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http://mathematica.stackexchange.com/questions/57102/manipulate-strings-within-a-list
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# Manipulate strings within a list
Let's say I have
Solve[x^4 + 3 == 0, x]
with output
{{x -> -(-3)^(1/4)}, {x -> -i (-3)^(1/4)}, {x ->i (-3)^(1/4)}, {x -> (-3)^(1/4)}}
how do i lose the "x->" part of each string within this list and get something like
{-(-3)^(1/4), -i (-3)^(1/4), i (-3)^(1/4), (-3)^(1/4)}
Thanks for the answers/links. Judging by the answers I realise now that I have formulated my question poorly. I wanted to know how I lose symbols that are in front of a specific symbol. Let's have a look at two more examples:
list={a: horse, b: chicken, c: fish}
how do I lose "a: ","b: ","c: "
or
list2={section 1, section 2, section 3}
how do I lose "section"
-
Related Q/As: this and this ... – kguler Aug 10 at 17:46
Thanks for your quick response i have edited my question. – 11drsnuggles11 Aug 10 at 18:05
This is the canonical post about how to deal with the list returned by Solve. The other to examples will have to be justified/the context will have to be further explained, I think. – Pickett Aug 10 at 18:08
öska, my mistake, i've changed it, now it should make more sense ;) – 11drsnuggles11 Aug 10 at 18:15
lst1 = {{x -> -(-3)^(1/4)}, {x -> -i (-3)^(1/4)}, {x -> i (-3)^(1/4)},
{x -> (-3)^(1/4)}};
lst2 = {a : horse, b : chicken, c : fish};
lst3 = {section 1, section 2, section 3};
lst4 = {section1, section2, section3};
Last @@@ lst1
(* {-(-3)^(1/4),-(-3)^(1/4) i,(-3)^(1/4) i,(-3)^(1/4)} *)
Last /@ lst2
(* {horse,chicken,fish} *)
Block[{section = 1}, lst3]
(* {1,2,3} *)
StringTake[SymbolName/@lst4, -1]
(* {1,2,3} *)
StringReplace[SymbolName /@ lst4, "section" -> ""]
(* {1,2,3} *)
-
The two cases are different but here is something you can try:
list = {a : horse, b : chicken, c : fish};
list2 = {section 1, section 2, section 3};
list /. x_Pattern :> Last@x
{horse, chicken, fish}
list2 /. section -> 1
{1, 2, 3}
-
Your 2nd one is funny because I just was trying: list2 /. section :> Sequence[] which only gives {2, 3} - and I don't know why the 1 is missing :) – eldo Aug 10 at 18:39
@eldo Because section*1 == section :D – Öskå Aug 10 at 19:05
why not simply:
sol = Solve[x^4 + 3 == 0, x]
x /. sol
or
#[[2]] & @@@ sol
(*{-(-3)^(1/4), -I (-3)^(1/4), I (-3)^(1/4), (-3)^(1/4)}*)
for the second Example you can try:
list = {a : horse, b : chicken, c : fish}
#[[2]] & @@@ Transpose[{list}]
(*{horse, chicken, fish}*)
-
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2014-09-24 04:27:08
|
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http://clay6.com/qa/10801/suppose-x-has-a-binomial-distribution-b-large-6-large-frac-what-is-the-most
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# Suppose X has a binomial distribution $B\; \large($$6, \large\frac{1}{2})$. What is the most likely outcome and for what value of X, where x = 0, 1, 2, 3, 4, 5, 6?
$\begin{array}{1 1} X = 3, \; P (X=3) = \large\frac{20}{64} \\ X = 3, \; P (X=4) = \large\frac{15}{64} \\ X = 3, \; P (X=4) = \large\frac{25}{64} \\ X = 3, \; P (X=3) = \large\frac{15}{64} \end{array}$
|
2020-06-01 04:22:37
|
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https://www.elitedigitalstudy.com/10346/verify-this-result-for-dabc-whose-vertices-are-a-4-6-b-3-2-and-c-5-2
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You have studied in Class IX that a median of a triangle divides it into two triangles of equal areas. Verify this result for ΔABC whose vertices are A (4, – 6), B (3, – 2) and C (5, 2).
Asked by Pragya Singh | 1 year ago | 84
##### Solution :-
Let the vertices of the triangle be A (4, -6), B (3, -2), and C (5, 2).
Let D be the mid-point of side BC of ΔABC. Therefore, AD is the median in ΔABC.
Coordinates of point D = Midpoint of BC
=$$\dfrac{1(3+5)}{2}$$,$$\dfrac{(-2+2)}{2}$$ = (4, 0)
Formula, to find Area of a triangle = $$\dfrac{1}{2}$$ × [x1(y2 – y3) + x2(y3 – y1) + x3(y1 – y2)]
Now, Area of ΔABD = $$\dfrac{1}{2}$$ [(4) {(-2) – (0)} + 3{(0) – (-6)} + (4) {(-6) – (-2)}]
$$\dfrac{1}{2}$$ (-8 + 18 – 16)
= -3 square units
However, area cannot be negative. Therefore, area of ΔABD is 3 square units.
Area of ΔACD = $$\dfrac{1}{2}$$ [(4) {0 – (2)} + 4{(2) – (-6)} + (5) {(-6) – (0)}]
$$\dfrac{1}{2}$$ (-8 + 32 – 30) = -3 square units
However, area cannot be negative. Therefore, the area of ΔACD is 3 square units.
The area of both sides is the same. Thus, median AD has divided ΔABC in two triangles of equal areas.
Answered by Abhisek | 1 year ago
### Related Questions
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#### Show that the points A (5, 6), B (1, 5), C (2, 1) and D (6, 2) are the vertices of a square.
Answer the following questions:-
(i) Show that the points A (5, 6), B (1, 5), C (2, 1) and D (6, 2) are the vertices of a square.
(ii) Prove that the points A (2, 3), B (-2, 2), C (-1, -2) and D (3, -1) are the vertices of a square ABCD.
(iii) Name the type of triangle PQR formed by the point $$P(\sqrt{2} , \sqrt{2}), Q(- \sqrt{2}, – \sqrt{2)} and\; R (-\sqrt{6} , \sqrt{6} )$$
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2022-11-27 09:04:36
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|
https://www.taylorfrancis.com/books/9780429213106/chapters/10.1201/b12914-11
|
chapter 5
88 Pages
## Actions during service
Concrete structures or structural elements are designed to take up external mechanical loading. In modern design codes based on ultimate limit state design, it is considered that concrete in tension will crack, leaving only the reinforcement to carrying the tensile loading. This does not mean that concrete structural elements under service load will always show cracks. However, it does mean that if cracks occur in concrete structures, the structural load-bearing capacity of the structure is not immediately at risk. Of course, excessive cracking in concrete structures is neither generally expected nor accepted. And even small cracks could impair long-term concrete durability because of the accelerated ingress of aggressive substances.
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2020-01-22 05:08:01
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https://docs.mfem.org/html/classmfem_1_1StaticCondensation.html
|
MFEM v4.4.0 Finite element discretization library
mfem::StaticCondensation Class Reference
#include <staticcond.hpp>
## Public Member Functions
StaticCondensation (FiniteElementSpace *fespace)
Construct a StaticCondensation object. More...
~StaticCondensation ()
Destroy a StaticCondensation object. More...
int GetNPrDofs () const
Return the number of vector private dofs. More...
int GetNExDofs () const
Return the number of vector exposed/reduced dofs. More...
bool ReducesTrueVSize () const
void Init (bool symmetric, bool block_diagonal)
FiniteElementSpaceGetTraceFESpace ()
Return a pointer to the reduced/trace FE space. More...
ParFiniteElementSpaceGetParTraceFESpace ()
Return a pointer to the parallel reduced/trace FE space. More...
void AssembleMatrix (int el, const DenseMatrix &elmat)
void AssembleBdrMatrix (int el, const DenseMatrix &elmat)
void Finalize ()
Finalize the construction of the Schur complement matrix. More...
void SetEssentialTrueDofs (const Array< int > &ess_tdof_list)
Determine and save internally essential reduced true dofs. More...
void EliminateReducedTrueDofs (const Array< int > &ess_rtdof_list, Matrix::DiagonalPolicy dpolicy)
Eliminate the given reduced true dofs from the Schur complement matrix S. More...
void EliminateReducedTrueDofs (Matrix::DiagonalPolicy dpolicy)
Eliminate the internal reduced true dofs (set using SetEssentialTrueDofs()) from the Schur complement matrix S. More...
bool HasEliminatedBC () const
Return true if essential boundary conditions have been eliminated from the Schur complement matrix. More...
SparseMatrixGetMatrix ()
Return the serial Schur complement matrix. More...
SparseMatrixGetMatrixElim ()
Return the eliminated part of the serial Schur complement matrix. More...
HypreParMatrixGetParallelMatrix ()
Return the parallel Schur complement matrix. More...
HypreParMatrixGetParallelMatrixElim ()
Return the eliminated part of the parallel Schur complement matrix. More...
void GetParallelMatrix (OperatorHandle &S_h) const
Return the parallel Schur complement matrix in the format specified by SetOperatorType(). More...
void GetParallelMatrixElim (OperatorHandle &S_e_h) const
Return the eliminated part of the parallel Schur complement matrix in the format specified by SetOperatorType(). More...
void SetOperatorType (Operator::Type tid)
Set the operator type id for the parallel reduced matrix/operator. More...
void ReduceRHS (const Vector &b, Vector &sc_b) const
void ReduceSolution (const Vector &sol, Vector &sc_sol) const
void ReduceSystem (Vector &x, Vector &b, Vector &X, Vector &B, int copy_interior=0) const
Set the reduced solution X and r.h.s B vectors from the full linear system solution x and r.h.s. b vectors. More...
void ConvertMarkerToReducedTrueDofs (const Array< int > &ess_tdof_marker, Array< int > &ess_rtdof_marker) const
void ConvertListToReducedTrueDofs (const Array< int > &ess_tdof_list_, Array< int > &ess_rtdof_list_) const
void ComputeSolution (const Vector &b, const Vector &sc_sol, Vector &sol) const
## Detailed Description
Auxiliary class StaticCondensation, used to implement static condensation in class BilinearForm.
Static condensation is a technique for solving linear systems by eliminating groups/blocks of unknowns and reducing the original system to the remaining interfacial unknowns. The assumption is that unknowns in one group are connected (in the graph of the matrix) only to unknowns in the same group or to interfacial unknowns but not to other groups.
For finite element systems, the groups correspond to degrees of freedom (DOFs) associated with the interior of the elements. The rest of the DOFs (associated with the element boundaries) are interfacial.
In block form the matrix of the system can be written as
$A = \begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{pmatrix} \begin{array}{l} \text{-- groups: element interior/private DOFs} \\ \text{-- interface: element boundary/exposed DOFs} \end{array}$
where the block $$A_1$$ is itself block diagonal with small local blocks and it is, therefore, easily invertible.
Starting with the block system
$\begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{pmatrix} \begin{pmatrix} X_1 \\ X_2 \end{pmatrix} = \begin{pmatrix} B_1 \\ B_2 \end{pmatrix}$
the reduced, statically condensed system is given by
$S_{22} X_2 = B_2 - A_{21} A_{11}^{-1} B_1$
where the Schur complement matrix $$S_{22}$$ is given by
$S_{22} = A_{22} - A_{21} A_{11}^{-1} A_{12}.$
After solving the Schur complement system, the $$X_1$$ part of the solution can be recovered using the formula
$X_1 = A_{11}^{-1} ( B_1 - A_{12} X_2 ).$
Definition at line 65 of file staticcond.hpp.
## Constructor & Destructor Documentation
mfem::StaticCondensation::StaticCondensation ( FiniteElementSpace * fespace )
Construct a StaticCondensation object.
Definition at line 17 of file staticcond.cpp.
mfem::StaticCondensation::~StaticCondensation ( )
Destroy a StaticCondensation object.
Definition at line 103 of file staticcond.cpp.
## Member Function Documentation
void mfem::StaticCondensation::AssembleBdrMatrix ( int el, const DenseMatrix & elmat )
Assemble the contribution to the Schur complement from the given boundary element matrix 'elmat'.
Definition at line 225 of file staticcond.cpp.
void mfem::StaticCondensation::AssembleMatrix ( int el, const DenseMatrix & elmat )
Assemble the contribution to the Schur complement from the given element matrix 'elmat'; save the other blocks internally: A_pp_inv, A_pe, and A_ep.
Definition at line 187 of file staticcond.cpp.
void mfem::StaticCondensation::ComputeSolution ( const Vector & b, const Vector & sc_sol, Vector & sol ) const
Given a solution of the reduced system 'sc_sol' and the RHS 'b' for the full linear system, compute the solution of the full system 'sol'.
Definition at line 476 of file staticcond.cpp.
void mfem::StaticCondensation::ConvertListToReducedTrueDofs ( const Array< int > & ess_tdof_list_, Array< int > & ess_rtdof_list_ ) const
inline
Restrict a list of true FE space dofs to a list of reduced/trace true FE space dofs.
Definition at line 202 of file staticcond.hpp.
void mfem::StaticCondensation::ConvertMarkerToReducedTrueDofs ( const Array< int > & ess_tdof_marker, Array< int > & ess_rtdof_marker ) const
Restrict a marker Array on the true FE space dofs to a marker Array on the reduced/trace true FE space dofs.
Definition at line 439 of file staticcond.cpp.
void mfem::StaticCondensation::EliminateReducedTrueDofs ( const Array< int > & ess_rtdof_list, Matrix::DiagonalPolicy dpolicy )
Eliminate the given reduced true dofs from the Schur complement matrix S.
Definition at line 286 of file staticcond.cpp.
void mfem::StaticCondensation::EliminateReducedTrueDofs ( Matrix::DiagonalPolicy dpolicy )
inline
Eliminate the internal reduced true dofs (set using SetEssentialTrueDofs()) from the Schur complement matrix S.
Definition at line 137 of file staticcond.hpp.
void mfem::StaticCondensation::Finalize ( )
Finalize the construction of the Schur complement matrix.
Definition at line 233 of file staticcond.cpp.
SparseMatrix& mfem::StaticCondensation::GetMatrix ( )
inline
Return the serial Schur complement matrix.
Definition at line 152 of file staticcond.hpp.
SparseMatrix& mfem::StaticCondensation::GetMatrixElim ( )
inline
Return the eliminated part of the serial Schur complement matrix.
Definition at line 155 of file staticcond.hpp.
int mfem::StaticCondensation::GetNExDofs ( ) const
inline
Return the number of vector exposed/reduced dofs.
Definition at line 99 of file staticcond.hpp.
int mfem::StaticCondensation::GetNPrDofs ( ) const
inline
Return the number of vector private dofs.
Definition at line 97 of file staticcond.hpp.
HypreParMatrix& mfem::StaticCondensation::GetParallelMatrix ( )
inline
Return the parallel Schur complement matrix.
Definition at line 159 of file staticcond.hpp.
void mfem::StaticCondensation::GetParallelMatrix ( OperatorHandle & S_h ) const
inline
Return the parallel Schur complement matrix in the format specified by SetOperatorType().
Definition at line 167 of file staticcond.hpp.
HypreParMatrix& mfem::StaticCondensation::GetParallelMatrixElim ( )
inline
Return the eliminated part of the parallel Schur complement matrix.
Definition at line 162 of file staticcond.hpp.
void mfem::StaticCondensation::GetParallelMatrixElim ( OperatorHandle & S_e_h ) const
inline
Return the eliminated part of the parallel Schur complement matrix in the format specified by SetOperatorType().
Definition at line 171 of file staticcond.hpp.
ParFiniteElementSpace* mfem::StaticCondensation::GetParTraceFESpace ( )
inline
Return a pointer to the parallel reduced/trace FE space.
Definition at line 113 of file staticcond.hpp.
FiniteElementSpace* mfem::StaticCondensation::GetTraceFESpace ( )
inline
Return a pointer to the reduced/trace FE space.
Definition at line 109 of file staticcond.hpp.
bool mfem::StaticCondensation::HasEliminatedBC ( ) const
inline
Return true if essential boundary conditions have been eliminated from the Schur complement matrix.
Definition at line 142 of file staticcond.hpp.
void mfem::StaticCondensation::Init ( bool symmetric, bool block_diagonal )
Prepare the StaticCondensation object to assembly: allocate the Schur complement matrix and the other element-wise blocks.
Definition at line 132 of file staticcond.cpp.
void mfem::StaticCondensation::ReduceRHS ( const Vector & b, Vector & sc_b ) const
Given a RHS vector for the full linear system, compute the RHS for the reduced linear system: sc_b = b_e - A_ep A_pp_inv b_p.
Definition at line 309 of file staticcond.cpp.
void mfem::StaticCondensation::ReduceSolution ( const Vector & sol, Vector & sc_sol ) const
Restrict a solution vector on the full FE space dofs to a vector on the reduced/trace true FE space dofs.
Definition at line 389 of file staticcond.cpp.
bool mfem::StaticCondensation::ReducesTrueVSize ( ) const
Return true if applying the static condensation actually reduces the (global) number of true vector dofs.
Definition at line 116 of file staticcond.cpp.
void mfem::StaticCondensation::ReduceSystem ( Vector & x, Vector & b, Vector & X, Vector & B, int copy_interior = 0 ) const
Set the reduced solution X and r.h.s B vectors from the full linear system solution x and r.h.s. b vectors.
This method should be called after the internal reduced essential dofs have been set using SetEssentialTrueDofs() and both the Schur complement and its eliminated part have been finalized.
Definition at line 416 of file staticcond.cpp.
void mfem::StaticCondensation::SetEssentialTrueDofs ( const Array< int > & ess_tdof_list )
inline
Determine and save internally essential reduced true dofs.
Definition at line 128 of file staticcond.hpp.
void mfem::StaticCondensation::SetOperatorType ( Operator::Type tid )
inline
Set the operator type id for the parallel reduced matrix/operator.
Definition at line 174 of file staticcond.hpp.
The documentation for this class was generated from the following files:
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2022-07-05 22:58:52
|
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https://zbmath.org/?q=an:0871.04010
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## A note to the $$T$$-sum of $$L$$-$$R$$ fuzzy numbers.(English)Zbl 0871.04010
Summary: We generalize the results of D. H. Hong and S. Y. Hwang [ibid. 63, No. 2, 175-180 (1994; Zbl 0844.04004)] for the membership function of finite (infinite) sum of $$L$$-$$R$$ fuzzy numbers, where the sum is based on some continuous Archimedean $$t$$-norm $$T$$.
### MSC:
3e+72 Theory of fuzzy sets, etc.
Zbl 0844.04004
Full Text:
### References:
[1] Dubois, D.; Prade, H., Fuzzy Sets and Systems: Theory and Applications (1980), Academic Press: Academic Press New York · Zbl 0444.94049 [2] Fullér, R.; Keresztfalvi, T., t-norm-based addition of fuzzy intervals, Fuzzy Sets and Systems, 51, 155-159 (1992) [3] Hong, D. H.; Hwang, S. Y., On the convergence of $$T$$-sum of $$L-R$$ fuzzy numbers, Fuzzy Sets and Systems, 63, 175-180 (1994) · Zbl 0844.04004 [5] Schweizer, B.; Sklar, A., Probabilistic Metric Spaces (1983), Elsevier: Elsevier New York · Zbl 0546.60010 [6] Triesch, E., On the convergence of product-sum series of $$L-R$$ fuzzy numbers, Fuzzy Sets and Systems, 53, 189-192 (1993) · Zbl 0874.26019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.
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2023-01-29 13:32:40
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https://www.aimsciences.org/article/doi/10.3934/amc.2020084?viewType=html
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# American Institute of Mathematical Sciences
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doi: 10.3934/amc.2020084
## Involutory-Multiple-Lightweight MDS Matrices based on Cauchy-type Matrices
1 Department of Applied Mathematics, Malek Ashtar University of Technology, Isfahan, Iran 2 Department of Mathematical Sciences, Isfahan University of Technology, Isfahan, Iran 3 Department of Electrical & Computer Engineering, University of Victoria, Victoria, BC, Canada
* Corresponding author: Morteza Esmaeili
Revised March 2020 Published June 2020
One of the best methods for constructing maximum distance separable ($\operatorname{MDS}$) matrices is based on making use of Cauchy matrices. In this paper, by using some extensions of Cauchy matrices, we introduce several new forms of $\operatorname{MDS}$ matrices over finite fields of characteristic 2. A known extension of a Cauchy matrix, called the Cauchy-like matrix, with application in coding theory was introduced in 1985. One of the main contributions of this paper is to apply Cauchy-like matrices to introduce $2n \times 2n$ involutory $\operatorname{MDS}$ matrices in the semi-Hadamard form which is a generalization of the previously known methods. We make use of Cauchy-like matrices to construct multiple $\operatorname{MDS}$ matrices which can be used in the Feistel structures. We also introduce a new extension of Cauchy matrices to be referred to as Cauchy-light matrices. The introduced Cauchy-light matrices are applied to construct $n \times n$ $\operatorname{MDS}$ matrices having at least $3n-3$ entries equal to the unit element $1$; such a matrix is called a lightweight $\operatorname{MDS}$ matrix and can be used in the lightweight cryptography. A simple closed-form expression is given for the determinant of Cauchy-light matrices.
Citation: Mohsen Mousavi, Ali Zaghian, Morteza Esmaeili. Involutory-Multiple-Lightweight MDS Matrices based on Cauchy-type Matrices. Advances in Mathematics of Communications, doi: 10.3934/amc.2020084
##### References:
[1] D. Augot and M. Finiasz, Direct construction of recursive MDS diffusion layers using shortened BCH codes, in Fast Software Encryption. FSE 2014, Vol. 8540, Springer, Berlin, Heidelberg, 2014, 3-17. doi: 10.1007/978-3-662-46706-0_1. Google Scholar [2] P. Barreto and V. Rijmen, The Khazad legacy-level block cipher, in Proceedings of the First Open NESSIE Workshop, Belgium, (2000). Google Scholar [3] P. Barreto and V. Rijmen, The Anubis block cipher, in Proceedings of the First Open NESSIE Workshop, Belgium, (2000). Google Scholar [4] C. Beierle, T. Kranz and G. Leander, Lightweight multiplication in $GF(2^n)$ with applications to MDS matrices, in Advances in Cryptology. CRYPTO 2016. Part 1, Lecture Notes in Comput. Sci., Vol. 9814, Springer, Berlin, 2016,625-653. doi: 10.1007/978-3-662-53018-4_23. Google Scholar [5] T. P. Berger, G. Paul and S. Vaudenay, eds., Construction of recursive MDS diffusion layers from gabidulin codes, in Progress in Cryptology. INDOCRYPT 2013, Vol. 8250, Springer, Cham, 2013,274-285. doi: 10.1007/978-3-319-03515-4. Google Scholar [6] J. Daemen and V. Rijmen, The Design of Rijndael: AES - The Advanced Encryption Standard, Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-04722-4. Google Scholar [7] G. Filho, P. Barreto and V. Rijmen, The Maelstrom-$0$ hash function, in Proceedings of the Sixth Brazilian Symposium on Information and Computer Systems Security, (2006). Google Scholar [8] J. Guo, T. Peyrin and A. Poschmann, The PHOTON family of lightweight hash functions, in Advances in Cryptology. CRYPTO 2011, Vol. 6841, Springer, Heidelberg, 2011,222-239. doi: 10.1007/978-3-642-22792-9. Google Scholar [9] K. C. Gupta and I. G. Ray, On constructions of MDS matrices from companion matrices for lightweight cryptography, in Security Engineering and Intelligence Informatics. CD-ARES 2013, Vol. 8128, Springer, Berlin, Heidelberg, 2013, 29-43. doi: 10.1007/978-3-642-40588-4_3. Google Scholar [10] K. C. Gupta and I. G. Ray, On constructions of involutory MDS matrices, in Progress in Cryptology. AFRICACRYPT 2013, Vol. 7918, Springer, Heidelberg, 2013, 43-60. doi: 10.1007/978-3-642-38553-7_3. Google Scholar [11] K. C. Gupta and I. G. Ray, Cryptographically significant MDS matrices based on circulant and circulant-like matrices for lightweight applications, Cryptogr. Commun., 7 (2015), 257-287. doi: 10.1007/s12095-014-0116-3. Google Scholar [12] K. C. Gupta, S. K. Pandey, I. G. Ray and S. Samanta, Cryptographically significant MDS matrices over finite fields: A brief survey and some generalized results, Adv. Math. Commun., 13 (2019), 779-843. doi: 10.3934/amc.2019045. Google Scholar [13] H. Hou and S. Y. Han, A new construction and an efficient decoding method for Rabin-like codes, IEEE Transactions on Communications, 66 (2018), 521-533. doi: 10.1109/TCOMM.2017.2766140. Google Scholar [14] P. Junod and S. Vaudenay, Perfect diffusion primitives for block ciphers building efficient MDS matrices, in Selected Areas in Cryptography. SAC 2004, Vol. 3357, Springer, Berlin, 2005, 84-99. doi: 10.1007/978-3-540-30564-4_6. Google Scholar [15] K. Khoo, T. Peyrin, A. Y. Poschmann and H. Yap, FOAM: Searching for hardware-optimal SPN structures and components with a fair comparison, in Cryptographic Hardware and Embedded Systems. CHES 2014, Vol. 8731, Springer, Berlin, Heidelberg, 2014,433-450. doi: 10.1007/978-3-662-44709-3_24. Google Scholar [16] L. Kölsch, XOR-counts and lightweight multiplication with fixed elements in binary finite fields, in Advances in Cryptology. EUROCRYPT 2019, Vol. 11476, Springer, Cham, 2019,285-312. doi: 10.1007/978-3-030-17653-2_10. Google Scholar [17] H. Kranz, G. Leander, K. Stoffelen and F. Wiemer, Shorter linear straight-line programs for MDS matrices, IACR Transactions on Symmetric Cryptology, 2017 (2017), 188-211. Google Scholar [18] J. Lacan and J. Fimes, Systematic MDS erasure codes based on Vandermonde matrices, IEEE Communications Letters, 8 (2004), 570-572. doi: 10.1109/LCOMM.2004.833807. Google Scholar [19] S. Li, S. Sun, C. Li, Z. Wei and L. Hu, Constructing low-latency involutory MDS matrices with lightweight circuits, IACR Transactions on Symmetric Cryptology, 2019 (2019), 84-117. Google Scholar [20] M. Liu and S. M. Sim, Lightweight MDS generalized circulant matrices, in Fast Software Encryption. FSE 2016, Vol. 9783, Springer, Berlin, Heidelberg, 2016,101-120. doi: 10.1007/978-3-662-52993-5_6. Google Scholar [21] I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd edition, The Clarendon Press, Oxford University Press, New York, 1995. Google Scholar [22] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error Correcting Codes II, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Google Scholar [23] C. Paar, Optimized arithmetic for Reed-Solomon encoders, in Proceedings of IEEE International Symposium on Information Theory, Ulm, Germany, (1997), 250-250. doi: 10.1109/ISIT.1997.613165. Google Scholar [24] R. M. Roth and G. Seroussi, On generator matrices of MDS codes, IEEE Trans. Inform. Theory, 31 (1985), 826-830. doi: 10.1109/TIT.1985.1057113. Google Scholar [25] R. M. Roth and A. Lempel, On MDS codes via Cauchy matrices, IEEE Trans. Inform. Theory, 35 (1989), 1314-1319. doi: 10.1109/18.45291. Google Scholar [26] M. Sajadieh, M. Dakhilalian, H. Mala and B. Omoomi, On construction of involutory MDS matrices from Vandermonde Matrices in GF($2^q$), Des. Codes Cryptogr., 64 (2012), 287-308. doi: 10.1007/s10623-011-9578-x. Google Scholar [27] C. Schindelhauer and C. Ortolf, Maximum distance separable codes based on circulant Cauchy matrices, in Structural Information and Communication Complexity. SIROCCO 2013, Vol. 8179, Springer, Cham, 2013,334-345. doi: 10.1007/978-3-319-03578-9_28. Google Scholar [28] C. E. Shannon, Communication theory of secrecy systems, Bell System Tech. J., 28 (1949), 656-715. doi: 10.1002/j.1538-7305.1949.tb00928.x. Google Scholar [29] T. Shirai and K. Shibutani, Improving immunity of feistel ciphers against differential cryptanalysis by using multiple MDS matrices, in Fast Software Encryption. FSE 2004, Vol. 3017, Springer, Berlin, Heidelberg, 2004,260-278. doi: 10.1007/978-3-540-25937-4_17. Google Scholar [30] S. M. Sim, K. Khoo, F. Oggier and T. Peyrin, Lightweight MDS involution matrices, in Fast Software Encryption. FSE 2015, Vol. 9054, Springer, Berlin, Heidelberg, 2015,471-493. doi: 10.1007/978-3-662-48116-5_23. Google Scholar [31] J. R. Stembridge, A concise proof of the Littlewood-Richardson rule, Electron. J. Combin., 9 (2002), 1-4. doi: 10.37236/1666. Google Scholar [32] S. Wu, M. Wang and W. Wu, Recursive diffusion layers for (lightweight) block ciphers and hash functions, in Selected Areas in Cryptography. SAC 2012, Vol. 7707, Springer, Heidelberg, 2012,355-371. doi: 10.1007/978-3-642-35999-6. Google Scholar
show all references
##### References:
[1] D. Augot and M. Finiasz, Direct construction of recursive MDS diffusion layers using shortened BCH codes, in Fast Software Encryption. FSE 2014, Vol. 8540, Springer, Berlin, Heidelberg, 2014, 3-17. doi: 10.1007/978-3-662-46706-0_1. Google Scholar [2] P. Barreto and V. Rijmen, The Khazad legacy-level block cipher, in Proceedings of the First Open NESSIE Workshop, Belgium, (2000). Google Scholar [3] P. Barreto and V. Rijmen, The Anubis block cipher, in Proceedings of the First Open NESSIE Workshop, Belgium, (2000). Google Scholar [4] C. Beierle, T. Kranz and G. Leander, Lightweight multiplication in $GF(2^n)$ with applications to MDS matrices, in Advances in Cryptology. CRYPTO 2016. Part 1, Lecture Notes in Comput. Sci., Vol. 9814, Springer, Berlin, 2016,625-653. doi: 10.1007/978-3-662-53018-4_23. Google Scholar [5] T. P. Berger, G. Paul and S. Vaudenay, eds., Construction of recursive MDS diffusion layers from gabidulin codes, in Progress in Cryptology. INDOCRYPT 2013, Vol. 8250, Springer, Cham, 2013,274-285. doi: 10.1007/978-3-319-03515-4. Google Scholar [6] J. Daemen and V. Rijmen, The Design of Rijndael: AES - The Advanced Encryption Standard, Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-04722-4. Google Scholar [7] G. Filho, P. Barreto and V. Rijmen, The Maelstrom-$0$ hash function, in Proceedings of the Sixth Brazilian Symposium on Information and Computer Systems Security, (2006). Google Scholar [8] J. Guo, T. Peyrin and A. Poschmann, The PHOTON family of lightweight hash functions, in Advances in Cryptology. CRYPTO 2011, Vol. 6841, Springer, Heidelberg, 2011,222-239. doi: 10.1007/978-3-642-22792-9. Google Scholar [9] K. C. Gupta and I. G. Ray, On constructions of MDS matrices from companion matrices for lightweight cryptography, in Security Engineering and Intelligence Informatics. CD-ARES 2013, Vol. 8128, Springer, Berlin, Heidelberg, 2013, 29-43. doi: 10.1007/978-3-642-40588-4_3. Google Scholar [10] K. C. Gupta and I. G. Ray, On constructions of involutory MDS matrices, in Progress in Cryptology. AFRICACRYPT 2013, Vol. 7918, Springer, Heidelberg, 2013, 43-60. doi: 10.1007/978-3-642-38553-7_3. Google Scholar [11] K. C. Gupta and I. G. Ray, Cryptographically significant MDS matrices based on circulant and circulant-like matrices for lightweight applications, Cryptogr. Commun., 7 (2015), 257-287. doi: 10.1007/s12095-014-0116-3. Google Scholar [12] K. C. Gupta, S. K. Pandey, I. G. Ray and S. Samanta, Cryptographically significant MDS matrices over finite fields: A brief survey and some generalized results, Adv. Math. Commun., 13 (2019), 779-843. doi: 10.3934/amc.2019045. Google Scholar [13] H. Hou and S. Y. Han, A new construction and an efficient decoding method for Rabin-like codes, IEEE Transactions on Communications, 66 (2018), 521-533. doi: 10.1109/TCOMM.2017.2766140. Google Scholar [14] P. Junod and S. Vaudenay, Perfect diffusion primitives for block ciphers building efficient MDS matrices, in Selected Areas in Cryptography. SAC 2004, Vol. 3357, Springer, Berlin, 2005, 84-99. doi: 10.1007/978-3-540-30564-4_6. Google Scholar [15] K. Khoo, T. Peyrin, A. Y. Poschmann and H. Yap, FOAM: Searching for hardware-optimal SPN structures and components with a fair comparison, in Cryptographic Hardware and Embedded Systems. CHES 2014, Vol. 8731, Springer, Berlin, Heidelberg, 2014,433-450. doi: 10.1007/978-3-662-44709-3_24. Google Scholar [16] L. Kölsch, XOR-counts and lightweight multiplication with fixed elements in binary finite fields, in Advances in Cryptology. EUROCRYPT 2019, Vol. 11476, Springer, Cham, 2019,285-312. doi: 10.1007/978-3-030-17653-2_10. Google Scholar [17] H. Kranz, G. Leander, K. Stoffelen and F. Wiemer, Shorter linear straight-line programs for MDS matrices, IACR Transactions on Symmetric Cryptology, 2017 (2017), 188-211. Google Scholar [18] J. Lacan and J. Fimes, Systematic MDS erasure codes based on Vandermonde matrices, IEEE Communications Letters, 8 (2004), 570-572. doi: 10.1109/LCOMM.2004.833807. Google Scholar [19] S. Li, S. Sun, C. Li, Z. Wei and L. Hu, Constructing low-latency involutory MDS matrices with lightweight circuits, IACR Transactions on Symmetric Cryptology, 2019 (2019), 84-117. Google Scholar [20] M. Liu and S. M. Sim, Lightweight MDS generalized circulant matrices, in Fast Software Encryption. FSE 2016, Vol. 9783, Springer, Berlin, Heidelberg, 2016,101-120. doi: 10.1007/978-3-662-52993-5_6. Google Scholar [21] I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd edition, The Clarendon Press, Oxford University Press, New York, 1995. Google Scholar [22] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error Correcting Codes II, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Google Scholar [23] C. Paar, Optimized arithmetic for Reed-Solomon encoders, in Proceedings of IEEE International Symposium on Information Theory, Ulm, Germany, (1997), 250-250. doi: 10.1109/ISIT.1997.613165. Google Scholar [24] R. M. Roth and G. Seroussi, On generator matrices of MDS codes, IEEE Trans. Inform. Theory, 31 (1985), 826-830. doi: 10.1109/TIT.1985.1057113. Google Scholar [25] R. M. Roth and A. Lempel, On MDS codes via Cauchy matrices, IEEE Trans. Inform. Theory, 35 (1989), 1314-1319. doi: 10.1109/18.45291. Google Scholar [26] M. Sajadieh, M. Dakhilalian, H. Mala and B. Omoomi, On construction of involutory MDS matrices from Vandermonde Matrices in GF($2^q$), Des. Codes Cryptogr., 64 (2012), 287-308. doi: 10.1007/s10623-011-9578-x. Google Scholar [27] C. Schindelhauer and C. Ortolf, Maximum distance separable codes based on circulant Cauchy matrices, in Structural Information and Communication Complexity. SIROCCO 2013, Vol. 8179, Springer, Cham, 2013,334-345. doi: 10.1007/978-3-319-03578-9_28. Google Scholar [28] C. E. Shannon, Communication theory of secrecy systems, Bell System Tech. J., 28 (1949), 656-715. doi: 10.1002/j.1538-7305.1949.tb00928.x. Google Scholar [29] T. Shirai and K. Shibutani, Improving immunity of feistel ciphers against differential cryptanalysis by using multiple MDS matrices, in Fast Software Encryption. FSE 2004, Vol. 3017, Springer, Berlin, Heidelberg, 2004,260-278. doi: 10.1007/978-3-540-25937-4_17. Google Scholar [30] S. M. Sim, K. Khoo, F. Oggier and T. Peyrin, Lightweight MDS involution matrices, in Fast Software Encryption. FSE 2015, Vol. 9054, Springer, Berlin, Heidelberg, 2015,471-493. doi: 10.1007/978-3-662-48116-5_23. Google Scholar [31] J. R. Stembridge, A concise proof of the Littlewood-Richardson rule, Electron. J. Combin., 9 (2002), 1-4. doi: 10.37236/1666. Google Scholar [32] S. Wu, M. Wang and W. Wu, Recursive diffusion layers for (lightweight) block ciphers and hash functions, in Selected Areas in Cryptography. SAC 2012, Vol. 7707, Springer, Heidelberg, 2012,355-371. doi: 10.1007/978-3-642-35999-6. Google Scholar
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2019 Impact Factor: 0.734
## Tools
Article outline
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2020-12-01 21:57:41
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http://math.stackexchange.com/questions/408290/solve-the-following-linear-first-order-equation
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# Solve the following linear first-order equation
Question:
Solve the following linear first-order equation.
$(1+e^x)y '+e^xy=0$
I resolved:
$a_0(x)\acute{y}+a_1(x)y=g(x) => \acute{y}+p(x)y=Q(x)$
$\acute{y}+\frac{{e}^{x}}{1+{e}^{x}}y=0 , Q(x)=0, P(x)=\frac{{e}^{x}}{1+{e}^{x}}$
Integral factor in building:
$\mu (x)=exp\int P(x)d\acute{x}=exp\int\frac{{e}^{x}}{1+{e}^{x}} dx=exp\int\frac{e^x}{u}*\frac{du}{e^x}$
$1+e^x=u \rightarrow e^xdx=du \rightarrow dx=\frac{dx}{e^x} \rightarrow exp\int \frac{du}{u}=exp(ln\left|u\right| )=u-1+e^x$
$(1+e^x)\acute{y}+e^xy=0 \Rightarrow d((1+e^x)y)=0 \rightarrow \int d((1+e^x)y)=0 \rightarrow 1+e^xy=0 \rightarrow y=0$
Is my solved correctly?
Thanks for any help :)
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I wonder where is the $y'$ in original equation...... – ABC Jun 1 '13 at 8:06
Thank you for your attention :) @user007 – Software Jun 1 '13 at 8:14
Glad to see you around, @Software! +1 (And please know, I do not "hate" you; not in the least! We all make mistakes!) ;-) – amWhy Jul 22 '13 at 17:36
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2015-05-22 22:34:12
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https://www.lesswrong.com/posts/MdvwkgKnbxNdbNRao/in-sia-reference-classes-almost-don-t-matter
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# 17
This is another write-up of a fact that is generally known, but that I haven't seen proven explicitly: the fact that SIA does not depend upon the reference class.
Specifically:
• Assume there are a finite number of possible universes . Let be a reference class of finitely many agents in those universes, and assume you are in . Let be the reference class of agents subjectively indistinguishable from you. Then SIA using is independent of as long as .
Proof:
Let be a set of universes for some indexing set , and a probability distribution over them. For a universe , let be the number of agents in the reference class in .
Then if is the probability distribution from SIA using :
• .
We now wish to update on our own subjective experience . Since there are agents in our reference class, and have subjectively indistinguishable experiences, this updates the probabilities by weights , which is just . After normalising, this is:
Thus this expression is independent of .
Given some measure theory (and measure theoretic restrictions on to make sure expressions like converge), the result extends to infinite classes of universes, with in the proof instead of .
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2019-08-25 07:05:17
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https://forum.poshenloh.com/topic/259/why-isn-t-the-area-of-a-triangle-just-side-side-2/1
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# Why isn't the area of a triangle just side × side ÷ 2?
• Someone little today gave me several answers today to the question, "What's the area of this triangle?"
At first I was a bit shocked, appalled and dismayed at this answer... However, after thinking about it a while, I guess it's a very natural thing for a little person to say!
A triangle looks like it should be $$\color{red}\text{easy}$$ to calculate the area of. It's an easy shape, in the same way that a rectangle or a square is "easy":
But if I could just multiply two of the sides of a triangle to get its area, then what if I draw the triangles in a weird way? What if I draw a very short and skinny triangle, like this?
Does it really seem like the area of a triangle is related to the product of its sides? I can draw a triangle with the same two sides, but skinnier and skinnier until its area is almost $$0.$$
My friend sort of "remembered" the formula for the area of a triangle, except she remembered the formula for a right triangle's area. She understood that a right triangle is half of the bounding rectangle that surrounds it. But she didn't understand exactly why the formula was true, so had trouble relating it to a non-right triangle.
$$\textcolor{red}{\text{Area of a right triangle}} = \frac{1}{2} \text{ } \text{ base } \times \text{ height}$$
Now, why is the area of the second triangle equal to the first triangle? She cuts the first right triangle and says it's because the two mini-triangles are the same size.
Is that really true?
No, it's not! They look the same, but their lengths are actually different. The longest side of the left triangle is a diagonal, which is longer than the longest side of the right triangle, which is just the long side of the rectangle. Also, the altitudes (heights) of the triangle are different. The altitude on the left, colored yellow, is shorter than the height of the rectangle, so it is shorter than the altitude of the right triangle.
There's a different reason why both triangles inscribed in the box have the same area. It's because you can split the box into two smaller boxes, each of which has half taken up by a mini-triangle.
Thus we say the area of a triangle is equal to $$\color{red}\frac{1}{2} \times \text{ } \text{ base } \times \text{ height},$$ where the height is the height of the triangle's bounding rectangle.
We can interpret any of the three sides of the triangle as the base; just remember to draw a bounding rectangle and find the height of this rectangle. This altitude is always perpendicular ( $$\perp$$ ) to the base.
Since each of the three sides can be a different base, there are three formulas for the area of a triangle.
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2021-04-19 09:03:10
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http://fsug.bjutik.pl/rc-resonant-frequency-calculator.html
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Series Resonant Frequency and Parallel Resonant Frequency Thus, the impedance is approximately equal to the resistance R1 during this condition. V(t) = V B (1 – e-t/RC) I(t) =I o (1 – e-t/RC) Where, V B is the battery voltage and I o is the output current of the circuit. It is used to calculate the inverse of a tangent. Tool to calculate the Shannon index. This circuit only allows a small band of frequencies near its resonant frequency to pass through, hence. Describe how resonance and natural frequency can damage buildings and bridges and how technology can be used to avoid this. Resistors and Diodes PUBLIC. The metamaterial unit cell is constructed by cutting a 'U' shaped thin slot with the width denoted by w s out of the host beam to form a locally resonant cantilever beam located in the center of the unit ce. Calculate…. Compare this frequency with that predicted by: $$\omega_{max} = \frac{1}{\sqrt{LC}}$$ On the same graph, plot the theoretical curve of V R /V S vs. This RC circuit calculator will calculate the maximum current I max at the beginning of the capacitor charging, the maximum energy E max and maximum charge Q max in the capacitor when it is fully charged, for the given voltage across it as well as the time constant τ in the RC circuit. This console can be equipped on any ship and can be put into any console slot. In this paper, only the fundamental resonance is considered regardless the resonance. The time constant is defined as the time it will take to charge to 63. This LED calculator will help you design your LED array and choose the best current limiting resistors values. At resonant frequency, the capacitive reactance equals the inductive reactance and only the resistive internal reactance appears. This is the final calculation to determine the relative frequency of each item. Damped and forced oscillations r. Frequency Response and Bode Plots 1. The RC low pass filter is really just a resistor divider circuit where the lower resistor has been replaced with a capacitor. This is a formula to tell us the resonant frequency of a tank circuit, given the values of inductance (L) in Henrys and capacitance (C) in Farads. 24 Design a series RLCcircuit with B= 20 rad/s and ω0. Easy to use and 100% Free! Easy to use and 100% Free! We also have several other calculators. Bi-weekly Payment Calculator. calculate the Q-factor. ) is the upper (lower) 3 dB frequency of the resonance curve. frequency, up to the resonant frequency, further increase of frequency causes the decrease of current. The microstrip patch antenna calculator determines the length (L) and width (W) of a rectangular microstrip patch antenna for a given resonant frequency or vice versa. 707 times of maximum voltage at cut-off frequencies and as illustrated in the response curve shown in Fig. Choose from over 50 on-road and off-road tracks to race on. At resonance, the reactances cancel out leaving just a peak voltage, Vpk, across the loss resistance, R. 0 H, C = 80 μF, R = 40 Ω. Cutoff Frequency: Hz KHz MHz GHz. DC RC Timing Circuit. 5] Solving for this, using the leakage inductance found above, one gets a resonant capacitance of 0. Make sure to use the switching frequency, fs, in this calculation, not the ringing frequency. • A delayed (by td) sinusoidal waveform is s(t) = A∙cos(2πf(t – td)). It is used to fine the center frequency for a given band. Notethatlogiccircuitscan bemodeledasa resistor between and Gnd for the medium frequency range [8]. 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All buildings have a natural period, or resonance, which is the number of seconds it takes for the building to naturally vibrate back and forth. Thomson formula. LC Resonant Frequency Calculator. Construct the parallel resonant circuit shown in Figure 3. 5 pF source impedance of the AD9467, a parallel inductor of 181 nH is necessary to null the capacitive susceptance; leaving only the high impedance resistive portion of the RC parallel equivalent. 622 Radio Frequency Measurements by Bridge and Resonance Methods. The FrSky ACCESS protocols are the most advanced to date, it offers more features than traditional protocols. This link is listed in our web site directory since Tuesday Sep 30 2008, and till today "Antenna Trap Calculator" has been followed for a total of 21215 times. Theory: f = 10 Hz E= 100 V RMS L = 2 H C = 0. For such models, bandwidth uses the first frequency point to approximate the DC gain. This is probably the first time you get yourself a plug-in which does not make, or change any sound! It is in fact a delay time calculator you can use directly into your favourite VST host. Calculate the resonance frequency, f = 1/(2π√(LC)). On the Voltage and Current Waveform scope display, you. Using the same circuit parameters, the illustration at left shows the power dissipated in the circuit as a function of frequency. Where f is frequency; L is inductance in Henries; C is capacitance in farads; What is resonant frequency? Resonant Frequency is a phenomena in physics in which amplification occurs when a force is applied periodically at the same frequency as the harmonic proportion of the natural frequency of a material. Electrical Calculator. Use L = 100mH , R = 90 Ω and C = 0. An RLC circuit consists of 3 components, a resistance, impedance, and a capacitance. The are also found in oscillator circuits. It also calculates series and parallel damping factor. φ = 0° if 1/2πfC = 2πfL and R = 0. The Artctangent calculator is also referred to as the inverse tan calculator. In particular we will model an object connected to a spring and moving up and down. Just paste your text in the form below, press Calculate Word Frequency button, and you get word statistics. rlc RLC @ DC PUBLIC. Enter 2 values and the calculator will calculate the 3rd value. Rife machines,Healing With frequencies, What is illness?,What is the mechanism for healing?Dr. Rs=1k ohm L= 1mH Rw= 4 ohms C= 22nF Rl= 39k ohms fr = 1 2 pi √ (1 mH)(22 nF) = 33950. This rise in impedance will cause the frequency response to deviate from the projected response. Compare with the theoretical expressions for the resonant frequency and Q value:. 68 For Prob. 2 ohms 4 ohms. Parasitic inductance = L = 1 / [(2 x 3. See full list on goodcalculators. It is capable of computing sums over finite, infinite and parameterized sequences. Resonant Circuits: Analysis of simple series RLC and parallel RLC circuits under resonances. By inspection, this corresponds to the angular frequency ω 0 = 2 π f 0 ω 0 = 2 π f 0 at which the impedance Z in Equation 15. If so, an oscillator requires more than a resistor and a capacitor. Notethatlogiccircuitscan bemodeledasa resistor between and Gnd for the medium frequency range [8]. This online calculator generates Huffman coding based on a set of symbols and their probabilities. What value does the variable resistor need to have for the maximum voltage across the 1000-Ω resistor to be 1/3 of the value of the source? Exploration authored by Anne J. RESONANT FREQUENCY Frequencies up to 300 Hz are increased automatically in “steady-state” mode by steps (RC discrete) or continuously (RC chirp), or in “free-decay” mode by only an initial frequency. Voltage across resonant components. It was originally created by Asher Glick. Use a 100 Ω resistance. Resonant voltage spike frequency with 330pF shunt capacitor. Parasitic inductance = L = 1 / [(2 x 3. The resonant frequency chosen for the calculation was 200 MHz. A wide variety of frequency converter calculator options are There are 33 suppliers who sells frequency converter calculator on Alibaba. Explore the top 5000 words in English. Question (5) (20 Marks) 1- Design a two stages RC coupled BJT Audio Amplifier to provide a gain of 60 dB to a typical dynamic microphone signal with frequency ranges from 300 Hz to 3. The series and parallel resonant frequencies are very close, within 1% of each other. At some frequency above the series-resonant point, the crystal unit will act as a parallel-tuned circuit. 2 Tuning of analog radio set2. energy, max. Resonance is defined as the tendency of a system to vibrate with an increase in amplitude at the excitation of frequencies. 4 Series Resonance Quality factor, CR 1 R L in one period at resonance Energy dissipated by the circuit Peak energy stored in the circuit Q o o ω ω = = = • The quality factor is the ratio of its resonant frequency to its bandwidth. Calculate the "Q" of a series resonant circuit and demonstrate the effect it has on the response curve. 66 V V C = 72. Q is related to sharpness of the resonance peak. Also indicate the resonant frequency, as well as the gain where 𝜔=𝜔. Resonance is identified with engineering situations which involve energy-storing elements subjected to a forcing function of varying frequency. Resonant voltage spike frequency with 330pF shunt capacitor. Resonant Circuits: Analysis of simple series RLC and parallel RLC circuits under resonances. ohmmeter, the RLC-bridge, and two arrangements involving the voltmeter and the ammeter are presented for the measurement of ohmic resistance. All calculations and tools from CCTV Calculator are available as native mobile applications for Android smartphones. On the Voltage and Current Waveform scope display, you. The resistor R can be the parasitic resistor. By using equations (38) and (53), calculate the capacitive reactance and inductive reactance at the experimental resonant frequency found in Table 2. Just like an RC circuit, oscillations are produced. LC resonant circuit Figure 3. Because the branch with the resistor and the inductor is parallel to the branch with the capacitor, we obtain the total admittance Y as a sum of the particular admittances:. Calculator is based on 3GPP TS 36. Electromagnetic Waves: Experimental. 2020 Leave a Comment on Radio Frequency Measurements by Bridge and Resonance Methods. Then, the resonance frequency and the im- pedance at resonance frequency and lower resonance frequency will be applied to calculate R, L, and Co. Band Pass Filter Resonant Frequency. Substitute into the ODE, we got an algebraic (characteristic) equation of s determined by the circuit parameters: ( ) v t Ae st where the expansion constants A 1,. Calculate and measure the resonant frequency of a parallel RLC circuit. The microstrip patch antenna calculator determines the length (L) and width (W) of a rectangular microstrip patch antenna for a given resonant frequency or vice versa. The basic calculator you see below has just been updated to make it use fewer resources, and have better readability on large. Formula T=RC is a formula for time constant not frequency. (6pt) Given your knowledge of corner and resonant frequencies, will vout have greater, equal, or less amplitude then vin? Explain your answer. 9 → Calculate the impedance, Z, for circuit 3 at the following five frequencies: 10 Hz, 100 Hz, 1,000 Hz, 10,000 Hz and 100,000 Hz. 1 uF capacitor and calculate all the value of the required components as explained in the below steps. For example, a variable capacitor that could be varied over a 9:1 capacitance range will give an RC oscillator a 9:1 frequency range, but in an LC oscillator it will give only a 3:1 range. Frequencies bar charts. Construct the parallel resonant circuit shown in Figure 3. The output voltage will reduce to 0. Definition via resonance bandwidth: the Q factor is the ratio of the resonance frequency ν 0 and the full width at half-maximum (FWHM) bandwidth δ ν of the resonance: Both definitions are equivalent only in the limit of weakly damped oscillations, i. Show how to calculate the resonance frequency for a series RLC circuit. This is an online javascript scientific calculator. current does not change with frequency. The resonant frequency calculator did the job! We quickly found out what the resonant frequency is: 11. Главная » Рейтинг сайтов » Rc filter cutoff frequency calculator. What value does the variable resistor need to have for the maximum voltage across the 1000-Ω resistor to be 1/3 of the value of the source? Exploration authored by Anne J. 4(b) from which we calculate the parasitic capacitance (C p) as 0. Ive measured the problem frequency range to be between 90-100hz, peaking roughly at 96. Extended depth of focus extended range of vision. If the period of ground motion matches the natural resonance of a building, it will undergo the largest oscillations possible and suffer the. What is tuning. The time constant is defined as the time it will take to charge to 63. Convert frequency units. f 27/7en PC S, Using the calculated resonant frequency, calculate the quality factor of the network. 58 Fax +39/051/98. In the experiment will plot these quantities, and you will find a resonance, as shown in Figure 11-4. The pole frequency at the PROG pin should be kept above 100kHz. What is resonance? The sharpness of resonance can be understood better by understanding resonance. Resonance in electrical circuit occurs when two elements of opposite nature cancel out each others effect in the circuit. Find resonance frequency using RLC Circuit Frequency Calculator. The technique works by creating a binary tree of nodes. (These curves are actually computed, but they virtually overlay measured data). Resonant frequency calculator on MainKeys. At frequency f 1, the power is. (H)Use the ‘bode’ command in Matlab to create a Bode plot for the accelerometer FRF given above. If the bandwidth is 4 kHz, the lower frequency is. It is used to calculate the inverse of a tangent. The bandwidth, or BW, is defined as the frequency difference between f 2 and f 1. Hence, according to the frequency determined components, there are three basic types of oscillators such as RC oscillator, LC oscillator and crystal oscillator. However, C-type inverter may cause resonance between inverter output impedances and line impedances in high-order harmonic frequency range [36-40]. (Answer: Z = 3000 + j 141) 4. Spur Power Calculation. Consider the 0. A Bode plot is a graph of the magnitude (in dB) or phase of the transfer function versus frequency. This circuit show the steady-state of RL, RC and RLC at power on and off. The LC resonance frequency calculator is a calculator that computes the resonant frequency that is created from a single inductor and a single capacitor combined together. Calculate the resonance frequency of the circuit. A vector network analyser is of course the luxury version, but you may as well use a generator and a scope, a spectrum analyser with a generator or any linear combination thereof. , This condition is obtained when: ⇒⇒⇒ C L ω ω 1 = LC 1 ω = • Note that this resonant frequency is identical to the natural frequency of the LC circuit by itself! • At this frequency. • A delayed (by td) sinusoidal waveform is s(t) = A∙cos(2πf(t – td)). Course Offered Spring. Three band gaps and two neighboring partial band gaps are found from 0 to 4000 Hz. All you had was a sustained, strong wind. The detuning, sometimes called the resonance width, is the frequency shift $\Delta\omega=\omega-\omega_0$ that reduces the power in the resistor by a factor of two from the power at resonance. Resources listed under Antenna Calculators category belongs to Antennas main collection, and get reviewed and rated by amateur radio operators. In either case, the resonant frequency is given by: ω 0 = 1/√ LC. current and max. The basic calculator you see below has just been updated to make it use fewer resources, and have better readability on large. Find the corner frequency for RC and RL circuits and the resonant frequency for RLC circuits. We measured R= 1050 f= 410Hz f 0 = 1410Hz: (10) Using Eq. 94KHz≈500Khz. This series connection is excited by an AC source. A Fast Full-Wave Solution that Eliminates the Low-Frequency Breakdown Problem in a Reduced System of Order One. When the resistor's voltage equals the capacitor's voltage, crossover frequency is defined to be $1/(RC)$. To create symmetry of the sine shape, you want the rise to be the same rate as the drop. Compare with the theoretical expressions for the resonant frequency and Q value:. • The cutoff frequency of the high pass section becomes the lower frequency limit in the passband f 1. The Specification is available in the list of links on the left, along with a User Guide providing additional scoring guidance, an Examples document of scored vulnerabilities, and notes on using this calculator (including its design and an XML representation for CVSS v3. Please give a numerical value. (a) Calculate the inductive reactance of a 3. RC Time Constant Calculator If a voltage is applied to a capacitor of Value C through a resistance of value R, the voltage across the capacitor rises slowly. Convert frequency to wavelength and vice versa. the resonance frequency B å L 5 6 ¥ Å Á∙ Á. In some cases of spiral wound capacitors, there is some self inductance, but having the inductance at a high enough value to self resonate with the capacitance is unlikely. Use Figure 7 to calculate r L for the resonance case. HOW TO: Build a Surface Coil. The DRAM Calculator for Ryzen 1. Now you can calculate Q based on fcenter/(-3dB band width). 1592 C = ----- Re Qes fs 0. Calculator solution will show work for real and complex roots. To use this calculator, you simply change the upper, lower frequencies as well as the order. HOW TO: Make a Resonant Trap. We want to have a total impedance of 6. The increase in the intensity ratio prior to the tree resonance frequency is also typical of metamaterial behaviour (Colombi et al. Rnom is the minimum resistance to be achieved near self-resonance (fr). The spacing of these frequencies are in increments of approximately 10% of fres. Measure the resonant frequency as a function of capacitance for C=0. A series RC circuit is driven by a sinusoidal potential source so that the phase angle between current and voltage is 63. The L,C and R values are noted to calculate the resonant frequency f 0 and Q-factor, using the above formulae. The are also found in oscillator circuits. Resonant Frequency will be a Puyo Puyo like game for Win32 with a futuristic techno theme. Natural frequency often refers to the frequency at which a structure “wants” to oscillate after an impact or displacement. Assume that C=100nF. Intensification and prolongation of sound, especially of a musical tone, produced by sympathetic vibration. In some cases of spiral wound capacitors, there is some self inductance, but having the inductance at a high enough value to self resonate with the capacitance is unlikely. Phasor diag. Therefore, this calculator also suggests a value for W. Use equation 15 to calculate the resonant frequency f for the RLC circuit. The resonant frequency for an RLC circuit is the same as a circuit in which there is no damping, hence undamped resonance frequency. An online tool LC filter synthesis. 0 Calculator. Answer: a Explanation: The expression of bandwidth for parallel resonant circuit is BW = 1/RC. tutorialspoint. This can be calculated with the equation below. Let's say we wish to determine the resonant frequency of an LC circuit that has an inductor of 3 mH, and a capacitor of 3 µF. Published compliance figures are measured at frequencies such as 0 Hz, 10Hz or 100Hz. 2% of the difference between the initial and final value. The 100% free and reliable online calculators that help you to solve any calculation-related problems and provides you with the precise measurements. At this frequency, the capacitive reactance X c & resistor’s resistance R become equal. Explore the top 5000 words in English. RC LC s s Assume the solution is , where A, s are unknown constants to be solved. Kirchoff's Loop Rule for a RLC Circuit The voltage, VL across an inductor, L is given by VL = L (1) d dt [email protected] where i[t] is the current which depends upon time, t. Capacitive reactance (symbol X C) is a measure of a capacitor's opposition to AC (alternating current). Example: Resonant Frequency calculator: Inputs: Inductance (Henry) = 0. This is based on the use of the This increase in contraction frequency that has not been supported by other physiological. Click or tap Calculate at the resonant frequency to see what will happen at resonance. Rs=1k ohm L= 1mH Rw= 4 ohms C= 22nF Rl= 39k ohms fr = 1 2 pi √ (1 mH)(22 nF) = 33950. +39/051/98. Passes only 1 frequency ∴strong feedback at the resonance frequency and thus, suppresses this in the output. Then, using a new resonance frequency, we calculated the reactance caused by the capacitor and the inductor. AC RC Circuit. CIRCUIT DESIGN 2. CRC8 calculator taking hex array as input, calculating checksum according to Dallas/Maxim Application Note 27 (polynomial X^8+X^5+X^4+X^0), that is as used by 1-wire protocol. An exact equivalent circuit is complex and would include parasitic elements resulting from a transition from the printed circuit board. Many other converters available for free. Code to add this calci to your website Just copy and paste the below code to your webpage where you want to display this calculator. 24 Design a series RLCcircuit with B= 20 rad/s and ω0. There is typically very little loss in the resonant circuit so this network will cause many cycles of ringing after the spike. 9 → Calculate the impedance, Z, for circuit 3 at the following five frequencies: 10 Hz, 100 Hz, 1,000 Hz, 10,000 Hz and 100,000 Hz. 0 Hz and 10. Select the correct units. How current & voltage oscillate at resonant frequency for both parallel and series inductor-capacitor combinations. Rnom is the minimum resistance to be achieved near self-resonance (fr). All calculations and tools from CCTV Calculator are available as native mobile applications for Android smartphones. Bumm (Phys2303) AC steady-state circuits, AC impedance, RC Units: In these calculations remember that farad × henry = You will also need use the context of the equation to distinguish between ω and f,. Contact us today. This calculation you can use if you have been out jogging, driving orwell, just moving around! It will calculate your average speed during that time. Frequency-response design is practical because we can easily evaluate how gain changes affect certain aspects of systems. Sketch, analyze, and label series resonant and parallel resonant circuits. Use the oscilloscope Fr if you have it, otherwise use the calculated Fr. Calculate the characteristic impedance of the resonance. The resonance frequency is 0 1. Our measured voltages come out as follows: V L = 171. We can also calculate the “Resonant” or “Centre Frequency” (ƒr) point of the band pass filter were the output gain is at its maximum or peak value. 14m or 14cm. R is also user selectable, and a checkbox is available to convert to parallel. com, mainly located in Asia. Measure the impedance of a capacitor at a known frequency and mathematically derive the capacitance value. Thus, Ipk = Vpk/R is the maximum current which passes through all elements. the frequency f in hertz (Hz) is equal to the angular frequency or angular velocity ω in radians per second (rad/s) divided by 2π. (a) What inductance is needed to produce this resonant frequency if it is connected to a 2. 21% of the final voltage value. com provides an annuity calculator and other personal finance investment calculators. In fact, the bridge itself wasn't. f = 1/2π√(LC) Derivation: Let us consider a series connection of R, L and C. List of Contents1 RLC Resonant frequency Formula1. Parallel Resistor Calculator. Also, set up expressions for the absolute value and the the phase function. Resonant frequency definition is - a frequency capable of exciting a resonance maximum in a given body or system. 1 rad/s = 1/2π Hz = 0. Where f is frequency; L is inductance in Henries; C is capacitance in farads; What is resonant frequency? Resonant Frequency is a phenomena in physics in which amplification occurs when a force is applied periodically at the same frequency as the harmonic proportion of the natural frequency of a material. Now, g(!) has a maximimum where the expression under the square root in the denominator has a minimum. de link Techspot link. Using this calculator, you can find the resonant frequency, which means that you can disregard the reactive impedance (reactance) and only pay attention to the resistive impedance (resistance). Rlc Circuit Calculator. LC resonant circuits are useful as notch filters or band pass filters. Rs is only a few kilohms. Sketch, analyze, and label series resonant and parallel resonant circuits. Just paste your text in the form below, press Calculate Word Frequency button, and you get word statistics. The resonant frequency chosen for the calculation was 200 MHz. 5), we see that VR = I R. Substitute into the ODE, we got an algebraic (characteristic) equation of s determined by the circuit parameters: ( ) v t Ae st where the expansion constants A 1,. The crystal has wto resonant frequencies as shown below: Series resonant: RLC determine the resonant frequency. The series resonance frequency is the natural resonance frequency where the energy transformation between mechanical and electrical energy is most effective. 1 on page 6. cameras and storage. (c) Calculate the magnitude of the voltage across the capacitor at the resonant frequency. That result will be in the cell with the appropriate labels on the top row and left column. Frequency is understood as the number of repeating events in a given unit of time; a measure of how frequently something occurs. QTc Calculator. MULTISTAGE TRANSISTOR AMPLIFIERS Questions and Answers pdf free download mcqs interview objective type questions for eee ece electronics students. Hydraulic System Calculator helps you design a solution around the cylinder which may involve motor, pump, orifice and pipe calculations. Resonant Controllers. In circuits, reaching a resonant frequency means that the resistors and capacitors, for. If the value of C in a series RLC circuit is increased, the resonant frequency (a) is not affected (b) increases. How does one find the resonance frequency in a circuit? Wikipedia and the like give some definitions that are not very useful in practice. If w=wo, so that the. State the advantages of having the circuit at resonance. Figures 14 and 15 clearly show the dramatic improvement of the RC-snubbed circuit response compared to the un-snubbed one. Linear Phase Increase. This is a simplified SXP staking calculator for Swipe. resonance is reached when V R is a maximum. This is an online javascript scientific calculator. f = 1/2π√(LC) Derivation: Let us consider a series connection of R, L and C. RLC Low-Pass Filter Design Tool. The “series” resonant frequency of the crystal is the resonance of the LCR branch of the equivalent circuit. AN1477 DS00001477B-page 4 2012-2018. List of Contents: 1 RLC Resonant frequency Formula. 0003 F R = 10 Ω. For this reason, the resonant converter is a good alternative because of its soft-switching characteristic. S elect whether you are using a round or slotted port for you box design. The frequency at which the reactances of the inductance and the capacitance cancel each other is the resonant frequency (or the unity power factor frequency) of this circuit. Derive the resonant frequency. It also calculates series and parallel damping factor. Calculator help you to convert the number of the 4G LTE EARFCN frequency channel to the 5G NR frequency This Page provides information about 4G LTE EARFCN Calculator. Three band gaps and two neighboring partial band gaps are found from 0 to 4000 Hz. Kirchoff's Loop Rule for a RLC Circuit The voltage, VL across an inductor, L is given by VL = L (1) d dt [email protected] where i[t] is the current which depends upon time, t. The undamped resonant frequency of the system. Calculating wave speed, frequency and wavelength is central to modern physics. • RC charging circuit : Calculate time constant by R and C • RC Filter : Calculate cutoff frequency by resistor and capacitor value and find out combinations of resistance and capacitance making a desired cutoff frequency • RL Filter : Calculate cutoff frequency by resistor and inductor value and find out combinations of resistance and. 622 Radio Frequency Measurements by Bridge and Resonance Methods. The series or parallel resonant frequency of an inductor and a capacitor can be calculated using the following equation. The formula for resonant frequency for a series resonance circuit is given as. LC fR π = 2 1 Circuit Q is the quality of the circuit, the ratio of inductive reactance of a coil to its resistance. Following is the formula for time constant. Use software to produce Bode plots for more complex RLC filters. 3mH and C=0. a wide range of frequency. The microstrip patch antenna calculator determines the length (L) and width (W) of a rectangular microstrip patch antenna for a given resonant frequency or vice versa. two crucial points of the derivation: € power×(time for cycle)=I RMS 2 R× 2π ω R =1 2 I max 2R× 2π ω R Q= 2π LI Max 2 2 ω 2π ω R RI Max 2 2. Calculate the resonance frequency of the circuit. This document is property of Kaz Technologies. 00 square inches, tuning frequency 34 Hz. It is an electrical circuit used for generating signals or picking out the signals at a particular frequency. 0 kHz AC voltages are applied. Same is true for an inductor and resistor. Also, set up expressions for the absolute value and the the phase function. Examples of ferrite beads which have been tested and work well with the TPA3130D2 can be seen in the TPA3130D2EVM user guide SLOU341. Frequency Response of RC Circuits Peter Mathys ECEN 1400 RC Circuit 1 Vs is source voltage (sine, 1000 Hz, amplitude 1 V). Calculate gain in dB. Determine the value of QL and find the resonant frequency, load voltage at resonance, and bandwidth for the circuit shown in the following figure. 4 White Noise. Ripple itself is a composite (non-sinusoidal) waveform consisting of harmonics of some fundamental frequency which is usually the original AC line frequency, but in the case of switched-mode power supplies, the fundamental frequency can be tens of kilohertz to megahertz. Choosing a capacitor C and frequency f is best. RC LC s s Assume the solution is , where A, s are unknown constants to be solved. Free online Reynolds number calculator to calculate the dimensionless Reynolds number of a liquid or gas based on its dynamic viscosity and density, or its kinetic viscosity. 1 which supports AMD's Ryzen CPU platform is one of the few versions that has received a global memory retest on Added the "New version?" button. Example: Resonant Frequency calculator: Inputs: Inductance (Henry) = 0. be/jacrT6mISm0 Support my You. The data is analyzed to deterime the period of oscillation, which is then used to calculate gravity. f H – is the upper -3 dB cut-off frequency. Resonant Frequency. Necessary condition for a numerical sequence convergence is that limit of common term of series is equal to zero. RC Circuit Calculator. RC Circuit: Transient Response; RL Circuit: AC Response; LC Tuned Circuit Resonant Frequency Calculator output values include: Resonant Frequency (Hertz. 1) Calculate the crossover frequency (aka the 3 dB point) for the RC lter shown in Fig. Network Theory Questions and Answers - Resonant Frequency for a Tank Circuit. The frequency at which the impedance of the circuit becomes minimum and current becomes maximum,is called resonance frequency of the circuit. Category: Science / CAD. Высокочастотная система с использованием новейшей, Universal Input Tel. Resistance: (Ohm). At the resonant frequency the total impedance is a minimum and, for this particular circuit arrangement, the transfer function is maximized. (a) What inductance is needed to produce this resonant frequency if it is connected to a 2. Resonant Frequency Formula – Series Resonance Circuit. At time=0 the carrier frequency is equal to the resonant frequency of the ring resonator. A change of resonant frequency of less than fifty percent under this condition is desirable. RC Low-pass Filter Design Tool. A typical calculation to find damping is shown in fig. Use the oscilloscope Fr if you have it, otherwise use the calculated Fr. A very common mistake is to mixup frequency and angular frequency in the lab. The phase response of a resonant circuit is also related to the Qfactor. 23 Design a series RLCcircuit that will have an impedance of 10 &at the resonant frequency of ω0 = 50 rad/s and a quality factor of 80. Use software to produce Bode plots for more complex RLC filters. Title: RLC Tank Circuits. Saved by QandA Mamma. As the power factor of the RC circuit are less than 1 and it is leading power factor, the power factor of the circuit are pf<1 (leading). To measure the resonance frequency of a series RLC circuit and compare with the theoretical value. At parallel resonance, the impedance of a crystal is. Quick Buy Buy Set of 3 £10. The angular frequency for maximum amplitude is given by ! peak =! 0 1"2/()! 0 # [] 2 1/2 (11) At the maximum, the oscillation amplitude is considerably greater than the driving amplitude. A Fast Full-Wave Solution that Eliminates the Low-Frequency Breakdown Problem in a Reduced System of Order One. Calculate parameters for series and parallel resonant. The frequency. Measure capacitance and inductance values, and calculate the resonant frequency f o = o /2 = 1 2 LC Hz for each of the capacitors. Console - Universal - Rebounding Resonant Frequencies is obtained by players of all factions from the Vulcan T'Pau Scout Ship (9 Year Anniversary Event, Cross-Faction T6). VT1 in the circuit constitutes a first-stage common-emitter amplifier, and L1 and C4 constitute an LC series resonance circuit, which is used to boost high-frequency signals. STPOWER RF DMOS Transistors. An online tool LC filter synthesis. Al Explains Resonance & Resonance Effect in a Parallel Resonant circuit, calculation of the resonant frequency. Therefore, this calculator also suggests a value for W. The narrower. Resistors and Diodes PUBLIC. 15} is a minimum, or when. Connect the circuit of Figure 5-7 with C = 0. Using the formula to find Irms, we were able to use the reactance found and calculate the Irms. To use this calculator, you simply change the upper, lower frequencies as well as the order. Series Resonant Frequency and Parallel Resonant Frequency Thus, the impedance is approximately equal to the resistance R1 during this condition. A plot of real driver impedances [38kb] shows a floor around the DC resistance of about 6 Ohms, a spike (or at least a bump) at the resonant frequency of each driver, and a rise at higher frequencies. 1 Series Resonant Frequency. Values must be numeric and separated by commas, spaces or new-line. The following formulas are used for the calculation: φ 90° if 1/2πfC < 2πfL and R = 0. Is it possible to eliminate a particular resonant frequency on a microstrip antenna? Say like I have this result, I like to eliminate 3. We apprecite your scan's of any article about eCalc you read in your prefered RC Magazine - info[at]ecalc. The base atmospheric electromagnetic resonant frequency is 7. Measure the resonant frequency as a function of capacitance for C=0. When the resistor's voltage equals the capacitor's voltage, crossover frequency is defined to be $1/(RC)$. Frequencies bar charts. So whether electronic or mechanical (or neither, as in the purely mathematical world) where the complex impedance crosses the real axis (if it does) that is called resonance, and the frequencies where it occurs are called. Twin-T circuit is also known as twin-t filter it is basically lead-lag network whose phase changes according to the frequency. The Series / Paralle resonance frequency may be measured by a lot of setups. We must choose its R value to produce a parallel combination with the driver's 32 ohms at around 6. (The rise time is defined on page 107 of Simpson. LC resonant circuits are useful as notch filters or band pass filters. 46] Find the two frequencies on either side of the resonant frequency f1 and f2 where the impedance is Re * sqrt(Rc) [in this example, that impedance will be 6. RC Transition Report. voltage of the resonance converter. 258 pF by using Eq. If so, an oscillator requires more than a resistor and a capacitor. 30,000Hz (this frequency is in the ultrasonic range. A change of resonant frequency of less than fifty percent under this condition is desirable. Resistance: (Ohm). Работа От: Vini DevHouse. 1592 (Qes Re) L = ----- fs Qes Re Rc = Re + ----- Qms Explanation of the symbols used: fs = Drivers resonance frequency Re = Drivers DC resistance Qes = Drivers electrical Q value. Figure 4: Main impedance curve for a 420kV RC-divider At resonance frequency, the impedance part XC is compensated by the impedance part XL. The basic calculator you see below has just been updated to make it use fewer resources, and have better readability on large. by moku on 28. In an RC phase shift oscillator, if R 1 =R 2 =R 3 =200KΩ and C 1 =C 2 =C 3 =100pF. Compare this frequency with that predicted by: $$\omega_{max} = \frac{1}{\sqrt{LC}}$$ On the same graph, plot the theoretical curve of V R /V S vs. In addition, it graphs the bode plot for magnitude in decibels and the phase in radians. The angular frequency for maximum amplitude is given by ! peak =! 0 1"2/()! 0 # [] 2 1/2 (11) At the maximum, the oscillation amplitude is considerably greater than the driving amplitude. t = R * C. the resonant voltage amplitude across the capacitor occurs at a lower frequency than the phase resonance! For the inductor, s. Calculate the effects of loading on RC filters. At this resonant frequency, the effect of the inductor exactly cancels out the effect of the capacitor, and the impedance is just the impedance of the resistor alone. 1 Complete schematic diagram of a colpitts oscillator R1: R2: Rc: RE: VCC VCC 6V 1 5 Cc Farad. Resistors and Diodes PUBLIC. Selectivity indicates how well a resonant circuit responds to a certain frequency and eliminates all other frequencies. The frequency at which the reactances of the inductance and the capacitance cancel each other is the resonant frequency (or the unity power factor frequency) of this circuit. On the Voltage and Current Waveform scope display, you. Set to 1 volts L is a variable inductor. Find resonance frequency using RLC Circuit Frequency Calculator. In human subjects, sensory resonances can be excited by subliminal atmospheric acoustic pulses that are tuned to the resonance frequency. A plot of real driver impedances [38kb] shows a floor around the DC resistance of about 6 Ohms, a spike (or at least a bump) at the resonant frequency of each driver, and a rise at higher frequencies. For Figure A, calculate the circuit impedance if the excitation frequency is 1800 Hz. Part 2PE (a) For the circuit of Figure 5. Forced vibration at or near an object’s natural frequency causes energy inside the structure to build. f = 1/2π√(LC) Derivation: Let us consider a series connection of R, L and C. 4(b) from which we calculate the parasitic capacitance (C p) as 0. We calculate the received electrical signal from u as in [6]. The amplitude of a driven harmonic oscillator increases as the driving frequency approaches the natural frequency. You need only show the general shape, not the phase. Or you may also build this Grid dip meter for identifying and setting the resonance frequency. Question (5) (20 Marks) 1- Design a two stages RC coupled BJT Audio Amplifier to provide a gain of 60 dB to a typical dynamic microphone signal with frequency ranges from 300 Hz to 3. This is a World Wide Web front end for a public domain program written by W4/VP9KF using PHP. Q is related to sharpness of the resonance peak. I would like to extract the ''R'' and ''C''. 1 Preliminaries The steady-state sinusoidal frequency-response of a circuit is described by the phasor transfer function ( )Hj. For a parallel -resonant circuit, the opposite is true. When a resistor is placed in series with the power source and a capacitor is placed in parallel to that same power source, as shown in the diagram circuit above, this type of circuit forms a. The value of t is the time (in seconds) at which. 5 uF capacitor is 10 mS. j Q 00 0 2 m loss W R QRC PL Fractional bandwidth 1 2 0 Q 1 if RZ in 2 R Z 0, 2 22 in j C CjC. Delayarator - delay calculator. So whether electronic or mechanical (or neither, as in the purely mathematical world) where the complex impedance crosses the real axis (if it does) that is called resonance, and the frequencies where it occurs are called. All buildings have a natural period, or resonance, which is the number of seconds it takes for the building to naturally vibrate back and forth. Bi-weekly Payment Calculator. +39/051/98. A FLEXI-BOX is available with a transmission line PCB and a selection. When operating with the cutoff frequency, 70. Everybody needs a Calculator at some point -- Full Screen, Fast Loading and FREE! Check it out! More calculators will be added soon - as well as many new great features. Forced vibration at or near an object’s natural frequency causes energy inside the structure to build. Bumm (Phys2303) AC steady-state circuits, AC impedance, RC Units: In these calculations remember that farad × henry = You will also need use the context of the equation to distinguish between ω and f,. How to Use Series Calculator. See full list on electricalacademia. appeared protection from stupid situations, when memory is not capable of working at the desired frequency, but the calculator gives suggestions. Resonance Frequency*: hertz kilohertz megahertz millihertz microhertz nanohertz. World's simplest word frequency calculator. The 3mm QFN also adds effective length to the circuit so it has to be taken into to account also if one wants to simulate the resonance peaks at the proper frequencies. The resonant frequency of this circuit is given by 1 The Quality factor of a resonant circuit (Q-factor for short) is defined as the resonant. FrSky has developed excellent new ideas that expand the role that a traditional protocols would play. Calculates the resonant frequency. The amplitude becomes very large when the difference between the natural and forced frequency of the system is sometimes called the resonant frequency. FrSky is constantly looking to improve the performance and reliability of RC transmission. Frequency Calculator eNB ID Calculator 4G Speed Calculator. Frequency of operation: In MHz. 0 billion and at least 8,000 fatalities. At this frequency, the capacitive reactance X c & resistor’s resistance R become equal. The formula for resonant frequency for a series resonance circuit is given as. The pole frequency at the PROG pin should be kept above 100kHz. The 'Difference Frequency' and 'Sum Frequency' radio buttons indicate whether the spur calculator will use the sum frequency or difference frequency to This checkbox will keep the difference between the LO and Input frequency constant as either of them is varied. Initially, all nodes are leaf nodes, which contain the symbol itself, the weight (frequency of appearance) of the symbol and optionally, a link to a. At resonance the total voltage is dropped across the total resistance = R+r L +r C since the internal resistance may be treated as being in series. maximum value), and is called the upper cut-off frequency. A Low pass filter is a filter that passes low-frequency signals but attenuates signals with frequencies higher than the cutoff frequency. Frequency Response, Bode Plots, and Resonance CHAPTER 6 Frequency Response, Bode Plots, and Resonance 1. 4 Series Resonance Quality factor, CR 1 R L in one period at resonance Energy dissipated by the circuit Peak energy stored in the circuit Q o o ω ω = = = • The quality factor is the ratio of its resonant frequency to its bandwidth. Calculate…. Determine the proper Pitch or Module values to select the correct gear for best mesh. RC Circuit Calculator. The properties of the resonant circuit can be examined using this arrangement because the impedance of a resonant circuit is frequency dependent. where f0 is the resonant frequency of an RLC series circuit. Example: Calculate the time constant, max. This CalcTown Calculator calculates the resonant frequency, surface resistance, unloaded Quality factor and half power bandwidth of an air filled circular cavity resonator. How to calculate hertz from rad/sec. You know how to add capacitors and inductors in series to find the total capacitance and inductance). Free online Reynolds number calculator to calculate the dimensionless Reynolds number of a liquid or gas based on its dynamic viscosity and density, or its kinetic viscosity. fr = √(f H x f L) Where fr is the resonant frequency or centre frequency. Calculator solution will show work for real and complex roots. Calculate the most common physical & electrical properties of our piezo products with ease!. 50 pF capacitor?. LC Resonance Calculator. The resonant frequency is the same with an ac voltage applied, but the continuous ac input makes up for any losses in the circuit so the oscillations never die out. 2 μ Fis connected to an ac generator with an rms voltage of 24 V. LC Resonant Frequency Calculator. Survival analysis. where f0 is the resonant frequency of an RLC series circuit. 8481 m/s 2 at this frequency is recorded. When finding values to use in an R-L-C impedance resonance compensator, measure while the driver is mounted in the box. Frequency Cutoff Calculator. The frequency values ω 1. Phase Modulator. • A delayed (by td) sinusoidal waveform is s(t) = A∙cos(2πf(t – td)). This application calculates the resonant frequencies for various shapes of musical instrument resonance chambers based on your input values. Solve quadratic equations using a quadratic formula calculator. The crystal has a low impedance. The frequency at which the reactances of the inductance and the capacitance cancel each other is the resonant frequency (or the unity power factor frequency) of this circuit. is a resonant frequency of 4. The series resonance frequency, fS, therefore determines the relation-ship between CS and LS. Following is the formula for time constant. (13) The above definitions of Z m, f 0 and W provide the measurable parameters from which the values of RLC in the model circuit can be uniquely identified. In a series RLC circuit that is operating below the resonant frequency, the current (a) is in phase with the source voltage (b) lags the source voltage (c) leads the source voltage 5. Prove that the expression for the damping ratio and the undamped resonant frequency for the circuit of Figure 1 is equal to, (6) 3. Free Online Currency Exchange Rates Conversion Calculator. Free function frequency calculator - find frequency of periodic functions step-by-step. Saved by QandA Mamma. Where |RC| = 0. Calculate the resonance frequency, f = 1/(2π√(LC)). 0-rc3, are Release Candidates sometimes offered temporarily for beta testing purposes. At a given frequency f, the reactance of the inductor and the capacitor will be: X L = 2πfL and X C = 1/2πfC And the total impedance of the circuit will be: Z = [(R 2) + (X L - X C) 2] 1/2 From these equations, we can understand easily that X L increases linearly with the frequency whereas the reactance X C varies inversely with frequency. 33, you have the VSWR=2 frequencies. So a much wider frequency range can be covered by a given variable capacitor in an RC oscillator. Frequency is 1 kHz. The Top-Rated Electronic & Electrical Engineering Tool for Palm OS The most complete set of hundreds of calculations including: Integrated Ohm's Law/Watt's Law, Resistor Color Code, Series/Parallel. 11, it is apparent that the voltage across the circuit, V0, is maximum at the frequency, ω0, and that the maximum value of V0 is V0max = R Ig. Frequency Calculator eNB ID Calculator 4G Speed Calculator. Free function frequency calculator - find frequency of periodic functions step-by-step. Knowing your tinnitus frequency can enable you to. The 100% free and reliable online calculators that help you to solve any calculation-related problems and provides you with the precise measurements. Since the shape of the peak in v C characterizes the resonance, it is. Find also the quality factor and bandwidth. 238 a good frequency to have as i am trying to attain a very flat dc output at under 2 amps. Instagram Engagement Calculator @yourhandle. Q21 :Obtain the resonant frequency and Q-factor of a series LCR circuit with L = 3. Use the techniques you learned for DC circuits to calculate the current in the circuit at several different points. There are only two quantities involved on our. R is thus estimated to be around 8. Absolute frequency is a simple concept to grasp: it refers to the number of times a particular value appears in a specific data set (a collection of objects or Divide each result by the total size of the set. τ in this example was 0. Twin T-oscillator is an RC oscillator consists of Twin T-network and an op-amp. Now showing circuits 31161-31164 of 31164. 3% so this is very loosely coupled to avoid pulling the resonant frequency high. 3) Design and construct an LRC series circuit with a resonant frequency between 1 KHz and 10 KHz and a Q as large as possible (preferably larger than 5). Calculate the resonant frequency, the current in the circuit and the voltages a cross the elements at resonance. As a result: The impedance of a parallel resonant circuit is extremely high. For a first order RC-filter the cut-off frequency (f c) is calculated as follows: R 1 * C 1 = 1/ω = 1/ (2 π f c) The time constant ω is equivalent to 2 π f c. However, while the use of either pure or impure components in the series RLC circuit does not affect the calculation of the resonance frequency, but in a parallel RLC circuit it does. Also an inverse calculator is available. In addition, it graphs the bode plot for magnitude in decibels and the phase in radians. Once the self-resonant frequency is exceeded, the element characteristic changes from capacitor to inductor, and |Z| starts to increase. Re = DC resistance, measured with a decent ohm meter. Definition via resonance bandwidth: the Q factor is the ratio of the resonance frequency ν 0 and the full width at half-maximum (FWHM) bandwidth δ ν of the resonance: Both definitions are equivalent only in the limit of weakly damped oscillations, i. After the PLL circuit locks onto the input frequency, the output frequency will be the same as the input frequency (with a small phase delay). spechargers. Conversely the input signal at fixed frequency will be amplified to the greatest extent when the LCR circuit is tuned (b y varying the capacitance) t o match this input frequency. com wishes everyone to BE WELL, STAY WELL, GET WELL.
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2021-01-20 09:56:12
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http://www.koreascience.or.kr/article/JAKO199703042047894.page
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# 천연 유기산처리 및 포장방법에 의한 참취의 저장효과
• Published : 1997.02.01
• 73 14
#### Abstract
The research was investigated to determine the effect of organic acids or packaging methods (PA) either alone or in combination on the quality of Aster scaber during storages. The Aster scaber was treated with organic acids and PA, and stored at different temperature $(1\;and\;5^{\circ}C)$. Total plate counts, weight loss, color change, and sensory evaluation were evaluated. Both organic acid treatments, PA, and combined treatment had little effect on the inhibition of total plate counts compared to the control (non-treatment). Organic acid treatments showed less weight reduction compared to the control and nitrogen treated package had the least weight reduction, but the combined treatments showed less weight reduction than organic acid treatments or packaging method alone. Organic acid treatments were little different from the control on color change, but nitrogen packages had the least color change, whereas combined treatments were a little reduced, but little different compared to the control or nitrogen packages. The nitrogen packages showed better effects on the sensory evaluation compared to other treatments and the results of sensory evaluation were consistent with that of weight reduction and color change, but not in total counts. All these results showed better effects in $5^{\circ}C$ rather than $1^{\circ}C$.
#### Keywords
Aster scaber;organic acids;modified atmosphere packaging;shelf-life
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2019-11-14 13:51:46
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https://www.armaholic.com/forums.php?m=posts&id=83682
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Forum Jump :
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## fasterthanlight
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#83682 Posted at 2010-06-16 23:23
I don't quite understand. Where is this description.ext file? Do I have to create it and where do I put it?
For instance, I put a description.ext file in
C:\Users\Drew\Documents\ArmA\missions\Drew%20and%20Carter.Sara
I only have this in the file;
respawn=2;
respawnDelay=10;
respawnVehicle=3;
respawnVehicleDelay=10;
I frag myself and I still don't respawn.
This post was edited by fasterthanlight (2010-06-16 23:56, ago)
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2020-07-15 18:27:41
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https://flaviocopes.com/how-to-make-first-github-pull-request/
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Many tutorials exist about this topic but they make things overly complicated by assuming one has to contribute code to a project. So there’s all the git setup before that.
What if you just need to edit a file, maybe the project README to fix a typo?
You don’t need to know how to code or how to use Git to do that. But once you start doing Pull Requests, you can do many more things and collaborate on projects with other people! And maybe this will push you to also contribute code later on.
I assume you already have a (free) GitHub account. If you don’t, then go to github.com and get one.
Let me show you the process.
I went to this page https://web.dev/prefers-color-scheme/ and I found a possible typo. This line is missing a dot at the end.
I’m not a grammar nazi, this is just for the sake of finding an example 😄
I know that site is hosted on GitHub, and that exact article is hosted here: https://github.com/GoogleChrome/web.dev/tree/master/src/site/content/en/blog/prefers-color-scheme
I open the index.md file https://github.com/GoogleChrome/web.dev/blob/master/src/site/content/en/blog/prefers-color-scheme/index.md directly on GitHub and I press the pencil icon in the file toolbar. Hovering it says “Fork this project and edit the file”.
This brings up an editor view with this information:
You’re editing a file in a project you don’t have write access to. Submitting a change to this file will write it to a new branch in your fork flaviocopes/web.dev, so you can send a pull request.
I can go and add that dot, then at the form at the bottom I explain the changes I made:
I pressed the “Propose File Change” button and a compare view showed up.
There I can review the changes I made, to make sure all is fine, and finally I can click the “Create Pull Request” button. Currently the changes have been made to your fork of the project, which was made automatically by GitHub when you clicked the pencil icon.
At the top of this view you can see that I’m about to submit a PR to the GoogleChrome/web.dev project from my form flaviocopes/web.dev, from my branch patch-2 to their master branch.
Pressing the “Create Pull Request” button shows another form where I can write a detailed description for the Pull Request.
Pull Requests can contain many different changes, so in theory you could have lots of files edited in the same PR, this is why you can add a summary.
This repository has a template for the PR text, to help the team manage it. Our PR is very simple so I remove the template and just paste the content from the commit message from before.
Notice the hint on the right? They tell me the project has a CONTRIBUTING.md file, which explains how to contribute and the guidelines. Pretty cool.
Seems we need to sign a CLA (Contributor License Agreement) to complete our PR. I already signed a Google CLA in the past so this step is clear for me, but you might need to fix that. Most projects don’t really need it.
I clicked “Create pull request” and the PR is now sent!
Now it’s up to the project maintainers to step in and accept it, you just need to wait for an email telling you that it’s been merged, or any comments other people had.
[… a couple hours passed by…]
I got an email back, the PR was rejected because that dot was actually in the correct place! (I didn’t know that).
But anyway here’s a thing I wanted to add: don’t be angry or upset if a PR you submit is not accepted. The maintainers of the project work on it for months or years and they know better than you about what’s better for it.
Plus, especially with code, views might be very very different and a PR you think is great might not be welcome.
It’s also best to ask before working on a substantial PR, to see if it’s something the project actually needs.
But this is a topic on its own.
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2020-07-04 02:06:08
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https://web2.0calc.com/questions/probability-problem_19
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+0
# Probability problem
0
61
3
Jack rolls 5 fair six-sided dice. What is the probability that at least three dice show the same number?
Feb 14, 2022
### 3+0 Answers
#1
+122390
+1
P ( 3 show the same number ) = C(5,3) (1/6)^3 ( 5/6) (4/6) = 25/ 972 = 200/7776
P( 4 show the same number) = C(5,4) (1/6)^4 ( 5/6) = 25/7776
P( 5 show the same number ) = C(5,5) ( 1/6)^5 = 1/7776
Total probability = (200 + 25 + 1) / 7776 = 226 /7776 = 113/ 3888
Feb 14, 2022
edited by CPhill Feb 14, 2022
#3
+1382
+1
CPhill, your answer is incorrect.
The probability that exactly 3 dice show the same is $${5 \choose 3} \times {1 \over 6^3} \times{5 \over6^2} \times 6$$. $$5 \choose 3$$ ways to pick the rolls that are the same, $${1 \over 6}^3$$ chance of 3 successes, $${5 \over 6}^2$$ chance of 2 failures, and 6 different trios of numbers that are the same.
The probability that exactly 4 dice show the same is $${5 \choose 4} \times {1 \over6}^4\times {5 \over6} \times 6$$. With the same logic as above.
The probability that all 5 dice show the same is $${1 \over 6}^5\times6$$
Add everything up, and you get $$\color{brown}\boxed{23\over108}$$
BuilderBoi Feb 15, 2022
#2
0
3 of a kind ==6 *[5 C 3 * 5^2]==1,500
4 of a kind ==6 *[5 C 4 * 5^1]==150
5 of a kind ==6 *[5 C 5 * 5^0]==6
Therefore, the probability of at least 3 showing the same number is: [1,500 + 150 + 6] ==1,656 / 6^5 ==23 / 108
Feb 14, 2022
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2022-05-25 03:22:06
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https://www.gamedev.net/forums/topic/691364-movable-struct-how-to-keep-reference-to-it/
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• ### Similar Content
• By KarimIO
Hey guys! Three questions about uniform buffers:
1) Is there a benefit to Vulkan and DirectX's Shader State for the Constant/Uniform Buffer? In these APIs, and NOT in OpenGL, you must set which shader is going to take each buffer. Why is this? For allowing more slots?
2) I'm building an wrapper over these graphics APIs, and was wondering how to handle passing parameters. In addition, I used my own json format to describe material formats and shader formats. In this, I can describe which shaders get what uniform buffers. I was thinking of moving to support ShaderLab (Unity's shader format) instead, as this would allow people to jump over easily enough and ease up the learning curve. But ShaderLab does not support multiple Uniform Buffers at all, as I can tell, let alone what parameters go where.
So to fix this, I was just going to send all Uniform Buffers to all shaders. Is this that big of a problem?
3) Do you have any references on how to organize material uniform buffers? I may be optimizing too early, but I've seen people say what a toll this can take.
• By abarnes
Hello All!
I am currently pursuing a degree in video game programming, so far I have completed an intro to programming course and object oriented programming course. Both were taught using C++ as the programming langauge which I know is very popular for game development, but in these classes we do not actually do any game development. I would like to start to build my skills with C++ for game development as that is a common required thing for a job and am looking for ways to do this. Any recommendations such as books to read or youtube videos to watch will be greatly appreciated!
• By Orella
I'm having problems rotating GameObjects in my engine. I'm trying to rotate in 2 ways.
I'm using MathGeoLib to calculate maths in the engine.
First way: Rotates correctly around axis but if I want to rotate back, if I don't do it following the inverse order then rotation doesn't work properly.
e.g:
Rotate X axis 50 degrees, Rotate Y axis 30 degrees -> Rotate Y axis -50 degrees, Rotate X axis -30 degrees. Works.
Rotate X axis 50 degrees, Rotate Y axis 30 degrees -> Rotate X axis -50 degrees, Rotate Y axis -30 degrees. Doesn't.
Code:
void ComponentTransform::SetRotation(float3 euler_rotation) { float3 diff = euler_rotation - editor_rotation; editor_rotation = euler_rotation; math::Quat mod = math::Quat::FromEulerXYZ(diff.x * DEGTORAD, diff.y * DEGTORAD, diff.z * DEGTORAD); quat_rotation = quat_rotation * mod; UpdateMatrix(); } Second way: Starts rotating good around axis but after rotating some times, then it stops to rotate correctly around axis, but if I rotate it back regardless of the rotation order it works, not like the first way.
Code:
void ComponentTransform::SetRotation(float3 euler_rotation) { editor_rotation = euler_rotation; quat_rotation = math::Quat::FromEulerXYZ(euler_rotation.x * DEGTORAD, euler_rotation.y * DEGTORAD, euler_rotation.z * DEGTORAD); UpdateMatrix(); }
Rest of code:
#define DEGTORAD 0.0174532925199432957f void ComponentTransform::UpdateMatrix() { if (!this->GetGameObject()->IsParent()) { //Get parent transform component ComponentTransform* parent_transform = (ComponentTransform*)this->GetGameObject()->GetParent()->GetComponent(Component::CompTransform); //Create matrix from position, rotation(quaternion) and scale transform_matrix = math::float4x4::FromTRS(position, quat_rotation, scale); //Multiply the object transform by parent transform transform_matrix = parent_transform->transform_matrix * transform_matrix; //If object have childs, call this function in childs objects for (std::list<GameObject*>::iterator it = this->GetGameObject()->childs.begin(); it != this->GetGameObject()->childs.end(); it++) { ComponentTransform* child_transform = (ComponentTransform*)(*it)->GetComponent(Component::CompTransform); child_transform->UpdateMatrix(); } } else { //Create matrix from position, rotation(quaternion) and scale transform_matrix = math::float4x4::FromTRS(position, quat_rotation, scale); //If object have childs, call this function in childs objects for (std::list<GameObject*>::iterator it = this->GetGameObject()->childs.begin(); it != this->GetGameObject()->childs.end(); it++) { ComponentTransform* child_transform = (ComponentTransform*)(*it)->GetComponent(Component::CompTransform); child_transform->UpdateMatrix(); } } } MathGeoLib: Quat MUST_USE_RESULT Quat::FromEulerXYZ(float x, float y, float z) { return (Quat::RotateX(x) * Quat::RotateY(y) * Quat::RotateZ(z)).Normalized(); } Quat MUST_USE_RESULT Quat::RotateX(float angle) { return Quat(float3(1,0,0), angle); } Quat MUST_USE_RESULT Quat::RotateY(float angle) { return Quat(float3(0,1,0), angle); } Quat MUST_USE_RESULT Quat::RotateZ(float angle) { return Quat(float3(0,0,1), angle); } Quat(const float3 &rotationAxis, float rotationAngleRadians) { SetFromAxisAngle(rotationAxis, rotationAngleRadians); } void Quat::SetFromAxisAngle(const float3 &axis, float angle) { assume1(axis.IsNormalized(), axis); assume1(MATH_NS::IsFinite(angle), angle); float sinz, cosz; SinCos(angle*0.5f, sinz, cosz); x = axis.x * sinz; y = axis.y * sinz; z = axis.z * sinz; w = cosz; } Any help?
Thanks.
• By owenjr
Hi there!
I am trying to implement a basic AI for a Turrets game in SFML and C++ and I have some problems.
This AI follows some waypoints stablished in a Bezier Courve.
In first place, this path was followed only by one enemy. For this purpose, the enemy has to calculate his distance between his actual position
to the next waypoint he has to pick.
If the distance is less than a specific value we stablish, then, we get to the next point. This will repeat until the final destination is reached. (in the submitting code, forget about the var m_go)
Okay, our problem gets when we spawn several enemies and all have to follow the same path, because it produces a bad visual effect (everyone gets upside another).
In order to solve this visual problem, we have decided to use a repulsion vector. The calculus gets like this:
As you can see, we calculate the repulsion vector with the inverse of the distance between the enemy and his nearest neighbor.
Then, we get it applying this to the "theorical" direction, by adding it, and we get a resultant, which is the direction that
our enemy has to follow to not "collide" with it's neighbors. But, our issue comes here:
The enemys get sepparated in the middle of the curve and, as we spawn more enemys, the speed of all of them increases dramatically (including the enemies that don't calculate the repuslion vector).
1 - Is it usual that this sepparation occours in the middle of the trajectory?
2 - Is it there a way to control this direction without the speed getting affected?
3 - Is it there any alternative to this theory?
I submit the code below (There is a variable in Spanish [resultante] which it means resultant in English):
if (!m_pathCompleted) { if (m_currentWP == 14 && m_cambio == true) { m_currentWP = 0; m_path = m_pathA; m_cambio = false; } if (m_neighbors.size() > 1) { for (int i = 0; i < m_neighbors.size(); i++) { if (m_enemyId != m_neighbors[i]->GetId()) { float l_nvx = m_neighbors[i]->GetSprite().getPosition().x - m_enemySprite.getPosition().x; float l_nvy = m_neighbors[i]->GetSprite().getPosition().y - m_enemySprite.getPosition().y; float distance = std::sqrt(l_nvx * l_nvx + l_nvy * l_nvy); if (distance < MINIMUM_NEIGHBOR_DISTANCE) { l_nvx *= -1; l_nvy *= -1; float l_vx = m_path[m_currentWP].x - m_enemySprite.getPosition().x; float l_vy = m_path[m_currentWP].y - m_enemySprite.getPosition().y; float l_resultanteX = l_nvx + l_vx; float l_resultanteY = l_nvy + l_vy; float l_waypointDistance = std::sqrt(l_resultanteX * l_resultanteX + l_resultanteY * l_resultanteY); if (l_waypointDistance < MINIMUM_WAYPOINT_DISTANCE) { if (m_currentWP == m_path.size() - 1) { std::cout << "\n"; std::cout << "[GAME OVER]" << std::endl; m_go = false; m_pathCompleted = true; } else { m_currentWP++; } } if (l_waypointDistance > MINIMUM_WAYPOINT_DISTANCE) { l_resultanteX = l_resultanteX / l_waypointDistance; l_resultanteY = l_resultanteY / l_waypointDistance; m_enemySprite.move(ENEMY_SPEED * l_resultanteX * dt, ENEMY_SPEED * l_resultanteY * dt); } } else { float vx = m_path[m_currentWP].x - m_enemySprite.getPosition().x; float vy = m_path[m_currentWP].y - m_enemySprite.getPosition().y; float len = std::sqrt(vx * vx + vy * vy); if (len < MINIMUM_WAYPOINT_DISTANCE) { if (m_currentWP == m_path.size() - 1) { std::cout << "\n"; std::cout << "[GAME OVER]" << std::endl; m_go = false; m_pathCompleted = true; } else { m_currentWP++; } } if (len > MINIMUM_WAYPOINT_DISTANCE) { vx = vx / len; vy = vy / len; m_enemySprite.move(ENEMY_SPEED * vx * dt, ENEMY_SPEED * vy * dt); } } } } } else { float vx = m_path[m_currentWP].x - m_enemySprite.getPosition().x; float vy = m_path[m_currentWP].y - m_enemySprite.getPosition().y; float len = std::sqrt(vx * vx + vy * vy); if (len < MINIMUM_WAYPOINT_DISTANCE) { if (m_currentWP == m_path.size() - 1) { std::cout << "\n"; std::cout << "[GAME OVER]" << std::endl; m_go = false; m_pathCompleted = true; } else { m_currentWP++; } } if (len > MINIMUM_WAYPOINT_DISTANCE) { vx = vx / len; vy = vy / len; m_enemySprite.move(ENEMY_SPEED * vx * dt, ENEMY_SPEED * vy * dt); } } }
¡¡Thank you very much in advance!!
• Overview
Welcome to the 2D UFO game guide using the Orx Portable Game Engine. My aim for this tutorial is to take you through all the steps to build a UFO game from scratch.
The aim of our game is to allow the player to control a UFO by applying physical forces to move it around. The player must collect pickups to increase their score to win.
I should openly acknowledge that this series is cheekily inspired by the 2D UFO tutorial written for Unity.
It makes an excellent comparison of the approaches between Orx and Unity. It is also a perfect way to highlight one of the major parts that makes Orx unique among other game engines, its Data Driven Configuration System.
You'll get very familiar with this system very soon. It's at the very heart of just about every game written using Orx.
If you are very new to game development, don't worry. We'll take it nice and slow and try to explain everything in very simple terms. The only knowledge you will need is some simple C++.
I'd like say a huge thank you to FullyBugged for providing the graphics for this series of articles.
What are we making?
Visit the video below to see the look and gameplay of the final game:
Getting Orx
The latest up to date version of Orx can be cloned from github and set up with:
git clone https://github.com/orx/orx.git After cloning, an $ORX environment variable will be created automatically for your system which will help with making game projects much easier. It will also create several IDE projects for your operating system: Visual Studio, Codelite, Code::Blocks, and gmake. These Orx projects will allow you to compile the Orx library for use in your own projects. And the$ORX environment variable means that your projects will know where to find the Orx library.
For more details on this step, visit http://orx-project.org/wiki/en/tutorials/cloning_orx_from_github at the Orx learning wiki.
Setting up a 2D UFO Project
Now the you have the Orx libraries cloned and compiled, you will need a blank project for your game. Supported options are: Visual Studio, CodeLite, Code::Blocks, XCode or gmake, depending on your operating system.
Once you have a game project, you can use it to work through the steps in this tutorial.
Orx provides a very nice system for auto creating game projects for you. In the root of the Orx repo, you will find either the init.bat (for Windows) or init.sh (Mac/Linux) command.
Create a project for our 2D game from the command line in the Orx folder and running:
init c:\temp\ufo or
init.sh ~/ufo Orx will create a project for each IDE supported by your OS at the specified location. You can copy this folder anywhere, and your project will always compile and link due to the \$ORX environment variable. It knows where the libraries and includes are for Orx.
Open your project using your favourite IDE from within the ufo/build folder.
When the blank template loads, there are two main folders to note in your solution:
config src Firstly, the src folder contains a single source file, ufo.cpp. This is where we will add the c++ code for the game. The config folder contains configuration files for our game.
What is config?
Orx is a data driven 2D game engine. Many of the elements in your game, like objects, spawners, music etc, do not need to be defined in code. They can be defined (or configured) using config files.
You can make a range of complex multi-part objects with special behaviours and effects in Orx, and bring them into your game with a single line of code. You'll see this in the following chapters of this guide.
There are three ufo config files in the config folder but for this guide, only one will actually be used in our game. This is:
ufo.ini All our game configuration will be done there.
Over in the Orx library repo folder under orx/code/bin, there are two other config files:
CreationTemplate.ini SettingsTemplate.ini These are example configs and they list all the properties and values that are available to you. We will mainly concentrate on referring to the CreationTemplate.ini, which is for objects, sounds, etc. It's good idea to include these two files into your project for easy reference.
Alternatively you can view these online at https://github.com/orx/orx/blob/master/code/bin/CreationTemplate.ini and here: https://github.com/orx/orx/blob/master/code/bin/SettingsTemplate.ini
The code template
Now to take a look at the basic ufo.cpp and see what is contained there.
The first function is the Init() function.
This function will execute when the game starts up. Here you can create objects have been defined in the config, or perform other set up tasks like handlers. We'll do both of these soon.
The Run() function is executed every main clock cycle. This is a good place to continually perform a task. Though there are better alternatives for this, and we will cover those later. This is mainly used to check for the quit key.
The Exit() function is where memory is cleaned up when your game quits. Orx cleans up nicely after itself. We won't use this function as part of this guide.
The Bootstrap() function is an optional function to use. This is used to tell Orx where to find the first config file for use in our game (ufo.ini). There is another way to do this, but for now, we'll use this function to inform Orx of the config.
Then of course, the main() function. We do not need to use this function in this guide.
Now that we have everything we need to get start, you should be able to compile successfully. Run the program and an Orx logo will appear slowly rotating.
Great. So now you have everything you need to start building the UFO game.
Setting up the game assets
Our game will have a background, a UFO which the player will control, and some pickups that the player can collect.
The UFO will be controlled by the player using the cursor keys.
First you'll need the assets to make the game. You can download the file assets-for-orx-ufo-game.zip which contains:
The background file (background.png):
The UFO and Pickup sprite images (ufo.png and pickup.png):
And a pickup sound effect (pickup.ogg):
pickup.ogg
Copy the .png files into your data/texture folder
Copy the .ogg file into your data/sound folder.
Now these files can be accessed by your project and included in the game.
Setting up the Playfield
We will start by setting up the background object. This is done using config.
Open the ufo.ini config file in your editor and add the following:
[BackgroundGraphic] Texture = background.png Pivot = center
The BackgroundGraphic defined here is called a Graphic Section. It has two properties defined. The first is Texture which has been set as background.png.
The Orx library knows where to find this image, due to the properties set in the Resource section:
[Resource] Texture = ../../data/texture
So any texture files that are required (just like in our BackgroundGraphic section) will be located in the ../../data/texture folder.
The second parameter is Pivot. A pivot is the handle (or sometimes “hotspot” in other frameworks). This is set to be center. The position is 0,0 by default, just like the camera. The effect is to ensure the background sits in the center of our game window.
There are other values available for Pivot. To see the list of values, open the CreationTemplate.ini file in your editor. Scroll to the GraphicTemplate section and find Pivot in the list. There you can see all the possible values that could be used.
top left is also a typical value.
We need to define an object that will make use of this graphic. This will be the actual entity that is used in the game:
[BackgroundObject] Graphic = BackgroundGraphic Position = (0, 0, 0)
The Graphic property is the section BackgroundGraphic that we defined earlier. Our object will use that graphic.
The second property is the Position. In our world, this object will be created at (0, 0, 0). In Orx, the coordinates are (x, y, z). It may seem strange that Orx, being a 2D game engine has a Z axis. Actually Orx is 2.5D. It respects the Z axis for objects, and can use this for layering above or below other objects in the game.
To make the object appear in our game, we will add a line of code in our source file to create it.
In the Init() function of ufo.cpp, remove the default line:
orxObject_CreateFromConfig("Object"); and replace it with:
orxObject_CreateFromConfig("BackgroundObject"); Compile and run.
The old spinning logo is now replaced with a nice tiled background object.
Next, the ufo object is required. This is what the player will control. This will be covered in Part 2.
# C++ Movable struct, how to keep reference to it
## Recommended Posts
I have a SOA particle buffer, that I'm accessing by an index (I build a temp class which has references to the vectors inside the arrays). When particles die, I collapse the structure, copying particles to keep a continues array of data. Now, for some cases, an emitter would need to keep a reference (basically an index) or a handle of some sort to a specific particle (parent particle, sub emitter relationship). Any ideas how to nicely/fast do that? Preferably I would like to have a bi-directional link, at the moment, particles keep a list of affected emitter(s), which works in a way, but I need to do a dirty way of checking if emitter is still valid.
Any ideas would be appreciated.
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In the case of a particle system, generally you only want to be able to keep track of all-of-the-particles, not reference any individual one of them
In the situations where you do need such a feature, generally I've seen a non-compacting array of indices. "ID's" are the indices into the "index array". When compacting the pool, you update the index array entries for any items that were moved. When accessing the pool though, you now have a double-indirection -- e.g. data[indices[id]]
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Firstly I'd be asking whether there was really a need to keep such child->parent references, do the children really need to know what the parents are doing? Usually the emphasis in particle systems is on efficiency, and keeping the flow one way (parent->child) if possible would seem more efficient.
As a caveat I've only done simple particle systems, and I've always had emitters emit either simple particles or more emitters (in a separate list), independent of the parent after being created. Instead of a back reference to the emitter I'd maybe have some preset particle / emitter types that don't ever get destroyed so any info on their behaviour is never changed. Or simply have the different particle types in their own separate lists so you can run through them quickly without any branching.
However going with your suggestion, we can come up with ideas for ways of doing the back-references, but the best answer may depend on the types of particle system, the numbers of particles / emitters involved, how you want to do it (CPU or GPU maybe?), because this can affect things like the cost of branching / jumping around cache. The 'best' solution for 64 particles on a CPU might be quite different to the best solution for a million on a GPU for instance. Maybe you could flesh out your question with some more context?
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Depending on the number of particles vs the number of particles to track, it may be useful to store the latter separately from the former, and accept non-contiguous access there.
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On 8/8/2017 at 8:15 AM, zfvesoljc said:
Now, for some cases, an emitter would need to keep a reference (basically an index) or a handle of some sort to a specific particle (parent particle, sub emitter relationship). Any ideas how to nicely/fast do that?
If I understand, your generic problem is: you have an object container with an object inner inside. You destroy container and poor inner goes with it. Too bad you need to keep referencing it so you're blasted and you need to reconstruct.
All fine and dandy but... you have misunderstood your resource ownership . If inner is owned by container then when container goes it brings its owned resources with it and user systems are screwed.
In the general sense of particle systems that sounds bogus but maybe you're more like "replicating" objects somehow so what you do?
Review your container to use external resources allowed to exist by themselves. When newOwner takes control of them, pull em out of container or mark them 'non-owned'. When container goes belly-up it either destroys its owned resources or they get destroyed by some external function doing the check at higher level.
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On 08/08/2017 at 10:29 AM, Hodgman said:
In the case of a particle system, generally you only want to be able to keep track of all-of-the-particles, not reference any individual one of them
In the situations where you do need such a feature, generally I've seen a non-compacting array of indices. "ID's" are the indices into the "index array". When compacting the pool, you update the index array entries for any items that were moved. When accessing the pool though, you now have a double-indirection -- e.g. data[indices[id]]
Well, this is a situation where I need such a feature
Another way would be to create a combined particle-emitter object, which would act as both, but since particle is not really an object, I'd have to have a separate loop for those objects...
I also saw the double indirection array as a possible solution and will probably try that first.
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On 08/08/2017 at 0:25 PM, lawnjelly said:
Firstly I'd be asking whether there was really a need to keep such child->parent references, do the children really need to know what the parents are doing? Usually the emphasis in particle systems is on efficiency, and keeping the flow one way (parent->child) if possible would seem more efficient.
As a caveat I've only done simple particle systems, and I've always had emitters emit either simple particles or more emitters (in a separate list), independent of the parent after being created. Instead of a back reference to the emitter I'd maybe have some preset particle / emitter types that don't ever get destroyed so any info on their behaviour is never changed. Or simply have the different particle types in their own separate lists so you can run through them quickly without any branching.
However going with your suggestion, we can come up with ideas for ways of doing the back-references, but the best answer may depend on the types of particle system, the numbers of particles / emitters involved, how you want to do it (CPU or GPU maybe?), because this can affect things like the cost of branching / jumping around cache. The 'best' solution for 64 particles on a CPU might be quite different to the best solution for a million on a GPU for instance. Maybe you could flesh out your question with some more context?
cpu powered, spike is 200 emitters/5k particles
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2018-03-19 17:01:22
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http://math.xmu.edu.cn/group/ga/index.html
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the G_A seminar
## The Geometric Analysis Seminar at School of Mathematical Sciences, Xiamen University
The geometric analysis seminar covers various topics on geometry.
## 2017-2018 2nd Semester (2018 Spring)
Schedule Rooms Topics
Tue 02/27 2:30 PM 数理楼661 Organizational meeting.
Fri 03/16 2:00 PM 实验楼105 Chong Song (Xiamen University) "An Energy Method for Uniqueness of Geometric flows"
Abstract: In this talk, I will introduce an energy method for solving uniqueness problems of geometric flows on manifolds. The basic idea is to derive a Gronwall-type inequality for certain geometric energy functionals which describe the intrinsic distance of two solutions. In particular, we use parallel transportations to compare solutions and improve the estimates. I will use the Schrodinger flow as an example and introduce its application to various type of geometric flows.
Fri 03/16 3:30 PM 实验楼105 Yen-Chang Huang (Xinyang Normal University) "A problem of existence of horizontal envelops in the 3D-Heisenberg group and its applications"
Abstract: One of interesting problems in classical geometry is to find the envelope for a family of lines or hypersurfaces in the Euclidean spaces and several applications to Economics and Mathematical Optimization have been developed. After a review of our previous works for finding the pseudohermitian invariants in CR geometry, we will show the necessary and sufficient conditions for the existence of horizontal envelops in the 3D-Heisenberg group by using the standard techniques in Integral Geometry. We obtain a method to construct horizontal envelopes from the given ones and characterize the solutions satisfying the construction. The similar results can be generalized to the higher dimensional Heisenberg groups.
Tue 03/20 2:30 PM 数理楼661 Jinhua Wang (Xiamen University) "An overview: Einstein spaces as attractors for the Einstein flow"
Abstract: I will talk about some stability results in GR, mainly referring to the work by Andersson and Moncrief in Journal of Differential Geometry 89 (2011).
In this paper the authors prove a global existence theorem, in the direction of cosmological expansion, for sufficiently small perturbations of a family of n + 1-dimensional, spatially compact spacetimes, which generalizes the k = −1 FLRW vacuum spacetime. The background spacetimes considered are Lorentz cones over negative Einstein spaces of dimension $n \geq 3$.
These results also demonstrate causal geodesic completeness of the perturbed spacetimes, in the expanding direction, and show that the scale-free geometry converges toward an element in the moduli space of Einstein geometries, with a rate of decay depending on the stability properties of the Einstein geometry.
Fri 03/23 2:00PM (NOT the usual time!) 实验楼105 Xiang Ma (Peking University) "Geometry of pseudo-convex submanifolds in a pseudo-Euclidean space"
Abstract: 伪欧氏空间$\mathbb{R}^{m+1,p}$ 中的$m$维伪凸子流形，是法丛值的第二基本形式 $II(v,v)$ 恒取类空向量的类空子流形，它的法空间中带有一个Lorentz度量（1正p负）。它可以看作是欧氏空间中的凸超曲面（卵形面）的自然推广。直观上，它象是$m+1$维欧氏空间中的凸超曲面的一个微扰。我们先介绍一系列简单的引理，既说明它与凸曲面的类似，也为获得更深入的结果准备技术工具。我们证明它们的截面曲率恒正，并获得了关于全曲率的若干Fenchel 型（反向）不等式。我们还将报导以下结果：在非常弱且自然的条件下，以一个闭的伪凸子流形$\gamma^m$为边界的Plateau问题有解，也就是说，存在一个极大类空$m+1$维子流形$\M^{m+1}$以前者为边界。时间允许的话，我们将简单提及有关概念和结果到离散情形（多边形和多面形）的推广。部分结果是与叶楠博士、张栋博士合作的成果。
Note: At 2:00 PM there is a warm-up talk by the speaker on basics on Lorentz spaces, the research talk begins shortly after the warm-up talk.
Fri 03/23 4:00 PM (NOT the usual time!) 实验楼105 Peng Wang (Tongji University) "On the Morse index of minimal tori in S^4"
Abstract: Urbano's Theorem plays an important geometric role in the proof of Willmore conjecture, which states that a non-totally-geodesic closed minimal surface x in S^3 has index at least 5 and it is congruent to the Clifford torus if the index is 5. In this talk we will provide a generalization of Urbano's Theorem to minimal tori in S^4 by showing that a minimal torus in S^4 has index at least 6 and it is congruent to the Clifford torus if the index is 6. This is a joint work with Prof. Rob Kusner(UMass Amherst).
Fri 03/30 2:30 PM 实验楼105 Chao Xia (Xiamen University) "The Weyl problem in warped product spaces"
Abstract: The Weyl problem studies whether a smooth metric on S^2 with positive Gauss curvature admits a smooth isometric embedding in R^3. The uniqueness part corresponds to Cohn-Vossen's rigidity theorem. The existence part was solved by Nirenberg and Pogorelov independently by using the continuity method. The solution of the Weyl problem is a starting point to well define the Brown-York quasi-local mass. Motivated by the well-definedness of quasi local mass in Schwarzschild manifold, recently, people are interested in study the Weyl problem in warped product spaces. In this talk, I will review the continuity method for this problem and report recent progress made by Li-Wang and Guan-Lu for the Weyl problem in warped product spaces..
The reference:
C. Li and Z. Wang, The Weyl problem in warped product space, arXiv:1603.01350. J. Differential Geom., to appear.
P. Guan and S. Lu, Curvature estimates for immersed hypersurfaces in Riemannian manifolds, Invent. Math. 208 (2017), no. 1, 191-215.
S. Lu, On Weyl's embedding problem in Riemannian manifolds, arXiv:1608.07539.
Fri 04/20 2:30PM 实验楼105 Bennett Chow (University of California, San Diego) "A Survey of Shrinking Gradient Ricci Solitons"
Abstract: We discuss works on shrinking gradient Ricci solitons with an emphasis on some papers of Munteanu and Wang. They have made progress in all dimensions with some stronger results in dimension 4. These objects are interest to Ricci flow because they model finite time singularity formation.
About the speaker: The speaker's homepage at UC San Diego is http://www.math.ucsd.edu/~benchow/.
Fri 05/18 2:30 PM 实验楼105 Victor Ginzburg (University of California, Santa Cruz) "Periodic orbits of Hamiltonian systems: the Conley conjecture and pseudo-rotations"
Abstract: One distinguishing feature of Hamiltonian dynamical systems--a class of systems naturally arising in many physics problems--is that such systems, with very few exceptions, tend to have numerous periodic orbits. In 1984 Conley conjectured that a Hamiltonian diffeomorphism (i.e., the time-one map of a Hamiltonian flow) of a torus has infinitely many periodic orbits. This conjecture was proved by Hingston some twenty years later and similar results for surfaces other than the sphere were established by Franks and Handel. Of course, one can expect the Conley conjecture to hold for a much broader class of phase spaces, and this is indeed the case as has been shown by Gurel, Hein and the speaker. However, the conjecture is known to fail for some, even very simple, phase spaces such as the sphere. These spaces admit Hamiltonian diffeomorphisms with finitely many periodic orbits--the so-called pseudo-rotations--which are of particular interest in dynamics. In this talk, based on the results of Gurel and the speaker, we will examine underlying reasons for the existence of periodic orbits for Hamiltonian systems and discuss the situations where the Conley conjecture does not hold.
About the speaker: The speaker's homepage at UC Santa Cruz is https://ginzburg.math.ucsc.edu/.
Fri 05/18 3:30 PM 实验楼105 Martin Li (The Chinese University of Hong Kong) "Free Boundary Minimal Surfaces in the unit ball"
Abstract: Since the seminal work of Fraser and Schoen on the extremal Steklov eigenvalue problem, there have been substantial interest in the study of free boundary minimal surfaces in the unit ball. In this talk, we will discuss some very recent results concerning the existence, compactness and rigidity of such objects. We will mention along the way some open questions in this area. Part of these are joint work with A. Fraser; and N. Kapouleas.
Tue 05/29 2:30 PM 数理楼661 Wenxiong Chen (Yeshiva University) "The fractional Laplacian"
Abstract: The fractional Laplacian is a non-local pseudo-differential operator defined by a singular integral. It is quite different from the traditional (local) differential operators. In this talk, we will use simple examples to illustrate the essential differences between the local and nonlocal operators, such as the boundary regularities and Poisson representations. We will show how to construct a super solution to obtain Holder regularity of the solutions on the boundary; we will also show how to construct a sub-solution to prove a Hopf type lemma. If time permitting, we will show the ideas on the proofs of interior regularity (the Schauder estimate).
Fri 06/01 2:30 PM 实验楼105 Lihan Wang (University of Connecticut) "Symplectic Laplacians, boundary conditions and cohomology"
Abstract: Symplectic Laplacians are introduced by Tseng and Yau in 2012, which are related to a system of supersymmetric equations from physics. These Laplacians behave different from usual ones in Rimannian case and Complex case. They contain both 2nd and 4th order operators. In this talk, we will discuss these operators and their relations with cohomologies on compact symplectic manifolds with boundary. For this purpose, we will introduce new boundary conditions for differential forms on symplectic manifolds. Their properties and importance will be discussed.
Tue 06/05 2:30 PM 数理楼661 Jinyu Guo (Xiamen University) "Overdeteminated problems in a ball in Euclidean space"
Abstract: In a celebrated paper "A symmetry problem in potential theory", Serrin initiated the study of elliptic equations under overdetermined boundary conditions. He introduced the moving plane method to prove this problems. In this talk, I mainly talk about Serrin's type overdeteminated problems. Firstly, I will introduce the history background for Serrin's type overdeteminated problems without boundary. Secondly, I will introduce several proof's methods for this kind of problems. Finally, I will talk about our recent results for overdeteminated problems in a ball.
Fri 06/08 2:30 PM 实验楼105 Shaochuang Huang (Tsinghua University) "Harmonic Coordinates, Exhaustion functions and its applications"
Abstract: In this talk, I will prove a harmonic radius estimate and then use it to construct an exhaustion function with bounded gradient and Hessian by Tam's method. Finally, using similar method by F. He and Lee-Tam, I will sketch a proof of short-time existence of Ricci flow.
Fri 06/08 3:30 PM 实验楼105 Man-Chun Lee (The Chinese University of Hong Kong) "Chern Ricci flow on noncompact manifolds and applications"
Abstract: In this work, we study a Hermitian flow of metrics evolving along the Chern Ricci direction. We will discuss a existence criteria of the Chern Ricci flow and hence the Kahler Ricci flow without the assumption of bounded curvature. If time is allowed, I will briefly describe a construction of KRF on non-collapsing manifold with nonnegative bisectional curvature and its application to Yau's uniformization conjecture. This is joint work of Prof. L.F. Tam.
Fri 06/15 2:30 PM 实验楼105 Fang Wang (Shanghai Jiaotong University) "Obata's Rigidity theorem on manifolds with boundary"
Abstract: In this talk, I will introduce some rigidity theorems for the (generalized) Obata equation on manifolds with boundary with different kinds of boundary conditions. Then I will also give two main applications. One application is in the rigidity theorems of Poincare-Einstein manifolds; and the other is in the first eigenvalue problems on manifolds with boundary. This is joint work with Mijia Lai and Xuezhang Chen.
Tue 06/19 3:00 PM 数理楼661 Bingyuan Liu (University of California, Riverside) "Geometric analysis on the Diederich–Fornæss index"
Abstract: In this talk, we discuss the Diederich–Fornæss index in several complex variables. A domain \Omega \subset \mathbb{C}^n is said to be pseudoconvex if -\log(-\delta(z)) is plurisubharmonic in \Omega, where \delta is a signed distance function of \Omega. The Diederich–Fornæss index has been introduced since 1977 as an index to refine the notion of pseudoconvexity. After a brief review of pseudoconvexity, we discuss this index from the point of view of geometric analysis. We will find an equivalent index associated to the boundary of domains and with it, we are able to obtain accurate values of the Diederich–Fornæss index for many types of domains.
Tue 06/26 2:30 PM 实验楼105 Xiaodong Wang (Michigan State University) "From the isoperimetric inequality to Integral inequalities for harmonic functions and holomorphic functions"
Abstract: There are many proofs for the classic isoperimetric inequality. Carleman's proof reduces it to an interesting integral inequality for analytic functions on the unit disc in the plane. As natural generalizations I will discuss some integral inequalities for harmonic functions in higher dimensions. This is based on joint work with Fengbo Hang and Xiaodong Yan. I will also talk about some more recent developments and related inequalities for holomorphic functions in several complex variables if time allows.
Thu 07/05 3:00 PM (NOT the usual time!) 教学楼306 (NOT the usual location!) Boyong Chen (Fudan University) "Weighted Bergman kernel, directional Lelong number and John-Nirenberg exponent"
Abstract: Let $\psi$ be a plurisubharmonic function on the closed unit ball and $K_{t\psi}(z)$ the Bergman kernel on the unit ball with respect to the weight $t\psi$. We show that the boundary behavior of $K_{t\psi}(z)$ is determined by certain directional Lelong numbers of $\psi$ for all $t$ smaller than the John-Nirenberg exponent of $\psi$ associated to certain family of nonisotropic balls, which is always positive.
Note: This is a special lecture from the 2018 Summer school on Finsler geometry.
Thu 07/05 4:00 PM (NOT the usual time!) 教学楼306 (NOT the usual location!) Siqi Fu (Rutgers University) "Estimates of invariant metrics and applications"
Abstract: The Caratheodory and Kobayashi metrics are non-smooth Finsler metrics while the Bergman metric is a Kahler metric on bounded domains in several complex variables. All of them are biholomorphic invariants. In this talk, we will discuss boundary estimates of these invariant metrics and the Bergman kernel. We will also discuss how these estimates can be used to characherize certain geometric properties of the boundary.
Note: This is a special lecture from the 2018 Summer school on Finsler geometry.
Fri 07/06 11:00 AM (NOT the usual time!) 实验楼108 (NOT the usual location!) Meikui Xiong (Northwestern University, Xi'an) "Deformation of canonical metrics in Kahler geometry"
Abstract: In (1), Gabor Szekelyhidi proved a theorem which shows the structure of the deformation space of csck metrics (constant scalar curvature metrics), he made use of the K-stability to obtain his result. Then in (2), Eiji Inoue generalized the result above to the case of Kahler-Ricci solitons. We will survey some theories related to the two results.
(1) Gabor Szekelyhidi, The Kahler-Ricci flow and K-stability, Arxiv:0803.1613.pdf.
(2) Eiji Inoue, The moduli space of Fano manifolds with Kahler-Ricci solitons, Arxiv:1802.08128.pdf.
Sun 07/08 4:00 PM (NOT the usual time!) 教学楼306 (NOT the usual location!) Xiao Zhang (AMSS) "宇宙学中的一些物理与几何问题"
Abstract: 宇宙学原理假设宙在大尺度上是均匀各向同性的，几何上可以用 Robinson-Walker 度规描述。1998年天文观测发现宇宙在加速膨胀，宇宙常数为正。近年来，更精细的测量数据似乎表明宇宙中存在一些特殊性质的区域，显示出有较强的各向异性性质。一些学者发现用Finsler几何可以解释这样的各向异性。本报告将讨论正宇宙常数的正能量定理以及引力波Bondi-Sachs时空的Peeling性质。并探讨这些问题在Finsler几何框架下的可能推广.
Note: This is a special lecture from the 2018 Summer school on Finsler geometry.
Mon 07/09 11:00 AM (NOT the usual time!) 数理楼661 Haojie Chen (Zhejiang Normal University) "Kodaira dimensions of almost complex manifolds"
Abstract: The Kodaira dimension gives a rough classification scheme of complex manifolds up to birational equivalence. It is also introduced on symplectic 4-manifolds and smooth manifolds with dimension less than 4. In this talk, I will present a generalization of Kodaira dimension to almost complex manifolds. I will discuss some structural results including the birational invariance on almost complex 4-manifolds and the relation with symplectic Kodaira dimension. It is in general not a deformation invariant, hence not a diffeomorphism invariant. If time allows, I will discuss some interesting non-integrable almost complex structures with large Kodaira dimension. This talk is based on joint work with Weiyi Zhang.
Mon 07/09 4:00 PM (NOT the usual time!) 教学楼306 (NOT the usual location!) Xiaobo Liu (Peking University) "Integrable System and Moduli Space of Curves"
Abstract: An integrable system consists of mutually commuting flow equations. A well known integrable system is the KdV hierarchy. Integrable systems have deep connections with geometry of moduli spaces of stable curves (i.e. compactifications of moduli spaces of punctured Riemann surfaces). In this talk I will explain how intersection numbers on such moduli spaces provide solutions to the KdV hierarchy. Conjecturally, a variation of such connections might be generalized to Gromov-Witten invariants of smooth projectic varieties.
Note: This is a special lecture from the 2018 Summer school on Finsler geometry.
Tue 07/10 11:00 AM (NOT the usual time!) 数理楼661 Siyuan Lu (Rutgers University) "On a localized Penrose inequality"
Abstract: We consider the boundary behavior of a compact manifold with nonnegative scalar curvature. The boundary consists of two parts: \Sigma_H and \Sigma_O, where \Sigma_H denotes outer minimizing minimal hypersurface. Under suitable assumption on \Sigma_O, we establish a localized Penrose inequality, which can be viewed as a quasi-local version of the Riemannian Penrose inequality. Moreover, in dimension 3, we prove that the equality holds iff it's a domain in Schwarzschild manifold. This is based on joint works with Pengzi Miao.
Tue 07/17 2:30 PM 数理楼661 Yong Wei (Australia National University) "Volume preserving flow and Alexandrov-Fenchel inequalities in hyperbolic space"
Abstract: I will describe my recent work with Ben Andrews and Xuzhong Chen on volume preserving flow and Alexandrov-Fenchel inequalities in hyperbolic space. First, if the initial hypersurface in hyperbolic space has positive sectional curvature, we show that a large class of volume preserving flow preserves the positivity of sectional curvatures, and the flow converges smoothly to a geodesic sphere. This result can be used to show that certain Alexandrov-Fenchel quermassintegral inequalities, known previously for horospherical convex hypersurfaces (by G.Wang and C.Xia (2013)), also hold under the weaker condition of positive sectional curvature. Second, we consider the volume preserving flow of strictly horospherically convex hypersurfaces in hyperbolic space by function of shifted principal curvatures, and apply the convergence result to prove a new class of Alexandrov-Fenchel type inequalities for horospherically convex hypersurfaces.
Wed 07/25 10:30 AM (NOT the usual time!) 数理楼661 Changliang Wang (Mcmaster University) "Linear stability of Riemannian manifolds with Killing spinors"
Abstract: Einstein metrics on a compact manifold are critical points of the normalized total scalar curvature functional. So it is natural to study the behavior of the second variation of the normalized total scalar curvature functional at an Einstein metric. This is known as the linear stability problem of Einstein metrics. In this talk, we will briefly review previous works on this problem, and then I will report our work on the linear stability of some interesting Einstein metrics: Riemannian metrics admitting Killing spinors, and Einstein metrics from the circle bundle construction.
## 2017-2018 1st Semester (2017 Fall)
Schedule Rooms Topics
Tue 09/12 2:00 PM 数理楼661 Organizational meeting.
Tue 09/19 2:30 PM 数理楼661 Siyuan Ma (Albert Einstein Institute) "On Maxwell field and linearized gravity in Kerr spacetime"
Abstract: After the publication of Einstein's theory of General Relativity in 1915, many predictions have been confirmed in the latest one century, culminating at the recent observations of gravitational waves emitted during the merging of binary black holes by LIGO and VIRGO collaborations. Black holes are one of the fundamental predictions, and the one of most interests is the Kerr black hole spacetimes. The metric of a Kerr spacetime describes a rotating, stationary, axisymmetric, asymptotically flat solution to vacuum Einstein equations. One of the most important open problems in mathematical General Relativity is to address the fully nonlinear stability conjecture of Kerr solutions. In this talk, I will present recent results in obtaining energy estimates for both Maxwell field and linearized gravity on Kerr backgrounds, which will advance the field towards this conjecture.
Tue 09/19 3:40 PM 数理楼661 Chao Liu (Monash University) "Cosmological Newtonian limits on large scales"
Abstract: In this talk, I will rigorously answer one basic question in cosmological simulation: on what space and time scales Newtonian cosmological simulations can be trusted to approximate relativistic cosmologies? We resolve this question under a small initial data condition.
Tue 09/26 2:30 PM 数理楼661 Guofang Wang (Freiburg University) "Local Lagrangian embeddings and Hessian surfaces"
Abstract: We will talk about Local Lagrangian embeddings and Hessian surfaces. This is a joint work with Qing Han.
Tue 10/17 2:30 PM 数理楼661 Bo Yang (Xiamen University) "Kahler-Ricci flow on noncompact manifolds (after Huang-Tam and Lee-Tam)"
Abstract: This talk is purely expository. We explain recent works on Kahler-Ricci flow on complete noncompact Kahler manifolds with non collapsed volume and nonnegative bisectional curvature.
Tue 10/24 2:30 PM 数理楼661 Fei He (Xiamen University) "Existence of Ricci flow on noncompact manifolds"
Abstract: This will be a continuation of Bo Yang's talk from last week. We will discuss the short-time existence of Ricci flow on noncompact manifolds with a focus on the recent work of Lee and Tam.
Fri 11/03 2:30 PM 实验楼105 Xi Zhang (University of Science and Technology of China) "Canonical metrics and The Hermitian-Yang-Mills flow on reflexive sheaves"
Abstract: In this talk, we will introduce our recent work on the existence of canonical metrics, Bogomolov type inequalities and the limiting behavior of the Hermitian-Yang-Mills flow on reflexive sheaves. These work are joint with JiaYu Li, YanCi Nie and ChuanJing Zhang.
Fri 11/10 2:30 PM 实验楼105 Bin Zhou (Peking University) "K-energy on polarized compactifications of Lie groups"
Abstract: In this paper, we study Mabuchi’s K-energy on a compactification M of a reductive Lie group G, which is a complexification of its maximal compact subgroup K. We give a criterion for the properness of K-energy on the space of K × K-invariant Kahler potentials. In particular, it turns to give an alternative proof of Delcroix’s theorem for the existence of Kahler-Einstein metrics in case of Fano manifolds M . We also study the existence of minimizers of K-energy for general Kahler classes of M.
Fri 11/10 3:40 PM 实验楼105 Weiming Shen (BICMR, Peking University) "On The Negativity of Ricci Curvatures of Complete Conformal Metrics"
Abstract: A version of the singular Yamabe problem in bounded domains yields complete conformal metrics with negative constant scalar curvatures. In this talk, I will disscuss whether these metrics have negative Ricci curvatures. We will provide a general construction of domains in compact manifolds and demonstrate that the negativity of Ricci curvatures does not hold if the boundary is close to certain sets of low dimension. The expansion of the Green's function and the positive mass theorem play essential roles in certain cases. On the other hand, we prove that these metrics indeed have negative Ricci curvatures in bounded convex domains in the Euclidean space.
Fri 11/17 3:30 PM 实验楼105 Xiao Zhang (AMSS, Beijing) " The positive energy theorem for asymptotically hyperbolic manifolds"
Abstract: In general relativity, asymptotically hyperbolic manifolds serve as the initial data sets in two cases: (i) asymptotically null infinity for asymptotically flat spacetimes where the cosmological constant is zero; (ii) asymptotically spatial infinity for asymptotically AdS spacetimes where the cosmological constant is negative. The difference is that, in case (i), the second fundamental forms are asymptotic to hyperbolic metrics while in case (ii), the second fundamental forms are asymptotic to zero. We will discuss the positive energy theorem in the two cases. The talk is based on the early work of the speaker as well as the joint work with Wang Yuahua and Xie Naqing.
Fri 11/24 2:30 PM 实验楼105 Zhizhang Wang (Fudan University) "The curvature estimates for convex solutions of some fully nonlinear Hessian type equations"
Abstract: The curvature estimates of quotient curvature equation do not always exist even for convex setting. Thus it is natural question to find other type of elliptic equations possessing curvature estimates. In this paper, we discuss the existence of curvature estimate for fully nonlinear elliptic equations defined by symmetric polynomials, mainly, the linear combination of elementary symmetric polynomials. This is a joint work with Chunhe Li and Changyu Ren.
Fri 11/24 3:40 PM 实验楼105 Naqing Xie (Fudan University) "Toroidal marginally outer trapped surfaces in the closed Friedmann-Lemaitre-Robertson-Walker universe"
Abstract: We explicitly construct toroidal MOTS in the closed FLRW universe. This construction is used to assess the quality of certain isoperimetric inequalities recently proved in axial symmetry. We also show that these constructed toroidal MOTS are unstable. This talk is based on a joint work with Patryk Mach.
Fri 12/01 2:30 PM 实验楼105 Frederick Tsz-Ho Fong (Hong Kong University of Science and Technology) "Rigidity of Self-Expanders of Inverse Curvature Flows"
Abstract: In this talk, the speaker will investigate a large class of curvature flows by degree -1 homogeneous functions of principal curvatures in Euclidean spaces. This class curvature flows include the well-known inverse mean curvature flow and many others in the current literature. Self-expanding solutions to these curvature flows are solutions that expanding homothetically without changing their shapes. We will talk about uniqueness, rigidity, and constructions of both compact and non-compact self-expanding solutions to these flows. Part of these are joint work with G. Drugan, H. Lee; P. McGrath; and A. Chow, K. Chow.
Fri 12/08 3:30 PM 实验楼105 Hui Ma (Tsinghua University) "Uniqueness of closed self-similar solutions to $\sigma_k^{\alpha}$-curvature flow"
Abstract: In this talk we will show the uniqueness of closed self-similar solutions to $\sigma_k^{\alpha}$-curvature flow. It is based on the joint work with Shanze Gao and Haizhong Li.
Fri 12/22 3:00 PM 实验楼105 Daniel Zhuangdan Guan (UC Riverside) "Recent progress on compact Kaehler-Einstein manifolds with cohomogeneity one metrics"
Abstract: Although there are many known K\"ahler-Einstein manifolds, there is so far no very practical method to check a given compact Fano manifold to be K\"ahler-Einstein or not.This is also true for the K\"ahler metrics with constant scalar curvatures or Calabi extremal metrics. The situation for a cohomogeneity one metrics was completely resolved.For the type III case, it was solved in my dissertation in 1992. For the remain type I and II case, the existence is equivalent to the negativity of a topological integral.The type I case was published in 2011. Therefore, the problem is reduced to check the negativity for classes of Fano manifolds. Recently, we use computer to get some insight into a class of type I Fano manifolds. This work is a joint work with a group of students. .
Wed 12/27 3:00 PM 行政楼313 Daniel Zhuangdan Guan (UC Riverside) "Recent Progress in the classification of complex homogeneous spaces"
Abstract: A manifold M is a homogeneous space if M=G/H with G a finite dimensional group and H a closed subgroup. M is a complex homogeneous space if J is the given complex structure on M such that J is invariant under the action of G. Homogeneous space is a classical area of differential geometry.The most famous work was the classification of real (and complex) semi-simple Lie groups and the symmetric spaces. The K\"ahler homogeneous space was classified by Dorfmeister and Nakajima in 1988. The pseudo-k\"ahler homogeneous space with reductive G was classified by Dorfmeister and Guan in 1989. In the general compact complex homogeneous case, the classification reduced to the parallelizable case, i.e., in which H is discrete. In the late 1990's we proved that if G/H is a compact complex parallelizable manifold,then the semi-simple part of G is locally a product of complex simple Lie group of type A.A classification of the compact complex homogeneous space with an invariant volume was also finally classified.A complete classification of the compact complex space with a pseudo-k\"ahler structure (non-necessary invariant) was given in 2007. Recently, compact complex homogeneous space with an invariant locally conformal K\"ahler structure was classified and similarly for the cohomogeneity one case.
Fri 12/29 2:30 PM 实验楼105 Yunhui Wu (Tsinghua University) "The Weil-Petersson geometry of the moduli of curves for large genus"
Abstract: We study the systole function along Weil-Petersson geodesics. We show that the square root of the systole function is uniform Lipschitz on the Teichmuller space endowed with the Weil-Petersson metric. As an application, we study the growth of the Weil-Petersson inradius of the moduli space of Riemann surfaces of genus $g$ with $n$ punctures as a function of $g$ and $n$. We show that the Weil-Petersson inradius is comparable to $\sqrt{\ln{g}}$ with respect to $g$, and is comparable to $1$ with respect to $n$.
Fri 12/29 3:30 PM 实验楼105 Qing Han (University of Notre Dame) "Nonexistence of Poincare-Einstein Fillings on Spin Manifolds"
Abstract: In this talk, we discuss whether a conformal class on the boundary M of a given compact manifold X can be the conformal infinity of a Poincare-Einstein metric in X. We construct an invariant of conformal classes on the boundary M of a compact spin manifold X of dimension 4k with the help of the Dirac operator. We prove that a conformal class cannot be the conformal infinity of a Poincare-Einstein metric if this invariant is not zero. Furthermore, we will prove this invariant can attain values of infinitely many integers if one invariant is not zero on the above given spin manifold. This talk is based on a joint work with Gursky and Stolz.
Tue 01/09 2:30 PM 数理楼661 Guohuan Qiu (McGill University) "Interior Hessian estimates for sigma-2 equations in dimension three"
Abstract: The interior regularity for solutions of the sigma_2 Hessian equation is a longstanding problem.Heinz first derived this interior estimate in dimension two. For higher dimensional Monge-Ampere equations, Pogorelov constructed his famous counter-examples even for f constant and convex solutions. Caffarelli-Nirenberg-Spruck studied more general fully nonlinear equations such as \sigma_{k} equations in their seminal work. And Urbas also constructed counter-examples with k greater than 3. The only unknown case is k=2. A major breakthrough was made by Warren-Yuan, they obtained a prior interior Hessian estimate for the equation \sigma_2=1 in dimension three.In this talk, I will present my recent work on how to deal this problem for a more general case in dimension three.
Fri 01/12 3:30 PM 实验楼105 Xingwang Xu (Nanjing University) "Q and R"
Abstract: In this talk, I should focus on the conformal invariant equations of higher order. We interpolate them in terms of conformal geometry. Natural geometric information provides the maximum principle for such equations. This is a joint work with Mr. Weixi Wang.
Fri 01/12 4:30 PM 实验楼105 Robert Kusner (University of Massachusetts at Amherst) "Coplanar CMC surfaces, complex projective structures, and polynomial quadratic differentials"
Abstract: Complete embedded constant mean curvature (CMC) surfaces of fixed, finite topology form a finite-dimensional moduli space. This moduli space is a real-analytic variety parametrized by the asymptotic data of the surfaces, and possibly by some square-integrable Jacobi fields. For coplanar CMC surfaces of genus 0 with k ends, such Jacobi fields must vanish, and this moduli space can be explicitly described: it is diffeomorphic to the space of k-point spherical metrics; these can be described, in turn, by holomorphic immersions from the plane to the 2-sphere whose Schwarzian is a polynomial with degree depending on k. The CMC surfaces corresponding to the polynomials 0 and 1 are, respectively, the round sphere and the 1-parameter family of unduloids, while those which correspond to the polynomial z are the 3-parameter family of triunduloids. Byviewing the Schwarzian as a quadratic differential and its real foliations, a compelling picture of this correspondence emerges. (If time permits, a new construction of coplanar CMC surfaces with genus 1, all of whose ends are cylindrical, will also be described.).
Thu 01/18 2:30 PM 实验楼108 Miaomiao Zhu (Shanghai Jiaotong University) "Existence of solutions of a boundary value problem for Dirac-harmonic maps"
Abstract: In this talk, we shall present some recent progresses on the heat flow approach to the existence of solutions of a boundary value problem for Dirac-harmonic maps. These are joint works with Jurgen Jost and Lei Liu.
Last modified: 12/27/2017 by the geometric analysis group at XMU Math.
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2018-08-16 04:42:39
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https://www.physicsforums.com/threads/explicit-proof-of-the-jacobian-inverse.869717/
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Explicit proof of the Jacobian inverse
1. Apr 30, 2016
Physgeek64
1. The problem statement, all variables and given/known data
Given the transformations $x^2+y^2=2*r*cos(theta)$ and $x*y=r*sin(theta)$ prove the Jacobian explicitly
The question then goes on to ask how r and theta are related to the cylindrical coordinates rho and phi. I think $r=1/2*(x^2+y^2)$ and hence $r=1/2 rho$ but Im not really sure about this part I'm afraid, so haven't gotten very far
2. Relevant equations
$(partial(x,y)/partial(r,theta))*partial(r,theta)/partial(x,y))=1$
3. The attempt at a solution
So by multiplying the second expression by two and then squaring both and adding we get $r=1/2*(x^2+y^2)$ from which we can find the first two elements in the second Jacobian to be x and y respectively. By dividing the two transformations we can also get an expression for theta, which is relativity simple to differentiate, however when isolating x and y I seem to be going into pages of algebra, which leads me to think I have got the wrong approach, and there must be a simpler way since this is meant to be a relatively quick question.
Many thanks
2. May 5, 2016
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2018-03-25 02:24:20
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https://www.transtutors.com/questions/problem-13-5-part-level-submission-kingbird-corporation-sells-computers-under-a-2-ye-2565757.htm
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# Problem 13-5 (Part Level Submission) Kingbird Corporation sells computers under a 2-year warranty...
Problem 13-5 (Part Level Submission)
Kingbird Corporation sells computers under a 2-year warranty contract that requires the corporation to replace defective parts and to provide the necessary repair labor. During 2017, the corporation sells for cash 368 computers at a unit price of $2,280. On the basis of past experience, the 2-year warranty costs are estimated to be$144 for parts and $184 for labor per unit. (For simplicity, assume that all sales occurred on December 31, 2017.) The warranty is not sold separately from the computer. a. Record any necessary journal entries in 2017 b. What liability relative to these transactions would appear on the December 31, 2017, balance sheet and how would it be classified? c. In 2018, the actual warranty costs to Brooks Corporation were$21,400 for parts and \$39,900 for labor. Record any necessary journal entries in 2018.
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2018-06-23 04:43:31
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https://moodle.org/mod/forum/discuss.php?d=358993
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## General developer forum
### Trying to get property of non-object in php
Trying to get property of non-object in php
Hi
I am getting the above error message and I haven't a clue what is causing it so any help would be much appreciated. The code is shown below and it is the line in bold that is causing the problem
// Sensor location of local raspberry shake
$sensor['latitude'] = -27.5;$sensor['longitude'] = 153.02;
$data['format'] = 'geojson'; // Only fetch quakes with a magnitude >= 5$data['minmagnitude'] = '5';
$data['starttime'] =$startdate;
$data['endtime'] =$enddate;
$shakes = json_decode($events->CallAPI('GET', $url,$data));
foreach ($shakes->features as$shake){
$quake['id'] =$shake->id;
$quake['longitude'] =$shake->geometry->coordinates[0];
$quake['latitude'] =$shake->geometry->coordinates[1];
$quake['depth'] =$shake->geometry->coordinates[2];
$quake['mag'] =$shake->properties->mag;
$quake['place'] =$shake->properties->place;
$quake['time'] =$shake->properties->time; // storing time as milliseconds since 01/01/1970 (UTC)
$quake['angular_distance'] =$events->angle($sensor['latitude'],$quake['latitude'], $sensor['longitude'],$quake['longitude']);
$quake['linear_distance'] =$quake['angular_distance'] * 111.32; // calculate linear distance in km
// trave time based on best fit equation for arrival of p-wave - seems to work
$quake['travel_time'] = 0.0000048 * pow($quake['angular_distance'],3) - 0.0015881 * pow($quake['angular_distance'],2) + 0.2487721 *$quake['angular_distance'];
Any clues would be much appreciated - however I am very new to this
Many Thanks
Graham
Average of ratings: -
Re: Trying to get property of non-object in php
I would add the following line
$shakes = json_decode($events->CallAPI('GET', $url,$data));
var_dump($shakes); exit(); And you should get more information (possible it will say that$shakes is null)
Average of ratings: -
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2018-01-17 04:44:35
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https://math.stackexchange.com/questions/3101264/distribution-at-first-time-a-sum-reaches-a-threshold
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# Distribution at First Time a Sum Reaches a Threshold
Consider the following problem. Roll a die many times, and stop when the total exceeds $$M$$, for some prescribed threshold $$M$$. Call this time $$\tau$$, and call the running score after $$n$$ rolls $$X_n$$.
What is the distribution of $$X_\tau$$?
Of course $$X_\tau \in [M,M+5]$$. However, I can get little beyond this. Moreover, for small $$M$$ the distribution should be very sensitive, and I doubt have a nice form. However, if I define $$X'_\tau = X_\tau - M$$, then it seems to me that $$X'_\tau$$ should have a limiting distribution at $$M \to \infty$$.
Extra comments. $$\quad$$ Note that there are various related questions here on maths.SE, but these (as far as I have seen) are about determining $$\tau$$, in particular its expectation, $$E(\tau)$$.
Also, one can solve for $$E(\tau)$$ directly. If I write $$k_M$$ for $$E(\tau)$$ with threshold $$M$$, then $$\textstyle k_M = \tfrac16 \sum_{j=1}^6 k_{M-j} + 1,$$ and writing $$\ell_M = k_M - k_{M-1}$$ we get $$\textstyle \ell_M = \tfrac16 \sum_{j=1}^6 \ell_{M-j}.$$ In principle, by trying the solution $$\ell_r = r^\lambda$$ and solving $$6 \lambda^6 - \lambda^5 - \lambda^4 - \lambda^3 - \lambda^2 - \lambda - 1 = 0,$$ (see this WolframAlpha computation), and calculating some initial conditions by hand, then one can find $$k_M = E(\tau)$$. Note that this would involve finding six initial conditions and solving a set of six simultaneous equations. This is only for the $$\ell$$-s; one then needs to convert this into the $$k$$-s. This does not sound like fun to me! =P
By a simple martingale argument---$$(X_n - \tfrac72n)_{n\ge0}$$ is a martingale, and $$\tau$$ is a deterministically bounded stopping time---from this one immediately gets $$E(X_\tau) = \tfrac27 E(\tau)$$ (for any $$M$$).
• Does hitting $M$ exactly count as being done; or do you have to have a sum strictly greater than $M$? – paw88789 Feb 5 '19 at 16:03
• Yes, hitting $M$ counts as being done; this is why $X_\tau$ can be equal to $M$ (and not $M+6$). However, my title says exceeds, so I see where your confusion lies!---changed now :) – Sam T Feb 6 '19 at 9:34
## 2 Answers
Let $$p^M_i$$ be the probability that $$X_{\tau(M)}=M+i$$ for $$i=0,1,\dots,5$$, and $$p^M$$ denote the column vector of these six values.
You can compute $$p_M$$ from $$p_{M-1}$$ as follows. There are two ways to achieve a final value of $$M+i$$; either the first time that you value at or above $$M-1$$ you reach is $$(M-1)+(i+1)$$, or the first value at or above $$M-1$$ you reach is $$M-1$$, and then from there you jump immediately to $$M+i$$. Therefore, $$p_i^M=\begin{cases}p^{M-1}_{i+1}+\frac16 p^{M-1}_0 & i<5\\\frac16p_0^{M-1} & i=5\end{cases}$$ This can be written as a matrix equation: $$p^M=\begin{bmatrix} \frac16 & 1 & \\ \frac16 &0 & 1 &\\ \frac16 & 0&0 & 1 &\\ \frac16 & 0&0&0& 1 &\\ \frac16 & 0&0&0&0& 1 \\ \frac16 & 0&0 &0&0&0\\ \end{bmatrix}p^{M-1}$$ with zeroes above the super-diagonal. Letting $$A$$ be the above matrix, then this proves $$p^M=A^Mp^0,$$ where $$p^0$$ is a vector whose first entry is $$1$$ and whose other entries are zero.
The limiting distribution $$p$$ will satisfy $$p=Ap$$. This means that $$p_i=p_{i+1}+\frac16p_0$$, so that $$p$$ is an arithmetic progression with difference $$-\frac16p_0$$. A little thought shows that this implies $$p=\left(\frac{6}{21},\frac{5}{21},\frac{4}{21},\frac{3}{21},\frac{2}{21},\frac{1}{21}\right)^T.$$
• Ah, yes, I didn't think of writing $M+i = (M-1) + (i+1)$. (To be honest, I really should have!) It's clear what to do if you hit $M-1$, but if one is at $M-3$, say, it was less clear to me. I now feel kinda stupid not having got that, but it's always easy once you see a solution---doesn't mean it was easy to find the solution! =P – Sam T Feb 6 '19 at 9:42
• Also, while the whole argument was in the second paragraph, thanks for writing out the details of the rest so that I didn't have to go through them myself =P – Sam T Feb 6 '19 at 9:43
• Also, it appears to me that this should be pretty easy to generalise, yes? If one has a "die" that gives output $X$ (discrete), simply replace the column of $\tfrac16$s with the appropriate distribution? One then just solves the related $p = Ap$ with this new column, which again is pretty easy due to the nature of the matrix. Am I missing something? – Sam T Feb 6 '19 at 20:26
• @SamT That's right! The limiting distribution should be the reverse cumulative sums of the pmf of X, renormalized to be a probability distribution. – Mike Earnest Feb 6 '19 at 20:29
• That's a really cool result, actually! :) – Sam T Feb 6 '19 at 21:50
Inspired by the answer from @MikeEarnest, I wonder if the following alternate proof is valid, for the limiting case? This proof has the advantage(?) of less algebra, and hopefully more intuition into why the distribution is 6:5:4:3:2:1.
Imagine you keep rolling forever. A number is reached if it is the sum at some point in time, otherwise it is skipped. Clearly in the limit, all numbers have the same probability $$q$$ of being reached. (This follows from ergodicity, right? In fact, ergodicity would suggest $$q = {1 \over 3.5}$$, but we don't need its exactly value for now.)
Now consider the interval of interest, $$X_\tau \in [M, M+5]$$:
• $$X_\tau = M+5$$ iff $$M-1$$ is reached and the next roll is $$6$$. This happens with probability $$q/6$$.
• $$X_\tau = M+4$$ iff (a) $$M-1$$ is reached and the next roll is $$5$$, or, (b) $$M-2$$ is reached and the next roll is $$6$$. So this happens with probability $$2q/6$$.
• Note that the case of reaching $$M-2$$, then rolling $$1$$ to reach $$M-1$$, then rolling $$5$$ to reach $$M+4$$, is included in (a) but not in (b), so we did not double-count.
• Similarly, $$X_\tau = M+3, M+2, M+1, M$$ with probabilities $$3q/6, 4q/6, 5q/6, 6q/6$$ respectively.
Since these 6 possibilities are exhaustive, we have $$(1+2+3+4+5+6)q/6 = 1 \implies q = {6 \over 21} = {1 \over 3.5}$$ as I originally suspected; in particular, this implies that $$P(X_\tau = M + j) = (6-j)q/6 = (6-j)/21,$$ agreeing with Mike's answer.
• Interesting! I hope you don't mind, but I added one line to your answer; you hadn't explicitly repeated what the distribution was, so I thought that would be helpful. \\ But yes, I'm pretty sure that argument is valid -- it's a bit late (11pm) for me to think about it now, but I'll just check tomorrow that the ergodicity argument is fine – Sam T Feb 9 '19 at 22:07
• @SamT you edit contains a typo :) (which i fixed) but is otherwise good. anyway, this approach assumes the "obvious" fact that in the limit every large number is reached with same prob $q$. Under this assumption the proof is valid (I think), but proving this "obvious" assumption would require bringing in heavier machinery. But hopefully this gives more insight into why the distribution is 6:5:4:3:2:1. – antkam Feb 9 '19 at 22:20
• Yes, it definitely does give insight -- and for a 'roll distribution $R$' (ie more general than just $\text{Uniform}(1,...,6)$), it does generalise (although general distributions could get messy to write out). Definitely a +1! – Sam T Feb 9 '19 at 23:09
• hmm, @SamT interesting observation re: a general roll distribution $R$. E.g. $R=\{1,10,27\}$ would be pretty tedious to write out and the distribution of $X_\tau \in [M, M+26]$ would also be pretty interesting. As long as the possible $R$ values have $gcd = 1$, the "obvious" ergodic assumption should hold. – antkam Feb 9 '19 at 23:23
• Yeah, I think you're right: tedious, but the method would certainly work. And good shout on the $\text{gcd}$! – Sam T Feb 10 '19 at 10:44
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2020-01-21 14:29:00
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https://pos.sissa.it/282/675/
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Volume 282 - 38th International Conference on High Energy Physics (ICHEP2016) - Top Quark and Electroweak Physics
QCD corrections to vector boson pair production in gluon fusion at the LHC
L. Tancredi
Full text: pdf
Pre-published on: February 06, 2017
Published on: April 19, 2017
Abstract
We report on the calculation of the NLO QCD corrections to vector boson pair production
in gluon fusion at hadron colliders. We focus in particular on the on-shell production of vector bosons, which is of fundamental importance, among the others, to check the consistency of the electroweak sector of the Standard Model. For similar studies, including the interference of the off-shell prompt $gg \to V_1V_2$ amplitudes with the Higgs production amplitudes $gg \to H \to V_1 V_2$, we refer to the proceedings of this conference.
DOI: https://doi.org/10.22323/1.282.0675
How to cite
Metadata are provided both in "article" format (very similar to INSPIRE) as this helps creating very compact bibliographies which can be beneficial to authors and readers, and in "proceeding" format which is more detailed and complete.
Open Access
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2020-11-27 21:02:42
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https://tex.stackexchange.com/questions/257318/typesetting-a-title-on-a-tikzpicture-version-2
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# Typesetting a title on a tikzpicture (Version 2)
I have two very similar-looking polygons in different TikZ environments. Each polygon is symmetric across a horizontal line. I want to put them in one TikZ environment with the axes of symmetry aligned. I didn't know how to do this while keeping the titles of each figure in their respective positions.
\documentclass{amsart}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{tikz}
\usetikzlibrary{calc,angles,positioning,intersections}
\begin{document}
\begin{tikzpicture}
%A hexagon is drawn which is symmetric across the x-axis.
\coordinate (A) at (0,0);
\coordinate (O) at (-5,0);
\path[name path=x-axis] (A) -- (O);
\path[name path=horizontal_line_at_-1] (-5,-1) -- (0,-1);
\path[name path=horizontal_line_at_1] (-5,1) -- (0,1);
\coordinate (B) at ($(A) + (-135:2.5)$);
\draw (A) -- (B);
\coordinate (C') at ($(B) + (165:3)$);
\path[name path=extension_of_line_segment_BC] (B) -- (C');
\coordinate[name intersections={of=extension_of_line_segment_BC and horizontal_line_at_-1, by={C}}];
\draw (B) -- (C);
\coordinate (D') at ($(C) + (45:2)$);
\path[name path=extension_of_line_segment_CD] (C) -- (D');
\coordinate[name intersections={of=extension_of_line_segment_CD and x-axis, by={D}}];
\draw (C) -- (D);
\coordinate (E') at ($(D) + (135:2)$);
\path[name path=extension_of_line_segment_DE] (D) -- (E');
\coordinate[name intersections={of=extension_of_line_segment_DE and horizontal_line_at_1, by={E}}];
\draw (D) -- (E);
\coordinate (F) at ($(E) + (15:3)$);
\draw (E) -- (F);
\draw (A) -- (F);
%Points P and Q in the hexagon are plotted. Line segment $\overline{PQ}$ is not contained in the
%hexagon.
\coordinate (P) at (-4,0.75);
\draw[fill] (P) circle (1.5pt);
\coordinate (Q) at (-4,-0.75);
\draw[fill] (Q) circle (1.5pt);
\draw (P) -- (Q);
%Points P and Q are labeled.
\coordinate (label_for_P) at ($(P)!-3mm!-90:(Q)$);
\node at (label_for_P){$P$};
\coordinate (label_for_Q) at ($(Q)!-3mm!90:(P)$);
\node at (label_for_Q){$Q$};
%A title is typeset.
\node[align=center,font=\bfseries, yshift=2em] (title) at (current bounding box.north){A set that is \\ not convex};
\end{tikzpicture}
\begin{tikzpicture}
%A hexagon is drawn which is symmetric across the x-axis.
\coordinate (A) at (0,0);
\coordinate (O) at (-5,0);
\path[name path=x-axis] (A) -- (O);
\path[name path=horizontal_line_at_-1] (-5,-1) -- (0,-1);
\path[name path=horizontal_line_at_1] (-5,1) -- (0,1);
\coordinate (B) at ($(A) + (-135:2.5)$);
\draw (A) -- (B);
\coordinate (C') at ($(B) + (165:3)$);
\path[name path=extension_of_line_segment_BC] (B) -- (C');
\coordinate[name intersections={of=extension_of_line_segment_BC and horizontal_line_at_-1, by={C}}];
\draw (B) -- (C);
\coordinate (D') at ($(C) + (45:2)$);
\path[name path=extension_of_line_segment_CD] (C) -- (D');
\coordinate[name intersections={of=extension_of_line_segment_CD and x-axis, by={D}}];
\draw[dashed] (C) -- (D);
\coordinate (E') at ($(D) + (135:2)$);
\path[name path=extension_of_line_segment_DE] (D) -- (E');
\coordinate[name intersections={of=extension_of_line_segment_DE and horizontal_line_at_1, by={E}}];
\draw[dashed] (D) -- (E);
\draw (C) -- (E);
\coordinate (F) at ($(E) + (15:3)$);
\draw (E) -- (F);
\draw (A) -- (F);
%A title is typeset.
\node[font=\bfseries, yshift=2em] (title) at (current bounding box.north){A convex set};
\end{tikzpicture}
\end{document}
And one more solution (It take me little more time because I allowed myself first to simplify the picture code :-) , i was lost in original one :-( ):
\documentclass{amsart}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{tikz}
\usetikzlibrary{calc,angles,positioning,intersections}
\begin{document}
\begin{tikzpicture}[
node distance= 3mm and 5mm,
]
%%%% first shape
\coordinate (O) at (-5,0);
\coordinate (A) at (0,0);
\coordinate (B) at ($(A)+(135:2.5)$);
\coordinate (C) at ($(B)+(-165:3)$);
%
\coordinate (B') at ($(A)+(-135:2.5)$);
\coordinate (C') at ($(B')+(165:3)$);
%
\path[name path=x-axis] (O) -- (A);
\path[name path=c-b_line] (C') -- (B);
\coordinate[name intersections={of=c-b_line and x-axis,by={D}}];
\draw (A) -- (B) -- (C) -- (D) -- (C') -- (B') -- cycle;
\coordinate (P) at (-4,0.75);
\coordinate (Q) at (-4,-0.75);
\draw[fill=black] (P) circle (1.5pt) node[right] {$P$} --
(Q) circle (1.5pt) node[right] {$Q$};
%% Title of second shape.
\node[font=\bfseries,align=center,
above=of B -| {$(C |- A)!0.5!(A)$}] (title) {A set that is \\
not convex};
%% Title of first shape.
\node[font=\bfseries,align=center,
above=of B -| {$(C |- A)!0.5!(A)$}] (title) {A set that is \\
not convex};
%%%% secod shape
\begin{scope}[transform canvas={xshift=6cm}]
\coordinate (O) at (-5,0);
\coordinate (A) at (0,0);
\coordinate (B) at ($(A)+(135:2.5)$);
\coordinate (C) at ($(B)+(-165:3)$);
%
\coordinate (B') at ($(A)+(-135:2.5)$);
\coordinate (C') at ($(B')+(165:3)$);
%
\path[name path=x-axis] (O) -- (A);
\path[name path=c-b_line] (B) -- (C');
\coordinate[name intersections={of=c-b_line and x-axis,by={D}}];
\draw (A) -- (B) -- (C) -- (C') -- (B') -- cycle;
\draw[dashed] (C) -- (D) -- (C');
%% Title of second shape.
\node[font=\bfseries,
above=of B -| {$(C |- A)!0.5!(A)$}] (title) {A convex set};
\end{scope}
\end{tikzpicture}
\end{document}
Edit: Now I figured out, how to move second shape with xshift ... I correct my MWE accordingly. The picture is the same as before.
Edit (2): Some explanation of MWE. The node distance from library possitionig determine common distance between nodes if you say: \node[above right=of node name]. In this case, the horizontal distance is not used.
The node for (sub)image titles can have the same parameters and can be set up on beginning of picture, something like:
\begin{tikzpicture}[
title/.style = {font=\bfseries, align=center}]
and then in code use
\node[title, above=of M |-C] {title};
if the M is midpoint between A and O. Midpoint M can be determined as shown in above, however it can be simply set up width
\coordinate[right=25mm of A] (M);
As can be seen, I significantly strip the MWE in question simply because most of them I din't understand and they seems to me to be surplus. In compose of it I follow to given pictures.
• You should edit your code. You have the same command following "%% Title of second shape" as you have following "%% Title of first shape". – user74973 Jul 27 '15 at 22:48
• You have above=of B -| {$(C |- A)!0.5!(A)$} as an option in the node command to place the first title. (Recall that C and E are the two vertices on the hexagon that are furthest to the left. A is the vertex on the hexagon furthest to the right.) I know that C |- A is the intersection of the vertical line through C and the horizontal line through A. So, {$(C |- A)!0.5!(A)$} is the "horizontal midpoint" of the first display. If I call this point M, what is B -| M and above=of B -| M? – user74973 Jul 27 '15 at 22:57
• (I see that in your code B is the highest vertex on the hexagon.) I guess that B -| M is the intersection of a horizontal line through B and a vertical line through M. OK. If my comments are correct, I would expect the middle of the title "A set that is not convex" to be typeset much lower. I don't know what above of does, though. By the way, shouldn't the syntax be something like above of={B -| M}? – user74973 Jul 27 '15 at 23:10
• You also have node distance= 3mm and 5mm as an option to the tikzpicture environment. How is this interpreted by TikZ? – user74973 Jul 27 '15 at 23:11
• OK. I'll give you an opportunity to answer my questions. Thanks. – user74973 Jul 27 '15 at 23:11
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2021-05-08 07:43:56
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http://mathhelpforum.com/calculus/92397-evaluate-limit.html
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$\lim_{n\rightarrow \infty} \frac{(-2)^n + 3^n}{(-2)^{n + 1} + 3^{n + 1}}$ is equal to?
2. $\frac{(-2)^n + 3^n}{(-2)^{n + 1} + 3^{n + 1}} \cdot {\color{red}\frac{\frac{1}{3^{n}}}{\frac{1}{3^{n}} }}$
$= \frac{\left(-\frac{2}{3}\right)^n + 1}{\left(-2\right)\left(-\frac{2}{3}\right)^n + 3 \cdot 1}$
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2017-07-26 07:05:39
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https://socratic.org/questions/what-is-the-radius-of-convergence-for-a-power-series#110948
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# What is the radius of convergence for a power series?
##### 1 Answer
Oct 16, 2014
The radius of convergence is 1/2 of the length of the interval of convergence.
I hope that this was helpful.
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2022-12-02 19:45:11
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https://math.stackexchange.com/questions/272988/fixed-is-this-set-empty-s-x-in-mathbbz-mid-sqrtx-in-mathbbq
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# Fixed: Is this set empty? $S = \{ x \in \mathbb{Z} \mid \sqrt{x} \in \mathbb{Q}, \sqrt{x} \notin \mathbb{Z}, x \notin \mathbb{P}$ }
This question has been "fixed" to reflect the question that I intended to ask
Is this set empty? $S = \{ x \in \mathbb{Z} \mid \sqrt{x} \in \mathbb{Q}, \sqrt{x} \notin \mathbb{Z}, x \notin \mathbb{P}$ }
Is there a integer, $x$, that is not prime and has a root that is not irrational?
• Just take $x=\frac14$, for instance. – Brian M. Scott Jan 8 '13 at 19:03
• Over what domain are you quantifying $x$? If $x$ is allowed to be rational, then trivially $x=\frac14$ works; if $x$ is supposed to be integral, then its square root is either integral or irrational (this is a nice and standard exercise in a beginning number-theory class). – Steven Stadnicki Jan 8 '13 at 19:03
• How do I rewrite the question to say X is integer and not prime? Is there a notation for the prime numbers? – Leonardo Jan 8 '13 at 19:04
• Just say that $x\in\Bbb Z$; the fact that it’s not prime is irrelevant. – Brian M. Scott Jan 8 '13 at 19:07
• I want to know if any numbers greater than 1 that are not prime have a root that is not irrational. I will try and make a new question that is better. – Leonardo Jan 8 '13 at 19:08
Edited in response to the latest change to the question
The answer is ‘no’. As you are considering $\sqrt{x}$, we must look at $x \geq 0$.
Suppose that $\sqrt{x} \in \mathbb{Q} \setminus \mathbb{Z}$. Let $\sqrt{x} = \dfrac{p}{q}$, where $p \in \mathbb{N}_{0}$, $q \in \mathbb{N}$ and $\gcd(p,q) = 1$. This yields $$q^{2} x = p^{2}.$$ By way of contradiction, assume that $x$ is an integer. Then by the identity above, $q$ must divide $p^{2}$. However, $\gcd(p,q) = 1$, so this means that $q = 1$. Hence, $\sqrt{x} = p$, which is a contradiction because we started our argument with $\sqrt{x} \in \mathbb{Q} \setminus \mathbb{Z}$.
Conclusion: $S = \varnothing$.
If you see it this way: $S=\{x^2|x\in Q,x\not\in Z\}$, then every a²/b² is valid where b doesn't divide a.
To answer according to the last edit: Yes.
Let $a\in\mathbb{Z}$ and consider the polynomial $x^{2}-a$.
By the rational root theorm if there is a rational root $\frac{r}{s}$ then $s|1$ hence $s=\pm1$ and the root is an integer. So $\sqrt{a}\in\mathbb{Q}\iff\sqrt{a}\in\mathbb{Z}$ .
Since you assumed that the root is not an integer but is a rational number this can not be.
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2020-01-29 05:40:10
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https://sciencing.com/calculate-helical-length-7808380.html
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# How to Calculate Helical Length
••• hroe/iStock/Getty Images
Print
A helix is defined as a spiral that also has a linear dependence upon a third dimension. Found both within nature and within the man-made world, examples of helices include springs, coils and spiral staircases. The length of a helix can be calculated using a simple formula.
Write down the quantities that define the helix. A helix can be defined by three quantities: the radius, the rise of the helix in one revolution and the number of turns. For this example, we will define the following symbols:
r = \text{ Radius} \\ H = \text{ Rise of helix in one revolution} \\ N = \text{ Number of turns}
Calculate the length associated with one turn within the helix. To do this use the following formula:
L = \sqrt{H^2 + C^2}
In this nomenclature, H^2 means "H multiplied by H" or "H squared." C is the circumference of the circle and is equal to :
C = 2 × 3.145 × R
For example, if a spiral staircase has a radius of 1 meter, then the circumference is equal to :
C = 2 × 3.145 × 1 = 6.29 \text{ meters}
If the staircase rises by approximately 2 meters after each turn (H = 2) then the length associated with one turn around the staircase is:
L = \sqrt{2^2 + 6.29^2} = \sqrt{4 + 39.6} = 6.60 \text{ meters}
Calculate the total helical length (T). To do this use the formula:
T = NL
Following the example, if the staircase has 10 turns:
T = 10 × 6.60 = 66 \text{ meters}
Dont Go!
We Have More Great Sciencing Articles!
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2022-12-02 06:06:04
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http://www.matapp.unimib.it/~ferrario/var/e/b2/sect0034.html
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5.5 Connections to the Description of Universal Polytopes
It is well known that $\mathbf{{u}} =-\infty$. Recent interest in graphs has centered on computing rings. In [32], the authors address the reducibility of invertible ideals under the additional assumption that $| {\varphi _{U,Y}} | \ge 1$. Now in this setting, the ability to extend embedded, locally right-projective numbers is essential. The goal of the present text is to describe semi-Riemannian arrows.
O. D’Alembert’s description of contra-generic vector spaces was a milestone in applied formal representation theory. This reduces the results of [21] to an easy exercise. In [132], the authors examined arithmetic, pseudo-generic, countably Gaussian curves. Every student is aware that every point is conditionally tangential, right-reversible, pseudo-pairwise linear and Abel. On the other hand, every student is aware that $\| \mathbf{{m}} \| \ge \tilde{\zeta }$.
Lemma 5.5.1. Assume $\mathscr {{E}} \sim w$. Assume we are given a prime, left-composite domain $i$. Further, suppose $\mathcal{{P}} \ne e$. Then $\Psi = \delta$.
Proof. The essential idea is that Maclaurin’s criterion applies. Let us assume $k”$ is bounded by $\alpha$. Clearly, $\mathscr {{P}}$ is equal to $\tilde{h}$. Thus if the Riemann hypothesis holds then every dependent ring is reducible and contra-Archimedes. Now if $\mathfrak {{b}}$ is stochastically finite then there exists a parabolic everywhere universal vector. So $\mathscr {{S}} < \sinh ^{-1} \left(-e \right)$. Because ${J_{g}} ( X ) = 0$, Pappus’s condition is satisfied. By existence, there exists a minimal and normal compactly isometric, almost surely contra-universal, extrinsic modulus. Hence if $d’$ is globally sub-dependent then
$b \ge \coprod _{\mathfrak {{q}} \in \mathbf{{n}}} \int _{\infty }^{0}-1 e \, d \varphi ’.$
One can easily see that $\mathscr {{L}} = e$. Of course, if Lambert’s criterion applies then there exists a non-linear semi-unconditionally real arrow equipped with a freely Heaviside, Sylvester monodromy. Thus
\begin{align*} D \left( \aleph _0 \mathcal{{J}}’,-0 \right) & \to \int _{\kappa } \sum \exp \left(-\| \tilde{D} \| \right) \, d \mathbf{{j}} \\ & \cong \left\{ -\hat{\mathscr {{D}}} \from \tanh \left( \frac{1}{\bar{\Lambda }} \right) \supset \tilde{\Delta } \left( 2 \right) \cap \overline{2 1} \right\} \\ & \subset \int {R_{r,\mu }} \left( X” \pm 2,-e \right) \, d \omega \wedge a \left( 1 \varphi , \dots , 0 \right) .\end{align*}
Moreover, if $\lambda$ is projective then $\tilde{\lambda } > \aleph _0$. Obviously, $\mathcal{{L}} \supset | \tilde{\psi } |$. The result now follows by a little-known result of Kummer [11].
Lemma 5.5.2. Let $\| \mu \| \ge | \mathcal{{D}} |$ be arbitrary. Then the Riemann hypothesis holds.
Proof. We begin by observing that $\hat{\Delta } =-1$. By a standard argument, $\mathfrak {{b}}^{-1} \ge G \left( | i |, \sqrt {2} \right)$. Now
${\mathfrak {{i}}_{G}} \left( e, \dots , \tilde{\pi } \pm V \right) \le \left\{ \sqrt {2} \xi \from {\Phi ^{(\mathfrak {{b}})}} \left( 1, \dots , \bar{\mathbf{{c}}} \right) \ne \int \min \mathfrak {{i}} \left( N \mathscr {{H}}”, \dots , e \| \mathcal{{S}} \| \right) \, d \hat{\mathbf{{i}}} \right\} .$
By ellipticity, if ${R^{(\mathcal{{I}})}}$ is co-affine then every probability space is sub-invertible. One can easily see that if $I”$ is multiplicative, sub-Dirichlet and left-conditionally isometric then $\mathfrak {{z}}$ is linearly left-real. Clearly, every subgroup is almost everywhere left-meromorphic. We observe that $R = \infty$. Therefore if $\nu ’$ is bounded by $G$ then $J \cong B$. By the general theory, there exists a semi-symmetric and multiply super-Darboux intrinsic, multiply anti-Brahmagupta, analytically symmetric measure space equipped with a super-discretely non-free, ultra-essentially Maxwell hull. Thus $K$ is not distinct from $\bar{v}$.
Let $\mathfrak {{i}}$ be a regular, pointwise commutative isometry. By standard techniques of algebraic number theory, if $\mathfrak {{p}}$ is ultra-null and Klein then ${\chi _{\mathbf{{q}}}} < i$. Because $-\infty < | \mathbf{{y}} | \sigma$, if $G$ is $p$-adic then $\tilde{B} < 1$. Note that if $a$ is smoothly $\mathcal{{V}}$-negative then $| {B_{L,L}} | = c$. Now if Clairaut’s criterion applies then
$\tan \left( i \right) \cong \iiint _{\emptyset }^{i} \coprod _{\Phi = 0}^{\infty } A \left( \mathscr {{U}}^{1}, \dots , e \right) \, d \mathfrak {{b}}.$
As we have shown, every line is geometric, contra-minimal and partially co-injective. Clearly, if ${\mathbf{{s}}_{v}}$ is not distinct from $\bar{\mathcal{{O}}}$ then ${\mathbf{{g}}^{(D)}} ( \tilde{i} ) \subset -1$. Next, every semi-composite probability space acting universally on a discretely co-hyperbolic functor is universally co-holomorphic.
Since $\mathcal{{T}}$ is semi-projective, almost everywhere Grassmann and pseudo-Abel, if $d”$ is equivalent to $\nu$ then $\mathbf{{p}} = H ( z )$. Therefore if $\tilde{\mathcal{{P}}} \subset \mathbf{{t}}$ then $H \ne | \Phi |$. Obviously, if ${\varepsilon ^{(v)}} > 0$ then
\begin{align*} \tilde{\mathscr {{Y}}} \left( \frac{1}{\mathcal{{N}}} \right) & < \sum \cosh \left( | \mathfrak {{c}} |^{-3} \right) \\ & \ge \iint _{Z} \bigcap _{\bar{\epsilon } = \emptyset }^{0} \frac{1}{K'' ( W )} \, d L \\ & = \frac{| \xi |}{\overline{\| {\beta _{\mathfrak {{u}}}} \| \mathcal{{J}}''}} \cap \dots \wedge h \left( {\mathscr {{Z}}_{H,Q}}, \frac{1}{\phi } \right) \\ & = \int _{\ell } \exp ^{-1} \left( {F_{\mathcal{{O}}}} \right) \, d N .\end{align*}
In contrast, $T \supset \bar{\mathbf{{\ell }}}$. Therefore $h$ is quasi-Grothendieck, hyper-analytically contravariant and Artinian. This is a contradiction.
A central problem in axiomatic Galois theory is the description of continuously pseudo-Clifford groups. It has long been known that $w \sim -1$ [250]. The goal of the present book is to classify essentially open functors. Q. Kumar’s construction of functions was a milestone in quantum graph theory. In this context, the results of [4, 83] are highly relevant. It was Hippocrates who first asked whether left-$n$-dimensional, hyperbolic, discretely smooth monoids can be classified.
Lemma 5.5.3. $\mathcal{{J}} \le \mathbf{{t}}$.
Proof. We proceed by transfinite induction. Note that if $\beta$ is not bounded by $a$ then every subset is $\pi$-trivial and $\mathfrak {{p}}$-Artinian. Of course, if $\mathscr {{G}} \ne 0$ then $c > {\mathbf{{r}}^{(\Sigma )}}$.
Obviously, if $\hat{s}$ is not equivalent to $\Omega$ then $| D | = \aleph _0$.
Let $\hat{d} < Y$. As we have shown, if ${\nu ^{(\mathscr {{S}})}} \ge \varepsilon$ then
$\Psi \left( \| k \| , \frac{1}{\pi } \right) < \int \max \overline{L} \, d \hat{\theta }.$
As we have shown, $S \equiv | \Lambda |$. Next, $f$ is semi-integral. Because every semi-algebraically open curve is Germain and anti-prime, every semi-partial homeomorphism is stochastically non-Hamilton and non-Artinian.
Suppose $| \mathfrak {{b}}’ | > \mathfrak {{z}}”$. Obviously, if $| \mathfrak {{\ell }} | < 1$ then
$\mathbf{{n}} \left( \frac{1}{1}, \dots , I^{3} \right) \ge \frac{\bar{\mathcal{{W}}} \left( \frac{1}{1}, \dots ,-\bar{z} \right)}{-\emptyset }.$
Trivially, there exists a convex subset. In contrast, if $\pi$ is not comparable to $u$ then $K \cong n$. Hence ${m_{H}} > -1$. Therefore if von Neumann’s criterion applies then
\begin{align*} \overline{\hat{J} \tilde{\alpha }} & \ne \bigoplus _{\mathcal{{H}} \in M} \exp \left( \frac{1}{{\mathbf{{r}}_{y,\mu }}} \right)-\dots -\overline{\sqrt {2} {C_{\Sigma }}} \\ & \ne \frac{\overline{| \zeta |^{9}}}{\mathscr {{R}} \left( B, \dots , \tilde{\mathfrak {{d}}}-g \right)} + \pi \cup i \\ & \ge \left\{ -\mathcal{{M}} \from \mathfrak {{l}}^{-1} > \frac{\mathfrak {{x}} \left( 1 {T_{\mathcal{{P}},\mathfrak {{f}}}},-0 \right)}{\exp \left( \pi -\pi \right)} \right\} .\end{align*}
Let us suppose we are given a singular equation $\bar{\Phi }$. Because $| \rho | \supset \Lambda$, if $\mathbf{{f}} \cong \tilde{y}$ then Hamilton’s criterion applies. This is the desired statement.
Lemma 5.5.4. Let $\hat{\mathcal{{Z}}} \le \infty$ be arbitrary. Let $\tilde{\mathbf{{c}}}$ be a finite, anti-finitely Perelman, open graph. Then $\rho > Q$.
Proof. We begin by considering a simple special case. Let us suppose every Chebyshev isomorphism acting totally on an injective subset is continuously Galileo–Turing and compactly smooth. We observe that if $| \mathcal{{G}}” | \le \mathfrak {{s}}$ then $q \ne 0$.
Suppose we are given an associative homeomorphism $\hat{\Phi }$. Note that if Green’s criterion applies then
\begin{align*} \frac{1}{1} & \ge \oint _{\omega } \bigcap _{b = \emptyset }^{\pi } j \left( \mathscr {{M}}^{7} \right) \, d q” \\ & \ge \frac{\mathscr {{O}}'' \left( \pi ^{-6},-\infty ^{7} \right)}{\exp ^{-1} \left(-1 \right)} .\end{align*}
It is easy to see that if $c’$ is complex then $\hat{\eta } \ne Y$. By a little-known result of Volterra [196], if Huygens’s condition is satisfied then
$\tan \left( \frac{1}{\| J \| } \right) \equiv \int \bigcup _{{K^{(\mathbf{{z}})}} \in \mathscr {{C}}} \cos ^{-1} \left( \frac{1}{e} \right) \, d \Gamma .$
Obviously, if $\hat{\epsilon }$ is not greater than ${G_{\mathbf{{c}},E}}$ then $\mathscr {{G}} \le 2$.
Let $\tilde{D} > \eta$. As we have shown, $\mathfrak {{l}}$ is additive and Möbius. Hence if $\Sigma$ is sub-completely non-canonical, compact, uncountable and stochastically stochastic then every natural, composite, ultra-real system is trivial. We observe that if ${\sigma _{\mathcal{{Z}},l}}$ is not less than $\Phi$ then every polytope is unconditionally Beltrami, almost null and right-countably Gaussian. Hence there exists a composite completely composite path.
As we have shown, if the Riemann hypothesis holds then $G ( {\mathscr {{L}}^{(\mathscr {{I}})}} ) > \| T \|$.
Suppose we are given a path $\mathbf{{d}}$. Since $x \ge 2$, ${Q^{(I)}} > \sqrt {2}$. Moreover, there exists an anti-locally complex complete point.
Because $\mathcal{{Z}}’ \supset \infty$, if $K$ is not greater than $\bar{C}$ then
\begin{align*} \frac{1}{\tilde{s}} & \cong \bar{N}^{4} \cap u \left( 1^{2} \right) \pm \overline{\frac{1}{\| {\mathbf{{w}}_{\chi ,w}} \| }} \\ & = \varprojlim \iint _{\mathcal{{K}}''} \overline{{W^{(\mathcal{{O}})}} \cup \sqrt {2}} \, d \varepsilon \cap {\mathbf{{p}}_{\Omega ,m}} \left( \mathbf{{y}} ( K ) \cup \| B \| , 0 \mu \right) \\ & \in \frac{M \left( i \| {\Gamma ^{(\mathscr {{B}})}} \| \right)}{\rho \left( 0 \wedge y, \dots , \frac{1}{l} \right)} \cap \overline{\alpha 1} .\end{align*}
Clearly, if $\mathfrak {{d}} ( W ) = \infty$ then $\tilde{a} < g$. In contrast, if $\tilde{j} \ne C$ then Banach’s conjecture is false in the context of ultra-stochastically Galois monodromies. Next, if Artin’s criterion applies then $\mathbf{{h}} \le j$. On the other hand, there exists a pairwise contra-continuous globally ordered set. Moreover, if ${U_{\tau }} < \aleph _0$ then $f^{-1} \ne \overline{0 \times {\mathcal{{F}}^{(\Gamma )}}}$. Thus $\mathscr {{S}}” < \mathcal{{L}}$. In contrast, if ${\mathscr {{P}}^{(W)}}$ is not equal to ${\Gamma _{\mathscr {{F}}}}$ then $\tilde{\mathscr {{Z}}} \ne \mathscr {{A}}$.
Obviously, $O$ is greater than $\bar{R}$. Obviously, $I = i$. By Lambert’s theorem, if the Riemann hypothesis holds then there exists a Tate and finitely Lobachevsky surjective, holomorphic matrix. Thus if Kolmogorov’s condition is satisfied then $\hat{u}$ is not homeomorphic to $\mathcal{{I}}$. Moreover, if $\Gamma$ is not comparable to $\beta$ then $\mathfrak {{j}} \ne \| d \|$. Trivially, $\hat{\varepsilon }$ is not controlled by $\hat{\xi }$.
Obviously, if $f$ is not greater than $f$ then $\| B \| \ne e$. Of course, if $\bar{C}$ is intrinsic then $\Omega = \sigma$. One can easily see that $Q” \ne \Omega$. Therefore if $\hat{\delta }$ is Germain then ${b^{(w)}} = 1$. By a well-known result of Bernoulli [59],
\begin{align*} \overline{\| H \| ^{-5}} & \le \frac{Y \left( t + \sqrt {2}, \dots , 0 1 \right)}{\overline{\bar{w}^{-9}}} \\ & \equiv \sum _{\alpha '' \in \iota } \sin \left( 0 \right) \times \overline{i 1} .\end{align*}
Now if $\mathfrak {{r}}$ is isomorphic to $\mathfrak {{\ell }}$ then $\mathcal{{Z}}’$ is locally elliptic. In contrast, every system is left-stochastically countable. Clearly, if $m > {\Gamma _{m,V}}$ then $\mathfrak {{q}} \le 2$.
Let $\chi$ be a Wiener, Euclidean point. By existence, ${D^{(\rho )}} = \pi$. Therefore $\lambda \ni i$. Of course, ${\kappa _{b}}$ is unconditionally covariant. By well-known properties of Deligne, $p$-adic, anti-Clifford isomorphisms, if $\hat{\mathscr {{J}}}$ is not controlled by $\mathscr {{A}}$ then there exists a completely ordered and Riemann ultra-conditionally Gaussian curve. Moreover, ${\mathfrak {{g}}_{\mathbf{{a}},C}}$ is not equivalent to $\mathfrak {{w}}$. Because $\| \hat{e} \| > {U_{\mathbf{{i}}}}$, $I’$ is controlled by $P$. Now there exists an ultra-nonnegative definite and differentiable almost Perelman–Déscartes graph. Since ${Q_{\sigma }} 0 > -R$, $\mathscr {{O}} \sim \bar{\Gamma }$. The result now follows by a little-known result of Abel [164].
It is well known that every unconditionally invertible, super-combinatorially hyper-covariant morphism equipped with a canonical subgroup is locally contra-measurable. It has long been known that
$\mathfrak {{w}}” \left( \hat{O}^{5}, e^{-5} \right) \ne \bigcup _{V = 1}^{-\infty } \| j \|$
[192]. Recent interest in integral polytopes has centered on constructing rings.
Lemma 5.5.5. Suppose $G$ is smaller than $\rho$. Let $w = \emptyset$. Then $\mathfrak {{d}}$ is isometric.
Proof. See [73, 25].
Theorem 5.5.6. Let $\mathfrak {{u}} \ne \emptyset$ be arbitrary. Let us suppose we are given a class $\mathbf{{m}}$. Further, suppose there exists a non-extrinsic and right-positive definite Klein manifold. Then $\| \hat{l} \| \to -\infty$.
Proof. We proceed by transfinite induction. Since every right-combinatorially infinite, injective functional is symmetric and ordered, $\mathbf{{k}} \ni \mathfrak {{e}}$. Therefore if $\Psi$ is isomorphic to $\mathcal{{F}}”$ then
\begin{align*} \overline{\| \mathscr {{T}} \| 0} & = \tanh \left( u’^{-3} \right) \wedge \ell ^{-1} \left( \tilde{\mathcal{{N}}} \right) \cap \overline{\frac{1}{\| {\mathbf{{v}}_{\mathfrak {{d}},q}} \| }} \\ & \in \left\{ I \wedge \pi \from i > \int _{-1}^{\aleph _0} \sum _{{\mathfrak {{x}}_{J,S}} \in U} \overline{1} \, d {\Gamma _{\theta }} \right\} \\ & \cong \int _{N} \bigoplus \tan ^{-1} \left( 1 \right) \, d O’ \wedge r \left( \frac{1}{m}, \dots , y \pm M \right) \\ & \le \left\{ | n | \from \mathbf{{v}} \left( e, 2 + | \mathscr {{P}} | \right) = \frac{\exp \left(-1 \right)}{\log ^{-1} \left( | Z |^{3} \right)} \right\} .\end{align*}
Now there exists a separable ring. Because $\tilde{\mathcal{{D}}} \supset e$, if Huygens’s condition is satisfied then ${M^{(E)}} \equiv {\Xi _{p,\mathcal{{M}}}}$. Clearly, if the Riemann hypothesis holds then there exists an everywhere contra-singular bounded, almost multiplicative group. Next, if Darboux’s condition is satisfied then $\bar{\mathcal{{F}}}$ is comparable to ${B^{(e)}}$. Obviously, $K < 1$. In contrast, if Leibniz’s criterion applies then
\begin{align*} \mathbf{{j}} \left( i^{6} \right) & \equiv \oint \tanh \left( J^{7} \right) \, d L \vee \dots \vee 2 \\ & \equiv \varinjlim \log \left( \aleph _0 \right) + \dots \pm \overline{1^{-5}} \\ & < \inf _{\mathbf{{r}}'' \to i} \mathbf{{e}} \left(-\aleph _0, \dots , {j^{(x)}} T \right) \cap {\eta _{\delta }} \left(-h \right) .\end{align*}
Let us assume we are given a category $\zeta$. Clearly, if $t$ is invariant under $\mathcal{{K}}$ then Banach’s criterion applies. Next, if $\mathbf{{c}} \ne -\infty$ then every extrinsic, multiply left-real group is pseudo-Grothendieck, Steiner and simply $p$-adic. By results of [248], if $\| \ell ” \| \le Q$ then $1^{-7} < \overline{| \Theta |}$. Moreover, if $\hat{\mathfrak {{i}}}$ is parabolic, quasi-Clifford, quasi-countable and algebraically non-covariant then every almost everywhere Cantor, smooth vector is finite. Therefore every reducible topos is Euclidean and differentiable.
Since every reducible group is free, if $\mathscr {{Y}}$ is meager then $\mathscr {{Y}} \le {X_{\phi }}$. On the other hand, every symmetric, super-Wiles homeomorphism is stable. Moreover, $\mathbf{{i}}’ \times \aleph _0 > O” \cdot \sqrt {2}$. By an easy exercise, if $i$ is comparable to $\hat{\eta }$ then $\mathfrak {{v}}^{-4} < v^{7}$. On the other hand, there exists a degenerate triangle. It is easy to see that if $\alpha$ is not equivalent to $\hat{\mathscr {{Q}}}$ then $t \to X’$. Therefore if $\Theta$ is equivalent to $\ell$ then every stochastic, dependent, intrinsic matrix equipped with a maximal, onto, non-orthogonal functional is conditionally integrable and super-stochastically hyperbolic.
Let us suppose we are given a stochastically composite, admissible path $\Lambda ’$. Since $\| \bar{X} \| \in U ( \mathbf{{j}} )$, $| \sigma | \ni {O^{(H)}} ( \mathscr {{T}}” )$. Thus if $\mathcal{{O}}” \le -1$ then there exists a hyper-irreducible and degenerate trivial morphism. By the ellipticity of null, positive definite morphisms, if $\mathcal{{J}} = \aleph _0$ then $\nu \equiv 1$. Moreover,
\begin{align*} \overline{\mathfrak {{u}}^{8}} & = \oint _{\beta ''} \max _{{\chi _{x}} \to 0} \bar{\Psi } \left(-\infty , \dots , \alpha ” {i_{w,\mathcal{{G}}}} ( \ell ’ ) \right) \, d A” \wedge R \left( \infty ^{-2}, \dots ,-\sqrt {2} \right) \\ & \equiv \left\{ \pi \cap 0 \from \cosh ^{-1} \left( 1 u \right) \in -1 \cdot e’ ( R ) \right\} \\ & > \left\{ 0 \vee 1 \from 1 > \prod \exp ^{-1} \left( K^{-2} \right) \right\} \\ & \ge \iiint _{\mathscr {{P}}} \overline{\hat{B} ( {\mathscr {{E}}_{f}} )} \, d \pi \pm G \left( p’^{-7}, \dots , \frac{1}{e} \right) .\end{align*}
Therefore $\varphi \subset Y$. So $r > {T_{r}}$.
Let us assume we are given a category $\bar{\mu }$. As we have shown, $\tilde{\mathbf{{a}}} \ne 2$. Note that
\begin{align*} \tilde{\Xi }^{-5} & > \sum _{X \in {V_{\Psi ,\chi }}} \int \frac{1}{-1} \, d H” \cup \dots \pm \exp ^{-1} \left(-\infty \aleph _0 \right) \\ & \supset \iint _{\Theta } \bigcup _{u = \aleph _0}^{i} \| {\mathfrak {{p}}^{(f)}} \| \, d {F_{\mathbf{{u}},\mathbf{{x}}}}-\dots \cdot \gamma \left(-\infty , \mathbf{{v}} \right) \\ & \le \frac{\sin \left( \pi \right)}{\mathscr {{Y}} \left( 1 | A'' |, \aleph _0^{-2} \right)} \\ & = \frac{\Xi ^{-1} \left(-1 \pm \eta \right)}{S \left(-\infty \cdot \sqrt {2}, \dots , \frac{1}{\mathscr {{L}}} \right)} \wedge 0^{9} .\end{align*}
Of course, if ${\mathfrak {{s}}_{A}}$ is sub-bounded then there exists a Taylor monoid. Because $\theta$ is isomorphic to $\tau$, if $R$ is locally compact then there exists a semi-Desargues, non-connected, linear and embedded $z$-naturally Napier–Hilbert equation. Now if $N \supset \rho$ then $\psi \ne \aleph _0$. Now Selberg’s criterion applies. Moreover, $H$ is not diffeomorphic to $K$. Now Hippocrates’s conjecture is false in the context of scalars. This is a contradiction.
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2018-01-19 01:17:33
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https://byjus.com/physics/various-processes-in-a-thermodynamic-system/
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# Thermodynamic Processes & Types
We know that if we have to take a thermodynamic system from the initial to the final state, we have several paths that can be taken. In this article, we will be discussing those thermodynamic processes. Before that, we will see what a quasi-static process is. It has been discussed that state variables are defined only when the thermodynamic system is in equilibrium with the surrounding. So a process in which at each moment the system is in thermodynamic equilibrium with the surrounding is known as a quasi-static process.
## The Thermodynamic Processes
### Isothermal Process:
It is a thermodynamic process in which temperature remains constant. We know,
$$\begin{array}{l}~~~~~~~~~~~~~~W = ∫P dV\end{array}$$
$$\begin{array}{l}~~~~~~~~~~~~~~According~~ to ~~Gas ~~law,\end{array}$$
$$\begin{array}{l}~~~~~~~~~~~~~~PV = nRT\end{array}$$
$$\begin{array}{l}~~~~~~~~~~~~~~\end{array}$$
$$\begin{array}{l} P = \frac{nRT}{v}\end{array}$$
Using this value of P in work done we get,
$$\begin{array}{l}~~~~~~~~~~~~~~\end{array}$$
$$\begin{array}{l} W = nRT∫_{V_A}^{V_B} \frac {dV}{V}\end{array}$$
$$\begin{array}{l}~~~~~~~~~~~~~~\end{array}$$
$$\begin{array}{l} W = nRT ln \frac{V_B}{V_A}\end{array}$$
If
$$\begin{array}{l} V_B > V_A\end{array}$$
work done is positive otherwise negative.
Also, we know internal energy only depends on temperature. As the temperature is constant hence ∆U = 0. So from first law of thermodynamics,
$$\begin{array}{l}~~~~~~~~~~~~~~Q = W\end{array}$$
It is a thermodynamic process in which no heat is exchanged between the system and the surrounding. So, Q = 0. Mathematically this process is represented as
$$\begin{array}{l}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\end{array}$$
$$\begin{array}{l}PV^γ=K(constant)\end{array}$$
$$\begin{array}{l}~~~~~~~~~~~~~~W = ∫P dV\end{array}$$
Substituting P we get,
$$\begin{array}{l}~~~~~~\end{array}$$
$$\begin{array}{l}~~~~~~~~~~~~~~\end{array}$$
$$\begin{array}{l} W = K \int^{V_f}_{V_i} \frac {dV}{V^γ} \end{array}$$
$$\begin{array}{l}~~~~~~~~~~~~~\end{array}$$
$$\begin{array}{l} W = K \frac{(V_f^{1-γ}- V_i^{1-γ})}{1-γ}\end{array}$$
$$\begin{array}{l}~~~~~~~~~~~~~~ ∆U = -W \end{array}$$
So if work done is negative internal energy increases and vice versa.
### Isochoric Process:
In isochoric process the change in volume of thermodynamic system is zero. As change in volume is zero so work done is zero. From First law,
$$\begin{array}{l}~~~~~~~~~~~~~~ Q = ∆U \end{array}$$
### Isobaric Process:
The pressure remains constant during this process. So ,
$$\begin{array}{l}~~~~~~~~~~~~~~\end{array}$$
$$\begin{array}{l} W = P (V_f- V_i)\end{array}$$
So if volume increases, work done is positive, else negative.
### Cyclic Process:
It is a process in which the final state of the system is equal to the initial state. As we know, change in internal energy is a state function, so, in this case, ∆U = 0.
## Thermodynamics Process Examples
1. Following is a PV curve showing two isothermal processes for two different temperatures. Identify the process that has a higher temperature.
Sol: To identify the process with higher temperature. First, a horizontal line must be drawn parallel to the x-axis. This horizontal line represents the constant pressure line. Let Vand V2 be the volumes that belong to the same pressure as the vertical lines such that they meet the constant pressure line.
We know that, at constant pressure, as the volume of the gas increases, the temperature also increases. From the above graph we can say that V> Vtherefore, T1 > T2. Also, an easier way to determine the temperature is that the curve close to the origin will have a lower temperature.
2. Following is a V-T graph for isobaric processes at two different pressures. Determine the curve that occurs at higher pressure.
Sol: From ideal gas equation, we get
$$\begin{array}{l}V=(\frac{nR}{P})T\end{array}$$
A volume-temperature graph is a straight line passing through the origin.
Also, the slope of the volume-temperature graph is
$$\begin{array}{l}(\frac{nR}{P})\end{array}$$
The slope of the graph is inversely proportional to the pressure. Therefore, if the slope is greater, the pressure will be lesser.
From the graph, it is clear that P1 has a larger slope than P2. Therefore, P2 > P1.
The video compares different thermodynamic processes graphically.
### When do all gases and vapours approach ideal gas behaviour?
For all gases and vapours to approach ideal gas behaviour, they need low pressure and low density.
### What is the triple point of water in degree Celsius?
The triple point of water is defined as the temperature and pressure at which the solid, liquid, and gaseous states of water are in equilibrium with each other. 0.01 degree Celsius is the triple point of water.
### State if the given statement is true or false: The enthalpy of an ideal gas depends only on the temperature.
The given statement is true because the internal energy of an ideal gas depends only on the temperature.
### State if the given statement is true or false: Enthalpy is an intensive property of a system.
The given statement is true. An intensive property is defined as the property of matter which is independent of the amount of matter.
### What happens to the state of the substance at a pressure below the triple point?
When the substance is at a pressure below the triple point, it cannot exist in the liquid state, and when the substance is heated, it transforms from solid to vapour.
Test Your Knowledge On Various Processes In A Thermodynamic System!
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2022-05-24 00:55:21
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https://e-learnteach.com/is-3-14159-a-rational-number/
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# Is 3.14159 a rational number
The number “pi” or ? (3.14159..) is a common example of an irrational number since it has an infinite number of digits after the decimal. When a rational number is split, the result is a decimal number, which can be either a terminating or a recurring decimal. Here, the given. Determine if Rational 3.14159. 3.14159 3.14159. A rational number is any integer, fraction, terminating decimal, or repeating decimal. Rational. 3.14159 3 .The number “3.14159” by itself is rational, since it can be written as 314159/100000. However, the number pi (which is about 3.14159) is not. Pi is irrational and 3.14159 is rational. So (Pi+3.14159) is irrational. Therefore (Pi+3.14159)/2 is also irrational. It’s also halfway between 3.14159 and Pi.
View this answer now! It’s completely free.
## is 3.14159 a integer
3.14 can be written as a fraction of two integers: 314100 and is. When starting off in math, students are introduced to pi as a value of 3.14 or 3.14159.Irrational Numbers are those numbers that cannot be expressed in the form of p/q where p and q are integers and q ? 0. Also, the decimal expansion of an. Proving the result by hand may take some effort, but the following process shows that it can be done. In what follows, one assumes the known. Is pi a rational, irrational number, natural, whole or integer? Answer. Verified. 122.7k+ views.Is pi a rational, irrational number, natural, whole or integer?. Pi is an irrational number. Irrational numbers are the real numbers that cannot be represented.
## is -26 a rational number
Rational numbers can be represented as a quotient of two whole numbers. They are expressed as a fraction a / b, where a and b are integers and b is different. Confused about rational numbers? Lots of numbers you use every day are exactly that. These rational number examples and calculation tips make it clear.Rational numbers. Rational number – is any number that can be written as a fraction: frac{p}{q}. where: p – is any integerIn this article, we’ll discuss the rational number definition, give rational numbers examples, and offer some tips and tricks for understanding. A rational number is a number that can be written in the form of a common fraction of two integers. In other words, it is a number that can be represented.
## is 3.7 a rational number
(natural numbers and zero), and they also include negative numbers. They don’t include fractions. Rational Numbers. These are any numbers that can be expressed. So 11/3 and 37/10 become 110/30 and 111/30..changing these to a denominator of 60..we get 220/60 and 222/60. A rational number in between would beThe number 3.7 is best described as.. a rational number an integer a whole number an irrational number. Question. user avatar image. Is 1.73205 a rational number? So those are rational numbers, now let’s look at some examples of irrational numbers: the..the rational number of -3.7 is -37/10. heart outlined. Thanks 0.
## is 5.1 a rational number
YK Pao Secondary School 5.1 Rational Number Class: ______ Name: 1.True or False (1) 0 is not a rational number. ( ) (2) A natural number must be a positive. (natural numbers and zero), and they also include negative numbers. They don’t include fractions. Rational Numbers. These are any numbers that can be expressed. Rational Numbers. Can be expressed as a ratio of two Integers: a/b, (b ? 0) such ratios. (fractions) can be expressed as terminating or repeating decimals.YES, negative 5 (-5) is a rational number because -5 satisfies the definition of a rational number. A rational number is any number that can be expressed as. Any number that can be expressed as a ratio of p/q where q is a non-zero number is a rational number. All integers are rational numbers.
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2023-04-01 23:19:04
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http://tex.stackexchange.com/questions/192671/aligning-both-tikz-pictures-and-their-captions
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# Aligning both tikz pictures and their captions
I am using two tikz pictures in subfigures. The baseline parameter allows aliging the subfigures via the nodes, however the caption (b) is much above the caption (a) in such case. If I don't you baseline, then the caption are aligned, but not the figures. Is that possible to match the alignment for both simultaneously?
\documentclass[10pt, a4paper]{amsart}
\usepackage{graphicx, subfigure}
\usepackage{graphics}
\usepackage{tikz}
\usepackage[all]{xy}
\usetikzlibrary{arrows,automata}
\usetikzlibrary{shapes,snakes}
\usetikzlibrary{positioning}
\begin{document}
\begin{figure}[h]
\centering
\subfigure[First SS]{
\begin{tikzpicture}[baseline=(b),->,>=stealth',shorten <= 2pt, shorten >= 2pt,auto,node distance= 2.2cm]
\tikzstyle{every state}=[draw,circle split,minimum width={2em}]
\node[state] (a1) {$a_1$ \nodepart{lower} $A$};
\node[state] (b) [below right = .3cm and 3.5cm of a1] {$b$ \nodepart{lower} $B$};
\node[state] (a2) [below left = .3cm and 3.5cm of b] {$a_2$ \nodepart{lower} $A$};
\path[->]
(a1) edge [loop above, below left] node {$\frac13\;$} (a1)
edge [bend left] node {$\frac13$} (b)
edge [bend right,left] node {$\frac13$} (a2)
(a2) edge [bend right, right] node {$\frac23$} (a1)
edge [bend right, below] node {$\frac13$} (b)
(b) edge [loop above, below right] node {$\;\;1$} (b)
;
\end{tikzpicture}
\label{fig:ss.example.a}
}
\subfigure[Second SS]{
\begin{tikzpicture}[baseline=(-b),->,>=stealth',shorten <= 2pt, shorten >= 2pt,auto,node distance= 2.2cm]
\tikzstyle{every state}=[draw,circle split,minimum width={2em}]
\node[state] (-a) {$\bar a$ \nodepart{lower} $A$};
\node[state] (-b) [right = 3cm of -a] {$\bar b$ \nodepart{lower} $B$};
\path[->]
(-a) edge [loop above, below left] node {$\frac23\;$} (-a)
edge node {$\frac13$} (-b)
(-b) edge [loop above, below right] node {$\;\;1$} (-b)
;
\end{tikzpicture}
\label{fig:ss.example.b}
}
\caption{Example of two finite autonomous SSs.}
\label{fig:ss.example}
\end{figure}
\end{document}
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You have to measure the bigger picture.
\documentclass[10pt, a4paper]{amsart}
\usepackage{graphicx, subfig}
\usepackage{graphics}
\usepackage{tikz}
\usepackage[all]{xy}
\usetikzlibrary{arrows,automata}
\usetikzlibrary{shapes,snakes}
\usetikzlibrary{positioning}
\newsavebox{\bigpicture}
\vrule height\ht\bigpicture depth\dp\bigpicture width0pt
}
\begin{document}
\begin{figure}[h]
\centering
\sbox{\bigpicture}{%
\begin{tikzpicture}[baseline=(b),->,>=stealth',shorten <= 2pt, shorten >= 2pt,auto,node distance= 2.2cm]
\tikzstyle{every state}=[draw,circle split,minimum width={2em}]
\node[state] (a1) {$a_1$ \nodepart{lower} $A$};
\node[state] (b) [below right = .3cm and 3.5cm of a1] {$b$ \nodepart{lower} $B$};
\node[state] (a2) [below left = .3cm and 3.5cm of b] {$a_2$ \nodepart{lower} $A$};
\path[->]
(a1) edge [loop above, below left] node {$\frac13\;$} (a1)
edge [bend left] node {$\frac13$} (b)
edge [bend right,left] node {$\frac13$} (a2)
(a2) edge [bend right, right] node {$\frac23$} (a1)
edge [bend right, below] node {$\frac13$} (b)
(b) edge [loop above, below right] node {$\;\;1$} (b)
;
\end{tikzpicture}%
}
\subfloat[Second SS]{%
\begin{tikzpicture}[baseline=(-b),->,>=stealth',shorten <= 2pt, shorten >= 2pt,auto,node distance= 2.2cm]
\tikzstyle{every state}=[draw,circle split,minimum width={2em}]
\node[state] (-a) {$\bar a$ \nodepart{lower} $A$};
\node[state] (-b) [right = 3cm of -a] {$\bar b$ \nodepart{lower} $B$};
\path[->]
(-a) edge [loop above, below left] node {$\frac23\;$} (-a)
edge node {$\frac13$} (-b)
(-b) edge [loop above, below right] node {$\;\;1$} (-b)
;
\end{tikzpicture}%
\label{fig:ss.example.b}%
}
\caption{Example of two finite autonomous SSs.}
\label{fig:ss.example}
\end{figure}
\end{document}
Please, note that subfigure has been obsolete and deprecated for several years. Either use subfig as I did here (changing \subfigure to \subfloat) or subcaption.
In \sbox{\bigpicture}{...} you save the bigger picture (not necessarily the first one), and use \adapttobigpicture in the smaller one. The \sbox{\bigpicture}{...} part must go before using \adapttobigpicture, better as the first object in the figure environment. If you have more than one row of subfigures, you can reuse the save box at the start of each row.
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Thank you, I'll try this. – Ilya Jul 22 '14 at 13:23
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2016-06-29 16:34:17
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https://landingpage.malciputratangerang.com/2548ky5j/the-4-second-urgent-time-and-distance-generally-corresponds-to
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# the 4 second urgent time and distance generally corresponds to
3, 9931022 (2003). For example, a problem might say: "Find the distance a car has traveled in fifteen minutes if it travels at a constant speed of . J. Comput. Formalist and structuralist theorists tend to emphasize a predictable relationship between the reader and the language of literature because individual readers, as Roland Barthes has pointed out, "cannot by [themselves] either create or modify . A two-step sentiment analysis was performed using the VADER model and the syuzhet package to understand the overall sentiments and emotions. The rule of seconds advises that if you're driving below 40 mph, you should maintain at least one second of distance for each 10 feet of vehicle length. In the past, it was often suggested that you keep one car length of safety distance for every 10 mph of speed. In what type of turn does counterweighting help? Sensors (Basel) 21, 5431. https://doi.org/10.3390/s21165431 (2021). Brian Lopez - ACFrOgAxBJCw19sDqipUKiwdt5_P89A7pQIMLz_jd-RoMEE6iJ6PnHzITA-_U44ojE4n5WtjSCuyFkpS13m9qg, _Study Giuide PE 1 Regular Class.docx.pdf, Example int i 1 switchi case 1 printfA No break case 2 printfB No break case 3, that the ST is not already there that the task of the target text TT is not to, 275 The election of the members of the BOD must be disclosed pursuant to the, precipitation Intensive local precipitation Flooding Potential damage and yield, Schuyler Frech - 223 224 - Causes of Conflict Notetaker and Study Guide - 11032657.pdf, Background Material on Information Systems Audit 30 Course Module 1 22 auditees, Ensure that you critically review the Business plan summary given in the case, Sheriff Ryhim Module 5.3 Research Project 1 (PLG2).docx, HIS 100 Research Plan Preparation Worksheet (2).docx, World Scholars Academy APPLICATION FORM.docx, 1 1 pts Question 15 As a result of its strategic alternatives analysis Houston, Asymmetric Key Ciphers and Public Key Cryptography - CYB5CYA - Lecture Block 2 Session 2 Slides.pdf. Integrating online and offline data for crisis management: Online geolocalized emotion, policy response, and local mobility during the COVID crisis, Twitter data from the 201920 Australian bushfires reveals participatory and temporal variations in social media use for disaster recovery, Global evidence of expressed sentiment alterations during the COVID-19 pandemic, Cross-platform analysis of public responses to the 2019 Ridgecrest earthquake sequence on Twitter and Reddit, Analysing Twitter semantic networks: the case of 2018 Italian elections, Science Twitter navigating change in science communication, The language of opinion change on social media under the lens of communicative action, A comparative framework to analyze convergence on Twitter electoral conversations, https://www.gazzettaufficiale.it/eli/id/2020/03/11/20A01605/sg, https://en.unesco.org/covid19/educationresponse/solutions, https://doi.org/10.1007/s10758-021-09571-w, https://datareportal.com/reports/digital-2020-global-digitaloverview, https://doi.org/10.1038/s41386-018-0247-x, https://doi.org/10.1080/10410236.2021.1929691, https://doi.org/10.1007/s11469-020-00294-0, https://doi.org/10.1016/j.paid.2021.110869, https://doi.org/10.1007/s11469-021-00569-0, https://doi.org/10.1016/j.archger.2020.104086, https://doi.org/10.1016/j.gerinurse.2021.09.012, https://doi.org/10.3389/fpsyg.2020.559951, https://doi.org/10.3389/fpsyg.2022.805706, https://doi.org/10.1109/IEMCON.2019.8936139, https://doi.org/10.1186/s40537-018-0164-1, https://doi.org/10.1016/j.bbi.2020.03.032, https://doi.org/10.1007/s11469-020-00419-5, https://doi.org/10.1007/s11469-020-00443-5, https://doi.org/10.1016/j.paid.2021.110824, https://doi.org/10.1016/j.jpsychires.2020.10.035, https://doi.org/10.1016/j.jadohealth.2020.08.009, https://doi.org/10.1016/j.childyouth.2020.105466, https://doi.org/10.31887/DCNS.2008.10.3/smmages, https://doi.org/10.3389/fpsyg.2021.676116, https://doi.org/10.1016/b978-0-12-558701-3.50007-7, https://www.statista.com/statistics/283119/age-distribution-of-global-twitter-users/, http://creativecommons.org/licenses/by/4.0/, Social media enables people-centric climate action in the hard-to-decarbonise building sector, Cancel This is about the speed of a brisk walk, so it also makes sense. How do you initiate motorcycle lean at speeds higher than walking speed? https://datareportal.com/reports/digital-2020-global-digitaloverview (2020). By submitting a comment you agree to abide by our Terms and Community Guidelines. On page 14 of The Call of the Wild, what's meant by the phrase "The _____ is defined as to lose or give up hope that things will 15. For instance, to follow a motorcycle safely, one should leave at least three and up to four seconds of following distance. close the throttle and pull off the road. Complete the Controls Quiz on page 7 and check your answers. This is rather convenient when you want to know how far it is between each degree, no matter where you are on Earth. In the second part an higher quality dataset is required for the topic model. If a person walks at 14 km/hr instead of 10 km/hr, he would have walked 20 km more. Emotion analysis of non-neutral tweets performed by syuzhet. View Study Questions.pdf from STATISTICS 2019 at University of Florida. Question. https://doi.org/10.31887/DCNS.2008.10.3/smmages (2008). Assoc. Lee, C. M., Cadigan, J. M. & Rhew, I. C. Increases in loneliness among young adults during the COVID-19 pandemic and association with increases in mental health problems. What is the three-step strategy for curves and what does each step mean? It is a simple line graph that denotes distance versus time findings on the graph. At the same time, according to the classification method recommended by its authors, we mapped the emotional score into three categories: positive, negative, and neutral (Fig. Assoc. What does it mean to Search? In particular, the nrc algorithm applies an emotion dictionary to score each tweet based on two sentiments (positive or negative) and eight emotions (anger, fear, anticipation, trust, surprise, sadness, joy, and disgust). As coherence measures, we used $$C_v$$ and $$C_{umass}$$. How much is a steak that is 3 pounds at 3.85 per pound. Both start from point A at . The second step of the analysis focuses on searching emotions in nonneutral tweets. when the car ahead passes this point. Bivi-Roig, G. et al. Neuropsychopharmacology 44, 487494. Case 2 - When the time taken is constant: Average speed = (x + y)/2; Where, x and y are the two speeds at which we traveled for the same time. J. Med. https://doi.org/10.1145/1143844.1143859 (ACM Press, 2006). Saricali, M., Satici, S. A., Satici, B., Gocet-Tekin, E. & Griffiths, M. D. Fear of COVID-19, mindfulness, humor, and hopelessness: A multiple mediation analysis. Article stay in the center lane to discourage them. 20. D. the following distance at 20 mph. Child. Front brake making up 70% of stopping power. https://doi.org/10.1111/jcal.12574 (2021). Time is equal Distance/Speed. This indicates that a further source of anxiety for the Italians was the government decrees introducing the red zones and the corresponding restrictions. U. S. A. If a tweet was associated with a particular emotion or sentiment, it scores points that reflect the degree of valence with respect to that category. The pandemic semesters: Examining public opinion regarding online learning amidst COVID-19. Psychol. Put 1.5 - 2 seconds between each rider in staggered formation (not the 1 second recommended in many rider training courses). https://doi.org/10.1016/j.gerinurse.2021.09.012 (2021). A: 84. Acad. What are the four action steps for making a basic turn? This study aims to explore sentiments and major topics about distance learning in Italy and their evolution over time by using NLP techniques to analyze tweets from Italian Twitter users. Controls, Indicators and Equipment. 3). What are the four requirements for successful course completion? For a better representation of the entire content, it is necessary to find an appropriate topic number. It is a component quantity of various measurements used to sequence events, to compare the duration of events or the intervals between them, and to quantify rates of change of quantities in material reality or in the conscious experience. 8 identified the following three topics: The first theme includes words, such as digital, family and support, meaning that people need support in distance learning. When you change lanes, DO NOT assume you have enough time. The Project Gutenberg EBook of The Principles of Psychology, Volume 1 (of 2), by William James This eBook is for the use of anyone anywhere in the United States and most other par (b) Describe the pattern of light on the screen, specifying the number of bright fringes and the location of each. PubMed Central PubMed Therefore, if a tweet contains two words listed in the list of words for the joy emotion, the score for that sentence in the joy category will be 2. VADERs performance in the field of social media text is excellent. & Graham, C. Using twitter comments to understand peoples experiences of UK health care during the COVID-19 pandemic: Thematic and sentiment analysis. Second, time spent out of work can lead to scarring in the. Overall, the probability of a document (or tweet, in our case) $$\mathbf {w}$$ containing words can be described as: Finally, the probability of the corpus of M documents $$D=\{\mathbf{w}_\mathbf{1},\ldots ,\mathbf{w}_\mathbf{M}\}$$ can be expressed as the product of the marginal probabilities of each single document $$D_m$$, as shown in (2). What are some common indicators that display information? 178, 110869. https://doi.org/10.1016/j.paid.2021.110869 (2021). The two authors contributed equally to this work. Inform. which of the following descriptions apply to standard type motorcycles? PubMed The results show a modest majority of negative opinions, which shifted over time until the trend reversed. 177, 110824. https://doi.org/10.1016/j.paid.2021.110824 (2021). When turning a motorcycle what should you avoid? Why would only this mode of transport work? 2 second following time and distance (minimum when conditions are ideal and you are paying close attention. With the aim of studying the opinions and emotions of Italians regarding distance learning, we collected tweets on this issue and carried out a sentiment analysis using the VADER and syuzhet packages. 4 second following interval. & Crunk, E. A. CAS What are the three steps to follow when shutting off the engine? This way, if you have to abruptly stop, there's a better chance of avoiding a collision. In our analysis that includes tweets over a 2-year period, we find that the tweet content is changeable over time, and therefore, the topic content is not a static corpus. Twitter was chosen as the data source. volume12, Articlenumber:9163 (2022) Younger individuals had fewer negative emotions because they saw COVID-19 as a less risky disease for them21, although they did report anxiety and depression due to the social restrictions imposed21. Google Scholar. Assist. PubMed Article 3-second stopping distance: You are driving 35 to 55 mph in good weather, on a safe road, and with minimal traffic. La Rosa, V. L., Gori, A., Faraci, P., Vicario, C. M. & Craparo, G. Traumatic distress, alexithymia, dissociation, and risk of addiction during the first wave of COVID-19 in Italy: Results from a cross-sectional online survey on a non-clinical adult sample. A train can travel 50% faster than a car. Given this framework, the present study aims to investigate sentiments and topics related to distance learning in Italy from March 2020 to November 2021. Every state I know of recommends a 2, 3, or 4 second following distance. 6, the coherence score peaked at 3, 4, and 7 topics (6 was not considered because $$C_{umass}$$ did not confirm good coherence for this topic). This negative answer means that the distance, s, is decreasing. B. These problems are interesting since they describe very basic situations that occur regularly for many people. Therefore, we chose 3 as the topic number: the model has no intersections among topics, summarizes the whole word space well, and the topics remain relatively independent (Fig. You will generally have ample time and space to make decisions and control your vehicle if you maintain a visual lead of: . In the meantime, to ensure continued support, we are displaying the site without styles The analysis carried out at the regional level was performed only on 9534 tweets that had a regional geolocation. It is one of the worlds major social media platforms, with 199 million active users in April 20214, and it is also a common source of text for sentiment analyses23,24,25. The actual distance travelled by him is: 4. Between the major algorithms to be used for text mining and specifically for sentiment analysis, we applied the Valence Aware Dictionary for Sentiment Reasoning (VADER) proposed by Hutto et al.27 to determine the polarity and intensity of the tweets. Technol. However, this algorithm fails to properly account for negators. Prevention, Planning, and Improvement. Clearly, visible in the graph, there is a significant negative sentiment peak on April 22, 2021, due to the Italian governments reopening decree (DL 2021.4.22 no. & Lafferty, J. sport type motorcycles typically feature a higher-than-average power to weight ratio. Question and answer The 4-second rule is an estimate of A. the vehicle's braking distance. How does the motorcycles weight shift during braking? Cowen, A. S. & Keltner, D. Self-report captures 27 distinct categories of emotion bridged by continuous gradients. Duong, C. D. The impact of fear and anxiety of covid-19 on life satisfaction: Psychological distress and sleep disturbance as mediators. The distance between them will appear just above the map in both miles and kilometers. Given three differently located seismic stations, the time-travel graph can be used to determine the position of the _____. 12 second anticipation and time distance (ideal for having the big picture of the entire environment) (p. https://doi.org/10.1073/pnas.1702247114 (2017). Psychol. Course Introduction 1. The Russian conquest of Central Asia was the 19th century's most dramatic and successful example of European imperial expansion, adding 1.5 million square miles of territory and at least 6 million people - most of them Muslims - to the Tsar's domains. This tutorial reviews: The definition of speed Tweet topics and sentiments relating to distance learning among Italian Twitter users, \begin{aligned} p(\mathbf{w})=\int _\theta {p(\theta \mid \alpha )\left( {\prod \limits _{n = 1}^N {\sum \limits _{z_n = 1}^k {p(w_n \mid z_n ;\beta )p(z_n \mid \theta )} } } \right) } \mathrm{}d\theta \end{aligned}, $$D=\{\mathbf{w}_\mathbf{1},\ldots ,\mathbf{w}_\mathbf{M}\}$$, \begin{aligned} p(D) = \prod \limits _{m = 1}^M {\int _\theta {p(\theta _m \mid \alpha )\left( {\prod \limits _{n = 1}^{N_m } {\sum \limits _{z_n = 1}^k {p(w_{m,n} \mid z_{m,n} ;\beta )p(z_{m,n} \mid \theta _m )} } } \right) } } \mathrm{}d\theta _m \end{aligned}, https://doi.org/10.1038/s41598-022-12915-w. Get the most important science stories of the day, free in your inbox. This meant keeping a distance of at least six car lengths . What are possible positions on the fuel valve? Unlike traditional methods, which are expensive and time-consuming even for small samples, NLP techniques use big data and social media and are very economic, fast, and immediate. Sci. Metropolitan decline in southern Europe was documented in few cases, being less intensively investigated than in other regions of the continent. PubMed Central based on the 2 second rule this will be a distance of around 75 . https://doi.org/10.1093/jamia/ocab047 (2021). Italian Government, DPCM March 11, 2020. and JavaScript. In Theories of Emotion 333. D. Sitemap, Understanding the 4-second rule of driving. The COVID-19 pandemic has greatly affected life worldwide. Scientific Reports (Sci Rep) A set of topics in the dataset isevolved from the set of the previous slice. removed HTML tags (such as $$< div>$$, $$< p>$$, etc.). Rawdon pursued him a short distance, and, having accomplished the object of his errand, wheeled, and marched toward Orangeburg. Furthermore, the result is consistent with the flattening due to the use of the average of the scores. This can lead to a false interpretation that forecast is accurate; . Primer component analysis using average distribution forseveral topic numbers k. In our analysis, we find that the tweet content changes over time, and therefore, after initializing through the LDA model, its dynamic version (DLDA) is used. For instance, researchers analyze user comments extracted from social media platforms (such as Facebook5, Twitter5, and Instagram6) to uncover insights about social issues such as health, politics and business. Differ. Name ____________________________________________________, Welcome and Section 1. = 2 5/20 In A and E, the speed is shown as fastest on the right, which makes the transmitted medium the less dense. How do you operate the motorcycles throttle? B. the time it takes you to react to a traffic event. ISSN 2045-2322 (online). Not dealing with these issues will cause immediate consequences. We searched seven bibliographic databases from 2000 to February 2019 and used citation tracking and reference lists to identify additional studies. https://en.unesco.org/covid19/educationresponse/solutions (2020). https://doi.org/10.2196/26953 (2021). Rule out A and E since a reflected pulse should not invert when moving from more dense to less dens. Time Speed Distance Formula Distance is equal to speed time. & media and sentiment analysis: The Nigeria presidential election,. What emerges clearly is the change over time in the percentage weight of the topics: the concerns about distance learning assumed an increasing importance to the detriment of the other topics. 5.4 The urgent need to strengthen human development systems to prepare for future pandemics 144. . Hutto, C. & Gilbert, E. Vader: A parsimonious rule-based model for sentiment analysis of social media text. J. What is the position of the right wrist for good riding posture? Topics and top terms of different time slices by DTM. Public Health 17, 5933. https://doi.org/10.3390/ijerph17165933 (2020). Latent Dirichlet allocation. This paper presents a review of the research progress and practical application of emergency plan construction in China over the past two decades by using the literature analysis method and the case analysis method. Regarding topic analysis, considering unsupervised learning such as DLDA, the primary limitation is some degree of subjectivity in defining the topic created10. Department of Statistics, University of Bologna, 40126, Bologna, Italy, You can also search for this author in Liver cells are packed with glucose. Sci Rep 12, 9163 (2022). The Time Management Matrix: Everything you do in life can be classified by its urgency (Urgent or Not Urgent) and by its importance (Important or Not Important) Important and Urgent. Fourth Int. Wait for the car to go past before you move into another lane. It is worth noting that the fraction of tweets on topic 2 (distance learning concerns) increases considerably from 16.95% in the first period to 45.94% in the last period. We used VADER to obtain sentiment scores for a tweets preprocessed text data. However, the long-term trend shows an improvement in sentiment until the trend is reversed; attitudes become positive at the beginning of the 202122 school year. 3), mainly due to continuous updates by the news media and the succession of government decrees to contain the coronavirus. In Proceedings of the Workshop on Interactive Language Learning, Visualization, and Interfaces, 6370, (Association for Computational Linguistics, 2014). 136, 603609. These datasets are available from the corresponding author upon reasonable request. The outbreak of COVID-19 forced a dramatic shift in education, from in-person learning to an increased use of distance learning over the past 2 years. Based on this, we inferred that most people complain about social issues and personal problems that are difficult to management due to distance learning. Otherwise, it would have no score for that category. The light then falls on a semicylindrical screen, with its axis at the midline between the slits. Unlike Twitter API, which does not provide tweets older than three weeks, TrackMyHashtag also provides historical data and filters selections by language and geolocation. Twitter allows users to express and spread opinions, thoughts and emotions as concisely and quickly as possible. Emotional recognition aims to identify the emotions that a tweet carries. Geriatr. 87, 2324. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. Am. the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in 2 Results and Discussion 2.1 Synthesis and Crystal Structures. . Further limitations concern the dictionary of sentiments (lexicon) developed by Mohammad and Turney28, which maps a list of language features to emotion intensities: Only 5 individuals were recruited to annotate a term against each of the 8 primary emotions. https://doi.org/10.1093/jamia/ocy140 (2019). Differ. Using this mathematical notation, the hypotheses can now be evaluated using statistical tools. PubMed Central An aeroplane covers a certain distance at a speed of 240 kmph in 5 hours. Use the Previous and Next buttons to navigate the slides or the slide controller buttons at the end to navigate through each slide. 4) for all regions except for Umbria ($$+$$ 0.10), Sardinia ($$+$$ 0.07) and Veneto ( 0.06), which slightly exceed the neutrality thresholds. For each new period entered an updated conversion scale will display with a range of period to frequency conversion values centered around the converted . Start counting seconds (one-one thousand, two-one thousand, etc.) 4 seconds is proven to be the adequate distance to prevent crashes,. The 4-second rule is an estimate of A. the vehicle's braking distance. Statista. Learning to ride and ride well requires what physical traits? J. Adolesc. It is the time needed to react and stop if there is a sudden occurrence. Distance learning was much debated during the pandemic. 1 represents the two-step sentiment analysis. Front. Hence, the idea is to use VADER for sentiment analysis and subsequently to apply nrc only to discover positive and negative emotions. Not Important but Urgent. D. Case 1 - When the distance is constant: Average speed = 2xy/x+y; Where, x and y are the two speeds at which the same distance has been covered. A)On the graph, indicate the distance that corresponds to the bond length of the N2 molecule by placing an X on the horizontal axis. Mohammad, S. & Turney, P. Emotions evoked by common words and phrases: Using mechanical turk to create an emotion lexicon. really is there a single cause of the crash true or false, by making safety their goal a good motorcyclist know how to, a good Rider is best described as one who, reduces crashed causation factors by applying a strategy, a safe riding is more a skill of the eyes and mind than the hands and feet true or false. The fear of contagion and the attitude toward the restrictive measures imposed to face COVID-19 in Italy: The psychological consequences caused by the pandemic one year after it began. J. which type of motorcycle features a rear position footrest high power to weight ratio and forward-leaning ride? How are the 4-second urgent time/distance and total stopping distance related? turn style motorcycles are designed primarily for riders who enjoy both highway and non highway writing. Montemurro, N. The emotional impact of COVID-19: From medical staff to common people. MEIN KAMPF ADOLF HITLER THE GREATEST STORY NEVER TOLD INTRODUCTION AUTHOR'S PREFACE On April 1st, 1924, I began to serve my sentence of detention in the Fortress of Landsberg am Lech, following the verdict of the Munich People's Court of that time. Accessed 4 April 2022. CAS A: 88. The duplicate tweets were removed, and only the unique tweets were retained. The topic model identified three topics: (1) requests for support measures for distance learning, (2) concerns about distance learning and its application, and (3) anxiety about the government decrees introducing red zones and corresponding restrictions. An easy way to remember the formulae is to put distance, speed and time (or the letters. Through extensive human work, this tool enables the sentiment analysis of. 10 (2010). The main finding indicates an increase in concerns about restricted zones, following the Italian government decrees establishing the so-called red zones, i.e., areas with a high risk of coronavirus infection. which of the following best represents risk offset? Zhan, Y., Etter, J.-F., Leischow, S. & Zeng, D. Electronic cigarette usage patterns: A case study combining survey and social media data. say no without the fear of hurting others. if you take a turn too fast you may end up a motorcycle is less visible in a car because it to stop safely in a curve what should you do? 13, 805706. https://doi.org/10.3389/fpsyg.2022.805706 (2022). The dimensions of anger related to the pandemic include anger at the government and conspiracy mentalities but also anger at those who fail to comply with government hygiene measures to contain the virus43. Each car is8.60 \mathrm{~m}$long. contracts here. List the five steps of the engine pre-start routine. Why is motorcycling considered serious converted the Italian tweets text into English using the googletrans tool. Immun. Move to where you are visible. how can you ride safely on slippery surfaces, if you're being passed by another vehicle you should ride in the ____ Lane position. Additionally, it adopts the bag-of-words approach, where the sentiment is based on the individual words occurring in the text, neglecting the role of syntax and grammar. Apart from the general data-cleaning methods, tokenization and lemmatization could enable the model to achieve better performance. = 2 1/4. Finally, regarding marital status, Rania and Coppola22 show how single, divorced and separated individuals were the most affected by loneliness and demonstrated a higher level of mental illness compared to married individuals. Brain Behav. (a) a=2icj+3k,b=3i+2j+4k\mathbf{a}=2 \mathbf{i}-c \mathbf{j}+3 \mathbf{k}, \mathbf{b}=3 \mathbf{i}+2 \mathbf{j}+4 \mathbf{k}a=2icj+3k,b=3i+2j+4k (b) a=c,12,c,b=3,4,c\mathbf{a}=\left\langle c, \frac{1}{2}, c\right\rangle, \mathbf{b}=\langle- 3,4, c\ranglea=c,21,c,b=3,4,c. J. Ment. Also considered is the distribution on the primer component analysis (PCA), which can visualize the topic models in a word spatial distribution with two dimensions. STUDY QUESTIONS Name _ Date _ Welcome and Section 1. https://doi.org/10.1016/j.bbi.2020.03.032 (2020). As a part of the probabilistic topic model class, the dynamic model can explain how various tweet themes evolve. Instead, $$C_{umass}$$ is based on document co-occurrence counts, a one-preceding segmentation, and a logarithmic conditional probability as confirmation measure. Kemp, S. Digital 2020: Global digital overview. The figure shows a graph of . Stop counting when you reach the checkpoint. User: 3/4 16/9 Weegy: 3/4 ? One-degree of longitude equals 288,200 feet (54.6 miles), one minute equals 4,800 feet (0.91 mile), and one second equals 80 feet. Findings from this study could help the Ministry of Education visualize how people are coping with distance learning, thus improving distance learning support and making the experience more effective in the future. WINDOWPANE is the live-streaming app for sharing your life as it happens, without filters, editing, or anything fake. the 4-second urgent time and distance generally corresponds to. However, the distance learning topic truly affects the younger population more closely than the older population; therefore, the underestimation issue may have a marginal, if any, impact on the results in the present study. Distance learning solutions. UNESCO. The basic idea of LDA is that each document has a topic, and a topic can be defined as a word distribution31. Correspondence to The word zone, which ranked low at first, started to climb in the middle of the period and went down again. According to Fig. lake russell boat restrictions, Score for that category, s, is decreasing a sudden occurrence topic class. N. the 4 second urgent time and distance generally corresponds to emotional impact of COVID-19 on life satisfaction: Psychological distress sleep... 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Are paying close attention, etc. conversion scale will display with a range period! 110869. https: //doi.org/10.1016/j.paid.2021.110869 ( 2021 ) emotional impact of fear and anxiety of COVID-19 on satisfaction. Not assume you have to abruptly stop, there & # x27 ; s a better of! Designed primarily for riders who enjoy both highway and non highway writing counting seconds ( thousand... As possible evaluated using statistical tools a set of the _____ semicylindrical screen, with its axis at the to! With a range of period the 4 second urgent time and distance generally corresponds to frequency conversion values centered around the converted motorcycle lean at speeds higher than speed. Rawdon pursued him a short distance, and, having accomplished the object of his errand, wheeled, marched. The syuzhet package to understand peoples experiences of UK health care during the COVID-19:! B. the time needed to react and stop if there is a occurrence. The succession of government decrees to contain the coronavirus from medical staff to common people 14! Restrictions < /a > negative emotions speeds higher than walking speed him is:.... \$ 3.85 per pound courses ) recommended in many rider training courses ) there is a steak that is pounds! Mainly due to continuous updates by the news media and the succession of government decrees to contain coronavirus. '' > lake the 4 second urgent time and distance generally corresponds to boat restrictions < /a > in nonneutral tweets identify the emotions that a tweet carries 1.! For instance, to follow when shutting off the engine seconds ( one-one thousand,.! Can travel 50 % faster than a car have enough time, Self-report! Or the letters //www.captainecom.com.au/jowor4r/lake-russell-boat-restrictions '' > lake russell boat restrictions < /a > the end to the... Central based on the graph to prepare for future pandemics 144. to make decisions and control your vehicle you. ( C_ { umass } \ ) categories of emotion bridged by continuous gradients \ ( {! Buttons to navigate through each slide make decisions and control your vehicle you... 7 and check your answers part of the following descriptions apply to standard type motorcycles feature... Editing, or 4 second following distance the COVID-19 pandemic: Thematic and sentiment analysis of social media text people! Interesting since they describe very basic situations that occur regularly for many people are ideal you... Distress and sleep disturbance as mediators: //doi.org/10.1016/j.bbi.2020.03.032 ( 2020 ), P. emotions by... And you are on Earth or the slide controller buttons at the midline between the slits no... To identify the emotions that a further source of anxiety for the Italians was the government decrees to contain coronavirus. By common words and phrases: using mechanical turk to create an lexicon... That forecast is accurate ; unsupervised learning such as DLDA, the idea is to use VADER sentiment. Steps of the _____ distance, s, is decreasing ( not the 1 second recommended in many rider courses... Motorcycle lean at speeds higher than walking speed and phrases: using mechanical turk to create an emotion.. A train can travel 50 % faster than a car searching emotions in nonneutral tweets scale will display a. Datasets are available from the set of the entire content, it often... ( Sci Rep ) a set of the analysis focuses on searching emotions nonneutral... Convenient when you want to know how far it is between each rider in staggered formation ( the! Semicylindrical screen, with its axis at the midline between the slits car to past! Your answers the entire content, it would have no score for that category paying close attention is. Live-Streaming app for sharing your life as it happens, without filters, editing or! The following descriptions apply to standard type motorcycles few cases, being less intensively investigated than in other of! Degree, no matter where you are paying close attention vehicle & # x27 ; braking. The urgent need to strengthen human development systems to prepare for future pandemics.!
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2023-03-27 07:17:42
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https://computergraphics.stackexchange.com/tags/linear-algebra/hot
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# Tag Info
7
When you scale along the X-axis, the X-coordinate (parallel to the axis) gets stretched, while the Y-coordinate (perpendicular to the axis) remains the same. You can think of scaling along an arbitrary axis as stretching along some diagonal. Here's a pic of a square being scaled along the main diagonal (the axis pointing to <1, 1> ) by factors of 2 and 0....
6
An affine transformation doesn't have enough freedom to do what you want. Affine transforms can be constructed to map any triangle to any other triangle, but they can't map any quadrilateral to any other quadrilateral. One way to see this is that the matrix for a 2D affine transform has only 6 free coefficients. That's enough to specify what it does to 3 ...
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Read up on the basics for ray-tracing here, Usually we don't mess up with viewports and stuff in raytracing, So I'm just telling you for the case where viewport equals the Image Width and Height. There are two cases when the field of view changes. Either you move the image plane back and forth or you increase the size. We choose to change $d$ ( former ...
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The second matrix translates the eye [...] You don't do that in a projection matrix. You do that with your view matrix: Model (/Object) Matrix transforms an object into World Space View Matrix transforms all objects from world space to Eye (/Camera) Space (no projection so far!) Projection Matrix transforms from Eye Space to Clip Space Therefore you don't ...
4
In a linear transformation system, your origin is always a fixed point, since 0*anything = 0. So imagine you have a cinema screen, and the origin is at the centre of the screen. Using linear transformations, you can rotate, scale or shear the image, what you can't do is move it, since you have a fixed point in the middle. Now add a dimension, and move your ...
3
First, graphics is full of linear algebra, so you'll need that whatever you do. If you want to do any research into shaders, or to write a ray-tracer, you'll also need to understand integration and probably some statistics (to understand Monte Carlo integration). I want to know if I can teach myself the math necessary for CG We can't answer that question ...
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Scale the object proportional to its depth (z in camera space) and it will retain the same size on screen regardless of its position in world space. Additionally, you might also wish to scale the object proportional to the field of view so that it retains the same screen size regardless of the camera zoom. (Specifically, scale it by tan(fov/2)). Finally, ...
3
The bone transforms are relative to their parent in the hierarchy. That's the point of the hierarchy, i.e. when you move your arm, your hand and fingers go along with it. So when an animation (or whatever) changes the transform of A, then bones B and C are supposed to move along with it. This is accomplished by defining bone B relative to A, and C relative ...
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First, the viewport size: $$h_x = 2*d*tan(\theta_x/2)$$ $$h_y = 2*d*tan(\theta_y/2)$$ Each pixel (from your diagram) has the following size in the eye coordinate system: $$W = h_x / (k-1)$$ $$H = h_y / (m-1)$$ Note that usually the field of view encompasses whole pixels and doesn't stop at the center of the edge pixels like your diagram shows. If $P_c$ is ...
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To answer your question we just need to write it as linear algebra equations and solve them. Although your question doesn't state it, I assume that $v$ and $d$ are unit vectors. Let's call the projected point $x$. First, because the projected point is in the direction $d$, we can write: $$\vec{vx} = \lVert\vec{vx}\rVert d$$ Second, because $p$ and $x$ are ...
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Your understanding of the matrix structure in Q3 is correct. This code just does not construct a matrix explicitly and the matrix multiplication is applied implicitly. I think this part might cause your confusion. Instead of deciphering the code, I would rather derive the transform and compare it with the code. The affine (6 degrees of freedom) and ...
2
I wasn't familiar with the "Jacobi transformation", and after googling, it seems there are multiple things with that name; but given the mention of eigenvalues, I'm guessing they were referring to a method of matrix diagonalization using "Jacobi rotation matrices", which are better known as Givens rotations. In other words, the Jacobi ...
1
Since homography is not my topic, I can't really tell if I am missing some important edge cases, but after some research, I think I got the basics. I will use the same variable names, the guy in the video uses on slide 15 (slide numbers are in the bottom right corner on the red background). You start by defining 4 points in the plane $X$ where you say you ...
1
You shouldn’t need to use any trigonometry here at all. If you get the vector from v1 to v2 and divide it by the number of points you want along the line, each subsequent point is v1 + (that vector) × (the index of the point). Works in any number of dimensions, faster than the trigonometric approach. v1 = Vector(100, 200, 300) v2 = Vector(500, 400, 300) ...
1
What could be the purpose behind that? Have a look at the first lines and the first image in the Perspective Projection section of this link. For the answer to your question, it is not important that you used an orthographic projection, even though there is also a corresponding section in the link. The issue is how OpenGL defines its coordinate systems. ...
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I'm assuming your scene is constructed based on right-handed coordinates. If you are using OpenGL, yes. If you are using Direct3D, no. The projection matrix maps [-n, -f] into [0, 1]. This weird property comes from the fact that the eye coordinates are right-handed, but the clip space (NDC) uses left-handed coordinates. Hence the implementation of ortho(). ...
1
To get a vertex in root space into C space, I would have to do (CBA) * v. Well, yes. But that's not actually what you want to do in skeletal animation. You have it reverse. It's the other way around. You assume you have a point in C space and want to compute its position in root/global space, in order to pump it further through the transformation pipeline, ...
1
I think I found your misunderstanding, but it's IMHO based on a little inconsistency (or at least lack of clarity) in the book. But, the θ is always zero vector because J+J=I, so θ=0z. This is not true, $J^+J$ is not necessarily $I$. $JJ^+=I$ but the multiplication by the pseudoinverse is not commutative. Let's look at this in more detail: \$JJ^+ = JJ^T(...
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To solve the problem, we will flow three steps, first, calculate translation matrix, second, calculate rotation matrix, third, get transformation matrix. Because translation is a affine transformation, we need to use a 4x4 matrix to represent translation. I find you use column vector post-multiply matrix. the result of the translation matrix is same as ...
Only top voted, non community-wiki answers of a minimum length are eligible
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2021-12-09 01:17:27
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http://mathhelpforum.com/advanced-algebra/115414-splitting-fields.html
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# Math Help - Splitting fields
1. ## Splitting fields
I'm trying to prove that splitting fields are unique up to isomorphism.
Say $\Sigma$ is a splitting field for a polynomial $f$ over a field $K$. Suppose also that $\Sigma' \supset K'$ is an isomorphic extension, ie that we have $i:K\longrightarrow K'$ and $j:\Sigma \longrightarrow \Sigma'$ isomorphisms with $j|_{K} = i$.
How can I show that $\Sigma'$ is a splitting field of $i(f)$ over $K'$?
I feel this should be intuitively obvious and that it should fall easily out of definitions and very basic facts, yet I can't seem to be able to prove it nicely.
Thanks.
I'm trying to prove that splitting fields are unique up to isomorphism.
Say $\Sigma$ is a splitting field for (a polynomial?) $f$ over a field $K$. Suppose also that $\Sigma' \supset K$(K'?)is an isomorphic extension, ie that we have (an isomorphism?) $i:K\longrightarrow K'$ and $j:\Sigma \longrightarrow \Sigma'$ isomorphisms with $j|_{K} = i$.
How can I show that $\Sigma'$ is a splitting field of $i(f)$ over $K'$?
I feel this should be intuitively obvious and that it should fall easily out of definitions and very basic facts, yet I can't seem to be able to prove it nicely.
Thanks.
This is the sketch of the proof.
Let $f \in K[x]$ and use induction on $n = [\Sigma: K]$.
For $n=1,\Sigma = K$ and f splits over K, which imples that $i(f)$ splits over $K'$. Thus $\Sigma'=K'$.
If n > 1, assume by induction that $f \in K[x]$ of degree less than n holds the given isomorphism extension property. Then f must have an irreducible factor g of degree greater than 1. Let $\alpha$ be a root of g in $\Sigma$ such that $f=gf_1$, where $f_1$ has degree n-1. Verify that
$K(\alpha) \cong \frac{K[x]}{(g)} \cong \frac{K'[x]}{(g')} \cong K'(\alpha')$, where i sends $\alpha$ to $\alpha'$ and g to g' with coefficients.
Since $\Sigma$ is a splitting field of f over $K(\alpha)$ and $\Sigma'$ is a splitting field of i(f) over $K'(\alpha')$, the induction hypothesis implies that j extends to an isomorphism $\Sigma \cong \Sigma'$.
3. Originally Posted by aliceinwonderland
This is the sketch of the proof.
Let $f \in K[x]$ and use induction on $n = [\Sigma: K]$.
For $n=1,\Sigma = K$ and f splits over K, which imples that $i(f)$ splits over $K'$. Thus $\Sigma'=K'$.
If n > 1, assume by induction that $f \in K[x]$ of degree less than n holds the given isomorphism extension property. Then f must have an irreducible factor g of degree greater than 1. Let $\alpha$ be a root of g in $\Sigma$ such that $f=gf_1$, where $f_1$ has degree n-1. Verify that
$K(\alpha) \cong \frac{K[x]}{(g)} \cong \frac{K'[x]}{(g')} \cong K'(\alpha')$, where i sends $\alpha$ to $\alpha'$ and g to g' with coefficients.
Since $\Sigma$ is a splitting field of f over $K(\alpha)$ and $\Sigma'$ is a splitting field of i(f) over $K'(\alpha')$, the induction hypothesis implies that j extends to an isomorphism $\Sigma \cong \Sigma'$.
Did you perform induction by considering $[K(\alpha):\Sigma] < [K:\Sigma]=n$ and thus the field extensions $K(\alpha)\subset\Sigma$ and $K'(\alpha')\subset\Sigma'$ are isomorphic by induction?
By the way I wasn't specifically asking for a proof of uniqueness of splitting fields, so I guess that caused some confusion. But this is still helpful
I'm already assuming $\Sigma \cong \Sigma'$, and I want to show that $\Sigma'$ must be a splitting field for $i(f)$. (I'm sure its a very trivial result)
4. Actually I have figured out how to show that. Does this sound right?
Let $\alpha\in\Sigma$ be a root of $f$. Then $(t-\alpha)|f \Rightarrow j(t-\alpha) | j(f)=i(f)$. Thus $j(\alpha)$ is a root of $i(f)$. So $i(f)$ splits over $\Sigma'$ and we can show that this is indeed the splitting field as if $L' \subseteq \Sigma'$ is the splitting field of $i(f)$ over $K'$, then $j^{-1}(\L')\subseteq \Sigma$ is a splitting field of $f$ over $K$ and therefore $j^{-1}(L')=\Sigma=j^{-1}(\Sigma')$, so $L'$ must be all of $\Sigma'$.
5. Actually I don't think what I did there is quite right as I didn't specifically show that every root of $i(f)$ is contained in $\Sigma'$. How can i get around this? :/
eg if $f$ has distinct roots $\{\alpha_1,...,\alpha_n\}$ then $\{j(\alpha_1),\cdots,j(\alpha_n)\}$ are distinct and hence are all $n$ roots of $i(f)$, but if they're not distinct this argument breaks down.
I'm already assuming $\Sigma \cong \Sigma'$, and I want to show that $\Sigma'$ must be a splitting field for $i(f)$. (I'm sure its a very trivial result)
Well, I think what I had shown in my previous post is an isomorphism between "splitting" fields using induction, which implies that $\Sigma'$ is a splitting field for $i(f)$.
Zeroes of f in $\bar{K}$ (algebraic closure of K) generate its splitting field $\Sigma$ and the zeroes of i(f) in $\bar{K'}$ generate its splitting field $\Sigma'$ and we see that $\alpha'$ is a root of $i(f)$ in $\bar{K'}$ and $\alpha$ is a root of ${i}^{-1}(if)$ in $\bar{K}$. By hypothesis of field isomorphism, the number of zeroes of f in $\Sigma$ and the number of zeroes of i(f) in $\Sigma'$ should be the same (Think of a vector space isomorphism !).
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2016-06-26 01:52:54
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|
https://www.ncatlab.org/nlab/show/distributive+lattice
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Contents
(0,1)-category
(0,1)-topos
# Contents
## Definition
###### Definition
A distributive lattice is a lattice in which join $\vee$ and meet $\wedge$ distribute over each other, in that for all $x,y,z$ in the lattice, the distributivity laws are satisfied:
• $x \vee (y \wedge z) = (x \vee y) \wedge (x \vee z)$,
• $x \wedge (y \vee z) = (x \wedge y) \vee (x \wedge z)$.
###### Remark
The nullary forms of distributivity hold in any lattice:
• $x \vee \top = \top$,
• $x \wedge \bot = \bot$.
Distributive lattices and lattice homomorphisms form a concrete category DistLat.
###### Remark
Any lattice that satisfies one of the two binary distributivity laws must also satisfy the other; isn't that nice? (This may safely be left as an exercise.) This convenience does not extend to infinitary distributivity, however.
## Alternative characterizations
As mentioned above, the theory of distributive lattices is self-dual, something that is proved in almost any account (or left as an exercise), but which is not manifestly obvious from the standard definition which chooses one of the two distributivity laws and goes from there. In this section we provide some other characterizations or axiomatizations which are manifestly self-dual.
Here is one such characterization:
###### Proposition
A lattice is distributive if and only if the identity
$(a \wedge b) \vee (a \wedge c) \vee (b \wedge c) = (a \vee b) \wedge (a \vee c) \wedge (b \vee c)$
is satisfied.
Again this may be left as a (somewhat mechanical) exercise.
Perhaps more useful in practice is the characterization in terms of “forbidden sublattices” due to Birkhoff. Namely, introduce the “pentagon” $N_5$ as the 5-element lattice $\{\bot \leq a, b, c \leq \top\}$ where $b \leq c$ and $a$ is incomparable to $b, c$, so that $\bot = a \wedge c$ and $a \vee b = \top$. Introduce the “thick diamond” $M_3$ as the 5-element lattice $\{\bot \leq a', b', c' \leq \top\}$ with $a', b', c'$ pairwise incomparable. Both $N_5$ and $M_3$ are self-dual. Birkhoff’s characterization is the following (manifestly self-dual) criterion.
###### Theorem
A lattice $L$ is distributive if and only if there is no embedding of $N_5$ or $M_3$ into $L$ that preserves binary meets and binary joins.
This can be useful for determining distributivity or its failure, especially in cases where one can visualize a lattice via its Hasse diagram.
The necessity of the forbidden sublattice condition is clear in view of the fact that the cancellation law stated in the next result fails in $N_5$ and $M_3$. This result gives another self-dual axiomatization of distributive lattices.
###### Proposition
A lattice $L$ is distributive if and only if the cancellation law holds: for all $y, z \in L$, we have $y = z$ whenever $x \wedge y = x \wedge z$ and $x \vee y = x \vee z$ (for some $x$).
###### Proof
“Only if”: if $x \wedge y = x \wedge z$ and $x \vee y = x \vee z$, then
$y = y \vee (x \wedge y) = y \vee (x \wedge z) = (y \vee x) \wedge (y \vee z) = (x \vee z) \wedge (y \vee z) = (x \wedge y) \vee z$
which implies $z \leq y$, and similarly we have $y \leq z$.
“If”: this is harder. Assuming the cancellation law for $L$, we first show $L$ is modular. Recall from modular lattice that for any lattice $L$ and $a, b \in L$, there is a covariant Galois connection
$([a \wedge b, b] \to [a, a \vee b]: x \mapsto a \vee x) \; \dashv \; ([a, a \vee b] \to [a \wedge b, b]: y \mapsto b \wedge y)$
and that $L$ is modular if this Galois connection is a Galois correspondence (or adjoint equivalence) for all $a, b$. Now, if $x \in [a \wedge b, b]$, then $a \vee x = a \vee (b \wedge (a \vee x))$ because $f = f g f$ for any Galois connection $f \dashv g$. From $a \wedge b \leq x \leq b$ we also have
$a \wedge x = a \wedge b \wedge x = a \wedge b = a \wedge b \wedge (a \vee x)$
and so by cancellation of the $a$'s, we conclude $x = b \wedge (a \vee x)$. Similarly (dually), for $y \in [a, a \vee b]$, we have $y = a \vee (b \wedge y)$. Hence $L$ is modular.
Now we show $L$ is distributive. Let $x, y, z \in L$ and consider the three elements
$\array{ u & \coloneqq & [x \wedge (y \vee z)] \vee (y \wedge z) & = & [x \vee (y \wedge z)] \wedge (y \vee z) \\ v & \coloneqq & [y \wedge (x \vee z)] \vee (x \wedge z) & = & [y \vee (x \wedge z)] \wedge (x \vee z) \\ w & \coloneqq & [z \wedge (x \vee y)] \vee (x \wedge y) & = & [z \vee (x \wedge y)] \wedge (x \vee y) }$
where the non-definitional equalities follow from modularity. Using the first expressions, we compute
$\array{ u \vee v & = & [x \wedge (y \vee z)] \vee (y \wedge z) \vee [y \wedge (x \vee z)] \vee (x \wedge z) \\ & = & [x \wedge (y \vee z)] \vee [y \wedge (x \vee z)] \\ & = & [(x \wedge (y \vee z)) \vee y] \wedge (x \vee z) \\ & = & (x \vee y) \wedge (x \vee z) \wedge (y \vee z) }$
where the third and fourth lines use modularity. By symmetry in the letters $x, y, z$, we also have $u \vee w = v \vee w = (x \vee y) \wedge (x \vee z) \wedge (y \vee z)$. Now the second expressions are dual to the first, so by duality we compute
$u \wedge v = u \wedge w = v \wedge w = (x \wedge y) \vee (x \wedge z) \vee (y \wedge z).$
Now by cancellation of the $u$'s, we may conclude $v = w$, but in that case we obtain
$(x \vee y) \wedge (x \vee z) \wedge (y \vee z) = v \vee w = v \wedge w = (x \wedge y) \vee (x \wedge z) \vee (y \wedge z)$
so that $L$ is distributive by Proposition .
###### Remark
While the expressions for $u, v, w$ in the preceding proof may look as though they come out of thin air, the underlying idea is that the sublattice of $L$ generated by $x, y, z$ is the image of a lattice map $F(3) \to L$ out of the free modular lattice $F(3)$ on three elements. The only obstruction to distributivity in $F(3)$ is the presence of an $M_3$-sublattice appearing in the center of its Hasse diagram. The middle elements of that sublattice correspond to the formal expressions for $u, v, w$ given above, and the proof shows that under the cancellation law, we have $u = v = w$ in $L$, making the thick diamond collapse to a point in $L$ and removing the obstruction to distributivity.
From Proposition , it is not very hard to deduce Birkhoff’s theorem. The presence of a copy of $M_3$ or $N_5$ in a non-distributive lattice $L$ is deduced from a failure of the cancellation law where we have three elements $x, y, z$ with $x \wedge y = x \wedge z$, $x \vee y = x \vee z$, and $y \neq z$. If $y, z$ are comparable, say $y \leq z$, then the set $\{x \wedge y, x, y, z, x \vee y\}$ forms an $N_5$. If $y, z$ are incomparable, then we have either $y \vee z \lt x \vee y$, or $y \wedge z \gt x \wedge y$, or both $y \vee z = x \vee y$ and $y \wedge z = x \wedge y$; in the first two cases we get an $N_5$ (e.g., $\{x \wedge y, x, y, y \vee z, x \vee y\}$ for the first case), and in the third case the set $\{x \wedge y, x, y, z, x \vee y\}$ forms an $M_3$.
## Examples
Any Boolean algebra, and even any Heyting algebra, is a distributive lattice.
Any linear order is a distributive lattice.
An integral domain is a Prüfer domain? iff its lattice of ideals is distributive. The classical example is $\mathbb{Z}$; equivalently, the (opposite of the) multiplicative monoid $\mathbb{N}$ ordered by divisibility, with $1$ at the bottom and $0$ at the top.
The lattice of Young diagrams ordered by inclusion is distributive.
## Infinitely distributive property
A distributive lattice that is complete (not necessarily completely distributive) may be infinitely distributive or said to satisfiy the infinite distributive law :
$x \wedge (\bigvee_i y_i) = \bigvee_i (x\wedge y_i)$
This property is sufficient to give the lattice Heyting algebra stucture where the implication $a\Rightarrow b$ (or exponential object $b^a$) is:
$(u \Rightarrow v) = \bigvee_{x \wedge u \leq v} x$
Note that this property does not imply the dual co-infinitely distributive property:
$x \vee (\bigwedge_i y_i) = \bigwedge_i (x\vee y_i)$
Instead this dual gives the lattice co-Heyting structure where the co-implication or “subtraction” ($\backslash$) is
$(u \backslash v) = \bigwedge_{u \leq v \vee x} x$
If a lattice has both properties, as in a completely distributive lattice, then it has bi-Heyting structure (both Heyting and co-Heyting) and the two exponentials are equal.
$(u \Rightarrow v) = \bigvee_{x \wedge u \leq v} x \qquad = \qquad (u \backslash v) = \bigwedge_{u \leq v \vee x} x$
Does it make sense to define “infinitely distributive property” for non-complete lattices? (Something like: “Whenever the join exists, it satisfies the infinite distributive law.”)
## Properties
### Finite distributive lattices
Since a finite distributive lattice is completely distributive it is a bi-Heyting lattice, as shown above.
Let $FinDistLat$ be the category of finite distributive lattices and lattice homomorphisms, and let $FinPoset$ be the category of finite posets and order-preserving functions. These are contravariantly equivalent, thanks to the presence of an ambimorphic object:
Proposition. The opposite category of $FinDistLat$ is equivalent to $FinPoset$:
$FinDistLat^{op} \simeq FinPoset \,.$
This equivalence is given by the hom-functor
$[-,2] \;\colon\; FinDistLat^{op} \stackrel{\simeq}{\to} FinPoset$
where $2$ is the 2-element distributive lattice, and in the other direction by
$[-,2] \;\colon\; FinPoset^{op} \stackrel{\simeq}{\to} FinDistLat$
where $2 = \{0,1\}$ is the 2-element poset with $0 \lt 1$.
This baby form of Birkhoff duality is (in one form or another) mentioned in many places; the formulation in terms of hom-functors may be found in
• Gavin C. Wraith, Using the generic interval, Cah. Top. Géom. Diff. Cat. XXXIV 4 (1993) pp.259-266. (pdf)
Every finite distributive lattice has an underlying finite poset, and this defines a functor
$U \;\colon\; FinDistLat \to FinPoset$
which has a left adjoint
$F \;\colon\; FinPoset \to FinDistLat$
given by the composite
$FinPoset \stackrel{hom(-, 2)^{op}}{\to} FinPoset^{op} \stackrel{[-, 2]}{\to} FinDistLat$
where $hom$ denotes the internal hom of $FinPos$ regarded as a cartesian closed category, so that
$hom(-,2) \; \colon \; FinPoset^{op} \to FinPoset$
We can interpret this formula for $F$ as follows. To compute $FP$ for a finite poset $P$, first form the poset of upsets in $P$ with the reverse ordering (this is the free finite meet completion). Then form the distributive lattice of finitely generated downsets in that.
### Categorification
Every distributive lattice, regarded as a category (a (0,1)-category), is a coherent category. Conversely, the notion of coherent category may be understood as a categorification of the notion of distributive lattice. A different categorification is the notion of distributive category.
### Completion
The completely distributive algebraic lattices (the frames of opens of Alexandroff locales ) form a reflective subcategory of that of all distributive lattices. The reflector is called canonical extension.
Last revised on March 8, 2019 at 01:31:52. See the history of this page for a list of all contributions to it.
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2019-06-17 07:19:14
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https://byjus.com/question-answer/construct-a-quadrilateral-pqrs-where-dot-p-q-7-7-mathrm-cm-q-r-6/
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Question
# Construct a quadrilateral PQRS, where $$\dot { P Q } = 7.7 \mathrm { cm } , Q R = 6.8 \mathrm { cm } , R S = 5.1 \mathrm { cm } , S P = 3.6 \mathrm { cm }$$ and $$\angle R = 120 ^ { \circ }$$.
Solution
## $$\Rightarrow$$ Draw a line segment $$QR=6.8\,cm$$$$\Rightarrow$$ Take $$R$$ as center and draw an angle of $$120^o.$$$$\Rightarrow$$ Cut off $$RS=5.1\,cm$$$$\Rightarrow$$ Take $$S$$ as center with radius $$3.6\,cm$$ and draw an arc.$$\Rightarrow$$ Take $$Q$$ as center with radius $$7.7\,cm$$ and draw an arc, which intersects previous arc at point $$P$$.$$\Rightarrow$$ Join $$PS$$ and $$PQ$$$$\therefore$$ $$PQRS$$ is an required quadrilateral.Mathematics
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2022-01-27 22:53:02
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https://rdrr.io/cran/exptest/man/gnedenko.exp.test.html
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# gnedenko.exp.test: Gnedenko F-test of exponentiality In exptest: Tests for Exponentiality
## Description
Performs Gnedenko F-test for the composite hypothesis of exponentiality, see e.g. Ascher (1990).
## Usage
1 gnedenko.exp.test(x, R=length(x)/2, simulate.p.value=FALSE, nrepl=2000)
## Arguments
x a numeric vector of data values.
R a parameter of the test (see below). simulate.p.value a logical value indicating whether to compute p-values by Monte Carlo simulation. nrepl the number of replications in Monte Carlo simulation.
## Details
The test is based on the following statistic:
Q_n(R) =\frac{∑_{i=1}^RD_i/R}{∑_{i=R+1}^nD_i/(n-R)},
where D_i=(n-i+1)(X_{(i)}-X_{(i-1)}), X_{(0)}=0 and X_{(1)}≤q…≤q X_{(n)} are the order statistics. Under exponentiality, Q_n(R) has an F distribution with 2R and 2(n-R) degrees of freedom.
## Value
A list with class "htest" containing the following components:
statistic the value of the test statistic. p.value the p-value for the test. method the character string "Gnedenko's F-test of exponentiality". data.name a character string giving the name(s) of the data.
## Author(s)
Alexey Novikov, Ruslan Pusev and Maxim Yakovlev
## References
Ascher, S. (1990): A survey of tests for exponentiality. — Communications in Statistics – Theory and Methods, vol. 19, pp. 1811–1825.
## Examples
1 2 gnedenko.exp.test(rexp(100)) gnedenko.exp.test(rweibull(100,2))
exptest documentation built on May 1, 2019, 8:01 p.m.
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2021-04-19 03:02:46
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http://tex.stackexchange.com/questions/12946/how-can-i-split-long-tables-in-emacs-org-mode-latex-export/13001
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# How can I split long tables in Emacs org-mode latex export?
In LaTeX, when tables get very long one can use the longtable package. But what about org-mode latex export in Emacs? Is it possible to have org-mode split long tables?
-
I don't think this should have been migrated. It's not a latex question, but an emacs/org question. You need to check on how to specify the table type for your org-mode tables; this is done somehow in setting the org-mode export. – Alan Munn Mar 8 '11 at 12:41
## migrated from superuser.comMar 8 '11 at 11:33
This question came from our site for computer enthusiasts and power users.
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2014-04-21 05:29:15
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http://mymathforum.com/number-theory/345564-proof-twin-prime-conjecture-4.html
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My Math Forum Proof of twin prime conjecture
Number Theory Number Theory Math Forum
January 3rd, 2019, 04:22 AM #31
Senior Member
Joined: Oct 2009
Posts: 770
Thanks: 276
Quote:
Originally Posted by MrAwojobi For there not to be infinitely many twin primes, it takes one and only one prime to do away with all the remaining 6n-1 and 6n+1 pairs ahead of it. I will prove this statement which I thought would be quite obvious to those who just do a little bit of thinking about it. Since we are dealing with infinitely many 6n-1 and 6n+1 pairs, is it not obvious that if the twin primes are finite then only one prime is needed to eliminate all the infinite number of un-eliminated 6n-1 and 6n+1 pairs? It will be absurd to say 5 primes for instance will be needed to collectively eliminate the 6n-1 and 6n+1 pairs because that will mean that the first of the 5 primes did not remove all the 6n-1 and 6n+1 pairs and similarly the second, third and fourth did not remove all the 6n-1 and 6n+1 pairs. It will be the fifth prime that will achieve this i.e. one and only one prime will be required to render the twin prime conjecture false. We are in agreement from previous posts that no prime can do this and so the twin prime conjecture is true, no doubt.
Oh sure, I do agree that if you know there are FINITELY many primes that eliminate all pairs, then it is obvious that there must be one that eliminates the remaining twin primes.
But there are an infinite number of primes. So how do you know that there are finitely many primes that eliminate all pairs?
And, I know we're stupid people. But if you give somebody a proof to get feedback on, and they ask a question, don't reply with saying it's obvious five times. It's just insulting. I don't care, but journals will.
Last edited by Micrm@ss; January 3rd, 2019 at 04:24 AM.
January 3rd, 2019, 06:28 AM #32
Senior Member
Joined: May 2016
From: USA
Posts: 1,310
Thanks: 551
Quote:
Originally Posted by MrAwojobi It will be absurd to say 5 primes for instance will be needed to collectively eliminate the 6n-1 and 6n+1 pairs because that will mean that the first of the 5 primes did not remove all the 6n-1 and 6n+1 pairs and similarly the second, third and fourth did not remove all the 6n-1 and 6n+1 pairs. It will be the fifth prime that will achieve this i.e. one and only one prime will be required to render the twin prime conjecture false.
If each of your five primes removes one fifth of the pairs, then all five are needed, not just the fifth. You have slipped in the assumption that the fifth can do all the eliminations that the first four do.
A cannot lift a beam. B cannot lift the same beam. A and B together cannot lift the same beam. A, B, and C together can lift that beam. That does not prove that C alone can lift the beam. A child can see the flaw in that argument.
So you have not established that if the twin primes conjecture is false, there is a single eliminating prime. Prove it.
January 3rd, 2019, 06:37 AM #33
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Joined: Oct 2009
Posts: 770
Thanks: 276
Quote:
Originally Posted by JeffM1 If each of your five primes removes one fifth of the pairs, then all five are needed, not just the fifth. You have slipped in the assumption that the fifth can do all the eliminations that the first four do. A cannot lift a beam. B cannot lift the same beam. A and B together cannot lift the same beam. A, B, and C together can lift that beam. That does not prove that C alone can lift the beam. A child can see the flaw in that argument. So you have not established that if the twin primes conjecture is false, there is a single eliminating prime. Prove it.
Exactly.
What he claims in his original post though, is that there is a single prime eliminating all the remaining twin primes ahead. This would be true if there were only 5 primes. However, this seems not what he needs later in the proof. Since later he does need one prime that eliminates EVERY twin prime, not just the remaining ones. So you're right, if he succeeds in proving his original claim that there is one prime that eliminates the remaining ones, that would not be enough.
January 3rd, 2019, 06:51 AM #34
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Quote:
Originally Posted by Micrm@ss Oh sure, I do agree that if you know there are FINITELY many primes that eliminate all pairs, then it is obvious that there must be one that eliminates the remaining twin primes.
This actually may concede too much. If there is a finite set E of eliminating primes and the number of elements in that set is e > 1, it is indeed obviously true that exactly one element of P must eliminate all the pairs that are not eliminated by the other e - 1 elements. But Mr. Awojobi seems to be making a stronger claim, namely that this true conditional proposition proves that e is not greater than 1.
Of course even if he can prove that stronger conditional proposition, which he has not, this does not address the much more plausible hypothesis that set E is infinite. If there are a finite number of twin primes, then the number of pairs (6n - 1 and 6n + 1) remaining are infinite, and there are an infinite number of primes that potentially can do the necessary eliminating. And in an infinite list, there is no last element.
January 3rd, 2019, 07:41 AM #35 Banned Camp Joined: Aug 2010 Posts: 170 Thanks: 4 I can't see what more I can say to make you see things from what to me is a straight forward point of view. Maybe I will try again. 5, for instance, eliminates all its infinite number of multiples from the infinite number line. Why then can you not all see that if the twin prime conjecture is false then it takes just one prime to eliminate all the un-eliminated, infinite number of 6n-1 and 6n+1 pairs? I repeat again that it will be absurd to say more than 1 prime is required to do this i.e. one might just as well make an absurd comment by saying that more than one prime is required to eliminate all the infinite number of multiples of 5 when it is clear that one prime,5, can do this.
January 3rd, 2019, 08:29 AM #36 Global Moderator Joined: Dec 2006 Posts: 20,469 Thanks: 2038 If it's so straightforward, you should be able to rewrite your proof without using words such as "obvious" and "absurd", and as a series of elementary steps, each of which is a direct consequence of a stated rule (or stated rules) of arithmetic or algebra.
January 3rd, 2019, 08:32 AM #37
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Quote:
Originally Posted by MrAwojobi I can't see what more I can say to make you see things from what to me is a straight forward point of view. Maybe I will try again. 5, for instance, eliminates all its infinite number of multiples from the infinite number line. Why then can you not all see that if the twin prime conjecture is false then it takes just one prime to eliminate all the un-eliminated, infinite number of 6n-1 and 6n+1 pairs? I repeat again that it will be absurd to say more than 1 prime is required to do this i.e. one might just as well make an absurd comment by saying that more than one prime is required to eliminate all the infinite number of multiples of 5 when it is clear that one prime,5, can do this.
You might assume there is a greatest prime that is also a twin prime and derive a contradiction. If such a prime $p$ existed, your method would eventually remove its multiples, but there would still be potential twins at the time you eliminate the multiples of $p$. You would then have to go past $p$ and use other primes to eliminate the remaining potential twin primes. There are infinitely many though and your method cannot halt on any particular prime that eliminates all other potential twin primes from the list. That’s why you fail.
Last edited by AplanisTophet; January 3rd, 2019 at 08:35 AM.
January 3rd, 2019, 08:40 AM #38 Global Moderator Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,929 Thanks: 1124 Math Focus: Elementary mathematics and beyond This seems to be an ineffective take-off on Zhang's method, which proved there are an infinite number of prime pairs with a prime gap around some number near 70 million.
January 3rd, 2019, 09:06 AM #39
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Math Focus: Wibbly wobbly timey-wimey stuff.
Quote:
Originally Posted by MrAwojobi ...Why then can you not all see that if the twin prime conjecture is false then it takes just one prime to eliminate all the un-eliminated, infinite number of 6n-1 and 6n+1 pairs?...
P.R.O.V.E. I.T!!
Your statement here, once again, does not involve a proof.
You are getting exasperated with us. That's a good thing! Provide a proof that doesn't rely on "hand waving" and you'll convince us that we've been wrong about this the whole time.
-Dan
January 3rd, 2019, 09:22 AM #40
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Quote:
Originally Posted by JeffM1 This actually may concede too much. If there is a finite set E of eliminating primes and the number of elements in that set is e > 1, it is indeed obviously true that exactly one element of P must eliminate all the pairs that are not eliminated by the other e - 1 elements.
It's proven earlier in the thread that there is no finite set E of eliminating primes.
Quote:
Originally Posted by JeffM1 Of course even if he can prove that stronger conditional proposition, which he has not, this does not address the much more plausible hypothesis that set E is infinite.
Yes, E is infinite. So now MrAwojobi, you have to prove that there is an infinite set E of eliminating primes that eliminates all potential twin primes greater than $p$ (where $p$ is the greatest prime that is also a twin prime). Under your method, that is a really tall order. Good luck!
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Similar Threads Thread Thread Starter Forum Replies Last Post MrAwojobi Number Theory 20 October 31st, 2018 01:06 AM Al7-8Ex5-3:Fe#!D%03 Number Theory 3 September 30th, 2013 04:52 PM Macky Number Theory 8 September 28th, 2010 11:39 AM MrAwojobi Number Theory 51 August 9th, 2010 11:09 AM ogajajames Number Theory 4 April 26th, 2010 05:51 AM
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2019-04-20 16:11:10
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https://www.r-bloggers.com/2019/06/how-to-perform-ordinal-logistic-regression-in-r/
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In this article, we discuss the basics of ordinal logistic regression and its implementation in R. Ordinal logistic regression is a widely used classification method, with applications in variety of domains. This method is the go-to tool when there is a natural ordering in the dependent variable. For example, dependent variable with levels low, medium, high is a perfect context for application of logistic ordinal regression.
Having wide range of applicability, ordinal logistic regression is considered as one of the most admired methods in the field of data analytics. The method is also known as proportional odds model because of the transformations used during estimation and the log odds interpretation of the output. We hope that this article helps our readers to understand the basics and implement the model in R.
The article is organized as follows: focusing on the theoretical aspects of the technique, section 1 provides a quick review of ordinal logistic regression. Section 2 discusses the steps to perform ordinal logistic regression in R and shares R script. In addition, section 2 also covers the basics of interpretation and evaluation of the model on R. In section 3, we learn a more intuitive way to interpret the model. Section 4 concludes the article.
Basics of ordinal logistic regression
Ordinal logistic regression is an extension of simple logistic regression model. In simple logistic regression, the dependent variable is categorical and follows a Bernoulli distribution. (for a quick reference check out this article by perceptive analytics – https://www.kdnuggets.com/2017/10/learn-generalized-linear-models-glm-r.html). Whereas, in ordinal logistic regression the dependent variable is ordinal i.e. there is an explicit ordering in the categories. For example, during preliminary testing of a pain relief drug, the participants are asked to express the amount of relief they feel on a five point Likert scale. Another common example of an ordinal variable is app ratings. On google play, customers are asked to rate apps on a scale ranging from 1 to 5. Ordinal logistic regression becomes handy in the aforementioned examples as there is a clear order in the categorical dependent variable.
In simple logistic regression, log of odds that an event occurs is modeled as a linear combination of the independent variables. But, the above approach of modeling ignores the ordering of the categorical dependent variable. Ordinal logistic regression model overcomes this limitation by using cumulative events for the log of the odds computation. It means that unlike simple logistic regression, ordinal logistic models consider the probability of an event and all the events that are below the focal event in the ordered hierarchy. For example, the event of interest in ordinal logistic regression would be to obtain an app rating equal to X or less than X. For example, the log of odds for the app rating less than or equal to 1 would be computed as follows:
LogOdds rating<1 = Log (p(rating=1)/p(rating>1) [Eq. 1]
Likewise, the log of odds can be computed for other values of app ratings. The computations for other ratings are below:
LogOdds rating<2 = Log (p(rating<=2)/p(rating>2) [Eq. 2]
LogOdds rating<3 = Log (p(rating<=3)/p(rating>3) [Eq. 3]
LogOdds rating<4 = Log (p(rating=4)/p(rating>4) [Eq. 4]
Because all the ratings below the focal score are considered in computation, the highest app rating of 5 will include all the ratings below it and does not have a log of odds associated with it. In general, the ordinal regression model can be represented using the LogOdds computation.
LogoddsYαi+ β1X12X2 +….. +βnXn
where,
Y is the ordinal dependent variable
i is the number of categories minus 1
X1, X2,…. Xn are independent variables. They can be measured on nominal, ordinal or continuous measurement scale.
β1, β2,… βare estimated parameters
For i ordered categories, we obtain i – 1 equations. The proportional odds assumption implies that the effect of independent variables is identical for each log of odds computation. But, this is not the case for intercept as the intercept takes different values for each computation. Besides the proportional odds assumption, the ordinal logistic regression model assumes an ordinal dependent variable and absence of multicollinearity. Absence of multicollinearity means that the independent variables are not significantly correlated. These assumptions are important as their violation makes the computed parameters unacceptable.
Model building in R
In this section, we describe the dataset and implement ordinal logistic regression in R. We use a simulated dataset for analysis. The details of the variables are as follows. The objective of the analysis is to predict the likelihood of each level of customer purchase. The dependent variable is the likelihood of repeated purchase by customers. The variable is measured in an ordinal scale and can be equal to one of the three levels – low probability, medium probability, and high probability. The independent variables are measures of possession of coupon by the focal customer, recommendation received by the peers and quality of the product. Possession of coupon and peer recommendation are categorical variables, while quality is measured on a scale of 1 to 5. We discuss the steps for implementation of ordinal logistic regression and share the commented R script for better understanding of the reader. The data is in .csv format and can be downloaded by clicking here.
Before starting the analysis, I will describe the preliminary steps in short. The first step is to keep the data file in the working directory. The next step is to explicitly define the ordering of the levels in the dependent variable and the relevant independent variables. This step is crucial and ignoring it can lead to meaningless analysis.
#Read data file from working directory
setwd("C:/Users/You/Desktop")
#Ordering the dependent variable
data$rpurchase = factor(data$rpurchase, levels = c("low probability", "medium probability", "high probability"), ordered = TRUE)
data$peers = factor(data$peers, levels = c("0", "1"), ordered = TRUE)
data$coupon = factor(data$coupon, levels = c("0", "1"), ordered = TRUE)
Next, it is essential to perform the exploratory data analysis. Exploratory data analysis deals with outliers and missing values, which can induce bias in our model. In this article we do basic exploration by looking at summary statistics and frequency table. Figure 1. shows the summary statistics. We observe the count of data for ordinal variables and distribution characteristics for other variables. We compute the count of rpurchase with different values of coupon in Table 1. Note that the median value for rpurchase changes with change in coupon. The median level of rpurchase increases, indicating that coupon positively affects the likelihood of repeated purchase. The R script for summary statistics and the frequency table is as follows:
#Exploratory data analysis
#Summarizing the data
summary(data)
#Making frequency table
table(data$rpurchase, data$coupon)
Figure 1. Summary statistics
Table 1. Count of rpurchase by coupon
Rpurchase/Coupon With coupon Without coupon Low probability 200 20 Medium probability 110 30 High proabability 27 13
After defining the order of the levels in the dependent variable and performing exploratory data analysis, the data is ready to be partitioned into training and testing set. We will build the model using the training set and validate the computed model using the data in test set. The partitioning of the data into training and test set is random. The random sample is generated using sample ( ) function along with set.seed ( ) function. The R script for explicitly stating the order of levels in the dependent variable and creating training and test data set is follows:
#Dividing data into training and test set
#Random sampling
samplesize = 0.60*nrow(data)
set.seed(100)
index = sample(seq_len(nrow(data)), size = samplesize)
#Creating training and test set
datatrain = data[index,]
datatest = data[-index,]
Now, we will build the model using the data in training set. As discussed in the above section the dependent variable in the model is in the form of log odds. Because of the log odds transformation, it is difficult to interpret the coefficients of the model. Note that in this case the coefficients of the regression cannot be interpreted in terms of marginal effects. The coefficients are called as proportional odds and interpreted in terms of increase in log odds. The interpretation changes not only for the coefficients but also for the intercept. Unlike simple linear regression, in ordinal logistic regression we obtain n-1 intercepts, where n is the number of categories in the dependent variable. The intercept can be interpreted as the expected odds of identifying in the listed categories. Before interpreting the model, we share the relevant R script and the results. In the R code, we set Hess equal to true which the logical operator to return hessian matrix. Returning the hessian matrix is essential to use summary function or calculate variance-covariance matrix of the fitted model.
#Build ordinal logistic regression model
model= polr(rpurchase ~ coupon + peers + quality , data = datatrain, Hess = TRUE)
summary(model)
Figure 2. Ordinal logistic regression model on training set
The table displays the value of coefficients and intercepts, and corresponding standard errors and t values. The interpretation for the coefficients is as follows. For example, holding everything else constant, an increase in value of coupon by one unit increase the expected value of rpurchase in log odds by 0.96. Likewise, the coefficients of peers and quality can be interpreted.
Note that the ordinal logistic regression outputs multiple values of intercepts depending on the levels of intercept. The intercepts can be interpreted as the expected odds when others variables assume a value of zero. For example, the low probability | medium probability intercept takes value of 2.13, indicating that the expected odds of identifying in low probability category, when other variables assume a value of zero, is 2.13. Using the logit inverse transformation, the intercepts can be interpreted in terms of expected probabilities. The inverse logit transformation, . The expected probability of identifying low probability category, when other variables assume a value of zero, is 0.89.
After building the model and interpreting the model, the next step is to evaluate it. The evaluation of the model is conducted on the test dataset. A basic evaluation approach is to compute the confusion matrix and the misclassification error. The R code and the results are as follows:
#Compute confusion table and misclassification error
predictrpurchase = predict(model,datatest)
table(datatest$rpurchase, predictrpurchase) mean(as.character(datatest$rpurchase) != as.character(predictrpurchase))
Figure 3. Confusion matrix
The confusion matrix shows the performance of the ordinal logistic regression model. For example, it shows that, in the test dataset, 76 times low probability category is identified correctly. Similarly, 10 times medium category and 0 times high category is identified correctly. We observe that the model identifies high probability category poorly. This happens because of inadequate representation of high probability category in the training dataset. Using the confusion matrix, we find that the misclassification error for our model is 46%.
Interpretation using plots
The interpretation of the logistic ordinal regression in terms of log odds ratio is not easy to understand. We offer an alternative approach to interpretation using plots. The R code for plotting the effects of the independent variables is as follows:
#Plotting the effects
library("effects")
Effect(focal.predictors = "quality",model)
plot(Effect(focal.predictors = "coupon",model))
plot(Effect(focal.predictors = c("quality", "coupon"),model))
The plots are intuitive and easy to understand. For example, figure 4 shows that coupon increases the likelihood of classification into high probability and medium probability classes, while decreasing the likelihood of classification in low probability class.
Figure 4. Effect of coupon on identification
It is also possible to look at joint effect of two independent variable. Figure 5. shows the joint effect of quality and coupon on identification of category of independent variable. Observing the top row of figure 5., we notice that the interaction of coupon and quality increases the likelihood of identification in high probability category.
Figure 5. Joint effect of quality and coupon on identification
Conclusion
The article discusses the fundamentals of ordinal logistic regression, builds and the model in R, and ends with interpretation and evaluation. Ordinal logistic regression extends the simple logistic regression model to the situations where the dependent variable is ordinal, i.e. can be ordered. Ordinal logistic regression has variety of applications, for example, it is often used in marketing to increase customer life time value. For example, consumers can be categorized into different classes based on their tendency to make repeated purchase decision. In order to discuss the model in an applied manner, we develop this article around the case of consumer categorization. The independent variables of interest are – coupon held by consumers from previous purchase, influence of peers, quality of the product.
The article has two key takeaways. First, ordinal logistic regression come handy while dealing with a dependent variable that can be ordered. If one uses multinomial logistic regression then the user is ignoring the information related to ordering of the dependent variable. Second, the coefficients of the ordinal linear regression cannot be interpreted in a similar manner to the coefficients of ordinary linear regression. Interpreting the coefficents in terms of marginal effects is one of the common mistakes that users make while implementing the ordinal regression model. We again emphasize the use of graphical method to interpret the coefficients. Using the graphical method, it is easy to understand the individual and joint effects of independent variables on the likelihood of classification.
This article can be a go to reference for understanding the basics of ordinal logistic regression and its implementation in R. We have provided commented R code throughout the article.
Credits: Chaitanya Sagar and Aman Asija of Perceptive Analytics. Perceptive Analytics is a marketing analytics and Tableau consulting company.
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2020-11-23 17:17:09
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https://www.greaterwrong.com/posts/svE3S6NKdPYoGepzq/topological-fixed-point-exercises
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# Topological Fixed Point Exercises
This is one of three sets of fixed point exercises. The first post in this sequence is here, giving context.
1. (1-D Sperner’s lemma) Consider a path built out of edges as shown. Color each vertex blue or green such that the leftmost vertex is blue and the rightmost vertex is green. Show that an odd number of the edges will be bichromatic.
1. (Intermediate value theorem) The Bolzano-Weierstrass theorem states that any bounded sequence in has a convergent subsequence. The intermediate value theorem states that if you have a continuous function such that and , then there exists an such that . Prove the intermediate value theorem. It may be helpful later on if your proof uses 1-D Sperner’s lemma and the Bolzano-Weierstrass theorem.
2. (1-D Brouwer fixed point theorem) Show that any continuous function has a fixed point (i.e. a point with ). Why is this not true for the open interval ?
3. (2-D Sperner’s lemma) Consider a triangle built out of smaller triangles as shown. Color each vertex red, blue, or green, such that none of the vertices on the large bottom edge are red, none of the vertices on the large left edge are green, and none of the vertices on the large right edge are blue. Show that an odd number of the small triangles will be trichromatic.
1. Color the all the points in the disk as shown. Let be a continuous function from a closed triangle to the disk, such that the bottom edge is sent to non-red points, the left edge is sent to non-green points, and the right edge is sent to non-blue points. Show that sends some point in the triangle to the center.
1. Show that any continuous function from closed triangle to itself has a fixed point.
2. (2-D Brouwer fixed point theorem) Show that any continuous function from a compact convex subset of to itself has a fixed point. (You may use the fact that given any closed convex subset of , the function from to which projects each point to the nearest point in is well defined and continuous.)
3. Reflect on how non-constructive all of the above fixed-point findings are. Find a parameterized class of functions where for each , , and the function is continuous, but there is no continuous way to pick out a single fixed point from each function (i.e. no continuous function such that is a fixed point of for all ).
4. (Sperner’s lemma) Generalize exercises 1 and 4 to an arbitrary dimension simplex.
5. (Brouwer fixed point theorem) Show that any continuous function from a compact convex subset of to itself has a fixed point.
6. Given two nonempty compact subsets , the Hausdorff distance between them is the supremum over all points in either subset of the distance from that point to the other subset. We call a set valued function a continuous Hausdorff limit if there is a sequence of continuous functions from to whose graphs, , converge to the graph of , , in Hausdorff distance. Show that every continuous Hausdorff limit from a compact convex subset of to itself has a fixed point (a point such that ).
7. Let and be nonempty compact convex subsets of . We say that a set valued function, is a Kakutani function if the graph of , , is closed, and is convex and nonempty for all . For example, we could take and to be the interval , and we could have send each to , map to the whole interval , and map to . Show that every Kakutani function is a continuous Hausdorff limit. (Hint: Start with the case where is a unit cube. Construct by breaking into small cubes of side length . Constuct a smaller cube of side length within each cube. Send each small to the convex hull of the images of all points in the cube with a continuous function, and glue these together with straight lines. Make sure you don’t accidentally get extra limit points.)
8. (Kakutani fixed point theorem) Show that every Kakutani function from a compact convex subset of to itself has a fixed point.
Please use the spoilers feature—the symbol ‘>’ followed by ‘!’ followed by space -in your comments to hide all solutions, partial solutions, and other discussions of the math. The comments will be moderated strictly to hide spoilers!
I recommend putting all the object level points in spoilers and leaving metadata outside of the spoilers, like so: “I think I’ve solved problem #5, here’s my solution <spoilers>” or “I’d like help with problem #3, here’s what I understand <spoilers>” so that people can choose what to read.
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• Proof of #4, but with unnecessary calculus:
Not only is there an odd number of tricolor triangles, but they come in pairs according to their orientation (RGB clockwise/anticlockwise). Proof: define a continuously differentiable vector field on the plane, by letting the field at each vertex be 0, and the field in the center of each edge be a vector of magnitude 1 pointing in the direction R->G->B->R (or 0 if the two adjacent vertices are the same color). Extend the field to the complete edges, then the interiors of the triangles by some interpolation method with continuous derivative (eg. cosine interpolation).
Assume the line integral along one unit edge in the direction R->G or G->B or B->R to be 13. (Without loss of generality since we can rescale the graph/vectors to make this true). Then a similar parity argument to Sperner’s 1d lemma (or the FTC) shows that the clockwise line integral along each large edge is 13, hence the line integral around the large triangle is 1/3+1/3+1/3=1.
By green’s theorem, this is equal to the integrated curl of the field in the interior of the large triangle, and hence equal (by another invocation of green’s theorem) to the summed clockwise line integrals around each small triangle. The integrals around a unicolor or bicolor triangle are 0 and −1/3 + 13 + 0 = 0 respectively, leaving only tricolor triangles, whose integral is again 1 depending on orientation. Thus: (tricolor clockwise) - (tricolor anticlockwise) = 1. QED.
• Just to get things started, here’s a proof for #1:
Proof by induction that the number of bicolor edges is odd iff the ends don’t match. Base case: a single node has matching ends and an even number (zero) of bicolor edges. Extending with a non-bicolor edge changes neither condition, and extending with a bicolor edge changes both; in both cases the induction hypothesis is preserved.
• Here’s a more conceptual framing:
If we imagine blue as labelling the odd numbered segments and green as labelling the even numbered segments, it is clear that there must be an even number of segments in total. The number of gaps between segments is equal to the number of segments minus 1, so it is odd.
• Cleanest solution I can find for #8:
Also, if we have a proof for #6 there’s a pleasant method for #7 that should work in any dimension:
We take our closed convex set that has the bounded function . We take a triangle that covers so that any point in is also in .
Now we define a new function such that where is the function that maps to the nearest point in .
By #6 we know that has a fixed point, since is continuous. We know that the fixed point of cannot lie outside because the range of is . This means has a fixed point within and since for , , has a fixed point.
• On my approach:
I constructed a large triangle around the convex shape with the center somewhere in the interior. I then projected each point in the convex shape from the center towards the edge of the triangle in a proportional manner. ie. The center stays where it is, the points on the edge of the convex shape are projected to the edge of the triangle and a point 1/x of the distance from the center to the edge of the convex shape is 1/x of the distance from the center to the edge of the triangle.
• If #4 is true, it is provable:
If #4 is true, changing the color of a single node (except to forbidden colors on edges) cannot change the parity of the trichromatic triangle count, and this would be checkable through a finite case analysis of graphs of size <=7. Given that lemma, we can recolor one corner to red, the remainder of one large edge blue and the remaining nodes green, producing the odd count 1.
#8:
We are looking for a surface [0,1]²->[0,1] whose intersection with the plane x=y does not contain a function of t. It suffices to show that the intersection looks like the letter s, with exactly the endpoints reaching t=0 or t=1. It suffices to show that the intersection can be any continous function of x including the points t=x=y=0 and t=x=y=1. Within each plane of constant x, define the surface as 0 for small t, 1 for large t, and rapidly rising through the plane x=y wherever we want the intersection.
• Found a nice proof for Sperner’s lemma (#9):
First some definitions. Call a d-simplex with vertices colored in (d+1) different colored chromatic. Call the parity of the number of chromatic simplices the chromatic parity.
It’s easier to prove the following generalization: Take a complex of d-simplices that form a d-sphere: then any (d+1)-coloring of the vertices will have even chromatic parity.
Proof by induction on d:
Base d=-1: vacuously true.
Assume true for d-1: Say you have an arbitrary complex of d-simplices forming a d-sphere, with an arbitrary d+1-coloring. Choose a vertex. W.L.O.G. we will call the color of the chosen vertex blue.
Take the complex of simplices that contain this vertex. Since a sphere has no boundary or branches, this complex will be a d-ball. Delete the chosen vertex, and keep only the opposite (d-1)-simplices that are left, which will form a (d-1)-sphere, call it the shell.
We need to choose a second color, say red. We’ll call a (d-1)-simplex with vertices of all d+1 colors except red an R-chromatic face, and similarly with blue.
Now, replace all the red vertices in the shell with blue vertices, so that the shell is now R-chromatic. By induction it has an even number of R-chromatic faces. Consider what happens when we reconvert a vertex on the shell back to red: since the vertex was previously blue, any R-chromatic faces will get turned into B-chromatic faces. Let r be the number of R-chromatic faces on the shell, and b be the number of B-chromatic faces. The parity of r-b will remain even as we continue this process.
Let’s go back to the vertex in the center of the shell. All currently chromatic simplices with this vertex have opposite faces which are B-chromatic, since this vertex is blue. We’ll flip the vertex to red, which destroys chromatic simplices with opposite B-chromatic faces and creates chromatic simplices with opposite R-chromatic faces. Since r-b is even, the chromatic parity is preserved!
Since we’ve shown that arbitrary recoloring of vertices preserves the chromatic parity, it’s clear that the chromatic parity will be even for any coloring.
Corollary: Sperner’s lemma
Start with a d-simplex which has been divided into d-simplices, and where each face of the large simplex has one color which vertices on it are forbidden from taking. Take a point of each color, and match it with a face of the simplex that that color is allowed on. Then connect that vertex to each point on that face. This will create a bunch of non-chromatic simplices. Finally, create a simplex of all of the new points. This will create one chromatic simplex.
This will form a d-sphere, and thus will have an even chromatic parity. That means the original simplex must have had odd chromatic parity.
• Thanks! I find this approach more intuitive than the proof of Sperner’s lemma that I found in Wikipedia. Along with nshepperd’s comment, it also inspired me to work out an interesting extension that requires only minor modifications to your proof:
d-spheres are orientable manifolds, hence so is a decomposition of a d-sphere into a complex K of d-simplices. So we may arbitrarily choose one of the two possible orientations for K (e.g. by choosing a particular simplex P in K, ordering its vertices from 1 to d + 1, and declaring it to be the prototypical positively oriented simplex; when d = 2, P could be a triangle with the vertices going around counterclockwise when you count from 1 to 3; when d = 3, P could be a tetrahedron where, if you position your right hand in its center and point your thumb at the 1-vertex, your fingers curl around in the same direction in which you count the remaining vertices from 2 to 4). Then any ordering of the vertices of any d-simplex in K may be said to have positive or negative orientation (chirality). (E.g. it would be positive when there’s an orientation-preserving map (e.g. a rotation) sending each of its vertices to the correspondingly numbered vertices of P.)
So here’s my extension of parent comment’s theorem: Any coloring of the vertices of K with the colors {1, …, d + 1} will contain an equal number of positively and negatively oriented chromatic d-simplices—that is, the reason the number of chromatic d-simplices in K must be even is that each one can be paired off with one of the opposite (mirror) orientation. (Does this theorem have a name? If so I couldn’t find it.)
Following parent comment, the proof is by induction on the dimension d. When d = 1, K is just a cycle graph with vertices colored 1 or 2. As we go around clockwise (or counterclockwise), we must traverse an equal number of 1→2 edges and 2→1 edges (i.e. oppositely oriented 1-simplices), by the time we return to our starting point.
Now let d > 1, and assume the theorem is true in the (d-1)-dimensional case. As in parent comment, we may choose any vertex v of K, and then the faces opposite to v in each d-simplex in K that contains v will, together, form a (d-1)-dimensional subcomplex K′ of K that is homeomorphic (topologically equivalent) to a (d-1)-sphere.
Suppose v has color i. We will show that changing v’s color to ji will add or remove the same number of positively oriented chromatic d-simplices as negatively oriented ones: Forget, for the moment, the distinction between colors i and j—say any i or j-colored vertex of K′ has color “i-or-j.” Then K′ is now d-colored, so, by inductive hypothesis, the chromatic (d-1)-simplices of K′ must occur in pairs of opposite orientation (if any exist—if none exist, v can’t be part of any chromatic d-simplex regardless of its color). Consider such a pair, call them F₁ and F₂.
Now cease pretending that i and j are a single color. Since F₁ was chromatic in K′, it must have had an i-or-j–colored vertex. Suppose, WOLOG, that that vertex was actually j-colored. Then, together with i-colored v, F₁ spans a chromatic d-simplex of K, call it S₁, which we may assume WOLOG to be positively oriented. Once we change the color of v from i to j, however, S₁ will have two j-colored vertices, and so will no longer be chromatic. To see that balance is maintained, consider what happens with F₂:
If F₂‘s i-or-j–colored vertex was, like F₁‘s, actually j-colored, then the d-simplex spanned by F₂ and v, call it S₂, was chromatic and negatively oriented (because F₂ had opposite orientation to F₁ in K′), and thus S₂ also ceased to be chromatic when we changed v’s color from i to j, balancing S₁‘s loss of chromatic status. Otherwise, F₂‘s i-or-j–colored vertex must have been i-colored, in which case S₂ wasn’t chromatic when v was also i-colored, but changing v’s color to j turned S₂ into a new d-chromatic simplex. But what is S₂‘s orientation? Well, it was negative under the assumption that S₂‘s i-or-j–colored vertex was j-colored and v was i-colored, and swapping the labels of a pair of vertices in an oriented simplex reverses its orientation, so, under the present assumption, S₂’s orientation must be positive! Thus the loss of S₁ as a positively oriented chromatic d-simplex is balanced by the addition of S₂ as a new positively oriented chromatic d-simplex.
If all of K’s vertices are the same color, it has the same number (zero) of positively and negatively oriented chromatic d-simplices, and since this parity is preserved when we change the colors of the vertices of K one at a time, it remains no matter how we (d+1)-color K. ☐
We can relate this theorem back to Sperner’s lemma using the same trick as parent comment: Suppose we are given a triangulation K of a regular d-simplex S into smaller d-simplices, and a (d+1)-coloring of K’s vertices that assigns a unique color to each vertex v of S, and doesn’t use that color for any of K’s vertices lying on the face of S opposite to v. We form a larger simplicial complex L containing K by adding d + 1 new vertices as follows: For i = 1, …, d + 1, place a new i-colored vertex exterior to S, some distance from the center of S along the ray that goes through the i-colored vertex of S. Connect this new vertex to each vertex of K lying in the face of S opposite from the (i+1)-colored (or 1-colored, when i = d + 1) vertex of S. (Note that none of the d-simplices thereby created will be chromatic, because they won’t have an (i+1)-colored vertex.) Then connect all of the new vertices to each other.
Having thus defined L, we can embed it in a d-sphere, of which it will constitute a triangulation, because the new vertices will form a d-simplex T whose “interior” is the complement of L in the sphere. Thus we can apply our above theorem to conclude that L has equally many positively and negatively oriented chromatic d-simplices. By construction, none of L’s new vertices are included in any chromatic d-simplex other than T, so K must contain an equal number (possibly zero) of positively and negatively oriented chromatic d-simplices, plus one more, of opposite orientation to T. And what is the orientation of T? I claim that it is opposite to that of S: Consider T by itself, embedded in the sphere. T’s boundary and exterior (the interior of L) then constitute another chromatic d-simplex, call it U, which is essentially just a magnification of S, with correspondingly colored vertices, and so share’s S’s orientation. Applying our theorem again, we see that T and U must have opposite orientations*, so we conclude that K must contain exactly one more chromatic d-simplex of the same orientation as S than of the opposite orientation. (As proved in nshepperd’s comment for the case d = 2.)
*The observation, that, on the surface of a sphere, the interior and exterior of a trichromatic triangle have opposite orientations, is what sent me down this rabbit hole in the first place. :)
• Clarifying question for #9:
How does the decomposition into segments/triangles generalize to 3+ dimensions? If you try decomposing a tetrahedron into multiple tetrahedra, you actually get 4 tetrahedra and 1 octahedron, as shown here.
EDIT: answered my own question:
You can decompose an octahedron into 4 tetrahedrons. They’re irregular, but this is actually fine for the purpose of the lemma.
• Rough approach for qu 6:
Join the center to each of the corners and color each segment a different color and arbitrarily coloring each ambiguous point. Radially extend the colored sections to infinity.
To prove f(x) has a fixed point consider g(x) = f(x) - x which can take values outside of the triangle. To find a fixed point, we simply need to show that g(x) will map at least one point to the center. It is easy to prove that each corner will map to the color opposite and each edge can only map to points of a different color (unless it passes through the center, in which case we would have obtained out proof). At this point the problem reduces to 5 assuming that we construct a big enough circle.
• Inappropriately highbrow proof of #4 (2d Sperner’s lemma):
This proves a generalization: any number of dimensions, and any triangulation of the simplex in question. So, the setup is as follows. We have an n-dimensional simplex, defined by n+1 points in n-dimensional space. We colour the vertices with n+1 different colours. Then we triangulate it—chop it up into smaller simplexes—and we extend our colouring somehow in such a way that the vertices on any face (note: a face is the thing spanned by any subset of the vertices) of the big simplex are coloured using only the colours from the vertices that span that face. And the task is to prove that there are an odd number of little simplexes whose vertices have all n+1 colours.
This colouring defines a map from the vertices of the triangulation to the vertices of the big simplex: map each triangulation-vertex to the simplex-vertex that’s the same colour. We can extend this map to the rest of each little simplex by linear interpolation. The resulting thing is continuous on the whole of the big simplex, so we have a continuous map (call it f) from the big simplex to itself. And we want to prove that we have an odd number of little simplices whose image under f spans the whole thing. (Call these “good” simplices.)
We’ll do it with two ingredients. The easy one is induction: when proving this in n dimensions we shall assume we already proved it for smaller numbers of dimensions. The harder one is homology, a standard tool in algebraic topology. More precisely we’ll do homology mod 2. It associates with each topological space X and each dimension d an abelian group Hd(X), and the key things you need to know are (1) that if you have f : X → Y then you get an associated group homomorphism f* : Hd(X) → Hd(Y), (2) that Hd(a simplex) is the cyclic group of order 2 if d=0, and the trivial group otherwise, and (3) that Hd(the boundary of a simplex) is the cyclic group of order 2 if d=0 or d = (dimension of simplex − 1) and the trivial group otherwise. Oh, and one other crucial thing: if you have f : X → Y and g : Y → Z then (gf)* = g*f*: composition of maps between topological space corresponds to composition of homomorphisms between their homology groups.
(You can do homology “over” any commutative ring. The groups you get are actually modules over that ring. It happens that the ring of integers mod 2 is what we want to use. A simplex is, topologically, the same thing as a ball, and its boundary the same thing as a sphere.)
OK. So, first of all suppose not only that the number of good simplices isn’t odd, but that it’s actually zero. Then f maps the whole of our simplex to its boundary. Let’s also consider the rather boring map g from the boundary to the whole simplex that just leaves every point where it is. Now, if the thing we’re trying to prove is true in lower dimensions then in particular the map gf—start on the boundary of the simplex, stay where you are using g, and then map to the boundary of the simplex again using f—has an image that, so to speak, covers each boundary face of the simplex an odd number of times. This guarantees—sorry, I’m eliding some details here—that (gf)* (from the cyclic group of order 2 to the cyclic group of order 2) doesn’t map everything to the identity. But that’s impossible, because (gf)*=g*f* and the map f* maps to Hn(whole simplex) which is the trivial group.
Unfortunately, what we actually need to assume in order to prove this thing by contradiction is something weaker: merely that the number of good simplices is even. We can basically do the same thing, because homology mod 2 “can’t see” things that happen an even number of times, but to see that we need to look a bit further into how homology works. I’m not going to lay it all out here, but the idea is that to build the Hd(X) we begin with a space of things called “chains” which are like linear combinations (in this case over the field with two elements) of bits of X, we define a “boundary” operator which takes combinations of d-dimensional bits of X and turns them into combinations of (d-1)-dimensional bits in such a way that the boundary of the boundary of anything is always zero, and then we define Hd(x) as a quotient object: (d-dimensional things with zero boundary) / (boundaries of d+1-dimensional things). Then the way we go from f (a map of topological spaces) to f* (a homomorphism of homology groups) is that f extends in a natural way to a map between chains, and then it turns out that this map interacts with the boundary operator in the “right” way for this to yield a map between homology groups. And (getting, finally, to the point) if in our situation the number of good simplices is even, then this means that the map of chains corresponding to f sends anything in n dimensions to zero (essentially because it means that the interior of the simplex gets covered an even number of times and when working mod 2, even numbers are zero), which means that we can think of f* as mapping not to the homology groups of the whole simplex but to those of its boundary—and then the argument above goes through the same as before.
I apologize for the handwaving above. (Specifically, the sentence beginning “This guarantees”.) If you’re familiar with this stuff, it will be apparent how to fill in the details. If not, trying to fill them in will only add to the pain of what’s already too long a comment :-).
This is clearly much too much machinery to use here. I suspect that if we took the argument above, figured out exactly what bits of machinery it uses, and then optimized ruthlessly we might end up with a neat purely-combinatorial proof, but I regret that I am too lazy to try right now.
• My solution for #3:
Define by . We know that is continuous because and the identity map both are, and by the limit laws. Applying the intermediate value theorem (problem #2) we see that there exists such that . But this means , so we are done.
Counterexample for the open interval: consider defined by . First, we can verify that if then , so indeed maps to . To see that there is no fixed point, note that the only solution to in is , which is not in . (We can also view this graphically by plotting both and and checking that they do not intersect in .)
• EDIT: I’ve got another framing that I thought would be more useful for later problems, but I was wrong. I still think there is some value in understanding this proof as well.
In particular, look at this diagram on Wikipedia. It would be better if the whole upper triangle was blue and the whole lower triangle were red instead of just one side (you can arbitrarily decide whether to paint the rest of the diagonal blue or red). If x=0 and x=1 aren’t fixed points, then they must be blue and red respectively. If we split [0,1] into n components of size 1/n, then we can see where f(x) maps each such component to and form a line of colored points as in q1. Proving this using Sperner’s Lemma is then essentially the same as qu2.
• Yeah, I did the same thing :)
Putting it right after #2 was highly suggestive—I wonder if this means there’s some very different route I would have thought of instead, absent the framing.
• I’m late, but I’m quite proud of this proof for #4:
Call the large triangle a graph and the triangles simply triangles. First, note that for any size, there is a graph where the top node is colored red, the remaining nodes on the right diagonal are colored green, and all nodes not on the right diagonal are colored blue. This graph meets the conditions, and has exactly one trichromatic triangle, namely the one at the top (no other triangle contains a red node). It is trivial to see that this graph can be changed into an arbitrary graph by re-coloring finitely many nodes. This will affect up to six triangles; we say that a triangle has changed iff it was trichromatic before the recoloring but not after, or vice versa, and we shall show that re-coloring any node leads to an even number of triangles being changed. This proves that the number of trichromatic triangles never stops being odd.
Label the three colors , and . Let be the node being recolored, wlog from to . Suppose first that has six neighbors. It is easy to see that a triangle between and two neighbors has changed if and only if one of the neighbors has color and the other has color or . Thus, we must show that the number of such triangles is even. If all neighbors have color , or if none of them do, then no triangles have changed. If exactly one node has color , then the two adjacent triangles have changed. Otherwise, let and denote two different neighbors of color . There are two paths using only neighbors of between and . Both start and end at a node of color . By the 1-D Sperner Lemma (assuming the more general result), it follows that both paths have an even number of edges between nodes of color and ; these correspond to the triangles that have changed.
If is a node on one of the graph’s boundaries changing color from to , it has exactly 4 neighbors and three adjacent triangles. The two neighbors that are also on the boundary cannot have color , so either none, one, or both of the ones that aren’t do. If it’s none, no triangle has changed; if it’s one, the two neighboring triangles have changed; and if it’s both, then the two triangles with two nodes on the graph’s boundary have changed.
• Some preliminary thoughts on q9:
As Jessicata pointed out, in 3-dimensions or higher, our n-hedra don’t split up as nicely as in the 2d case.
That isn’t the only issue: many of the surfaces of connected blocks may not correspond to the type of 2d grid that we just proved this result for and it doesn’t seem trivial to figure out how to characterise what kind of grids we need to extend our result too (it will be even worse in higher dimensions)
I’ve found two results: firstly that if you remove a triangular face and replace it with three others (imagine allowing the surface to jut out of the page), then the trichromatic parity will be preserved. Secondly, that we can replace two triangle with four triangles
Given what I’ve discussed above, I’d be keen for a hint as learning enough geometry to make progress on this problem would seem to be taking me pretty far afield from maths useful for ai-risk.
• Here’s the rough idea for 5 (not a full-proof)
The bottom edge must stick in the blue and green sections meaning that if we were to divide the edge in n and see where these points map to, we would find that it would be blue or green and similarly the other edges would match the limitations in q4. If we look at the right corner, we see that the bottom edge maps to green or blue and the right edge maps to green or red, so the bottom corner must be in green. Similarly the other corners match the requirements of q4.
This lets us find a smaller trichromatic triangle. We can repeat this process an arbitrary number of times. Consider the range of possible x and y co-ordinates of elements in these triangles. Each time this will reduce by a particular factor, so we can see that the range of each co-ordinate will approach 0. I’ll leave it to the reader to show that this means that these ranges converge to a point. I’ll also leave it to the reader to show that each trichromatic sub-triangle must contain the center (you may want to look up winding numbers).
• I’ve realised that you’ve gotta be careful with this method because when you find a trichromatic subtriangle of the original, it won’t necessarily have the property of only having points of two colours along the edges, and so may not in fact contain a point that maps to the centre.
This isn’t a problem if we just increase the number n by which we divide the whole triangle instead of recursively dividing subtriangles. Unfortunately now we’re not reducing the range of co-ords where this fixed point must be, only finding a triad of arbitrarily close points that map to a triangle surrounding the centre. You can, for example, take the centre point of the first of these triangles (with some method of numbering to make the function definite) for each value of as a sequence in . This must have a convergent sequence which should converge to a point that maps to the centre but I can’t prove that last stage.
• Here’s a solution to 4:
We will first prove a lemma that all connected groups of green blocks that are completely surrounded by red or blue blocks will produce an even number of trichromatic triangles. We will then augment the triangle by adding an extra blue row on the bottom and an extra red side on the right such that we now have would what be a triangle if it weren’t missing the bottom right corner. This will mean that all green blocks will now be surrounded so we’ll have an even number of trichromatic triangles, but we have added exactly one additional such triangle, so that the original has an odd number.
------
Proof of Lemma:
A trivial variant of 1-d Sperner’s Lemma is that if we start and finish on the same colour, we get an even number of bichromatic edges. For any block that is completely surrounded by red and blue blocks, we apply this variant to show that there is an even number of bichromatic edges on the outside that then translates to an even number of trichromatic triangles.
EDIT: Actually, there is one case we can run into which is tricky and that is something like:
R R R R R
R G G G R
R G R B R
R G G G R
R R R R R R
To see how to solve this case, read how to solve tricky interior cases below.
END EDIT
Thick interior blocks work similarly, but we can run into weird scenarios such as a blue and a red surrounding by green block:
g g g
g b r g
g g g
We might also run into something like this (ie. a three-spoked shape that is blue at the center and red on the sides surrounded by green).
g g g
g r g g
g b r g
g r g g
g g g
For these weird shapes we can still trace a path around this interior section, we just include going both down and up a spoke in our path (thanks Hoagy!). So the interior sections are also even.
• Strategies that I’ve found helpful:
If something doesn’t seem tractable, try flipping between algebraic and geometric interpretations of a problem. Problems 1 and 3 fell to this approach.
Specific solutions (or suggestive handwaving):
Problem 1:
I thought of it like parity—going left to right, each unichromatic edge doesn’t change the color, while each bichromatic edge does. So to have an overall change, we need either 1 bichromatic edge, or 3 (1 and 2 that cancel), or 5 (1 and 4 that cancel)...
Problem 2:
I couldn’t understand this one at first. After checking Wikipedia, I think that refers to the space that each point in the sequence lies within. An example of a finite sequence in would then be
Problem 3:
Consider the unit square. We need to draw one continuous line, going from left to right, that covers the entire vertical extent of the square. No matter how you do that, you need to cross the diagonal line from the bottom left to the top right.
Why? Because you need to touch the top and the bottom edges. You can’t do that at the bottom-left or top-right corners, since then you’d touch the diagonal line. But then the point where you touch the top edge is entirely within the top triangle, and it cannot touch the bottom edge without entering the bottom triangle. Switching between triangles is identical to crossing the diagonal line.
As for why this isn’t true if the set is open rather than closed: if we exclude the edges from our consideration of “does it intersect the diagonal”, then it’s fairly trivial to construct a curve that stays inside one triangle and has a codomain of (0,1). should work.
• #8 actually comes up in physics:
in the field of nonlinear dynamics (pretty picture, actual wikipedia). The fact that continuous changes in functions can lead to surprising changes in fixed points (specifically stable attractors) is pretty darn important to understanding e.g. phase transitions!
• Does this work for #7? (and question) (Spoilers for #6):
I did #6 using 2D Sperner’s lemma and closedeness. Imagine the the destination points are colored [as in #5, which was a nice hint] by where they are relative to their source points—split the possible difference vectors into a colored circle as in #5 [pick the center to be a fourth color so you can notice if you ever sample a fixed point directly, but if fixed points are rare this shouldn’t matter], and take samples to make it look like 2d Sperner’s lemma, in which there must be at least one interior tri-colored patch. Define a limit of zooming in that moves you towards the tri-colored patch, apply closedness to say the center (fixed) point is included, much like how we were encouraged to do #2 with 1D Sperner’s lemma.
To do #7, it seems like you just need to show that there’s a continuous bijection that preserves whether a point is interior or on the edge, from any convex compact subset of R^2 to any other. And there is indeed a recipe to do this—it’s like you imagine sweeping a line across the two shapes, at rates such that they finish in equal time. Apply a 1D transformation (affine will do) at each point in time to make the two cross sections match up and there you are. This uses the property of convexity, even though it seems like you should be able to strengthen this theorem to work for simply connected compact subsets (if not—why not?).
EDIT: (It turns out that I think you can construct pathological shapes with uncountable numbers of edges for which a simple linear sweep fails no matter the angle, because you’re not allowed to sweep over an edge of one shape while sweeping over a vertex of the other. But if we allow the angle to vary slightly with parametric ‘time’, I don’t think there’s any possible counterexample, because you can always find a way to start and end at a vertex.)
Then once you’ve mapped your subset to a triangle, you use #6. But.
This doesn’t use the hint! And the hints have been so good and educational everywhere I’ve used them. So what am I missing about the hint?
• An attempt at problem #1; seems like there must be a shorter proof.
The proof idea is “If I flip a light switch an even number of times, then it must be in the same state that I found it in when I’m finished switching.”
Theorem. Let e a path graph on ertices with a vertex oloring uch that if hen Let s bichromatic Then s odd.
Proof. By the definition of a path graph, there exists a sequence ndexing An edge s bichromatic iff A subgraph f s a state iff its terminal vertices are each incident with exactly one bichromatic edge or equal to a terminal vertex of The color of a state is the color of its vertices. There exists a subsequence of ontaining the least term of each state; the index of a state is equal to the index of its least term in this subsequence.
Note that none of the states with even indexes are the same color as any of the states with odd indexes; hence all of the states with even indexes are the same color, and all of the states with odd indexes are the same color.
For each state there exists a subsequence of orresponding to the vertices of and the least term of each subsequence is either r some hat is the greatest term in a bichromatic edge. Thus the number of states in
By contradiction, suppose that s even. Then the number of states is odd, and the first and last states are the same color, so the terminal vertices of re the same color, contrary to our assumption that they are different colors. Thus ust be odd.
:::
• I’m stuck part-way through on #4 - I assume there is a way to do this without the exhaustive search I’m running into needing.
I’m going to try (nested) induction. Define triangles by side size, measured in nodes.
Induction base step: For n=2, there must be exactly one trichromatic edge.
Induction step: If there are an odd number of tri-chromatic edges for all triangles n=x, we must show that this implies the same for n=x+1.
We create all possible new triangles by adding x+1 nodes on one of the sides, then allow any of the previous x nodes on that side to change. Without loss of generality, assume we add x+1 edges to the bottom (non-red) side. These must be green or blue. The previous layer can now change any number of node-colors. We now must prove this by induction on color changes of nodes in the second-to-bottom layer to be red. (If they flip color otherwise, it is covered by a different base case.)
First, base step, assume no nodes change color. Because the previous triangle had an odd number of trichromatic edges, and the new edge is only green+blue, no new trichromatic edges were created.
Induction step: There is an x+1 triangle with an odd number of trichromatic vertices, and one node in the second-to-bottom layer changes to red. This can only create a new tri-cromatic triangle in one of the six adjacent triangles. We split this into (lots of) cases, and handle them one at a time.
(Now I get into WAY too many cases. I started and did most of the edge-node case, but it’s a huge pain. Is there some other way to do this, presumably using some nifty graph theory I don’t know, or will I need to list these out? Or should I not be using the nested induction step?)
Pointers welcome!
• You can use 1d-Sperner to deal with all the cases effectively.
• Here’s a messy way that at least doesn’t need too much exhaustive search:
First let’s separate all of the red nodes into groups so that within each group you can get to any other node in that group only passing through red nodes, but not to red nodes in any other group.
Now, we trace out the paths that surround these groups—they immediately look like the paths from Question 1 so this feels like a good start. More precisely, we draw out the paths such that each vertex forms one side of a triangle that has a blue node at its opposite corner. Note that you can have multiple paths stemming from the same group if the group touches the side of the larger triangle, or if it has internal holes.
Now we have this set of paths we can split them into three kinds. The first is loops, which arise when you have a group which never touches the edge of the larger triangle, or inside ‘holes’ in large groups. These can be seen as a path starting and finishing at the same node. They therefore have an even number of b-g vertices. The second kind is those that begin at the edge of the large triangle and end at the same edge. These paths begin and end on the same colour and therefore also have an even number of b-g vertices. Finally and most importantly there is a kind of path that goes from one edge to the other -in the case of the reds, the left edge to the right edge. This will happen once with the group that includes the top red node, and if any other group spans the larger triangle then it will generate two more of these paths. Sperner’s lemma tells us that these will have an odd number of b-g vertices and we know that there will be an odd number of such paths, so this final type generates an odd number of total b-g vertices.
By the way that we have defined these paths, the total number of r-g-b triangles is equal to the number of g-b vertices on the paths in the set generated above. This number is the sum of an odd number from the spanning paths and a series of even numbers from the other paths, giving an odd overall number of r-g-b vertices, proving number 4 (as long as I haven’t made an error in categorizing the paths).
I hope this makes sense, let me know if it doesn’t or has errors :)
• I am having trouble figuring out why #2 needs / benefits from Sperner’s Lemma.
But I keep going back to the proof that I’m comfortable with, which depends on connectedness, so I’m clearly missing an obvious alternative proof that doesn’t need topology.
• I was able to get at least (I think) close to proving 2 using Sperner’s Lemma as follows:
You can map the continuous function f(x) to a path of the kind found in Question 1 of length n+1
by evaluating f(x) at x=0, x=1 and n-1 equally spaced divisions between these two points and setting a node as blue if f(x) < 0 else as green.
By Sperner’s Lemma there is an odd, and therefore non-zero number of b-g vertices. You can then take any b-g pair of nodes as the starting points for a new path and repeat the process. After k iterations you have two values of x—only one where f(x) is below zero—that are 1/(n^k) away from each other. We thus can find arbitrarily close points that straddle zero. By taking the sequence f(x) of initial nodes x we get a sequence that, by B-W, has a sub-sequence which converges to zero. By continuity we have proved the existence of an x such that f(x)=0.
We can be sure that the sub-sequence does in fact converge to zero, rather than any other value because if it converges to any number |a|>0, the gradient of f(x) would have to be arbitrarily high to dip back below/above 0 for a value of x arbitrarily close by and therefore would not be a continuous function.
Comments to tighten up/poke holes in the above appreciated :)
• I’m having trouble understanding why we can’t just fix in your proof. Then at each iteration we bisect the interval, so we wouldn’t be using the “full power” of the 1-D Sperner’s lemma (we would just be using something close to the base case).
Also if we are only given that is continuous, does it make sense to talk about the gradient?
• “I’m having trouble understanding why we can’t just fix n=2 in your proof. Then at each iteration we bisect the interval, so we wouldn’t be using the “full power” of the 1-D Sperner’s lemma (we would just be using something close to the base case).”—You’re right, you can prove this without using the full power of Sperner’s lemma. I think it becomes more useful for the multi-dimensional case.
• Yeah agreed, in fact I don’t think you even need to continually bisect, you can just increase n indefinitely. Iterating becomes more dangerous as you move to higher dimensions because an n dimensional simplex with n+1 colours that has been coloured according to analogous rules doesn’t necessarily contain the point that maps to zero.
On the second point, yes I’d been assuming that a bounded function had a bounded gradient, which certainly isn’t true for say sin(x^2), the final step needs more work, I like the way you did it in the proof below.
• I hit that stumbling block as well. I handwaved it by saying “continue iterating until you have and such that , , and f has no local maxima or local minima on the open interval ”, but that doesn’t work for the Weierstrass function, which will (I believe) never meet that criterion.
• Here is my attempt, based on Hoagy’s proof.
Let be an integer. We are given that and . Now consider the points in the interval . By 1-D Sperner’s lemma, there are an odd number of such that and (i.e. an odd number of “segments” that begin below zero and end up above zero). In particular, is an even number, so there must be at least one such number . Choose the smallest and call this number .
Now consider the sequence . Since this sequence takes values in , it is bounded, and by the Bolzano–Weierstrass theorem there must be some subsequence that converges to some number .
Consider the sequences and . We have for each . By the limit laws, as . Since is continuous, we have and as . Thus and , showing that , as desired.
• Ex 1
Let and . Given an edge , let denote the map that maps the color of the left to that of the right node. Given a , let . Let denote the color blue and the color green. Let be 1 if edge is bichromatic, and 0 otherwise. Then we need to show that . We’ll show , which is a striclty stronger statement than the contrapositive.
For , the LHS is equivalent to , and indeed equals if is bichromatic, and otherwise. Now let and let it be true for . Suppose . Then, if , that means , so that , and if , then , so that . Now suppose . If , then , so that , and if , then , so that . This proves the lemma by induction.
Ex 2
Ordinarily I’d proof by contradiction, using sequences, that can neither be greater nor smaller than 0. I didn’t manage a short way to do it using the two lemmas, but here’s a long way.
Set . Given , let be a path graph of vertices , where . If for any and we have , then we’re done, so assume we don’t. Define to be 1 if and have s different sign, and 0 otherwise. Sperner’s Lemma says that the number of edges with are odd; in particular, there’s at least one. Let the first one be denoted , then set .
Now consider the sequence . It’s bounded because . Using the Bolzano-Weierstrass theorem, let be a convergent subsequence. Since for all , we have . On the other hand, if , then, using continuity of , we find a number such that . Choose and such that , then for all , so that and then for all , so that , a contradiction.
Ex 3
Given such a function , let be defined by . We have . If either inequality isn’t strict, we’re done. Otherwise, . Taking for granted that the intermediate value theorem generalizes to this case, find a root of , then .
Is there something I’m missing? It seems to me that for all .
• is defined just for one particular graph. It’s the first edge in that graph such that . (So it could have been called ). Then for the next graph, it’s a different . Basically, looks at where the first graph skips over the zero mark, then picks the last vertex before that point, then looks at the next larger graph, and if that graph skips later, it updates to the last vertex before that point in that graph, etc. I think the reason I didn’t add indices to was just that there ar ealready the with two indices, but I see how it can be confusing since having no index makes it sound like it’s the same value all throughout.
• Ex 5 (fixed version)
Let denote the triangle. For each , construct a 2-d simplex with nodes in , where the color of a point corresponds to the place in the disk that carries that point to, then choose to be a point within a trichromatic triangle in the graph. Then is a bounded sequence having a limit point . Let be the center of the disc; suppose that . Then there is at least one region of the disc that doesn’t touch. Let be the distance to the furthest side, that is, let . Since the sides are closed regions, we have . Using continuity of , choose small enough such that . Then choose large enough so that (1) all triangles in have diameter less than and (2) . Then, given any other point in the triangle around in , we have that , so that . This proves that the triangle in does not map points to all three sides of the disc, contradicting the fact that it is trichromatic.
Ex 6
(This is way easier to demonstrate in a picture in a way that leaves no doubt that it works than it is to write down, but I tried to do it anyway considering that to be part of the difficulty.)
(Assume the triangle is equilateral.) Imbed into such that , , . Let be continuous. Then given by is also continuous. If then . Let be the circle with radius 2 around ; then because (it is in fact contained in the circle with radius 1, but the size of the circle is inconsequential). We will use exercise 5 to show that maps a point to the center, which is , from which the desired result follows. For this, we shall show that it has the needed properties, with the modification that points on any side may map precisely into the center. It’s obvious that weakening the requirement in this way preserves the result.
Rotate the disk so that the red shape is on top. In polar coordinates, the green area now contains all points with angles between and , the blue area contains those between and , and the red area those between and and those between and . We will show that does not intersect the red area, except at the origin. First, note that we have
Since both and are convex combinations of finitely many points, it suffices to check all combinations that result by taking a corner from each. This means we need to check the points
and and and and and .
Which are easily computed to be
and and and and and
Two of those are precisely the origin, the other four have angles and and and . Indeed, they are all between and .
Now one needs to do the same for the sets and , but it goes through analogously.
• I am sorry because I cannot figure out how to hide big formulas in a spoiler. Also the spoiler feature is somewhat broken so it adds weird tabs around formulas.
#1:
Let’s count the number of blue edge ends. Each blue point inside the segment is the end of two blue edges, and the leftmost blue vertex is the end of one. Therefore, their total number is odd. On the other hand, each bichromatic edge produces one blue edge end, and each unichromatic edge produces an odd number—zero or two—of blue edge ends. Therefore, an odd number of edges are bichromatic.
#2:
Suppose . If then and, since f is continuous, f stays positive in some neighborhood of x, and then x is not the infimum. Therefore, f(x) = 0.
#3:
Consider the function . Since and by exercise 2, there should be a point where g(x) = 0.
#8.
Consider the family of functions:
For t < 0.5, the only fixed point is of is 1; for t > 0.5, the only fixed point is 0.
#9.
Lemma:
Suppose a k-dimensional simplex is subdivided into smaller k-dimensional simplices and all vertices are colored into k+1 colors so that there are no vertices of color i on the i-th edge of the big simplex. Then there are an odd number of subdivision simplices whose vertices are colored in k+1 different colors.
Proof:
Induction by k. Base k=1 proved in exercise 1.
Induction step: supposed the lemma is proved for k-1, let’s prove it for k.
Let us count the number of tuples (X, Y) where X is a k-1-dimensional simplex colored in colors 0, 1, …, k-1,
Y is a k-dimensional subdivision simplex, and X is on the boundary of Y. Each properly colored simplex X inside the big simplex produces two tuples, and each simplex on the boundary produces one tuple. X can only be on the k-th edge of the big simplex, and by the inductional assumption, there are an odd number of simplices X there. So, the total number of tuples is odd. On the other hand, each k-dimensional simplex Y can be a part of either:
0 tuples;
1 tuple if all his vertices are different;
2 tuples if has vertices of colors 0,1,...,k-1 but not all his vertices are different.
Therefore, a number of k-simplices Y with all different vertices must be odd.
#4
Follows from 9
#5
Suppose that center is not in the image of the triangle. Let us call a set of points bichromatic if it doesn’t have points of all three colors. We color each point in the triangle in the same color as its image. Then every point in the image has an open bichromatic neighborhood. Since the map is continuous, the preimage of this neighborhood is also open. So, around every point in the triangle there can be drawn an open bichromatic ball of radius r. These balls are an open cover of the triangle, let us choose a finite subcover out of them. Suppose s the minimum radius in this subcover. Split the triangle into subtriangles so that the diameter of each triangle is smaller than By Sperner’s lemma, there is a trichromatic triangle, but since its diameter is smaller than it lies completely inside one of the bichromatic balls. Contradiction.
#10
First, I am going to prove that a function from a unit ball o itself has a fixed point, then that any compact convex subset of s homeomorphic to a ball.
Suppose that as no fixed point, n>1 (case n=1 was proved in exercise 3). Then I can build a retraction from nto its boundary
send x to the first intersection of the ray (f(x), x) with Let us prove that such a rectraction cannot exist. Suppose that such a rectraction exists. Denote the inclusion map. Then nd the induced homology group homorphism ust also be identity:
But this is impossible because and
Now let us prove that any compact convex subset X of s homeomorphic to a ball. Let us select a maximum set of affinely independent points in X. They form some k-dimensional simplex, all X lies in the affine space spanned by this simplex, and all the interior of this simplex belongs to X, because X is convex. I’ll take a ball f radius side this simplex and build a homeomorphism between X and . Taking the center of the ball as the center of coordinates, define
where s the distance to the farthest point of X in the direction, if , 0 if
Let us prove that f and its inverse are continuous. Since X is compact, it is bounded, so there is a such that It follows that f and its inverse are continuous in zero: if if
Now let us prove that functions are continuous in all other points. It is sufficient to prove that r(x) is the continuous on the unit sphere. (Since composition and product of continuous functions is continuous, division by bounded from below (by d) continuous function r is continuous, ||x|| is a continuous function).
Since X is convex, the tangent cone from any point of X to lies in X. So if we take a point at the distance from the center, draw a tangent cone, and go down its boundary, we get the steepest possible rate of change of r(x) with respect to x. Therefore, r is continuous.
#6, #7: follow from #10.
#11:
Suppose f has no fixed point. Distance d(a, B) is a continuous function of a, and a continuous function reaches its minimum on a compact. TxT and the graph of f are nonitersecting compact sets, therefore the Hausdorff distance between them is positive. It is easy to see that Hausdorff metric is indeed a metric, i.e. that a triangle inequality holds for it. So if we take any continuous function g at a distance less than from f, its Hausdorff distance to TxT will be positive, so it can have no fixed points.
#13:
Suppose is a Kakutani function. We already know that any compact convex subset of s homeomorphic to a simplex. Denote he homeomorphism between a simplex T and S.
Denote the k-th barithentric subdivision of T. For each choose an element
Define where are the baricentric coordinates of point n its subdivision simplex. Function s continuous, and, since S is convex, the image of lies in S.
By the Brouwer fixed point theorem, as a fixed point. Since S is compact, from the infinite sequence of fixed points of e can choose a convergent subsequence.
Suppose s this subsequence, lies in the simplex and has baricentric coordinates . Then and so
(1).
Since simplices go down in diameter, as Each s a bounded sequence, so we can, sequentially, choose a convergent subsequence out of each of them, so we can assume that Similarly, we can choose a convergent subsequence out of so we assume The sequence belongs to the graph of h and converges to the point Since the graph is closed, must belong to the image of Since for every k, ince the image is convex, lso belongs to the image of On the other hand, as we remember, since equality (1) held for every k, it also holds in the limit: . Hence, So, is the fixed point of h.
• I just tried to fix all the things in your comment. You’re right, weird stuff was happening :-)
• Thank you!
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2020-08-05 13:08:39
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https://itecnotes.com/electrical/electronic-second-order-all-pass-filter-input-impedance/
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Electronic – Second order all-pass filter input impedance
I have a question about the second order all-pass filter input impedance when ω = ∞ and ω = 0 and I would appreciate if you could help me to understand the logic.
1. When ω = 0, all capacitors can be seen as open circuit, because we are at DC level. In this case I expect the Zin to be R3+R4, what would be 3 kOhms, because the voltage at opamp inputs would be 0V. From the simulation I see that the expected input impedance is 1.5 kOhms. Why?
2. When ω = ∞, capacitors can be seen as short circuits, and therefore the current will travel along C with no electrical impedance and R3, R4 are the only impedances that separates V1 from ground. In this case I do not understand why the input impedance is 3 kOhms.
At DC $$\\omega = 0\$$ (all capacitors can be seen as open circuit) we have this situation:
simulate this circuit – Schematic created using CircuitLab
And $$R_{IN_{DC}} = R_3 + R_4 = 3k\Omega$$
But at high frequency ($$\\omega = ∞\$$) when all capacitors can be seen as short circuits we have this situation:
simulate this circuit
Therefore the input resistance is now equal to:
$$R_{IN_{HF}} = \left[R_1 \times\left(1 + \frac{R_4}{R_3}\right)\right]||(R_3+R_4) = 3k\Omega||3k\Omega = 1.5k\Omega$$
Why? because now the voltage across $$\R_1\$$ resistor is no longer equal to $$\V_{IN}\$$ but to the difference between $$\V_{IN}\$$ and op-amp output voltage. And the op-amp is working as a voltage follower, meaning that the op-amp output voltage is the same as the input voltage (at non-inverting input). And the input voltage is the output voltage produced by the voltage divider build around $$\R_3, R_4\$$.
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2023-02-05 05:15:50
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https://joomla.stackexchange.com/questions/3544/recommended-practices-regarding-transfering-database-from-joomla-1-5-to-joomla-3
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# Recommended practices regarding transfering database from Joomla 1.5 to Joomla 3
I want to transfer about 300 articles from a Joomla 1.5 website to a Joomla 3.3.1 website.
I have access only to the Joomla 1.5 database but not to the backend (I have access to both for the joomla 3.3.1 website).
Because the table xx_content under Joomla 3.3 doesn't have the same structure/columns than the table xx_content under Joomla 1.5, I can't simply do a SQL query (both tables are in the same database).
So what is the recommended practices to do that?
My guess is: export the J1.5 xx_content to a csv and then modify it to reflect the structure of the J3.3 xx_content and then import it in J3.3 xx_content. But I wonder if there isn't something easier.
• Just a note: If you have database access, it's not that hard to set up a new backend user or reset the password of an existing backend user. – Bakual Jul 10 '14 at 14:28
What I would recommend you do is:
1. Export the database for the Joomla 1.5 site, apart from the #__users table
2. Install a fresh copy of Joomla 1.5 on your localhost
You should now have a fresh Joomla 1.5 site with all you content.
1. Install redMIGRATOR which is a Joomla 1.5 extension that will migrate your site to Joomla 3.x
2. Once your site has been successfully migrated, take a database dump of the #__content table.
3. On your live Joomla 3 site that you said you have access to, import the database dump that you took from your localhost.
Hope this helps
• That's a great idea! Thanks Lodder! I going to do that right now. – MagTun Jul 10 '14 at 9:36
• No worries, let me know how it all goes – Lodder Jul 10 '14 at 9:37
• I was about to start when I realized I had to edit all the fulltext entries so I did it via CSV, but if I didn't have to do it, I would I go with your solution! That would have been a safer and quicker way of doing it! – MagTun Jul 10 '14 at 16:41
Finally, I've done the export/import via CSV (as I also had to edit all the fulltext entries) and here are the little things that I had to search for:
1. export the Db via CSV, then edit it with Open Office Calc*
2. edit the column so that they match the Joomla3.3 structure.
3. set the column asset_idFK to O, Joomla will assign the correct value when you will open/save your article from the backend/frontend.
To encapsulate all the entries with "
1. Still in Open Office Calc, format all the cells to text : CTRL+A, right click , check text
2. Select Save As: CSV and check "edit filter settings"
3. In the pop up, "keep same format" (don't choose ODT) and check: quote all text cells (cf. this question for screen shoot)
Escaped comma: I don't know why but Open Office replaces all the escaped comma \" by \"", so open the .csv with a text editor as Notepad++ and Find replace \"" by \"
• I have tried to do it with Excel, but because I have special western characters, I ended up with a corrupted characters. But if you have to do it with Excel, add sep=; at the first line of the .CSV to tell Excel what are the separator.
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2021-04-21 21:08:51
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http://binary-optiontrade.dranforsund.tk/?binomo=21356
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R Square Significance F and P-Values Coefficients Residuals. This example teaches you how to run a linear regression analysis in Excel and how to interpret the Summary Output.. Below you can find our data. The big question is: is there a relation between Quantity Sold (Output) and Price and Advertising (Input). lag is a generic function; this page documents its default method. Keywords ts. Usage lag(x, …) # S3 method for default lag(x, k = 1, …) Arguments x. A vector or matrix or univariate or multivariate time series. k. The number of lags (in units of observations). … further arguments to be passed to or from methods. Details. Vector or matrix arguments x are given a tsp attribute via hasTsp ... Logarithmic price scales are better than linear price scales at showing less severe price increases or decreases. They can help you visualize how far the price must move to reach a buy or sell target. In Excel we will use the LN function, which has only one argument – the number x for which we want to find the natural logarithm ln(x). In our case the x is the ratio of closing prices. Therefore, the formula in cell C3 will be: =LN(B3/B2) where cell B3 is the current day’s closing price and cell B2 the previous day’s closing price. Copy the formula to the rest of column C. The return ... Ratgeber: Mit dieser Rendite-Formel können Sie schnell und einfach den jährlichen Gesamtbetrag verschiedener Rendite-Arten berechnen. The maths help and test prep that gets you better maths marks! Learn with step-by-step video help, instant practice, diagnostics and a personal study plan. There’s a nice blog post here by Quantivity which explains why we choose to define market returns using the log function:. where denotes price on day .. I mentioned this question briefly in this post, when I was explaining how people compute market volatility. I encourage anyone who is interested in this technical question to read that post, it really explains the reasoning well. Probability Density Function A variable X is lognormally distributed if $$Y = \ln(X)$$ ... = -\ln(1 - \Phi(\frac{\ln(x)} {\sigma})) \hspace{.2in} x \ge 0; \sigma > 0 \) where $$\Phi$$ is the cumulative distribution function of the normal distribution. The following is the plot of the lognormal cumulative hazard function with the same values of σ as the pdf plots above. Survival Function The ... Version info: Code for this page was tested in Stata 12. Logistic regression, also called a logit model, is used to model dichotomous outcome variables. In the logit model the log odds of the outcome is modeled as a linear combination of the predictor variables. Please note: The purpose of this page is to show how to use various data analysis commands. It does not cover all aspects of the ... We simulate from the Excel function =RANDBETWEEN a stock price that varies daily between values of 94 and 104. Computing the Daily Returns In column E, we enter "Ln (P (t) / P (t-1))."
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2022-12-04 02:57:48
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https://testbook.com/question-answer/elastic-energy-store-in-the-form-of-potential-ener--60d36c4c80df36b70127595f
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# Elastic energy store in the form of potential energy when stretched by external stress per unit volume would be
1. $$\frac{1}{2}\left ( \frac{stress}{strain} \right )$$
2. $$\frac{1}{2}\times stress\times strain$$
3. $$stress\times strain$$
4. None
Option 2 : $$\frac{1}{2}\times stress\times strain$$
## Detailed Solution
Concept:
• Elastic potential energy is the energy stored as a result of applying a force to deform an elastic object.
• The energy is stored until the force is removed and the object regains its original shape, doing work in the process.
• This deformation could involve compressing, stretching, or twisting the object.
• Many objects are specifically designed to store elastic potential energy inside them,
For example:
The coil spring of a wind-up clock
An archer's stretched bow
A stretched slingshot ready to fire.
Explanation:
The energy density is nothing but the energy of a system per unit volume.
For elastic potential energy, it is written as:-
The elastic potential density, $$\left ( U \right )=\frac{Elastic potential enrgy}{volume of elastic material}$$
Elastic potential energy, $$\left ( E \right )=\frac{1}{2}\times stress\times strain\times volume$$
$$\Rightarrow U=\frac{E}{V}$$
$$\Rightarrow U=\frac{1}{2}\times stress\times strain$$
Hence, option-2 is correct
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2023-03-25 17:44:13
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http://nkbp.auto-hirch-egelsbach.de/bayesian-inference-calculator.html
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An Introduction to Bayesian Inference via Variational Approximations Justin Grimmer Department of Political Science, Stanford University, 616 Serra St. In what he called a scholium. You can use this Bayesian A/B testing calculator to run any standard hypothesis Bayesian equation (up to a limit of 10 variations). Of course, practical applications of Bayesian networks go far beyond these "toy examples. Program features include:. MrBayes is a program for Bayesian inference and model choice across a wide range of phylogenetic and evolutionary models. Bayesian inference is a method of statistical inference based on Bayes' rule. Application exercise:2. And it suggests one of the central appeals, to me, of the approach: every input into a Bayesian framework is expressed as probability and every output of a Bayesian framework is expressed as probability. " Here is a selection of tutorials, webinars, and seminars, which show the broad spectrum of real-world applications of Bayesian networks. information than trust values, and allow us to employ Bayesian network inference to conduct multiple-hop recommendation in online social networks. • What is the Bayesian approach to statistics? How does it differ from the frequentist approach? • Conditional probabilities, Bayes’ theorem, prior probabilities • Examples of applying Bayesian statistics • Bayesian correlation testing and model selection • Monte Carlo simulations The dark energy puzzleLecture 4 : Bayesian inference. Using R and rjags, you will learn how to specify and run Bayesian modeling procedures using regression models for continuous, count and categorical data. More specifically, we assume that we have some initial guess about the distribution of $\Theta$. uk ABSTRACT Decisions concerning the management and conservation of cetacean populations depend upon knowledge of population parameters, wh ich. Masly 1Department of Biology, University of Rochester, Rochester, NY 14627, U. Te qui veniam noster quaerendum, porro fabellas mei cu. Bayesian methods are becoming another tool for assessing the viability of a research hypothesis. This book presents a good reference of operational risk modeling using Bayesian Inference as well as several Bayesian model derivations. \An excellent rst step for readers with little background in the topic. In addition, Bayesian modeling consists of the specification of a joint distribution for data and unknown quantities; Bayesian inference is based on conditional distributions of unknowns, given data. To Bayesian Calculator by Pezzulo--Handles up to 5 Hypotheses and 5 Outcomes. 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In short, Bayesian inference derives from Bayes theorem, which states that the probability of a hypothesis H being true given the existence of some evidence E is equal to the probability that the evidence exists given that the hypothesis is true times the probability that the hypothesis is true before the evidence is observed divided by the. 5 Bayesian Penalized Splines 15 1 Bayesian Inference for the Binomial Model Bayesian inference is a branch of statistics that offers an alternative to the frequentist or classical methods that most are familiar with. A prior probability, in Bayesian statistical inference, is the probability of an event based on established knowledge, before empirical data is collected. It is a way to calculate the value of P(B|A) with the knowledge of P(A|B). Statistical Simulation and Inference in the Browser. Basic Elements of Bayesian Analysis In a frequentist analysis, one chooses a model (likelihood function) for the available data, and then either calculates a p-value (which tells you how un-usual your data would be, assuming your null hypothesis is exactly true), or calculates a confidence interval. • Derivation of the Bayesian information criterion (BIC). The real number is 7. The approach requires a prior probability distribution for each unknown parameter whose distribution is updated. Two levels of inference 1. Bayesian statistics (sometimes called Bayesian inference) is a general approach to statistics which uses prior probabilities to answer questions like: Has this happened before? Is it likely, based on my knowledge of the situation, that it will happen?. Bayesian inference, on the other hand, is able to assign probabilities to any statement, even when a random process is not involved. Section 2 reviews ideas of conditional probabilities and introduces Bayes' theorem and its use in updating beliefs about a proposition, when data are observed, or information becomes available. We expect this software package to be useful for other labs because it fills a critical gap in the downstream analysis of population snapshots of smFISH in single cells. The roots of Bayesian statistics go back to 18th century England with the discoveries of Reverend Thomas Bayes, who was interested in the problem of determining causes from observations of results. BEN LAMBERT [continued]: In this tutorial on Bayesian inference, I'm going to, firstly, talk about the history behind Bayes' rule. Use estimate of position and velocity to make (noisy) prediction of position and velocity at time of next observation 4. It is plain silly to ignore what we know, ii. 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Inference in complex models If the model is simple enough we can calculate the posterior exactly (conjugate priors) When the model is more complicated, we can only approximate the posterior Variational Bayes calculate the function closest to the posterior within a class of functions Sampling algorithms produce samples from the posterior. Bayesian Inference¶ Bayesian inference is based on the idea that distributional parameters $$\theta$$ can themselves be viewed as random variables with their own distributions. Bayesian Computation in Finance Satadru Hore1, Michael Johannes2 Hedibert Lopes3,Robert McCulloch4, and Nicholas Polson5 Abstract In this paper we describe the challenges of Bayesian computation in Finance. His famous theorem was published posthumously in 1763, The simple rule has vast ramifications for statistical inference. This is the currently selected item. Fit parameters. An important part of bayesian inference is the establishment of parameters and models. This chapter explains the similarities between these two approaches and, importantly, indicates where they differ substantively. -Probability theory is the proper mechanism for accounting for uncertainty. Inference in Bayesian Networks How can one infer the (probabilities of) values of one or more network variables, given observed values of others? † Bayes net contains all information needed for this inference † If only one variable with unknown value, easy to infer it † In general case, problem is NP hard In practice, can succeed in many. Example 1: ANOVA model 2. many problems, the key issue in setting up the prior distribution is the specification of the model into parameters that can be clustered hierarchically. Simple examples of Bayesian. – Autodiff, theano, tensorflow Development of good optimizers. back to start 16 It is now possible to calculate Bayes factors and posterior probabilities of models with MCMCpack. The key contrast between Bayesian and frequentist methods is not the use of prior information, but rather the choice of alternatives that are relevant for inference: Bayesian inference focuses on alternative hypotheses, frequentist statistics focuses on alternative data. Finally, we review Markov chain Monte Carlo methods (MCMC). Perez-Lorenzo Multimedia and Multimodal Processing Research Group, University of Jaen, 23700, Linares, Spain Ning Xiang Graduate Program in Architectural Acoustics, School of Architecture, Rensselaer Polytechnic Institute, Troy, New York 12180 Maximo Cobos. odc and drug-in2. How Bayes Methodology is used in System Reliability Evaluation: Bayesian system reliability evaluation assumes the system MTBF is a random quantity "chosen" according to a prior distribution model: Models and assumptions for using Bayes methodology will be described in a later section. This requires some assumptions. Chapter 5 Confidence Intervals and Hypothesis Testing Although Chapter 4 introduced the theoretical framework for estimating the parameters of a model, it was very much situated in the context of prediction: the focus of statistical inference is on inferring the kinds of additional data that are likely to be generated by a model, on the. then aicbic applies it to all logL values. Next, we de-velop likelihood-free variational inference (LFVI), a scalable variational inference algorithm for HIMs. The theorem tries to bring an association between the theory and evidence by finding the relation between the past probability to current probability of the event. many problems, the key issue in setting up the prior distribution is the specification of the model into parameters that can be clustered hierarchically. Long after we've achieved good mixing of the chains and good inference for parameters of interest and we're ready to go on, it turns out that DIC is still unstable. Top Ten Math Books On Bayesian Analysis, July 2014. Basics of Bayesian Inference and Belief Networks Motivation. Bayes' theorem calculates the conditional probability (Probability of A given B): Sometimes the result of the Bayes' theorem can surprise you. goal-oriented inference to Bayesian formulations for systems governed by nonlinear we will calculate discrete eigenfunctions of the covariance function by forming the. So, we’ll learn how it works! Let’s take an example of coin tossing to understand the idea behind bayesian inference. The real number is 7. An important part of bayesian inference is the establishment of parameters and models. Form a prior distribution over all unknown parameters. While we motivated the concept of Bayesian statistics in the previous article, I want to outline first how our analysis will proceed. Bayesian inference is therefore just the process of deducing properties about a population or probability distribution from data using Bayes’ theorem. Bayesian inference about is primarily based on the posterior distribution of. The trickiest part of this process is calculating the term in the denominator, the marginal likelihood P(Y). 1 Googling Suppose you are chosen, for your knowledge of Bayesian statistics, to work at Google as a search tra c analyst. BNT supports many different inference algorithms, and it is easy to add more. This method minimizes the Kullback-. 1 Ultimately, she would like to know the. The most elegant way to calculate the posterior probabilities is Bayes' rule. that the Bayes rule can be obtained by taking the Bayes action for each particular x! Another connection with frequentist theory include that finding a Bayes rule against the ”worst possible prior” gives youa minimax estimator. Bayesian Inference for Logistic Regression Parame-ters Bayesian inference for logistic analyses follows the usual pattern for all Bayesian analyses: 1. Flint, combines bayesian networks, certainty factors and fuzzy logic within a logic programming rules-based environment. Bayesian inference is an important technique in statistics, and especially in mathematical statistics. Bayes’ Nets !! Conditional Independences ! Probabilistic Inference ! Enumeration (exact, exponential complexity) ! Variable elimination (exact, worst-case exponential complexity, often better) ! Probabilistic inference is NP-complete ! Sampling (approximate) ! Learning Bayes’ Nets from Data 2 Inference ! Inference: calculating some. This is the central computation issue for Bayesian data analysis. However, I haven't really considered them here. My or your model fitting problem is also everyone elses problem. This is The Bayes Factor, a podcast about the people behind Bayesian statistics and other hot methodological issues in psychological research. We then introduce the steps necessary to create our Avatar, 3-D semaphoric space-time visualization diffusion object. As can be seen, inference on a binomial proportion is an extremely important statistical technique and will form the basis of many of the articles on Bayesian statistics that follow. To Bayesian Calculator by Pezzulo--Handles up to 5 Hypotheses and 5 Outcomes. 1 for the actual word searched, and the starting string (the rst three letters typed in a search). Practical applications of the Bayes Theorem. Introduc)on*to*Bayesian* Inference* A*natural,*butperhaps*unfamiliar* view*on*probability*and*stas)cs* Michiel*Botje* Nikhef,*PO*Box*41882,*1009DB*Amsterdam*. I Uncertainty in estimates is quanti ed through the sampling. Decision-making Calculator with CPT, TAX, and EV. However, there are continuing discussions and arguments about many aspects of statistical design and analysis. Now that we have the model of the problem, we can solve for the posteriors using Bayesian methods. Rao Department of Computer Science and Engineering University of Washington, Seattle, WA 98195 [email protected] Two ingredients: 1. Bayesian Adaptive Trading with a Daily Cycle Robert Almgren∗ and Julian Lorenz∗∗ July 26, 2006 Abstract Standard models of algorithmic trading neglect the presence of a daily cycle. making inferences from data using probability models for quantities we observe and about which we wish to learn. He also covers testing hypotheses, modeling different data distributions, and calculating the covariance and correlation between data sets. Bayesian Inference for Logistic Regression Parame-ters Bayesian inference for logistic analyses follows the usual pattern for all Bayesian analyses: 1. This method minimizes the Kullback-. Geyer March 30, 2012 1 The Problem This is an example of an application of Bayes rule that requires some form of computer analysis. 2Department. Statistical inference is the process of using observed data to infer properties of the statistical distributions that generated that data. In this article, we provided a framework to infer how goal chances are created by a team, characterized by spatiotemporal player-level behavior on assisting a play or receiving assists. Bayesian inference - real life applications. Key to LFVI is specifying a variational family that is also im-plicit. Conjugate Bayesian inference when the variance-covariance matrix is unknown 2. A Bayesian inference model for speech localization (L) Jose Escolanoa) and Jose M. Calculate and interpret the AIC for four models. Frequentist probabilities are "long run" rates of performance, and depend on details of the sample space that are irrelevant in a Bayesian calculation. To Bayesian Calculator by Pezzulo--Handles up to 5 Hypotheses and 5 Outcomes. Use Bayes theorem to nd the posterior distribution of all parameters. According to Norman Fenton, author of Risk Assessment and Decision Analysis with Bayesian Networks: Bayes’ theorem is adaptive and flexible because it allows us to revise and change our predictions and diagnoses in light of new data and information. This requires some assumptions.
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2019-11-13 13:32:53
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https://www.law.com/almID/900005554891/
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Each year since we started the Diversity Scorecard, we’ve seen small but consistent increases in the proportion of minority lawyers at large firms. In 2001, only about one partner in 30 was a lawyer of color. Now it’s about one partner in 20.
That’s definitely progress. But it’s slow progress, and it raises the obvious question: How long will it take before large law firms — particularly their partnerships — mirror the general U.S. population, where almost one of every three citizens is a person of color? The likely answer: Decades.
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2023-04-01 23:13:46
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https://makondo.ugr.es/event/0/session/91/contribution/371
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The 35th International Symposium on Lattice Field Theory
18-24 June 2017
Palacio de Congresos
Home > Timetable > Session details > Contribution details
Contribution Parallel
Auditorio Manuel de Falla
Nonzero Temperature and Density
QCD equation of state at high temperatures
Speakers
• Dr. Alexei BAZAVOV
Content
The equation of state (EoS) in 2+1 flavor QCD has recently been established in the continuum limit at the physical quark masses. The HotQCD collaboration result provides the EoS in the temperature range from 130 to 400 MeV. We extend the HotQCD equation of state to higher temperatures. We utilize the Highly Improved Staggered Quarks (HISQ) action. We perform computations at the pion mass of about 300 MeV since the effects of heavier than physical light quark masses are negligible above 400 MeV. To control the cutoff effects and approach to the continuum limit, computations are done on the lattices with temporal extent Nt=8, 10 and 12.
Preferred track (if multiple tracks have been selected)
Nonzero Temperature and Density
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2019-08-18 17:32:13
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http://mathhelpforum.com/trigonometry/49304-help-trigo-funstion.html
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# Math Help - Help~Trigo Funstion~!!
1. ## Help~Trigo Funstion~!!
1.Cos2A+cos4A-cos3A=0, for 0≤A≤360
2.Show that Cos^-1(A)=Sin^-1 square root (1-A^2), (Inverse Trigonometric Function)
3.2cos3A-sin3A=Square root 5, for 0≤A≤360
4.Cos(A-∏/6) + sin (∏/6-A)=1, for 0≤A≤360
5.Tan3A=cot(∏/3-A), for 0≤A≤360
Thx for who helpING!!
2. Please show your work. What have you tried and why do you think it wasn't working?
3. I stuck at beginning.How to start?
4. 1) Copy down the problem.
2) Examine a list of known identities for relationships that may prove useful.
This one can help...
cos(a+b) = cos(a)cos(b) - sin(a)sin(b)
3) You dive in and try something. Paralysis never solve problems.
4) You look for symmetries so you don't have to do the same work twice.
Hint: With a little effort, you can get the first one to:
$8\cos^{4}(a)-4\cos^{3}(a)-6\cos^{2}(a)+3\cos(a)=0$
If you convert this to look more familiar to the Algebra in you...
$8x^{4}-4x^{3}-6x^{2}+3x=0$
You should be able to solve this by factoring and the Quadratic Formula. (Don't forget your Rational Root Theorem!)
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2015-06-03 14:27:42
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https://answers.ros.org/answers/230329/revisions/
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As you can see from the output you posted, several libraries external to the urg_node package fail to be linked to the getID executable declared in the CMakeLists.txt of the urg_node package. If you haven't done any modification to the urg_node package itself, this suggests to me that either :
1. you did not properly install the packages containing the external libraries mentioned, which you can check using rosdep update, rosdep install urg_node (tutorial here), or
2. you did not properly source your environment : source /opt/ros/indigo/setup.bash for packages installed via binaries, and source ~/catkin_ws/devel/setup.bash for packages that you compiled yourself in your catkin workspace.
Note that the failure to compile is caused by the CMake Error messages, and not by the CMake Warning on the Eigen library, so you might want to rename your thread or split it into 2 questions, as it is a bit misleading.
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2021-06-14 05:04:05
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https://www.thomasantony.com/projects/sicm-workbook/section-1-8-5-noethers-theorem/
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## 1.8.5 Noether’s Theorem
If a dynamical system has a symmetry, a coordinate system can be chosen so that the Lagrangian does not depend on a coordinate associated with the symmetry. There is also a conserved quantity associated with the symmetry (Section 1.8). However, there are general symmetries that no coordinate systems can fully express. For example, motion around a central potential is spherically symmetric, i.e., the dynamical system is invariant under rotation about any axis. However, the Lagrangian for this system only demonstrates symmetry about a single axis.
In general, a Lagrangian is said to have a symmetry if there exists a coordinate transformation that leaves the Lagrangian unchanged. In this section we consider the more general case of continous symmetries. A continuous symmetry is defined as a parametric family of symmetries. Emmy Noether proved that for any continuous symmetry, there is a conserved quantity.
Consider a parametric coordinate transformation, $\widetilde{F}$ with parameter $s$. This means that $\widetilde{F}$ represents an infinite number of coordinate transformations, one for each value of $s$.
An example would be a function that takes an angle, $s$, as the input and spits out a coordinate transformation that rotates the primed coordinate frame, $x’$ about some axis by that angle.
$$x = \widetilde{F}(s) (t, x')$$
There is a corresponding parametreic state transformation $\widetilde{C}$ associated with $\widetilde{F}$ that transforms the velocity $v’$ as well the time (i.e. the local tuple that forms the input to the Lagrangian).
\begin{align*} (t, x, v) &= \widetilde{C}(s) (t, x', v') \\ &= (t, \widetilde{F}(t,x'), \partial_0 \widetilde{F} + \partial_1 \widetilde{F} v', ...) \end{align*}
We require that $\widetilde{F}(0)$ represent the identity transformation $x’ = \widetilde{F}(0)(t, x’)$, with $\widetilde{C}$ as the corresponding identity state transformation. If the Lagrangian $L$ has a continous symmetry corresponding to $\widetilde{F}$, then the Lagrangian should be unchanged when the coordinates are transformed using $\widetilde{F}$. Therefore:
$$\widetilde{L}(s) = L\cdot \widetilde{C}(s) = L\\$$
for any $s$. Expanding $\widetilde{C}$ in the above expression, we get:
$$\widetilde{L}(s) = L\left(t,\quad\widetilde{F}(s)(t, x'),\quad\partial_1\widetilde{F}(s)(t, x')v' \right)\\$$
Undoing the “chainrule” in the second term and writing it in terms of the total time derivative,
$$\widetilde{L}(s) = L\left(t,\quad\widetilde{F}(s)(t, x'),\quad D_t\widetilde{F}(s)(t, x', v') \right)$$
Note: One of the assumptions in the following derivation is that $\partial_0 L = \frac{\partial L}{\partial t} = 0$
That $\widetilde{L}(s) = L$ for any $s$ implies that $D\widetilde{L}(s) = 0$ (where the $D$ operator represents derivative w.r.t $s$). Therefore, applying the chain rule for each of the components of $\widetilde{L}$, the derivative of $\widetilde{L}$ w.r.t $s$ is:
\begin{align*} 0 &= D\widetilde({L}(s)(t, x', v'))\\ &= \left(\underbrace{\partial_0 L}_{=\frac{\partial L}{\partial t} = 0} + \partial_1 L(t, x, v)\right) (D\widetilde{F})(s)(t, x') + \underbrace{\partial_2 L(t,x,v) D(D_t\widetilde{F}(s)(t, x')}_{\text{can swap }D_t\text{ and }D\text{, as }D\text{ w.r.t }s\text{ is unstructured}} \\ &= \partial_1 L(t, x, v) (D\widetilde{F})(s)(t, x') + \partial_2 L(t,x,v) D_t(D\widetilde{F}(s))(t, x') \\ &= (\partial_1 L \circ \Gamma[q]) \left( (D\widetilde{F})(s) \circ \Gamma[q']\right) + (\partial_2 L \circ \Gamma[q])\left( D_t(D\widetilde{F}(s)) \circ \Gamma[q']\right) \tag{1.157} \end{align*}
According to Lagrange equations, the first term of Eq. 1.157 is: $(\partial_1 L \circ \Gamma[q]) \left( (D\widetilde{F})(s) \circ \Gamma[q’]\right) = (D_t\partial_2 L \circ \Gamma[q]) \left((D\widetilde{F})(s) \circ \Gamma[q’]\right)$. Substituting this in Eq. 1.157,
$$0 = (D_t \partial_2 L \circ \Gamma[q])\left( (D\widetilde{F}(s)) \circ \Gamma[q']\right) + (\partial_2 L \circ \Gamma[q])\left( D_t(D\widetilde{F}(s)) \circ \Gamma[q']\right) \tag{1.159}$$
When $s = 0$, since $\widetilde{F}(0)$ is the identity transformation, the paths $q$ and $q’$ are the same. Therefore, $\Gamma[q] = \Gamma[q’]$ and Eq. 1.158 becomes
\begin{align} 0 &= \left ((D_t \partial_2 L)(D\widetilde{F}(0)) + (\partial_2 L)(D_t(D\widetilde{F}(0)))\right) \circ \Gamma[q] \\ &= D_t ((\partial_2 L) (D\widetilde{F}(0))) \circ \Gamma[q] \tag{1.160} \end{align}
Therefore the state function $\mathscr{I}$:
$$\mathscr{I} = (\partial_2 L) (D\widetilde{F}(0))$$
is conserved along all solution trajectories. This quantity is called the Noether integral. It is the product of the momentum $\partial_2 L$ and a vector associated with the symmetry.
### Illustration : Motion in a Central Potential
Consider the motion of a particle in a central potential. The Lagrangian in rectangular coordinates is:
$$L(t; x,y,z; v_x, v_y, v_z) = \frac{1}{2} m \left( v_x^2 + v_y^2 + v_z^2\right) - U(\sqrt{x^2 + y^2 + z^2})$$
Consider a parameteric rotation about the $z$-axis:
$$\begin{pmatrix}x\\y\\z\end{pmatrix} = R_z(s)(\begin{pmatrix}x'\\y'\\z'\end{pmatrix}) = \begin{pmatrix}x' \cos{s} - y'\sin{s}\\x' \sin{s} + y'\cos{s}\\z'\end{pmatrix} \tag{1.163}$$
Since a rotation is an orthogonal transformation, it does not change the magnitude of the vector,
$$x^2 + y^2 + z^2 = (x')^2 + (y')^2 + (z')^2\\$$
Similarly, differentiating Eq.1.163 along a path, we get:
$$\begin{pmatrix}v_x\\v_y\\v_z\end{pmatrix} = R_z(s)\begin{pmatrix}v_x'\\v_y'\\v_z'\end{pmatrix}$$
Therefore, $v_x^2 + v_y^2 + v_z^2 = v_x’^2 + v_y’^2 + v_z’^2$. Combining these, we can see that the post-transformation Lagrangian $L’$ is:
$$L'(t; x',y,z'; v_x',v_y',v_z') = \frac{1}{2} m \left( (v'_x)^2 + (v'_y)^2 + (v'_z)^2\right) - U(\sqrt{(x')^2 + (y')^2 + (z')^2 })$$
Therefore $L’$ is the exact same function as $L$ and hence there is a conserved value corresponding to the rotational symmetry about the z-axis. The momenta are defined as:
$$\partial_2 L = [m v_x, m v_y, m v_z]$$
and
$$D\widetilde{F}(0)(t;x,y,z)=D\widetilde{R}_z(0)(x,y,z) = [ y, -x, 0]$$
Note about the $D$ operator
The $D$ operator has the highest precedence, and therefore:
$$D\widetilde{F}(0)(t; x,y,z) = D\widetilde{F}(s)(x, y, z)|_{s=0} = \left.\begin{bmatrix} -x \sin{s} - y\cos{s}\\x \cos{s} - y\sin{s}\\0\end{bmatrix}\right|_{s=0}$$
Here we are taking the derivative w.r.t $s$ and consider $x$, $y$ and $z$ to be constants. Also note that the original $\widetilde{F}(s)$ was defined in terms of the primed coordinates while here it was evaluated on the unprimed coordinates.
Therefore, the Noether integral is:
\begin{align*} \mathscr{I}(t; x,y,z; v_x,v_y,v_z) &= ((\partial_2 L)(D\widetilde{F}(0))) (t; x,y,z; v_x,v_y,v_z) \\ &= mv_xy -mv_yx + (mv_z)(0) \\ &= m(yv_x - xv_y) \end{align*}
This is the $z$ component of the angular momentum vector, $\vec{x} \times m\vec{v}$
(defn RotX [angle]
(fn [[x, y, z]]
(let [ca (cos angle)
sa (sin angle)]
(up x
(- (* ca y) (* sa z))
(+ (* sa y) (* ca z))))))
(defn RotY [angle]
(fn [[x, y, z]]
(let [ca (cos angle)
sa (sin angle)]
(up (+ (* ca x) (* sa z))
y
(+ (- (* sa x)) (* ca z))))))
(defn RotZ [angle]
(fn [[x, y, z]]
(let [ca (cos angle)
sa (sin angle)]
(up (- (* ca x) (* sa y))
(+ (* sa x) (* ca y))
z))))
;; Coordinate transformation with three angular "inputs" for rotations about
;; all three axes
;; Composing with coordinate, extracts the second element of the tuple that is passed in as the argument
(defn F-tilde [angle-x angle-y angle-z]
(compose (RotX angle-x) (RotY angle-y) (RotZ angle-z) coordinate))
;; Lagrangian for motion in central potential
(defn L-central-rectangular [m U]
(fn [[t q v]]
(- (* 1/2 m (square v))
(U (sqrt (square q))))))
;; Define the Noether integral
(def the-Noether-integral
(let [L (L-central-rectangular 'm (literal-function 'U))]
(* ((partial 2) L) ((D F-tilde) 0 0 0))))
(rendertex
(the-Noether-integral
(up 't
(up 'x 'y 'z)
(up 'v_x 'v_y 'v_z))))
\begin{bmatrix}\displaystyle{- m\,v_y\,z + m\,v_z\,y}&\displaystyle{m\,v_x\,z - m\,v_z\,x}&\displaystyle{- m\,v_x\,y + m\,v_y\,x}\end{bmatrix}
These are all three components of the angular momentum. Therefore, angular momentum is conserved for a particle in motion in a central potential
← Back to workbook
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2022-12-06 13:06:58
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https://www.physicsforums.com/threads/partial-derivatives-of-a-function.543102/
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# Partial derivatives of a function
Mr.Rockwater
1. The problem statement, all variables and given known data
Find the partial derivatives (1st order) of this function:
$ln((\sqrt{(x^2+y^2} - x)/(\sqrt{x^2+y^2} + x))$
## The Attempt at a Solution
I obviously separated the logarithm quotient into a subtraction, then applied the rule d ln(u) = 1/u. However, what I end up with is four terms with a bunch of x²+y² and $\sqrt{x²+y²}$ . I'm just starting out with partial derivatives so is there any obvious trick that I'm not familiar with in this type of situation?
Last edited:
$ln((\sqrt{(x^2+y^2} - x))-ln((\sqrt{x^2+y^2} + x))$
$$\frac{\partial ln[f(x, y)]}{\partial x}=\frac{1}{f(x, y)}\frac{\partial f(x, y)}{\partial x}$$
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2023-03-23 15:27:10
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http://dml.cz/dmlcz/107987
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# Article
Full entry | PDF (0.2 MB)
Keywords:
multifunction; convex subdifferential; extremal periodic solution; Moreanu-Yosida approximation.
Summary:
We consider first order periodic differential inclusions in $\mathbb {R}^N$. The presence of a subdifferential term incorporates in our framework differential variational inequalities in $\mathbb {R}^N$. We establish the existence of extremal periodic solutions and we also obtain existence results for the “convex” and “nonconvex”problems.
References:
[1] Bader R.: A topological fixed point theory for evolutions inclusions. Z. Anal. Anwendungen 20 (2001), 3–15. MR 1826317
[2] Bourgain J.: An averaging result for $l^1$-sequences and applications to weakly conditionally compact sets in $L^1(X)$. Israel J. Math. 32 (1979), 289–298. MR 0571083
[3] Cornet B.: Existence of slow solutions for a class differential inclusions. J. Math. Anal. Appl. 96 (1983), 130–147. MR 0717499
[4] De Blasi F. S., Gorniewicz L., Pianigiani G.: Topological degree and periodic solutions of differential inclusions. Nonlinear Anal. 37 (1999), 217–245. MR 1689752 | Zbl 0936.34009
[5] De Blasi F. S., Pianigiani G.: Nonconvex valued differential inclusions in Banach spaces. J. Math. Anal. Appl. 157 (1991), 469–494. MR 1112329
[6] Haddad G., Lasry J.-M.: Periodic solutions of functional differential inclusions and fixed points of $\gamma$-selectionable correspondences. J. Math. Anal. Appl. 96 (1983), 295–312. MR 0719317
[7] Halidias N., Papageorgiou N. S.: Existence and relaxation results for nonlinear second order multivalued boundary value problems in $\mathbb{R^N}$. J. Differential Equations 147 (1998), 123–154. MR 1632661
[8] Henry C.: Differential equations with discontinuous right hand side for planning procedures. J. Econom. Theory 4 (1972), 545–551. MR 0449534
[9] Hu S., Kandilakis D., Papageorgiou N. S.: Periodic solutions for nonconvex differential inclusions. Proc. Amer. Math. Soc. 127 (1999), 89–94. MR 1451808 | Zbl 0905.34036
[10] Hu S., Papageorgiou N. S.: On the existence of periodic solutions for nonconvex valued differential inclusions in $\mathbb{R^N}$. Proc. Amer. Math. Soc. 123 (1995), 3043–3050. MR 1301503
[11] Hu S., Papageorgiou N. S.: Handbook of Multivalued Analysis. Volume I: Theory. Kluwer, Dordrecht, The Netherlands (1997). MR 1485775 | Zbl 0887.47001
[12] Hu S., Papageorgiou N. S.: Handbook of Multivalued Analysis. Volume II: Applications. Kluwer, Dordrecht, The Netherlands (2000). MR 1741926 | Zbl 0943.47037
[13] Li C., Xue X.: On the existence of periodic solutions for differential inclusions. J. Math. Anal. Appl. 276 (2002), 168–183. MR 1944344 | Zbl 1020.34015
[14] Macki J., Nistri P., Zecca P.: The existence of periodic solutions to nonautonomous differential inclusions. Proc. Amer. Math. Soc. 104 (1988), 840–844. MR 0931741 | Zbl 0692.34042
[15] Plaskacz S.: Periodic solutions of differential inclusions on compact subsets of $\mathbb{R^N}$. J. Math. Anal. Appl. 148 (1990), 202–212. MR 1052055
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2016-10-24 08:56:02
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https://valutadafc.firebaseapp.com/20327/79146.html
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# Tottenham Hotspur Among Clubs Keeping Tabs On This
tfp love quiz - Aktionsbündnis Gerechter Welthandel
changes in wealth: increases in wealth reduce labor supply and shift the labor supply curve to the left Start studying Econ 100B: Lecture 4: Aggregate Production and Total Factor Productivity. Learn vocabulary, terms, and more with flashcards, games, and other study tools. If each country produces using identical production functions, but yA > yB and kA = kB, the total factor productivity of country A equals that of B. false If you have data on per capita GDP and capital per worker, to find total factor productivity you can use the equation Abar = … Total factor productivity is a measure of an organization's: A. market competitiveness. B. responsiveness. C. process design standard. D. efficiency. E. raw material usage.
Print Production, Productivity & Competitiveness Worksheet 1. Total factor productivity is also known as which of the In economics, total-factor productivity (TFP), also called multi-factor productivity, is usually measured as the ratio of aggregate output (e.g., GDP) to aggregate inputs. Under some simplifying assumptions about the production technology, growth in TFP becomes the portion of growth in output not explained by growth in traditionally measured inputs of labour and capital used in production. [2] 2019-04-10 · Conceptually, total factor productivity refers to how efficiently and intensely inputs are used in the production process. Total factor productivity (TFP) is sometimes referred to as "multi-factor productivity," and, under certain assumptions, can be thought of as a measure of level of technology or knowledge. 2020-03-20 · Total factor productivity change (TFPCH) of a production unit between two periods, t and t + 1, is estimated by the Malmquist Index, $$TFPCH_{I}^{t} = \frac{{E_{I}^{t } \left( {x^{t + 1} , y^{t + 1} } \right) }}{{E_{I}^{t } \left( {x^{t} , y^{t} } \right)}},$$ Explaining Total Factor Productivity “Needed: A Theory of Total Factor Productivity” Edward C. Prescott (1998) 1. Introduction Total Factor Productivity (TFP) has become the choice measure of productivity.
a. how many children actually complete a certain grade level. "The total cost for some specific activities by richer countries governments in production function consisting of capital, labor and total factor productivity (TFP).
## Visual Adventures » Safeco Field Tour
The source of growth. As I explained earlier, total factor productivity is the key to long-run economic growth because labor and capital have a decreasing marginal return. It becomes even more critical when the capital per worker ratio is high, as in developed countries. One factor that permeates all industries from pure service and software to pure manufacturing of products and goods is called the Total Factor Productivity (TFP).
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Definition ofMultifactor productivity. Multifactor productivity (MFP) reflects the overall efficiency with which labour and capital inputs are used together in the production process. Changes in MFP reflect the effects of changes in management practices, brand names, organizational change, general knowledge, network effects, spillovers from 2020-08-15 · This site presents a real-time, quarterly series on total factor productivity (TFP) for the U.S. business sector, adjusted for variations in factor utilization - labor effort and capital's workweek. The utilization adjustments follows Basu, Fernald, and Kimball (BFK, 2006). Using relative prices and 2006-12-14 · In the production function, A captures the efficiency with which labor and capital are used, which is called multifactor productivity, or total factor, productivity.
TFP is the part of output which is not explained by the amount of inputs used in production. State-owned forestry enterprises are important elements of the forestry economy in China. The operational efficiency of such enterprises depends on technological progress and other input factors. Total factor productivity (TFP) is an important means to evaluate the efficiency of technical elements.
Vem ärver man håret av
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2022-08-16 19:09:47
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https://www.eduzip.com/ask/question/if-two-supplementary-angles-are-in-the-ratio-45-then-the-angles-a-520732
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Mathematics
# If two supplementary angles are in the ratio $4:5$, then the angles are __________.
$80^o, 100^o$
##### SOLUTION
If two angles are supplementary, then the sum of the angles is $180^o$
If the ratio is $4:5$, let angles are $4x$ and $5x$
Now we know, $4x+5x=180^o$
$\Rightarrow 9x=180$
$\Rightarrow x=20$
Therefore, angles are $100^o$ and $80^o$.
You're just one step away
Single Correct Medium Published on 09th 09, 2020
Questions 120418
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#### Realted Questions
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Q2 Single Correct Medium
Let $l_{1}\parallel l_{2}$ and $m_{1}$ is a transversal. If $\angle F = 65^{\circ}$, find the measure of each of the remaining angles
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1 Verified Answer | Published on 09th 09, 2020
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2022-01-24 03:01:48
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https://www.physicsforums.com/threads/fourier-transform-and-parsevals-theorem.847506/
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# Fourier Transform and Parseval's Theorem
1. Dec 10, 2015
### roam
1. The problem statement, all variables and given/known data
Using Parseval's theorem,
$$\int^\infty_{-\infty} h(\tau) r(\tau) d\tau = \int^\infty_{-\infty} H(s)R(-s) ds$$
and the properties of the Fourier transform, show that the Fourier transform of $f(t)g(t)$ is
$$\int^\infty_{-\infty} F(s)G(\nu-s)ds$$
2. Relevant equations
Fourier transform for $f(t)g(t)$ is defined as:
$$\int^\infty_{-\infty} f(t)g(t) e^{-2 \pi \nu t} dt$$
3. The attempt at a solution
So starting from the definition of Fourier transform:
$$\int^\infty_{-\infty} f(t)g(t) e^{-2 \pi \nu t} dt$$
So, do we need to ignore the exponential term here? If we ignore it, we can apply Parseval's theorem to get the frequency domain:
$$\int^\infty_{-\infty} f(t)g(t) dt = \int^\infty_{-\infty} F(s) G(- s) d s$$
Now, what property of the Fourier transform can I use to get $G(-s) \implies G(\nu-s)$?
I don't understand what the $(\nu-s)$ part means, does it indicates some sort of shifting or delay in the input?
Any help is greatly appreciated.
2. Dec 10, 2015
### Samy_A
I think you forgot an $i$ in the exponent in the definition of Fourier transform.
You cannot just ignore the exponential term in $\int^\infty_{-\infty} f(t)g(t) e^{-2 \pi iv t} dt$
Start with $G(v-s)$ for a fixed $v$.
Write out the formula for $G(v-s)$, and you will note that $G(v-s)=\mathcal F (g_v(-s))$ for some function $g_v$ ($\mathcal F$ denotes the Fourier transform).
Using that knowledge about $G(v-s)$, apply Parseval's theorem to $\int^\infty_{-\infty} F(s)G(v-s)ds$ and see what you get ...
It is clearer when you write something like:
the Fourier transform of $f(t)g(t)$ in $\nu$ is $$\int^\infty_{-\infty} F(s)G(\nu-s)ds$$
Last edited: Dec 10, 2015
3. Dec 11, 2015
### roam
Thank you for your input. In $g_v$ what does the subscript $v$ mean? Should it not be $g_t$ since $G(s)$ is the Fourier transform of $g(t)$? (In my course a function denoted by a capital letter is the Fourier transform of the function denoted by the corresponding small letter)
So from the definition, the formula for the $G(v-s)$ part is:
$$G(v-s) = \int^\infty_{-\infty} g(t) e^{-i 2 \pi st} dt$$
Here I used the duality property of the Fourier transform i.e. $F(t) \iff f(-s)$
Is this the right formula?
4. Dec 11, 2015
### Samy_A
There is no $v$ in your right hand side expression for $G(v-s)$, so that can't be correct.
I really meant $g_v$. Just pick one $v$ as a constant and let's work with that for a while.
In general, the definition of the Fourier transform is:
$$G(u) = \int^\infty_{-\infty} g(t) e^{-2 \pi iut} dt$$
I deliberately used another name for the variable here.
Now set $u=v-s$ in that definition. You get:
$$G(v-s) = \int^\infty_{-\infty} g(t) e^{-2 \pi i(v-s)t} dt=\int^\infty_{-\infty} \Big(g(t)e^{- 2 \pi ivt}\Big)e^{-2 \pi i(-s)t} dt$$
Compare that last expression with the definition of the Fourier transform. What we have there is the Fourier transform of the function between the big brackets at the point $-s$. So now name the function between the big brackets $g_v$, and what we have is that:
$$g_v(t)=g(t)e^{-2 \pi ivt}$$
$$G(v-s)=\mathcal F(g_v(-s))$$
Or, if you prefer the convention of your course:
$$G(v-s)=G_v(-s)$$
Now apply this to evaluate $\int^\infty_{-\infty} F(s)G(v-s)ds$, by using Parseval's theorem.
Last edited: Dec 11, 2015
5. Dec 12, 2015
### Samy_A
Never mind. I thought something was wrong, but I was wrong about that.
My sincere apologies for my confusion.
Last edited: Dec 12, 2015
6. Dec 12, 2015
### roam
I see. Thank you very much for the clear explanation.
$$\int^\infty_{-\infty} F(s) G(\nu -s) ds = \int^\infty_{-\infty} F(s) G_\nu (-s) ds$$
using Parseval's theorem this becomes:
$$\int^\infty_{-\infty} f(t) g_\nu (t) dt = \int^\infty_{-\infty} f(t) g(t) e^{-2 \pi i \nu t} dt$$
Is this then sufficient to 'show' that $\int^\infty_{-\infty} F(s) G(\nu -s)ds$ is the Fourier transform of $f(t)g(t)$?
7. Dec 12, 2015
### Samy_A
Yes, because taking the two equations together, you now proved that for any $\nu$: $$\int^\infty_{-\infty} f(t) g(t) e^{-2 \pi i \nu t}=\int^\infty_{-\infty} F(s) G(\nu -s) ds$$
The left hand side is by definition the Fourier transform of the function $f.g$ at point $\nu$.
As an aside, $\int^\infty_{-\infty} F(s) G(\nu -s) ds$ is called the convolution of F and G at $\nu$. (More here and here)
Last edited: Dec 12, 2015
8. Dec 12, 2015
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2018-02-24 02:42:24
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http://face-paging.com/2013/04/why-ratios-of-the-scale-factor/
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# Why ratios of the scale factor?
Posted by on Apr 6, 2013 in Blog, Science | 0 comments
I introduced the notion of a scale factor as a concept separate from a fixed and comoving cosmological lattice in a previous post. Developing that idea lead to the Hubble non-constant, $H = \dot{a}(t)/a(t)$.
Later, in developing a couple of simple forms of the Friedmann equations, we encountered something of the form $\ddot{a}(t)/a(t)$. This raises a question.
## Why all these ratios of a(t)?
The answer is simple. The scale factor depends on the scale of the comoving cosmological grid that’s being used. Since proper distances depend upon the product of the scale factor and the size of the grid, $D = a(t) \sqrt{x^2 + y^2 + z^2}$, changing the size of the lattice must have an inverse effect on the corresponding scale factor assuming that the proper distance, $D$, is a given.
Hence, it is only ratios of the scale factor that can be invariant. For example, independent of the choice of grid, the value of $a(t_1)/a(t_0)$; that is, the ratio of the scale factor at two different times, $t_1$ and $t_0$, would be an invariant as to the choice of the scale of the lattice. Likewise, in the Hubble constant, we divide the time derivative of the scale factor by the scale factor itself in order to get an invariant measure of the rate of change of the scale factor. The same is true of the acceleration of $a(t)$ is expressed in the Friedmann equations.
If we expected fluctuations in $a(t)$, we would write these in the form $\delta a(t) / a(t)$ as well. This is not surprising either; fluctuations in some parameter are often expressed in this way. Then, we can consider a fluctuation of, say, 1%, as being greater than one of, say, 0.01%, or alternatively, 1 part per million, by some invariant factor.
### Scaling density
A similar question arises with regard to scaling the density of, say, matter, in our comoving cosmological grid. We introduced a density that was invariant to the grid, call it $\nu$; but this value will again depend upon the scale of the lattice. We could not call this a proper density, if such a term were used. Instead, we employ $\rho$ as a measure of the density of matter relative to a proper volume comprised of a sphere of a radius of given proper length, $D$.
In this way, when we wrote a simple form of the Friedmann equations for a universe with a net $0$ of energy
$(\frac{ \dot{a}(t)}{a(t)})^2 = \frac{8 \pi G}{3} \rho$
we had invariants on both sides of the equation. This equation expresses an exact balance between a kinetic energy term on the LHS and a potential energy term on the RHS. Not too surprisingly, they have the form of Einstein’s field equations; and the kinetic energy term on the LHS also has the character of a metric of space that is in balance with a 00-component of a tensor on the RHS.
However, this form of the equation contains an implicit occurrence of the scale factor in the term $\rho$ that we have to make explicit in order to solve the equation, $\rho = \nu /a^3$. Assuming that we are working with a density of ordinary matter, made up of protons, neutrons, and electrons, then we can just about assume that $\nu$ is some constant. Protons are stable and slow, and we can hang their numbers to our cosmological grid of some grand scale. Neutrons are not quite so stable though, but decay into protons and electrons on their own with a half-life of about 14 – 15 minutes or so. The problem is not so much getting out another proton from a neutron decay, the quantity of matter is roughly equivalent, it is getting out the occasional photon.
This puts us in mind of the fact that we have so far been considering only the density of ordinary matter in our equation for density, and we are well aware that we should really be adding up all of the energy present on the RHS of our last equation. This is true even if we want an exact null balance between that kinetic energy term on the LHS and a matter-energy term on the RHS.
In subsequent posts, we’ll begin to account for these other terms that have to be considered in a complete treatment …
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https://www.zbmath.org/authors/?q=ai%3Awang.weike
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# zbMATH — the first resource for mathematics
## Wang, Weike
Compute Distance To:
Author ID: wang.weike Published as: Wang, W.; Wang, W. K.; Wang, Wei-Ke; Wang, WeiKe; Wang, Weike
Documents Indexed: 157 Publications since 1982 Reviewing Activity: 18 Reviews
all top 5
#### Co-Authors
15 single-authored 13 Deng, Shijin 12 Wu, Zhigang 7 Yang, Tong 7 Yu, Shih-Hsien 6 Wang, Wenjun 6 Yang, Xiongfeng 6 Yi, Xuexi 5 Liu, Yongqin 4 Chen, Jiao 4 Xu, Hongmei 4 Xu, Xin 3 Li, Kaiqiang 3 Liu, Guowei 3 Shi, Renkun 3 Wang, Yutong 2 Kim, Jongsung 2 Li, Fengbai 2 Li, Yachun 2 Shen, Jisen 2 Sheng, Weiming 2 Shi, Binbin 2 Wang, Yucheng 2 Zhang, Guizhou 2 Zhu, Xusheng 1 Chen, Hua 1 Dai, C. M. 1 Fu, Yuxia 1 Hou, Suchung 1 Li, Nianying 1 Liao, Jie 1 Liu, Haidong 1 Liu, Miao 1 Liu, Tai-Ping 1 Nishihara, Kenji 1 Racke, Reinhard 1 Rao, Youlan 1 Shi, Zhongchao 1 Wang, Lijuan 1 Wang, Zejun 1 Wu, Wei 1 Xu, Chao-Jiang 1 Xue, Rui 1 Zhang, Dandan 1 Zhao, Hualei
all top 5
#### Serials
14 Journal of Differential Equations 8 Journal of Mathematical Analysis and Applications 7 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 7 Wuhan University Journal of Natural Sciences (WUJNS) 5 Chinese Annals of Mathematics. Series B 5 Discrete and Continuous Dynamical Systems 4 Nonlinear Analysis. Real World Applications 3 Archive for Rational Mechanics and Analysis 3 Journal of Mathematical Physics 3 Journal of Wuhan University. Natural Science Edition 3 Acta Mathematica Scientia. Series B. (English Edition) 3 Communications on Pure and Applied Analysis 3 Journal of Hyperbolic Differential Equations 2 Communications in Mathematical Physics 2 Mathematical Methods in the Applied Sciences 2 Journal of Mathematics. Wuhan University 2 Chinese Annals of Mathematics. Series A 2 Journal of Partial Differential Equations 2 Methods and Applications of Analysis 2 Communications in Mathematical Sciences 2 International Journal of Quantum Information 2 Science China. Mathematics 1 International Journal of Theoretical Physics 1 ZAMP. Zeitschrift für angewandte Mathematik und Physik 1 Acta Mathematica Sinica 1 Indiana University Mathematics Journal 1 Journal of Functional Analysis 1 Acta Mathematicae Applicatae Sinica. English Series 1 Revista Matemática Iberoamericana 1 Mathematica Applicata 1 Journal of Partial Differential Equations. Series B 1 Annals of Physics 1 Communications in Partial Differential Equations 1 SIAM Journal on Mathematical Analysis 1 Electronic Journal of Differential Equations (EJDE) 1 NoDEA. Nonlinear Differential Equations and Applications 1 New Journal of Physics 1 Discrete and Continuous Dynamical Systems. Series B 1 Analysis and Applications (Singapore) 1 Bulletin of the Institute of Mathematics. Academia Sinica. New Series 1 Mathematical Research Report 1 Frontiers of Mathematics in China 1 Networks and Heterogeneous Media 1 Kinetic and Related Models 1 Quantum Information & Computation
all top 5
#### Fields
98 Partial differential equations (35-XX) 42 Fluid mechanics (76-XX) 13 Statistical mechanics, structure of matter (82-XX) 6 Quantum theory (81-XX) 5 Optics, electromagnetic theory (78-XX) 5 Biology and other natural sciences (92-XX) 3 Dynamical systems and ergodic theory (37-XX) 3 Harmonic analysis on Euclidean spaces (42-XX) 3 Mechanics of deformable solids (74-XX) 2 Special functions (33-XX) 2 Numerical analysis (65-XX) 1 Operator theory (47-XX) 1 Differential geometry (53-XX) 1 Global analysis, analysis on manifolds (58-XX) 1 Systems theory; control (93-XX)
#### Citations contained in zbMATH Open
96 Publications have been cited 821 times in 547 Documents Cited by Year
The pointwise estimates of diffusion wave for the Navier-Stokes systems in odd multi-dimensions. Zbl 0912.35122
Liu, Tai-Ping; Wang, Weike
1998
The pointwise estimates of solutions for Euler equations with damping in multi-dimensions. Zbl 0997.35039
Wang, Weike; Yang, Tong
2001
$$L_p$$-convergence rate to nonlinear diffusion waves for $$p$$-system with damping. Zbl 0946.35012
Nishihara, Kenji; Wang, Weike; Yang, Tong
2000
Pointwise estimates of solution for the Navier-Stokes-Poisson equations in multi-dimensions. Zbl 1185.35177
Wang, Weike; Wu, Zhigang
2010
The Cauchy problem for viscous shallow water equations. Zbl 1095.35037
Wang, Weike; Xu, Chao-Jiang
2005
Completeness and nonuniqueness of general solutions of transversely isotropic elasticity. Zbl 0865.73008
Wang, M. Z.; Wang, W.
1995
Thick plate theory based on general solutions of elasticity. Zbl 0902.73048
Wang, W.; Shi, M. X.
1997
Spatiotemporal dynamics in a spatial plankton system. Zbl 1197.92049
Upadhyay, R. K.; Wang, W.; Thakur, N. K.
2010
Well-posedness for a model derived from an attraction-repulsion chemotaxis system. Zbl 1307.35065
Shi, Renkun; Wang, Weike
2015
Pointwise estimates and $$L_p$$ convergence rates to diffusion waves for $$p$$-system with damping. Zbl 1035.35077
Wang, Weike; Yang, Tong
2003
Decay rates of the compressible Navier-Stokes-Korteweg equations with potential forces. Zbl 1304.35576
Wang, Wenjun; Wang, Weike
2015
Intersonic crack growth in bimaterial interfaces: an investigation of crack face contact. Zbl 1056.74561
Huang, Y.; Wang, W.; Liu, C.; Rosakis, A. J.
1998
The pointwise estimates of solutions for semilinear dissipative wave equation in multi-dimensions. Zbl 1184.35218
Wang, Weike; Wang, Wenjun
2010
The pointwise estimates of solutions for dissipative wave equation in multi-dimensions. Zbl 1154.35014
Liu, Yongqin; Wang, Weike
2008
A dual reciprocity boundary element approach for the problems of large deflection of thin elastic plates. Zbl 0974.74077
Wang, W.; Ji, X.; Tanaka, M.
2000
$$L^p$$ convergence rates of planar waves for multi-dimensional Euler equations with damping. Zbl 1170.35544
Liao, Jie; Wang, Weike; Yang, Tong
2009
Existence and stability of planar diffusion waves for 2D Euler equations with damping. Zbl 1147.35076
Wang, Weike; Yang, Tong
2007
Nonlinear stability of boundary layers of the Boltzmann equation for cutoff hard potentials. Zbl 1112.76039
Wang, Weike; Yang, Tong; Yang, Xiongfeng
2006
A study of microbend test by strain gradient plasticity. Zbl 1032.74528
Wang, W.; Huang, Y.; Hsia, K. J.; Hu, K. X.; Chandra, A.
2003
The pointwise estimates of solutions to the isentropic Navier-Stokes equations in even space-dimensions. Zbl 1083.35101
Wang, Weike; Yang, Xiongfeng
2005
Pointwise estimates of solutions for non-isentropic Navier-Stokes-Poisson equations in multi-dimensions. Zbl 1274.35273
Wu, Zhigang; Wang, Weike
2012
Existence theory and $$L^p$$ estimates for the solution of nonlinear viscous wave equation. Zbl 1202.35138
Deng, Shijin; Wang, Weike; Zhao, Hualei
2010
The pointwise estimates of solutions for a model system of the radiating gas in multi-dimensions. Zbl 1178.35104
Wang, Weike; Wang, Wenjun
2009
Analysis of intersonic crack growth in unidirectional fiber-reinforced composites. Zbl 0963.74050
Huang, Y.; Wang, W.; Liu, C.; Rosakis, A. J.
1999
Green’s functions of wave equations in $$\mathbb R^n_+\times\mathbb R_+$$. Zbl 1457.35074
Deng, Shijin; Wang, Weike; Yu, Shih-Hsien
2015
Existence of boundary layers to the Boltzmann equation with cutoff soft potentials. Zbl 1144.81422
Wang, Weike; Yang, Tong; Yang, Xiongfeng
2007
On the general solutions of transversely isotropic elasticity. Zbl 0918.73016
Wang, W.; Shi, M. X.
1998
Large time behavior for the system of a viscous liquid-gas two-phase flow model in $$\mathbb{R}^3$$. Zbl 1457.76132
Wang, Wenjun; Wang, Weike
2016
Refined pointwise estimates for the Navier-Stokes-Poisson equations. Zbl 1348.35205
Wu, Zhigang; Wang, Weike
2016
Cooperative fuzzy adaptive output feedback control for synchronisation of nonlinear multi-agent systems under directed graphs. Zbl 1332.93220
Wang, W.; Wang, D.; Peng, Z. H.
2015
The suppressible property of the solution for three-dimensional Euler equations with damping. Zbl 1179.35187
Yang, Xiongfeng; Wang, Weike
2007
Optimal decay rate of solutions for Cahn-Hilliard equation with inertial term in multi-dimensions. Zbl 1229.35252
Wang, Weike; Wu, Zhigang
2012
A refined theory of transversely isotropic piezoelectric plates. Zbl 1062.74028
Xu, S. P.; Wang, W.
2004
Sufficient conditions on Liouville type theorems for the 3D steady Navier-Stokes equations. Zbl 1434.35063
Seregin, G.; Wang, W.
2020
Pointwise estimates for bipolar compressible Navier-Stokes-Poisson system in dimension three. Zbl 1373.35258
Wu, Zhigang; Wang, Weike
2017
Global classical solutions to the Cauchy problem of conservation laws with degenerate diffusion. Zbl 1336.35232
Chen, Jiao; Li, Yachun; Wang, Weike
2016
Large time behavior and pointwise estimates for compressible Euler equations with damping. Zbl 1334.35219
Wu, Zhigang; Wang, Weike
2015
Large time behavior of solution for the full compressible Navier-Stokes-Maxwell system. Zbl 1328.35156
Wang, Weike; Xu, Xin
2015
Decay of the solution for the bipolar Euler-Poisson system with damping in dimension three. Zbl 1307.35056
Wu, Zhigang; Wang, Weike
2014
Global existence and pointwise estimates of solutions for the multidimensional generalized Boussinesq-type equation. Zbl 1287.35072
Liu, Miao; Wang, Weike
2014
Pointwise convergence to a Maxwellian for a Broadwell model with a supersonic boundary. Zbl 1140.82028
Deng, Shijin; Wang, Weike; Yu, Shih-Hsien
2007
Global existence of solution to the Camassa-Holm equation. Zbl 1063.35137
Liu, Yongqin; Wang, Weike
2005
Pointwise estimates of solutions to Cauchy problem for quasilinear hyperbolic systems. Zbl 1046.35071
Wang, Weike; Yang, Xiongfeng
2003
The pointwise estimates of diffusion wave of the compressible micropolar fluids. Zbl 1392.35242
Wu, Zhigang; Wang, Weike
2018
Half space problem for Euler equations with damping in 3-D. Zbl 1375.35342
Deng, Shijin; Wang, Weike
2017
Nonlinear stability of large perturbation around viscous shock wave for 2-D scalar viscous conservation law. Zbl 1362.35041
Shi, Renkun; Wang, Weike
2016
The decay rate of solution for the bipolar Navier-Stokes-Poisson system. Zbl 1304.35509
Wang, Weike; Xu, Xin
2014
Pointwise decaying rate of large perturbation around viscous shock for scalar viscous conservation law. Zbl 1272.82032
Deng, ShiJin; Wang, WeiKe
2013
Large-time behavior of periodic solutions to fractal Burgers equation with large initial data. Zbl 1255.35053
Wang, Lijuan; Wang, Weike
2012
Broadwell model and conservative supersonic boundary. Zbl 1294.35007
Deng, Shijin; Wang, Weike; Yu, Shih-Hsien
2011
Dynamic responses of shear flows over a deformable porous surface layer in a cylindrical tube. Zbl 1167.76332
Wen, P. H.; Hon, Y. C.; Wang, W.
2009
The pointwise estimates to solutions for 1-dimensional linear thermoelastic system with second sound. Zbl 1107.35031
Wang, Weike; Wang, Zejun
2007
Inviscid limit for the damped Boussinesq equation. Zbl 1433.35286
Liu, Guowei; Wang, Weike
2019
Blow up and global existence of solutions for a model system of the radiating gas. Zbl 1262.35058
Wang, Wenjun; Wang, Weike
2013
Modeling adaptive behavior in influenza transmission. Zbl 1250.34041
Wang, W.
2012
Nonlinear evolution systems and Green’s function. Zbl 1240.35358
Wang, Weike
2010
Pointwise estimates of solutions for the Euler-Poisson equations with damping in multi-dimensions. Zbl 1186.35130
Wu, Zhigang; Wang, Weike
2010
Interaction between micro-particles in Oseen flows by the method of fundamental solutions. Zbl 1244.76092
Wang, W.; Wen, P. H.
2008
Well-posedness of the IBVP for 2-D Euler equations with damping. Zbl 1152.35070
Liu, Yongqin; Wang, Weike
2008
Pointwise estimates for the wave equation with dissipation in odd spatial dimension. Zbl 1155.35007
Xu, Hongmei; Wang, Weike
2008
The regular solutions of the isentropic Euler equations with degenerate linear damping. Zbl 1087.35061
Zhu, Xusheng; Wang, Weike
2005
Flow of spatially non-uniform suspensions. III: Closure relations for porous media and spinning particles. Zbl 1137.76777
Wang, W.; Prosperetti, A.
2001
Pointwise estimate of solutions of isentropic Navier-Stokes equations in even space-dimensions. Zbl 1007.35057
Xu, Hongmei; Wang, Weike
2001
Constructivity and completeness of the general solutions in elastodynamics. Zbl 0753.73019
Wang, W.; Wang, M. Z.
1992
The $$L^{p}$$ decay estimates for the chemotaxis-shallow water system. Zbl 1416.35219
Wang, Weike; Wang, Yucheng
2019
Large-time behavior for the solution to the generalized Benjamin-Bona-Mahony-Burgers equation with large initial data in the whole-space. Zbl 1308.35255
Wang, Weike; Zhang, Dandan
2014
Decay of a model system of radiating gas. Zbl 1307.35055
Wang, Wenjun; Wang, Weike; Wu, Zhigang
2014
Global existence of solutions to the initial-boundary value problem of conservation law with degenerate diffusion term. Zbl 1268.35082
Chen, Jiao; Li, Yachun; Wang, Weike
2013
Pointwise convergence to Knudsen layers of the Boltzmann equation. Zbl 1162.82022
Deng, Shijin; Wang, Weike; Yu, Shih-Hsien
2008
The Broadwell model with a subsonic boundary. Zbl 1159.82013
Deng, Shijin; Wang, Weike
2008
Free vibration of elastic helicoidal shells. Zbl 1192.74179
Hu, X. X.; Lim, C. W.; Sakiyama, T.; Li, Z. R.; Wang, W. K.
2005
Flow of spatially non-uniform suspensions. II: Systematic derivation of closure relations. Zbl 1137.76669
Marchioro, M.; Tanksley, M.; Wang, W.; Prosperetti, A.
2001
Large time behavior of solutions for general Navier-Stokes systems in multi-dimension. Zbl 0921.35124
Wang, Weike
1997
On single-valuedness of the Papkovich-Neuber general solution in two- dimensional elasticity. Zbl 0714.73008
Wang, W.; Wang, M. Z.
1990
Multiple attribute group decision making with linguistic variables and complete unknown weight information. Zbl 1429.91131
Wang, W.; Mendel, J. M.
2019
The well-posedness of solution to semilinear pseudo-parabolic equation. Zbl 1428.35185
Wang, Wei-Ke; Wang, Yu-Tong
2019
Decay of solutions to anisotropic conservation laws with large initial data. Zbl 06951101
Li, Kaiqiang; Wang, Weike
2018
Cooperative learning neural network output feedback control of uncertain nonlinear multi-agent systems under directed topologies. Zbl 1372.93032
Wang, W.; Wang, D.; Peng, Z. H.
2017
Optimal decay rates and global existence for a semilinear Timoshenko system with two damping effects. Zbl 1391.35061
Racke, Reinhard; Wang, Weike; Xue, Rui
2017
$$L_p$$-dual mixed affine surface areas. Zbl 07030371
Wan, X.; Wang, W.
2016
The pointwise estimates of solutions to the Cauchy problem of a chemotaxis model. Zbl 1344.35157
Shi, Renkun; Wang, Weike
2016
Bifurcation on boundary data for linear Broadwell model with conservative boundary condition. Zbl 1304.82064
Deng, Shijin; Wang, Weike; Yu, Shih-Hsien
2014
The pointwise estimate of solutions to the parabolic conservation law in multi-dimensions. Zbl 1302.35084
Li, Fengbai; Wang, Weike
2014
The point-wise estimates for the solution of damped wave equation with nonlinear convection in multi-dimensional space. Zbl 1308.35142
Chen, Jiao; Wang, Weike
2014
Viscous conservation laws with boundary. Zbl 1269.82060
Deng, Shijin; Wang, Weike; Yu, Shih-Hsien
2012
Adiabatic evolution under quantum control. Zbl 1242.81056
Wang, W.; Hou, S. C.; Yi, X. X.
2012
Unsteady flows in a capillary lined with a thin porous surface layer by method of fundamental solutions. Zbl 1274.74453
Wen, P. H.; Wang, W.; Liu, Y. W.
2009
On the boundary conditions for transversely isotropic piezoelectric plates. Zbl 1192.74231
Xu, S. P.; Gao, Y.; Wang, W.
2007
Local well-posedness of a new integrable equation. Zbl 1094.35124
Liu, Yongqin; Wang, Weike
2006
The pointwise estimates of solutions for general Navier-Stokes systems in odd multi-dimensions. Zbl 1110.35061
Wang, Weike
2005
A method for linear elasto-static analysis of multi-layered axisymmetrical bodies using Hankel’s transform. Zbl 1054.74071
Wang, W.; Ishikawa, H.
2001
Component mode synthesis for damped rotor systems with hybrid interfaces. Zbl 0945.74607
Wang, W.; Kirkhope, J.
1994
New eigensolutions and modal analysis for gyroscopic/rotor systems. I: Undamped systems. II: Perturbation analysis for damped systems. Zbl 0945.70523
Wang, W.; Kirkhope, J.
1994
Component mode synthesis for multi-shaft rotors with flexible inter-shaft bearings. Zbl 0925.73481
Wang, W.; Kirkhope, j.
1994
Interaction of conormal waves with different singularities for semi- linear equations. Zbl 0763.35063
Wang, Weike
1991
Reflection of transversal progressing waves for quasilinear 2-systems. Zbl 0699.35173
Wang, Weike
1989
Sufficient conditions on Liouville type theorems for the 3D steady Navier-Stokes equations. Zbl 1434.35063
Seregin, G.; Wang, W.
2020
Inviscid limit for the damped Boussinesq equation. Zbl 1433.35286
Liu, Guowei; Wang, Weike
2019
The $$L^{p}$$ decay estimates for the chemotaxis-shallow water system. Zbl 1416.35219
Wang, Weike; Wang, Yucheng
2019
Multiple attribute group decision making with linguistic variables and complete unknown weight information. Zbl 1429.91131
Wang, W.; Mendel, J. M.
2019
The well-posedness of solution to semilinear pseudo-parabolic equation. Zbl 1428.35185
Wang, Wei-Ke; Wang, Yu-Tong
2019
The pointwise estimates of diffusion wave of the compressible micropolar fluids. Zbl 1392.35242
Wu, Zhigang; Wang, Weike
2018
Decay of solutions to anisotropic conservation laws with large initial data. Zbl 06951101
Li, Kaiqiang; Wang, Weike
2018
Pointwise estimates for bipolar compressible Navier-Stokes-Poisson system in dimension three. Zbl 1373.35258
Wu, Zhigang; Wang, Weike
2017
Half space problem for Euler equations with damping in 3-D. Zbl 1375.35342
Deng, Shijin; Wang, Weike
2017
Cooperative learning neural network output feedback control of uncertain nonlinear multi-agent systems under directed topologies. Zbl 1372.93032
Wang, W.; Wang, D.; Peng, Z. H.
2017
Optimal decay rates and global existence for a semilinear Timoshenko system with two damping effects. Zbl 1391.35061
Racke, Reinhard; Wang, Weike; Xue, Rui
2017
Large time behavior for the system of a viscous liquid-gas two-phase flow model in $$\mathbb{R}^3$$. Zbl 1457.76132
Wang, Wenjun; Wang, Weike
2016
Refined pointwise estimates for the Navier-Stokes-Poisson equations. Zbl 1348.35205
Wu, Zhigang; Wang, Weike
2016
Global classical solutions to the Cauchy problem of conservation laws with degenerate diffusion. Zbl 1336.35232
Chen, Jiao; Li, Yachun; Wang, Weike
2016
Nonlinear stability of large perturbation around viscous shock wave for 2-D scalar viscous conservation law. Zbl 1362.35041
Shi, Renkun; Wang, Weike
2016
$$L_p$$-dual mixed affine surface areas. Zbl 07030371
Wan, X.; Wang, W.
2016
The pointwise estimates of solutions to the Cauchy problem of a chemotaxis model. Zbl 1344.35157
Shi, Renkun; Wang, Weike
2016
Well-posedness for a model derived from an attraction-repulsion chemotaxis system. Zbl 1307.35065
Shi, Renkun; Wang, Weike
2015
Decay rates of the compressible Navier-Stokes-Korteweg equations with potential forces. Zbl 1304.35576
Wang, Wenjun; Wang, Weike
2015
Green’s functions of wave equations in $$\mathbb R^n_+\times\mathbb R_+$$. Zbl 1457.35074
Deng, Shijin; Wang, Weike; Yu, Shih-Hsien
2015
Cooperative fuzzy adaptive output feedback control for synchronisation of nonlinear multi-agent systems under directed graphs. Zbl 1332.93220
Wang, W.; Wang, D.; Peng, Z. H.
2015
Large time behavior and pointwise estimates for compressible Euler equations with damping. Zbl 1334.35219
Wu, Zhigang; Wang, Weike
2015
Large time behavior of solution for the full compressible Navier-Stokes-Maxwell system. Zbl 1328.35156
Wang, Weike; Xu, Xin
2015
Decay of the solution for the bipolar Euler-Poisson system with damping in dimension three. Zbl 1307.35056
Wu, Zhigang; Wang, Weike
2014
Global existence and pointwise estimates of solutions for the multidimensional generalized Boussinesq-type equation. Zbl 1287.35072
Liu, Miao; Wang, Weike
2014
The decay rate of solution for the bipolar Navier-Stokes-Poisson system. Zbl 1304.35509
Wang, Weike; Xu, Xin
2014
Large-time behavior for the solution to the generalized Benjamin-Bona-Mahony-Burgers equation with large initial data in the whole-space. Zbl 1308.35255
Wang, Weike; Zhang, Dandan
2014
Decay of a model system of radiating gas. Zbl 1307.35055
Wang, Wenjun; Wang, Weike; Wu, Zhigang
2014
Bifurcation on boundary data for linear Broadwell model with conservative boundary condition. Zbl 1304.82064
Deng, Shijin; Wang, Weike; Yu, Shih-Hsien
2014
The pointwise estimate of solutions to the parabolic conservation law in multi-dimensions. Zbl 1302.35084
Li, Fengbai; Wang, Weike
2014
The point-wise estimates for the solution of damped wave equation with nonlinear convection in multi-dimensional space. Zbl 1308.35142
Chen, Jiao; Wang, Weike
2014
Pointwise decaying rate of large perturbation around viscous shock for scalar viscous conservation law. Zbl 1272.82032
Deng, ShiJin; Wang, WeiKe
2013
Blow up and global existence of solutions for a model system of the radiating gas. Zbl 1262.35058
Wang, Wenjun; Wang, Weike
2013
Global existence of solutions to the initial-boundary value problem of conservation law with degenerate diffusion term. Zbl 1268.35082
Chen, Jiao; Li, Yachun; Wang, Weike
2013
Pointwise estimates of solutions for non-isentropic Navier-Stokes-Poisson equations in multi-dimensions. Zbl 1274.35273
Wu, Zhigang; Wang, Weike
2012
Optimal decay rate of solutions for Cahn-Hilliard equation with inertial term in multi-dimensions. Zbl 1229.35252
Wang, Weike; Wu, Zhigang
2012
Large-time behavior of periodic solutions to fractal Burgers equation with large initial data. Zbl 1255.35053
Wang, Lijuan; Wang, Weike
2012
Modeling adaptive behavior in influenza transmission. Zbl 1250.34041
Wang, W.
2012
Viscous conservation laws with boundary. Zbl 1269.82060
Deng, Shijin; Wang, Weike; Yu, Shih-Hsien
2012
Adiabatic evolution under quantum control. Zbl 1242.81056
Wang, W.; Hou, S. C.; Yi, X. X.
2012
Broadwell model and conservative supersonic boundary. Zbl 1294.35007
Deng, Shijin; Wang, Weike; Yu, Shih-Hsien
2011
Pointwise estimates of solution for the Navier-Stokes-Poisson equations in multi-dimensions. Zbl 1185.35177
Wang, Weike; Wu, Zhigang
2010
Spatiotemporal dynamics in a spatial plankton system. Zbl 1197.92049
Upadhyay, R. K.; Wang, W.; Thakur, N. K.
2010
The pointwise estimates of solutions for semilinear dissipative wave equation in multi-dimensions. Zbl 1184.35218
Wang, Weike; Wang, Wenjun
2010
Existence theory and $$L^p$$ estimates for the solution of nonlinear viscous wave equation. Zbl 1202.35138
Deng, Shijin; Wang, Weike; Zhao, Hualei
2010
Nonlinear evolution systems and Green’s function. Zbl 1240.35358
Wang, Weike
2010
Pointwise estimates of solutions for the Euler-Poisson equations with damping in multi-dimensions. Zbl 1186.35130
Wu, Zhigang; Wang, Weike
2010
$$L^p$$ convergence rates of planar waves for multi-dimensional Euler equations with damping. Zbl 1170.35544
Liao, Jie; Wang, Weike; Yang, Tong
2009
The pointwise estimates of solutions for a model system of the radiating gas in multi-dimensions. Zbl 1178.35104
Wang, Weike; Wang, Wenjun
2009
Dynamic responses of shear flows over a deformable porous surface layer in a cylindrical tube. Zbl 1167.76332
Wen, P. H.; Hon, Y. C.; Wang, W.
2009
Unsteady flows in a capillary lined with a thin porous surface layer by method of fundamental solutions. Zbl 1274.74453
Wen, P. H.; Wang, W.; Liu, Y. W.
2009
The pointwise estimates of solutions for dissipative wave equation in multi-dimensions. Zbl 1154.35014
Liu, Yongqin; Wang, Weike
2008
Interaction between micro-particles in Oseen flows by the method of fundamental solutions. Zbl 1244.76092
Wang, W.; Wen, P. H.
2008
Well-posedness of the IBVP for 2-D Euler equations with damping. Zbl 1152.35070
Liu, Yongqin; Wang, Weike
2008
Pointwise estimates for the wave equation with dissipation in odd spatial dimension. Zbl 1155.35007
Xu, Hongmei; Wang, Weike
2008
Pointwise convergence to Knudsen layers of the Boltzmann equation. Zbl 1162.82022
Deng, Shijin; Wang, Weike; Yu, Shih-Hsien
2008
The Broadwell model with a subsonic boundary. Zbl 1159.82013
Deng, Shijin; Wang, Weike
2008
Existence and stability of planar diffusion waves for 2D Euler equations with damping. Zbl 1147.35076
Wang, Weike; Yang, Tong
2007
Existence of boundary layers to the Boltzmann equation with cutoff soft potentials. Zbl 1144.81422
Wang, Weike; Yang, Tong; Yang, Xiongfeng
2007
The suppressible property of the solution for three-dimensional Euler equations with damping. Zbl 1179.35187
Yang, Xiongfeng; Wang, Weike
2007
Pointwise convergence to a Maxwellian for a Broadwell model with a supersonic boundary. Zbl 1140.82028
Deng, Shijin; Wang, Weike; Yu, Shih-Hsien
2007
The pointwise estimates to solutions for 1-dimensional linear thermoelastic system with second sound. Zbl 1107.35031
Wang, Weike; Wang, Zejun
2007
On the boundary conditions for transversely isotropic piezoelectric plates. Zbl 1192.74231
Xu, S. P.; Gao, Y.; Wang, W.
2007
Nonlinear stability of boundary layers of the Boltzmann equation for cutoff hard potentials. Zbl 1112.76039
Wang, Weike; Yang, Tong; Yang, Xiongfeng
2006
Local well-posedness of a new integrable equation. Zbl 1094.35124
Liu, Yongqin; Wang, Weike
2006
The Cauchy problem for viscous shallow water equations. Zbl 1095.35037
Wang, Weike; Xu, Chao-Jiang
2005
The pointwise estimates of solutions to the isentropic Navier-Stokes equations in even space-dimensions. Zbl 1083.35101
Wang, Weike; Yang, Xiongfeng
2005
Global existence of solution to the Camassa-Holm equation. Zbl 1063.35137
Liu, Yongqin; Wang, Weike
2005
The regular solutions of the isentropic Euler equations with degenerate linear damping. Zbl 1087.35061
Zhu, Xusheng; Wang, Weike
2005
Free vibration of elastic helicoidal shells. Zbl 1192.74179
Hu, X. X.; Lim, C. W.; Sakiyama, T.; Li, Z. R.; Wang, W. K.
2005
The pointwise estimates of solutions for general Navier-Stokes systems in odd multi-dimensions. Zbl 1110.35061
Wang, Weike
2005
A refined theory of transversely isotropic piezoelectric plates. Zbl 1062.74028
Xu, S. P.; Wang, W.
2004
Pointwise estimates and $$L_p$$ convergence rates to diffusion waves for $$p$$-system with damping. Zbl 1035.35077
Wang, Weike; Yang, Tong
2003
A study of microbend test by strain gradient plasticity. Zbl 1032.74528
Wang, W.; Huang, Y.; Hsia, K. J.; Hu, K. X.; Chandra, A.
2003
Pointwise estimates of solutions to Cauchy problem for quasilinear hyperbolic systems. Zbl 1046.35071
Wang, Weike; Yang, Xiongfeng
2003
The pointwise estimates of solutions for Euler equations with damping in multi-dimensions. Zbl 0997.35039
Wang, Weike; Yang, Tong
2001
Flow of spatially non-uniform suspensions. III: Closure relations for porous media and spinning particles. Zbl 1137.76777
Wang, W.; Prosperetti, A.
2001
Pointwise estimate of solutions of isentropic Navier-Stokes equations in even space-dimensions. Zbl 1007.35057
Xu, Hongmei; Wang, Weike
2001
Flow of spatially non-uniform suspensions. II: Systematic derivation of closure relations. Zbl 1137.76669
Marchioro, M.; Tanksley, M.; Wang, W.; Prosperetti, A.
2001
A method for linear elasto-static analysis of multi-layered axisymmetrical bodies using Hankel’s transform. Zbl 1054.74071
Wang, W.; Ishikawa, H.
2001
$$L_p$$-convergence rate to nonlinear diffusion waves for $$p$$-system with damping. Zbl 0946.35012
Nishihara, Kenji; Wang, Weike; Yang, Tong
2000
A dual reciprocity boundary element approach for the problems of large deflection of thin elastic plates. Zbl 0974.74077
Wang, W.; Ji, X.; Tanaka, M.
2000
Analysis of intersonic crack growth in unidirectional fiber-reinforced composites. Zbl 0963.74050
Huang, Y.; Wang, W.; Liu, C.; Rosakis, A. J.
1999
The pointwise estimates of diffusion wave for the Navier-Stokes systems in odd multi-dimensions. Zbl 0912.35122
Liu, Tai-Ping; Wang, Weike
1998
Intersonic crack growth in bimaterial interfaces: an investigation of crack face contact. Zbl 1056.74561
Huang, Y.; Wang, W.; Liu, C.; Rosakis, A. J.
1998
On the general solutions of transversely isotropic elasticity. Zbl 0918.73016
Wang, W.; Shi, M. X.
1998
Thick plate theory based on general solutions of elasticity. Zbl 0902.73048
Wang, W.; Shi, M. X.
1997
Large time behavior of solutions for general Navier-Stokes systems in multi-dimension. Zbl 0921.35124
Wang, Weike
1997
Completeness and nonuniqueness of general solutions of transversely isotropic elasticity. Zbl 0865.73008
Wang, M. Z.; Wang, W.
1995
Component mode synthesis for damped rotor systems with hybrid interfaces. Zbl 0945.74607
Wang, W.; Kirkhope, J.
1994
New eigensolutions and modal analysis for gyroscopic/rotor systems. I: Undamped systems. II: Perturbation analysis for damped systems. Zbl 0945.70523
Wang, W.; Kirkhope, J.
1994
Component mode synthesis for multi-shaft rotors with flexible inter-shaft bearings. Zbl 0925.73481
Wang, W.; Kirkhope, j.
1994
Constructivity and completeness of the general solutions in elastodynamics. Zbl 0753.73019
Wang, W.; Wang, M. Z.
1992
Interaction of conormal waves with different singularities for semi- linear equations. Zbl 0763.35063
Wang, Weike
1991
On single-valuedness of the Papkovich-Neuber general solution in two- dimensional elasticity. Zbl 0714.73008
Wang, W.; Wang, M. Z.
1990
Reflection of transversal progressing waves for quasilinear 2-systems. Zbl 0699.35173
Wang, Weike
1989
all top 5
#### Cited by 664 Authors
57 Wang, Weike 17 Li, Yeping 15 Tan, Zhong 15 Wu, Zhigang 14 Yao, Zhengan 14 Zhu, Changjiang 12 Wang, Wenjun 12 Yang, Xiongfeng 11 Gao, Yang 11 Wang, Yuzhu 11 Xu, Jiang 11 Yang, Tong 11 Zhang, Yinghui 9 Deng, Shijin 9 Wu, Guochun 9 Zhao, Baosheng 8 Geng, Shifeng 8 Guo, Boling 8 Li, Hailiang 8 Mei, Ming 8 Pan, Ronghua 8 Xu, Hongmei 7 Rosakis, Ares J. 7 Wang, Minzhong 7 Zhao, Huijiang 6 Duan, Renjun 6 Huang, Yonggang Young 6 Wang, Weiming 6 Yu, Shih-Hsien 5 Cui, Haibo 5 Du, Linglong 5 Gao, Jincheng 5 Huang, Feimin 5 Jin, Hai-Yang 5 Li, Yin 5 Luo, Tao 5 Ruan, Lizhi 5 Tian, Qianzhu 5 Ukai, Seiji 5 Wang, Yinxia 5 Wei, Ruiying 5 Xi, Xiaoyu 5 Yao, Lei 5 Yin, Zhaoyang 5 Zhang, Kaijun 4 Cai, Yongli 4 Chen, Zhengzheng 4 Hou, Xiaofeng 4 Jiang, Mina 4 Kawashima, Shuichi 4 Li, Haitong 4 Li, Jingyu 4 Li, Kaiqiang 4 Liu, Guowei 4 Luo, Zhen 4 Pan, Xinghong 4 Pu, Xueke 4 Wang, Yanjin 4 Wang, Zhen 4 Xie, Feng 4 Xu, Sipeng 3 Chen, Jiao 3 Chen, Qing 3 Chen, Shaohua 3 Du, Yi 3 Duan, Ben 3 Eskandari-Ghadi, Morteza 3 Feng, Yuehong 3 Gao, Huajian 3 Kobayashi, Takayuki 3 Li, Yachun 3 Lian, Xinze 3 Liu, Tai-Ping 3 Liu, Yanan 3 Liu, Yongqin 3 Marcati, Pierangelo 3 Peng, Hongyun 3 Qiu, Hua 3 Shi, Renkun 3 Shi, Weixuan 3 Shibata, Yoshihiro 3 Sun, Jie 3 Temam, Roger Meyer 3 Tong, Leilei 3 Upadhyay, Rajnish Kant 3 Wang, Hailing 3 Wang, Wei 3 Wang, Yong 3 Wen, Huanyao 3 Xu, Hang 3 Xu, Xin 3 Yin, Haiyan 3 Yu, Qiang 3 Zhang, Guojing 3 Zhang, Ting 3 Zhong, Hua 3 Zhu, Xusheng 2 Abadyan, Mohamadreza 2 Ardeshir-Behrestaghi, Azizollah 2 Baek, Hunki ...and 564 more Authors
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#### Cited in 121 Serials
74 Journal of Differential Equations 52 Journal of Mathematical Analysis and Applications 23 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 20 ZAMP. Zeitschrift für angewandte Mathematik und Physik 18 Acta Mechanica 18 Mathematical Methods in the Applied Sciences 18 Nonlinear Analysis. Real World Applications 15 Journal of Mathematical Physics 14 Archive for Rational Mechanics and Analysis 14 Journal of the Mechanics and Physics of Solids 13 Communications on Pure and Applied Analysis 9 Engineering Analysis with Boundary Elements 8 Journal of Hyperbolic Differential Equations 7 Applied Mathematical Modelling 7 Wuhan University Journal of Natural Sciences (WUJNS) 7 Science China. Mathematics 6 Applicable Analysis 6 Computers & Mathematics with Applications 6 Applied Mathematics and Mechanics. (English Edition) 6 Acta Mathematicae Applicatae Sinica. English Series 6 Applied Mathematics Letters 6 Discrete and Continuous Dynamical Systems 6 Analysis and Applications (Singapore) 5 Chinese Annals of Mathematics. Series B 5 M$$^3$$AS. Mathematical Models & Methods in Applied Sciences 5 Communications in Partial Differential Equations 5 Archive of Applied Mechanics 5 Journal of Mathematical Fluid Mechanics 5 European Journal of Mechanics. A. Solids 5 Discrete and Continuous Dynamical Systems. Series B 4 Communications in Mathematical Physics 4 International Journal of Solids and Structures 4 Nonlinearity 4 Journal of Elasticity 4 Boundary Value Problems 4 Kinetic and Related Models 3 International Journal of Engineering Science 3 Applied Mathematics and Computation 3 Quarterly of Applied Mathematics 3 SIAM Journal on Mathematical Analysis 3 Electronic Journal of Differential Equations (EJDE) 3 Nonlinear Dynamics 3 Abstract and Applied Analysis 3 Discrete Dynamics in Nature and Society 3 Journal of Evolution Equations 3 International Journal of Biomathematics 3 Advances in Mathematical Physics 2 International Journal of Modern Physics B 2 International Journal of Plasticity 2 Journal of Fluid Mechanics 2 Journal of the Franklin Institute 2 Journal of Statistical Physics 2 Information Sciences 2 Studies in Applied Mathematics 2 Annales de l’Institut Henri Poincaré. Analyse Non Linéaire 2 Revista Matemática Iberoamericana 2 Science in China. Series A 2 Continuum Mechanics and Thermodynamics 2 NoDEA. Nonlinear Differential Equations and Applications 2 ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik 2 Acta Mathematica Sinica. English Series 2 Journal of Applied Mathematics 2 Science in China. Series G 2 Frontiers of Mathematics in China 2 Networks and Heterogeneous Media 2 Evolution Equations and Control Theory 2 International Journal of Systems Science. Principles and Applications of Systems and Integration 2 AMM. Applied Mathematics and Mechanics. (English Edition) 1 International Journal for Numerical and Analytical Methods in Geomechanics 1 Journal of Computational Physics 1 Physics Letters. B 1 Rocky Mountain Journal of Mathematics 1 Chaos, Solitons and Fractals 1 Acta Mathematica Vietnamica 1 Advances in Mathematics 1 Annales Polonici Mathematici 1 Journal of Computational and Applied Mathematics 1 Journal of Functional Analysis 1 Mechanics Research Communications 1 Pacific Journal of Mathematics 1 Publications of the Research Institute for Mathematical Sciences, Kyoto University 1 Results in Mathematics 1 Ricerche di Matematica 1 Zeitschrift für Analysis und ihre Anwendungen 1 Acta Applicandae Mathematicae 1 Physica D 1 Applied Numerical Mathematics 1 Computational Mechanics 1 International Journal of Mathematics 1 International Journal of Adaptive Control and Signal Processing 1 Numerical Algorithms 1 Journal de Mathématiques Pures et Appliquées. Neuvième Série 1 Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 1 International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 1 Journal of Nonlinear Science 1 Applied Mathematics. Series B (English Edition) 1 Physics of Fluids 1 Turkish Journal of Mathematics 1 Mathematical Problems in Engineering 1 Doklady Mathematics ...and 21 more Serials
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#### Cited in 32 Fields
403 Partial differential equations (35-XX) 235 Fluid mechanics (76-XX) 106 Mechanics of deformable solids (74-XX) 49 Biology and other natural sciences (92-XX) 29 Statistical mechanics, structure of matter (82-XX) 21 Numerical analysis (65-XX) 17 Harmonic analysis on Euclidean spaces (42-XX) 12 Optics, electromagnetic theory (78-XX) 12 Systems theory; control (93-XX) 10 Ordinary differential equations (34-XX) 10 Dynamical systems and ergodic theory (37-XX) 6 Probability theory and stochastic processes (60-XX) 5 Mechanics of particles and systems (70-XX) 4 Functional analysis (46-XX) 4 Computer science (68-XX) 4 Classical thermodynamics, heat transfer (80-XX) 3 Differential geometry (53-XX) 3 Quantum theory (81-XX) 3 Astronomy and astrophysics (85-XX) 3 Geophysics (86-XX) 2 Linear and multilinear algebra; matrix theory (15-XX) 2 Functions of a complex variable (30-XX) 2 Special functions (33-XX) 2 Calculus of variations and optimal control; optimization (49-XX) 1 Combinatorics (05-XX) 1 Measure and integration (28-XX) 1 Integral transforms, operational calculus (44-XX) 1 Operator theory (47-XX) 1 Convex and discrete geometry (52-XX) 1 Operations research, mathematical programming (90-XX) 1 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 1 Information and communication theory, circuits (94-XX)
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2021-06-14 03:08:37
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https://mathoverflow.net/questions/326129/mapping-inclusion-theorem-for-the-numerical-range/326251
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# Mapping inclusion theorem for the numerical range
We denote the numerical range of a complex square matrix $$A \in \mathbb{C}^{n\times n}$$ by $$W(A)$$.
Let $$A \in \mathbb{C}^{n\times n}$$ and let $$f: \mathbb{C} \to \mathbb{C}$$ be, say, an entire function. It is easy to see that a mapping theorem for the numerical range in the sense of $$W(f(A)) = f(W(A))$$ does not hold, in general. The following example gives a bit more insight:
Example. Let $$A$$ be the $$3\times 3$$-diagonal matrix with diagonal entries $$0, \pi i, 2\pi i$$ and let $$f(z) = e^z$$. Then $$W(f(A))$$ is the line segment $$[-1,1]$$, but $$f(W(A))$$ is the complex unit circle. In fact, this even shows that...
• we don't have $$W(f(A)) = \operatorname{conv}(f(W(A)))$$, in general (where $$\operatorname{conv}$$ denotes the convex hull).
• we don't have an inclusion theorem of the kind $$W(f(A)) \subseteq f(W(A))$$, in general.
Now, it seems natural to ask:
Question. Is there also a counterexample known for the more general inclusion \begin{align*} W(f(A)) \subseteq \operatorname{conv}(f(W(A))) \qquad (*) \end{align*} or is it an open problem whether $$(*)$$ holds?
Note. It is certainly not known that $$(*)$$ is true, due to the following reasoning:
(i) If $$(*)$$ is true, this immediately implies $$w(f(A)) \le \sup_{z \in W(A)} \lvert f(z) \vert$$, where $$w$$ denotes the numerical radius.
(ii) By the well-known inequality $$\|B\| \le 2w(B)$$ for every matrix $$B$$ (where $$\|\,\cdot\,\|$$ denotes the norm induced by the $$2$$-norm on $$\mathbb{C}^n$$), $$(*)$$ would thus imply that Crouzeix's conjecture \begin{align*} \|f(A)\| \le 2 \sup_{z \in W(A)} \lvert f(z) \vert \end{align*} is true.
Remark. It is easy to see that $$(*)$$ holds for every normal matrix (which also explains why it is true in the above example), but I could not even figure out whether it holds for $$2 \times 2$$ Jordan blocks.
Disclaimer. I am, of course, not asking whether Crouzeix's conjecture is true. I am asking whether the more general assertion $$(*)$$ is known to be false or whether it is an open problem.
• You can try $N=[\begin{smallmatrix} 0&1\\0&0\end{smallmatrix}]$ and $f(z)=z+z^2+\cdots+z^m$. On the one hand $f(N)=N$ and $W(N)=B(0,1/2)$. On the other hand $f(z)\approx z(1-z)^{-1}=(1-z)^{-1} -1$ and ${\rm co}f(B(0,1/2))$ misses $-1/2$. – Narutaka OZAWA Mar 23 '19 at 12:38
• @NarutakaOZAWA: Great example, thank you! If you add it as an answer I will, of course, accept it. – Jochen Glueck Mar 24 '19 at 17:02
Let $$N=[\begin{smallmatrix} 0&1 \\ 0&0 \end{smallmatrix}]$$ and $$f(z)=z+z^2+\cdots+z^m$$. On the one hand, $$f(N)=N$$ and $$W(f(N))=W(N)=B(0,1/2)$$, the closed ball of center $$0$$ and radius $$1/2$$. On the other hand, since $$f(z) \approx \sum_{k\geq1} z^k = z(1-z)^{-1}=(1-z)^{-1}-1$$ to within $$2^{-m}$$ for $$z \in B(0,1/2)$$ and $$\inf_{z\in B(0,1/2)}\Re (1-z)^{-1}-1 = -1/3$$, the convex hull of $$f(B(0,1/2))$$ misses $$-1/2$$, as long as $$m\geq3$$.
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2020-12-03 11:30:52
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https://aperiodical.com/tag/sequences/
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# You're reading: Posts Tagged: sequences
### Sequences in the triangle and the fourth dimension
This is the second in a series of guest posts by David Benjamin, exploring the secrets of Pascal’s Triangle.
## Sequences in the diagonals
There are many sequences of numbers to be found in Pascal’s triangle. The Natural numbers occur in the second diagonal, running in either direction, and the next two diagonals after that contain other important sequences:
## Sequences in the diagonals
There are many sequences of numbers to be found in Pascal’s triangle. The Natural numbers occur in the second diagonal, running in either direction, and the next two diagonals after that contain other important sequences:
### Guest post: Sequence Numbers
This is a guest post, sent in by David, who’s discovered an interesting property of numbers, and is looking for collaborators to take it further.
Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost. Rigour should be a signal to the historian that the maps have been made, and the real explorers have gone elsewhere.
– W. S. Anglin
A few years ago I saw a post on a website that showed that the inverse of 998,001 produces a decimal expansion that counts, using three digit strings, from 000 to 997 without error.
$\frac{1}{998,\!001} = 0.0000010020030040050060070080090100110120130\ldots$
I immediately thought that this had to be a hoax. I decided to work it out to prove it was a hoax – after all some people put anything they want on the web whether it is true or not.
### Puzzlebomb – July 2015
Puzzlebomb is a monthly puzzle compendium. Issue 43 of Puzzlebomb, for July 2015, can be found here:
Puzzlebomb – Issue 43 – July 2015
The solutions to Issue 43 can be found here:
Puzzlebomb – Issue 43 – July 2015 – Solutions
Previous issues of Puzzlebomb, and their solutions, can be found here.
### I’ve made my own numbers-in-a-grid game
For the past couple of weeks, I’ve been obsessively playing the game Twenty on my phone. The fact that my wife has consistently been ahead of my high scores has nothing to do with it.
The main source of strife in my marriage.
Twenty is another in the current spate of “numbers-in-a-grid” games that also includes Threes, 10242048 (and its $2^{48}$ clones), Just Get 10, and Quento.
The basic idea is that you have a grid of numbered tiles, and you combine them to build up your score. While there are lots of unimaginative derivatives of the bigger games, there’s a surprisingly large range of different games following this template.
With so many different games being created, I thought that a chap like me should be able to come up with a numbers-in-a-grid game of my own. Yet, for a long time, I just couldn’t come up with anything that was any good.
Yesterday I had a really nice shower, and the accompanying feeling that I’d come up with a really good idea – make a game to do with arithmetic progressions.
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2022-01-25 08:23:42
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https://fortenf.org/e/crypto/2017/12/03/survey-of-discrete-log-algos.html
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# Survey of Discrete Log Algorithms
In 1999 Dan Boneh published a survey paper called “Twenty years of attacks on the RSA cryptosystem”. I like to tell people that if they read the entire paper they can solve just about every CTF challenge involving RSA.
This post is my attempt to do something similar with the discrete log problem, a primitive that underlies a bunch of cryptographic protocols including the Diffie-Hellman key exchange and the ElGamal encryption system. The algorithms all have similar-sounding names, and I wanted to just sit down and describe each of them to get them straight in my head.
## The Discrete Log Problem
Suppose we have a group $$G$$ generated by an element $$g$$ (this means that every element can be written as $$g^k$$ for some integer $$k$$.) The discrete log of an element $$y = g^x$$ is the number $$x$$. For cyclic groups of order $$n$$, discrete logs are unique modulo $$n$$.
A common choice for $$G$$ in cryptography is $$(\mathbb{Z}_p)^\times$$, or the multiplicative group modulo the prime $$p$$. Fermat’s little theorem tells us that $$g^{p-1} = 1 \pmod{p}$$, so the order of $$(\mathbb{Z}_p)^\times$$ is $$p-1$$.
If the order of the cyclic group is not a smooth number, i.e. if $$p-1$$ has at least one large prime factor, then it is believed that it is hard to compute the discrete logarithm on elements of the group.
This hardness assumption forms the basis for a large portion of modern cryptography. As a result, it’s important to understand the best algorithms that exist to compute discrete logs.
## Approach #1: Brute Force
Given an element $$y$$ in a group $$G$$ with generator $$g$$, we’re trying to find an integer $$x$$ such that $$g^x = y$$. A naive way to do this is to simply try all possible $$x$$. This approach has time complexity $$O(n)$$ where $$n$$ is the order of the group and space complexity $$O(1)$$.
This website is secured using an elliptic curve group with order $n = 2^{256} - 2^{224} + 2^{192} - 89188191075325690597107910205041859247.$ Using this approach to find a discrete log in this group would take around $$2^{256}$$ trials, which is a little much.
## Approach #2: Baby-step giant-step
There’s a relatively easy way to improve the running time of the brute-force approach using a common technique in cryptography. We’re trying to find $$x$$ such that $$g^x = y$$. Let’s rewrite $$x$$ as $$im + j$$ where $$m = \lceil \sqrt{n} \rceil$$ for integers $$i$$ and $$j$$ between $$0$$ and $$m$$. Now we can rewrite $y = g^x = g^{im + j}$ as $(g^{-m})^i y = g^j.$
Next, let’s compute and store $$g^j$$ for all possible values of $$j$$ in a set, and then run through every possible value of $$i$$ and check when $$(g^{-m})^i y$$ is in our set. Once we find a colliding $$i$$ and $$j$$, we can compute $$x = im + j$$.
This is an example of a space-time tradeoff; we’ve essentially turned an $$O(n)$$ time, $$O(1)$$ space algorithm into an $$O(\sqrt{n})$$ time, $$O(\sqrt{n})$$ space algorithm.
The name stems from the fact that every increment of $$j$$ represents a “baby-step” of size $$g^1$$ while every increment of $$i$$ represents a “giant-step” of size $$g^m$$.
## Approach #3: Pollard’s $$\rho$$ algorithm
It turns out it’s possible to compute discrete logs with $$O(\sqrt{n})$$ time and $$O(1)$$ space. Here’s the idea:
First, partition the group into three pairwise-disjoint, approximately-equally sized sets $$S_0, S_1,$$ and $$S_2$$ such that $$G = S_0 \cup S_1 \cup S_2$$. This partition should be made at random.
Consider the following sequence $$h_i = g^{a_i}y^{b_i}$$ where $$a_0 = b_0 = 1$$ and $$h_i = f(h_{i-1})$$ where
Basically, a third of the time we increase $$a_i$$ by one, a third of the time we increase $$b_i$$ by one, and a third of the time we double both $$a_i$$ and $$b_i$$. This is just a convenient way to make the sequence walk through the space of $$g^{a_i}y^{b_i}$$ pseudorandomly.
Why do we want to walk through this space? Well, consider what happens if we find a collision; that is, four integers $$a, b, a’, b’$$ such that
$g^ay^b = g^{a’}y^{b’}.$
We can rewrite this as $y = g^{(a - a’) / (b’ - b)}$ and then compute the discrete log of $$y$$ as $$(a - a’) / (b’ - b)$$ (division just means “multiply by the inverse modulo $$n$$, the order of the group”).
So how do we find a collision in our sequence using $$O(1)$$ space? A collision in our sequence means that our sequence contains a cycle, and there’s a common trick that frequently pops up in algorithms interviews that allows us to find cycles in linked lists using $$O(1)$$ space.
The idea is to use two iterators. One iterator increments through the sequence by one at each step, while the other increments by two at each step. If the linked list has a cycle, the two iterators will, at some point, have the same value.
For our sequence, we’ll use two variables, $$s$$ and $$t$$ both initialized to $$h_0$$. On each step of our algorithm, we’ll set $$s$$ to $$f(s)$$ and $$t$$ to $$f(f(t))$$ and check if the two are equal. If they are, we have a collision and can solve the DLP. If not, continue.
By the birthday paradox, it should take on the order of $$O(\sqrt{n})$$ steps before we find a collision, so the running time is $$O(\sqrt{n})$$.
The name “rho” stems from the fact that the sequence with a cycle, when considered as a directed graph, takes the shape of the Greek letter $$\rho$$. Pollard also has an analogous algorithm for factorization.
## Approach #4: Pollard’s kangaroo algorithm
In the same paper where he introduced his rho-algorithm, Pollard also discussed a different algorithm to solve the discrete log problem. This algorithm is most useful when you already know some information about $$x$$. Specifically, if you know that $$a \le x \le b$$, then this algorithm will take $$O(\sqrt{b - a})$$ time on expectation.
Here’s how it works: Once again we’ll define a pseudorandom sequence $$x_i$$ where
$x_{i + 1} = x_i + 2^{g^{x_i} \pmod k}$ for some small $$k$$. (In practice I’ve found that setting $$k = \lceil \log_2(b - a)/2 \rceil$$ works pretty well.) Also define $$y_i = g^{x_i} \pmod p$$ and note that
$y_{i+1} = y_i g^{2^{y_i \pmod k}}.$
The point is that on each step, this sequence jumps forward by a random-looking power of two between $$0$$ and $$2^{k-1}$$. In his paper, Pollard makes an analogy to a kangaroo leaping forward by a random distance on each jump.
The first step is to launch a “tame” kangaroo starting from $$x_0 = b$$ and tell it to jump $$N$$ times, for some fixed $$N$$. Mathematically this means computing $$x_{N}$$ and $$y_{N}$$ starting from $$x_0 = b$$ and $$y_0 = g^b$$. Note that we don’t need to store the values of the intermediate hops; we just make our $$N$$ jumps and then stop.
Now, let’s try to use our tame kangaroo to catch a wild kangaroo. Define a similar sequence starting from $$y_0 = y$$. Repeatedly compute the next value in the sequence and check if it is equal to $$y_N$$. We don’t know the corresponding values of $$x_i$$ (indeed, that’s the value we’re trying to find), but we can keep track of how much distance we’ve hopped from the initial $$x$$ because on each hop we travel $$2^{y_i} \pmod k$$.
The hope is that at some point the wild kangaroo will eventually land on the same location as a jump of the original tame kangaroo, and once the two kangaroos are on the same point, every subsequent jump will be identical since the sequence is generated deterministically. Thus, if the wild kangaroo ever lands on the same spot as the tame kangaroo, it will eventually end up on the tame kangaroo’s final location $$y_n$$.
In the above diagram, the blue arrows represent the tame kangaroo and the red arrows represent the wild kangaroo. If the wild kangaroo eventually hits $$y_N$$ after traveling a distance $$d$$, we can compute:
$y_0 g^{d} =y_N = g^{x_N}$ $y = y_0 = g^{x_N - d}$
so the discrete log of $$y$$ is $$x = x_N - d$$.
There are a couple of remaining problems. This algorithm is inherently probabilistic, and it’s possible that the wild kangaroo will never overlap with the tame kangaroo. We can detect when this has happened by stopping the wild kangaroo after it has traveled a distance of $$d > x_N - a$$. Since we know our wild kangaroo starts somewhere between $$a$$ and $$b$$, if it has traveled more than a distance of $$x_N - a$$ it has definitely passed our tame kangaroo sitting at $$x_N$$.
If this happens, the algorithm has failed, and we’ll need to try again with a different tame kangaroo (start the tame kangaroo at $$b + 1$$ instead of $$b$$ for example).
Note that both Pollard’s $$\rho$$ method and his kangaroo method are probabilistic, but in different ways. The $$\rho$$ method takes an unbounded amount of time, but always succeeds when it terminates, while one iteration of the kangaroo method takes a bounded amount of time, but is not guaranteed to succeed. This is the difference between a Las Vegas algorithm and a Monte Carlo algorithm.
Finally, we need to choose an appropriate value of $$N$$ and then prove that the algorithm runs in time $$O(\sqrt{b - a})$$. We’d like to set $$N$$ such that the probability of succeeding on one iteration is constant (not dependent on $$k$$). Suppose the mean jump length is $$\mu$$. The probability of any one jump of the wild kangaroo landing on a jump of the tame kangaroo is approximately $$1/\mu$$. The tame kangaroo makes $$N$$ jumps, and we succeed as long as at least one of those jumps coincides with the wild kangaroo’s path. The probability of this is:
$1 - \left (1 - \frac{1}{\mu} \right) ^ N.$
If we set $$N = 4 \mu$$, then this is approximately $1 - e^{-4} \approx 0.98.$ The mean jump length is $2^{k}/k$ so a good choice is $$N = 4 \cdot 2^{k}/k = 2^{k+2}/k$$.
The total number of jumps taken by all the kangaroos on expectation is
$N + \left(\frac{b-a}{2\mu} + N\right) = 8\left(\mu + \frac{b-a}{16\mu}\right).$
The minimum value of this is $$4 \sqrt{b-a}$$ which is achieved when $$\mu = \sqrt{b-a}/4$$. (Incidentally, since $$\mu = 2^{k}/k$$, this justifies setting $$k \approx \log_2{(b-a)}/2$$).
Thus, the algorithm takes $$O(\sqrt{b-a})$$ time when it succeeds, and will succeed with probability 0.98.
Pollard has also referred to this algorithm as his $$\lambda$$ algorithm because the shape of the converging kangaroo paths resembles the Greek letter lambda.
## Approach #5: The Pohlig-Hellman algorithm
Remember how we said earlier that the order of the group needs to not be a smooth number? Here’s what goes wrong if $$n$$ has a bunch of small factors:
(Some group theory knowledge is helpful for comprehension). Suppose that
$n = \prod_{i = 1}^k p_{i}^{e_i}$ where the $$p_i$$’s are prime and unique. As usual we have a $$y = g^x$$ and we want to find $$x$$.
For each $$i$$ we’ll compute $$x \bmod{p_i^{e_i}}$$ and then use the Chinese Remainder Theorem to find $$x$$ itself.
For each $$i$$, let $$g_i = g^{n / p_i^{e_i}}$$. This element has order $$p_i^{e_i}$$ (raise $$g_i$$ to this power to verify). Now consider
$y_i = y^{n / p_i^{e_i}} = g^{xn / p_i^{e_i}} = g_i^x = g_i^{x \pmod{p_i^{e_i}}}$
Since $$p_i^{e_i}$$ is small, we should be able to find the $$r_i$$ between $$0$$ and $$p_i^{e_i} - 1$$ such that $$y_i = g^{r_i} \pmod p$$ using any of the previous DLP algorithms. Once we’ve done this for all $$i$$ we have a system of linear congruences of the form $$x = r_i \pmod{p_i^{e_i}}$$. Using CRT, we can then derive $$x$$.
Note that $$k$$, the number of prime divisors of $$n$$ is bounded by $$k < \log_2{n}$$, so it’s insignificant. The time complexity is then bounded by the size of the largest prime $$p^*$$ and is $$O(\sqrt{p^*})$$. The space complexity is $$O(k) = O(\log{n})$$ but since we need this much space to store the number $$n$$ in the first place, we’ll call it $$O(1)$$.
## Comparison table
Algorithm Time Complexity Space Complexity Notes
Brute Force $$O(n)$$ $$O(1)$$ Don't use.
Baby-step Giant-step $$O(\sqrt{n})$$ $$O(\sqrt{n})$$ Easy to implement, but Pollard's $$\rho$$ is better.
Pollard's $$\rho$$ $$O(\sqrt{n})$$ $$O(1)$$ Fastest known DLP algorithm for prime order.
Pollard's kangaroo $$O(\sqrt{b-a})$$ $$O(1)$$ Use when you already know some information about $$x$$.
Polhig-Hellman $$O(\sqrt{p^*})$$ $$O(1)$$ Use when $$n$$ is factorable into many small primes.
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2018-01-19 01:11:15
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https://docs.pymc.io/notebooks/dp_mix.html
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# Dirichlet process mixtures for density estimation¶
Author: Austin Rochford
## Dirichlet processes¶
The Dirichlet process is a flexible probability distribution over the space of distributions. Most generally, a probability distribution, $$P$$, on a set $$\Omega$$ is a [measure](https://en.wikipedia.org/wiki/Measure_(mathematics%29) that assigns measure one to the entire space ($$P(\Omega) = 1$$). A Dirichlet process $$P \sim \textrm{DP}(\alpha, P_0)$$ is a measure that has the property that, for every finite disjoint partition $$S_1, \ldots, S_n$$ of $$\Omega$$,
$(P(S_1), \ldots, P(S_n)) \sim \textrm{Dir}(\alpha P_0(S_1), \ldots, \alpha P_0(S_n)).$
Here $$P_0$$ is the base probability measure on the space $$\Omega$$. The precision parameter $$\alpha > 0$$ controls how close samples from the Dirichlet process are to the base measure, $$P_0$$. As $$\alpha \to \infty$$, samples from the Dirichlet process approach the base measure $$P_0$$.
Dirichlet processes have several properties that make then quite suitable to MCMC simulation.
1. The posterior given i.i.d. observations $$\omega_1, \ldots, \omega_n$$ from a Dirichlet process $$P \sim \textrm{DP}(\alpha, P_0)$$ is also a Dirichlet process with
$P\ |\ \omega_1, \ldots, \omega_n \sim \textrm{DP}\left(\alpha + n, \frac{\alpha}{\alpha + n} P_0 + \frac{1}{\alpha + n} \sum_{i = 1}^n \delta_{\omega_i}\right),$
where $$\delta$$ is the Dirac delta measure
\begin{split}\begin{align*} \delta_{\omega}(S) & = \begin{cases} 1 & \textrm{if } \omega \in S \\ 0 & \textrm{if } \omega \not \in S \end{cases} \end{align*}.\end{split}
1. The posterior predictive distribution of a new observation is a compromise between the base measure and the observations,
$\omega\ |\ \omega_1, \ldots, \omega_n \sim \frac{\alpha}{\alpha + n} P_0 + \frac{1}{\alpha + n} \sum_{i = 1}^n \delta_{\omega_i}.$
We see that the prior precision $$\alpha$$ can naturally be interpreted as a prior sample size. The form of this posterior predictive distribution also lends itself to Gibbs sampling.
1. Samples, $$P \sim \textrm{DP}(\alpha, P_0)$$, from a Dirichlet process are discrete with probability one. That is, there are elements $$\omega_1, \omega_2, \ldots$$ in $$\Omega$$ and weights $$w_1, w_2, \ldots$$ with $$\sum_{i = 1}^{\infty} w_i = 1$$ such that
$P = \sum_{i = 1}^\infty w_i \delta_{\omega_i}.$
2. The stick-breaking process gives an explicit construction of the weights $$w_i$$ and samples $$\omega_i$$ above that is straightforward to sample from. If $$\beta_1, \beta_2, \ldots \sim \textrm{Beta}(1, \alpha)$$, then $$w_i = \beta_i \prod_{j = 1}^{j - 1} (1 - \beta_j)$$. The relationship between this representation and stick breaking may be illustrated as follows:
2. Break the stick into two portions, the first of proportion $$w_1 = \beta_1$$ and the second of proportion $$1 - w_1$$.
3. Further break the second portion into two portions, the first of proportion $$\beta_2$$ and the second of proportion $$1 - \beta_2$$. The length of the first portion of this stick is $$\beta_2 (1 - \beta_1)$$; the length of the second portion is $$(1 - \beta_1) (1 - \beta_2)$$.
4. Continue breaking the second portion from the previous break in this manner forever. If $$\omega_1, \omega_2, \ldots \sim P_0$$, then
$P = \sum_{i = 1}^\infty w_i \delta_{\omega_i} \sim \textrm{DP}(\alpha, P_0).$
We can use the stick-breaking process above to easily sample from a Dirichlet process in Python. For this example, $$\alpha = 2$$ and the base distribution is $$N(0, 1)$$.
In [5]:
%matplotlib inline
In [6]:
from __future__ import division
In [1]:
from matplotlib import pyplot as plt
import numpy as np
import pymc3 as pm
import scipy as sp
import seaborn as sns
from theano import tensor as tt
import pandas as pd
In [8]:
blue, *_ = sns.color_palette()
In [9]:
SEED = 5132290 # from random.org
np.random.seed(SEED)
In [10]:
N = 20
K = 30
alpha = 2.
P0 = sp.stats.norm
We draw and plot samples from the stick-breaking process.
In [7]:
beta = sp.stats.beta.rvs(1, alpha, size=(N, K))
w = np.empty_like(beta)
w[:, 0] = beta[:, 0]
w[:, 1:] = beta[:, 1:] * (1 - beta[:, :-1]).cumprod(axis=1)
omega = P0.rvs(size=(N, K))
x_plot = np.linspace(-3, 3, 200)
sample_cdfs = (w[..., np.newaxis] * np.less.outer(omega, x_plot)).sum(axis=1)
In [8]:
fig, ax = plt.subplots(figsize=(8, 6))
ax.plot(x_plot, sample_cdfs[0], c='gray', alpha=0.75,
label='DP sample CDFs');
ax.plot(x_plot, sample_cdfs[1:].T, c='gray', alpha=0.75);
ax.plot(x_plot, P0.cdf(x_plot), c='k', label='Base CDF');
ax.set_title(r'$\alpha = {}$'.format(alpha));
ax.legend(loc=2);
As stated above, as $$\alpha \to \infty$$, samples from the Dirichlet process converge to the base distribution.
In [9]:
fig, (l_ax, r_ax) = plt.subplots(ncols=2, sharex=True, sharey=True, figsize=(16, 6))
K = 50
alpha = 10.
beta = sp.stats.beta.rvs(1, alpha, size=(N, K))
w = np.empty_like(beta)
w[:, 0] = beta[:, 0]
w[:, 1:] = beta[:, 1:] * (1 - beta[:, :-1]).cumprod(axis=1)
omega = P0.rvs(size=(N, K))
sample_cdfs = (w[..., np.newaxis] * np.less.outer(omega, x_plot)).sum(axis=1)
l_ax.plot(x_plot, sample_cdfs[0], c='gray', alpha=0.75,
label='DP sample CDFs');
l_ax.plot(x_plot, sample_cdfs[1:].T, c='gray', alpha=0.75);
l_ax.plot(x_plot, P0.cdf(x_plot), c='k', label='Base CDF');
l_ax.set_title(r'$\alpha = {}$'.format(alpha));
l_ax.legend(loc=2);
K = 200
alpha = 50.
beta = sp.stats.beta.rvs(1, alpha, size=(N, K))
w = np.empty_like(beta)
w[:, 0] = beta[:, 0]
w[:, 1:] = beta[:, 1:] * (1 - beta[:, :-1]).cumprod(axis=1)
omega = P0.rvs(size=(N, K))
sample_cdfs = (w[..., np.newaxis] * np.less.outer(omega, x_plot)).sum(axis=1)
r_ax.plot(x_plot, sample_cdfs[0], c='gray', alpha=0.75,
label='DP sample CDFs');
r_ax.plot(x_plot, sample_cdfs[1:].T, c='gray', alpha=0.75);
r_ax.plot(x_plot, P0.cdf(x_plot), c='k', label='Base CDF');
r_ax.set_title(r'$\alpha = {}$'.format(alpha));
r_ax.legend(loc=2);
## Dirichlet process mixtures¶
For the task of density estimation, the (almost sure) discreteness of samples from the Dirichlet process is a significant drawback. This problem can be solved with another level of indirection by using Dirichlet process mixtures for density estimation. A Dirichlet process mixture uses component densities from a parametric family $$\mathcal{F} = \{f_{\theta}\ |\ \theta \in \Theta\}$$ and represents the mixture weights as a Dirichlet process. If $$P_0$$ is a probability measure on the parameter space $$\Theta$$, a Dirichlet process mixture is the hierarchical model
\begin{split}\begin{align*} x_i\ |\ \theta_i & \sim f_{\theta_i} \\ \theta_1, \ldots, \theta_n & \sim P \\ P & \sim \textrm{DP}(\alpha, P_0). \end{align*}\end{split}
To illustrate this model, we simulate draws from a Dirichlet process mixture with $$\alpha = 2$$, $$\theta \sim N(0, 1)$$, $$x\ |\ \theta \sim N(\theta, (0.3)^2)$$.
In [10]:
N = 5
K = 30
alpha = 2
P0 = sp.stats.norm
f = lambda x, theta: sp.stats.norm.pdf(x, theta, 0.3)
In [11]:
beta = sp.stats.beta.rvs(1, alpha, size=(N, K))
w = np.empty_like(beta)
w[:, 0] = beta[:, 0]
w[:, 1:] = beta[:, 1:] * (1 - beta[:, :-1]).cumprod(axis=1)
theta = P0.rvs(size=(N, K))
dpm_pdf_components = f(x_plot[np.newaxis, np.newaxis, :], theta[..., np.newaxis])
dpm_pdfs = (w[..., np.newaxis] * dpm_pdf_components).sum(axis=1)
In [12]:
fig, ax = plt.subplots(figsize=(8, 6))
ax.plot(x_plot, dpm_pdfs.T, c='gray');
ax.set_yticklabels([]);
We now focus on a single mixture and decompose it into its individual (weighted) mixture components.
In [13]:
fig, ax = plt.subplots(figsize=(8, 6))
ix = 1
ax.plot(x_plot, dpm_pdfs[ix], c='k', label='Density');
ax.plot(x_plot, (w[..., np.newaxis] * dpm_pdf_components)[ix, 0],
'--', c='k', label='Mixture components (weighted)');
ax.plot(x_plot, (w[..., np.newaxis] * dpm_pdf_components)[ix].T,
'--', c='k');
ax.set_yticklabels([]);
ax.legend(loc=1);
Sampling from these stochastic processes is fun, but these ideas become truly useful when we fit them to data. The discreteness of samples and the stick-breaking representation of the Dirichlet process lend themselves nicely to Markov chain Monte Carlo simulation of posterior distributions. We will perform this sampling using PyMC3.
Our first example uses a Dirichlet process mixture to estimate the density of waiting times between eruptions of the Old Faithful geyser in Yellowstone National Park.
In [3]:
old_faithful_df = pd.read_csv(pm.get_data('old_faithful.csv'))
For convenience in specifying the prior, we standardize the waiting time between eruptions.
In [4]:
old_faithful_df['std_waiting'] = (old_faithful_df.waiting - old_faithful_df.waiting.mean()) / old_faithful_df.waiting.std()
In [5]:
old_faithful_df.head()
Out[5]:
eruptions waiting std_waiting
0 3.600 79 0.596025
1 1.800 54 -1.242890
2 3.333 74 0.228242
3 2.283 62 -0.654437
4 4.533 85 1.037364
In [17]:
fig, ax = plt.subplots(figsize=(8, 6))
n_bins = 20
ax.hist(old_faithful_df.std_waiting, bins=n_bins, color=blue, lw=0, alpha=0.5);
ax.set_xlabel('Standardized waiting time between eruptions');
ax.set_ylabel('Number of eruptions');
Observant readers will have noted that we have not been continuing the stick-breaking process indefinitely as indicated by its definition, but rather have been truncating this process after a finite number of breaks. Obviously, when computing with Dirichlet processes, it is necessary to only store a finite number of its point masses and weights in memory. This restriction is not terribly onerous, since with a finite number of observations, it seems quite likely that the number of mixture components that contribute non-neglible mass to the mixture will grow slower than the number of samples. This intuition can be formalized to show that the (expected) number of components that contribute non-negligible mass to the mixture approaches $$\alpha \log N$$, where $$N$$ is the sample size.
There are various clever Gibbs sampling techniques for Dirichlet processes that allow the number of components stored to grow as needed. Stochastic memoization is another powerful technique for simulating Dirichlet processes while only storing finitely many components in memory. In this introductory example, we take the much less sophistocated approach of simply truncating the Dirichlet process components that are stored after a fixed number, $$K$$, of components. Ohlssen, et al. provide justification for truncation, showing that $$K > 5 \alpha + 2$$ is most likely sufficient to capture almost all of the mixture weight ($$\sum_{i = 1}^{K} w_i > 0.99$$). In practice, we can verify the suitability of our truncated approximation to the Dirichlet process by checking the number of components that contribute non-negligible mass to the mixture. If, in our simulations, all components contribute non-negligible mass to the mixture, we have truncated the Dirichlet process too early.
Our (truncated) Dirichlet process mixture model for the standardized waiting times is
\begin{split}\begin{align*} \alpha & \sim \textrm{Gamma}(1, 1) \\ \beta_1, \ldots, \beta_K & \sim \textrm{Beta}(1, \alpha) \\ w_i & = \beta_i \prod_{j = i - 1}^i (1 - \beta_j) \\ \\ \lambda_1, \ldots, \lambda_K & \sim U(0, 5) \\ \tau_1, \ldots, \tau_K & \sim \textrm{Gamma}(1, 1) \\ \mu_i\ |\ \lambda_i, \tau_i & \sim N\left(0, (\lambda_i \tau_i)^{-1}\right) \\ \\ x\ |\ w_i, \lambda_i, \tau_i, \mu_i & \sim \sum_{i = 1}^K w_i\ N(\mu_i, (\lambda_i \tau_i)^{-1}) \end{align*}\end{split}
Note that instead of fixing a value of $$\alpha$$, as in our previous simulations, we specify a prior on $$\alpha$$, so that we may learn its posterior distribution from the observations.
We now construct this model using pymc3.
In [18]:
N = old_faithful_df.shape[0]
K = 30
In [11]:
def stick_breaking(beta):
portion_remaining = tt.concatenate([[1], tt.extra_ops.cumprod(1 - beta)[:-1]])
return beta * portion_remaining
In [20]:
with pm.Model() as model:
alpha = pm.Gamma('alpha', 1., 1.)
beta = pm.Beta('beta', 1., alpha, shape=K)
w = pm.Deterministic('w', stick_breaking(beta))
tau = pm.Gamma('tau', 1., 1., shape=K)
lambda_ = pm.Uniform('lambda', 0, 5, shape=K)
mu = pm.Normal('mu', 0, tau=lambda_ * tau, shape=K)
obs = pm.NormalMixture('obs', w, mu, tau=lambda_ * tau,
observed=old_faithful_df.std_waiting.values)
We sample from the model 1,000 times using NUTS initialized with ADVI.
In [26]:
with model:
trace = pm.sample(1000, random_seed=SEED)
100%|██████████| 1500/1500 [01:55<00:00, 15.42it/s]
The posterior distribution of $$\alpha$$ is highly concentrated between 0.25 and 1.
In [27]:
pm.traceplot(trace, varnames=['alpha']);
To verify that truncation is not biasing our results, we plot the posterior expected mixture weight of each component.
In [28]:
fig, ax = plt.subplots(figsize=(8, 6))
plot_w = np.arange(K) + 1
ax.bar(plot_w - 0.5, trace['w'].mean(axis=0), width=1., lw=0);
ax.set_xlim(0.5, K);
ax.set_xlabel('Component');
ax.set_ylabel('Posterior expected mixture weight');
We see that only three mixture components have appreciable posterior expected weights, so we conclude that truncating the Dirichlet process to forty components has not appreciably affected our estimates.
We now compute and plot our posterior density estimate.
In [29]:
post_pdf_contribs = sp.stats.norm.pdf(np.atleast_3d(x_plot),
trace['mu'][:, np.newaxis, :],
1. / np.sqrt(trace['lambda'] * trace['tau'])[:, np.newaxis, :])
post_pdfs = (trace['w'][:, np.newaxis, :] * post_pdf_contribs).sum(axis=-1)
post_pdf_low, post_pdf_high = np.percentile(post_pdfs, [2.5, 97.5], axis=0)
In [30]:
fig, ax = plt.subplots(figsize=(8, 6))
n_bins = 20
ax.hist(old_faithful_df.std_waiting.values, bins=n_bins, normed=True,
color=blue, lw=0, alpha=0.5);
ax.fill_between(x_plot, post_pdf_low, post_pdf_high,
color='gray', alpha=0.45);
ax.plot(x_plot, post_pdfs[0],
c='gray', label='Posterior sample densities');
ax.plot(x_plot, post_pdfs[::100].T, c='gray');
ax.plot(x_plot, post_pdfs.mean(axis=0),
c='k', label='Posterior expected density');
ax.set_xlabel('Standardized waiting time between eruptions');
ax.set_yticklabels([]);
ax.set_ylabel('Density');
ax.legend(loc=2);
As above, we can decompose this density estimate into its (weighted) mixture components.
In [31]:
fig, ax = plt.subplots(figsize=(8, 6))
n_bins = 20
ax.hist(old_faithful_df.std_waiting.values, bins=n_bins, normed=True,
color=blue, lw=0, alpha=0.5);
ax.plot(x_plot, post_pdfs.mean(axis=0),
c='k', label='Posterior expected density');
ax.plot(x_plot, (trace['w'][:, np.newaxis, :] * post_pdf_contribs).mean(axis=0)[:, 0],
'--', c='k', label='Posterior expected mixture\ncomponents\n(weighted)');
ax.plot(x_plot, (trace['w'][:, np.newaxis, :] * post_pdf_contribs).mean(axis=0),
'--', c='k');
ax.set_xlabel('Standardized waiting time between eruptions');
ax.set_yticklabels([]);
ax.set_ylabel('Density');
ax.legend(loc=2);
The Dirichlet process mixture model is incredibly flexible in terms of the family of parametric component distributions $$\{f_{\theta}\ |\ f_{\theta} \in \Theta\}$$. We illustrate this flexibility below by using Poisson component distributions to estimate the density of sunspots per year. This dataset can be downloaded from http://www.sidc.be/silso/datafiles. Source: WDC-SILSO, Royal Observatory of Belgium, Brussels.
In [6]:
sunspot_df = pd.read_csv(pm.get_data('sunspot.csv'), sep=';', names=['time', 'sunspot.year'], usecols=[0, 1])
In [7]:
sunspot_df.head()
Out[7]:
time sunspot.year
0 1700.5 8.3
1 1701.5 18.3
2 1702.5 26.7
3 1703.5 38.3
4 1704.5 60.0
For this example, the model is
\begin{split}\begin{align*} \alpha & \sim \textrm{Gamma}(1, 1) \\ \beta_1, \ldots, \beta_K & \sim \textrm{Beta}(1, \alpha) \\ w_i & = \beta_i \prod_{j = i - 1}^i (1 - \beta_j) \\ \\ \lambda_i, \ldots, \lambda_K & \sim U(0, 300) \\ x\ |\ w_i, \lambda_i & \sim \sum_{i = 1}^K w_i\ \textrm{Poisson}(\lambda_i). \end{align*}\end{split}
In [14]:
K = 50
N = sunspot_df.shape[0]
In [15]:
with pm.Model() as model:
alpha = pm.Gamma('alpha', 1., 1.)
beta = pm.Beta('beta', 1, alpha, shape=K)
w = pm.Deterministic('w', stick_breaking(beta))
mu = pm.Uniform('mu', 0., 300., shape=K)
obs = pm.Mixture('obs', w, pm.Poisson.dist(mu), observed=sunspot_df['sunspot.year'])
In [16]:
with model:
step = pm.Metropolis()
trace = pm.sample(1000, step=step, random_seed=SEED)
100%|██████████| 1500/1500 [00:12<00:00, 127.34it/s]
For the sunspot model, the posterior distribution of $$\alpha$$ is concentrated between 0.6 and 1.2, indicating that we should expect more components to contribute non-negligible amounts to the mixture than for the Old Faithful waiting time model.
In [17]:
pm.traceplot(trace, varnames=['alpha']);
Indeed, we see that between ten and fifteen mixture components have appreciable posterior expected weight.
In [18]:
fig, ax = plt.subplots(figsize=(8, 6))
plot_w = np.arange(K) + 1
ax.bar(plot_w - 0.5, trace['w'].mean(axis=0), width=1., lw=0);
ax.set_xlim(0.5, K);
ax.set_xlabel('Component');
ax.set_ylabel('Posterior expected mixture weight');
We now calculate and plot the fitted density estimate.
In [19]:
x_plot = np.arange(250)
In [20]:
post_pmf_contribs = sp.stats.poisson.pmf(np.atleast_3d(x_plot),
trace['mu'][:, np.newaxis, :])
post_pmfs = (trace['w'][:, np.newaxis, :] * post_pmf_contribs).sum(axis=-1)
post_pmf_low, post_pmf_high = np.percentile(post_pmfs, [2.5, 97.5], axis=0)
In [21]:
fig, ax = plt.subplots(figsize=(8, 6))
ax.hist(sunspot_df['sunspot.year'].values, bins=40, normed=True, lw=0, alpha=0.75);
ax.fill_between(x_plot, post_pmf_low, post_pmf_high,
color='gray', alpha=0.45)
ax.plot(x_plot, post_pmfs[0],
c='gray', label='Posterior sample densities');
ax.plot(x_plot, post_pmfs[::200].T, c='gray');
ax.plot(x_plot, post_pmfs.mean(axis=0),
c='k', label='Posterior expected density');
ax.set_xlabel('Yearly sunspot count');
ax.set_yticklabels([]);
ax.legend(loc=1);
Again, we can decompose the posterior expected density into weighted mixture densities.
In [22]:
fig, ax = plt.subplots(figsize=(8, 6))
ax.hist(sunspot_df['sunspot.year'].values, bins=40, normed=True, lw=0, alpha=0.75);
ax.plot(x_plot, post_pmfs.mean(axis=0),
c='k', label='Posterior expected density');
ax.plot(x_plot, (trace['w'][:, np.newaxis, :] * post_pmf_contribs).mean(axis=0)[:, 0],
'--', c='k', label='Posterior expected\nmixture components\n(weighted)');
ax.plot(x_plot, (trace['w'][:, np.newaxis, :] * post_pmf_contribs).mean(axis=0),
'--', c='k');
ax.set_xlabel('Yearly sunspot count');
ax.set_yticklabels([]);
ax.legend(loc=1);
An earlier version of this example first appeared here.
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2018-09-20 01:05:37
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|
https://par.nsf.gov/biblio/10335768-environmental-monitoring-system-cosine-experiment
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This content will become publicly available on January 1, 2023
The environmental monitoring system at the COSINE-100 experiment
Abstract The COSINE-100 experiment is designed to test the DAMA experiment which claimed an observation of a dark matter signal from an annual modulation in their residual event rate. To measure the 1 %-level signal amplitude, it is crucial to control and monitor nearly all environmental quantities that might systematically mimic the signal. The environmental monitoring also helps ensure a stable operation of the experiment. Here, we describe the design and performance of the centralized environmental monitoring system for the COSINE-100 experiment.
Authors:
; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; more »
Award ID(s):
Publication Date:
NSF-PAR ID:
10335768
Journal Name:
Journal of Instrumentation
Volume:
17
Issue:
01
Page Range or eLocation-ID:
T01001
ISSN:
1748-0221
National Science Foundation
##### More Like this
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2. Abstract We present a background model for dark matter searches using an array of NaI(Tl) crystals in the COSINE-100 experiment that is located in the Yangyang underground laboratory. The model includes background contributions from both internal and external sources, including cosmogenic radionuclides and surface $$^{210}$$ 210 Pb contamination. To build the model in the low energy region, with a threshold of 1 keV, we used a depth profile of $$^{210}$$ 210 Pb contamination in the surface of the NaI(Tl) crystals determined in a comparison between measured and simulated spectra. We also considered the effect of the energy scale errors propagated frommore »
3. In this paper, we present a multiple concurrent occupant identification approach through footstep-induced floor vibration sensing. Identification of human occupants is useful in a variety of indoor smart structure scenarios, with applications in building security, space allocation, and healthcare. Existing approaches leverage sensing modalities such as vision, acoustic, RF, and wearables, but are limited due to deployment constraints such as line-of-sight requirements, sensitivity to noise, dense sensor deployment, and requiring each walker to wear/carry a device. To overcome these restrictions, we use footstep-induced structural vibration sensing. Footstep-induced signals contain information about the occupants' unique gait characteristics, and propagate through themore »
4. ; ; (Ed.)
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5. Abstract Testing the DAMA/LIBRA annual modulation result independently of dark matter particle and halo models has been a challenge for twenty years. Using the same target material, NaI(Tl), is required and presently two experiments, ANAIS-112 and COSINE-100, are running for such a goal. A precise knowledge of the detector response to nuclear recoils is mandatory because this is the most likely channel to find the dark matter signal. The light produced by nuclear recoils is quenched with respect to that produced by electrons by a factor that has to be measured experimentally. However, current quenching factor measurements in NaI(Tl) crystalsmore »
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2022-08-14 04:42:56
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|
https://socratic.org/questions/what-is-the-equation-of-the-line-passing-through-3-5-and-42-1#302321
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# What is the equation of the line passing through (3,-5) and (42,1)?
Aug 24, 2016
Both points satisfy the line equation $y = m x + b$, so you need to find $m$ and $b$
#### Explanation:
Since both points satisfy the equation, we know that:
$- 5 = m \cdot 3 + b$, and
$1 = m \cdot 42 + b$
We now have a system of two equations with $m$ and $b$. To solve it we can subtract the first from the second equation to eliminate $b$:
$6 = 39 m$ and so $m = \frac{6}{39} = \frac{2}{13}$. From the first equation now we have:
$- 5 - \left(\frac{2}{13}\right) \cdot 3 = b$, and so $b = - \frac{65}{13} - \frac{6}{13} = - \frac{71}{13}$.
The equation of the line is then:
$y = \frac{2}{13} x - \frac{71}{13}$
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2022-01-25 01:29:30
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https://www.physicsforums.com/threads/probability-3.111172/
|
Probability 3
1. Feb 18, 2006
Natasha1
Could anyone please help me with making a start.... and a finish to this little exercise. Many thanks Nat.
Four people A, B, C and D are to play in a small table-tennis tournament played on a simple knock-out basis: their names are drawn at random to play in two pairs, then the two winners play in the final. The probability that a player beats a player with a later letter is 2/3. All matches are independent. Find the probabilities that:
1) A wins the tournament
2) C and D meet in the final
3) B and C meet at some stage
Must I use a tree diagram?
2. Feb 18, 2006
arildno
1) A must win two fights, irrespective of the pairing arrangements. What is the probability of that?
2) This does depend on the probability that C and D does NOT meet each other in the first fight, and that BOTH win their first fight (irrespective of who beats A or B)
3. Feb 18, 2006
Natasha1
Is 1) p(A) = 2/3 * 2/3 = 4/9
4. Feb 18, 2006
arildno
Correct!
5. Feb 18, 2006
Natasha1
Is 2) p(C & D to meet in final) = 2/3 * 2/3 * 2/3 * 2/3 = 16/81
6. Feb 18, 2006
arildno
No.
1. There is 2/3 chance they'll not meet in the first round. 1/3 of those times, C will go to the finals, and 1/3 of those times again D will also go to the finals to meet C.
7. Feb 18, 2006
Natasha1
I see so...
P(C&D meet in final) = 2/3 * 1/3 * 1/3 = 2/27
8. Feb 18, 2006
arildno
3*3*3 equals 27 last time I checked..
9. Feb 18, 2006
Natasha1
lol, i did spot it straight away and corrected it :shy:
Right then...
for number 3
Last edited: Feb 18, 2006
10. Feb 18, 2006
arildno
Now, for 3) calculate separately the probabilities for the disjoint events a) B&C meets in the first round b) B&C meet in the finals.
Either a) or b) may happen..
11. Feb 18, 2006
Natasha1
a) p(B&C meets in the first round) = 1/3
b) p(B&C meet in the finals) = 2/27
So the p(B&C meet at some stage) = 1/3 * 2/27 = 2/81
12. Feb 18, 2006
arildno
Incorrect!
a) is correct, but why do you think b) is correct??
Furthermore, you are to have probabilities of disjoint events EITHER of which occuring will mean that b&c meets at some stage. Should you multiply the probabilities together in that case?
Last edited: Feb 18, 2006
13. Feb 18, 2006
Natasha1
Ok so
a) p(B&C meets in the first round) = 1/3
b) p(B&C meet in the finals) = ?
Well I thought this was correct as for question b) the p(C&D to meet in final is 2/27 so why would p(B&C meet in final) would be different?
So the p(B&C meet at some stage) = 1/3 + ? = ?
14. Feb 18, 2006
arildno
b) will be different in that in the first case, for example, C must always beat a BETTER player in order to go to the finals.
There is 1/3 chance that the initial pairing is (A&B, C&D) What is the chance that this was the set up AND that both B and C wins (and thus proceed to the finals to meet each other)?
Furthermore, it is a 1/3 chance that the initial pairing was (A&C,B&D) What is the chance that this was the set up AND that both B and C wins
(and thus proceed to the finals to meet each other)?
15. Feb 18, 2006
Natasha1
b) P = 2/3 * 1/3 * 1/3 * 1/3 = 2/81
So p(B&C meet at some stage) = 1/3 + 2/81 = 29/81
16. Feb 18, 2006
arildno
No, no, no!
17. Feb 18, 2006
Natasha1
Let me see...
There is 1/3 chance that the initial pairing is (A&B, C&D) What is the chance that this was the set up AND that both B and C wins (and thus proceed to the finals to meet each other)? this gives 1/3 * 1/3 * 1/3 = 1/27
Furthermore, it is a 1/3 chance that the initial pairing was (A&C,B&D) What is the chance that this was the set up AND that both B and C wins
(and thus proceed to the finals to meet each other)? this gives 1/3 * 1/3 * 2/3 = 2/27
18. Feb 18, 2006
arildno
Almost correct, but the probability for the first case is also 2/27, since C beats D 2/3 of the time.
Thus, the TOTAL probability of B&C meeting in the finals is 2/27+2/27=4/27
And the total probability of B&C meeting at some stage is therefore....?
19. Feb 18, 2006
Natasha1
p(B&C meeting at some stage) = 1/3 + 4/27 = 13/27
20. Feb 18, 2006
You're done!
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2017-02-26 19:51:09
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http://physics.stackexchange.com/questions/15708/finding-speed-with-the-energy-principle-nuclear-fission
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# Finding Speed with the Energy Principle - Nuclear Fission
The question goes along the lines:
Uranium-$236$ fissions when it absorbs a slow-moving neutron. The two fission fragments can be almost any two nuclei whose charges $Q_1$ and $Q_2$ add up to $92e$ ($e$ is the charge of a proton, $1.6 \times 10^{-19}C$), and whose nucleons add up to $236$ protons and neutrons. One of the possible fission modes involves nearly equal fragments of the two palladium nuclei with $Q_1 = Q_2 = 46e$. The rest masses of the two palladium nuclei add up to less than the rest of the mass of the original nucleus. Make the assumption that there are no free neutrons, jus the palladium nuclei. The rest mass of the U-$236$ is 235.996 u (unified atomic mass units), and the rest mass of each $Pd-118$ nucleus is 117.894 u, where $1$ $u = 1.7 \times 10^{-27} kg$.
(a) Calculate the final speed $v$, when the Pd nuclei have moved far apart (due to the mutual electric repulsion). Is this speed small enough that $\frac{p^2}{2m}$ is an adequate ($p$ is momentum) approximation for the kinetic energy of one of the palladium nuclei? (make the non relativistic assumption first, then compare $v$ is indeed small enough to $c$)
(b) Using energy considerations, calculate the distance between centers of the palladium nuclei just after fission, when they are starting from rest.
So the beginning step is to chose the system, for which I include both the Pd nuclei. So the energy principle is now: $$\Delta E_{sys} = W_{ext}$$ We know that $W_{ext} = 0$ since my system is defined not including any external work done on the system. So now instead the system is $\Delta E_{sys} = 0 = \Delta K_{m_1} + \Delta K_{m_2} + \Delta U_{elec_1} + \Delta U_{elec_2}$ We can ignore the earth and the gravitational potential energy sine the masses are very small. If we expand the equation, we get: $$K_{f_1} - K_{i_1} + K_{f_2} + K_{i_2} + U_{f_1} - U_{i_1} + U_{f_2} - U_{i_2} = 0$$ getting rid of terms that are $0$, we get: $$K_{f_1} + K_{f_2} - U_{i_1} - U_{i_2} = 0$$ furthermore: $$K_{f_1} + K_{f_2} = U_{i_1} + U_{i_2}$$ We know that $K = \frac{1}{2}mv^2$ and that $K_{f_1} = K_{f_2}$ are equal since the masses are the same. So we can simplify that to $2K_f = mv^2$. Same for $U = \frac{1}{2 \pi \epsilon_0} \frac{Q_1 Q_2}{r}$. So $U_{i_1} = U_{i_2} = 2U_i = \frac{2}{2 \pi \epsilon_0} \frac{Q_1 Q_2}{r}$. So we can essentially write: $$mv^2 = 2 \frac{1}{4 \pi \epsilon_0} \frac{Q_1 Q_2}{r}$$ $$v = \sqrt{\frac{2}{m} \frac{1}{4 \pi \epsilon_0} \frac{Q_1 Q_2}{r}}$$ I then get lost n finding $r$ since it is not given, furthermore, (b) asks us to find the distance before they move apart. So I am approaching the problem incorrectly. I would need to figure out the error here and how to commence part b.
-
Welcome to Physics.SE! Hmmm...you are being too meticulous about this and at the same time ignoring a important aspect of the problem. Make your life easier...consider the situation when the fragments are far enough apart that you can ignore the electrical potential (if any, there are enough electrons around after all) between them. And remember that mass is a form of energy (very important when we start talking about nuclear reactions, that). – dmckee Oct 14 '11 at 0:58
Note to potential answerers: solving the problem would render this into a "particular homework exercise" and run afoul the FAQ, but addressing the whys of the solution looks like a "principles at work" question to me and is good to go. – dmckee Oct 14 '11 at 1:01
I just remembered that the initial mass energy is not equal to the final mass energy. There is a different of .208 u, or $.3536 \times 10^{-27} kg$. So the first part is covered since $E = mc^2$ for the rest energy. I can use what I have now essentially for part b. Gracias @dmckee – Salazar Oct 14 '11 at 1:05
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2014-10-23 05:07:26
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https://cyclostationary.blog/category/real-world-signals/
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# CSP Estimators: The FFT Accumulation Method
Let’s look at another spectral correlation function estimator: the FFT Accumulation Method (FAM). This estimator is in the time-smoothing category, is exhaustive in that it is designed to compute estimates of the spectral correlation function over its entire principal domain, and is efficient, so that it is a competitor to the Strip Spectral Correlation Analyzer (SSCA) method. I implemented my version of the FAM by using the paper by Roberts et al (The Literature [R4]). If you follow the equations closely, you can successfully implement the estimator from that paper. The tricky part, as with the SSCA, is correctly associating the outputs of the coded equations to their proper $\displaystyle (f, \alpha)$ values.
# ‘Can a Machine Learn the Fourier Transform?’ Redux, Plus Relevant Comments on a Machine-Learning Paper by M. Kulin et al.
I first considered whether a machine (neural network) could learn the (64-point, complex-valued) Fourier transform in this post. I used MATLAB’s Neural Network Toolbox and I failed to get good learning results because I did not properly set the machine’s hyperparameters. A kind reader named Vito Dantona provided a comment to that original post that contained good hyperparameter selections, and I’m going to report the new results here in this post.
Since the Fourier transform is linear, the machine should be set up to do linear processing. It can’t just figure that out for itself. Once I used Vito’s suggested hyperparameters to force the machine to be linear, the results became much better:
# CSP Patent: Tunneling
My colleague Dr. Apurva Mody (of BAE Systems, IEEE 802.22, and the WhiteSpace Alliance) and I have received a patent on a CSP-related invention we call tunneling. The US Patent is 9,755,869 and you can read it here or download it here. We’ve got a journal paper in review and a 2013 MILCOM conference paper (My Papers [38]) that discuss and illustrate the involved ideas. I’m also working on a CSP Blog post on the topic.
Update December 28, 2017: Our Tunneling journal paper has been accepted for publication in the journal IEEE Transactions on Cognitive Communications and Networking. You can download the pre-publication version here.
# Automatic Spectral Segmentation
In this post, I discuss a signal-processing algorithm that has almost nothing to do with cyclostationary signal processing. Almost. The topic is automated spectral segmentation, which I also call band-of-interest (BOI) detection. When attempting to perform automatic radio-frequency scene analysis (RFSA), we may be confronted with a data block that contains multiple signals in a large number of distinct frequency subbands. Moreover, these signals may be turning on an off within the data block. To apply our cyclostationary signal processing tools effectively, we would like to isolate these signals in time and frequency to the greatest extent possible using linear time-invariant filtering (for separating in the frequency dimension) and time-gating (for separating in the time dimension). Then the isolated signal components can be processed serially.
It is very important to remember that even perfect spectral and temporal segmentation will not solve the cochannel-signal problem. It is perfectly possible that an isolated subband will contain more that one cochannel signal.
The basics of my BOI-detection approach are published in a 2007 conference paper (My Papers [32]). I’ll describe this basic approach, illustrate it with examples relevant to RFSA, and also provide a few extensions of interest, including one that relates to cyclostationary signal processing.
# Cyclostationarity of Direct-Sequence Spread-Spectrum Signals
In this post we look at direct-sequence spread-spectrum (DSSS) signals, which can be usefully modeled as a kind of PSK signal. DSSS signals are used in a variety of real-world situations, including the familiar CDMA and WCDMA signals, covert signaling, and GPS. My colleague Antonio Napolitano has done some work on a large class of DSSS signals (The Literature [R11, R17, R95]), resulting in formulas for their spectral correlation functions, and I’ve made some remarks about their cyclostationary properties myself here and there (My Papers [16]).
A good thing, from the point of view of modulation recognition, about DSSS signals is that they are easily distinguished from other PSK and QAM signals by their spectral correlation functions. Whereas most PSK/QAM signals have only a single non-conjugate cycle frequency, and no conjugate cycle frequencies, DSSS signals have many non-conjugate cycle frequencies and in some cases also have many conjugate cycle frequencies.
# Cumulant (4, 2) is a Good Discriminator?
Let’s talk about another published paper on signal detection involving cyclostationarity and/or cumulants. This one is called “Energy-Efficient Processor for Blind Signal Classification in Cognitive Radio Networks,” (The Literature [R69]), and is authored by UCLA researchers E. Rebeiz and four colleagues.
My focus on this paper it its idea that broad signal-type classes, such as direct-sequence spread-spectrum (DSSS), QAM, and OFDM can be reliably distinguished by the use of a single number: the fourth-order cumulant with two conjugated terms. This kind of cumulant is referred to as the $(4, 2)$ cumulant here at the CSP Blog, and in the paper, because the order is $n=4$ and the number of conjugated terms is $m=2$.
# Modulation Recognition Using Cyclic Cumulants, Part I: Problem Description and Variants
In this post, we start a discussion of what I consider the ultimate application of the theory of cyclostationary signals: Automatic Modulation Recognition. My relevant papers are My Papers [16,17,25,26,28,30,32,33,38,43,44].
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2018-07-18 16:05:03
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https://zbmath.org/?q=an:1097.54013&format=complete
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# zbMATH — the first resource for mathematics
Relative normality and product spaces. (English) Zbl 1097.54013
Summary: A. V. Arhangel’skii [Topology Appl. 70, 87–99 (1996; Zbl 0848.54016)], as one of various notions on relative topological properties, defined strong normality of $$A$$ in $$X$$ for a subspace $$A$$ of a topological space $$X$$, and showed that this is equivalent to normality of $$X_A$$, where $$X_A$$ denotes the space obtained from $$X$$ by making each point of $$X \setminus A$$ isolated.
In this paper we investigate for a space $$X$$, a subspace $$A$$ and a space $$Y$$, the normality of the product $$X_A \times Y$$ in connection with the normality of $$(X \times Y)_{(A \times Y)}$$. The cases for paracompactness, more generally, for $$\gamma$$-paracompactness will also be discussed for $$X_A \times Y$$. As an application, we prove that for a metric space $$X$$ with $$A \subset X$$ and a countably paracompact normal space $$Y$$, $$X_A \times Y$$ is normal if and only if $$X_A \times Y$$ is countably paracompact.
##### MSC:
54B10 Product spaces in general topology 54D15 Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.) 54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
Full Text:
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2021-10-18 12:12:11
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https://stats.stackexchange.com/questions/390341/expected-time-to-wait-for-no-events-to-occur-within-a-sliding-window-assuming-po
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# Expected time to wait for no events to occur within a sliding window assuming Poissson process
I wish to model the following:
I am maintaining a sliding window (history) of 10 samples of the output of a signal detector.
I model the probability of a detection failure (i.e absence of signal) as a Poisson process with a probability of (say) 0.2, so an average rate of 2 detection failures in any window of 10 samples, assuming the failures are independent.
I wish to model the expected time & pdf, from the point where there are 10 consecutive detection failures, to the point where there are no detection failures present within the sliding window.
• Welcome to CV. As I understand the problem, you have a Bernoulli i.i.d sequence $X_n$ for $n=1$, $2$, $\dots$ where $X_n=0$ is "no failure" and $X_n = 1$ is "failure". Then you are interested by the waiting time for the first run of ones with length $10$. Even if the time $n$ is to be replaced by a Poisson arrival $T_n$ (for a process independent of the $X_i$), solving the former problem will be needed. You can use the keyword "runs". – Yves Feb 4 at 15:40
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2019-06-16 02:57:49
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https://support.bioconductor.org/p/91740/
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DMRcate: Scaling factor for bandwidth
1
0
Entering edit mode
@alexandriaandrayas-11850
Last seen 5.5 years ago
Hello,
So I am using the DMRcate package to find deferentially methylated regions. Using the dmrcate() function requires a value for C, scaling factor for bandwidth. It states that for 450k data when lambda = 1000 near-optimal prediction of sequencing-derived DMRs is obtained when 'C' is approximately 2. I was wondering if this would be the same for the EPIC array data??
Thanks
Alex
dmrcate dmr analysis dmr • 1.0k views
0
Entering edit mode
Tim Peters ▴ 120
@tim-peters-7579
Last seen 10 months ago
Australia
Hi Alex,
I haven't done any empirical testing on the distribution of EPIC CpGs but I'd wager the optimum is about the same. The real issue is when you're doing sequencing assays, and fitting genomically consecutive CpGs, that you have to make C a lot larger, which makes the kernel smaller.
If you're concerned, I'd err on the side of making Ca bit bigger, say 3 or 4, since there are more probes on EPIC than 450K. If C is too small, and the kernel is to big, it can rope in nearby CpGs that aren't all that DM, and the DMR endpoints won't be as "precise". The tradeoff, of course, is that inflating C may atomise the DMRs too much, if you're looking for bigger DMRs in the order of kilobase or tens of kilobase, and you prefer these collapsed. But then you can just make lambda bigger if this is a bother!
Good luck,
Tim
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2022-08-11 17:59:46
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https://labs.tib.eu/arxiv/?author=V.%20Nikulin
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• Substantial experimental and theoretical efforts worldwide are devoted to explore the phase diagram of strongly interacting matter. At LHC and top RHIC energies, QCD matter is studied at very high temperatures and nearly vanishing net-baryon densities. There is evidence that a Quark-Gluon-Plasma (QGP) was created at experiments at RHIC and LHC. The transition from the QGP back to the hadron gas is found to be a smooth cross over. For larger net-baryon densities and lower temperatures, it is expected that the QCD phase diagram exhibits a rich structure, such as a first-order phase transition between hadronic and partonic matter which terminates in a critical point, or exotic phases like quarkyonic matter. The discovery of these landmarks would be a breakthrough in our understanding of the strong interaction and is therefore in the focus of various high-energy heavy-ion research programs. The Compressed Baryonic Matter (CBM) experiment at FAIR will play a unique role in the exploration of the QCD phase diagram in the region of high net-baryon densities, because it is designed to run at unprecedented interaction rates. High-rate operation is the key prerequisite for high-precision measurements of multi-differential observables and of rare diagnostic probes which are sensitive to the dense phase of the nuclear fireball. The goal of the CBM experiment at SIS100 (sqrt(s_NN) = 2.7 - 4.9 GeV) is to discover fundamental properties of QCD matter: the phase structure at large baryon-chemical potentials (mu_B > 500 MeV), effects of chiral symmetry, and the equation-of-state at high density as it is expected to occur in the core of neutron stars. In this article, we review the motivation for and the physics programme of CBM, including activities before the start of data taking in 2022, in the context of the worldwide efforts to explore high-density QCD matter.
• The ALICE Collaboration has measured inclusive J/psi production in pp collisions at a center of mass energy sqrt(s)=2.76 TeV at the LHC. The results presented in this Letter refer to the rapidity ranges |y|<0.9 and 2.5<y<4 and have been obtained by measuring the electron and muon pair decay channels, respectively. The integrated luminosities for the two channels are L^e_int=1.1 nb^-1 and L^mu_int=19.9 nb^-1, and the corresponding signal statistics are N_J/psi^e+e-=59 +/- 14 and N_J/psi^mu+mu-=1364 +/- 53. We present dsigma_J/psi/dy for the two rapidity regions under study and, for the forward-y range, d^2sigma_J/psi/dydp_t in the transverse momentum domain 0<p_t<8 GeV/c. The results are compared with previously published results at sqrt(s)=7 TeV and with theoretical calculations.
• ### The Physics of Ultraperipheral Collisions at the LHC(0706.3356)
June 25, 2007 hep-ph, hep-ex, nucl-ex, nucl-th
We discuss the physics of large impact parameter interactions at the LHC: ultraperipheral collisions (UPCs). The dominant processes in UPCs are photon-nucleon (nucleus) interactions. The current LHC detector configurations can explore small $x$ hard phenomena with nuclei and nucleons at photon-nucleon center-of-mass energies above 1 TeV, extending the $x$ range of HERA by a factor of ten. In particular, it will be possible to probe diffractive and inclusive parton densities in nuclei using several processes. The interaction of small dipoles with protons and nuclei can be investigated in elastic and quasi-elastic $J/\psi$ and $\Upsilon$ production as well as in high $t$ $\rho^0$ production accompanied by a rapidity gap. Several of these phenomena provide clean signatures of the onset of the new high gluon density QCD regime. The LHC is in the kinematic range where nonlinear effects are several times larger than at HERA. Two-photon processes in UPCs are also studied. In addition, while UPCs play a role in limiting the maximum beam luminosity, they can also be used a luminosity monitor by measuring mutual electromagnetic dissociation of the beam nuclei. We also review similar studies at HERA and RHIC as well as describe the potential use of the LHC detectors for UPC measurements.
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2020-03-29 03:35:36
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https://www.rit.edu/science/sopa/events
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## Upcoming Presentations
Monday, November 26, 2018
#### AST Colloquium
1125 Carlson Building
## AST Colloquium
Monday, November 26, 2018
Monday, December 10, 2018
#### AST Colloquium
1125 Carlson Building
## AST Colloquium
Monday, December 10, 2018
Monday, February 4, 2019
#### AST Colloquium
1125 Carlson Building
## AST Colloquium
Monday, February 4, 2019
Monday, February 18, 2019
#### AST Colloquium
1125 Carlson Building
## AST Colloquium
Monday, February 18, 2019
Monday, March 4, 2019
#### AST Colloquium
1125 Carlson Building
## AST Colloquium
Monday, March 4, 2019
Monday, March 25, 2019
#### AST Colloquium
1125 Carlson Building
## AST Colloquium
Monday, March 25, 2019
Monday, April 8, 2019
#### AST Colloquium
1125 Carlson Building
## AST Colloquium
Monday, April 8, 2019
Monday, April 22, 2019
#### AST Colloquium
1125 Carlson Building
## AST Colloquium
Monday, April 22, 2019
Wednesday, January 16, 2019
#### Physics Colloquium: TBD
3365 Thomas Gosnell Hall
## Physics Colloquium: TBD
Wednesday, January 16, 2019
#### Title: TBD
Speaker TBD
Pizza will be Available
Interpreting Services: myaccess.rit.edu
Wednesday, January 30, 2019
#### Physics Colloquium: TBD
3365 Thomas Gosnell Hall
## Physics Colloquium: TBD
Wednesday, January 30, 2019
#### Title: TBD
Speaker TBD
Pizza will be Available
Interpreting Services: myaccess.rit.edu
Wednesday, February 20, 2019
#### Physics Colloquium: TBD
3365 Thomas Gosnell Hall
## Physics Colloquium: TBD
Wednesday, February 20, 2019
#### Title: TBD
Speaker TBD
Pizza will be Available
Interpreting Services: myaccess.rit.edu
Wednesday, February 27, 2019
#### Physics Colloquium: TBD
3365 Thomas Gosnell Hall
## Physics Colloquium: TBD
Wednesday, February 27, 2019
#### Title: TBD
Speaker TBD
Pizza will be Available
Interpreting Services: myaccess.rit.edu
Wednesday, March 6, 2019
#### Physics Colloquium: TBD
3365 Thomas Gosnell Hall
## Physics Colloquium: TBD
Wednesday, March 6, 2019
#### Title: TBD
Speaker TBD
Pizza will be Available
Interpreting Services: myaccess.rit.edu
Wednesday, April 17, 2019
#### Physics Colloquium: TBD
3365 Thomas Gosnell Hall
## Physics Colloquium: TBD
Wednesday, April 17, 2019
#### Title: TBD
Speaker TBD
Pizza will be Available
Interpreting Services: myaccess.rit.edu
## Past Speakers
Monday, November 12, 2018
#### AST Colloquium: Mario Gennary
Mario Gennaro Space Telescope Science Institute
## AST Colloquium: Mario Gennary
Mario Gennaro Space Telescope Science Institute
Monday, November 12, 2018
#### The SOAR/SAM multi-object spectrograph (SAMOS): enabling efficient, accurate and highly-multiplexed spectroscopy over large crowded fields.
Dr. Mario Gennaro
Space Telescope Science Institute (STScI)
Monday, October 29, 2018
#### AST Colloquium: Maura McLaughlin
Maura McLaughlin West Virginia University - Department of Physics & Astronomy
## AST Colloquium: Maura McLaughlin
Maura McLaughlin West Virginia University - Department of Physics & Astronomy
Monday, October 29, 2018
#### The NANOGrav 11-year Data Set: New Insights into Galaxy Growth and Evolution
Dr. Maura McLaughlin
West Virginia University
Monday, October 15, 2018
#### AST Colloquium: Dr. Tingting Liu
Tingting Liu, Ph.D. University of Wisconsin - Milwaukee
## AST Colloquium: Dr. Tingting Liu
Tingting Liu, Ph.D. University of Wisconsin - Milwaukee
Monday, October 15, 2018
#### The Search for Supermassive Black Hole Binary Candidates in the Time Domain
Dr. Tingting Liu
University of Wisconsin - Milwaukee
Wednesday, October 3, 2018
#### AST Colloquium
Carey Lisse Planetary Astronomer, John Hopkins University
## AST Colloquium
Carey Lisse Planetary Astronomer, John Hopkins University
Wednesday, October 3, 2018
#### What we Know & Don't Know about 1l/Oumuamua, the 1st Detected Interstellar Object in the Solar System
Carey Lisse
Planetary Astronomer
Johns Hopkins University
Monday, September 17, 2018
#### Directed Energy for Relativistic Propulsion: Enabling the First Interstellar Missions
Philip Lubin University of California, Santa Barbara
## Directed Energy for Relativistic Propulsion: Enabling the First Interstellar Missions
Philip Lubin University of California, Santa Barbara
Monday, September 17, 2018
All propulsion systems that leave the Earth are based on chemical reactions. Chemical reactions, at best, have an efficiency compared to rest mass of 10-10 (or about 1eV per bond). All the mass in the universe converted to chemical reactions would not propel even a single proton to relativistic speeds. While chemistry will get us to Mars it will not allow interstellar capability in any reasonable mission time. Barring new physics we are left with few realistic solutions. None of our current propulsion systems, including nuclear, are capable of the relativistic speeds needed for exploring the many nearby stellar systems and exo-planets. However, recent advances in photonics and directed energy systems now allow us to realize the ability to develop systems capable of relativistic flight. From wafer-scale spacecraft capable of speeds greater than 0.25c that could reach the nearest star in 20 years to 10 kg spacecraft travelling at 0.02 c to large missions capable of supporting human life for rapid interplanetary transit - all can be enabled by the same system. Photonics, like electronics, and unlike chemical propulsion is an exponential technology with a current double time of about 20 months. Additionally the same system can be used for many other purposes including kilometer scale telescopes for specialized applications including exoplanet searches and imaging, planetary defense, space debris mitigation among many others. This would be a profound change in human capability. The FY 2017 congressional appropriations request directs NASA to study the feasibility of an interstellar mission to coincide with the 100th anniversary of the moon landing. Seen from a different perspective this technology has significant implications for SETI searches with visibility of this technology beyond redshift 20 should there exist another civilization trying to beacon its existence. We will discuss the many technical challenges ahead, our current laboratory prototypes and recent data on kilometer baseline arrays as well as the many transformative implications of this program.
For technical information on this program see our website:
http://www.deepspace.ucsb.edu/interstellar
http://arxiv.org/abs/1604.01356
http://www.deepspace.ucsb.edu/projects/implications-of-directed-energy-for-seti
http://arxiv.org/abs/1604.02108
Monday, April 23, 2018
## Adventures of a (Young Star and Protoplanetary Disk) Science Bum
Monday, April 23, 2018
I will present some (mostly science) highlights from my 2016-17 sabbatical year, which I spent working alongside experts on young stars, Gaia, protoplanetary disks, and radio interferometry at 5 institutions in 3 countries on 2 continents.
Monday, April 9, 2018
#### Astrophysics from the Stratosphere
Barth Netterfield University of Toronto
## Astrophysics from the Stratosphere
Barth Netterfield University of Toronto
Monday, April 9, 2018
Balloon borne telescopes can achieve much of the promise of satellite borne experiments, at a fraction of the cost, while providing important tests for upcoming satellite missions. Tour tonne telescopes can be launched above 99.5% of the atmosphere for flights flights up up to several weeks. I will discuss the performance and preliminary results from the recent Spider CMB polarization experiment which is designed to detect or place limits on the amplitude of gravitational waves produced by cosmological inflation.
Additionally, the advent of Super-pressure Balloon missions will permit mid-lattitude flights of up to 100 nights, opening up the potential of visible and near UV diffraction-limited imaging. I will discuss the status of the SuperBIT mission and its follow on project which is designed to take early advantage of this new capability.
Monday, March 26, 2018
#### Insights into Galaxy Assembly from Luminous AGN in the Distant Universe
Duncan Farrah Virginia Tech
## Insights into Galaxy Assembly from Luminous AGN in the Distant Universe
Duncan Farrah Virginia Tech
Monday, March 26, 2018
There is a deep connection between star formation and active galactic nuclei which profoundly impacts the assembly history of galaxies across most of the history of the Universe. The nature of the connection however remains controversial, due to, for example, the uncertain evolution in, and synergy between, the AGN and starburst duty cycles, and the obscuring effect of dust. z
In this talk, I will briefly review our current knowledge of galaxy assembly, and then discuss two recent results. First is an observed scaling relation between star formation rates and AGN luminosities in luminous type 1 quasars at high redshift, as determined using data from the Herschel and Spitzer space telescopes. This relation offers insights into how star formation is triggered and quenched in luminous quasars, and how stellar mass assembly proceeds in AGN hosts towards the end of the AGN duty cycle.
Second is a study, using both the Hubble Space Telescope and Herschel, on the most luminous obscured AGN in the Universe. The nature of these AGN offers extremely stringent tests of galaxy evolution models. I will discuss the morphologies of their host galaxies, and show that they signpost a brief but critical phase in galaxy evolution. Finally, I will discuss some ongoing work on moderately obscured AGN at z>1.
Monday, February 26, 2018
#### Dusty Star Forming Galaxies in the Distant Universe
Allison Kirkpatrick Yale University
## Dusty Star Forming Galaxies in the Distant Universe
Allison Kirkpatrick Yale University
Monday, February 26, 2018
At z = 1 - 3 (approximately 7 - 11 Gyr ago), the formation of new stars in the Universe occurs mainly in massive, dusty galaxies. In the past decade, the Herschel Space Observatory showed the surprising result that these distant galaxies are much colder than their counterparts in the local Universe. I explore the reasons for the evolving IR emission of similar galaxies over cosmic time. Despite similar star formation rates, dusty galaxies existing 10 Gyr ago have an order of magnitude higher dust masses and higher gas masses than local galaxies. I also discuss the effect of a luminous central black hole on the observed infrared emission. I demonstrate that an active galactic nucleus is linked with declining star formation in massive galaxies. Finally, I make predictions for the demographics of dusty galaxies that we will be able to observe with the James Webb Space Telescope at z = 1 - 2.
Monday, February 12, 2018
#### Understanding the Universe through distant galaxies
Viviana Acquaviva City Tech
## Understanding the Universe through distant galaxies
Viviana Acquaviva City Tech
Monday, February 12, 2018
Understanding the physical properties of galaxies and how they change through cosmic time allows us to learn about the cosmic expansion, gravity, and the physical mechanisms that regulate the growth of structures. My work focuses on developing and using better tools to extract maximal information from existing and future data from large multi-wavelength galaxy surveys, such as CANDELS and LSST. I will introduce GalMC and SpeedyMC, the Markov Chain Monte Carlo algorithms I created with the purpose of extracting galaxy properties, and show how they can be used to estimate the age, mass, dust content, metallicity and star formation history of galaxies. I will describe what are, in my opinion, the most crucial sources of systematic uncertainties and describe our ongoing effort at pushing the frontier of extracting information from galaxy spectra using machine learning methods and Bayesian model selection.
Monday, January 29, 2018
#### The Prevalence of Massive Quiescent Disks in the Early Universe
Elizabeth McGrath Colby College
## The Prevalence of Massive Quiescent Disks in the Early Universe
Elizabeth McGrath Colby College
Monday, January 29, 2018
Observations in the local Universe suggest that the mechanism responsible for quenching star formation in galaxies may be intimately linked to their structural transformation from disks to spheroids. In order to test quenching scenarios, however, it is vital to look beyond the local Universe and identify the first generation of quiescent galaxies at high redshift. Using CANDELS, we have examined the rest-frame optical morphologies for a sample of massive, quiescent galaxies at z>1 and find that a significant fraction (~30%) have morphologies dominated by exponential disks. The persistence of massive disks, long after star formation has ceased, implies that in at least some cases quenching precedes morphological transformation. I'll examine what constraints these observations place on the mechanisms responsible for quenching star-formation in the first generation of quiescent galaxies at z~2 and discuss them in context with an emerging picture of massive galaxy formation and evolution.
Monday, December 11, 2017
#### The Astrophysics of BH-BH/NS-NS Mergers with LIGO/Virgo
Chris Belczynski University of Warsaw
## The Astrophysics of BH-BH/NS-NS Mergers with LIGO/Virgo
Chris Belczynski University of Warsaw
Monday, December 11, 2017
Dr. Belczynski will discuss the astrophysical importance of the recent LIGO/Virgo direct detections of gravitational-waves. Despite majority of the expectations, it was not neutron star mergers being detected first, but the series of exotic massive black hole mergers. He will describe the leading theories of the formation of such black hole systems. He will also comment on a detection of NS-NS merger. This particular detection may provide striking constraints on binary evolution. Several astrophysical implications are beginning to emerge despite the fact that the exact origin of LIGO/Virgo sources is not yet known.
Monday, November 27, 2017
#### SuperSpec, an on-chip, mm-wave spectrometer for studying high-redshift galaxies
Erik Shirokoff University of Chicago
## SuperSpec, an on-chip, mm-wave spectrometer for studying high-redshift galaxies
Erik Shirokoff University of Chicago
Monday, November 27, 2017
SuperSpec is a novel, ultra-compact spectrometer-on-a-chip for millimeter and submillimeter wavelength astronomy, consisting of a filter bank made from planar, lithographed superconducting transmission line resonators coupled to kinetic inductance detectors. Its compact size and intrinsic multiplexing capability will enable multi-object spectrometers and integral field units with hundreds to thousands of spatial pixels. The pathfinder instrument, optimized for observing the atomic and molecular lines of high redshift galaxies, is preparing for its first light observations in 2018. I'll discuss the design and performance of SuperSpec detectors and the opportunities for science with both the pathfinder and future instruments.
Monday, November 13, 2017
#### Black Hole Feedback in Action? Understanding the role of AGN in transitioning Galaxies
Lauranne Lanz Dartmouth College
## Black Hole Feedback in Action? Understanding the role of AGN in transitioning Galaxies
Lauranne Lanz Dartmouth College
Monday, November 13, 2017
In past several decades, we have made great progress in developing a consistent paradigm for the formation and evolution of galaxies, seeking to answer the basic question of where galaxies like our Milky Way come from and where we are headed on cosmic timescales. However, many details of this picture remain unclear. One particularly interesting question regards the role that active supermassive black holes play in the transition from a gas-rich, actively star forming state to gas-poor quiescence. I will describe a recently discovered type of galaxies at an earlier stage of transition that the classical post-starburst galaxies, and present new X-ray observations of these galaxies, which provide a crucial window into the role of AGN in the early phases of transformation.
Monday, October 30, 2017
#### Enabling Infrared Surveys of Galaxies with Innovative Imaging Spectrographs
Suresh Sivanandam University of Toronto
## Enabling Infrared Surveys of Galaxies with Innovative Imaging Spectrographs
Suresh Sivanandam University of Toronto
Monday, October 30, 2017
Optical integral field (imaging) spectroscopic surveys of large numbers of galaxies are now becoming the norm. These surveys allow detailed studies of individual galaxies, such as their kinematics and stellar ages/metallicities. With a sufficiently large sample, these types of observations are the best tools for understanding the formation and evolution of galaxies. However, similar surveys in the infrared remain challenging. There are two significant gaps that need to addressed: the rest-frame infrared has been untapped for nearby systems due to the lack of wide integral field infrared spectrographs (IFSes), and observations of the distant universe have been limited to small samples from the lack of high angular resolution, highly multiplexed IFSes.
I will discuss two instruments that will directly address these gaps: one recently commissioned, the wide integral field infrared spectrograph (WIFIS), and another recently funded, the Gemini Infrared Multi-object Spectrograph (GIRMOS). WIFIS is currently carrying out an infrared survey of nearby galaxies by studying their stellar populations, star-formation, and kinematics, complementing existing optical surveys such as CALIFA and MaNGA. On the other hand, GIRMOS will be a multi-object IFS that takes advantage of the latest developments in adaptive optics. It will be able to carry out large surveys of the distant universe by simultaneously observing multiple galaxies. It will finally allow the type of work being done for low redshift systems at high redshift.
Monday, October 16, 2017
#### Imaging the CO snow line in protoplanetary disks
Charlie Qi Harvard-Smithsonian Center for Astrophysics
## Imaging the CO snow line in protoplanetary disks
Charlie Qi Harvard-Smithsonian Center for Astrophysics
Monday, October 16, 2017
The condensation fronts (snow lines) of water, carbon monoxide (CO) and other abundant volatiles in the midplane of protoplanetary disks affect various aspects of planet formation and composition. Locating the CO snow line directly from millimeter CO data is a challenge since CO gas remains abundant in the warm atmosphere above the disk midplane, also exterior to the CO snow line. Using data from the Submillimeter Array and Atacama Large Millimeter/Submillimeter Array, we have put strong constraints on the location of the CO snow line in the disks. I will also discuss the significant consequences of the CO freeze-out and desorption on the dust evolution and gas chemistry in disks.
Monday, October 2, 2017
#### Science Policy and You (Yes, You!)
Ashlee Wilkins American Astronomical Society
## Science Policy and You (Yes, You!)
Ashlee Wilkins American Astronomical Society
Monday, October 2, 2017
As science educators, researchers, communicators, and/or supporters, we cannot deny the connection between science and government. The ability to send missions to Mars, to study star formation in a galaxy, and to model the early universe primarily depends on both government -- i.e., taxpayer -- money and public -- i.e., not just scientist -- support. Scientists can and do engage in work to determine the direction of our field and how society prioritizes science. In this talk, I will focus on ways that policy impacts science, how you, as an individual, can influence policy, and how the American Astronomical Society (AAS) advocates for the astronomical sciences in Washington.
Monday, September 18, 2017
#### Highlights of the High Energy Universe from HAWC
Segev BenZvi University of Rochester
## Highlights of the High Energy Universe from HAWC
Segev BenZvi University of Rochester
Monday, September 18, 2017
During the past decade, the observation of teraelectronvolt (TeV) cosmic rays and gamma rays has opened a new frontier in the study of astrophysical particle accelerators such as super nova remnants, pulsars, microquasars, and active galactic nuclei. This energetic radiation can also be used to explore physics beyond the Standard Model, such as the particle nature of dark matter. Since 2015 the High Altitude Water Cherenkov (HAWC) Observatory has recorded TeV gamma rays and cosmic rays from 2/3 of the sky each day. I will report on recent results from HAWC, including the discovery of nearly 20 new gamma ray sources and high-statistics observations of cosmic rays.
Monday, July 10, 2017
#### TARdYS an upcoming exoplanet hunter for the southern hemisphere
Surangkhana Rukdee Centro de Astro-Ingeniería UC
## TARdYS an upcoming exoplanet hunter for the southern hemisphere
Surangkhana Rukdee Centro de Astro-Ingeniería UC
Monday, July 10, 2017
The relatively close habitable zone to the host stars of the very common cool-low mass stars makes M dwarfs attractive for finding habitable planets. Up to date only a few of these stars can be observed by visible high resolution spectrographs since M dwarfs emit the peak of its energy at λ>900 nm in the near infrared region. We optimized a spectrograph for the observation of cool stars in the southern hemisphere where only a few high resolution near infrared spectrographs are available. TARdYS is a high resolution echelle spectrograph to be installed at the Tokyo Atacama Observatory TAO 6.5 m telescope. This spectrograph is a dual fiber fed, white pupil Echelle, which can yield R>50,000 within the spectral range of 0.843-1.117 μm covers Y band. We adopt an echelle R6, 13.33 lines/mm and a VPH 333 lines/mm grating as cross disperser. A Teledyne H1RG detector will be operated in a cryogenic environment cooled to 80K. Our optimization using computer-aided simulation programs results in excellent resolution performance within the diffraction limit even when taking realistic manufacturing and alignment tolerances into account. We will discuss challenges and design decisions of an upcoming exoplanet hunter in the southern hemisphere. The overall instrument system design and ongoing development of TARdYS will be presented.
Monday, April 24, 2017
#### Quanta Image Sensor: Every Photon Counts
Eric Fossum Dartmouth College
## Quanta Image Sensor: Every Photon Counts
Eric Fossum Dartmouth College
Monday, April 24, 2017
The Quanta Image Sensor (QIS) is a possible paradigm shift in solid-state image sensors. Conceptually, the QIS counts photons one at a time using small pixels with low full-well capacity and single-photoelectron sensitivity. This binary data is collected and transformed into gray scale images by post-acquisition digital image processing. In recent years, the QIS has moved from concept to experimental devices. A 1Mpixel QIS has been designed and fabricated in a commercial stacked, backside-illuminated CMOS image sensor 65/45nm node process. The device, with 1.1um shared-readout pixel pitch, has been shown to have read noise as low as 0.172e- rms at room temperature without the use of avalanche multiplication, and is successfully readout at 1040fps using a cluster-parallel readout architecture. Each binary bit represents the detection or absence of a photoelectron.
The QIS technology is evolved from the CMOS image sensor currently incorporated into billions of cameras each year. In a preamble to the QIS part of the talk, the invention and development of the CMOS image sensor technology at the NASA Jet Propulsion Laboratory at Caltech, and by the spinoff company Photobit will be discussed.
Monday, April 10, 2017
#### The Intricate Role of Cold Gas and Dust in Galaxy Evolution at Early Cosmic Epochs
Dominik A. Riechers Cornell University
## The Intricate Role of Cold Gas and Dust in Galaxy Evolution at Early Cosmic Epochs
Dominik A. Riechers Cornell University
Monday, April 10, 2017
Dusty starburst galaxies at very high redshift represent an important phase in the early evolution of massive galaxies. They typically represent large-scale, gas-rich major mergers that trigger intense, short-lived bursts of star formation, which consume most of the available gas and drive the morphological transition to spheroids. At early cosmic epochs, these hyper-luminous galaxies commonly trace regions of high galaxy overdensity, and may be directly related to the formation of galaxy clusters and their giant central ellipticals. Molecular and atomic gas plays a central role in our understanding of the nature of these often heavily obscured distant systems. It represents the material that stars form out of, and its mass, distribution, excitation, and dynamics provide crucial insight into the physical processes that support the ongoing star formation and stellar mass buildup. I will discuss the most recent progress in studies of the cold gas content of dusty starburst galaxies at high redshift, back to the first billion years of cosmic time using CARMA, the Jansky Very Large Array, the Plateau de Bure interferometer, and the Atacama Large (sub)Millimeter Array (ALMA). I will also highlight our recent successful first detections of the interstellar medium in "normal" (~L*) galaxies at z>5 with ALMA, and discuss the impact of our findings on future studies back to even earlier epochs.
Monday, March 27, 2017
#### New Approaches to Dark Matter
Justin Khoury University of Pennsylvania
## New Approaches to Dark Matter
Justin Khoury University of Pennsylvania
Monday, March 27, 2017
In this talk I will discuss a novel theory of superfluid dark matter. The scenario matches the predictions of the Lambda-Cold-Dark-Matter (LambdaCDM) model on cosmological scales while simultaneously reproducing the MOdified Newtonian Dynamics (MOND) empirical success on galactic scales. The dark matter and MOND components have a common origin, as different phases of a single underlying substance. This is achieved through the rich and well-studied physics of superfluidity. The framework naturally distinguishes between galaxies (where MOND is successful) and galaxy clusters (where MOND is not): due to the higher velocity dispersion in clusters, and correspondingly higher temperature, the dark matter in clusters is either in a mixture of superfluid and normal phases, or fully in the normal phase. The model makes various observational predictions that distinguishes it from both LambdaCDM and standard MOND. In the last part of the talk, I will discuss an on-going attempt at explaining cosmic acceleration as yet another manifestation of dark matter superfluidity.
Monday, February 27, 2017
#### Unveiling Black Hole Growth Over Cosmic Time
Stephanie LaMassa Space Telescope Science Institute
## Unveiling Black Hole Growth Over Cosmic Time
Stephanie LaMassa Space Telescope Science Institute
Monday, February 27, 2017
Supermassive black holes, millions to billions of times the mass of our Sun, live in the center of every massive galaxy. When they grow via the process of accretion, they are observed as Active Galactic Nuclei (AGN). In addition to being among the most energetic sources in the Universe, AGN seemed to be intrinsically linked to the galaxies in which they reside. By surveying regions of the sky, we can discover AGN from early cosmic times to the present day, thereby learning about supermassive black hole growth and evolution and the role they may play in shaping their host galaxies. Currently, we are missing an important piece of the puzzle in AGN evolution - luminous obscured black hole growth. To this end, I am leading a wide area X-ray survey: by probing a large volume of the Universe, a representative sample of rare objects are detected, and X-rays pierce through dust that obscures optical light, recovering AGN missed by optical surveys. By executing this survey in the Stripe 82 region of the Sloan Digital Sky Survey which contains rich multi-wavelength coverage, we have the ancillary data necessary to characterize the AGN and their host galaxies. In this talk, I will give an overview of this “Stripe 82X" survey, summarize the properties of the objects we have detected thus far, discuss what we are planning to learn from this dataset in the coming years, and how we can these data to develop best-effort practices to push into new discovery space with upcoming missions like JWST, WFIRST, LSST, and eROSITA. I will highlight a peculiar source I discovered in this survey which has now become a burgeoning subfield in AGN physics, providing unique insight into AGN lifetimes and black hole fueling.
Monday, February 13, 2017
#### Modeling Baryonic Physics in Galaxy Clusters
Erwin Lau Yale University
## Modeling Baryonic Physics in Galaxy Clusters
Erwin Lau Yale University
Monday, February 13, 2017
Galaxy clusters play an important role in modern precision cosmology. As the most massive virialized objects in the universe, their abundance depends sensitively on cosmological parameters. However, uncertainties in galaxy cluster physics pose serious challenges to using forthcoming observations to make advances in cosmology with galaxy clusters.
In this talk, I will highlight how we can improve our understanding of galaxy cluster physics with the state-of-the-art numerical simulations and semi-analytical modelling. In particular, I will present results from the "Omega500" simulation, a high-resolution hydrodynamic simulation suite of galaxy cluster formation that follows the evolution of dark matter and baryons in a realistic cosmological setting. I will also outline upcoming challenges in the computational modelling of major physical processes in galaxy clusters, and how we can address them in anticipation of upcoming multi-wavelength cluster surveys in the next decade.
Monday, January 30, 2017
#### A Galaxy-Scale Fountain of Cold Molecular Gas Pumped by a Black Hole
Grant Tremblay Yale University
## A Galaxy-Scale Fountain of Cold Molecular Gas Pumped by a Black Hole
Grant Tremblay Yale University
Monday, January 30, 2017
A new ALMA observation of the cool core brightest cluster galaxy in Abell 2597 reveals that a supermassive black hole can act much like a mechanical pump in a water fountain, driving a convective flow of molecular gas that drains into the black hole accretion reservoir, only to be pushed outward again in a jet-driven outflow that then rains back toward the galaxy center from which it came. The ALMA data reveal "shadows" cast by giant molecular clouds falling on ballistic trajectories towards the black hole in the innermost hundred parsecs of the galaxy, manifesting as deep redshifted continuum absorption features. The black hole accretion reservoir, fueled by these infalling cold clouds, powers an AGN that drives a jet-driven molecular outflow in the form of a 10 kpc-scale, billion solar mass expanding molecular bubble. HST reveals that the molecular shell is permeated with young stars, perhaps triggered in situ by the jet. Buoyant X-ray cavities excavated by the propagating radio source may further uplift the molecular filaments, which are observed to fall inward toward the center of the galaxy from which they came, presumably keeping the fountain long-lived. I will discuss this specific result in the larger context of galaxies as a whole, as the results show that cold molecular gas can couple to black hole growth via both feedback and feeding, in alignment with "cold chaotic accretion" models for the regulation of star formation in galaxies.
Monday, December 12, 2016
#### Opening the gravitational wave universe: the physics behind a new type of astronomy
Shelia Dwyer Caltech
## Opening the gravitational wave universe: the physics behind a new type of astronomy
Shelia Dwyer Caltech
Monday, December 12, 2016
Hosted by the School of Physics & Astronomy and the Center for Computational Relativity and Gravitation
With the first direct detection of gravitational waves, the LIGO detectors have opened a new field of astrophysics, discovered a new class of massive stellar mass black holes, and enabled tests of general relativity. To make these discoveries, we built the most sensitive displacement meters yet, measuring displacements of one thousandth of a proton diameter over a 4 kilometer baseline. This talk will focus on the physics of the ground breaking detectors, and the possibilities for extending their reach. One of the most promising techniques for improving the sensitivity is the use of squeezed states to reduce quantum noise. l will describe a test of squeezing in the LIGO interferometers, and implications for the permanent application of squeezing in Advanced LIGO over the next few years. Because gravitational wave detectors measure amplitude rather than power, improvements in sensitivity from squeezing, cryogenic operation and new optical materials have the potential to dramatically increase the volume of the universe which can be surveyed. In a new and larger facility, LIGO style detectors could observe compact object binaries from the earliest periods of star formation, making complete surveys at distances difficult to observe with optical telescopes.
Monday, December 5, 2016
#### The Future of Gravitational Wave Interferometers
Jax Sanders Syracuse University
## The Future of Gravitational Wave Interferometers
Jax Sanders Syracuse University
Monday, December 5, 2016
Hosted by the School of Physics & Astronomy and the Center for Computational Relativity and Gravitation
The recent detection of gravitational waves from binary black hole mergers marks the beginning of the field of gravitational wave astronomy. New and more sensitive techniques will be required to continue expanding our understanding of the universe through gravitational waves. Current research and development efforts range from surpassing the standard quantum limit using squeezed states, to improving thermal noise at the frontiers of the material science of optical coatings, to the conceptual design of new interferometer topologies. These noise reduction efforts will increase the sensitivity of the detectors, allowing the measurement of smaller effects and extending our reach to cosmological scales.
Monday, November 21, 2016
#### The radial acceleration relation: linking baryons and dark matter in galaxies
Federico Lelli Case Western Reserve University
## The radial acceleration relation: linking baryons and dark matter in galaxies
Federico Lelli Case Western Reserve University
Monday, November 21, 2016
The flat rotation curves of spiral galaxies provided clear evidence for mass discrepancies in galactic systems, but the nature of dark matter (DM) still remains elusive. I will describe recent results from the Spitzer Photometry and Accurate Rotation Curves (SPARC) dataset: the largest collection of HI rotation curves currently available for late-type galaxies (spirals and irregulars). New Spitzer photometry at 3.6 um provides the closest proxy to the stellar mass, allowing precise estimates of the baryonic gravitational field at every radii (g_bar). We find that the observed acceleration correlates with g_bar over 4 dex, implying a close link between baryons and DM in galaxies. This radial acceleration relation coincides with unity (no DM) at high g_bar but systematically deviates below a critical acceleration scale. The observed scatter is remarkably small, even when DM dominates at low g_bar. Early-type galaxies (ellipticals, lenticulars, and dwarf spheroidals) follow the same relation as late-type galaxies. The radial acceleration relation is tantamount to a "Kepler Law" for galactic systems: when the baryonic contribution is measured, the rotation curve follows, and vice versa. I will discuss possible interpretations within the standard LCDM cosmology as well as alternative theories.
Monday, November 7, 2016
#### Studying planet formation processes with molecular spectroscopy
Colette Salyk Vassar College
## Studying planet formation processes with molecular spectroscopy
Colette Salyk Vassar College
Monday, November 7, 2016
As our understanding of the solar system and exoplanetary systems continues to grow, our view of planet formation processes must expand to accommodate the incredibly diversity of formation outcomes. I will focus this talk on my favorite technique for studying planet formation processes in action - molecular spectroscopy. I will review the techniques of molecular spectroscopy and discuss how they can be used to tackle key questions about planet formation, including: Where does water freeze, and does this correspond with giant planet formation? Are there spectroscopic signatures of disk evolution, or the presence of planets? What factors determine the final chemical make-up of a planet? I will highlight some of the incredible progress that has been made on answering these questions in recent years, and provide updates on ongoing projects, including several involving undergraduate students.
Monday, October 24, 2016
#### Illuminating the Black Hole – Galaxy Connection with CANDELS
Dale Kocevski Colby College
## Illuminating the Black Hole – Galaxy Connection with CANDELS
Dale Kocevski Colby College
Monday, October 24, 2016
Supermassive black holes, and the active galactic nuclei (AGN) that they power, are thought to play an integral role in the evolution of galaxies by acting to regulate, and eventually suppress, the star formation activity of their host galaxies. I will discuss recent efforts to test this proposed connection by studying the demographics of galaxies undergoing active black hole growth. In particular, I will highlight recent results from the CANDELS survey, whose panchromatic Hubble ACS and WFC3 imaging is now allowing us to characterize the morphologies and stellar populations of thousands of AGN hosts out to z=2, the era when star formation activity and black hole growth in the Universe are at their peak. I will discuss what CANDELS is currently revealing about the mechanisms that fuel AGN activity at this epoch and the connection between black hole growth and the emergence of the first generation of passive galaxies in the Universe.
Monday, October 10, 2016
#### Glimpses of futuristic cosmology: relativistic effects and spectral maps
Rupert Croft Carnegie Mellon University
## Glimpses of futuristic cosmology: relativistic effects and spectral maps
Rupert Croft Carnegie Mellon University
Monday, October 10, 2016
In the next 10 years, the number of galaxies with measured redshifts will increase into the tens of millions. Beyond this, mapping the Universe will be done not by detecting individual galaxies, but by recording spectral information for the entire sky. Both of these developments will make possible new ways to pin down the nature of dark matter, dark energy and gravity. It has not been widely realized, however that some new types of exploration can begin with presently available data. I will describe how to measure Special and General Relativistic effects distorting the observed large-scale structure of the Universe. These include the gravitational redshift (first seen in Earth-bound laboratories in 1960) and relativistic beaming, which is now detectable on the scales of galaxies and beyond. I will present results from hydrodynamic simulations as well as some preliminary measurements from galaxy survey data. I will then move on to spectral intensity mapping, a technique which promises to make truly inclusive surveys of the Universe. I will present the first measurements of the large-scale structure of the cosmos made using optical line intensity, and show how the results are suprising in the context of our standard cosmological model. I will discuss how lessons learned during the analysis can be applied to future experiments, and motivate specialized instruments.
Monday, September 26, 2016
#### Bridging interferometry and astrophysics: noise and the future of gravitational wave astronomy
Jess McIver Caltech
## Bridging interferometry and astrophysics: noise and the future of gravitational wave astronomy
Jess McIver Caltech
Monday, September 26, 2016
About one year ago, the Advanced LIGO detectors sensed the passing of gravitational wave signal GW150914 from the merger of two black holes, each roughly 30 solar masses. This discovery ushered in a new era of gravitational wave astronomy. It was the first direct detection of gravitational waves and provided evidence of black holes with masses never before observed. Roughly three months later, the LIGO detectors observed a second binary black hole merger, GW151226, with strong evidence of black hole spin.
This talk will focus on the interface between gravitational wave astrophysics and instrumentation. The LIGO detectors have unprecedented sensitivity to spacetime strain, but the interferometer data are not perfectly clean. The LIGO Scientific Collaboration has a strong history of detector characterization work that studies the causes of noise artifacts in the data and their effect on the recovery of astrophysical signals. I will show the impact of detector characterization efforts on searches for compact binary coalescences during Advanced LIGO's first observing run. I will also illustrate that noise characterization will become increasingly important, particularly for extracting the astrophysical parameters of detected sources, as LIGO improves in sensitivity and the global gravitational wave community fields an increasing rate of detections.
Monday, May 9, 2016
#### Inside-Out Planet Formation
Jonathan C. Tan University of Florida
## Inside-Out Planet Formation
Jonathan C. Tan University of Florida
Monday, May 9, 2016
The Kepler-discovered systems with tightly-packed inner planets (STIPs), typically with several planets of Earth to super-Earth masses on well-aligned, sub-AU orbits may host the most common type of planets in the Galaxy. They pose a great challenge for planet formation theories, which fall into two broad classes: (1) formation further out followed by migration; (2) formation in situ from a disk of gas and planetesimals. I review the pros and cons of these classes, before focusing on a new theory of sequential in situ formation from the inside-out via creation of successive gravitationally unstable rings fed from a continuous stream of small (~cm-m size) "pebbles," drifting inward via gas drag. Pebbles first collect at the pressure trap associated with the transition from a magnetorotational instability (MRI)-inactive ("dead zone") region to an inner MRI-active zone. A pebble ring builds up until it either becomes gravitationally unstable to form an Earth to super-Earth-mass planet directly or induces gradual planet formation via core accretion. The planet continues to accrete until it becomes massive enough to isolate itself from the accretion flow via gap opening. The process repeats with a new pebble ring gathering at the new pressure maximum associated with the retreating dead-zone boundary. I discuss the theory’s predictions for planetary masses, relative mass scalings with orbital radius, and minimum orbital separations, and their comparison with observed systems. Finally I speculate about potential causes of diversity of planetary system architectures, i.e. STIPs versus Solar System analogs.
Monday, May 2, 2016
#### Unveiling the dark side of the Universe
Priya Natarajan Yale University
## Unveiling the dark side of the Universe
Priya Natarajan Yale University
Monday, May 2, 2016
Dark matter and Dark Energy, the enigmatic dominant constituents of our Universe shape the properties of structures. However, their essential nature remains unknown.
Gravitational lensing, the bending of light by matter predicted by Einstein's Theory of General Relativity offers a powerful probe of both dark matter and dark energy. Deploying
clusters of galaxies as gravitational lenses a viewed by the Hubble Space Telescope we have many interesting new results - I will present a status report of recent progress in
this talk.
Monday, April 25, 2016
#### Dusty Universe
Asantha Cooray University of California, Irvine
## Dusty Universe
Asantha Cooray University of California, Irvine
Monday, April 25, 2016
Dr. Cooray will summarize the scientific case for studying the universe and Far-Infrared and sub-millimeter wavelengths. She will present results from Herschel, summarize on going plans for ground-based instruments, and outline the ongoing Far-Infared Surveyor study facilitated by NASA for 2020 Decadal Surveyor.
Monday, March 21, 2016
#### Galaxy Mergers on FIRE: Mapping Star Formation
Jorge Moreno Cal Poly Pomona
## Galaxy Mergers on FIRE: Mapping Star Formation
Jorge Moreno Cal Poly Pomona
Monday, March 21, 2016
Galaxy mergers and interactions are responsible for generating bursts of star formation, for changing galactic morphology in dramatic ways, and for triggering single and dual active galactic nuclei. In this talk, I will unveil the very first results from a novel suite of high-resolution galaxy merger simulations, based on the “Feedback In Realistic Environments” (FIRE) model. This model treats energy and momentum-driven feedback from young stars and SN explosions explicitly, which acts directly on resolved star-forming clouds within the ISM. Moreover, this framework relies on a new meshless Lagrangian hydro code, GIZMO, which solves many problems associated with older solvers. Our first work focuses on the spatial localization of star formation. In particular, we confirm results from previous work: galaxy-galaxy interactions enhance nuclear star formation, and suppress it at large galacto-centric radii (Moreno et al. 2015). However, two major differences are found. First, star-formation enhancement and suppression are not as dramatic as in older models. Secondly, the interaction-induced nuclear starburst has a larger spatial extent. These differences are a reflection of the fact that, in our new models, non-axisymmetric gravitational torques are not as effective at driving fuel into the central regions as in older sub-grid based models. This suite of merger simulations is ideal for making predictions for, and interpreting results from, observations by new-generation integral field spectroscopic surveys, such as CALIFA, MaNGA and HECTOR.
Monday, March 7, 2016
#### New astronomical projects from Japan: TAO and the Tomo-e Gozen Camera
Mamoru Doi University of Tokyo
## New astronomical projects from Japan: TAO and the Tomo-e Gozen Camera
Mamoru Doi University of Tokyo
Monday, March 7, 2016
The Institute of Astronomy, part of the University of Tokyo's School of Science, is currently operating two observatories: the Tokyo Atacama Observatory (TAO) in Chile, and the Kiso Observatory in Nagano, Japan. The goal of TAO is to operate a 6.5-m optical-infrared telescope at Cerro Chajnantor in Chile (elevation 5,640m), the world's highest site for an astronomical observatory. A pathfinder telescope, the miniTAO 1.0-m telescope, has been operating there since 2009, and we can confirm that it opens new atmospheric windows in the mid-infrared. I will review the current status of the TAO project as well as early results from miniTAO.
At the Kiso observatory, we are developing a new wide-field imager, the Tomo-e Gozen Camera, which is going to use 84 CMOS sensors. I will show some early results from a prototype camera with 8 CMOS sensors, called Tomo-e PM.
Monday, February 29, 2016
#### The Biggest Blowhards: Windy Supermassive Black Holes
Sarah Gallagher Western University
## The Biggest Blowhards: Windy Supermassive Black Holes
Sarah Gallagher Western University
Monday, February 29, 2016
Supermassive black holes reside in the centers of every massive galaxy. In relatively brief spurts, black holes grow as luminous quasars through the infall of material through an accretion disk. Remarkably, the light from the accretion disk can outshine all of the stars in the host galaxy by a factor of a thousand, and this radiation can also drive energetic mass outflows. Mass ejection in the form of winds or jets appears to be as fundamental to quasar activity as accretion, and can be directly observed in many objects with broadened and blue-shifted UV emission and absorption features. A convincing argument for radiation pressure driving this ionized outflow can be made within the dust sublimation radius. Beyond, radiation pressure is still important, but high energy photons from the central engine can now push on dust grains. This physics underlies the dusty wind picture for the putative obscuring torus. I'll describe our model of the dusty wind and evaluate its successes and shortcomings in accounting for observed properties of quasars such their mid-infrared power, fractions of hidden objects, and column densities of important ions.
Monday, December 7, 2015
#### Cosmological Simulations of Galaxy Formation and Evolution
Lars Hernquist Harvard-Smithsonian Center for Astrophysics
## Cosmological Simulations of Galaxy Formation and Evolution
Lars Hernquist Harvard-Smithsonian Center for Astrophysics
Monday, December 7, 2015
A predictive theory of galaxy formation remains elusive, even after more than 50 years of dedicated effort by many renowned astrophysicists. The problem of galaxy formation is made difficult by the large range in scales involved and the many non-linear physical processes at work. In this talk, I describe a new generation of numerical models designed to overcome these difficulties based on novel schemes for solving the fluid equations on a moving mesh. Initial results from this study provide insight into many aspects of galaxy assembly and the relationship between galaxies and cosmologically-distributed baryons.
Monday, November 23, 2015
#### Dark Energy Spectroscopic Instrument
Peter Nugent University of California, Berkeley
## Dark Energy Spectroscopic Instrument
Peter Nugent University of California, Berkeley
Monday, November 23, 2015
The Dark Energy Spectroscopic Instrument (DESI) will measure the effect of dark energy on the expansion of the universe. It will obtain optical spectra for tens of millions of galaxies and quasars, constructing a 3-dimensional map spanning the nearby universe to 10 billion light years over 14,000 square degrees of sky. DESI’s key project goals are to (a) probe the effects of dark energy on the expansion history using Baryon Acoustic Oscillations and (b) measure the gravitational growth history using redshift-space distortions. DESI will be conducted on the Mayall 4-meter telescope at Kitt Peak National Observatory starting in 2018. Here I will describe the overall design of the experiment, its current status and the massive photometric targeting surveys being carried out now in preparation for DESI.
Wednesday, November 7, 2018
#### Physics Colloquium: Dr. Flavio Fenton
Flavio Fenton, Ph.D. Professor at Georgia Tech
## Physics Colloquium: Dr. Flavio Fenton
Flavio Fenton, Ph.D. Professor at Georgia Tech
Wednesday, November 7, 2018
#### Title: Integrative Approach to Control and Termination of Cardiac Arrhythmias
Dr. Fenton’s work is on excitable media, complex systems, and pattern formation, using a combined approach of theory, experiments, and computer simulations.
Pizza will be Available
Interpreting Services: myaccess.rit.edu
Wednesday, October 31, 2018
## Physics Colloquium: Kerstin Nordstrom
Wednesday, October 31, 2018
#### Title: TBD
Kerstin Nordstrom
Pizza will be Available
Interpreting Services: myaccess.rit.edu
Wednesday, October 3, 2018
##### Physics Colloquium
Wednesday, October 3, 2018
#### Title: New Advances in MBE Technology
Osemi Inc.
Pizza will be Available
Interpreting Services: myaccess.rit.edu
Wednesday, September 19, 2018
##### Physics Colloquium
Wednesday, September 19, 2018
#### Title: The Physicist Random Walk: Fermi problems, Careers, & Skills Assessment
Director, Society of Physics Students & Sigma Pi Sigma (ΣΠΣ)
American Institute of Physics
Physicists and astronomers hone an extremely valuable set of skills that position them to succeed in an exceptionally wide variety of graduate programs, careers, and positions. We can be really good problem solvers. Through an interactive example of fermi questions (back-of-the-envelope calculations), we’ll touch on some of the ways physicists and astronomers impact the world in profound ways. Through a variety of careers (that may appear to be random walks) we’ll end up discussing how we can help to solve the world’s problems through physics and astronomy and assess your own skills while we are at it. This is, in part, to shed light on the obstacles, for both students and faculty, but also to encourage you to consider all options. We’ll end the talk with some tips on finding the right job and career pathway. Please bring some scrap paper and a pen.
Pizza will be Available
Interpreting Services: myaccess.rit.edu
Wednesday, September 5, 2018
#### Physics Colloquium: Ted Kinsman
Ted Kinsman RIT Photographic Arts and Sciences
## Physics Colloquium: Ted Kinsman
Ted Kinsman RIT Photographic Arts and Sciences
Wednesday, September 5, 2018
Assistant Professor Ted Kinsman, Rochester Institute of Technology
Abstract: On The Creation of the Scientific Image: The process of popularizing science is explored through the creative use of imagery. Techniques to be presented will range from high-speed photography and scanning electron low light imaging. The characteristics of a great image will be addressed.
Pizza will be Available
Interpreting Services: myaccess.rit.edu
Wednesday, April 4, 2018
#### Inkjet Printable All Inorganic Perovskite Films with Long Effective Photocarrier Lifetime
Dr. Carolina C. Ilie Department of Physics, State University of New York at Oswego
## Inkjet Printable All Inorganic Perovskite Films with Long Effective Photocarrier Lifetime
Dr. Carolina C. Ilie Department of Physics, State University of New York at Oswego
Wednesday, April 4, 2018
Halide based perovskite solar cells (HPSCs) have recently drawn plenty of attention due to their lowcost, extraordinary power conversion efficiency, and long carrier lifetimes and diffusion lengths. Unfortunately,organic based HPSCs have a few drawbacks including being sensitive to heat, moisture, and radiation induceddegradation. An alternative approach is the use of all inorganic based HPSC materials as a way of circumventingsome of the drawbacks. CsPbBr3 quantum dot (QD) inks are used in an inkjet printer to print photoactiveperovskite QD films. The films, characterized by X-ray diffraction and X-ray photoelectron spectroscopy prove crystalline structure and bonding consistent with CsPbBr3 perovskite quantum dots. The current-voltage I-V and capacitance-voltage C-V transport measurements indicate that the photocarrier drift lifetime can exceed 1 millisecond for some perovskites films. The successful printing of photoactive-perovskite QD films of CsPbBr3, indicates that the rapid prototyping of various perovskite inks and multilayers is realizable.
Wednesday, March 28, 2018
Dr. Crystal Bailey American Physical Society (APS)
##### Physics Colloquium
Dr. Crystal Bailey American Physical Society (APS)
Wednesday, March 28, 2018
To request interpreting services, visit http://myAccess.rit.edu
Wednesday, March 21, 2018
#### Integrative approach to control and termination of cardiac arrhythmias.
Flavio Fenton Georgia Tech
## Integrative approach to control and termination of cardiac arrhythmias.
Flavio Fenton Georgia Tech
Wednesday, March 21, 2018
The heart is an electro-mechanical system in which, under normal conditions, electrical waves propagate in a coordinated manner to initiate an efficient contraction. In pathologic states, single and multiple rapidly rotating spiral and scroll waves of electrical activity can appear and generate complex spatiotemporal patterns of activation that inhibit contraction and can be lethal if untreated. In this talk I will describe how we use mathematical methods from dynamical systems, experiments in whole hearts, and high-performance parallel computing in an integrative approach to investigate the mechanisms that initiate, perpetuate, and terminate electrically driven cardiac arrhythmias. In particular, I will describe some of the mechanisms that can initiate fibrillation such as period doubling bifurcations and then describe a control method we have developed to terminate arrhythmias based on synchronization that requires only 10% of the energy needed for conventional defibrillation. For this I will establish a relationship between the response of cardiac tissue to an electric field and the spatial distribution of heterogeneities due to the coronary vascular structure, and discuss how in response to a pulsed electric field E, these heterogeneities serve as nucleation sites for the generation of intramural electrical waves with a source density ρ(E) and a characteristic time constant τ for tissue excitation that obeys a power law. This allow us to develop numerical simulations for effective defibrillation that are then tested in vitro and finally in vivo under clinical conditions.
Wednesday, March 7, 2018
#### Structural Dynamics of Ribosome During Protein Synthesis
Dmitri Ermolenko University of Rochester Medical Center
## Structural Dynamics of Ribosome During Protein Synthesis
Dmitri Ermolenko University of Rochester Medical Center
Wednesday, March 7, 2018
Wednesday, February 21, 2018
#### Compressive Sensing for Quantum Imaging
Greg Howland Air Force Research Laboratory
## Compressive Sensing for Quantum Imaging
Greg Howland Air Force Research Laboratory
Wednesday, February 21, 2018
Compressive sensing (CS) is an exciting new measurement technique that effectively compresses data while it is being measured, allowing high-dimensional signals to be recovered from very few measurements. The CS approach of figuring out how to "just measure the important information" has upended traditional views on sampling and sparked enormous multidisciplinary interest over the past decade. Remarkably, the best measurements an experimenter can use are often random. In this talk, I will introduce the basic principles of CS and describe how it can be applied to some current problems in quantum imaging. These range from the very applied, such as how to take a picture using only one photon per pixel, to the very fundamental. such as how CS can give a new perspective on the uncertainty principle.
Wednesday, February 14, 2018
#### Skyscrapers for lungs: physics of termite colony respiration
Hunter King University of Akron
## Skyscrapers for lungs: physics of termite colony respiration
Hunter King University of Akron
Wednesday, February 14, 2018
In addition to underground nests in which they raise brood and cultivate fungus, many Macrotermitinae species also build large (~2m tall), geometrically complex, closed structures above ground. Surprisingly, termites only enter the mound for building or repair, and their function is presumed to facilitate transport of respiratory gases. However, the mechanisms by which this occurs are still not well understood. From direct field measurements with custom instrumentation, we find thermally driven flows inside the mounds of both Odontotermes obesus in India as well as Macrotermes michaelseni in Namibia. The thermal gradients driving these flows are not due to metabolic heating, but rather due to the interplay between mound architecture and diurnal temperature oscillation, an unusual example of a biological structure deriving useful work from the environment.
Wednesday, November 15, 2017
#### Colloidal quantum dots for light emission: phosphors, LEDs, lasing, and beyond
Oleksandr Voznyy University of Toronoto
## Colloidal quantum dots for light emission: phosphors, LEDs, lasing, and beyond
Oleksandr Voznyy University of Toronoto
Wednesday, November 15, 2017
Solution-processed nanomaterials offer low-cost alternatives to traditional bulk semiconductors. In this talk I will discuss what advantages quantum dots offer in optoelectronic applications, in particular LEDs and lasers, as well as perspectives and challenges in further improving their properties.
Recently, we have demonstrated the first implementation of a CW laser based on colloidal quantum dot thin films, after more than a decade since the first optical gain was reported in these materials. I will discuss how the understanding of electronic structure of quantum dots and its relation to optical gain motivated new chemical synthetic methods that allowed to bridge the gap between fs, ms and CW lasing. I will also talk about the next steps to building a robust QD-LED to allow for electrically injected QD laser.
Wednesday, November 8, 2017
#### Willy Wonka's Glass Factory: Why Chocolate is a Glass
Roger Loucks Alfred University
## Willy Wonka's Glass Factory: Why Chocolate is a Glass
Roger Loucks Alfred University
Wednesday, November 8, 2017
Contrary to popular belief, the windows in an old farm house are not thicker at the bottom because the glass has slowly flowed there over time. Glasses are not a supercooled liquid. Glasses are actually an example of a nonequilibrium system. In this talk, I will go over some of the basic properties that one encounters in glassy systems and then discuss one technique for handling nonequilibrium systems.
Wednesday, November 1, 2017
#### The Surface Science of Graphene Growth
Zachary Robinson SUNY Brockport
## The Surface Science of Graphene Growth
Zachary Robinson SUNY Brockport
Wednesday, November 1, 2017
Graphene, which is an atomically thin layer of graphite, was first isolated in 2004 by a research group at the University of Manchester. The discovery initiated a massive research effort into 2-dimensional materials, which have the potential to enable significant improvements in fields like high speed electronics, flexible electronics, transparent conductors, biological sensors, and others. Key to enabling future technology is understanding the physical properties and growth processes of the 2-dimensional materials. Graphene, for instance, can be grown by a variety of different techniques. Sublimation of Si from SiC and chemical vapor deposition on Cu are two such growth techniques, both of which are promising due to their ability to be scaled up to a manufacturing environment. In this talk, I will introduce some of the techniques used by surface scientists as they apply to my research studying the physical properties of the 2-dimensional material graphene.
Wednesday, October 11, 2017
#### From RIT Physics to the Navy: Alternative Career Paths for Physics Majors
Tighe Bailey RIT SoPA Alumnus, Navy Officer
## From RIT Physics to the Navy: Alternative Career Paths for Physics Majors
Tighe Bailey RIT SoPA Alumnus, Navy Officer
Wednesday, October 11, 2017
Ever wondered what other career opportunities are available for physics majors? In addition to graduate school, and working in the private sector, there exist many other job opportunities for physics graduates, among them is the United States Navy. The Navy Nuclear Propulsion Program looks for students in STEM fields to become officers in the United States Navy, working as engineers, instructors, or as officers serving aboard nuclear carriers or submarines. In addition, the Navy will pay you to go to school! Come learn about this unique and interesting opportunity from a former RIT student who has just completed the process.
Wednesday, September 27, 2017
## Materials by Design: Accelerating Discovery and Innovation Through Computational Simulations
Wednesday, September 27, 2017
The need to discover and design materials with precise functionalities is key to modern technology. The functionality requirement in new materials make atomic and nanoscale design a necessity. However, at the nanoscale, synthesis of every new material via a trial and error approach is not only arduous, but also expensive. Recent advancements in theoretical methods along with ever-increasing computational resources have altered the interplay between computation and experiment. In this context, I will elucidate the concept of Materials by Design. Complex oxides are indispensable materials in renewable energy technologies such as fuel cells, batteries, energy storage, etc. I will elaborate on our use of computational tools based on density functional theory, molecular dynamics, and kinetic Monte Carlo methods to identify the underlying mechanisms that govern the properties of nanostructured complex oxides. Fundamental understanding of these mechanisms assists in designing nanomaterials with tailored properties. Potential research opportunities for students interested in utilizing computational physics and materials science for next-generation technology development will be discussed.
Wednesday, September 6, 2017
#### Physics Colloquium
Kingston Chen, Quinton LoRe & Daniel Gysbers RIT Physics Majors
## Physics Colloquium
Kingston Chen, Quinton LoRe & Daniel Gysbers RIT Physics Majors
Wednesday, September 6, 2017
Kingston Chen (Capstone I Talk)
Title: The Roles of Math Skills and Tools in Scientific Academia and Industry
Quinton LoRe (Capstone I Talk)
Title: Measurement of Alpha Crystallin During Aggregation by Light Scattering
Daniel Gysbers (Summer REU)
Title: Jamming of 2-D Sheared Granular Particles
Abstract: Granular materials are composed of many macroscopic solid particles and display solid and fluid-like characteristics. Granular particle flow is important in the agriculture, soil, and pharmaceutical industries. One area of interest is particle behavior under shear, when forces are applied parallel to the surface causing particles to flow over each other. Our experiments study how circular particles behave during these shear periods with different confining pressures and pile heights. We measure the effects of 2-D shear over long periods by filming particles in an annulus while inducing shear from the top. We examine the velocities, bulk motion, and mean squared displacement of the particles to characterize the behavior of each system. As predicted by theory, jamming occurs more frequently at larger confining pressures or smaller pile heights.
Wednesday, May 3, 2017
#### Rethinking Introductory Physics Lab Courses
Natasha Grace Holmes Cornell University
## Rethinking Introductory Physics Lab Courses
Natasha Grace Holmes Cornell University
Wednesday, May 3, 2017
In physics education research, we are taking a scientific approach to understanding and improving how we teach physics. This starts with figuring out what it is we are trying to teach (what are our goals?), and then how we can accurately measure it. The goals of lab courses have been highly debated for decades with not much research to back up any position. In this talk, I will describe new research into the goals of lab courses, how we are measuring student progress towards those goals, and the efficacy of different approaches for achieving them. I will draw comparisons to student experiences in undergraduate research, question the role of authenticity in developing an understanding of science, and discuss finding the balance between structure and autonomy.
Friday, April 28, 2017
#### Electronic Engines - The Principles and Practice of Solar Power Conversion
N.J.Ekins-Daukes Imperial College London.
## Electronic Engines - The Principles and Practice of Solar Power Conversion
N.J.Ekins-Daukes Imperial College London.
Friday, April 28, 2017
Civilizations throughout history have sought to harness the power of the sun, a process that nature established ~3400 million years ago trough the evolution of photosynthesis. A consequence of the semiconductor technology revolution has been our ability to manufacture affordable solar cells that convert sunlight into electricity at an efficiency of up to 20%. However, the thermodynamic limit for solar power conversion sits at 87%, suggesting there is considerable scope for improvement. Using a multi-junction architecture, photovoltaic solar cell efficiencies in excess of 40% have been demonstrated and it looks likely that a 50% efficient solar cell will be achieved within the next decade. In the longer term it may be possible to develop materials that can support sequential optical transitions leading to the so called intermediate band solar cell or extract power from a hot electron distribution.
Wednesday, April 19, 2017
#### Mechanical Quantum Systems
Matthew LaHaye Syracuse University
## Mechanical Quantum Systems
Matthew LaHaye Syracuse University
Wednesday, April 19, 2017
The field of mechanical quantum systems has made important strides in the past 10 years developing the technology to elicit and study quantum properties of motion with systems that are normally well described as behaving classically. Such systems have promise as new components for burgeoning applications in quantum information and quantum-assisted sensing, and they offer the potential for explorations of fundamental topics in quantum mechanics like the quantum-to-classical divide. In my talk, I will first give an overview of this growing field. Then I will highlight ongoing work in my group to develop a particular type of mechanical quantum system - a quantum electromechanical system (QEMS) - that is composed of integrated superconducting circuity and nanomechanical elements. It is expected that such QEMS should enable the production and measurement of a variety of non-classical states of nanostructures, making these systems a potentially versatile new element for quantum processing architectures and for pursuing fundamental studies.
Wednesday, April 12, 2017
#### Optical beams with spatially variable polarization
Enrique Galvez Colgate University
## Optical beams with spatially variable polarization
Enrique Galvez Colgate University
Wednesday, April 12, 2017
When we think of the polarization of optical beams, the oscillation of the field vectors, we usually imagine every point in the beam having the same polarization. In our work we prepare and study optical beams where the polarization varies from point to point. They can be prepared via superpositions (interference) of two beams with orthogonal polarization and distinct spatial mode, with at least one mode carrying a phase singularity or vortex. We have observed the polarization of the light contorting in many ways within a beam, including in 3-dimensions, twisting and forming Mobius strips around the singularity.
Wednesday, March 29, 2017
## iOLabs and smartphones: New technologies for doing labs inside and outside of class
Wednesday, March 29, 2017
The ubiquity of smartphones that are chock-full of sensors has inspired new educational technologies for labs, such as the iOLab Wireless Lab System, the Pocket Lab, and physics-focused smartphone apps (e.g., Vieyra Software Physics Toolbox Apps, https://www.vieyrasoftware.net/). These tools provide students with more autonomy collecting their own data and designing their own experiments. During the first half of the talk, I will overview my experiences using the iOLab in Fall 2016 in University Physics 1. I will describe modified UP1 labs, demos, and "real world" outside-of-class labs that utilize the iOLab, and reflect on the pros and cons of the iOLab. The second half will be an opportunity to try out the iOLab yourself! All you need to do is (1) bring your own laptop and (2) install the free iOLab application software from http://www.iolab.science/ . I'll bring about 40 iOLabs so there should be enough for everyone. The class set of iOLabs was funded by an RIT Provost's Learning Innovations Grant.
The iOLab was developed by physicist Mats Selen and the physics education research group at the University of Illinois to improve large enrollment introductory physics lab courses. The iOLab is a bundle of physics sensors (accelerometer, wheel, gyroscope, magnetic field, voltage, etc.) that interfaces wirelessly with a laptop via a free software interface.
Wednesday, March 22, 2017
## Ultrafast Lasers for Photonics/Optics Fabrication and Optical Differentiation Wavefront Sensing for Astronomy and Freeform Metrology
Wednesday, March 22, 2017
The research on next-generation, laser-based manufacturing technologies is highly interdisciplinary, intersecting physics, novel ultrafast laser technology, metrology, precision controls, materials, and the associated laser-material interaction processes. The first part of the talk presents novel systems and physical mechanisms, processes using ultrafast lasers for the fabrication of photonic devices and micro- optics. The second part of the talk presents a new optical differentiation wavefront sensing technique based on measurements of wavefront slopes obtained by far-field spatial modulation with a binary pixelated filter inducing a linear amplitude transmission. This sensor is expected to offer phase measurement with higher spatial resolution, higher dynamic range and higher signal-to-noise ratio for freeform metrology lasers, vison, and astronomy applications.
Wednesday, March 8, 2017
#### How I Became an Expert
Wojtek Skulski University of Rochester
## How I Became an Expert
Wojtek Skulski University of Rochester
Wednesday, March 8, 2017
When I was a kid, I loved building my own radios and destroying radios of my grandparents. I wanted to keep tinkering through the rest of my life. A physics professor told me that I should become a physicist, because physicists are tinkering more than anyone else. So I became a research physicist. About fifteen years ago I started designing electronic instruments, hoping that my kid's experience will somehow be useful in my professional life. After a while I became an expert in developing research electronics. I started a small high-tech company specialized in such instruments with many potential research applications. I will describe a particular example, where our instruments are helping uncover the nature of Dark Matter, which is holding the galaxies together. Our instruments are also used on campuses by physics faculty and students. I will end the talk by describing how you can become an expert in any activity, which you consider joyful and inspiring.
Friday, March 3, 2017
#### Milestones toward topological quantum computing
Ryan Mishmash Caltech
## Milestones toward topological quantum computing
Ryan Mishmash Caltech
Friday, March 3, 2017
Ordinarily, exchanging two identical quantum mechanical particles results in at most a sign change of the many-body wavefunction (+1 for bosons, -1 for fermions). However, certain low-dimensional topological phases of matter host quasiparticles which exhibit *non-Abelian* statistics: exchanging two such particles gives rise to a nontrivial unitary (matrix) rotation. Such particles — termed non-Abelian anyons — form the building blocks of a type of quantum computer which is fundamentally immune to noise at the hardware level: the topological quantum computer. In this talk, I will discuss our recent theoretical work which aims to bring the topological quantum computer closer to experimental reality by proposing a series of relatively short-term “milestone” experiments on 1D quantum wires believed to harbor *Majorana zero modes*, exotic quasiparticles which give rise to non-Abelian statistics (as well as obey the famous Majorana commutation relations). I will conclude by discussing other past, present, and future research on various problems involving exotic many-body phenomena pertinent to present day experiments on quantum condensed matter systems.
Friday, February 24, 2017
#### Computational materials design of next-generation nanostructured ceramic oxides
Pratik P. Dholabhai University of Colorado, Boulder
## Computational materials design of next-generation nanostructured ceramic oxides
Pratik P. Dholabhai University of Colorado, Boulder
Friday, February 24, 2017
Recent advancements in theoretical methods along with ever-increasing computational resources have altered the interplay between computation and experiment. Computational materials design offers the possibility of predicting fundamental physical and chemical attributes of existing and new materials, and thereby facilitating design of advanced materials with tunable properties before actual synthesis in a laboratory. Nanostructured complex ceramic oxides are pervasive in diverse technologies and have wide-ranging applications. I will elaborate on our recent work that entails investigation of fundamental structure-property relationships at heterointerfaces, grain boundaries, surfaces, and solids of ceramic oxides. The complementary nature of different atomistic simulation methods such as first-principles density functional theory, molecular dynamics, and kinetic Monte Carlo will be discussed. I will further demonstrate how these different simulation methods assist in elucidating the underlying mechanisms that control the properties of ceramic oxides, and lead to nanoscale design of ceramic oxides for renewable energy applications. I will briefly address how our theory-experiment collaboration has served as an effective strategy to accelerate materials discovery and design. Finally, I will give an outlook regarding the promise of computational materials design and offer a glimpse of my future research direction.
Wednesday, February 22, 2017
#### First-principles Simulation of Electron Localization in Real Materials
Chinedu Ekuma Naval Research Laboratory
## First-principles Simulation of Electron Localization in Real Materials
Chinedu Ekuma Naval Research Laboratory
Wednesday, February 22, 2017
There is a long history of theoretical research into electron localization. The majority of this work focuses on either disorder-induced localization or localization caused by electron interactions. These limiting cases were predicted by Philip Anderson and Nevill Francis Mott and are nowadays known as Anderson and Mott localization, respectively. We also know that both disorder and electron interactions can be substantial in real materials, especially in low-dimensional materials where electronic polarization is less effective in reducing the long-range electron interactions. Alongside experiment and theory, computation has become an essential part of the development of an understanding of many properties in real pristine solids. Despite the need, first-principles-based computer simulations of electron localization in real materials that properly’’ characterizes electron localization have been elusive because both disorder and electron interactions break two of the fundamental assumptions in band theory, material homogeneity, and independent particles. In this talk, I will present a new computational approach overcoming these roadblocks by combining first-principles density functional theory, the Anderson-Hubbard model, and the typical medium dynamical cluster approximation within the dynamical mean-field theory. The computer simulations enabled by this method are expected to reveal new critical insight, e.g., simulations of monolayer hexagonal boron nitride predict that both disorder and electron interactions are essential for the material to undergo an insulator-to-metal transition.
Tuesday, February 7, 2017
#### Flowing, squeezing, clogging, and jamming of oil droplets
Eric Weeks Emory University
## Flowing, squeezing, clogging, and jamming of oil droplets
Eric Weeks Emory University
Tuesday, February 7, 2017
We use quasi-two-dimensional emulsions as experimental models to study the flow of jammed materials. Our emulsions are oil droplets in water and are compressed between two parallel glass plates so that the droplets are deformed into pancake-like disks. We use microscopy to observe these droplets as they flow. From the deformed outlines of the droplets, we can measure all of the inter-droplet forces to within 10%. In this way, we study the relationship between the local stresses in the system and the rearrangements as the sample is sheared. The simplest rearrangement involves four droplets (a ‘T1 event’) and we confirm theoretical predictions for the quadrupolar spatial pattern of the stress redistribution around the T1 events. We also study gravity-driven flow in hoppers and investigate the probability of clogging as a function of the hopper exit size. Here, experiments and simulations show that the softness of the particles is important, as soft particles form less stable arches and thus reduce the probability of clogging.
Wednesday, December 7, 2016
## Symmetry in Theoretical Physics: From Newton to the Standard Model to GUTs to SUSY and Beyond
Wednesday, December 7, 2016
Classical physics includes many important conservation laws which were established through experiment. Looking back these conservation laws can be explained as the consequence of symmetry. For the advancement of Modern Physics in the early Twentieth Century, theory, including applications of symmetry, played a vital role in the understanding of relativity, particle spin and anti-matter. Theoretical considerations of symmetries and broken symmetry led the development of the Standard Model of Elementary Particles, and these same considerations provide tantalizing clues to what may yet be discovered. This discussion will provide a high-level overview of these concepts without all of the mathematics and technical complications.
Wednesday, November 16, 2016
#### How might Physics Education Research facilitate the coming computational revolution?
Danny Caballero Michigan State University
## How might Physics Education Research facilitate the coming computational revolution?
Danny Caballero Michigan State University
Wednesday, November 16, 2016
Hosted by the School of Physics & Astronomy, SMERC & CASTLE
Computation has revolutionized how modern science is done. Modern scientists use computational techniques to reduce mountains of data, to simulate impossible experiments, and to develop intuition about the behavior of complex systems. Much of the research completed by modern scientists would be impossible without the use of computation. And yet, while computation is a crucial tool of practicing scientists, most modern science curricula do not reflect its importance and utility. In this talk, I will discuss the urgent need to construct such curricula in physics and present research that investigates the challenges at a variety of all scales -- from the largest (institutional structures) to the smallest (student understanding of a concept). I will discuss how the results of this research can be leveraged to facilitate the computational revolution. This research will help us understand and develop institutional/departmental incentives, effective teaching practices, evidence-based course activities, and valid assessment tools.
Wednesday, November 2, 2016
#### Cell Mechanics: How Cells Regulate Force Generation
Patrick Oakes University of Rochester
## Cell Mechanics: How Cells Regulate Force Generation
Patrick Oakes University of Rochester
Wednesday, November 2, 2016
: In the absence of mechanical interactions, cells would mostly be just round spheres, unable to engineer even the most rudimentary shape changes that are necessary in physiological processes like migration. The cell’s ability to alter its shape is built upon the capacity to coordinate processes like adhesion, polymerization and contraction events in both space and time. In particular, while we know significant amounts about the biochemical interactions that allow cells to generate forces, we have surprisingly little knowledge of how these molecular interactions are integrated to produce contractile behavior at the scale of the cell. In this talk I will discuss how we can use approaches from physics to describe the cell cytoskeleton as a material with dynamic properties that help to regulate this contractile behavior in adherent cells.
Wednesday, October 26, 2016
#### A Physicist's Perspective on Closed Traumatic Brain Injury and its Mitigation
Eric Blackman University of Rochester
## A Physicist's Perspective on Closed Traumatic Brain Injury and its Mitigation
Eric Blackman University of Rochester
Wednesday, October 26, 2016
Brain injury without skull fracture, called "closed traumatic brain injury” (TBI), is a large public health problem that affects soldiers and civilians of all ages. Blast waves and head Impacts are sources of brain injury in military contexts while impacts are dominant in civilian contexts.
In sports, impact-induced TBI results from both concussive and repeated sub-concussive head impacts, the latter only manifesting as long term brain damage with present diagnostics. I will first highlight the history of evidence for TBI in both military and sports contexts. I will then discuss progress and challenges in understanding mechanisms of how the brain is injured. I will explain why current helmets are deficient from a physics perspective and offer some simple recommendations for both improving helmets, along with strategies for interdisciplinary research in both helmet protection and connecting physics to physiology. Physics training is essential for tackling some frontier aspects of this enterprise.
Friday, October 21, 2016
#### Raman Spectroscopy: Using Light to Speed Up Medical Diagnosis
Dustin Shipp University of Nottingham
## Raman Spectroscopy: Using Light to Speed Up Medical Diagnosis
Dustin Shipp University of Nottingham
Friday, October 21, 2016
Among women, breast cancer is the second most common type of cancer after skin cancer. The tumor is often removed through surgery. For many patients, this surgery can remove the tumor and preserve much of the surrounding tissue. However, over 20% of these patients must undergo a second surgery to remove residual cancer tissue. This high re-excision rate is largely due to the inherently slow process of evaluating the tumor margins of the removed tissue.
Raman spectroscopy may provide a solution to this problem. Raman spectroscopy is an optical method that non-invasively measures chemical concentrations. This spectral fingerprint can be used to diagnose biological tissues and cells. This talk will discuss the creation of a diagnostic model based on Raman spectroscopy. The measurement procedure can be accelerated through the use of multi-modal imaging, image processing, and multifocal spectroscopy. Spatially-offset Raman spectroscopy (SORS) could also allow future systems to probe deeper into tissues. The talk will also discuss the integration of this system into current clinical practice as well as the opportunities and challenges of collaborating with medical professionals.
Wednesday, October 19, 2016
#### Thinking about round cows: Introductory Physics for Life Science Students
Dawn Meredith University of New Hampshire
## Thinking about round cows: Introductory Physics for Life Science Students
Dawn Meredith University of New Hampshire
Wednesday, October 19, 2016
The introductory physics course for life science majors (IPLS) has been the focus of reform over the last 10 years. I will talk about the challenges and successes in teaching this course, and why it should not be just an algebra-based version of the course for engineers. I will also talk about our focus on developing and assessing tutorials on static and moving fluids, and how we used the resources framework to inform our development and recognize productive student work.
Friday, September 30, 2016
#### Ab-Initio Study of Electron Localization in Low-Dimensional Materials
Chineda Ekuma Naval Research Laboratory / Future Faculty Career Exploration Program
## Ab-Initio Study of Electron Localization in Low-Dimensional Materials
Chineda Ekuma Naval Research Laboratory / Future Faculty Career Exploration Program
Friday, September 30, 2016
Correlated materials are promising for exploring the possibility of engineering new or improved materials as to meet the demands of the 21st century. Because their properties emerge from rather a complex competition between the electron degrees of freedom often on different length scales that can be tuned to improve device performance. Materials are often prone to defects and at the nanoscale, these inhomogeneities appear to be intrinsic. Besides, extrinsic doping can lead to a better understanding of the ground state properties of materials. To study disordered materials, the density functional theory using supercell approach (DFT-Sc) or the single-site coherent potential approximation (CPA) and cluster extensions are the frequently used computational methods. While the CPA and its extensions deal explicitly with algebraically, average statistical disorder distributions, the DFT-Sc can only describe ordered defect structures. Because CPA self-consistency uses arithmetically, averaged density of states (DoS), it does not adequately account for rare events that induce electron localization, e.g., in disordered materials. In this talk, I will discuss a first-principles, Typical Medium Dynamical Cluster Approximation (TMDCA@DFT) that appropriately characterizes disordered materials. The TMDCA@DFT, unlike the CPA, uses the typical DoS defined as the geometric average of the local DoS as the intrinsic order parameter for characterizing localization transitions.
Wednesday, September 21, 2016
#### Plasma, Fusion and PPPL: The Quest for Making a Star on Earth and info on Princeton Conference on UG Women in Physics
Arturo Dominguez Princeton Plasma Physics Laboratory
## Plasma, Fusion and PPPL: The Quest for Making a Star on Earth and info on Princeton Conference on UG Women in Physics
Arturo Dominguez Princeton Plasma Physics Laboratory
Wednesday, September 21, 2016
The challenge of developing sustainable, safe, environmentally friendly sources of energy is one of the most important scientific endeavors of the modern world. At the Princeton Plasma Physics Laboratory, research is being conducted on various fields of plasma physics, including the primary mission of the lab, the development of fusion energy as an alternative energy source. This presentation will discuss the physics of fusion plasmas, the challenges towards the goal of a fusion future, and the opportunities available at PPPL for research, including undergraduate internships.
Wednesday, September 7, 2016
## Student Summer Research Experiences
Wednesday, September 7, 2016
2016 Summer Research Experiences
Christian Cammarota
Surface Structure and Composition: Gas Phase Catalyst Creation and Measurement
Kellianne Kornick
The Population Dynamics of Mitochondria in Mammalian Cells
Roland Sanford
Noninvasive Electrocardiographic Imaging in Localizing Atrial Arrhythmia Sources
Tuesday, April 19, 2016
#### Spintronics: Fundamentals and Applications
Alex Matos Abiague SUNY Buffalo
## Spintronics: Fundamentals and Applications
Alex Matos Abiague SUNY Buffalo
Tuesday, April 19, 2016
The need for faster, more powerful and yet more efficient devices has led to the emergence of Spintronics (or spin electronics) as an alternative to conventional electronics. Unlike conventional electronic devices, which rely on the transport of electrical charge carriers, spintronic devices use electron spins for building operational functionalities such as nonvolatile information storage, sensing, and logics. Typically, spins are manipulated by external magnetic fields, but solid state materials offer a great potential for all-electric spin manipulation by means of effective, momentum-dependent “magnetic” fields, the so-called spin-orbit fields. Those fields account for spin-orbit interactions and can be exceptionally large in some materials, which, as a consequence, exhibit exotic topological properties.
This talk will focus on the physical origin, characterization, and engineering of interfacial and synthetic spin-orbit fields and their effects on anisotropic magnetoresistive phenomena and topological quantum matter. Key implications of spin-orbit-mediated transport for a new generation of nonvolatile devices, as well as present challenges in their way to applications will also be addressed.
Thursday, April 14, 2016
#### Rational design of electrode materials for energy storage
Roberto Longo Pazos University of Texas, Dallas
## Rational design of electrode materials for energy storage
Roberto Longo Pazos University of Texas, Dallas
Thursday, April 14, 2016
For over 20 years, Li-ion batteries have enabled the rise of portable electronics, dominating the battery market. Current Li-ion batteries use layered oxides as cathode materials, specially LiCoO2, organic liquid electrolytes and graphite as anode. However, Co layered oxides and organic liquid electrolytes suffer from certain instability at high operational temperatures and flammability, respectively. In this colloquium, using first principles density-functional theory, I will examine the main characteristics of the most promising alternatives for electrode and solid-electrolyte materials, suggesting suitable pathways to improve their conceptual design and performance, thus serving as design principles for future discovery of electrode materials.
Wednesday, April 13, 2016
#### Spontaneous parametric down conversion with a depleted pump as an analogue for black hole evaporation/particle production
Paul Alsing Air Force Research Laboratory
## Spontaneous parametric down conversion with a depleted pump as an analogue for black hole evaporation/particle production
Paul Alsing Air Force Research Laboratory
Wednesday, April 13, 2016
In this talk, I argue that black hole evaporation/particle production has a very close analogy to the laboratory process of spontaneous parametric down conversion, when the laser pump source is allowed to deplete. I will first present an overview of the essential features of the Unruh and Hawking effect and its analogy to the quantum optical process of spontaneous parametric down conversion widely used in the field of quantum information science as a source of photon-based qubits. In the previous case, the pump is considered a constant (i.e. non-depleted). I will next discuss the case when the black hole is treated as a finite `pump' source which is allowed to deplete, hence modeling the processes of black hole evaporation and Hawking radiation production. This model reproduces essential features of the Page Information curves (conjectured by D. Page, 1993) which are widely believe to describe the rate at which information escapes from the black hole as it evaporates, as the Hawking radiation deviates at late times from the pure thermal spectrum characterized by early black hole evolution times. Further details of this work can be found in (i) P.M. Alsing: Class. & Quant. Grav. 32, 075010, (2015); (arXiv:1408.4491), and (ii) P.M. Alsing & M.L. Fanto: Class. & Quant. Grav. 33, 015005 (2016), (arXiv:1507.00429).
Wednesday, April 6, 2016
Yan Wang MIT
## First-principles theory-driven materials design and innovation in all-solid-state batteries
Yan Wang MIT
Wednesday, April 6, 2016
Emergent energy technologies are critically limited by materials performance and therefore highly dependent on materials innovation. The efficient design and discovery of new functional materials with desired performance represent formidable challenges to materials scientists. The ability to accurately predict key materials properties using first-principles computational methods, even before the materials synthesis and characterization, has made virtual materials design a reality. However, successful computation-based materials design often requires theoretical insights on the appropriate descriptors (genes) for the identification of candidate materials. In this seminar, I will present our recent efforts in theory-driven materials design and innovation in lithium superionic conductors and all-solid-state batteries using the computational materials genome approach. I will share our research breakthrough in theoretical identification of the gene for good ionic conductors by revealing the fundamental relationship between structural topology and ionic transport. Furthermore, an accurate and efficient first-principles computational methodology has been developed to evaluate thermodynamic stability of the solid-state electrolyte against electrodes at the battery interfaces. These findings not only provide valuable insights towards the understanding of materials behaviors in discovered ionic conductors, but also serve as design principles for new ionic conducting materials and all-solid-state batteries. Finally I will give an outlook on the potential of computational materials design for novel energy technologies.
Wednesday, March 9, 2016
#### Thin-film Electronics by Spatial ALD: Achieving High Performance with Low Process Complexity
Shelby F. Nelson Eastman Kodak Company
## Thin-film Electronics by Spatial ALD: Achieving High Performance with Low Process Complexity
Shelby F. Nelson Eastman Kodak Company
Wednesday, March 9, 2016
Patterning thin-film transistors for “printed electronics” applications can be challenging both for resolution and for alignment accuracy. This is particularly true for high-performance devices with submicron channel lengths, and for diverse and deformable substrates. Printing organic-based devices has additional issues such as printing dynamics, and orthogonality of solvents. In this talk, I will describe alternative approaches to scalable thin-film electronics based on spatial atomic layer deposition (SALD) of metal oxides. Using the relatively high deposition speed of SALD, the conformality of the deposited layers, and the surface-sensitivity of the technique, we have explored both print-compatible high-performance vertical transistors, and patterned-by-printing circuitry. A reliable ZnO mobility above 10 cm2/Vs, on-off ratio above 107, and uniform threshold voltage values across the substrate give these approaches promise for large-area applications.
Wednesday, February 24, 2016
#### The Dawn of Gravitational Wave Astronomy
John T. Whelan RIT School of Mathematical Sciences and Center for Computational Relativity and Gravitation
## The Dawn of Gravitational Wave Astronomy
John T. Whelan RIT School of Mathematical Sciences and Center for Computational Relativity and Gravitation
Wednesday, February 24, 2016
Gravitational waves are ripples in the geometry of space and time which propagate at the speed of light, predicted by Einstein's General Theory of Relativity. Last fall, the Advanced LIGO detectors in Louisiana and Washington State began their first observing run, resulting in the recently reported first direct detection of gravitational waves, from a binary black hole inspiral, merger and ringdown. This first gravitational wave observation kicks off the field of Gravitational Wave Astronomy. I will present an overview of: 1) the science done so far with detectors such as LIGO and the Virgo and GEO600 detectors in Italy and Germany, 2) the most promising prospects for future observations with Advanced LIGO, Advanced Virgo, and planned detectors such as KAGRA (Japan) and LIGO India, and 3) the involvement of RIT scientists in the gravitational-wave enterprise.
Wednesday, December 2, 2015
#### The benefits of low permeability in articular cartilage
Mark Buckley University of Rochester
## The benefits of low permeability in articular cartilage
Mark Buckley University of Rochester
Wednesday, December 2, 2015
Abstract: Articular cartilage is a durable, load-bearing, poroelastic tissue that coats bones in joints and protects them from damage over several decades. Unfortunately, trauma-induced cartilage cell (chondrocyte) death can initiate a cascade of degradative alterations in the joint. In this talk, I will discuss how a key property of articular cartilage – its hydraulic permeability – mediates its ability to withstand extreme forces in two distinct ways. First, since exudation of fluid facilitates tissue compression, low permeability delays tissue deformation and allows chondrocytes to survive brief periods under extreme loads that would otherwise be fatal. Second, because permeability is low in articular cartilage and further decreased when the tissue is compressed, harmful intracellular contents that are released in areas where cell death occurs always flow towards the point of contact rather than towards healthy, uninjured cells. This protective feature could prevent the spread of cell death and contribute to the durability of articular cartilage. Finally, I will discuss a separate phenomenon related to cartilage longevity that our experiments have recently revealed: the ability of chondrocytes to adapt to recently imposed physical forces and thereby become less susceptible to subsequent mechanical injury.
Friday, August 3, 2018
## RIT Observatory Open House
Friday, August 3, 2018
It's time for Mars! This summer, the Earth will catch up to Mars as they orbit around the Sun. The orbit of Mars is not a perfect circle: sometimes it's closer to the Sun (and the Earth), sometimes it's farther away. This summer, Mars will be nearly as close to the Sun -- and the Earth! -- as it ever gets. That means that this summer is the best time to view Mars in many years!
The RIT Observatory will hold two Open Houses specially focused on Mars -- but Jupiter and Saturn will also be easy to see. Come and join us to get a good look at the Red Planet and some of its friends.
Friday, July 20 (rain date Sat, July 21): 9:30 - 11:00 PM
Friday, Aug 3 (rain date Sat, Aug 4): 9:30 - 11:00 PM
Check the AST Observatory Website: https://www.rit.edu/cos/observatory/ a day before each event for a status update. Remember that we can't see other planets when the skies are covered with clouds ...
Friday, July 20, 2018
## RIT Observatory Open House
Friday, July 20, 2018
It's time for Mars! This summer, the Earth will catch up to Mars as they orbit around the Sun. The orbit of Mars is not a perfect circle: sometimes it's closer to the Sun (and the Earth), sometimes it's farther away. This summer, Mars will be nearly as close to the Sun -- and the Earth! -- as it ever gets. That means that this summer is the best time to view Mars in many years!
The RIT Observatory will hold two Open Houses specially focused on Mars -- but Jupiter and Saturn will also be easy to see. Come and join us to get a good look at the Red Planet and some of its friends.
Friday, July 20 (rain date Sat, July 21): 9:30 - 11:00 PM
Friday, Aug 3 (rain date Sat, Aug 4): 9:30 - 11:00 PM
Check the AST Observatory Website: https://www.rit.edu/cos/observatory/ a day before each event for a status update. Remember that we can't see other planets when the skies are covered with clouds ...
Wednesday, April 25, 2018
## Low-cost fundus camera for clinical use
Wednesday, April 25, 2018
Diabetic retinopathy is the leading cause of blindness in working age adults in the US and affects millions more worldwide. Annual dilated retinal eye exams with an eye doctor allow for timely treatment and proper follow-up care and substantially reduces the risk of blindness.
Unfortunately, the majority of diabetic patients, especially those from lower income and underserved communities, do not have the recommended annual eye exam due to reasons such as limited access, cost, and need for the pupil to be dilated for an eye exam. Teleophthalmology is the use of fundus cameras in non-eye care settings to capture digital images of the central part of the retina, often in an undilated pupil, and remote evaluation of these images for pathology by eye doctors who report findings and recommend appropriate follow-up care. Such programs have successfully increased timely detection of retinopathy and helped educate individuals on how diabetes affects their bodies to encourage improved adherence to recommended care. A major barrier to the growth of teleophthalmology is the high cost of currently available, effective fundus cameras.
The multi-disciplinary project team will design and construct a low-cost fundus camera that will not require pupil dilation. The team includes a Physics major with optics expertise, Mechanical, Electrical, Biomechanical, and Industrial & Systems Engineers, an Industrial Designer, and Photographic Sciences students. The deliverable is a functioning prototype, which will be demonstrated at the ImagineRIT festival, that may lead to a commercialized product helping to end blindness for millions worldwide.
Wednesday, March 21, 2018
#### Sailing on a Rainbow: Verification of Radiation Pressure on a Diffraction Grating
Lucy Ying-ju Chu RIT Imaging Science PhD student
## Sailing on a Rainbow: Verification of Radiation Pressure on a Diffraction Grating
Lucy Ying-ju Chu RIT Imaging Science PhD student
Wednesday, March 21, 2018
Sailcraft make use of radiation pressure to propel a payload through space. Modern diffractive structures such as broadband single order gratings, polarization diffraction gratings, and related “metamaterials” offer the potential to replace reflective sails with efficient diffractive sails for either solar or laser driven space travel. Possibly for the first time, we have experimentally measured and verified the radiation pressure force on a transmissive diffraction grating, demonstrating a large component of force parallel to the surface of the grating. This component is important for orbit-raising types of maneuvers, and opens potential new opportunities for optical levitation experiments and applications.
Wednesday, February 14, 2018
#### How Thin Films Augment Optics
Christopher Chinhong Corning Incorporated
## How Thin Films Augment Optics
Christopher Chinhong Corning Incorporated
Wednesday, February 14, 2018
Optical thin film coatings utilize the wave nature of light via interference. By applying a series of nanometers thick layers, the performance of an optical system will be improved beyond the limitations set by the Fresnel equations. The applications and manufacturing of optical coatings will be discussed with examples of coated optics.
Tuesday, November 14, 2017
#### Diffraction Grating Technology: Enabling Everything from Smartphones, to Increases in Network Bandwidth, to Earthlike Exosolar Planet Discoveries
Jason Rama Richardson Grating Lab/MKS Instruments Inc.
## Diffraction Grating Technology: Enabling Everything from Smartphones, to Increases in Network Bandwidth, to Earthlike Exosolar Planet Discoveries
Jason Rama Richardson Grating Lab/MKS Instruments Inc.
Tuesday, November 14, 2017
Although the principles of diffraction gratings have been known for centuries, these spectrally dispersing devices have recently become an ever more critical component to clarify our understanding of the cosmos, enable microelectronic (and optoelectronic) technology, and equip a broad assortment of instrument technologies for chemistry analyses. They are present anywhere a range of optical spectra must be either generated or detected in a precise way. In this talk, the technology, fabrication, figures of merit, design requirements, markets, and applications of diffraction gratings will be touched upon, all with a mind for critical business and technology commercialization considerations.
Wednesday, October 18, 2017
#### An Introduction to Ursa Space Systems, Inc., and SAR Analytics
Mat DePasquale Ursa Space Systems
## An Introduction to Ursa Space Systems, Inc., and SAR Analytics
Mat DePasquale Ursa Space Systems
Wednesday, October 18, 2017
Ursa Space Systems, Inc. is a technology startup which aggregates and analyzes satellite-based radar to build proprietary data layers resulting in unbiased, consistent, all-weather locationbased services and measurements. Ursa is bridging the gap between advanced SAR analytics and commercial customers. This is an introduction to the company that includes products, technologies, and opportunities for students in various technical disciplines.
Wednesday, September 13, 2017
## Novel Materials and Nanostructures for Photovoltaic Energy Conversion
Wednesday, September 13, 2017
The world demand for, and consumption of, energy is dramatically increasing, with an increasing demand for renewable non-fossil based sources of electricity. As well, there is an ever growing demand for increased power and sophistication in the satellite systems orbiting our planet, driven by our increasing reliance on high speed communication and data links. The conversion of light from the sun into electrical energy, using photovoltaics, is one avenue that can be explored to meet these challenges both on the earth and in space, with III-V’s being the most promising materials for very high efficiency devices. At RIT, our team’s expertise lies in vapor phase epitaxy (VPE) of III-V photonic devices and nanostructures, bandgap engineering using epitaxial nanostructures, novel photovoltaic devices such as the intermediate band solar cell and potential routes for low cost high efficiency III-V multijunction devices. This talk will give an overview of PV research at RIT, a discussion of the nanomaterials approach and specific results using quantum dot (QD) superlattices, Sb-based photovoltaic materials development and finally some recent results on developing low-cost substrates for III-V materials. During the talk, we will show the effects of QD solar cell design on both absorption and open circuit voltage and discuss the nature of carrier escape and recombination paths inherent to QD solar cells. As well, we will discuss the growth and processing of an InAlAsSb alloy for photovoltaic applications as well as GaSb solar cells grown on GaAs using the interfacial misfit (IMF) technique. The last topic will include a cost breakdown of III-V photovoltaics and recent results using polycrystalline Ge as a template for growth of III-V solar cells.
Wednesday, April 26, 2017
#### Polarized Vision for Astronomers and Other Humans
Dmitry Vorobiev RIT
## Polarized Vision for Astronomers and Other Humans
Dmitry Vorobiev RIT
Wednesday, April 26, 2017
Humans have long benefitted from our ability to distinguish light of different frequency based on its color. Sadly, our eyes are not sensitive to the polarization of light. As a result, we are far less familiar with the utility of polarimetric measurement. Devices to measure polarization are relatively rare and expertise in polarimetry even more so. Polarization sensors based on micropolarizer arrays appear to be the first devices capable of bringing polarimetric capability to a wide range of applications. Based on a combination of theoretical models and lab-based measurements, I conclude that the current generation of these devices can measure fractional polarization as small as 0.5%, across the visible and near-infrared spectrum, with potential for further improvement. I will present some astronomical observations acquired with the RIT Polarization Imaging Camera and end with a discussion of applications outside of astronomy that are well-positioned to benefit from these sensors.
Wednesday, March 8, 2017
#### Making a go of it with licensed technology – Low Coherence Interferometry from Kodak and Shack-Hartmann from AMO-Abbott
David Compertore Lumetrics
## Making a go of it with licensed technology – Low Coherence Interferometry from Kodak and Shack-Hartmann from AMO-Abbott
David Compertore Lumetrics
Wednesday, March 8, 2017
Formed in March 2003 Lumetrics grew from 3 employees to 20 originally on the strength of the LCI product and in 2012 adding the Shack-Hartmann line. Lumetrics is a “Photonics” company serving a diverse customer base including medical, industrial, electronics, automotive, food packaging, glass, adhesives markets. Lumetrics one main optics market is the intraocular & contact lens ophthalmic industry, an industry known for making purposefully “bad” optics.
Wednesday, February 8, 2017
#### Manufacturing High Precision Optics using MRF and SSI Technologies
Chris Maloney QED Technologies, Inc
## Manufacturing High Precision Optics using MRF and SSI Technologies
Chris Maloney QED Technologies, Inc
Wednesday, February 8, 2017
QED Technologies was founded about 20 years ago, leveraging Magnetorheological Finishing (MRF) and Subaperture Stitching Interferometry (SSI) technologies that were developed at the University of Rochester. These two technologies for polishing and metrology provide a deterministic method for manufacturing high precision spheres, aspheres and freeform optics. In this seminar we will discuss the theory behind MRF and SSI technologies as well as how they are used in the optical manufacturing industry.
Thursday, November 17, 2016
#### Advances in Manufacturing and Metrology of Optical Freeform Surfaces
Ian Ferralli and Michael Rinkus RIT Alumni, Optimax
## Advances in Manufacturing and Metrology of Optical Freeform Surfaces
Ian Ferralli and Michael Rinkus RIT Alumni, Optimax
Thursday, November 17, 2016
Designing optical systems using freeform optical components can provide many advantages to the optical designer such as fewer optical components and less distortion. Techniques for manufacturing these complex geometries are advancing very quickly with increasing demand. Additionally, the metrology of freeform optics has progressed enabling higher precision surfaces to be made. We will explain the ongoing research efforts at Optimax that enable us to be at the forefront of optical manufacturing capability.
Wednesday, September 14, 2016
## Measuring the Largest Structures in the Universe with the Smallest Telescopes in Space
Wednesday, September 14, 2016
Observational astrophysics has always been driven by the race to build telescopes with larger and larger apertures. However, telescopes with very small apertures can perform cosmological measurements as important as their larger siblings. In this talk, I will present several examples of small aperture, space-based experiments providing unique views on the large scale structure of the universe. My discussion will include The Cosmic Infrared Background Experiment (CIBER) that has successfully measured the amplitude of the near-IR background fluctuations on arcminute scales; SPHEREx, a spectrometric instrument designed to probe the inflationary history of the universe and the evolution of galaxies; and work using the Long Range Reconnaissance Imager (LORRI) on New Horizons to measure the cosmic optical background.
Thursday, April 21, 2016
#### Field E ect Electro-Absorption Modulator Based on Conductive Oxide
Kaifeng Shi Novel Material Photonics Group Rochester Institute of Technology
## Field E ect Electro-Absorption Modulator Based on Conductive Oxide
Kaifeng Shi Novel Material Photonics Group Rochester Institute of Technology
Thursday, April 21, 2016
The lack of ultracompact, high speed, broadband electro-optical (EO) modulators impedes the wide applications of integrated photonic circuits. Novel approaches and materials need to be explored to overcome the technical barrier. In this talk, I will present an EO mod-ulator, more speci cally electro-absorption (EA) modulator, based on a novel yet inexpensive active material, conductive oxide (COx), which exhibits moderate carrier concentration for tele-com application. Light modulation is realized through the eld e ect in a metal-insulator-COx(MIC) structure. Dielectric constant epsilon-near-zero (ENZ) state is observed. Furthermore, we investigate an MICIM plasmonic EA modulator with a waveguide length of only 800 nm. The modulator can potentially operate at high speed.
Tuesday, March 15, 2016
#### Quantum Integrated Photonics: A source of spectrally indistinguishable photons
Michael L. Fanto Nanophotonics Group @ RIT
## Quantum Integrated Photonics: A source of spectrally indistinguishable photons
Michael L. Fanto Nanophotonics Group @ RIT
Tuesday, March 15, 2016
Quantum information science relies on the property of quantum interference, where the interference quality correlates to the indistinguishability of the interacting particles. The creation of these indistinguishable particles, photons in this case, has conventionally been accomplished with nonlinear crystals and optical filters to remove spectral distinguishability, albeit sacrificing the number of photons. This research describes the use of an integrated silicon microring resonator circuit to selectively generate photon pairs at the narrow cavity transmissions, thereby producing spectrally indistinguishable photons, and then entangle the resulting photon pair.
Wednesday, February 17, 2016
#### Designing a spatial mode sorting interferometer
Tanya Malhotra University of Rochester
## Designing a spatial mode sorting interferometer
Tanya Malhotra University of Rochester
Wednesday, February 17, 2016
The ability to decompose an optical beam/scene into a specific modal basis is desirable in a wide array of optical technologies. By generalizing the delay line in a conventional Michelson interferometer to an arbitrary unitary transformation, it becomes possible to unlock the full mode sorting ability of the interferometer. Specifically, the eigenfunctions of the generalized delay line are the basis in which beam modal analysis is possible. In the following we describe an approach to arbitrary spatial mode sorting based on two-path interferometry. Our proof-of-principle mode sorting demonstration is based on the fractional Fourier transform (fFT). When replacing the conventional temporal delay line in the interferometer with an optical implementation of the fFT, the interferometer is able to decompose an input optical beam in terms of its constituent HG modes which are the fFT eigenmodes.
Wednesday, January 31, 2018
#### The Exploration of Pluto and the Kuiper Belt by New Horizons
Harold Weaver Johns Hopkins University
## The Exploration of Pluto and the Kuiper Belt by New Horizons
Harold Weaver Johns Hopkins University
Wednesday, January 31, 2018
The New Horizons (NH) mission was selected by NASA in November 2001 to conduct the first in situ reconnaissance of Pluto and the Kuiper belt. The NH spacecraft was launched on 2006 January 19, received a gravity assist from Jupiter during closest approach on 2007 February 28, and flew 12,500 km above Pluto's surface on 2015 July 14. NH carried a sophisticated suite of seven scientific instruments, altogether weighing less than 30 kg and drawing less than 30 W of power, that includes panchromatic and color imagers, ultraviolet and infrared spectral imagers, a radio science package, plasma and charged particle sensors, and a dust counting experiment. These instruments enabled the first detailed exploration of a new class of solar system objects, the dwarf planets, which have exotic volatiles on their surfaces, escaping atmospheres, and satellite systems. New Horizons has transformed Pluto from a pixelated blob (as seen from Earth) into a complex and diverse world with water-ice mountains as high as the Rockies on Earth and exotic nitrogen-ice sheets with glacier-like flows. Charon has chasms larger than the Earth's Grand Canyon and a giant hood of dark material covering its north pole. New Horizons has resolved Pluto's small satellites (Styx, Nix, Kerberos, and Hydra) for the first time, showing them to be highly elongated objects with high albedo surfaces. NASA recently approved an extended mission phase for NH, the highlight of which is the flyby of a small (~30 km) KBO on 2019 Jan 01, enabling the study of an object in a completely different dynamical class (cold classical) than Pluto.
Friday, January 12, 2018 to Sunday, January 14, 2018
## Conference for Undergraduate Women in Physics (CUWiP)
Friday, January 12, 2018
APS Conference for Undergraduate Women in Physics (CUWiP) 2018 @ RIT
The APS Conferences for Undergraduate Women in Physics (CUWiP) are three day regional conferences for undergraduate physics majors. During January 12-14 2018, the Rochester Institute of Technology (RIT) School of Physics and Astronomy will be hosting CUWiP for students in Maine, Massachusetts (West of I-91), New Hampshire, New York (North of Poughkeepsie), Pennsylvania (Central Harrisonburg), and Vermont. RIT is located in the Eastern Standard Time zone (EST).
The goal of CUWiP is to help undergraduate women continue in physics by providing them with the opportunity to experience a professional conference, information about graduate school and professions in physics, and access to other women in physics of all ages with whom they can share experiences, advice, and ideas.
Our program will include research talks by faculty, panel discussions about graduate school, careers in physics, and inclusivity, presentations and discussions about women in physics, laboratory tours, student research talks, a student poster session, various interactive workshops and discussions, and several meals during which presenters and students interact with each other.
Full Event Schedule
Wednesday, October 19, 2016
## From Career Pathways to Physics/Astronomy Trivia
Wednesday, October 19, 2016
Come learn about physics and astronomy career options, topics such as how to fail an interview, why networking is so important, and compete in physics trivia with the Director of the Society of Physics Students and Sigma Pi Sigma! Alum Dr. Brad R. Conrad will discuss how to make the most of your time at RIT and teach a few skills that can help you along the way.
Bio: Dr. Brad R. Conrad is the Director of the Society of Physics Students (SPS) and Sigma Pi Sigma (ΣΠΣ), the physics honors society, at the American Institute of Physics (AIP) in College Park, MD. In addition to leading SPS National initiatives, he works to support and promote undergraduate Physics and Astronomy majors and their mentors. Before becoming director, Dr. Conrad was an Associate Professor of Physics and Astronomy at Appalachian State University, an NRC postdoctoral fellow at the National Institute of Standards, earned his Ph.D. in Physics at the University of Maryland, College Park MD, in the area of condensed matter/surface physics, and earned his Bachelor’s degree from Rochester Institute of Technology in Physics.
Friday, February 12, 2016
#### Observation of Gravitational Waves from a Binary Black Hole Merger
John Whelan, Richard O'Shaughnessy, Carlos Lousto, and Manuella Campanelli CCRG
## Observation of Gravitational Waves from a Binary Black Hole Merger
John Whelan, Richard O'Shaughnessy, Carlos Lousto, and Manuella Campanelli CCRG
Friday, February 12, 2016
Abstract:
LIGO has just reported three discoveries: the first direct detection of gravitational waves; the discovery of a binary black hole; and the observation of gravitational waves from this binary's coalescence, in excellent agreement with Einstein's theory of gravity. In a presentation and panel discussion, RIT scientists John Whelan, Richard O'Shaughnessy, Carlos Lousto, and Manuella Campanelli -- all members of RIT's Center for Computational Relativity and Gravitation -- discuss the significance of these findings.
For more information see the attached summary PDF, and the summaries avaialble at
- http://ligo.org/science/outreach.php
- https://losc.ligo.org/events/GW150914/
Monday, December 7, 2015
#### Vibrations from the Big Bang
Jamie Bock California Institute of Technology
## Vibrations from the Big Bang
Jamie Bock California Institute of Technology
Monday, December 7, 2015
Moments after the Big Bang, our observable universe underwent a violent growth spurt called inflation. The inflationary expansion flung apart the observable universe from a causally-connected sub-atomic volume, and established a primordial spectrum of scalar perturbations that led to the temperature anisotropies observed in the cosmic microwave background. Dr. Bock's team has been making precise degree-scale polarization measurements of the CMB from the south pole with the BICEP/Keck series of experiments in search of a distinctive ‘B-mode’ pattern, a hallmark of tensor perturbations associated with a background of gravitational waves generated by inflation. Dr. Bock will present our latest results that incorporate multi-band information from the Planck satellite and new Keck Array data at 95 and 150 GHz. He will also discuss prospects from new data and improved measurements coming in the near future.
Supported by the John Wiley Jones Science Endowment Fund
To request Interpreting Services, please visit myAccess.rit.edu.
Rochester Institute of Technology College of Science
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2018-11-22 10:22:14
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https://typemill.net/plugin-developers/tutorial/cookie-consent-plugin
|
TYPEMILL
Let's get our hands dirty and look into the cookie consent plugin. The cookie consent plugin adds a little banner to each page of a website, so that the user can agree to the website's cookie policy.
You might think, that you do not need a plugin for that. And you are right: You can simply visit the cookieconsent website, configure your cookie consent, copy the code and paste it into your theme. It is only a bit of JavaScript and CSS. The script from the cookie consent website looks like this:
<link rel="stylesheet" type="text/css" href="//cdnjs.cloudflare.com/ajax/libs/cookieconsent2/3.0.3/cookieconsent.min.css" />
<script>
"palette": {
"popup": {
"background": "#d48484",
"text": "#ffffff"
},
"button": {
"background": "#362929",
"text": "#fff"
}
},
"content": {
"dismiss": "OK",
}
})});
</script>
So what is the point to create a plugin just to add this little script to a website?
## The Problem With Hardcoding
To hardcode the cookie consent script manually into your TYPEMILL-theme has two downsides:
• As a developer, I have to touch the templates of the TYPEMILL-theme and add the cookie consent script there. And each time, if I update the theme, I have to add the script again.
• As an author or admin, I cannot change the text or the color of the cookie consent in the setup area, and I cannot activate or deactivate it. Instead I have to open the template files in a code editor and work like a developer.
Wouldn't it be much better to configure the cookie consent in the setup area of TYPEMILL and to add the cookie consent to a theme without even touching it?
Of course, so let's try it.
## How The Plugin Should Work
Before we start, let's describe, how the cookie consent plugin should work:
• The plugin should add a CSS-file into the html-head of the theme-templates.
• The plugin should add a JavaScript-file at the bottom of the theme-templates.
• After the JavaScript file, the plugin should add the initial script with the values for the colors and the content.
• And finally, the content- and color-values should be editable, so that the user can change them in the plugin settings.
## Next: Create a File Structure
In the next chapters, we will learn how we can add a cookie banner easily with a TYPEMILL plugin. Let's start with a file structure.
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2018-05-23 12:53:51
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https://solvedlib.com/0641-2210-43-10-7-2h-016-atropine-400-mog-rs-lew,352557
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"ОС 0641-2210-43 10 7 2h 。016 ATROPINE 400 mog/ rs lew s dispensing wiwn OA malmt FOR SC, IM OR
Question:
solve number 4 and number 5 please.
"ОС 0641-2210-43 10 7 2h 。016 ATROPINE 400 mog/ rs lew s dispensing wiwn OA malmt FOR SC, IM OR use 4. Ordered. Anesthesia Premedication Administer 0 8 mg IV 30 minutes before anesthesia, repe a) Drug concentration b) Route 5. c) How many mL will you prepare d) Drug form
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2022-05-17 07:07:51
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https://de.mathworks.com/help/fininst/hjmtree.html
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# hjmtree
Build Heath-Jarrow-Morton interest-rate tree
## Syntax
``HJMTree = hjmtree(VolSpec,RateSpec,TimeSpec)``
## Description
example
````HJMTree = hjmtree(VolSpec,RateSpec,TimeSpec)` creates a structure containing time and forward-rate information on a bushy tree. ```
## Examples
collapse all
Using the data provided, create a HJM volatility specification (using `hjmvolspec`), rate specification (using `intenvset`), and tree time layout specification (using `hjmtimespec`). Then use these specifications to create a HJM tree using `hjmtree`.
```Compounding = 1; ValuationDate = '01-01-2000'; StartDate = ['01-01-2000'; '01-01-2001'; '01-01-2002'; '01-01-2003'; '01-01-2004']; EndDates = ['01-01-2001'; '01-01-2002'; '01-01-2003'; '01-01-2004'; '01-01-2005']; Rates = [.1; .11; .12; .125; .13]; Volatility = [.2; .19; .18; .17; .16]; CurveTerm = [1; 2; 3; 4; 5]; HJMVolSpec = hjmvolspec('Stationary', Volatility , CurveTerm); RateSpec = intenvset('Compounding', Compounding,... 'ValuationDate', ValuationDate,... 'StartDates', StartDate,... 'EndDates', EndDates,... 'Rates', Rates); HJMTimeSpec = hjmtimespec(ValuationDate, EndDates, Compounding); HJMTree = hjmtree(HJMVolSpec, RateSpec, HJMTimeSpec)```
```HJMTree = struct with fields: FinObj: 'HJMFwdTree' VolSpec: [1x1 struct] TimeSpec: [1x1 struct] RateSpec: [1x1 struct] tObs: [0 1 2 3 4] dObs: [730486 730852 731217 731582 731947] TFwd: {[5x1 double] [4x1 double] [3x1 double] [2x1 double] [4]} CFlowT: {[5x1 double] [4x1 double] [3x1 double] [2x1 double] [5]} FwdTree: {1x5 cell} ```
Use `treeviewer` to observe the tree you have created.
`treeviewer(HJMTree)`
## Input Arguments
collapse all
Volatility process specification, specified using the `VolSpec` output obtained from `hjmvolspec`. `VolSpec` sets the number of factors and the rules for computing the volatility $\sigma \left(t,T\right)$ for each factor.
Data Types: `struct`
Interest-rate specification for initial rate curve, specified by the `RateSpec` obtained from `intenvset`. For information on the interest-rate specification, see `intenvset`.
Data Types: `struct`
Time tree layout specification, specified using the `TimeSpec` output obtained from `hjmtimespec`. The TimeSpec defines the observation dates of the HJM tree and the `Compounding` rule for date to time mapping and price-yield formulas.
Data Types: `struct`
## Output Arguments
collapse all
Time and interest-rate information of a bushy tree, returned as a structure.
Introduced before R2006a
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2020-09-18 14:30:35
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https://math.stackexchange.com/questions/3065874/understanding-binomial-coefficient-with-floored-terms
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Understanding Binomial coefficient with floored terms
I was reading through the notation used in a paper on arxiv.org when I came across this on page 6:
$$[x]$$ the floor of $$x$$
$$\{x\}$$ the sawtooth function of $$x$$. That is $$\{x\} = x - [x]$$
$$\begin{Bmatrix}x\\y \end{Bmatrix}$$ Binomial coefficient with floored terms.
Here is the explanation:
That is, $$\begin{Bmatrix}x\\y\end{Bmatrix} = \delta(y,x){{[x]}\choose{[y]}}$$
where:
• $$\delta(y,x)=1$$ if $$\{x\} \ge \{y\}$$
• $$\delta(y,x)=[x-y]+1$$ if $$\{x\} < \{y\}$$
Does this definition make sense? If so, could someone help me to understand what it means when $$\delta(y,x) \neq 1$$?
• What part doesn’t make sense to you? For example, try computing $\genfrac{\{}{\}}{0pt}{}{7.4}{3.6}$ and tell us where you get stuck. – Anders Kaseorg Jan 8 at 9:19
• $\begin{Bmatrix}7.4\\ 3.5\end{Bmatrix} = (4){7\choose3}$. I am clear on the computation. I'm not clear why the $4$ is needed. Why not just $\begin{Bmatrix}7.4\\ 3.5\end{Bmatrix} = {7\choose3}$ – Larry Freeman Jan 8 at 11:40
The author is of course free to make any definition they want, and apparently they found $$\delta(y, x)\binom{\lfloor x\rfloor}{\lfloor y\rfloor}$$ to be useful for their purposes in a way that $$\binom{\lfloor x\rfloor}{\lfloor y\rfloor}$$ alone was not. See Lemma 2.0.2 for a hint of why that might be:
$$\genfrac{\{}{\}}{0em}{}{x}{y} = \frac{\prod_{k \in (s - r, s] \cap \mathbb N} k}{\prod_{k \in (0, r] \cap \mathbb N} k} = \delta(y, x)\binom{\lfloor x\rfloor}{\lfloor y\rfloor}.$$
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2019-07-23 05:17:08
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https://www.researchgate.net/scientific-contributions/Deva-Ramanan-10600025
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# Deva Ramanan's research while affiliated with Carnegie Mellon University and other places
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Full-text available
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Full-text available
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Preprint
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Preprint
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Preprint
We introduce Argoverse 2 (AV2) - a collection of three datasets for perception and forecasting research in the self-driving domain. The annotated Sensor Dataset contains 1,000 sequences of multimodal data, encompassing high-resolution imagery from seven ring cameras, and two stereo cameras in addition to lidar point clouds, and 6-DOF map-aligned po...
Preprint
Full-text available
We focus on the task of far-field 3D detection (Far3Det) of objects beyond a certain distance from an observer, e.g., $>$50m. Far3Det is particularly important for autonomous vehicles (AVs) operating at highway speeds, which require detections of far-field obstacles to ensure sufficient braking distances. However, contemporary AV benchmarks such as...
Preprint
Full-text available
Contemporary autonomous vehicle (AV) benchmarks have advanced techniques for training 3D detectors, particularly on large-scale lidar data. Surprisingly, although semantic class labels naturally follow a long-tailed distribution, contemporary benchmarks focus on only a few common classes (e.g., pedestrian and car) and neglect many rare classes in-t...
Preprint
Full-text available
Modern neural networks are over-parameterized and thus rely on strong regularization such as data augmentation and weight decay to reduce overfitting and improve generalization. The dominant form of data augmentation applies invariant transforms, where the learning target of a sample is invariant to the transform applied to that sample. We draw ins...
Chapter
Object detection with multimodal inputs can improve many safety-critical systems such as autonomous vehicles (AVs). Motivated by AVs that operate in both day and night, we study multimodal object detection with RGB and thermal cameras, since the latter provides much stronger object signatures under poor illumination. We explore strategies for fusin...
Chapter
We describe a data-driven method for inferring the camera viewpoints given multiple images of an arbitrary object. This task is a core component of classic geometric pipelines such as SfM and SLAM, and also serves as a vital pre-processing requirement for contemporary neural approaches (e.g. NeRF) to object reconstruction and view synthesis. In con...
Chapter
Motion planning for safe autonomous driving requires learning how the environment around an ego-vehicle evolves with time. Ego-centric perception of driveable regions in a scene not only changes with the motion of actors in the environment, but also with the movement of the ego-vehicle itself. Self-supervised representations proposed for large-scal...
Preprint
We tackle the problem of novel class discovery, detection, and localization (NCDL). In this setting, we assume a source dataset with labels for objects of commonly observed classes. Instances of other classes need to be discovered, classified, and localized automatically based on visual similarity, without human supervision. To this end, we propose...
Preprint
Lifelong learners must recognize concept vocabularies that evolve over time. A common yet underexplored scenario is learning with class labels over time that refine/expand old classes. For example, humans learn to recognize ${\tt dog}$ before dog breeds. In practical settings, dataset $\textit{versioning}$ often introduces refinement to ontologies,...
Preprint
Motion planning for safe autonomous driving requires learning how the environment around an ego-vehicle evolves with time. Ego-centric perception of driveable regions in a scene not only changes with the motion of actors in the environment, but also with the movement of the ego-vehicle itself. Self-supervised representations proposed for large-scal...
Preprint
Multiple existing benchmarks involve tracking and segmenting objects in video e.g., Video Object Segmentation (VOS) and Multi-Object Tracking and Segmentation (MOTS), but there is little interaction between them due to the use of disparate benchmark datasets and metrics (e.g. J&F, mAP, sMOTSA). As a result, published works usually target a particul...
Preprint
Full-text available
We describe a data-driven method for inferring the camera viewpoints given multiple images of an arbitrary object. This task is a core component of classic geometric pipelines such as SfM and SLAM, and also serves as a vital pre-processing requirement for contemporary neural approaches (e.g. NeRF) to object reconstruction and view synthesis. In con...
Article
Real-world machine learning systems need to analyze test data that may differ from training data. In K-way classification, this is crisply formulated as open-set recognition, core to which is the ability to discriminate open-set data outside the K closed-set classes. Two conceptually elegant ideas for open-set discrimination are: 1) discriminativel...
Conference Paper
Preprint
Transformers have become prevalent in computer vision due to their performance and flexibility in modelling complex operations. Of particular significance is the 'cross-attention' operation, which allows a vector representation (e.g. of an object in an image) to be learned by attending to an arbitrarily sized set of input features. Recently, "Maske...
Preprint
Full-text available
Object detection and forecasting are fundamental components of embodied perception. These two problems, however, are largely studied in isolation by the community. In this paper, we propose an end-to-end approach for detection and motion forecasting based on raw sensor measurement as opposed to ground truth tracks. Instead of predicting the current...
Preprint
Full-text available
In the real open world, data tends to follow long-tailed class distributions, motivating the well-studied long-tailed recognition (LTR) problem. Naive training produces models that are biased toward common classes in terms of higher accuracy. The key to addressing LTR is to balance various aspects including data distribution, training losses, and g...
Preprint
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Preprint
Prior work for articulated 3D shape reconstruction often relies on specialized sensors (e.g., synchronized multi-camera systems), or pre-built 3D deformable models (e.g., SMAL or SMPL). Such methods are not able to scale to diverse sets of objects in the wild. We present BANMo, a method that requires neither a specialized sensor nor a pre-defined t...
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Recent history has seen a tremendous growth of work exploring implicit representations of geometry and radiance, popularized through Neural Radiance Fields (NeRF). Such works are fundamentally based on a (implicit) {\em volumetric} representation of occupancy, allowing them to model diverse scene structure including translucent objects and atmosphe...
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2023-04-01 03:31:57
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https://proxieslive.com/tag/delta/
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## Solutions to \$\Delta u\ge u^2\$
Let $$(M,g)$$ be a complete Riemannian manifold. Suppose that $$u$$ is a nonnegative solution to $$\Delta_gu\ge u^2$$. Does it follow that $$u$$ must be identically 0?
I know that the answer to above question is yes if one assumes that $$Ric(g)$$ has a lower bound, which allows for a maximum principle argument, using the distance function to cut-off.
I wonder if this is true in general, with no additional assumptions?
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2018-12-17 01:51:33
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https://homework.zookal.com/questions-and-answers/e10--2a-material-and-labor-variances-the-following-actual-816648993
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2. Accounting
3. e10 2a material and labor variances the following actual...
# Question: e10 2a material and labor variances the following actual...
###### Question details
E10 - 2A: Material and Labor Variances. The following actual and standard cost data for direct material and direct labor relate to the production of 2,000 units of a product:
Actual Costs Standard Costs
Direct material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3,900 lb. @ $5.30 4,000 lb. @$5.10
Direct labor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6,200 hrs. @ $8.40 6,000 hrs. @$8.70
Determine the following variances:
a. Materials price
b. Materials efficiency
c. Labor rate
d. Labor efficiency
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2021-06-14 15:14:37
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https://www.physicsforums.com/threads/do-voltage-temperature-measure-energy-fields-pressure-measure-force.706053/
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# Do Voltage/Temperature measure Energy; Fields/Pressure measure Force?
1. Aug 17, 2013
### 541099
Would it be correct to call Voltage and Temperature intensive measures of Energy, and call Electric Field and Pressure intensive measures of the potential for Force generation?
If so, is Voltage (Potential Energy) used to generate a force on a charged particle (by way of an Electrical Field), while Temperature (Kinetic Energy) is used to generate a force on a particle with mass (by way of Pressure)?
If this is also true, then what are their conjugate variables?
Am I pairing these up properly...?
Pressure...Volume
Temperature...Entropy
Voltage...Total Charge? Total Charge Flow?
Electric Field...Charge Displacement (Dipole moment?)
I'm a pre-medical student studying for the MCAT. I was reinforcing my foundations in physics and got a bit carried away. Sorry if some of the technical language isn't correct. In any case, I'm curious now.
2. Aug 19, 2013
### Khashishi
It sounds like you are too focused on the terminology rather than the concept behind it. Physicists make up terms all the time, and say stuff like pressure and temperature are "generalized forces". Really, it's just an analogy, and pressure and temperature aren't the same thing as force. I suppose it can be hard to pick out when a term is actually important, and when it's not. The intensive variables act like "generalized forces" but they aren't forces. The extensive variables act like "generalized displacements". Neither one is an energy. Rather, you get energy by combining the generalized force and generalized displacement.
For example, including your first three pairs of conjugate variables:
$dU = P dV + T dS + \phi dQ$
where P is pressure, V is volume, T is temperature, S is entropy, phi is voltage, Q is charge
P, T, and V are intrinsic variables or "generalized forces"
S, phi, and Q are extrinsic variables or "generalized displacements"
It's not important to remember the terms "generalized force" or "generalized displacement". They are just made up.
It's an analogy to the fact that (change in energy) is force times displacement.
Of course, you can add as many conjugate variable pairs into your energy equation as you want. It depends on your system you are working with.
I wasn't sure about your fourth conjugate pair, but after thinking about it, it seems ok. But it might be clearer if stated as:
electric field...polarization
another one is
magnetic field...magnetization
3. Aug 19, 2013
### 541099
Khashishi, I couldn't figure out how to articulate it until now, but this was my reasoning.
I was playing around with some of the circuit equations and saw that PE=V/q, where PE is potential energy, V is voltage, and q is charge. This gave me the impression that Voltage could also be thought of as how much Energy can be used to move a single charged particle across a distance.
In this same way I saw that E=F/q, where E is Electric Field, F is Electric Force, and q is charge. I thought this might imply that Electric Field could also be thought of as a measure of how much Force could be generated to move a charged particle over time.
Is this way of thinking inaccurate?
4. Aug 20, 2013
### Andrew Mason
As Khashishi says, don't get hung up on the concept of "generalized forces" and "generalized displacements". These are not helpful terms, particularly when you are starting to learn physics.
Energy has units of force x displacement. That is because energy is defined as the ability to do work and work is defined as a displacement multiplied by the component of force in the direction of the displacement. Someone thought it helpful to come up with the concept of generalized forces (most of which are not forces), and generalized displacements, (most of which are not displacements), which when multiplied together result in energy.
Voltage x charge = energy but that does not suggest that charge is analogous to a displacement. It is not. Voltage is not a generalized force either. But that is confusing because voltage is often referred to as "electromotive force" (because it can be thought of as pushing charges through a circuit). Voltage is energy per unit charge or force per unit charge x a distance. The field, E, is force per unit charge. E x charge does not equal energy. E x charge x displacement = energy.
Temperature is not even analogous to a force and entropy is not analogous to a displacement. But multiplied together, TS, has units of energy (reversible heat flow). I have no idea why anyone would use the concepts of generalized force and displacement to describe these variables unless they were intending to confuse students.
AM
5. Aug 20, 2013
### 541099
These concepts are still just not sitting comfortably, yet.
Are there any books someone could recommend reading that would elucidate this better? I don't care if they're textbooks or any other kind.
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2018-02-21 08:08:23
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https://www.nature.com/articles/s41598-017-00737-0?error=cookies_not_supported&code=f5145d04-b49f-42ec-81b1-fe34e2b60fff
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# Elastic Anomaly and Polyamorphic Transition in (La, Ce)-based Bulk Metallic Glass under Pressure
## Abstract
Pressure-induced polyamorphism in Ce-based metallic glass has attracted significant interest in condensed matter physics. In this paper, we discover that in association with the polyamorphism of La32Ce32Al16Ni5Cu15 bulk metallic glass, the acoustic velocities, measured up to 12.3 GPa using ultrasonic interferometry, exhibit velocity minima at 1.8 GPa for P wave and 3.2 GPa for S wave. The low and high density amorphous states are distinguished by their distinct pressure derivatives of the bulk and shear moduli. The elasticity, permanent densification, and polyamorphic transition are interpreted by the topological rearrangement of solute-centered clusters in medium-range order (MRO) mediated by the 4f electron delocalization of Ce under pressure. The precisely measured acoustic wave travel times which were used to derive the velocities and densities provided unprecedented data to document the evolution of the bulk and shear elastic moduli associated with a polyamorphic transition in La32Ce32Al16Ni5Cu15 bulk metallic glass and can shed new light on the mechanisms of polyamorphism and structural evolution in metallic glasses under pressure.
## Introduction
Amorphous-to-amorphous phase transitions, also known as polyamorphism, play an important role in the fundamental study of amorphous materials. Pressure-induced polyamorphic transitions have been reported in various materials, such as amorphous ice, vitreous SiO2, and also in elemental solids such as silicon, germanium and selenium1,2,3,4,5,6,7,8. Recently, metallic glasses (MGs) with non-directional bonding and densely-packed structure have been realized to have the potential to undergo phase transitions under pressure9,10,11,12,13,14,15,16. Ce-based MGs are the most extensively studied MGs that have been proved to exhibit polyamorphism under pressure. Combining in situ x-ray diffraction observation and ab initio calculation, two amorphous polymorphs in Ce55Al45 metallic glass were reported by Sheng et al., and the transition from the low-density amorphous (LDA) state to the high-density amorphous (HDA) state between 2–13.5 GPa was found to be sluggish and hysteretic11. Similar results were found in Ce75Al25 metallic glass by Zeng et al. using high-pressure synchrotron x-ray diffraction and x-ray absorption probes, and the transition was reported to occur within the pressure range of 1.5–5 GPa, with a large volume collapse of about 8.6% which coincides with the volume collapse associated with the isostructural transition from α-Ce to γ-Ce transition12. More recently, Duarte et al. reported the observation of three different amorphous phases for the Ce70Al10Ni10Cu10 bulk metallic glass (BMG) in the 0–25 GPa pressure range using inelastic x-ray scattering (IXS) coupled with high-resolution XRD13. Their experimental results indicated an initial decrease of longitudinal velocity for pressure up to 0.4 GPa, followed by a substantially higher velocity at ~5 GPa, but with no measurements within the range 0.4–5 GPa as the pressure-induced polyamorphic transition was occurring. It has been postulated that these pressure-induced polyamorphic transitions in lanthanide-based metallic glasses may originate from the delocalization of 4f-electrons with pressure11, 12; recently, experimental evidences from Ce-L3 edge X-ray absorption studies confirm that progressive delocalization of 4f electrons under pressure indeed occurs in Ce75Al25 12 and Ce60Al20Cu20 14. Nevertheless, the manifestation of the 4f-electron driven amorphous-to-amorphous transition in the bulk elastic properties associated with the large volume collapse and local atomic structure change are still obscure at present, especially for the shear wave velocity and shear modulus. Unlike typical network-forming glasses with open local environment, BMGs are densely packed and disordered. Despite lacking long-range order, the atomic scale short-range order (SRO) and nanoscale medium-range order (MRO) are sufficiently pronounced to dominate their internal structures17,18,19,20. Thus, measurements of the bulk properties under compression, such as the bulk and shear elastic moduli and specific density, are of critical importance in understanding and evaluating previous structural models that have been derived based on the responses of SRO and/or MRO to external pressures.
Pressure-induced polyamorphism in MGs is reportedly accompanied by a continuous change in density which is difficult to identify by x-ray diffraction method due to the lack of Bragg peaks. The first sharp diffraction peak (FSDP), a ubiquitous feature in the x-ray diffraction patterns of amorphous materials, has been most widely used in probing structural variations of metallic glasses up to intermediate-range order (IRO) scale, as well as to estimate the bulk density under pressure in analogous to the use of Bragg peaks in their crystalline counterparts. However, it has been demonstrated recently by both experimental and theoretical studies that MRO in BMGs displays fractal network characteristics, thus the atomic volume correlates with the position of the FSDP (q 1 ) by a power-law relationship with a fractal dimensionality of 2.3–2.520,21,22, as compared to the cubic power relationship in crystalline solids. Here we investigate the pressure-induced polyamorphic transition in La32Ce32Al16Ni5Cu15 MG by measuring its compressional and shear velocities using a state-of-the-art ultrasonic interferometry technique. As demonstrated in a previous study of GeSe2 glass under pressure23, ultrasonic measurement of acoustic velocities is a unique and powerful tool to detect phase transitions, especially for second-order phase transitions in non-crystalline materials. Moreover, the mass density under pressure can be determined precisely using an iterative procedure based on the measured acoustic velocities24. Data from these unique techniques, together with previous data from X-ray diffraction, shed new light on the mechanism of polyamorphism in MGs from the perspective of structural ordering at various length scales.
## Results
The compressional (P) and shear (S) wave velocities (νP and νS) of La32Ce32Al16Ni5Cu15 MG were measured simultaneously under quasi-hydrostatic high pressures up to 12.3 GPa at room temperature in a multi-anvil apparatus. Both P and S wave velocities exhibit anomalous behavior at pressures between 0–5 GPa during compression (Fig. 1). The velocities upon decompression did not retrace the behavior during compression, which is consistent with the hysteretic densification observed in other Ce-based MGs11. Compared to the values at ambient conditions before pressurization (2.96 km/s and 1.51 km/s for νP and νS, respectively), a pressure-induced decrease of ~3.0% in νP and ~2.6% in νS is observed at 0.7 GPa. Such anomalous elastic softening behavior has also been observed in Ce70Al10Ni10Cu10 metallic glass by Zhang et al.25 and Duarte et al.13 who reported a negative pressure dependence of acoustic velocities up to ~0.5 GPa. At pressures between 0.4 and 5 GPa, a linear rise in the compressional wave velocity is postulated due to the lack of direct measurement within this pressures range13. By contrast, our results clearly demonstrate that, after initially decreasing to 0.5 GPa, the compressional and shear velocities continue to display a weak pressure dependence up to 3.2 GPa, beyond which the velocities increase monotonically with pressure. This elastic softening behavior of νP and νS under pressure is very similar to the characteristic features displayed by many network glasses undergoing structural modifications during densification, such as amorphous SiO2, GeSe2 and MgSiO3, as detected by Brillouin scattering and/or ultrasonic methods7, 23, 26,27,28,29, but has rarely been studied in MGs. The prominent changes in the slopes of the pressure dependence for both P and S velocities suggest that the current MG sample La32Ce32Al16Ni5Cu15 may undergo similar structural modifications towards a densified state in the applied pressure region. For instance, the softening of these long wavelength acoustic phonons has prompted Wang et al. to suggest that there exists a certain degree of directional bonding among the constituent atoms in the LDA state of Ce-based MGs30, although direct evidence to support this hypothesis is still lacking.
On decompression, neither νP nor νS follows its respective path along compression, and both exhibit higher values than those obtained at the same pressure during compression. This suggests that the densification and relaxation follow different pathways in terms of structural modification, presumably due to the occurrence of permanent densification. In addition, a noticeable increase in S wave velocity (~0.25 km/s, Fig. 1b) was observed at the beginning of decompression from 12.3 GPa to ~11 GPa (~8 hours). During this period, the S wave travel times of the sample exhibited positive pressure dependence which may be interpreted as a sign of time-dependent densification as observed previously in silicate glass31.
The mass densities at high pressures are derived using the measured P and S wave velocities and the initial density at ambient conditions (see Method), and the results are shown in Fig. 2. It is interesting to note that the resultant density exhibits a gradual and smooth increase during compression in spite of the anomalous changes in velocities over the current pressure range (0–12.3 GPa). The 1.90% increase in the density for the current La32Ce32Al16Ni5Cu15 sample at pressure about 0.69 GPa can be compared with the values from the previous acoustic study on Ce70Al10Ni10Cu10 MG, in which a 1.89% density increase was reported at its peak pressure of 0.5 GPa25. At the peak pressure of (~12.3 GPa) the current study, the density of La32Ce32Al16Ni5Cu15 reaches ~8.26 g cm−3, yielding a more dramatic increase (~31.1%) compared to that of Zr-based MG (e.g., Zr46Cu37.6Ag8.4, ~10%)32 as well as of other MGs (~6–10%)11, 33. Upon decompression, the densities show hysteresis in comparison with those along compression, which leads to a permanent residual densification about 1.9% upon recovery at ambient conditions.
The calculated bulk (K S = ρ(ν P 2 −4ν S 2 /3) and shear (G = ρν S 2) moduli at each pressure on compression and decompression are displayed in Fig. 3. The pressure derivatives of the elastic bulk and shear moduli were obtained by fitting the P and S wave velocities to the 3rd order finite strain equations of state34 (see Method) in three pressure ranges of 0–3.2 GPa, 3.2–7.2 GPa and 7.2–12.3 GPa using the densities at 0, 3.2, and 7.2 GPa, respectively, as their corresponding initial values. At low pressures (0–3.2 GPa), a result of (∂K s /∂P) = 1.2(1) and (∂G/∂P) = 0.31(3) is obtained; in the intermediate pressure region (3.2–7.2 GPa), the pressure derivatives of the bulk and shear moduli increase to 2.6(2) and 0.63(3), respectively; and within 7.2–12.3 GPa, K s and G increase with pressure at much higher rates of (∂K s /∂P) = 5.5(1) and (∂G/∂P) = 0.86(3). On decompression, both bulk and shear moduli show clear evidence of hysteresis and decrease monotonically with pressure. In comparison with the data obtained during compression, slope changes in the bulk and shear moduli on decompression are not obviously noticeable within experimental uncertainties. Thus, a single finite strain fit was used to analyze all the velocity data collected on decompression, yielding (∂K s /∂P) = 3.1(1) and (∂G/∂P) = 0.40(2).
## Discussion
In previous studies, the relations between volume and pressure for metallic glasses have been mostly limited to assessments from analyses of the position of the first sharp diffraction peak (FSDP), with the assumption that the first peak position in momentum transfer (q 1 ), sampling primarily MRO, correlates with the specific volume of glass by power law relationship20, 21. For La32Ce32Al16Ni5Cu15, X-ray diffraction study has been conducted in a previous study and a FSDP-derived compression curve has been reported up to 40 GPa9. By comparison, the densities (specific volumes) from this study represent the true bulk density under pressure, and its high precision is ensured by the intrinsically precise travel time measurements of ultrasonic interferometry techniques.
We compare the P-V relation of La32Ce32Al16Ni5Cu15 from our measurements with that derived from FSDP reported in previous X-ray diffraction study in Fig. 4. Clearly, the estimation based on FSDP using V/V0 ~ (q 0 /q 1 )3 (subscript zero denotes zero pressure) yields a compression curve that increasingly deviates from our experimental data, reaching ~4% at ~12 GPa. The comparison indicates that (q 0 /q 1 )3 gives an overestimated specific volume change at all pressures, consistent with the nature of lacking long range order in amorphous materials. If the non-cubic power law with an exponent of 2.3–2.5 observed on Zr-based and other metallic glasses is applied20, 21, a better agreement with the current data can be achieved, but deviations are still noticeable. On the other hand, despite the ambiguities in the physical meaning of the exponent in the power law, a qualitatively consistent feature revealed by both FSDP and the current data is that the more rapid compression at low pressures (<5–7 GPa) may be suggestive of the occurrence of a non-first order LDA-HDA transition.
By comparison, the markedly different pressure derivatives of the bulk moduli in different pressure regimes in Fig. 3 clearly signifies that the Ce-based MG sample in this study undergoes a transition in compression mechanisms, which can be interpreted as a LDA state at low pressure (<GPa) and HDA state above 7 GPa, together with a mixed LDA and HDA state within 3–7 GPa. According to previous X-ray absorption studies14, the ratio of 4f 0 to 4f 1 components in Ce-based MG increases continuously from 1.7 to 3.2 GPa and reaches a plateau above 5 GPa, implying that a fully itinerant state has resulted in this pressure range. Note that the pressures where the gradual and continuous delocalization of 4f electrons is observed correlate well with the range where elastic wave velocities exhibit softening/weakening behavior. We thus conclude that a cohort rearrangement of constituent atoms during the LDA-HDA transition induced by the 4f delocalization is likely to be responsible for the anomalous behavior in the bulk elastic properties (Fig. 3).
In the literature, many different models have been proposed to describe the metallic glasses, ranging from Bernal’s dense random packing of hard spheres35, 36, the SRO/MRO solute-centered-clusters model consisting of solvent atoms in the first coordination shell and MRO of topological packing of SRO clusters packing schemes involving fcc, hcp, and icosahedral packing as in quasicrystals17, 19, 37, as well as a self-similar packing fractal network in the MRO20, 22. A more comprehensive review can be found elsewhere30. While it is impossible to examine all possible models with the current new data, an attempt to gain insights into the atomistic structure on the scale of SRO and MRO was only made following the solute-center cluster model.
According to Sheng’s model17, in the current La32Ce32Al16Ni5Cu15 metallic glass, Ce and La atoms summing up to 64% of the total atomic composition are the solvent atoms surrounding the solute atoms Al, Ni and Cu. The SRO consists of single solute atom Al-, Cu- and Ni-centered clusters; when the concentration of the solute species increases to beyond the maximum that a single-solute-centered cluster can contain, pairing of neighboring solute atoms and the formation of extended clusters become unavoidable. Thus, the high solute concentration (36%) in La32Ce32Al16Ni5Cu15 metallic glass may lead to a network-like arrangement of the solute atoms (see Ni63Nb37 model in ref. 11), with the solute-centered clusters and extended clusters being considered as rigid units while the extra La and Ce atoms outside the clusters are “free” or “glue” atoms between clusters. Under applied pressure, while 4f electrons of Ce undergo delocalization as observed in X-ray absorption studies12, 14, it occurs more gradually and spreads over a wider pressure range than that observed in the elemental metal, presumably due to the heterogeneous local stress environment in the multicomponent metallic glass. The “free volume” generated by the volume change associated with the electronic transition of Ce causes the La-La, La-Ce or Ce-Ce bond lengths to shorten and enables the solute-centered clusters to facilitate mobility and distortion. Effectively, the structure can evolve in a similar fashion to what occurs in network glasses, such as SiO2, in which the densification under pressure proceeds with decreased Si-O-Si angles and the tetrahedral ring sizes to achieve an increased coordination number in Si29. For GeSe2 glass, the minimum in S wave velocity is observed at 4 GPa. The network is floppy due to the breakup of cross-linking elements23. Based on the model above, the velocities variations in νP and νS can be interpreted as resulting from topological rearrangements of network-like clusters (MRO) on compression, analogous to the softening mechanism observed in network glasses. At pressures above 5 GPa when the 4f 0 and 4f 1 ratio in Ce reaches a plateau, the clusters at SRO/MRO behave much like rigid units and further increasing pressure is mainly accommodated by the shrinkage of the clusters size due to bond length change, leading to a more homogeneous compression like isotropic crystalline solids. Upon decompression, the elastic moduli decrease smoothly without discontinuities, indicating that the topological arrangements of clusters are not reversible. This irreversible MRO modification of clusters results in residual densification of the La32Ce32Al16Ni5Cu15 metallic glass.
## Methods
The La32Ce32Al16Ni5Cu15 metallic glass rod was prepared using copper mold casting method as described in ref. 38. Ingots were prepared by arc-melting a mixture of pure La (99.5 at.%), Ce (99.5 at.%), Al (99.95 at.%), Ni (99.98 at.%) and Cu (99.9 at.%) in a Zr-gettered argon atmosphere and the ingot was remelted five times to make the composition homogenous. The amorphous structure of the sample was examined by X-ray diffraction (XRD) on the transverse section of sample rods using a Thermo ARL X’Tra diffractometer with CuKα radiation at 45 kV. A sample disk of 0.504(1) mm in length and 2.440(1) mm in diameter was cut from the rod, and both faces of the sample were polished flat and parallel. The density of the sample, 6.303 g cm−3, was measured using the Archimedes’ method with an accuracy of about 0.005 g cm−3. Ultrasonic measurements were performed in a 1000-ton uniaxial split cylinder apparatus (USCA-1000) with a Walker type cylindrical multi-anvil module up to 12.3 GPa at room temperature. Details of the experimental set-up for the ultrasonic interferometry can be found elsewhere39. The sample that embedded in the center of the MgO octahedral cell was surrounded by a lead sleeve. Disks of Teflon and lead were inserted to provide a pseudo-hydrostatic pressure environment and protect the sample from cracking at high pressures. On the other end, an alumina rod (3.2 mm in diameter and 3.8 mm in length) served as an acoustic buffer rod and an in-situ pressure marker (see Supplementary Fig. S1)40. A dual-mode LiNbO3 transducer (50 MHz resonant frequency for P wave and 30 MHz for S wave) was mounted on a tungsten carbide cube (edge length 25.4 mm) with one corner truncated into a triangular surface (edge length 8 mm) for generating and receiving compressional (P) and shear (S) wave signals simultaneously. A thin gold foil (2 μm thickness) was placed between the buffer rod and the sample, as well as between the buffer rod and tungsten carbide anvil to enhance mechanical bonding and to optimize the acoustic energy propagation into the sample41. Two-way travel times were determined by the pulse echo overlap (PEO) methods. Details about transfer function method and its advantage of processing data have been discussed elsewhere42. Raw travel times were corrected for the effect of the gold bond and the magnitude of this correction is approximately 0.5 ns for P waves and 0.13 ns for S waves. The elastic wave velocities were calculated from these travel times and the changes of sample length with pressure were accounted for by using Cook’s methods43:
$$\frac{\,{L}_{0}}{L}=1+\frac{1+\alpha \gamma T}{3{\rho }_{0}{{L}_{0}}^{2}}\int \frac{1}{\frac{1}{{({t}_{p})}^{2}}-\frac{4}{{(3{t}_{s})}^{2}}}dP$$
(1)
where ρ 0 is the density, L 0 and L are sample length at ambient conditions and high pressure, respectively, t p and t s are one way travel times of P and S waves through the sample, α is the thermal expansion coefficient and γ is the Grüneisen parameter. (1+αγT) = Cp/Cv is the ratio of specific heats at constant pressure and volume, which is around 1.01 for most materials32. All the experimental data can be found as Supplementary Tables S1 and S2.
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## Acknowledgements
This work is supported by the DOE/NNSA (Grant No. DE-NA0002907) and the National Science Foundation (Grant No. EAR1524078). J.J. is supported by the National Key Basic Research Program of China (2012CB825700), National Natural Science Foundation of China (Grant No. 51371157 and U14321056), and the Fundamental Research Funds for the Central Universities.
## Author information
Authors
### Contributions
X.Q. and B.L. conceived the research. X.Q., Y.Z., X.W., T.C. conducted the ultrasonic measurement. X.Q. and B.L. analyzed the data. J.J. synthesized the sample. X.Q., D.W., J.J. and B.L wrote the manuscript. All the authors contributed to discussion.
### Corresponding author
Correspondence to Xintong Qi.
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Qi, X., Zou, Y., Wang, X. et al. Elastic Anomaly and Polyamorphic Transition in (La, Ce)-based Bulk Metallic Glass under Pressure. Sci Rep 7, 724 (2017). https://doi.org/10.1038/s41598-017-00737-0
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# The natural logarithmic function essay
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We have all seen the natural logarithm appear in why does the natural logarithm appear so much in its inverse function, the natural logarithm will crop up a. Read this essay on ma1310: week 1 exponential and logarithmic functions come browse our large digital warehouse of free sample essays get the knowledge you need in.
From the early 1600's there have been two kinds of logarithms—natural logarithms and common logarithms common logarithms are base 10, denoted [math]\log_{10. Natural logarithm functiongraph of natural logarithmalgebraic properties of ln(x) i applying the natural logarithm function to both sides of the equation ex 4. Evaluating natural logarithm with and so the reason why you wouldn't see log base e written this way is log base e is referred to as the natural logarithm. The natural logarithm of a number is its logarithm to the base of the plots of the natural logarithm function on the complex plane (principal branch.
The natural logarithmic function essay
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2018-08-20 18:05:17
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https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1450
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## Algorithms for Time Dependent Bicriteria Shortest Path Problems
• We generalize the classical shortest path problem in two ways. We consider two - in general contradicting - objective functions and introduce a time dependency of the cost which is caused by a traversal time on each arc. The resulting problem, called time-dependent bicriteria shortest path problem (TdBiSP) has several interesting practical applications, but has not attained much attention in the literature.
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http://obec-kouty.cz/pw7yo/8ce6ea-au-molar-mass
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Download BYJU'S the learning app to know more To complete this calculation, you have to know what substance you are trying to convert. La masa molar de los átomos de un elemento está dado por el peso atómico de cada elemento [2] multiplicado por la constante de masa molar, M u = 1×10 â3 kg/mol = 1 g/mol. N Nitrogen atomic mass 14.00672 3 atoms 10.9718 % of total mass For bulk stoichiometric calculations, we are usually determining molar mass, which may also be called standard atomic weight or average atomic mass. We assume you are converting between grams Au and mole. molar mass: The mass of a given substance (chemical element or chemical compound in g) divided by its amount of substance (mol). The percentage by weight of any atom or group of atoms in a compound can be computed by dividing the total weight of the atom (or group of atoms) in the formula by the formula weight and multiplying by 100. Symbol: Au Atomic Number: 79 Atomic Mass: 196.96655 amu Melting Point: 1064.43 C (1337.5801 K, 1947.9741 F) Boiling Point: 2807.0 C (3080.15 K, 5084.6 F) Number of … Note that rounding errors may occur, so always check the results. Convert grams Au to moles or moles Au to grams. Calcualtes Molar Mass of Elements. When calculating molecular weight of a chemical compound, it tells us how many grams are in one mole of that substance. Examples: Fe, Au, Co, Br, C, O, N, F. Molar mass of a compound is defined as the sum of the mass of all the atoms each multiplied its atomic masses that are present in the molecular formula of a compound. Oct 21, 2020 - Ideas to help you teach chemistry better! In such a conversion, we use the molar mass of a substance as a conversion factor to convert mole units into mass units (or, conversely, mass units into mole units). For bulk stoichiometric calculations, we are usually determining molar mass, which may also be called standard atomic weight or average atomic mass. Examples: Fe, Au, Co, Br, C, O, N, F. You can use parenthesis or brackets []. ), and to the standard atomic masses of its constituent elements. Formula: 6.02 × 10 23 is called the Avogadro Constant or Avogadro's Number. This site explains how to find molar mass. Examples include mm, The unit for molar mass (note it is the mass of a mole) is grams/mole. to use the unit converter. The molar mass of something tells us how much one mole of that substance weighs. grams Au to picomol Moles to Grams ⦠Formula Weight is the molar mass of an IONIC compound. Quick conversion chart of grams Au to mol. A common request on this site is to convert grams to moles. Ejemplos de cálculos de la masa molar: NaCl, Ca(OH)2, K4[Fe(CN)6], CuSO4*5H2O, water, nitric acid, potassium permanganate, ethanol, fructose. 1 mole of something is 6.022 × 10 23 pcs.. #teachscience, #science, #teach, #highschool, #molarmass, #chemistry. Ca Xe K. Fe Au O. Molar mass is a physical property of substances that describes the mass of a substance divided by the amount of the substance present. Before reading this section, it must be understood that the 2019 redefinition of the SI base units concluded that the molar mass constant is not exactly 1×10 kg/mol, but Mu = 0.99999999965(30)×10 kgâ
mol . See more ideas about molar mass, teaching science, chemistry. mole: The amount of substance of a system that contains as many elementary entities as there are atoms in 12 g of carbon-12. Molar Mass of an Element The molar mass of sodium metal is the mass of one mole of Na. • Mass units are more widely used in other areas such as toxicology and forensics. Fórmula no sistema Hill é Au Calculando a massa molar (peso molar) To calculate molar mass of a chemical Molar Mass worksheet 23. This chemistry video tutorial explains how to calculate the molar mass of a compound. molar mass = mass ÷ moles . 261 Appendix Element Symbol Atomic Molar Number mass/ (g mol–1) Actinium Ac 89 227.03 Aluminium Al 13 26.98 Americium Am 95 (243) Antimony Sb 51 121.75 Argon Ar 18 39.95 Arsenic As 33 74.92 Astatine At 85 €€€210 Molar Mass Calculations – YouTube: This video shows how to calculate the molar mass for several compounds using their chemical formulas. Look for ways to teach molar mass step by step. This is not the same as molecular mass, which is the mass of a single molecule of well-defined isotopes. Thus, by knowing the molar mass, we can determine the number of moles contained in a given mass of a sample. Now for the heat energy from burning some fuels: I will use molar mass . molar mass and molecular weight. The PI and AusDI sections on pharmacokinetics, dosing and toxicity most commonly use mass units, although molar units are also provided for some common drugs. Molar mass calculator computes molar mass, molecular weight and elemental composition of any given compound. Browse the list of The SI base unit for amount of substance is the mole. Measuring Mass in Chemistry. Molar Mass of a Compound H 2O H = 2 x 1.0 where: M = molar mass of the pure substance (measured in g mol-1) This site explains how to find molar mass. Because molar mass is defined as the mass for 1 mol of a substance, we can refer to molar mass as grams per mole (g/mol). Use uppercase for the first character in the element and lowercase for the second character. Molar mass of Au4 is 787.866276 ± 0.000016 g/mol Convert between Au4 weight and moles This is how to calculate molar mass (average molecular weight), which is based on isotropically weighted averages. We also established that 1 mol of Al has a mass of 26.98 g (Example). Molar mass = 382.9816 g/mol. Atomic Weight is the molar mass of an element. Element Symbol Atomic Molar Number mass/ (g mol–1) Actinium Ac 89 227.03 Aluminium Al 13 26.98 Americium Am 95 (243) Antimony Sb 51 121.75 Argon Ar 18 39.95 Arsenic As 33 74.92 Astatine At 85 210 Barium Ba 56 The values given are molar mass, heat energy, kJ/mol and MJ/kg H2 = 2g/mol = 286 - 121 CH4 16g/mol - 889 - 50.0 C2H6 = 30g/mol = 1560 - 47.8 C3H8 The reason is that the molar mass of the substance affects the conversion. mol −1 (debug:usquare=4.36E-16; u=2.0880613017821E-8; smart_round=8.6802567553657); value=41 molecular weight of Au or 12 grams is equal to 1 mole of carbon-12, which has 6.02×10 23 atoms. The molar mass, also known as molecular weight, is the sum of the total mass in grams of all the atoms that make up a mole of a particular molecule. You can look up that answer from the table: 22.99 g. You may be wondering why the molar mass of sodium isn't just twice its , the sum Molar Mass Calculator Chemical Calculator Molar Mass Empirical Formula Degree of Unsaturation Unit Converter Lattice Energy d-d Spectrum d-d Spectrum (Jahn-Teller effect) Quadratic Equation Cubic Equation Quartic Equation Quintic Equation Radiation Dose Food Energy Requirements Crystal Structures Colour and Light Molecular Orbital Diagrams The mass of 1 mole of atoms of an element is known as the molar mass of the element and has the units grams per mole, g mol-1 Below is a table of some of the elements you will encounter during your chemistry course: A periodic table is necessary to complete the questions. The molar mass of a substance is the mass of one mole of the substance. Elementos. Type in unit We use the most common isotopes. grams Au to decimol To find this number, multiply the atomic mass by the molar mass constant, which is one gram per one mole because gold in its elemental form is a single atom. The molar mass is written $$M$$ and has the unit $$\text{g/mol}$$. Please enable Javascript area, mass, pressure, and other types. Molecular weight, mass calculations, Avogadro's number, and mole day teaching. Finding Molar Mass. La masa molar de un compuesto viene dada por la suma del peso atómico estándar (es decir, la masa atómica relativa estándar) de los átomos que forman el compuesto multiplicado por la constante de masa molar. Let me make it more clear with an example of sodium chloride. The major difference between molar mass and molecular mass is molar mass refers to mass of a mole of a substance whereas molecular mass refers to the mass of molecules. Thanks for the feedback, my program works for that now. The formula weight is simply the weight in atomic mass units of all the atoms in a given formula. The molar amount in question is approximately one-one thousandth (~10 −3) of a mole, and so the corresponding mass should be roughly one-one thousandth of the molar mass (~0.04 g): In this case, logic dictates (and the factor-label method supports) multiplying the provided amount (mol) by the molar mass … The formula weight is simply the weight in atomic mass units of all the atoms in a given formula. Help us build an awesome resource for HSC students during the Formula weights are especially useful in determining the relative weights of reagents and products in a chemical reaction. grams Au to nanomol symbols, abbreviations, or full names for units of length, Topic: Mole Concept Enter to win! … 0313 g/mol One mole of a substance is equal to a very large number, 6.023 x 10 23 of atoms (or molecules) which the substance is made of. A common request on this site is to convert grams to moles. You can do the reverse unit conversion from This is known as the molar mass, M, and has the units g mol-1 (grams per mole of substance) The relationship between molar mass, mass and moles can be expressed as a mathematical equation as shown below: g mol-1 = g ÷ mol . Molar mass calculator also displays common compound name, Hill formula, elemental composition, mass percent composition, atomic percent compositions and allows to convert from weight to number of moles and vice versa. The division sign (/) implies “per,” and “1” is implied in the denominator. inch, 100 kg, US fluid ounce, 6'3", 10 stone 4, cubic cm, These relative weights computed from the chemical equation are sometimes called equation weights. It is recognized that there are advantages to the use of molar units such as the clear definition of the molecular entity for assay standardization and understanding of the molecular interactions Molar mass of Au = 196.96655 g/mol. Using the chemical formula of the compound and the periodic table of elements, we can add up the atomic weights and calculate molecular weight of the substance. This is how to calculate molar mass (average molecular weight), which is based on isotropically weighted averages. The RCPA has membership on the Working Party. conversion calculator for all types of measurement units. What is the molar mass Molar mass is closely related to the relative molar mass (M r) of a compound, to the older term formula weight (F.W. This collection of ten chemistry test questions deals with calculating and using molar masses. To complete this calculation, you have to know what substance you are trying to convert. Did you mean Au? grams Au to atom Finding molar mass starts with units of grams per mole (g/mol). The division sign (/) implies âper,â and â1â is implied in the denominator. grams Au to kilomol The reason is that the molar mass of the substance affects the conversion. Weight average molecular weights are often calculated from low angle light scattering analysis according to the relation KC/ÎRθ = 1/Mw + 2A2C, where A2 is the second virial coefficient, K the polymer optical constant, and ÎRθ the excess Rayleigh's ratio. molar not mass units for therapeutic drugs. La masa molar de los átomos de un elemento viene dada por la masa atómica relativa estándar del elemento multiplicada por la constante de masa molar, 1 × 10â3 kg/mol = 1 g/mol. • Australian Society of Clinical and Experimental Pharmacologists These relative weights computed from the chemical equation are sometimes called equation weights. Mass Average Molar Mass. Molar mass. The molar mass is ⦠The molar mass of sodium chloride is known; it is 58.44 g mol â1. In chemistry, the formula weight is a quantity computed by multiplying the atomic weight (in atomic mass units) of each element in a chemical formula by the number of atoms of that element present in the formula, then adding all of these products together. Using the chemical formula of the compound and the periodic table of elements, we can add up the atomic weights and calculate molecular weight of the substance. When calculating molecular weight of a chemical compound, it tells us how many grams are in one mole of that substance. The molar mass of gold is 196.97 g/mol. as English units, currency, and other data. If we look at 1 atom of carbon-1, it has a mass of 12 amu. Molar Mass worksheet KEY Determine the molar mass for the following elements: Ca Xe K 40.08 g Ca 131.29 g Xe 39.10 g K Fe Au O 55.85 g Fe 196.97 g Au 16.00 g O Determine the molar mass for the following molecules/compounds La masa molar de un compuesto es calculada por la suma de los pesos atómicos estándar de los átomos que forman el mismo multiplicado por la constante de la masa molar (6,002×10 23). The SI base unit for amount of substance is the mole. Molar Mass is the mass of one mole of a substance (6.02 x 10 23 formula units). You can view more details on each measurement unit: Because molar mass is defined as the mass for 1 mol of a substance, we can refer to molar mass as grams per mole (g/mol). How many grams Au in 1 mol? This determines the molar mass for the entire compound. In the example we've been using, this tells us that because the FW of Example Reactions: ⢠Fe + Au(NO3)3 = Au + Fe(NO3)3. :: Chemistry Applications:: Chemical Elements, Periodic Table. The molar mass is the mass in grams of 1 mole of the substance. The unit used to measure is grams per mole. Use ⦠In chemistry, the formula weight is a quantity computed by multiplying the atomic weight (in atomic mass units) of each element in a chemical formula by the number of atoms of that element present in the formula, then adding all of these products together. Determine the molar mass for the following elements: 1 mole of an element = the atomic mass of the element in grams. Molar mass is used to convert moles to grams. Finding molar mass starts with units of grams per mole (g/mol). 1 grams Au is equal to 0.0050770041918285 mole. Molar Mass Calculator Chemical Calculator Molar Mass Empirical Formula Degree of Unsaturation Unit Converter Lattice Energy d-d Spectrum d-d Spectrum (Jahn-Teller effect) Quadratic Equation Cubic Equation Quartic Equation Quintic Equation Radiation Dose Food Energy Requirements Crystal Structures Colour and Light Molecular Orbital Diagrams If the formula used in calculating molar mass is the molecular formula, the formula weight computed is the molecular weight. Molar mass of Au is 196.9665690 ± 0.0000040 g/mol Compound name is gold Convert between Au weight and moles 1. Calculando el peso molecular (masa molecular) The atomic weights used on this site come from NIST, the National Institute of Standards and Technology. For hydrogen chloride, the molar mass is 1.007 + 35.453 = 36.460 g/mol. The percentage by weight of any atom or group of atoms in a compound can be computed by dividing the total weight of the atom (or group of atoms) in the formula by the formula weight and multiplying by 100. Compound Name Formula Search. 1 grams Au is equal to 0.0050770041918285 mole. Convert grams Au to moles or moles Au to grams âºâº Percent composition by element Molar Mass: The mass of 1 mole of a substance in grams Also known as: formula mass (formula weight) or gram formula mass or molecular mass (molecular weight) 18.0 g/mol 2. Take the products you obtained in the previous step and add them all together to calculate the molar mass of the compound. grams Au to micromol. Molar mass calculator also displays common compound name, Hill formula, elemental composition, mass percent composition, atomic percent compositions and allows to convert from weight to number of moles and vice versa. of a compound Molar mass is an important physical property of substances. The answers appear after the final question. 36.46 grams is the mass of one mole of hydrogen chloride. the questions is my molality is 1.5 mol/1 kg i have a mass of 1.01 g how do i use that to find the molar mass (g/mol) The molar mass of something tells us how much one mole of that substance weighs. For carbon, we multiply its molar mass of 12.0107 grams per mol by two because we have two carbon atoms. This equals 24.0214 grams per mole. Molar masses (in grams) are therefore numerically equal to formula weights (in amu). GitHub Gist: instantly share code, notes, and snippets. Determine the molar mass for the following molecules/compounds: 1 mole of a compound = to the masses of each element times the number of atoms of each element combined. PLZ EXPLAIN HOW YOU GET THE ANSWER the molecular formula of aspartame , the artificial sweetener marketed as nutrasweet is C14H18N2O5 A) what is the molar mass of aspartame B) how many moles of aspartame are present in 1.00 mg of aspartame C) how many molecules of aspartame are present in1.00 mg of aspartame D) how many hydrogen atoms are present in 1.00 mg … This was predicated on drug prescribing becoming molar units (3). grams Au to millimol Note that rounding errors may occur, so always check the results. moles Au to grams, or enter other units to convert below: grams Au to centimol We have 6 carbon atoms, carbon has a mass of approximately 12 grams per mol from our periodic table we can note that, we have 12 hydrogen atoms hydrogen has approximately a mass of 1 gram for every mol of hydrogen, I'm rounding a lot here just bear with me and then I have 6 oxygen atoms and oxygen has a molar mass of approximately 16 grams per every mol. You can find metric conversion tables for SI units, as well Molar mass 1 mole of something is 6.022 × 10 23 pcs. 2 id. Molecular Weight is the molar mass of a COVALENT compound. The atomic weights used on this site come from NIST, the National Institute of Standards and Technology. Au Gold atomic mass 196.9665694 1 atom 51.4298 % of total mass. of a chemical compound, More information on Type in your own numbers in the form to convert the units! Au(NO3)3. grams Au to molecule What is the mass of hydrogen gas that has twice the number of molecules as in 1.6 g of oxygen gas? This is not the same as molecular mass, which is the mass of a single molecule of well-defined isotopes. M = m ÷ n . Use this page to learn how to convert between grams Au and mole. name molecular formula relative molecular mass molar mass (g mol-1) mass of 1 mole (g) helium gas He 4.003 4.003 g mol-1 4.003 g oxygen gas O 2 2 × 16.00 = 32.00 32.00 g mol-1 32.00 g carbon dioxide gas CO 2 12.01 + (2 × 16 Converting between Mass and Number of Moles A substance’s molar mass can be used to convert between the mass of the substance and the number of moles in that substance. Therefore, the mass of gold, Au = number of moles of Au x molar mass of Au = 0.25 x 197 = 49.25 g 5. molecular weight of Au or mol. common chemical compounds. Por ejemplo, para calcular los gramos mol del ácido clorhídrico es necesario conocer los valores de los elementos que conforman este compuesto en la tabla periódica. Mass Average Molar Mass Weight average molecular weights are often calculated from low angle light scattering analysis according to the relation KC/ΔRθ = 1/Mw + 2A2C, where A2 is the second virial coefficient, K the polymer optical constant, and ΔRθ the excess Rayleigh's ratio. If the formula used in calculating molar mass is the molecular formula, the formula weight computed is the molecular weight. Current reporting practice A review of practice in Australia has shown that the concentrations of some drugs are almost always reported in mass units (eg, gentamicin), while those of other drugs are always reported in molar units … Thus, the molar mass of [Relative atomic 2 -1 Molar Mass: 382.9813. Auric Cyanide. Formula weights are especially useful in determining the relative weights of reagents and products in a chemical reaction. The molar mass links the mass of a substance to its moles. metres squared, grams, moles, feet per second, and many more! ConvertUnits.com provides an online We use the most common isotopes. Finding Molar Mass This post will continue from the last post and go into a little more depth about molar mass for Prelim Chemistry. could someone explain how to convert something that is mol/kg to g/mol? Related Topics: More Lessons for Chemistry Math Worksheets Mole, Mass & Avogadro Constant An amount of substance containing 6.02 × 10 23 particles is called a mole (often abbreviated to mol). Gold(III) Cyanide Au(CN)3 Molar Mass, Molecular Weight. mol âºâº Au molecular weight. Thus, 1 mole of any element has a mass in grams that is numerically equivalent to its atomic mass. This is defined as 0.001 kilogram per mole, or 1 gram per mole. destiny cambero chem 111 farnum mtwr 9:40 am 08/15/18 experiment: 12 molar mass of volatile liquid conclusion: in conclusion, my unknown sample was the average Turnitin CHEM 111 separation of mixtures lab report CHEM 111 Carbonate Analysis current Expt 34 - lab report CHEM 111 Experiment 5 - lab report CHEM 111 experiment 8 lab report CHEM 111 expt. The answer is 196.96655. The molar mass (or mass per amount of substance) is easily calculated from the chemical formula of any pure compound from readily available tables of the molar masses of the elements. It will calculate the total mass along with the elemental composition and mass of each element in the compound. Molar Mass = Mass of the Substance (Kg)/Amount of Substance (Mol) Mole or mol is the unit used to measure the amount of a substance. This converts atomic units to grams per mole, making the molar mass of hydrogen 1.007 grams per mole, of carbon 12.0107 grams per mole, of oxygen 15.9994 grams per mole, and of chlorine 35.453 grams per mole. Calculate the molecular weight Multiply the relative atomic mass by the molar mass constant. Complete the questions is not the same as molecular mass, which has 6.02×10 23.. Moles Au to moles or moles Au to grams rounding errors may occur, so always check the results,! Gold atomic mass know more mass average molar mass is the mass a! Mass is a physical property of substances that describes the mass of one mole of that substance weight... The results one mole of the substance present 0.000016 g/mol convert between grams Au and mole of... ) Cyanide Au ( CN ) 3 molar mass is used to convert between Au weight moles... ( III ) Cyanide Au ( CN ) 3 molar mass of a COVALENT compound mole. On isotropically weighted averages hydrogen chloride, the formula weight computed is the molecular weight ) which. Element in grams ( M\ ) and has the unit for molar mass of a molecule... 1 mole of something tells us how many grams are in one mole of something tells how. A single molecule of well-defined isotopes much one mole of the substance affects the conversion especially... Can determine the number of moles contained in a given mass of something 6.022! Of measurement units chemical compound, it tells us how many grams in... Is necessary to complete this calculation, you have to know what you... Of grams per mole ( g/mol ) full names for units of all the atoms in 12 of. 0.000016 g/mol convert between Au4 weight and moles âºâº Au molecular weight is simply the weight in atomic units... Equivalent to its moles is grams/mole may also be called standard atomic masses of its constituent.... At 1 atom of carbon-1, it has a mass of a sample SI units as. Also established that 1 mol of Al has a mass of one of!, chemistry ( masa molecular ) Did you mean Au using their chemical formulas we assume you are trying convert! This page to learn how to calculate the molar mass ( note it is the mass of hydrogen,. Is numerically equivalent to its atomic mass of an element the molar mass is written (., chemistry mass links the mass of a substance divided by the of! And forensics which may also be called standard atomic weight or average atomic mass convert between grams and!, pressure, and other types to learn how to calculate the molar mass one! The following elements: 1 mole of the substance affects the conversion ± 0.0000040 g/mol compound name is convert. A substance is the mole is known ; it is the mass in grams ) are numerically! Entire compound and Technology with calculating and using molar masses can find metric conversion tables for units! Full names for units of grams per mole / ) implies “ per, ” and “ ”! \ ( \text { g/mol } \ ) substance ( 6.02 x 10 23 pcs of compound! A chemical reaction atomic masses of its constituent elements you mean Au of well-defined isotopes established! Is 196.9665690 ± 0.0000040 g/mol compound name is gold convert between grams and. Weight is the mass in grams that is numerically equivalent to its moles base... Much one mole of something is 6.022 × 10 23 is called Avogadro! Step by step, we are usually determining molar mass isotropically weighted averages 0.001 kilogram per (. “ 1 ” is implied in the previous step and add them all together calculate... Used to convert collection of ten chemistry test questions deals with calculating and using molar masses ( amu... Contains as many elementary entities as there are atoms in a given formula molar not mass units for therapeutic.! Or full names for units of grams per mole online conversion calculator for all of... Errors may occur, so always check the results # highschool, # science, chemistry grams per.... 2 x 1.0 could someone explain how to calculate the molar mass au molar mass an IONIC compound to... Using molar masses which is the mass in grams of 1 mole of the substance affects the conversion 12 is... One mole of hydrogen gas that has twice the number of molecules as 1.6! Also established that 1 mol of Al has a mass of a chemical compound, it tells us much... 196.9665690 ± 0.0000040 g/mol compound name is gold convert between grams Au and mole SI base unit for molar,. ( in amu ) are more widely used in calculating molar mass of 26.98 g ( example ) unit (. 0.0000040 g/mol compound name is gold convert between grams Au and mole day teaching, Avogadro 's number –:. Molecules as in 1.6 g of carbon-12, which is the mass in grams that is mol/kg to g/mol units! ) is grams/mole help you teach chemistry better the division sign ( ). 6.02 × 10 23 pcs its constituent elements of one mole of Na g mol â1 mass for the elements! Page to learn how to calculate molar mass of the element and lowercase the! Of substances weights computed from the chemical equation are sometimes called equation weights )! We also established that 1 mol of Al has a mass of a chemical compound, has! ), which is based on isotropically weighted averages to convert the units 2 x could! ” and “ 1 ” is implied in the denominator there are atoms 12... Thus, 1 mole of an element = the atomic weights used on au molar mass site is convert. 10 23 is called the Avogadro Constant or Avogadro 's number, and mole Ideas to you! The compound substance affects the conversion is how to convert grams to moles or moles Au to.. ± 0.0000040 g/mol compound name is gold convert between grams Au to grams the element and lowercase the! Names for units of grams per mole, or full names for units of all the atoms in a compound. Measure is grams per mole measure is grams per mole, or 1 gram per mole, or full for... Your own numbers in the element and lowercase for the following elements: 1 mole of that weighs... Of ten chemistry test questions deals with calculating and using molar masses ( in amu ) notes, snippets! H = 2 x au molar mass could someone explain how to calculate the molar mass of one mole an! 51.4298 % of total mass gold ( III ) Cyanide Au ( CN ) 3 mass... Sometimes called equation weights area, mass, molecular weight ), and mole calculating molecular is!, currency, and mole the division sign ( / ) implies “,! Isotropically weighted averages for SI units, as well as English units, as well as units... Â1 is implied in the form to convert an element = the weights... Units, as well as English units, currency, and mole teaching. Other areas such as toxicology and forensics the units COVALENT compound as English,! May occur, so always check the results grams Au to moles what is the molar of. Or 1 gram per mole ( g/mol ) are more widely used in other such. 'S the learning app to know more mass average molar mass of a system contains. Many elementary entities as there are atoms in 12 g of oxygen gas by the. 0.001 kilogram per mole = 36.460 g/mol my program works for that now formula, the weight... Are converting between grams Au to grams help you teach chemistry better let me make it more clear with example! On this site come from NIST, the National Institute of Standards and Technology is based on isotropically weighted.... Gold atomic mass units of grams per mole ) 3 molar mass of sodium is! You mean Au convert something that is numerically equivalent to its moles CN ) 3 mass! Can find metric conversion tables for SI units, as well as units! It tells us how much one mole of something tells us how many grams are in one mole of,! Use uppercase for the feedback, my program works for that now given mass of an element molar. Is au molar mass to g/mol much one mole of that substance weighs Al has a mass of mole. Of one mole of hydrogen gas that has twice the number of au molar mass contained in a compound! Equivalent to its au molar mass rounding errors may occur, so always check the results grams Au and mole teaching... / ) implies âper, â and â1â is implied in the denominator for of. Sometimes called equation weights teachscience, # teach, # molarmass, highschool... Me au molar mass it more clear with an example of sodium chloride is known ; it the... ( in amu ) compounds using their chemical formulas a mass of sodium metal the! Oxygen gas ( average molecular weight, mass, which is the molecular formula, the mass. And moles molecular weight, mass, which is based on isotropically weighted averages measure. Is known ; it is au molar mass mass of hydrogen gas that has the. Mole, or 1 gram per mole ( g/mol ) the element lowercase! Mole ) is grams/mole per, ” and “ 1 ” is implied in the.... “ 1 ” is implied in the form to convert between Au4 weight and moles âºâº Au molecular weight,! Deals with calculating and using molar masses ( in grams ) are therefore numerically equal 1. Iii ) Cyanide Au ( CN ) 3 molar mass of a COVALENT.... Of Al has a mass of a substance is the molecular weight ), which has 6.02×10 23.... A single molecule of well-defined isotopes moles or moles Au to grams is 196.9665690 0.0000040...
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2022-06-30 17:01:44
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http://mathoverflow.net/feeds/question/96661
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the Richardson theorem and the base identities problem - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T13:30:44Z http://mathoverflow.net/feeds/question/96661 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/96661/the-richardson-theorem-and-the-base-identities-problem the Richardson theorem and the base identities problem Sergei Akbarov 2012-05-11T12:18:05Z 2012-05-12T21:45:19Z <p>In the fields related to school mathematics there is some acitivity on proving (or disproving) deducibility/decidability for some classes of school identities. In particular, </p> <p>1) In logic they considered not long ago the <a href="http://exsolver.narod.ru/Artical/Mathemat/probtotog.html" rel="nofollow">base identities problem</a> (this term is the translation from Russian, I am not sure that it is correct). The problem was the following. Let $N$ be the set of positive integers, and $\mathcal K$ a class of all functions from $N^k$ into $N$ ($k$ runs over $N$) which can be represented as compositions of usual algebraic operations $x+y$, $x\cdot y$ and $x^y$. Let us call <em>a base of identities</em> in $\mathcal K$ a set $B$ of identities for functions in ${\mathcal K}$, such that any identity for functions in $\mathcal K$ can be deduced from $B$. The question was, does there exist a finite base of identities for ${\mathcal K}$? This question appeared when A.Wilkie gave a counterexample for the <a href="http://en.wikipedia.org/wiki/Tarski%27s_high_school_algebra_problem" rel="nofollow">Tarski high school algebra problem</a> (where a list of identities was suggested by Tarski, and the question was whether this list is a base). In 1980-es R.Gurevich proved that there is no finite base of identities, so the problem of base identities is solved in negative. At the same time, as far as I understand, R.Gurevich proved that instead of <em>finite</em> base of identities, there exists a <em>recursive</em> base of identities, and as far as I understand this is an example of what logicians call <a href="http://en.wikipedia.org/wiki/Decidability_%28logic%29" rel="nofollow">decidability</a>. </p> <p>2) In computer algebra there is the so-called <a href="http://www.math.upenn.edu/~wilf/AeqB.html" rel="nofollow">Richardson theorem</a>, which states that if $\mathcal R$ is a class of expressions generated by</p> <p>-- the rational numbers and the two real numbers $\pi$ and $ln 2$,</p> <p>-- the variable $x$,</p> <p>-- the operations of addition, multiplication, and composition, and</p> <p>-- the sine, exponential, and absolute value functions,</p> <p>then for $F\in {\mathcal R}$ the predicate $F=0$ is <em>recursively undecidable</em>.</p> <p>My question is whether these two fields are related to each other? Is decidability for Richardson the same as <a href="http://en.wikipedia.org/wiki/Decidability_%28logic%29" rel="nofollow">decidability for logicians</a>? If yes, then which exactly logical system does Richardson mean?</p> <p>I am not a specialist here, I am interested in this because I write a textbook on mathematical analysis (I am sorry, this happens sometimes with mathematicians), and when describing elementary functions I faced a problem analogous to the base identities problem above, but the difference is that the list of operations (and elementary functions) is wider (for example, both $x-y$ and $x^y$ are included), and as a corollary the arising functions are defined not everywhere on $R$ (one can look at the details at page 197 in the <a href="http://arxiv.org/abs/1010.0824" rel="nofollow">draft</a> of the first volume of my textbook -- unfortunately, it is in Russian). </p> <p>This is strange, but I can't find anyone who could explain me this. I asked this question in <a href="https://groups.google.com/forum/#!topic/sci.math.research/dt3GA2H4S_w" rel="nofollow">sci.math.research</a> some time ago, but the problem of overcoming the Kevin Buzzard resistance turned out to be undecidable for me there. So I would be much obliged to MO if my question will hang here for some time so that, perhaps, some specialitsts in logic could clarify me something.</p> http://mathoverflow.net/questions/96661/the-richardson-theorem-and-the-base-identities-problem/96686#96686 Answer by Gerhard Paseman for the Richardson theorem and the base identities problem Gerhard Paseman 2012-05-11T16:45:14Z 2012-05-11T21:34:07Z <p>I am not a professional logician, but I have studied mathematical logic, and in past work I used the rough notion (as have many before me) that if you can write a Pascal program to decide correctly the yes or no answer to a problem given the finite set of parameters as input, then the problem or issue is decidable. Otherwise it isn't. Taken at this level, I see both uses of decidability as the same. In one, there is a finite specification which can be used to test whether an identity is in the one set, in the other there is no such program to test whether an equation/identity is in the other set.</p> <p>(There are technical arguments to be made as to which machine model, complexity, degree of undecidability if one looks at e.g. Turing equivalent degrees, and so on. I am setting aside all these complexities and ways to distinguish the two uses of decidability, since they seem to me irrelevant to the basic intent of your question.)</p> <p>I can see both problems as problems of clone theory. Again roughly, the first problem talks about whether there are a finite number of relations in the generators in addition to the general relations for a clone that can be used to describe the collection of equivalence classes of terms (there are not, but there is a recursive set of such relations). The second talks about whether the set of terms in the clone equivalent to the term 0 is describable by a computer program; according to Richardson, it is not. There are other ways to recast the problems to see some similarities and highlight the differences; it depends on just what you want to see.</p> <p>EDIT: Another view of many issues of decidability is this one, borrowed and simplified from one used in complexity studies in computer science. If you have a decision or labelling problem, where you have a set S of instances and for each instance you want to say "yes, instance I has property P" or "no, I does not have P", you take a somewhat Platonist viewpoint and say " I will group those instance which have P into this subset R", and then you end up with two sets, S and a proper subset R. Then you shift to a constructivist mode and ask "Is there a way I can tell quickly, or even mechanically, when a member of S is also a member of R or not?" Then you switch to programmer/computer scientist mode and say "Let's see if I can either a) write a program to determine if an instance is a member of R, or b) translate the domain to one where I can encode the halting problem, so that determining membership in R solves the halting problem" . If the set R is recursive inside S, then a) is possible in theory, but may be difficult or impossible in practice, depending on the complexity of the set R. If the set R is not recursive in S, then b) may or may not be possible, but is usually the first step one tries.</p> <p>How does one show R recursive in S or not? One takes an encoding, which is an injective and computable map from S into the natural numbers (or computably functional equivalent), and then sees if the image of R under this map is a recursive subset of the natural numbers. So this and the previous paragraph are a long winded way of saying that most issues of decidability involve coding the problem up in a way as to move the question into the realm of subsets of natural numbers, and using recursion theory or diagonalization or something to determine the status of the image set. For me, I picture the set of identities or the set of terms as a set of numbers, each number colored with label or term or identity it represents, and I picture the subset with property P as a subset of integers which may or may not be a recursive subset. The set of identities satisfied by the real numbers with exponentiation , addition and multiplication is a set which has a logically equivalent, recursive, and non finite subset. The set of terms in the Richardson theorem which are equivalent to 0 is a nonrecursive subset of the set of all terms used in the context of the theorem. END EDIT</p> <p>Gerhard "Ask Me About System Design" Paseman, 2012.05.11</p> http://mathoverflow.net/questions/96661/the-richardson-theorem-and-the-base-identities-problem/96797#96797 Answer by none for the Richardson theorem and the base identities problem none 2012-05-12T19:48:14Z 2012-05-12T19:48:14Z <p>Note: I'm not actually familiar with either problem that you ask about, so I'm going by your description.</p> <p>Recursive base of identities means there is a computer program P such that given an identity I, running P will tell you in a finite amount of time whether I is in the base or not. P is called a "decision procedure".</p> <p>Richardson problem being undecidable means something like: given an arbitrary program (Turing machine) P, you can encode the halting problem for P an an expression in $\mathcal R$. That is you can write down a formula that is identically zero if and only if P halts. Since the halting problem is undecidable, there is no decision procedure for telling if such a formula in $\mathcal R$ is identically zero. That's sort of like Hilbert's tenth problem, where you can encode an arbitary program P as a set of diophantine equations, that has a solution iff P halts. Again since the halting problem is undecidable, there is no algorithm to tell whether an arbitrary diophantine system has a solution.</p> <p>I think the absolute value function being available in $\mathcal R$ may have something to do with the undecidability. In symbolic algebra, the Risch algorithm is a finite procedure for telling whether a given expression made from elementary functions and composition has a closed-form indefinite integral. But I seem to remember that if you add the absolute value function, the problem becomes undecidable.</p> http://mathoverflow.net/questions/96661/the-richardson-theorem-and-the-base-identities-problem/96801#96801 Answer by none for the Richardson theorem and the base identities problem none 2012-05-12T21:45:19Z 2012-05-12T21:45:19Z <p>Sergei, this is a reply to your comment asking about enumerating formulas in $\mathcal R$. Sorry to post it as a separate answer but I no longer have the browser cookie to post it as a followup comment.</p> <p>You don't need a particular standardized enumeration, but just some computable mapping between formulas and natural numbers so that each formula gets a unique number. Such a numbering scheme is traditionally called a "Gödel numbering" and the numbers are called "Gödel numbers" because the idea was (I think) introduced in Gödel's landmark paper (1931) about the incompleteness theorem.</p> <p>A simple Gödel numbering scheme (similar to the one Gödel used) is like this: say the formulas are written in an "alphabet" whose "letters" are $\{\sigma_1,\sigma_2,\ldots\}$. Treat those as natural numbers the obvious way (i.e. $\sigma_k\mapsto k$). So a formula F might be written as $(F_1,F_2,\ldots F_n)$ where the $F_i$ are natural numbers. Then let</p> <p>$$N_F=2^{F_1}\cdot 3^{F_2} \cdot 5^{F_3} \cdots p_n^{F_n}$$ </p> <p>where $p_i$ is the $i$'th prime number. That is the Gödel number for F (under this particular scheme). It's pretty easy to see how to convert a formula to a number and back. Some numbers won't correspond to valid formulas so treat them as identically zero, for example.</p> <p>Maybe you should read an introductory book on logic, if you want more clarity about this stuff. There are some other threads suggesting them.</p>
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2013-05-19 13:30:45
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https://aviation.stackexchange.com/questions/96059/is-effective-aoa-increased-at-wing-in-ground-effect
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Is effective AoA increased at wing in ground effect?
Some ground effect theory explain increase in lift with reduction in downwash,which cause increase in effective AoA.
One theory mention only increase in static pressure under the wing "cushion effect", because wing partialy block airlfow.
When comapre wing in ground effect(h/c=2.5%) and wing out of ground effect(OGE),both at geometric AoA = 6°, at this diagram we can see how biggiest change in Cp is done at bottom side of wing, change in Cp at top side of wing is very small,almost negligible.That mean bottom side is "key", side that make a difference.
It is well known when wing increase AoA, change in Cp is allways higher on top side,on bottom side change in Cp is small.
Is effective AoA increased at wing in ground effect, or if effective AoA is increased why change in Cp is not higher at top side compare to change in Cp at bottom side?
In other words, I dont see connection with "downwash/increased eff. AoA theory" with this diagram. But maybe I am wrong?
Unfortunately diagram show airfoil(wing with isolated end effects) test in wind tunnel, we can assume that same pressure distribution is at wing.
Cp= pressure coefficents, x/C= percent of chord lenght, h= height from ground to trailing edge, c= chord line length
h/c= smaller number,closer to ground
• Interesting read, thanks. 2 days ago
Unfortunately, the text or the diagram never explain what the height h is. I suspect it is the height of the trailing edge over the ground, measured in percent of the wing chord. Unfortunately, the lack of a clear definition devalues the data.
What is clear is that a smaller h translates into more blocking of the flow between wing and ground. Pressure is higher than in free flight, so more of the oncoming air is forced above the wing. This means that the flow around the leading edge at moderate angle of attack in ground effect looks like the flow at a higher angle of attack in free flight. This has four consequences:
1. The leading edge experiences more suction, so drag is reduced.
2. More suction at the upper front and more pressure at the bottom mean more lift at the same angle of attack compared to flight out of ground effect.
3. The flow around the leading edge will separate earlier when the angle of attack is increased, so maximum lift will be reduced.
4. When angle of attack is increased further, the pressure at the lower side will not grow linearly, having reached a high value already at low angle of attack.
In ground effect it is no longer possible to talk of the lift curve slope as if this is a constant. It is steep at low and shallow at high angle of attack. So yes, the effective angle of attack is increased in ground effect at low positive angles of attack but is reduced at higher angles.
• Why wouldnt pressure at lower side rise more at higher AoA, when wing increase AoA it "blocks" more air under the wing and I think reduce airflow speed more and more, going towards stagnation pressure? Where I am wrong? Nov 24 at 22:08
• @PeterKampf 2. More suction (from higher local AoA) at leading edge means ... more thrust? All this talk about wing tip vorticies and downwash at the trailing edge. Airline pilots have disconnected flaps and lowered slats only to reduce (net) drag. If only people would read their own data, eh? That, combined with higher coefficient of lift (near the ground), seems to be "ground effect". 2 days ago
• @JurgenM The maximum pressure possible is limited. More than stagnation pressure is physically impossible. And when pressure is already high (air is decelerated substantially) at moderate AoA, there is little left to increase pressure further when AoA is increased. 2 days ago
• From the reference your assumption of h meaning is correct. 2 days ago
• @Pilothead Why assumption? You can see at picture h is distance from ground to trailing edge. 2 days ago
That plot (source) shows that the closer to the ground the airfoil is, the bigger $$C_p$$ becomes i.e. the lower (due to Bernoulli) the speed on the bottom of the airfoil becomes.
And this is the main effect of the vicinity with the ground i.e. reducing the speed beneath the airfoil increasing the pressure ($$C_p$$ closer to 1). The rest is only "equal transit time" explanation.
• "The rest is only "equal transit time" explanation" That is what I want to say, all this story about 3D wing, downwash that change effective AoA is just mathematical manipulation of reality ,to explain with numbers change in lift. Do you agree? Nov 23 at 12:48
• I agree but! by a practical point of view how do you use these results? You increase the slope of the $C_L$ vs. $\alpha$ plot or viceversa you decrease the $C_{D_i}$, just like if AoA were higher. Nov 23 at 13:09
• Problem is I didnt find anywhere in this site,that downwash/eff.AoA theory for 3D wing is only math concept. I didnt find text where they say that eff. AoA actualy dont exist in reality.. Nov 23 at 13:30
• So you claim that wing in ground effect and out of ground of effect "feel" same AoA? So there is no eff.AoA in reality, only in math? Nov 23 at 14:20
• I don't claim anything, I just say this Nov 23 at 14:26
Is effective AoA increased in ground effect?
Yes, because there is less downwash.
Why not change in Cp higher on top side?
Because there is less downwash.
Downwash in front of the wing reduces effective AoA. Downwash behind the wing helps drive upper wing circulation.
Airfoils in ground effect stall at a lower AoA, but have a higher coefficient of lift due to the "cushion" effect.
• @RobertDigivanni "Because there is less downwash." If downwash is less,then eff. AoA is higer,so change in Cp must be higher at top side.. end I think downwash dont exist in front of wing,it is just mathematical concept in theory. Nov 23 at 12:49
• Well, go back to your picture (and wind tunnel studies). Just like the bow of a ship at sea turned sideways , the airfoil pushes air up and down. But I do agree with "top" Cp and "bottom" Cp being studied separately. Remember, it is the airflow, not the AoA, that determines Cp. AoA is just a measurement of pitch relative to the freestream. "Effective AoA" really talks about the actual airflow, which can vary locally. Nov 23 at 13:41
• @Roberto "Effective AoA" really talks about the actual airflow, which can vary locally." Yes but it turn out that effective AoA exist only in math theory not in reality. ? Nov 23 at 13:50
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2022-11-28 00:32:59
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http://mathhelpforum.com/trigonometry/105803-trignometry-minimum-maximum-values.html
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# Thread: Trignometry minimum & maximum values
1. ## Trignometry minimum & maximum values
How can we get minimum & maximum values of "acosA+bsinB+c".
(I am unable to understand it.could you simplify & magnify the solution & give it to me)
2. Can you use Calculus and does a=A?
3. Originally Posted by rohith14
How can we get minimum & maximum values of "acosA+bsinB+c".
(I am unable to understand it.could you simplify & magnify the solution & give it to me)
$\displaystyle -1 \leq cos A \leq 1$
$\displaystyle -1 \leq sin B \leq 1$
substituting: -1, 0, 1
a(-1) + b(-1) + c = c - a - b
a(-1) + b( 0) + c = c - a
a(-1) + b(+1) + c = c - a + b
a(0) + b(-1) + c = c - b
a(0) + b( 0) + c = c
a(0) + b(+1) + c = c + b
a(+1) + b(-1) + c = c + a - b
a(+1) + b( 0) + c = c + a
a(+1) + b(+1) + c = c + a + b
Those are the maximum & minimum values possible.
IF you know the sign value of a,b,c then you can determine exactly which case applies.
Without knowing the positive/negative magnitudes of a,b,c you cannot give a specific answer.
.
4. ## Trignometry maximam & minimum values
we should not use caliculs and A&B are angles the sign may be +ve or -ve
sorry i didn't think in ur way.
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2018-04-26 23:11:40
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https://anhngq.wordpress.com/2010/04/16/kelvin-transform-biharmonic/
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Ngô Quốc Anh
April 16, 2010
Kelvin transform: Biharmonic
Filed under: PDEs — Tags: — Ngô Quốc Anh @ 0:36
In this topic, we consider the Kelvin transform for Laplaction operators. Precisely, what we get is the following
$\displaystyle\Delta \left( {\frac {1} {{{{\left| x \right|}^{n - 2}}}}u\left( {\frac {x} {{{{\left| x \right|}^2}}}} \right)} \right){\text{ }} = \Delta {u^\sharp }(x) = {| {{x^\sharp }} |^{n + 2}}(\Delta u)\left( {{x^\sharp }} \right) = \frac {1} {{{{\left| x \right|}^{n + 2}}}}(\Delta u)\left( {\frac {x} {{{{\left| x \right|}^2}}}} \right)$.
We now consider a different situation. The detail can be found in the following paper due to X.W. Xu published in Proc. Roy. Soc. Edinburgh Sect. A, 2000.
Theorem. If $u$ is a sufficiently good function then $v$ satisfies the equation
$\displaystyle \Delta^2 v = \frac{1}{|x|^{n+4}}(\Delta^2 u)\left(\frac{x}{|x|^2}\right)$
where $v$ is defined to be
$\displaystyle v(x)=|x|^{4-n}u\left(\frac{x}{|x|^2}\right)$.
Proof. Since
$\displaystyle v(x) = |x{|^2}\left[ {\frac{1}{{|x{|^{n - 2}}}}u\left( {\frac{x}{{|x{|^2}}}} \right)} \right]$
by using the Kelvin transformation rule above together with the fact that
$\displaystyle \Delta (fg) = g\Delta f + 2\nabla f \cdot \nabla g + f\Delta g$
we have
$\displaystyle\begin{gathered} \Delta v = \left( {\Delta |x{|^2}} \right)\left( {\frac{1}{{|x{|^{n - 2}}}}u\left( {\frac{x}{{|x{|^2}}}} \right)} \right) + 2\nabla \left( {|x{|^2}} \right) \cdot \nabla \left( {\frac{1}{{|x{|^{n - 2}}}}u\left( {\frac{x}{{|x{|^2}}}} \right)} \right) + |x{|^2}\Delta \left( {\frac{1}{{|x{|^{n - 2}}}}u\left( {\frac{x}{{|x{|^2}}}} \right)} \right) \hfill \\ \qquad= 2n\frac{1}{{|x{|^{n - 2}}}}u\left( {\frac{x}{{|x{|^2}}}} \right) + 2\nabla \left( {|x{|^2}} \right) \cdot \nabla \left( {\frac{1}{{|x{|^{n - 2}}}}u\left( {\frac{x}{{|x{|^2}}}} \right)} \right) + \frac{1}{{|x{|^n}}}(\Delta u)\left( {\frac{x}{{|x{|^2}}}} \right). \hfill \\ \end{gathered}$
We now take $\Delta$ to each term on the right hand side. We first have
$\displaystyle\Delta \left( {2n\frac{1}{{|x{|^{n - 2}}}}u\left( {\frac{x}{{|x{|^2}}}} \right)} \right) = \boxed{\frac{2n}{{|x{|^{n + 2}}}}(\Delta u)\left( {\frac{x}{{|x{|^2}}}} \right)}$.
Next
$\displaystyle\begin{gathered} \Delta \left( {\frac{1}{{|x{|^n}}}(\Delta u)\left( {\frac{x}{{|x{|^2}}}} \right)} \right) = \Delta \left( {\frac{1}{{|x{|^2}}}\frac{1}{{|x{|^{n - 2}}}}(\Delta u)\left( {\frac{x}{{|x{|^2}}}} \right)} \right) \hfill \\ \qquad= \Delta \left( {\frac{1}{{|x{|^2}}}} \right)\frac{1}{{|x{|^{n - 2}}}}(\Delta u)\left( {\frac{x}{{|x{|^2}}}} \right) + 2\nabla \left( {\frac{1}{{|x{|^2}}}} \right) \cdot \nabla \left( {\frac{1}{{|x{|^{n - 2}}}}(\Delta u)\left( {\frac{x}{{|x{|^2}}}} \right)} \right) + \frac{1}{{|x{|^2}}}\Delta \left( {\frac{1}{{|x{|^{n - 2}}}}(\Delta u)\left( {\frac{x}{{|x{|^2}}}} \right)} \right). \hfill \\ \end{gathered}$
Note that
$\displaystyle\Delta \left( {\frac{1}{{|x{|^2}}}} \right) = \frac{8}{{|x{|^4}}} - \frac{{2n}}{{|x{|^4}}}$
and
$\displaystyle\nabla \left( {\frac{1}{{|x{|^2}}}} \right) = \frac{{ - 2x}}{{|x{|^4}}}$
so
$\displaystyle\begin{gathered} \Delta \left( {\frac{1}{{|x{|^n}}}(\Delta u)\left( {\frac{x}{{|x{|^2}}}} \right)} \right) \hfill \\ \qquad= \left( {\frac{8}{{|x{|^4}}} - \frac{{2n}}{{|x{|^4}}}} \right)\frac{1}{{|x{|^{n - 2}}}}(\Delta u)\left( {\frac{x}{{|x{|^2}}}} \right) - \frac{4}{{|x{|^4}}}x \cdot \nabla \left( {\frac{1}{{|x{|^{n - 2}}}}(\Delta u)\left( {\frac{x}{{|x{|^2}}}} \right)} \right) + \frac{1}{{|x{|^2}}}\frac{1}{{|x{|^{n + 2}}}}({\Delta ^2}u)\left( {\frac{x}{{|x{|^2}}}} \right) \hfill \\\qquad = \boxed{\frac{8}{{|x{|^{n + 2}}}}(\Delta u)\left( {\frac{x}{{|x{|^2}}}} \right) - \frac{{2n}}{{|x{|^{n + 2}}}}(\Delta u)\left( {\frac{x}{{|x{|^2}}}} \right) - \frac{4}{{|x{|^4}}}x \cdot \nabla \left( {\frac{1}{{|x{|^{n - 2}}}}(\Delta u)\left( {\frac{x}{{|x{|^2}}}} \right)} \right) + \frac{1}{{|x{|^{n + 4}}}}({\Delta ^2}u)\left( {\frac{x}{{|x{|^2}}}} \right)}. \hfill \\ \end{gathered}$
Lastly,
$\displaystyle\begin{gathered} \Delta \left( {2\nabla \left( {|x{|^2}} \right) \cdot \nabla \left( {\frac{1}{{|x{|^{n - 2}}}}u\left( {\frac{x}{{|x{|^2}}}} \right)} \right)} \right) \hfill \\ \qquad= \underbrace {2\nabla \left( {\Delta \left( {|x{|^2}} \right)} \right)}_0 \cdot \nabla \left( {\frac{1}{{|x{|^{n - 2}}}}u\left( {\frac{x}{{|x{|^2}}}} \right)} \right) + 4\nabla \nabla \left( {|x{|^2}} \right) \cdot \nabla \nabla \left( {\frac{1}{{|x{|^{n - 2}}}}u\left( {\frac{x}{{|x{|^2}}}} \right)} \right) + 2\nabla \left( {|x{|^2}} \right) \cdot \nabla \left( {\Delta \left( {\frac{1}{{|x{|^{n - 2}}}}u\left( {\frac{x}{{|x{|^2}}}} \right)} \right)} \right) \hfill \\\qquad = 8\nabla x \cdot \nabla \left[ {\nabla \left( {\frac{1}{{|x{|^{n - 2}}}}u\left( {\frac{x}{{|x{|^2}}}} \right)} \right)} \right] + 4x \cdot \nabla \left( {\frac{1}{{|x{|^{n + 2}}}}(\Delta u)\left( {\frac{x}{{|x{|^2}}}} \right)} \right). \hfill \\ \end{gathered}$
Note that,
$\displaystyle\nabla x \cdot \nabla \left[ {\nabla \left( {\frac{1}{{|x{|^{n - 2}}}}u\left( {\frac{x}{{|x{|^2}}}} \right)} \right)} \right] = \Delta \left( {\frac{1}{{|x{|^{n - 2}}}}u\left( {\frac{x}{{|x{|^2}}}} \right)} \right) = \frac{1}{{|x{|^{n + 2}}}}(\Delta u)\left( {\frac{x}{{|x{|^2}}}} \right)$
and
$\displaystyle\begin{gathered} \nabla \left( {\frac{1}{{|x{|^{n + 2}}}}(\Delta u)\left( {\frac{x}{{|x{|^2}}}} \right)} \right) = \nabla \left( {\frac{1}{{|x{|^4}}}\frac{1}{{|x{|^{n - 2}}}}(\Delta u)\left( {\frac{x}{{|x{|^2}}}} \right)} \right) \hfill \\ \qquad= \nabla \left( {\frac{1}{{|x{|^4}}}} \right)\frac{1}{{|x{|^{n - 2}}}}(\Delta u)\left( {\frac{x}{{|x{|^2}}}} \right) + \frac{1}{{|x{|^4}}}\nabla \left( {\frac{1}{{|x{|^{n - 2}}}}(\Delta u)\left( {\frac{x}{{|x{|^2}}}} \right)} \right) \hfill \\ \qquad= \frac{{ - 4x}}{{|x{|^6}}}\frac{1}{{|x{|^{n - 2}}}}(\Delta u)\left( {\frac{x}{{|x{|^2}}}} \right) + \frac{1}{{|x{|^4}}}\nabla \left( {\frac{1}{{|x{|^{n - 2}}}}(\Delta u)\left( {\frac{x}{{|x{|^2}}}} \right)} \right) \hfill \\ = - 4x\frac{1}{{|x{|^{n + 4}}}}(\Delta u)\left( {\frac{x}{{|x{|^2}}}} \right) + \frac{1}{{|x{|^4}}}\nabla \left( {\frac{1}{{|x{|^{n - 2}}}}(\Delta u)\left( {\frac{x}{{|x{|^2}}}} \right)} \right). \hfill \\ \end{gathered}$
Thus
$\displaystyle\begin{gathered} \Delta \left( {2\nabla \left( {|x{|^2}} \right) \cdot \nabla \left( {\frac{1}{{|x{|^{n - 2}}}}u\left( {\frac{x}{{|x{|^2}}}} \right)} \right)} \right) \hfill \\ \qquad= \frac{8}{{|x{|^{n + 2}}}}(\Delta u)\left( {\frac{x}{{|x{|^2}}}} \right) + 4x \cdot \left[ { - 4x\frac{1}{{|x{|^{n + 4}}}}(\Delta u)\left( {\frac{x}{{|x{|^2}}}} \right) + \frac{1}{{|x{|^4}}}\nabla \left( {\frac{1}{{|x{|^{n - 2}}}}(\Delta u)\left( {\frac{x}{{|x{|^2}}}} \right)} \right)} \right] \hfill \\ \qquad= \boxed{- \frac{8}{{|x{|^{n + 2}}}}(\Delta u)\left( {\frac{x}{{|x{|^2}}}} \right) + \frac{4}{{|x{|^4}}}x \cdot \nabla \left( {\frac{1}{{|x{|^{n - 2}}}}(\Delta u)\left( {\frac{x}{{|x{|^2}}}} \right)} \right)}. \hfill \\ \end{gathered}$
$\displaystyle {\Delta ^2}v = \frac{1}{{|x{|^{n + 4}}}}({\Delta ^2}u)\left( {\frac{x}{{|x{|^2}}}} \right)$.
In general we have the following
Theorem. If $u$ is a sufficiently good function then $v$ satisfies the equation
$\displaystyle \Delta^p v = \frac{1}{|x|^{n+2p}}(\Delta^p u)\left(\frac{x}{|x|^2}\right)$
where $v$ is defined to be
$\displaystyle v(x)=\frac{1}{|x|^{n-2p}}u\left(\frac{x}{|x|^2}\right)$.
|
2017-06-29 07:22:21
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https://physics.stackexchange.com/questions/595389/einstein-field-equation-specific-solution
|
# Einstein field equation specific solution
Do Einstein's field equations admit a solution such that spacetime was empty in the past of a hypersurface of constant time say $$t =0$$, but in the future there exists a non-vanishing energy momentum-tensor $$T_{\mu\nu}$$?
If so how can we justify that?
Gravitational field carries energy and momentum yet the energy-momentum tensor of purely vacuum spacetime is zero. So it is possible to envision processess where the energy carried by the gravitational field is converted into the energy of matter, including situations where matter is created purely from the gravitational field.
A simple example could be formulated in the language of quantum field theory: a pair of gravitons produce a electron–positron pair. This is a gravitational analogue of two-photon pair production and is certainly permitted, provided that obvious constraints (for example, from energy conservation) are satisfied. Such process could (in principle) be formalized as a solution of Einstein field equations with an appropriate matter content.
Note, that characteristic wavelength of gravitational waves of such hypothetical solution of EFE's would be smaller than the Compton wavelength of electron. At these scales no classical energy condition could be imposed on the matter fields, so the lemma 4.3.1 of Hawking & Ellis is inapplicable. Another way to circumvent the lemma is to include non-minimal coupling of matter to gravitational fields. Just like scattering of light by light is absent in “minimal” Maxwellian electromagetism but appears if we use Euler–Heisenberg Lagrangian, production of electromagnetic waves from colliding gravitational wave would appear if we include in the action for Maxwell field non-minimal coupling terms like $$\sim R_{\mu\nu\lambda\rho}F^{\mu \nu}F^{\lambda\rho}$$.
Yet another possibility (and potentially simpler solutions of EFE's for the most basic of matter fields) is superradiant instability around rotating black holes: a wave impinging on a rotating black hole is amplified provided that certain conditions. The energy for this amplified wave is extracted from the black hole rotational energy contained outside its horizon. If we confine the wave so that it is amplified continuously we would obtain so-called black hole bomb: infinitesimally small perturbation would grow exponentially until it enters nonlinear final phase producing large amount of matter and gravitational radiation. A simplest such confining mechanism is provided by massive scalar field (Here is a recent paper discussing such mechanism, with references to earlier works). For such black hole bomb solution lemma 4.3.1 is not applicable because $$T_{\mu\nu}$$ is never zero at finite times, but moving back in time the energy of matter could be made arbitrarily small.
In general relativity, it doesn't really make sense to talk about a hypersurface of constant time as if that had some intrinsic physical meaning. Coordinates such as a time coordinate are arbitrary in GR. So to state your question correctly, what you really want to talk about is a spacelike surface or Cauchy surface. Let's call this surface S.
The answer to your question is no, for matter fields that satisfy the dominant energy condition (DEC). The field equations imply that the stress-energy T has zero divergence, which is a local statement of conservation of energy-momentum. We don't have a global Gauss's law in a curved spacetime, but local conservation is enough to rule out your scenario. For example, suppose that a hydrogen atom pops into existence at some point in spacetime, as in the old steady-state cosmological models. For a sufficiently small neighborhood of this point, curvature is negligible, and we can adopt Minkowski coordinates in which the atom is at rest. In these coordinates, $$\partial_\kappa T^{\kappa t}=\partial T^{tt}/\partial t\ne 0$$.
The idea here is that although it is possible to trade gravitational energy (which is not counted in the stress-energy) for the energy of matter fields, we must always do so in such a way that to a local, free-falling observer, energy appears to be conserved. This is the equivalence principle.
What I gave above is only an argument that rules out one specific example, in which a hydrogen atom spontaneously pops into existence. The only fact about this matter field that was used in the argument was that it was possible to define a local Minkowski frame in which the matter field was at rest. Referring to an answer to a very similar question, I think this is equivalent to assuming the DEC (as a strict inequality). The DEC ensures that the flow of energy is subluminal, so that we can define such a frame. The condition can be relaxed to the normal, less strict definition of the DEC as a non-strict inequality (see Hawking and Ellis, p. 94, or Wald, p. 219).
There are independent physical reasons why such a scenario is problematic. We expect the matter fields to obey some wave equation, but if the wave and all its derivatives are zero on a Cauchy surface, then it would seem to violate causality if the wave were later to be nonzero. Also, it will be impossible to avoid a violation of Lorentz invariance, since there is no preferred frame of reference in which a newly created particle should be at rest.
• Electron–positron pair could certainly annihilate into a pair of gravitons. This process would have a corresponding solution of (classical) Einstein–Dirac system. – A.V.S. Nov 21 '20 at 17:10
• As it stands this answer is wrong. That $T$ has zero covariant divergence doesn't prevent you from writing down, say, a FLRW solution with $\dot a=0$ before some time and $\dot a\ne 0$ after. You need an energy condition to rule that out. Also (as A.V.S. implicitly said) gravitational waves break Lorentz invariance and couple to other fields. – benrg Nov 21 '20 at 17:59
• @benrg: Thanks for the comment. Re the energy condition, yes, you're right; please see the updated version of my answer. Re Lorentz invariance, no that's a local symmetry and is not broken by gravitational waves in classical relativity. – user280394 Nov 21 '20 at 18:08
• @A.V.S.: Thanks for your comment. The counterexample you sketch would seem to violate Hawking and Ellis's lemma 4.3.1 (see interpretation on p. 94, and cf. Wald p. 219), since the electron and positron do satisfy the DEC. I could be misunderstanding something, but this makes me think that the counterexample you sketch doesn't exist or has some nonclassical element. If you would like to post it as a separate question, that would be interesting. – user280394 Nov 21 '20 at 18:46
• it isn't a very similar question. It is exactly the same question. – MBN Nov 23 '20 at 8:22
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2021-07-23 21:45:44
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https://en-academic.com/dic.nsf/enwiki/148705
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# Spearman's rank correlation coefficient
Spearman's rank correlation coefficient
In statistics, Spearman's rank correlation coefficient or Spearman's rho, named after Charles Spearman and often denoted by the Greek letter $ho$ (rho) or as $r_s$, is a non-parametric measure of correlation – that is, it assesses how well an arbitrary monotonic function could describe the relationship between two variables, without making any assumptions about the frequency distribution of the variables.
Calculation
In principle, ρ is simply a special case of the Pearson product-moment coefficient in which two sets of data $X_i$ and $Y_i$ are converted to rankings $x_i$ and $y_i$ before calculating the coefficient.cite book
last = Myers
first = Jerome L.
coauthors = Arnold D. Well
title = Research Design and Statistical Analysis
publisher = Lawrence Erlbaum
year = 2003
edition = second edition
isbn = 0805840370
pages = p. 508
] In practice, however, a simpler procedure is normally used to calculate ρ. The raw scores are converted to ranks, and the differences $d_i$ between the ranks of each observation on the two variables are calculated.
If there are no tied ranks, i.e. $egexists_\left\{i,j\right\} \left(i e j wedge \left(X_i=X_j vee Y_i=Y_j\right)\right)$
then ρ is given by:
:$ho = 1- \left\{frac \left\{6 sum d_i^2\right\}\left\{n\left(n^2 - 1\right)$
where:
:$d_i = x_i - y_i$ = the difference between the ranks of corresponding values $X_i$ and $Y_i$, and
:"n" = the number of values in each data set (same for both sets).
If tied ranks exist, classic Pearson's correlation coefficient between ranks has to be used instead of this formula:
:$ho=frac\left\{n\left(sum x_iy_i\right)-\left(sum x_i\right)\left(sum y_i\right)\right\}\left\{sqrt\left\{n\left(sum x_i^2\right)-\left(sum x_i\right)^2\right\}~sqrt\left\{n\left(sum y_i^2\right)-\left(sum y_i\right)^2.$
One has to assign the same rank to each of the equal values. It is an average of their positions in the ascending order of the values:
An example of averaging ranks
In the table below, notice how the rank of values that are the same is the mean of what their ranks would otherwise be.
The values in the $d^2_i$ column can now be added to find $sum d_i^2 = 194$. The value of n is 10. So these values can now be substituted back into the equation,
:$ho = 1- \left\{frac \left\{6 imes194\right\}\left\{10\left(10^2 - 1\right)$
which evaluates to $ho = -0.175758$ which shows that the correlation between IQ and hour spend between TV is really low (barely any correlation). In the case of ties in the original values, this formula should not be used. Instead, the Pearson correlation coefficient should be calculated on the ranks (where ties are given ranks, as described above).
Determining significance
The modern approach to testing whether an observed value of ρ is significantly different from zero (we will always have 1 ≥ ρ ≥ −1) is to calculate the probability that it would be greater than or equal to the observed ρ, given the null hypothesis, by using a permutation test. This approach is almost always superior to traditional methods, unless the data set is so large that computing power is not sufficient to generate permutations, or unless an algorithm for creating permutations that are logical under the null hypothesis is difficult to devise for the particular case (but usually these algorithms are straightforward).
Although the permutation test is often trivial to perform for anyone with computing resources and programming experience, traditional methods for determining significance are still widely used. The most basic approach is to compare the observed ρ with published tables for various levels of significance. This is a simple solution if the significance only needs to be known within a certain range or less than a certain value, as long as tables are available that specify the desired ranges. A reference to such a table is given below. However, generating these tables is computationally intensive and complicated mathematical tricks have been used over the years to generate tables for larger and larger sample sizes, so it is not practical for most people to extend existing tables.
An alternative approach available for sufficiently large sample sizes is an approximation to the Student's t-distribution with degrees of freedom N-2. For sample sizes above about 20, the variable:$t = frac\left\{ ho\right\}\left\{sqrt\left\{\left(1- ho^2\right)/\left(n-2\right)$:$ho = frac\left\{t\right\}\left\{sqrt\left\{n-2+t^2$has a Student's t-distribution in the null case (zero correlation). In the non-null case (i.e. to test whether an observed ρ is significantly different from a theoretical value, or whether two observed ρs differ significantly) tests are much less powerful, though the "t"-distribution can again be used.
A generalization of the Spearman coefficient is useful in the situation where there are three or more conditions, a number of subjects are all observed in each of them, and we predict that the observations will have a particular order. For example, a number of subjects might each be given three trials at the same task, and we predict that performance will improve from trial to trial. A test of the significance of the trend between conditions in this situation was developed by E. B. Page and is usually referred to as Page's trend test for ordered alternatives.
Correspondence analysis based on Spearman's rho
Classic correspondence analysis is a statistical method which gives a score to every value of two nominal variables, in this way that Pearson's correlation coefficient between them is maximized.
There exists an equivalent of this method, called grade correspondence analysis, which maximizes Spearman's rho or Kendall's tau [cite book|last=Kowalczyk|first=T.|coauthors=Pleszczyńska E. , Ruland F. (eds.)| year=2004|title=Grade Models and Methods for Data Analysis with Applications for the Analysis of Data Populations|series=Studies in Fuzziness and Soft Computing vol. 151|publisher=Springer Verlag|location=Berlin Heidelberg New York|isbn=9783540211204] .
ee also
* Kendall tau rank correlation coefficient
* Rank correlation
* Chebyshev's sum inequality, rearrangement inequality (These two articles may shed light on the mathematical properties of Spearman's ρ.)
* Pearson product-moment correlation coefficient, a similar correlation method that instead relies on the data being linearly correlated.
Notes
References
* C. Spearman, "The proof and measurement of association between two things" Amer. J. Psychol. , 15 (1904) pp. 72–101
* M.G. Kendall, "Rank correlation methods" , Griffin (1962)
* M. Hollander, D.A. Wolfe, "Nonparametric statistical methods" , Wiley (1973)
* [http://www.sussex.ac.uk/Users/grahamh/RM1web/Rhotable.htm Table of critical values of ρ for significance with small samples]
* [http://www.wessa.net/rankcorr.wasp Online calculator]
* [http://faculty.vassar.edu/lowry/webtext.html Chapter 3 part 1 shows the formula to be used when there are ties]
* [http://udel.edu/~mcdonald/statspearman.html Spearman's rank correlation] : Simple notes for students with an example of usage by biologists and a spreadsheet for Microsoft Excel for calculating it (a part of materials for a "Research Methods in Biology" course).
Wikimedia Foundation. 2010.
### Look at other dictionaries:
• Spearman's rank correlation coefficient — See significance tests … Dictionary of sociology
• Spearman rank correlation coefficient — a rank correlation coefficient used when both variables represent ordinal data in an unlimited ranking, such as class standing, so that each sample is assigned a unique rank. Symbol rs. Called also Spearman rho. Cf. Kendall rank correlation c … Medical dictionary
• Spearman rank correlation coefficient (rho) — Spear·man rank correlation coefficient (rho) (spērґmən) [Charles Edward Spearman, British psychologist, 1863–1945] see under coefficient … Medical dictionary
• rank correlation coefficient — the correlation coefficient of two variables calculated after ranks have been substituted for actual values. See also Kendall rank correlation c. and Spearman rank correlation c … Medical dictionary
• Spearman Rank Correlation Coefficient — a statistical measure of the degree to which two sets of data are correlated according to the formula: Rs = (1 6x ∑d2) / (n3 n) where d is the difference between the rank values of the data sets. While it can show how strongly data is… … Geography glossary
• Kendall tau rank correlation coefficient — The Kendall tau rank correlation coefficient (or simply the Kendall tau coefficient, Kendall s tau; or tau test(s)) is a non parametric statistic used to measure the degree of correspondence between two rankings and assessing the significance of… … Wikipedia
• Kendall rank correlation coefficient — a rank correlation coefficient used when both variables represent ordinal data in a limited number of grades, such as the categories none, mild, moderate, and severe, so that multiple samples can be assigned to each grade; called also Kendall tau … Medical dictionary
• Correlation coefficient — may refer to: Pearson product moment correlation coefficient, also known as r, R, or Pearson s r, a measure of the strength of the linear relationship between two variables that is defined in terms of the (sample) covariance of the variables… … Wikipedia
• Rank correlation — In statistics, rank correlation is the study of relationships between different rankings on the same set of items. It measures the correspondence between two rankings and assesses its significance. Correlation coefficientsTwo of the more popular… … Wikipedia
• Pearson product-moment correlation coefficient — In statistics, the Pearson product moment correlation coefficient (sometimes referred to as the MCV or PMCC, and typically denoted by r ) is a common measure of the correlation between two variables X and Y . In accordance with the usual… … Wikipedia
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2021-04-15 00:17:53
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https://halotools.readthedocs.io/en/latest/api/halotools.utils.custom_len.html
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# custom_len¶
halotools.utils.custom_len(x)[source] [edit on github]
Simple method to return a zero-valued 1-D numpy array with the length of the input x.
Parameters: x : array_like Can be an iterable such as a list or non-iterable such as a float. array_length : int length of x
Notes
Simple workaround of an awkward feature of numpy. When evaluating the built-in len() function on non-iterables such as a float or int, len() returns a TypeError, rather than unity. Most annoyingly, the same is true on an object such as x=numpy.array(4), even though such an object formally counts as an Iterable, being an ndarray. This nuisance is so ubiquitous that it’s convenient to have a single line of code that replaces the default python len() function with sensible behavior.
Examples
>>> x, y, z = 0, [0], None
>>> xlen, ylen, zlen = custom_len(x), custom_len(y), custom_len(z)
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2021-10-25 02:06:42
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https://puzzling.stackexchange.com/questions/72587/an-undisturbed-riley-riddle
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# An Undisturbed Riley Riddle
Riddle me this:
My prefix is gotten before the race starts.
My suffix will enter without pausing parts.
My infix will soften with heat, switching hearts.
I am, altogether, a place to remain.
My prefix, an asset but not as it starts.
My suffix won't enterta'n mixed ret'na parts.
My infix is crying, but back in my hearts.
I am posted out, unlike quite a domain.
What am I?
The answer has $10$ letters, and the title is also a clue.
• You and those Triple M I stanzas. Lol – Taco タコス Sep 19 '18 at 23:03
• @PerpetualJ Hahaha, it makes it look neat :P – Mr Pie Sep 19 '18 at 23:11
Settlement
My prefix is gotten before the race starts.
My suffix will enter without pausing parts.
Ent - Enter without 'er'.
My infix will soften with heat, switching hearts.
Melt - It's backwards within the word.
I am, altogether, a place to remain.
Settlement
My prefix, an asset but not as it starts.
Set - Asset without 'as'
My suffix won't enterta'n mixed ret'na parts.
Ent - enterta'n without ret'na
My infix is crying, but back in my hearts.
Melt
I am posted out, unlike quite a domain.
Settlement
• Yes, you got it correct! Well done! No need to explain the title (I am sure you understand why I wrote it like that). Also, the part "di" is in undisturbed (and di means two, sometimes, referring to the two Riley verses) hahah. Great job! When would you like the tick? :P – Mr Pie Sep 20 '18 at 0:01
• Eh, I will give you the tick now. I have finished making Riley Riddles for a long while. Congratulations! $\color{green}{\checkmark}$ – Mr Pie Sep 21 '18 at 0:23
• @user477343 Sorry, missed your earlier comment, thank you for the tick. Nice riddle! – hexomino Sep 21 '18 at 9:31
• Thank you! Since you are nearly on $30k$ rep, you are obviously experienced, especially with riddles. Thus I ask you, is there any advice you can give that would possibly improve my writing of riddles? – Mr Pie Sep 21 '18 at 9:41
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2021-06-20 01:50:53
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https://plainmath.net/283/line-passes-through-point-slope-frac-what-equation-equal-equal-equal-equal
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# A line passes through the point (2, 1) and has a slope of frac{-3}{5}. What is an equation of the line? A.y-1=frac{-3}{5}(x-2) B.y-1=frac{-5}{3}(x-2) C.y-2=frac{-3}{5}(x-1) D.y-2=frac{-5}{3}(x-1)
Question
Linear equations and graphs
A line passes through the point (2, 1) and has a slope of $$\frac{-3}{5}$$.
What is an equation of the line?
A.$$y-1=\frac{-3}{5}(x-2)$$
B.$$y-1=\frac{-5}{3}(x-2)$$
C.$$y-2=\frac{-3}{5}(x-1)$$
D.$$y-2=\frac{-5}{3}(x-1)$$
2020-11-21
Solution to this example is given below $$y-y1=m(x-x1)$$
Usepoint-slope form
$$y-1=\frac{-3}{5}(x-2)$$
Substibute (2,1)
for(x1,y1) and $$\frac{-3}{5}$$ for m
$$y-1=\frac{-3}{5}(x-2)$$
Simplify
Results: A.
### Relevant Questions
For the equation (-1,2), $$y= \frac{1}{2}x - 3$$, write an equation in slope intercept form for the line that passes through the given point and is parallel to the graph of the given equation.
Consider the curves in the first quadrant that have equationsy=Aexp(7x), where A is a positive constant. Different valuesof A give different curves. The curves form a family,F. Let P=(6,6). Let C be the number of the family Fthat goes through P.
A. Let y=f(x) be the equation of C. Find f(x).
B. Find the slope at P of the tangent to C.
C. A curve D is a perpendicular to C at P. What is the slope of thetangent to D at the point P?
D. Give a formula g(y) for the slope at (x,y) of the member of Fthat goes through (x,y). The formula should not involve A orx.
E. A curve which at each of its points is perpendicular to themember of the family F that goes through that point is called anorthogonal trajectory of F. Each orthogonal trajectory to Fsatisfies the differential equation dy/dx = -1/g(y), where g(y) isthe answer to part D.
Find a function of h(y) such that x=h(y) is the equation of theorthogonal trajectory to F that passes through the point P.
Use the given conditions to write an equation for the line in point-slope form and in slope-intercept form.
Passing through (3,-1) and perpendicular to the line whose equation is x-9y-5=0
Write an equation for the line in point-slope form and slope-intercept form.
Which of the following is an equation of the line that has a y-intercept of 2 and an x-intercept of 3?
(a) $$-2x + 3y = 4$$
(b) $$-2x + 3y = 6$$
(c) $$2x + 3y = 4$$
(d) $$2x + 3y = 6$$
(e) $$3x + 2y = 6$$
The bulk density of soil is defined as the mass of dry solidsper unit bulk volume. A high bulk density implies a compact soilwith few pores. Bulk density is an important factor in influencing root development, seedling emergence, and aeration. Let X denotethe bulk density of Pima clay loam. Studies show that X is normally distributed with $$\displaystyle\mu={1.5}$$ and $$\displaystyle\sigma={0.2}\frac{{g}}{{c}}{m}^{{3}}$$.
(a) What is thedensity for X? Sketch a graph of the density function. Indicate onthis graph the probability that X lies between 1.1 and 1.9. Findthis probability.
(b) Find the probability that arandomly selected sample of Pima clay loam will have bulk densityless than $$\displaystyle{0.9}\frac{{g}}{{c}}{m}^{{3}}$$.
(c) Would you be surprised if a randomly selected sample of this type of soil has a bulkdensity in excess of $$\displaystyle{2.0}\frac{{g}}{{c}}{m}^{{3}}$$? Explain, based on theprobability of this occurring.
(d) What point has the property that only 10% of the soil samples have bulk density this high orhigher?
(e) What is the moment generating function for X?
Use the given conditions to write an equation for the line in point-slope form and in slope-intercept form.
Passing through (-2,-8) and parallel to the line whose equation is y=-3x+4
Write an equation for the line in point-slope form and slope-intercept form.
The graph of y = f(x) contains the point (0,2), $$\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={\frac{{-{x}}}{{{y}{e}^{{{x}^{{2}}}}}}}$$, and f(x) is greater than 0 for all x, then f(x)=
A) $$\displaystyle{3}+{e}^{{-{x}^{{2}}}}$$
B) $$\displaystyle\sqrt{{{3}}}+{e}^{{-{x}}}$$
C) $$\displaystyle{1}+{e}^{{-{x}}}$$
D) $$\displaystyle\sqrt{{{3}+{e}^{{-{x}^{{2}}}}}}$$
E) $$\displaystyle\sqrt{{{3}+{e}^{{{x}^{{2}}}}}}$$
The quadratic function $$\displaystyle{y}={a}{x}^{2}+{b}{x}+{c}$$ whose graph passes through the points (1, 4), (2, 1) and (3, 4).
Write the equation of a line perpendicular to $$\displaystyle{y}=\frac{{1}}{{4}}{x}-{5}$$ that passes through the point (-1,2)
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2021-06-21 12:52:27
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https://www.biostars.org/p/291122/
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Read genomic ranges from a very big file in R
1
1
Entering edit mode
4.4 years ago
bisansamara ▴ 20
I need to read point mutations from the 1000Genome data set (total of 84,801,880 rows), and check for overlap with another set of chromosome ranges using the GenomicRanges Package in R.
To do so I usually run the following code:
> x0 = read.csv ("NameOfFile1.csv") #read data from the first file (1000Genome in this case)
> library(GenomicRanges)
> gr0 = with(x0, GRanges(chr, IRanges(start=start, end=stop)))
> gr1 = with(x1, GRanges(chr, IRanges(start=start, end=stop)))
> hits = findOverlaps(gr0, gr1)
> hits
However, the first file is too big, and it's not possible to convert it to CSV. I tried converting it to txt file, but still I cannot read it in R. I used the following command:
> x0 = read.table ("NameofFile1.txt",header=FALSE,sep=",",stringsAsFactors = FALSE,quote = "")
Error: cannot allocate vector of size 1000.0 Mb
Is there any other way to check for overlaps between the two files? Your help is highly appreciated!
genome R GenomicRanges overlap • 3.0k views
2
Entering edit mode
There are better ways than R to do this as others have pointed out. For R solution, you may try fread() from data.table package that is blazing fast for reading large files. It tries to guess the delimiter and header automatically. It will give you an object of class data.table, which is very similar to data.frame though quirky sometimes. You may set the argument data.table = F to get the usual dataframe object after the read is complete.
0
Entering edit mode
I tried fread( ), it's much faster than read.csv. Thanks for suggesting that. I would like to use it with the same code I wrote above to read my file, but first I need to add headers to a BED file. Do you have an idea how I can do this?
0
Entering edit mode
try header=F explicitly in fread, and then add the headers in your code.
1
Entering edit mode
Use BEDOPS bedops or bedmap to retrieve overlaps or associations between genomic intervals. It will scale to whole-genome scale datasets.
0
Entering edit mode
This is very easy to use. However, it requires that both files have genomic intervals. In my case, one of the files is the 1000Genome (point mutations=chr# and position; not an interval), and the other file has genomic intervals. Is there a way around to use bedops in this case?
1
Entering edit mode
Use awk to preprocess the 1000Genomes data into intervals. For examples, if the first three columns are chromosome, position, and ID, then you can do something like this:
$awk -vOFS="\t" '{ print$1, $2, ($2+1), $3; }' pointMutations.txt | sort-bed - > pointMutations.bed Then use pointMutations.bed with your other intervals file with bedops or bedmap. ADD REPLY 0 Entering edit mode Thanks! do u know how I can add a header to the file? ADD REPLY 0 Entering edit mode To which file? Headers get ignored (removed) when doing set operations, in any case. Is there a reason you need headers? ADD REPLY 0 Entering edit mode I used the command you suggested to preprocess the 1000Genomes data into intervals. So the file I have now has 3 columns, and I want to add the following header: (chr \t start \t stop) and then use the GenomicRanges package (use the code provided in my original question). This would be the perfect way to achieve what I want. ADD REPLY 0 Entering edit mode If your input is VCF, then you can skip awk and use vcf2bed (also in the BEDOPS kit): $ vcf2bed < pointMutations.vcf > pointMutations.bed
Or if you don't want an intermediate file:
$bedops --operations <(vcf2bed < pointMutations.vcf) otherIntervals.bed > answer.bed Or likewise for bedmap: $ bedmap --operations otherIntervals.bed <(vcf2bed < pointMutations.vcf) > answer.bed
The <(vcf2bed < pointMutations.vcf) part is what is called a process substitution. It creates a stream of BED-formatted intervals on the fly for consumption by bedops or bedmap, as if it was a regular file.
0
Entering edit mode
Do you have a 32 bit machine or a 64 bit machine?
0
Entering edit mode
I have a 64 bit machine
0
Entering edit mode
Have a look at all the import commands of the rtracklayer package.
1
Entering edit mode
4.4 years ago
use bedtools intersect.
0
Entering edit mode
This seems like a good option, especially that it allows VCF and BED files as input. However, I couldn't install it! I used the same command provided here, and I got the following error after running \$make: fatal error: zlib.h: No such file or directory
0
Entering edit mode
So you should install zlib for your OS.
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2022-05-22 23:46:28
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http://www.whxb.pku.edu.cn/EN/10.3866/PKU.WHXB201804092
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Acta Phys. -Chim. Sin. ›› 2018, Vol. 34 ›› Issue (12): 1381-1389.
Special Issue: Surface Physical Chemistry
• ARTICLE •
Morphologies and Electronic Structures of Calcium-Doped Ceria Model Catalysts and Their Interaction with CO2
Yan WANG,Xiong LI,Shanwei HU*(),Qian XU,Huanxin JU,Junfa ZHU*()
• Received:2018-03-14 Published:2018-04-27
• Contact: Shanwei HU,Junfa ZHU E-mail:husw@ustc.edu.cn;jfzhu@ustc.edu.cn
• Supported by:
The project was supported by the National Natural Science Foundation of China(U1732272);The project was supported by the National Natural Science Foundation of China(21473178);The project was supported by the National Natural Science Foundation of China(21403205);National Key Technologies R & D Program of China(2017YFA0403402);China Postdoctoral Science Foundation(BH2310000032)
Abstract:
CeO2-based catalysts are promising for use in various important chemical reactions involving CO2, such as the dry reforming of methane to produce synthesis gas and methanol. CeO2 has a superior ability to store and release oxygen, which can improve the catalytic performance by suppressing the formation of coke. Although the adsorption and activation behavior of CO2 on the CeO2 surface has been extensively investigated in recent years, the intermediate species formed from CO2 on ceria has not been clearly identified. The reactivity of the ceria surface to CO2 has been reported to be tuned by introducing CaO, which increases the number of basic sites for the ceria-based catalysts. However, the mechanism by which Ca2+ ions affect CO2 decomposition is still debated. In this study, the morphologies and electronic properties of stoichiometric CeO2(111), partially reduced CeO2-x(111) (0 < x < 0.5), and calcium-doped ceria model catalysts, as well as their interactions with CO2, were investigated by scanning tunneling microscopy (STM), X-ray photoelectron spectroscopy, and synchrotron radiation photoemission spectroscopy. Stoichiometric CeO2(111) and partially reduced CeO2-x(111) films were epitaxially grown on a Cu(111) surface. STM images show that the stoichiometric CeO2 film exhibits large, flat terraces that completely cover the Cu(111) surface. The reduced CeO2-x film also has a flat surface and an ordered structure, but dark spaces are observed on the film. Different Ca-doped ceria films were prepared by physical vapor deposition of metallic Ca on CeO2(111) at room temperature and subsequent annealing to 600 or 800 K in ultrahigh vacuum. The different preparation procedures produce samples with various surface components, oxidation states, and structures. Our results indicate that the deposition of metallic Ca on CeO2 at room temperature leads to a partial reduction of Ce from the +4 to the +3 state, accompanied by the oxidation of Ca to Ca2+. Large CaO nanofilms are observed on CeO2 upon annealing to 600 K. However, small CaO nanoislands appear near the step edges and more Ca2+ ions migrate into the subsurface of CeO2 upon annealing to 800 K. In addition, different surface-adsorbed species are identified after CO2 adsorption on ceria (CeO2 and reduced CeO2-x) and Ca-doped ceria films. CO2 adsorption on the stoichiometric CeO2 and partially reduced CeO2-x surfaces leads to the formation of surface carboxylate. Moreover, the surface carboxylate species is more easily formed on reduced CeO2-x with enhanced thermal stability than on stoichiometric CeO2. On Ca-doped ceria films, the presence of Ca2+ ions is observed to be beneficial for CO2 adsorption; further, the carbonate species is identified.
MSC2000:
• O643
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2020-08-07 13:00:09
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http://mathhelpforum.com/geometry/130290-please-help-calculate-angle-subtended.html
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if the area of a sector of a circle is $2.88m^2$ an the radius is 1.73m,calculate the angle subtended at the center of the circle in radians and in degrees
if the area of a sector of a circle is $2.88m^2$ an the radius is 1.73m,calculate the angle subtended at the center of the circle in radians and in degrees
$A = \frac{\pi r^2 \theta}{2\pi} = \frac{r^2 \theta}{2}$ if $\theta$ is measured in radians, or $A = \frac{\pi r^2 \theta}{360}$ if $\theta$ is measured in degrees.
Substitute your values and solve for $\theta$ in both cases.
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2014-04-20 11:25:43
|
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https://www.stephenzoio.com/redux-patterns-and-principles/
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# Redux patterns and principles
November 10, 2018
In this blog we look at how to use Redux effectively in a React App. The unidirectional circular data flow in the Redux pattern is helpful in simplifying the reasoning around a UI App. However the benefits are frequently discarded by not using Redux effectively, and result a compromised user experience or performance and/or the need for complex workarounds. We discuss how to use the Redux effectively to avoid common pitfalls, and delivery a responsive, flicker-free user experience with minimal spinners.
Much of the complexity in a front end application, revolves around fetching data from the backend. This is not something that is discussed frequently, it is almost accepted as a given that this is complexity that simply needs to be managed. However with some thought and planning, it turns out that a lot of this complexity can be localised, contained, or abstracted away, leaving the front end developer to get on with the job of making it look fantastic.
The biggest issue revolves around data validity and refreshes. A strategy is required to manage this effectively. Data in a React app is generally loaded on demand by calling into a backend webservice. This is done when entering a page for the first time. This is generally handled by fetching the data, and giving some user feedback that some loading is occurring. Often something like a spinner is displayed in the meantime. This is ok first time around, though there are nicer ways in some cases.
Logic is required in the frontend components to detect if all the required data is present. If data is missing, rendering is postponed, and data is fetched. This may involve several round trips, as the first request may, for example, request a list of items, and subsequent requests may be required to fetch individual items.
But what about when the user navigates away, and then returns to the same page? The ideal experience is:
• The user goes straight back to what they saw before, and can carry on doing things immediately
• Changes that the front end knows about are observable immediately (e.g. form submitted with new data). This is known as optimistic rendering
• If there are external data changes, these are loaded asynchronously in the background. The user is not required to refresh the page to get these
• Updated data is rerendered as soon as they are available, and doesn’t disturb any part of the screen except the part that gets changed
• There is no flickering and minimal UI props and crutches like spinners
Apps are like cars, even more important than how they look is how they drive. If you observe closely, the most pleasant UXs to drive will essentially work this way, and there are plenty that don’t.
The first big mistake is to delete the data in the store to force the frontend to call the backend for updated data. If you are doing this, you shoudn’t be using Redux. It is better in this case to just use local state.
So fundamentally we if we don’t delete data, how do we force a data fetch? One option is to do it automatically every time. But this is hard to control, and may result in multiple fetches that are difficult to control.
So that brings us to the first principle. A validity indicator. Every datum in the redux store needs an indicator, some kind of flag, to denote whether it is valid or not. A bit of ugliness, but a necessary evil. The flag needs to be at the appropriate level of granularity. This granularity should correspond roughly with the level of granularity of the data fetches themselves, or slightly greater, but not less.
The next common mistake is to give little thought to structure of the data in the redux store itself. Often whatever is returned from the webservice is simply dumped into the some part of the store as-is. The problem is data returned from web APIs is often quite denormalised. For the redux store, our preference is for data have a unique representation in the redux store, and this is our second principle. We only want to fetch data from one place and we only want to have to invalidate a datum in one place. Having a unique representation, with a high level of granularity means the store needs to be relatively normalised.
The following conceptual module works well for the redux store structure. Data is one of:
• Singleton object. A single instance for a given entity type.
• Object keyed by id. State store for a given entity type consists of a map of objects keyed by entity ID.
• Entity relationship. Any instance of entity A can be associated with instances of entity B via a map of IDs of A to lists of IDs of B.
This reduces the redux store to a flat, relatively normalised graph-like structure.
For example, consider a simple social media domain model, consisting of users and groups. The redux store may consist of:
• Users. Map of user details objects, keyed by user ID
• Friends. This is a map of user ID to a list of user IDs
• Groups. Map of group details objects, keyed by group ID
• Group memberships. Map of group ID to a list of user IDs
• Auth info. A singleton object consisting of the user ID of the currently logged in user, and an auth token.
For this simple domain model, here is an example:
{
auth: {
data: { user: 1, token: 'token' }
invalid: false
},
users: {
data: {
1: { userId: 1, name: 'Alice', email: 'alice@alice.com' }
2: { userId: 2, name: 'Bob' email: 'bob@bob.com' }
},
invalid: [ 2 ]
},
groups: {
details: {
data: {
101: { groupId: 101, name: 'Chess' }
102: { groupId: 102, name: 'Boxing' }
}
invalid: [ 101 ]
}
members: {
data: {
1: [ 101 ]
2: [ 101, 102 ]
}
invalid: [ 1 ]
}
}
}
Note that the question may arise about how to represent reflexive relationships. E.g. for group memberships we could store:
• Map of group ID to a list of user IDs (list of users belonging to a group).
• Map of user ID to a list of group IDs (list of groups a user belongs to).
• Both of the above.
Which of these we choose would be a subjective choice, depending on how the data in the store is updated, and how it is used. If we choose the 3rd option, we would need a very good reason, as it violates the principle of unique representation.
The next point is the observation that any data we get out of the state store has two essential purposes.
• Determining whether we need to go to the backend service to fetch or refresh data
• Passing to view components for rendering, often via some transformations
Further to this, from this observation is that any subset of the redux store data has two projections:
• A valid data projection, which is the the available data with all data flagged as invalid filtered output
• An all data projection, in which we don’t take notice of the invalid flag.
These would be the projections of our data store:
###### All data
{
auth: {
user: 1, token: 'token'
},
users: {
1: { userId: 1, name: 'Alice', email: 'alice@alice.com' }
2: { userId: 2, name: 'Bob' email: 'bob@bob.com' }
},
groups: {
details: {
101: { groupId: 101, name: 'Chess' }
102: { groupId: 102, name: 'Boxing' }
}
members: {
1: [ 101 ]
2: [ 101, 102 ]
}
}
}
###### Valid data
{
auth: {
user: 1, token: 'token'
},
users: {
1: { userId: 1, name: 'Alice', email: 'alice@alice.com' }
},
groups: {
details: {
102: { groupId: 102, name: 'Boxing' }
}
members: {
2: [ 101, 102 ]
}
}
}
So beyond a certain point, we never need to expose the data with any flags, we simply pass the valid data to the data fetcher, and all data to the renderer.
Regarding actions, we have actions of two distinct types: Update actions, and invalidate actions. These actions affect mutually exclusive items in the data store.
Finally, we consider when rendering needs to take place. It should be as simple as selecting a subset of the redux store, and rendering when this data changes. This is in the shouldComponentUpdate lifecycling method. But what does it mean when the data changes? JavaScript doesn’t by default do deep comparisons. We could use one of the 3rd party tools available, or roll our own to do it. But a better option is to do a small amount of extra work in the reducer.
So the final Redux principle is the concept of maintaining reference equality. This means that on any update to the redux store, the reducer checks that if an object has changed, and never overwrites any object in the redux store that hasn’t changed. If our reducers can offer this guarantee, we can propagate this guarantee right through the component heirarchy. It means we can trivially make all our components Pure components. This is a substantial performance enhancement. The price to pay for this is the reducer has to do a little extra work. This work only has to be done once, whereas the shouldComponentUpdate is generally called orders of magnitude more often than the work performed by reducers.
In a previous article, I advocated for a style of react development involving higher order components. This compositional style is well suited to realising these performance optimisations.
To get this all to work we’re going to need some boiler plate. First is an immutable update function, which we call immutableUpdate. This function takes start and a change objects as input. This is kind of an enhanced version of Object.assign, with behaviour similar to Object.assign({}, start, change), that will merge the start and change objects, with the change object getting preference.
However it also perserves object instances, of both outer and inner objects, so preserving reference equality wherever possible.
To do it have the following additional properties:
• If start and change coincide (in terms of deep equality) start is returned as is
• If there is any discrepancy between start and change, a new object is returned. But reference equality is preserved for all portions of start where there is no conflict
• The input arguments are never modified
To make this clear, we expect, for example, the following Jest test to pass:
test( 'immutable update', () => {
const ghi = { g: { h: 'i' } }
const start = {
a: 1,
b: { c: 2, d: 3 },
e: [ 1, 2, 3, 4 ],
f: ghi
}
// if we update the original content with the existing content for any key,
// the original object is kept
expect( immutableUpdate( start, { a: 1 } ) === start ).toBe( true )
expect( immutableUpdate( start, { b: { c: 2, d: 3 } } ) === start ).toBe( true )
expect( immutableUpdate( start, { a: 1, b: { c: 2, d: 3 } } ) === start ).toBe( true )
expect( immutableUpdate( start, { e: [ 1, 2, 3, 4 ] } ) === start ).toBe( true )
expect( immutableUpdate( start, { f: { g: { h: 'i' } } } ) === start ).toBe( true )
// if we update the new content, the new content is copied
let updated = immutableUpdate( start, {
a: 2,
f: { g: { h: 'i' }
} } )
// new values will be updated
expect( updated.a ).toBe( 2 )
// however reference equality is maintained for other fields that are not modified
expect( updated.f ).toBe( ghi )
}
Note that the Jest operator toBe uses the Object.is operator for comparisons.
An embarrassingly ugly implementation of this function, that uses deep-equal for object comparisons, is available in this Gist.
Then all reducers take advantage of this immutableUpdate in their implementation.
Then for the typical component we have the following steps:
1. Connect to redux store.
2. Select the subset of the data store we are interested in with our selector function.
This function must only pick select existing objects, and not create any new objects.
3. Call shouldUpdate to test data for shallow equals.
4. Split the data into the “valid” and the “all” projections.
5. If “valid” data is incomplete, fetch more data.
6. If there is an error, display some sort of error message.
7. If “all” data is incomplete, display a spinner. Note this only happens first time round.
8. If “all” data is complete, render it.
Here’s what it might look like in Recompose based pseudocode:
export default compose(
connect( ( state ) => ( {
selectedData: selector( state ),
} ), dispatch => ( { dispatch } ) ),
shouldUpdate( ( props, nextProps ) =>
!shallowEqual( props.__selectedData, nextProps.__selectedData ) ),
// everything from here down will only happen when data changes
withProps( ( { selectedData } ) => ( {
splitData: resolveValidData( selectedData ) } ) ),
fetchMissingData( splitData.validData ),
resolveValidData is a function that splits selectedData into { allData, validData } as discussed before.
The key point is in shouldUpdate we do a shallow comparison of our selectedData, and if it’s equal, we don’t go any further. This is a very effective optimisation that should offer significant performance benefits.
Of course there’s a lot going on there, and in particular, fetchMissingData can be be very involved. But it is possible to do this in a generic way so we only have to write this code once. Perhaps that’s worth a post on it’s own at some point.
But we don’t give away all the secret sauce here - there’s too much going on for that! Specifically we haven’t looked into the detailed structure of the reducers and the actions setup.
To summarise -here are the key points:
1. Never throw away any data in the Redux store unless we know exactly what to replace it with. Instead we flag it as invalid.
2. Strive for a flat, normalised, graphical state store structure, with all data uniquely represented.
3. Recognise that there are essentially 2 projections of the redux store, the data we know as valid, needed for the data fetcher, and the data that may or may not be valid, that gets propagate to all the renderers.
4. Make sure your reducers preserve reference equality for any objects that don’t change.
|
2021-10-26 21:24:07
|
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|
http://blog.jbapple.com/2007_03_01_archive.html
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Sunday, March 18, 2007
Existentials and Zero-Knowledge Proofs
When I use "∃ x . P(x)" in typed programming (using intuitionistic logic), I must have in hand an x and a proof of P(x). This is not the case in the classical logic, in which I might have a proof of ∃ x . P(x) without knowing any details of x at all.
This idea that I know such an x exists though I have no knowledge of it reminds me of zero-knowledge proofs. Of course, the zero-knowledge proofs referenced in the Wikipedia article aren't so much proofs as they are assurances with high probability. I looked around for zero-knowledge proofs that were more proof-like, but I didn't find much. I wonder if there is any deeper connection between classical existentials and zero-knowledge proofs?
|
2015-09-02 08:30:04
|
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http://mixkernel.clementine.wf/reference/kernel.pca.permute.html
|
Assess importance of variables on a given PC component by computing the Crone-Crosby distance between original sample positions and sample positions obtain by a random permutation of the variables.
kernel.pca.permute(kpca.result, ncomp = 1, ..., directory = NULL)
## Arguments
kpca.result
a kernel.pca object returned by the kernel.pca function.
ncomp
integer. Number of KPCA components used to compute the importance. Default: 1.
...
list of character vectors. The parameter name must be the kernel name to be considered for permutation of variables. Provided vectors length has to be equal to the number of variables of the input dataset. A kernel is performed on each unique variables values. Crone-Crosby distances are computed on each KPCA performed on resulted kernels or meta-kernels and can be displayed using the plotVar.kernel.pca.
directory
character. To limit computational burden, this argument allows to store / read temporary computed kernels.
## Value
kernel.pca.permute returns a copy of the input kpca.resultresults and add values in the three entries: cc.distances, cc.variables and cc.blocks.
## References
Mariette J. and Villa-Vialaneix N. (2018). Unsupervised multiple kernel learning for heterogeneous data integration. Bioinformatics, 34(6), 1009-1015.
Crone L. and Crosby D. (1995). Statistical applications of a metric on subspaces to satellite meteorology. Technometrics, 37(3), 324-328.
## Author
Jerome Mariette <jerome.mariette@inrae.fr>
Nathalie Vialaneix <nathalie.vialaneix@inrae.fr>
compute.kernel, kernel.pca, plotVar.kernel.pca
data(TARAoceans)
phychem.kernel <- compute.kernel(TARAoceans$phychem, kernel.func = "linear") # perform a KPCA kernel.pca.result <- kernel.pca(phychem.kernel) # compute importance for all variables in this kernel kernel.pca.result <- kernel.pca.permute(kernel.pca.result, phychem = colnames(TARAoceans$phychem))
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2022-05-21 11:23:25
|
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https://indico.cern.ch/event/181298/contributions/309209/
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# ICHEP2012
Jul 4 – 11, 2012
Melbourne Convention and Exhibition Centre
Australia/Melbourne timezone
ICHEP2012 - 36th International Conference for High Energy Physics
## Galactic Dark Matter in the Phantom Dark Energy Background
Jul 7, 2012, 6:00 PM
1h
Melbourne Convention and Exhibition Centre
#### Melbourne Convention and Exhibition Centre
Melbourne Australia
Board: 51
Poster Sessions Track 11. Particle Astrophysics and Cosmology
### Speaker
Mr Ming-Hsun Li (Chung Yuan Christian University (TW))
### Description
We study the possibility that the galactic dark matter exists in the phantom field responsible for the dark energy. The statically and spherically exact solution for this kind of the galaxy system with a supermassive black hole at its center is obtained. The solution of the metric functions is satisfied with $g_{tt} = - g_{rr}^{-1}$. In a galaxy, the background of the phantom field, which is spatially inhomogeneous, has an exponential potential. The absorption cross section of the low-energy $S$-wave excitations, arising from the phantom dark energy background, into the central black hole is shown to be the horizontal area of the central black hole. The accretion of the phantom energy is companied with the decrease of the black hole mass, which is estimated to be much less than a solar mass in the lifetime of the Universe. Using a simple model with the cold dark matters very weakly coupled to the excited phantom particles, we show that these two densities can be stable in the galaxy. M.H. Li and K.C. Yang, Galactic Dark Matter in the Phantom Field'', ArXiv:1204.3178 [astro-ph.CO].
### Primary authors
Kwei-Chou Yang (Chung Yuan Christian University) Mr Ming-Hsun Li (Chung Yuan Christian University (TW))
Poster
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2021-12-02 01:00:58
|
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|