Officially, talks start at 4:30 pm sharp, but this room (and coffee, tea and pastries) will be available for general (fairly random) arguments and discussions from 4 to 4:30. Sometimes these can be as entertaining as the talk itself.

Note to speakers: try very hard to make all your points by 5:20 pm, or save them for the next meeting in the 4-4:30 pre-meeting period. Remember, a number of our participants teach at 5:30!Link to the New Series, Mathematical Methods for Quantum Mechanics and General Relativity

This is the "notes" page for a colloquium series in which we will discuss the main theorems and definitions of those purely mathematical results necessary to phrase and understand elementary quantum mechanics in the modern style. These are standard results, whose proofs are readily found in any book on functional analysis. Frequently, there will be some feature of the proof itself which is useful to emphasize here and in that case we do give a proof, but otherwise reference to a proof will usually suffice.

Our goal is to approach these matters with sufficient generality so that the vector spaces used in the applications we have in mind can be seen in their natural

mathematicalcontext. We will strive to be very clear about what the theorems actually state: what the assumptions are, what the conclusions are ...exactly.This is not possible to do during a basic QM class; with all detail included, the amazing "physical" facts of quantum mechanics would not be revealed until a year of "non-physical" mathematical wrangling had played itself out.

But there is a danger in this informal approach. Theorems are not (usually) carefully stated. But this is not really the issue, since it is only rarely verified that the conditions required in the (unstated) statements of these theorems actually pertain.

While this may be appropriate for a first course, to accustom folks to usage of vocabulary, it can't be pushed beyond carefully staged examples organized by those who have actually checked these things and do know what they mean.

We want to be able to do this for ourselves, and that is the goal.

We will also include historical discussions, from both the math and physics side, so we can get some understanding of how this all came about and who the main players were. It is interesting to note (J. Dieudonné,

History of Functional Analysis, 1981) thatfinitedimensional Linear Algebra was not part of the standard Physicist's toolkit as late as 1926. In 1910, working with infinite dimensional spaces and operators on them was arcane, even shocking.Along the way, I personally would

everything about the following snippet, what it means and how it is implemented, taken from a Wikipedia article on Creally really like to understand^{*}-algebra:In quantum mechanics, one typically describes a physical system using a C^{*}-algebraAwith unit element; the self-adjoint elements ofA(elements x with x^{*}= x) are thought of as the observables, the measurable quantities, of the system.A state of the system is defined as a positive functional

φdefined onA(that is, a complex-linear functionalφfor whichφ(xx^{*}) ≥ 0 for all x inA) such thatφ(1) = 1.The expected value of the observable x, if the system is in state

φ, is thenφ(x).Regards,

Larry Susanka

Walter Eduard Thirring 1927-2014 Last modified on 03/5/15 at 08:04.

Notes from the May 15, 2014 talk by Susanka on Unbounded Operators Last modified on 05/15/14 at 14:14.

David Mermin's slides from the 2008 Oppenheimer Lecture at Berkeley and the 2011 Hofstadter Lecture at Stanford entitled "Spooky actions at a distance?". Last modified on 04/25/14 at 07:58.

Bell on the Foundations of Quantum Mechanics: The Early Hidden Variables Papers, the Einstein-Podolsky Paradox etc. Last modified on 03/15/14 at 11:14.

Here are some interesting thoughts about I.M. Gel'fand, a very famous Russian functional analyst (among other things). These articles are from the notices of the AMS, after his death in 2009. First part. Second part.

Nine formulations of Quantum Mechanics.

Simmons' "Introduction to Topology and Modern Analysis" This is a really good introduction to the kind of analysis we are dealing with in our talks.

Pietsch' "History of Banach Spaces and Linear Operators"

Narici and Beckenstein's "The Life and Times of the Hahn-Banach Theorem" A discussion of the history and variants of this theorem.

An interesting paper on Hilbert, a discussion of his relationship with Einstein

Victor Polinger's Notes Last modified on 04/17/13 at 08:53.

Wheelock's Notes Part Three Last modified on 02/15/13 at 06:57.

Wheelock's Notes Part Two Last modified on 02/15/13 at 06:57.

Wheelock's Notes Last modified on 02/15/13 at 06:57.

Curnutt's One-Page Preview of QM with some interesting problems Last modified on 02/26/13 at 20:49.

A few problems suggested by Dale Hoffman to firm up understanding of metric ideas. Last modified on 02/3/13 at 20:57.

Sets and Strings and Sealing Wax: An Epistle To Our Brothers and Sisters From the Physical Sciences Last modified on 02/3/13 at 20:36.

Susanka's HM for QM Notes Part One (Up to Hilbert space definition and its basic properties.) Last modified on 05/19/13 at 16:50.

Be aware that many of the sections in these last notes are included because I expect or fear their contents will be needed ...

later.They are there because I want a clear statement of the results on record with unified notation. I fully expect that many of these results and comments, as charming as they may be individually, will be excised in a slimmed-down second pass if they end up not contributing substantially to our goal.The order of the results is determined by the hierarchy of assumptions required to prove them, not their importance for us or other esthetic considerations. Early results depend only on vector space structures. Then we establish results requiring norms, then complete norms, then complete inner products and so on.

We will not spend much time on a very abstract result until an application requires its use.In other words, we will progress through the notes linearly but skipping a lot till we get to the first applications, then go back to pick up what we need.

Link to reference books that can be examined on short notice. A search on "functional" or "Hilbert" or "quantum" will turn up the most pertinent.

List of Talks in This Series, 87 meetings from January 2013 to December 2015

Our first meeting was held January 10, 2013 at 4:30 P.M. in Room L-210. (Susanka)

Handout for the First and Second MeetingsOur second meeting was held January 17, 2013 at 4:30 P.M. in Room L-210. (Susanka) Our third meeting was held January 24, 2013 at 4:30 P.M. in Room L-210. (DeVun on Metric Spaces) Our fourth meeting was held January 31, 2013 at 4:30 P.M. in Room L-210. (DeVun)

Handout for Dr. DeVun's discussionOur fifth meeting was held February 7, 2013 at 4:30 P.M. in Room L-210. Butch DeVun finished talking about Metric Spaces, Larry Curnutt then gave an overview of the relationship between the physics and the Hilbert space vocabulary we are building.

Handout for Curnutt's discussionOur sixth meeting was held February 14, 2013 at 4:30 P.M. in Room L-210 and featured

donutsandcoffee,and physicist Kevin Wheelock began the discussion from the "physicist's viewpoint."Our seventh meeting was held February 28, 2013 at 4:30 P.M. in Room L-210. Wheelock continued. Our eighth meeting was held March 7, 2013 at 4:30 P.M. in Room L-222. Note Permanent Room Change!Victor Polinger spoke, with further remarks on the topic from a physicist's POV.Our ninth meeting was held March 14, 2013 at 4:30 P.M. in Room L-222. Victor Polinger continued to speak, with further remarks on the topic from a physicist's POV. Our tenth meeting was held April 11, 2013 at 4:30 P.M. in Room L-222. Victor Polinger concluded (for now.) Our eleventh meeting was held April 18, 2013 at 4:30 P.M. in Room L-222. Larry Susanka brought us back to the mathematical side of all this, for a while: in this case, mathematical history. The slides for this talk are at: Notes for Susanka's History talk. Our twelfth meeting was held April 25, 2013 at 4:30 P.M. in Room L-222. Larry Susanka continued. Our thirteenth meeting was held May 2, 2013 at 4:30 P.M. in Room L-222. Larry Susanka finished the historical discussion. On to mathematics again! Our fourteenth meeting was held May 9, 2013 at 4:30 P.M. in Room L-222. Larry Susanka talked about functionals and duality. Kevin Wheelock had a surprise birthday cake. Our fifteenth meeting was held May 16, 2013 at 4:30 P.M. in Room L-222. Larry Susanka talked about functionals and duality. Our sixteenth meeting was held May 23, 2013 at 4:30 P.M. in Room L-222. Larry Susanka talked about functionals and duality. We arrived at the statement of the Hahn-Banach Theorem. Our seventeenth meeting was held May 30, 2013 at 4:30 P.M. in Room L-222. Larry Susanka proved the Hahn-Banach Theorem. Along the way we spent a bit of time discussing the Axiom of Choice, upon which Hahn-Banach depends. Then we spent some time talking about the ontology of mathematics, the furniture in the mathematical house. What exactly isan abstract object? Here are some ruminations written by me (Larry S.) on both topics: The Axiom of Choice and Foundational Issues in Mathematics. The first of these is an introductory section to a book on Analysis. The second is therapy, written to calm feelings of existential angst.Our eighteenth meeting was held June 6, 2013 at 4:30 P.M. in Room L-222. Larry Susanka talked norms and Continuity. Here are some detailed notes for the last few talks by Susanka.

Here are the slides which synopsize the most relevant points.

Our nineteenth meeting was held June 12, 2013 at 4:30 P.M. in Room L-222. Larry Curnutt talked about Inner Product and Hilbert Spaces. Our twentieth meeting was held Wednesday June 19, 2013 at 4:30 P.M. in Room L-222. Larry Curnutt continued to discuss Hilbert Space and orthogonality. Our twenty-first meeting was held Wednesday July 25, 2013 at 4:00 P.M. in Room L-222. Larry Curnutt continued to discuss issues involving orthogonal bases. Notes can be found here. Our twenty-second meeting was held Tuesday July 30 at 4 pm in L-222. Curnutt continued to discuss properties of orthonormal bases. Our twenty-third meeting was held Wednesday, October 2, 2013, at 4 pm in the new room, Room L-212. Larry Curnutt talked about Spectral Theory, primarily on finite dimensional spaces. Our twenty-fourth meeting was held Wednesday, October 9, 2013, at 4 pm in Room L-212. Curnutt talked about Spectral Theory. Our twenty-fifth meeting was held Wednesday, October 16, 2013, at 4 pm in Room L-212. Curnutt talked about Spectral Theory for Hilbert spaces. Here is a link to his slides. Our twenty-sixth meeting was held Wednesday, October 23, 2013, at 4 pm in Room L-212. Victor Polinger began with a discussion of the new discovery (October, 2013) of soliton-type solutions to Maxwell's equations, his slides are here, a rather shocking discovery that might have (if realized in physical form) important application. Apparently these are related to the Hopf fibration of the 3-sphere. A brief discussion about thatcan be found on page 37 of my notes on quaternions.

Victor provided links to movies of three of these solutions, here (trefoil) and here and here.

Curnutt then continued talking about Spectral Theory for Hilbert spaces, using these notes.

Our twenty-seventh meeting was held Wednesday, October 30, 2013, at 4 pm in Room L-212. Curnutt continued talking about Spectral Theory for Hilbert spaces, using these notes.Our twenty-eighth meeting was held Wednesday, November 6, 2013, at 4 pm in Room L-212. Curnutt continue to talk about Spectral Theory for Hilbert spaces. Our twenty-ninth meeting was held Wednesday, November 13, 2013, at 4 pm in Room L-212. Curnutt finished his introduction to Spectral Theory for Hilbert spaces. His final notes are here. Our thirtieth meeting was held Wednesday, November 20, 2013, at 4 pm in Room L-212. Larry Susanka recalled our original charge in these talks and surveyed the progress we have made.

Our thirty-first meeting was held Wednesday, December 4, 2013, at 4 pm in Room L-212. This was the final meeting of the quarter. Larry Susanka continued to survey our progress and talk about convergence issues so we will be able make sense of spectral measures. Here are the slides from that talk. Our thirty-second meeting was held Thursday, January 16, 2014, at 4 pm, in Room L-213, where we will meet this quarter. Larry Susanka talked about convergence issues and Banach and C* algebras. Slides are here. Our thirty-third meeting was held Thursday, January 23, 2014, at 4 pm in Room L-213. Larry Susanka talked about the remaining " Mega-Theorems" of functional analysis, and started talking about closed, self-adjoint and essentially self-adjoint operators. Notes here. Our thirty-fourth meeting was held Thursday, January 30, 2014, at 4 pm in Room L-213. Kevin Wheelock shared some ideas about the Heisenberg uncertainty relation. The title: "Bring certainty to the Uncertainty Principle." Our thirty-fifth meeting was held Thursday, February 6, 2014, at 4 pm in Room L-213. Kevin Wheelock finished his presentation on the uncertainty relation. Our thirty-sixth meeting was held Thursday, February 13, 2014, at 4 pm in Room L-213. Our guest speaker, Dan Curtis, gave a very interesting talk on aspects of Newtonian Mechanics. He provided a Mathematica notebook at the link here. I ran the workbook and created a pdf (obviously, not "live") which can be found here. Our thirty-seventh meeting was held Thursday, February 20, 2014, at 4 pm in Room L-213. After some comments by Victor, Larry Susanka talked a little about self-adjoint, closed, symmetric and essentially self-adjoint operators. Our thirty-eighth meeting was held Thursday, February 27, 2014, at 4 pm in Room L-213. Larry Susanka talked more about self-adjoint, closed, symmetric and essentially self-adjoint operators. Our thirty-ninth meeting was held Thursday, March 6, 2014, at 4 pm in Room L-213. Larry Susanka finished talking (for a while) about self-adjoint, closed, symmetric and essentially self-adjoint operators. Next week we hand things over to a physicist! Our fortieth meeting was held Thursday, March 13, 2014, at 4 pm in Room L-213. Kevin Wheelock spoke on spin in QM. Our forty-first meeting was held Thursday, March 20, 2014, at 4 pm in Room L-213. Kevin Wheelock continued to speak on spin and related topics, introducing the Pauli spin matrices. Our forty-second meeting was held Thursday, April 10, 2014, at 4 pm in Room L-213. Victor Polinger made historical remarks that will lead to a discussion of hidden variable theories. Essentially, a hidden variable theory attempts to resurrect deterministic reality in some form. Such theories postulate that all measurable quantities do actually have a fixed value, though that value may be impossible for us to ascertain, at least through the lens of the currentversions of QM. They leave open the possibility that better theories might allow for violations of the uncertainty relation, for instance. J. S. Bell's stunning result precludes such theories except under conditions that many people (i.e. me) find unpalatable. So it really seems, for example, that a particledoes not have, at a given instant, both a velocity and a position.Actual experiments using a pair of non-commuting observables demonstrate that, not only is it not possible to measure them simultaneously, the assumption that they have a simultaneous value at all contradicts observation.Our forty-third meeting was held Thursday, April 17, 2014, at 4 pm in Room L-213. Victor Polinger continued his discussion of hidden variable theories. Our forty-fourth meeting was held Thursday, April 24, 2014, at 4 pm in Room L-213. Victor Polinger finished his presentation of hidden variable theories. Our forty-fifth meeting was held Thursday, May 1, 2014, at 4 pm in Room L-213. Larry Susanka gave a description of an experiment which is an application of what Victor did with his more general inequality and counterexample to hidden variable theories, where he demonstrated that there seem to be unresolvable problems when you take the point of view that the qualities of an object supposedly evaluated by pairs of non-commuting operators do have exact values regardless of whether those values are actually measured, and regardless of whether those values can be determined within the framework of QM. This brings into disturbing doubt our comfortable notions about "reality" itself! Larry has excised and will discuss some slides from a beautiful talk by Physicist David Mermin at the 2008 Oppenheimer Lecture at Berkeley, in which Mermin discussed hidden variables theories and a clever 1989 reformulation of the original Bell experiment. Our forty-sixth meeting was held Thursday, May 8, 2014, at 4 pm in Room L-213. Victor led the discussion which involved calculations related to the previous talks on hidden variables. Our forty-seventh meeting was held Thursday, May 15, 2014, at 4 pm in Room L-213. Many symmetric operators, common and important, are NOT self-adjoint and it is only to self-adjoint operators that the main theorems we need apply. So the issue of self-adjoint extensions, if any, can't be avoided in a careful treatment of our material. Larry Susanka continued with some of the main theorems and constructions pertinent to QM. Our forty-eighth meeting was held Thursday, May 22, 2014, at 4 pm in Room L-213. Larry Susanka talked about the Cayley and Inverse Cayley transform which associates unbounded symmetric operators with partial isometries. The key result is that a symmetric operator is self-adjoint exactly when its associated Cayley transform is unitary. Our forty-ninth meeting was held Thursday, May 29, 2014, at 4 pm in Room L-213. Larry Susanka gave examples of unbounded operators related to the extension problems that concern us most: position, momentum, etc. Our FIFTIETH meeting was held Thursday, June 5, 2014, at 4 pm in Room L-213. Larry Susanka talked about preliminary facts needed to give versions of the spectral theorem for normal operators i.e. those BOUNDED operators which commute with their adjoint. We learned, in the last weeks, that self-adjoint operators, bounded or not, are each connected to a unique unitary operator, its Cayley transform, which is of course normal. Spectral properties of this unitary operator transform immediately to the the original through the inverse Cayley transform. Victor had a few words to say concerning the physical facts of the---fascinating---Aharanov-Bohm effect.

Susanka's Notes

Polinger's NotesOur fifty-first meeting was held Thursday, June 12, 2014, at 4 pm in Room L-213. Larry Susanka talked about versions of the spectral theorem for normal operators and Stone's theorem. slides here. Our fifty-second meeting was be held Thursday, October 2, 2014 in Room L-219. We were scheduled to talk about normal operators and Stone's theorem, but in fact we had a highly entertaining (to me) open philosophical discussion/organizational meeting.

Our fifty-third meeting was held Thursday, October 9, 2014 in Room L-220. Larry Susanka talked talk about versions of the spectral theorem for normal operators and Stone's theorem. Our fifty-fourth meeting was held Thursday, October 16, 2014 in Room L-220. We discussed seminar organization and a little bit of finite-dimensional spectral theory. Our fifty-fifth meeting was held Thursday, October 23, 2014 in Room L-220. We discussed some finite-dimensional spectral theory. Our fifty-sixth meeting was held Thursday, October 30, 2014 in Room L-220. Larry Susanka carried on with the reformulation/translation of our self-adjoint operator decomposition (spectral theory) to the finite dimensional case. Our fifty-seventh meeting was held Thursday, November 6, 2014 in Room L-220. Larry Susanka (virtually) concluded comments on the reformulation/translation of our self-adjoint operator decomposition (spectral theory) to the finite dimensional case. Our fifty-eighth meeting was held Thursday, November 13, 2014 in Room L-220. Larry Susanka spent a few minutes on the reformulation/translation of our self-adjoint operator decomposition (spectral theory) to the finite dimensional case. Here are some final notes on this.

Then Larry Curnutt took over, and will continue to speak for the remainder of the quarter.Our fifty-ninth meeting was held Thursday, November 20, 2014 in Room L-220. Larry Curnutt continued to speak on important questions about quantum mechanics that (one would think) we should easily be able to answer now. And yet .... Our sixtieth meeting was held Thursday, December 4, 2014 in Room L-220. Larry Curnutt continued to speak on important but elementary issues related to quantum mechanics. His notes are here.Our sixty-first meeting was held Thursday, January 15, 2015, Room L-213. Larry Curnutt continued to speak on the Axioms of Quantum Mechanics and the Heisenberg Uncertainty Relation. Our sixty-second meeting was held Thursday, January 22, 2015. Larry Curnutt finished (for now) his discussion on the Axioms of QM. Curnutt's's amended notes are here.Our sixty-third meeting was held Thursday, January 29, 2015, Room L-213. Mike Ulrey has some things to say about the interpretation of QM among other things. (Check herefor an article on on one of these, Qbism.)Our sixty-fourth meeting was held Thursday, February 5, 2015. Mike Ulrey had some things to say about the interpretation of QM among other things. (Check here for his notes in pdf formatand checkhere for the original Mathematica notebook.Mike will return in a week or two to carry on with the program he has outlined.)Our sixty-fifth meeting was held Thursday, February 12, 2015. Victor Polinger started his presentation on perturbation theory and will, ultimately, discuss several topics including Green's Functions and Feynman Diagrams.

Our sixty-sixth meeting was held Thursday, February 19, 2015. Victor Polinger continued.

Our sixty-seventh meeting was held Thursday, February 26, 2015. Victor Polinger continued.

Our sixty-eigth meeting was held Thursday, March 5, 2015. Victor Polinger continued his presentation on perturbation theory and Green's Functions and Feynman Diagrams. His latest talk slides are here. He is not finished with this discussion, and will continue after Mike Ulrey talks for a week or two. Our sixty-ninth meeting was held Thursday, March 12, 2015. Mike Ulrey carried on with his discussion from the sixty-third and sixty-fourth meetings on foundational issues in QM. Our seventieth meeting was held Thursday, March 19, 2015. Mike Ulrey carried on with his discussion on foundational issues in QM. Our seventy-first meeting was held Thursday, April 16, 2015. Mike Ulrey carried on with his discussion on various foundational matters from QM, and notes can be found here in pdf format and here as a live Mathematica notebook. Our seventy-second meeting was held Thursday, April 23, 2015. Victor Polinger talked about QFT, preparatory to a discussion of Feynman Diagrams. Our seventy-third meeting was held Thursday, May 7, 2015. Victor Polinger talked about QFT and Feynman Diagrams. Our seventy-fourth meeting was held Thursday, May 14, 2015,in room R-208. Larry Susanka talked about Orthogonal Sums and Tensor Products of two Hilbert Spaces. Our seventy-fifth meeting was held Thursday, May 21, 2015. Victor Polinger talked about QFT. Our seventy-sixth meeting was held Thursday, May 28, 2015. Robert Hobbs was the speaker. He presented a somewhat expanded version of a talk he is to give at a conference this summer. Here is what he says about his talk:

The Recently adopted Laboratory guidlines for four year university and college physics say (as the fourth of six objectives)

"Through laboratory work, students should gain the awareness that they are able to do science; that is, students should be able to collect, analyze, and interpret real measured data in an ethical manner as responsible scientists and draw meaningful conclusions from personal observations of the physical world. The laboratory curriculum should get students to start thinking like physicists by constructing knowledge that does not rely on an outside authority, should explicitly make them aware that they can construct knowledge in this way, and should build confidence in their ability to do so."

I have been asked to give a talk at this summer's national meeting on "...constructing knowledge that does not rely on an outside authority..."

Our seventy-seventh meeting was held Thursday, June 4, 2015. Robert Hobbs continued.

Our seventy-eighth meeting was held Thursday, June 11, 2015. Victor Polinger continued in his effort to bring us to the point where we can understand the meaning of, and make a prediction based on, a Feynman diagram. Our seventy-ninth meeting was held Thursday, October 1, 2015. Larry Susanka started us off this year with a discussion of Emmy Noether and an introduction to differentiable manifolds. Emmy Noether herself was an interesting person. She was a protégé of David Hilbert and Felix Klein at Göttingen and was described by many (including Einstein) as the greatest female mathematician who ever lived. She had numerous students who revered her, in spite of her eccentricities, and is widely acknowledged as the "mother" of modern abstract algebra. Her unfortunate early death at age 53 deprived the world of a great genius. This is a new arc, a second one in addition to but in support of our ongoing QM arc, which will lead us eventually to several forms of Noether's Theorem which associates symmetry in physical law, a vital component of QM as well as classical physics, with conserved quantities of that law. The low-hanging fruit here is in Newtonian mechanics with its Euclidean geometry, and special relativistic mechanics with Minkowski geometry. In Newtonian mechanics rotational symmetry of the Lagrangian is associated with conservation of angular momentum. Symmetry of the Lagrangian under time shifts is associated with conservation of energy. Symmetry of the Lagrangian under spacial translation is associated with conservation of linear momentum. To apply this in relativistic mechanics and other more general settings we need to use the language of Riemannian and Semi-Riemannian Manifolds and Lie Groups and Algebras, and that is our goal. Ultimately we will see how symmetry and invariants come into QM.

Our eightieth meeting was held Thursday, October 8, 2015. Larry Susanka began a mathematical discussion of differentiable manifolds, a key structure in many areas of mathematical physics including the relativity theories and Lie theory, important for QM. Our eighty-first meeting was held Thursday, October 15, 2015. Larry Susanka continued a mathematical discussion of differentiable manifolds. Our eighty-second meeting was held Thursday, October 22, 2015. Larry Susanka continued talking about differentiable manifolds. Our eighty-third meeting was held Thursday, October 29, 2015. Larry Susanka continued talking about differentiable manifolds. Our eighty-fourth meeting was held Thursday, November 5, 2015. Larry Susanka was still talking about differentiable manifolds. Our eighty-fifth meeting was held Thursday, November 12, 2015. Larry Susanka yet again talked about differentiable manifolds. Our eighty-sixth meeting was held Thursday, November 19, 2015. Though we never made it past the first slide, we did discuss issues of vocabulary, so that is progress. ;-) Our eighty-seventh and last meeting-of-the-quarter was held Thursday, December 3, 2015. We talked about manifolds.

I can be contacted by e-mail at WebContact@Susanka.org.My PGP public key can be found on the public keyservers and also here.

Join the Blue Ribbon Online Free Speech Campaign!This page was last modified on 06/13/19 at 19:09.