. What follows is an extended transcript of the talk I gave in Paris. It is divided into two posts. The present one reviews the basic theory of algebras for a signature and how they are related to computational effects. The impatient readers can skip ahead to the second part, which is about the programming language.

A side remark: I have updated the blog to WordPress to 3.0 and switched to MathJax for displaying mathematics. Now I need to go through 70 old posts and convert the old ASCIIMathML notation to MathJax, as well as fix characters which got garbled during the update. Oh well, it is an investment for the future.