text:
## Compiler IDE API

As I am working (slowly) on a compiler at the moment, I have started thinking about IDE integration. It occurs to me (as I am sure it has to everyone else) that a compiler should not be a command-line program, but a library that can be integrated with applications and IDEs. The command line interface for the compiler should be just another client of the library.

So my question is, in an ideal environment (starting from scratch) what should an API for a compiler (targeting IDE integration as well as other runtime uses) look like? Obviously some method to build the code, some methods for auto-completion, type checking, type inference.

Other thoughts are that all the above functions need to operate in a dynamic environment, where incomplete code fragments exist, without breaking the compiler, and hopefully still offering useful feedback.

## Logic Production Systems (LPS)

A new mixed logic/imperative programming language by Bob Kowalski and Fariba Sadri:

“Logic-based Production System" is a new computer language that combines the characteristics of an imperative programming language with those of a declarative database and knowledge representation language. It is the result of over a decade of research led by Bob Kowalski and Fariba Sadri at Imperial College London.

The kernel of LPS consists of a database, together with reactive rules of the form if antecedent then consequent. The database changes destructively in response to actions and external events, according to a domain-specific causal theory. Computation consists in making the reactive rules true, by performing actions to make the consequent true whenever the antecedent becomes true. In addition, LPS includes Prolog-like logic programs both to recognise when antecedents become true and to generate plans of actions that make consequents true.

The page links to papers and slides that describe the language in more detail and there is an online playground to try it out.

## Rust's language ergonomics initiative

Aaron Turon has a nice blog post about Rust's language ergonomics initiative. It contains a description of how the Rust language design team currently thinks about "implicit" features (those that do something without the user having to write anything explicitly at this point in the program), how to design such features or detect issues with existing features.

There are three dimensions of the reasoning footprint for implicitness:

Applicability.
Where are you allowed to elide implied information? Is there any heads-up that this might be happening?
Power.
What influence does the elided information have? Can it radically change program behavior or its types?
Context-dependence.
How much of do you have to know about the rest of the code to know what is being implied, i.e. how elided details will be filled in? Is there always a clear place to look?

Various existing or proposed Rust features are then discussed and critiqued along these evaluation axes. I found the blog post interesting on several levels and thought it could be a nice starting point for discussion.

One thing that is important to keep in mind is that, despite the generality of the claims made about programming language design, this text is mostly about a set of design tools that has been conceived to solve a specific category of ergonomics problem. LtU regulars will be quick to point out that the interaction with programming tools, for example, is not discussed or considered at all while it is a central aspect of ergonomics in many cases (including, I think, implicitness decision). I would personally recommend trying to stay within the approximate scope of the Rust discussion (generalizing to other languages and systems), rather than discussing all aspects of "Language Ergonomics" at once, which would result in discussions all over the place (but with less people interested in each sub-discussion).

A parting remark on the end paragraph which I foud interesting:

And, perhaps most importantly, we need empathy for each other. Transformative insights can be fragile; they can start out embedded in ideas that have lots of problems. If we’re too quick to shut down a line of thought based on those problems, we risk foreclosing on avenues to something better. We’ve got to have the patience to sit with ideas that are foreign and uncomfortable, and gain some new perspective from them.

Clearly Aaron Turon is getting at the tone of online discussions which can become heated, even for details of programming language feature design. But what I think of reading those lines is that such discussion are helped by one aspect of our work as language designers, which is to produce tests and criterions to evaluate language features (and, yes, shut down some proposals more quickly than others). I fully agree that, in a foreign land of creativity, one should not judge too quickly or too harshly. But I think that the development and wider adoption of these design tests would not only result in seeing the flaws in some proposals more quickly, it would also let people iterate more quickly on their own designs, applying the tests themselves to recognize the issues without having to go through public feedback (and potential harshness).

I see similarities with the notion that mechanized proof assistants made it much easier to teach how to write rigorous mathematical proofs to beginners, as they can get direct feedback (from the computer) on whether their proof attempts are correct, without having to ask for the explicit approval of another human (their teacher or their student colleagues), with the various inhibition effects that it created. Having this internal feedback is what the design principles, laws, and various meta-theorems that our community developed enable, and it is one of the side-benefits of formalism.

## notes on a C-ish memory manager design

I am (re)designing a C runtime memory manager for hosting fibers, DSLs, and lightweight daemons. I started teaching my early-20s sons to program (in C++ and C), and the idea is to give them a library allowing progressive elaboration of tools written in Unix philosophy: i.e. simple and composeable processes. The idea is to use interfaces that allow embedding, so pipelines can run in threads at first, then later fibers within a thread, without code rewrites beyond automatic translation for CPS (continuation passing style) to run in fibers. The "app" would be a daemon hosting a user space operating system, with services like an http agent on a port for a browser-based UI. The idea is to give my sons an undergraduate CS education in terms of what happens here, and in stuff they write to execute inside.

My only aim in this post is to briefly describe its memory management, and the relation to programming language implementation, since a couple different (Smalltalk-ish) Lisp interpreters will run inside, as DSL examples, one simple and one fancy. The paragraph above was just context, since otherwise you might wonder "why in the world I want to do that?" about each detail. This is for your entertainment only, since I would find a helpful comment surprising. My current version of this sort of thing is much nicer than any I did before. I was trying to solve the problem of making everything refcounted, without something terribly awkward happening. (Embedded sub-processes can garbage collect as they like without refcounting, though; my fancy interpreter would use stop-and-copy.)

Each runtime instance uses interfaces defined in the library only. The world outside is abstracted as an environment, so all calls to the outside world go through an env instance. This helps later automatic code rewriting, so blocking calls can be handled as async calls to a worker thread pool to avoid blocking a fiber. (Such calls park instead of blocking.)

The idea is to have multiple independent runtimes in one host operating system process, so there is no global memory manager unless glue code for the environment does that (whatever mechanism gets memory from native interfaces). There is at least one instance of environment E in the OS process, which has function pointers for every external interface supported. For memory management purposes, it must have a memalign() function pointer, to allocate large blocks of aligned memory. (This can be simulated with malloc() by allocating very big blocks, then chopping off misaligned start and end parts, using those for other purposes.)

Each runtime instance has at least one vat, acting as the heap, which allocates only aligned memory blocks. For example, each thread would have its own dedicated vat, which is single-threaded, so no races exist between threads. (All dataflow between threads is copied from one vat to another, mediated by thread-safe queues in the runtime.) Each vat allocates only large uniform blocks from the env, perhaps several at once for efficiency, where every block is 1MiB in size, or 2^20 bytes, and aligned to a 1MiB address. (The standard block size could be another power of two instead of 1MiB, but I will speak as if it could never change.) Since there will be fragmentation, this design only makes sense when allocating a lot of total memory, on the order of gigabytes. So you would not want this runtime on a system with limited resources.

Each vat describes its address space as a book, a set of 1MiB pages, where each 2^20-aligned block of 2^20 bytes is called a bp for book page -- or just page for short when book is understood. The main purpose of each bp is usually sub-allocation of smaller aligned memory blocks to satisfy vat allocation requests, but a garbage-collected vat would use its book any way it liked (an address space with page granularity). A vat used for C style allocation would return blocks that are powers-of-two in size, each aligned to the same power-of-two bytes, as small as 16 bytes and as large as, say, 256K. You can cheaply test any address for membership in a vat by seeing if the aligned bp address is in the book hashmap of bp addresses. Each allocated aligned block is called a rod, short for refcounted, because each one is reference counted by metainfo at the start of each book page.

Each bp begins with a standard bp header, describing page format and detail common to the page, followed by as many rh rod headers as needed to describe each rod sub-block in the rest of the page. (A garbage collected vat would have only the bp header for each bp.) Each rod head is 16 bytes in size and 16-byte aligned, with a 32-bit refcount a 32-bit checksum, and several descriptive fields, including a 16-bit generation number and a pair of 16-bit ids characterizing format and optional allocation context (naming a backtrace for memory usage analysis). Starting from the address of a rod -- some block inside a bp -- getting its refcount touches two cache lines in the general case, one for the bp head and one for the rh. Except for very small rods, this is only one cache line worse than co-locating the refcount inside the block itself. And that would only happen when the refcount was actually needed. (In my model, refcount is NOT linear in aliases; it is only linear in official handles which other aliases can borrow when their scope is clearly inside the handle scope.)

The purpose of a checksum in the rh for each rod is dynamic support for immutable values. When a rod is frozen to immutability, so it cannot be changed again before deallocation, the checksum must still be unchanged when the last reference goes away. You might only audit this in debug mode, but you an ask a rod if it is immutable, and enforce absence of mutation by accident.

Now here's the interesting part: each bp header has a pointer to the parent vat, and to the env. So the cost of those pointers is amortized across all the rods in the same page. In effect, every object allocated from a vat is a subclass of rod, and every rod has a pointer to the allocating vat and to the runtime env interface at zero space cost. It becomes unnecessary to pass vat and env as separate function arguments, since they are always available from any object in a vat. You can also add a handle reference to any sub-object inside a rod, because you can find the associated rh for any address in the page that is part of the rod space. So you can hold references to sub-fields which keep the parent object alive, without any problem, as long as you avoid accidental fragmentation cost (keeping something larger alive only to use a small sub-field).

What about details related to programming languages?

My overall objective is to write tools converting content into different forms, including source code rewrite transformations. (I tell my sons that most of programming is turning inputs into different outputs.) But that is little affected much by memory management per se. Except the i/o model of embedded languages can stream refcounted immutable rods between agents in a process pipeline. That helps get operating system style features into programming language semantics, where a language can also have a process model.

I expect to write a garbage collected Lisp using a vat as the heap, which allocates new bp instances for stop-and-copy before releasing old instances. The uniform use of standard book pages makes re-use efficient without fragmentation across multiple vats.

For bootstrapping, I probably want a simpler and dumber Lisp first, just to perform operations on S-expressions used to express whatever high level policies are used by the system in configuration, or in manifests describing large scale transformations. This can be done with a primitive AST interpreter over a refcounted tree of nodes allocated by C-oriented code. Graph cycles would be pathological unless manually handled or automatically detected. But it's not worse than other C manual memory schemes.

One field in each rh with metainfo describing each rod is the ID of a schema descriptor: the plan of any struct inside the rod. Among other things, it would exactly describe the location of embedded handles in the rod, so reference releasing can be automated. The idea is to be able to release all resources when a lightweight process is killed, without requiring as much manual intervention by a user. A secondary purpose is to make releasing large graphs incremental and uniform, managed by a background rod releaser, which incrementally walks a graph using the rod metainfo as a guide.

A second purpose in putting a plan ID in the rh for each rod is generic debug printing support: I want to be able to print everything. But manually writing a print method for everything is a pain. I would rather write a standard metainfo description of everything, especially since this can be emitted from a source code analysis tool. Then a single generic printer util can process this standard format guide when walking graphs. (Note: to stop huge graphs from being printed, a field in a plan can be marked as best cited briefly, rather than printed in full, and this also helps prevent following cycles due to back pointers.)

Note I am not describing handles, which are like smart pointers, as they little affect memory management and PL topics. But each rod refcount comes from an alias in a handle, which associated metainfo with the rod pointer, including a copy of the plan ID and generation number, which must match when the ref is released. (That detects failures to keep refcounted objects alive as expected.) Refcounting is not manual. It occurs as a side effect of aliasing via handles. When auditing is enabled, each handle can hold the ID of a backtrace that added a reference, and this finds refcount leaks. On a 64-bit system, the size of a handle is twice the size of a pointer: the metainfo is pointer-sized in total. This requires a library provide standard collection classes that work with handles too, when ownership of objects is expressed as collection membership. Each handle includes a bitflag field which, among other things, allows a reference to be denoted readonly, even if the rod is mutable, for copy-on-write data structures like btrees that share structure. (This is useful for transient collections expressing "the original collection plus any patches that might occur" without altering the original collection.)

## Controlling Reductions

Haskell has an option for user defined reductions. I have no idea how it works.

Felix also has two ways to do this.

The first way is complete but hard to use: the parser accepts user defined patterns in the form of grammar productions, and then the user writes Scheme code to manipulate the parse tree to produce any combination of a fixed set of AST terms they like.

The second way is simpler but typed, and looks roughly like this:

reduce strings
| (a:string,b:string) : a + b=> scat ([b,a]) ;
reduce scat2
| (x:list[string], y:string) : scat ([y, scat x]) => scat (Snoc (x,y)) ;

Its polymorphic though that's not seen in this example. The vertical bar indicates you can provide several alternatives which are tried in sequence. The algorithm applies each reduction, and each is applied top down. The above example doesn't work! The intent was to replace string concatenation using a left associative binary operator with a faster function scat, operating on a list. Clearly I needed the algo to work bottom up.

The contrast between these two methods is that the first method has a complete general purpose programming language available to perform the reductions, the pattern matching uses a fixed algorithm (the GLR+ parser). The second method recognises terms and performs the reductions using a fixed algorithm so is much weaker.

What I want is something less complex than having to hand write the whole reduction machinery in a general purpose language (I could do that, using dynamically compiled and loaded Ocaml modules but it's too hard for end users).

So I'm looking for a *compromise* which is reasonably simple, but also reasonably capable. I added alternatives for this reason and am thinking to add an adjective which specifies if a reduction should be applied top down or bottom up. A more complex option is to allow for nested reductions (so that if one reduction succeeds it triggers another one).

Any ideas?

## CFL parsing, and another way to look at the CNF...

I'll implement it and try to break it when I get a chance, but it seems the below should exhibit O(|G| |U|2 n2 log n) worst case behavior (in time), where |G| is the total size of the grammar, |U| is the number of distinct nonterminals in the subset of unary productions U -> w in G, and n the input's length.

Or, what am I missing?

We consider only unambiguous CFGs in Chomsky Normal Form (CNF).

We devise a sort of "type system" over rewritings of the input as we process it from right to left, from end to beginning.

We'll look at the algorithm's work environment as being made of a left context, say Lc, a current symbol, say w (talking about a terminal) and a right context, say Rc.

The left context (Lc) is always a (length-decreasing*) prefix string w0...wi...w of terminals of the actual input (equal to the entire input s = w0...wn-1 initially, and empty at the end, when no more production type rewritings can be applied).

(* albeit, non-strictly)

The right context (Rc) is a mutating list of computed S-expressions storing what has been consumed and "typed" by reading the input from right to left, from end to beginning.

In this scheme, it is constructed so as to be, also, the recorded value of the rightmost-derivative of the parse tree, at the "point" Lc \ w, before Lc gets empty and reduction to the grammar start symbol (if possible) happens, eventually finalizing Rc as the single S-expr (S w0...wn-1) (or failing to do so).

Initially, Rc is empty, and Lc equals the entire input.

In the CNF grammar, we treat the two distinct types of productions thusly:

Productions of the form U -> w (aka rule R1)

Rule R1: U -> w  will be said to "type" w between Lc and Rc as... w: U ("the utterance of w is of type U between any Lc and Rc")

that is, whenever Lc ends with w

rewrite...

Rewriting via R1

Lc Rc ~> (Lc \ w) (U w) Rc
new Lc> \------/ \------/ <new Rc

(takes a list of S-exprs as Rc and prepends the new S-expr (U w) to it, making a new Rc longer by one S-expr, while Lc gets shorter by one terminal, w)

Productions of the form U -> P R (aka rule R2)

Rule R2: U -> P R  will be said to "type" P followed by R in Rc as... P: R -> U ("the utterance of P applied to Rc typed as R promotes Rc to U")

that is, whenever head(Rc) = (P x) and head(tail(Rc)) = (R y)

rewrite...

Rewriting via R2

\----------------------------------------/ <new Rc

(takes a list of S-exprs as Rc and turns it into a new Rc shorter by one S-expr, Lc left unchanged)

Notice the one to one correspondence between mutually ambiguous grammar productions and mutually unsound production types, btw:

E.g., having, in the grammar, the mutually ambiguous

A -> BC
D -> BC

would yield, in this "type system", attempts at applying the mutually unsound (if only signature-wise)

B: C -> A
B: C -> D

Which would obviously be intrinsically problematic from either / both perspectives, and no matter what the input may be, as soon as said input will contain utterances of (a substring already reduced to) B, followed by (a substring already reduced to) C.

Anyway.

Now for a concrete example (full parse / derivation); consider the grammar in CNF (informal meaning given in parentheses):

(nonterminal-only RHS)

S  -> NP  VP  ("a noun phrase followed by a verb phrase reduces as a sentence")
VP -> Vt  NP  ("a transitive verb followed by a noun phrase reduces as a verb phrase")
NP -> Det N   ("a determinative followed by a noun reduces as a noun phrase")
N  -> Adj N   ("an adjective followed by a noun reduces as a noun")

and

(terminal-only RHS)

Det-> a       ("'a' reduces as a determinative")
Det-> the     ("'the' reduces as a determinative")
N  -> boy     ("'boy' reduces as a noun")
N  -> dragon  ("'dragon' reduces as a noun")
Vt -> saw     ("'saw' reduces as a transitive verb")

Which we'll use to "type" thru rewritings of S-exprs in Rc, as:

NP : VP -> S  ("the utterance of 'NP' applied to Rc typed as 'VP' promotes Rc to 'S'")
Vt : NP -> VP ("the utterance of 'Vt' applied to Rc typed as 'NP' promotes Rc to 'VP'")
Det: N  -> NP ("the utterance of 'Det' applied to Rc typed as 'N' promotes Rc to 'NP'")
Adj: N  -> N  ("the utterance of 'Adj' applied to Rc typed as 'N' promotes Rc to 'N'")

and

a     : Det   ("the utterance of 'a' is of type 'Det' between any Lc and Rc")
the   : Det   ("the utterance of 'the' is of type 'Det' between any Lc and Rc")
young : Adj   ("the utterance of 'young' is of type 'Adj' between any Lc and Rc")
boy   : N     ("the utterance of 'boy' is of type 'N' between any Lc and Rc")
dragon: N     ("the utterance of 'dragon' is of type 'N' between any Lc and Rc")
saw   : Vt    ("the utterance of 'saw' is of type 'Vt' between any Lc and Rc")

After iteration 0 (init)

   (the          young          boy         saw         a         dragon)
Lc>\--------------------------------------------------------------------/ ( ) <initial Rc

(NB: Rc = empty list)

After iteration 1, via R1

   (the          young          boy         saw         a)        dragon
Lc>\-----------------------------------------------------/       (N dragon)1
\--------/ <new Rc
(NB: Rc = list of one root S-expression)

After iteration 2, via R1

   (the          young          boy         saw)        a          dragon
Lc>\-------------------------------------------/       (Det a)1   (N dragon)2
\-------------------/ <new Rc

(NB: Rc = list of two root S-expressions)

After iteration 3, via R2

   (the          young          boy         saw)        a         dragon
Lc>\-------------------------------------------/   (NP (Det a)   (N dragon))1
\-----------------------/ <new Rc
(NB: Rc = list of one root S-expression)

After iteration 4, via R1

   (the          young          boy)        saw          a         dragon
Lc>\-------------------------------/       (Vt saw)1(NP (Det a)   (N dragon))2
\--------------------------------/ <new Rc

(NB: Rc = list of two root S-expressions)

After iteration 5, via R2

   (the          young          boy)        saw         a         dragon
Lc>\-------------------------------/
the          young          boy    (VP (Vt saw)(NP (Det a)   (N dragon)))1
\------------------------------------/ <new Rc

(NB: Rc = list of one root S-expression)

After iteration 6, via R1

   (the          young)         boy          saw         a         dragon
Lc>\------------------/
the          young       (N boy)1   (VP (Vt saw)(NP (Det a)   (N dragon)))2
\-----------------------------------------------/ <new Rc

(NB: Rc = list of two root S-expressions)

After iteration 7, via R1

   (the)         young           boy          saw         a         dragon
Lc>\---/
the         (Adj young)1  (N boy)2   (VP (Vt saw)(NP (Det a)   (N dragon)))3
\-------------------------------------------------------------/ <new Rc

(NB: Rc = list of three root S-expressions)

After iteration 8, via R2

   (the)         young          boy          saw         a         dragon
Lc>\---/
the      (N (Adj young)  (N boy))1  (VP (Vt saw)(NP (Det a)   (N dragon)))2
\---------------------------------------------------------------/ <new Rc

(NB: Rc = list of two root S-expressions)

After iteration 9, via R1 (final Lc is empty)

    the           young          boy          saw         a         dragon
(Det the)1 (N (Adj young)  (N boy))2  (VP (Vt saw)(NP (Det a)   (N dragon)))3
\--------------------------------------------------------------------------/ <new Rc

(NB: Rc = list of three root S-expressions)

After iteration 10, via R2

    the          young          boy          saw         a         dragon
(NP(Det the) (N (Adj young)  (N boy)))1 (VP (Vt saw)(NP (Det a)   (N dragon)))2
\----------------------------------------------------------------------------/ <new Rc

(NB: Rc = list of two root S-expressions)

After iteration 11, via R2

       the          young          boy         saw         a         dragon
(S (NP(Det the) (N (Adj young)  (N boy))) (VP (Vt saw)(NP (Det a)   (N dragon))))1
\-------------------------------------------------------------------------------/ <new Rc

(NB: Rc = list of one root S-expression)

Success, with the expected parse binary tree:

       the          young          boy         saw         a         dragon
Det          Adj            N           Vt          Det       N
N                                   NP
NP                                     VP
S

Disclaimer edit:

this is just genuine amateur research from a practitioner who's never been in academia (that would be me). I've obviously posted it here on a friendly LtU only because I'm (still) wondering if I'm on to something / or what am I missing. Bear with me (and thank you in advance).

## Refining Structured Type System

This is my first more serious paper on Structured Type System.

[Abstract]
...
As every theory needs some syntax form to express its elements, a road to a theory about theories leads through a syntax defining land, so structured type system, in the first place, provides a flexible generalized text parser that builds up internal abstract syntax trees (AST) from input data. The other aspect of theory about theories inevitably covers the meaning of input data. This is called semantics, and this is the point where structured type system provides a possibility to define deeper connections between syntactic elements of AST-s. For this purpose, structured type system uses a kind of functions known from functional programming paradigm. These functions are able to process any data corpus, being natural or artificial language translation, which in turn happens to be just enough for running any complexity task used to analyze existing and calculate new data from an input.
...

In short, we use BNF-ish grammars as types for function parameters and function results. Some nice constructions can be made by combining grammars and functions. One of the most important properties of structured type system is its ability to additionally extend grammars outside the grammars definitions, all based on function result types. It is fairly simple: where a certain type of expression is expected, there a grammar that results with the same type can be used, and there goes syntax extensibility. Conveniently, we can combine grammar definitions and their inputs in the same source code file.

I was hoping to get some feedback and critics from this community before attempting to get more publicity to the paper. This is an important milestone to me and I want to thank You all for being so inspirational community during my research.

## Cool stuff from recent conferences

I heard of some good stuff. How about someone who was there post the headline worthy papers?

## Céu: Structured Synchronous Reactive Programming (SSRP)

Céu is a Esterel-based synchronous language:

It appeared in LtU in the past in an announcement of the "SPLASH: Future of Programming Workshop" program.

In this new public version, we are trying to surpass the academic fences with a more polished work (docs, build, etc).

In summary:

• Reactive: code executes in reactions to events
• Synchronous: reactions run to completion in discrete logical units of time (there's no implicit preemption nor real parallelism)
• Structured: programs use structured/imperative control mechanisms, such as "await" and "par" (to combine multiple awaiting lines of execution)

Structured programming avoids deep nesting of callbacks letting programmers code in direct/sequential/imperative style. In addition, when a line of execution is aborted, all allocated resources are safely released.

The synchronous model leads to deterministic execution and simpler reasoning, since it does not demand explicit synchronization from the programmer (e.g., locks and queues). It is also lightweight to fit constrained embedded systems.

We promote SSRP as a complement to classical structured/imperative programming like FRP is now to functional programming.

## Archaeological dig to find the first Lisp example of the Y-combinator

I'm trying to find the first Lisp examples of the Y-combinator. Beyond that I am also trying to find the first time the Y-combinator was demonstrated using the factorial function and the mutually recursive definition of odd/even.

What works should I be looking at? The first Scheme paper references fixed-point combinators at page 16 and also shows the familiar LISP definition of the factorial function. But, it does not express the factorial function using a fixed-point operator.


text:
• Editorial Board +
• For Contributors +
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Journal Search Engine
ISSN : 1598-7248 (Print)
ISSN : 2234-6473 (Online)
Industrial Engineering & Management Systems Vol.19 No.1 pp.3-13
DOI : https://doi.org/10.7232/iems.2020.19.1.003

# Low-Carbon Economy as the Main Factor of Sustainable Development of Energy Security

Ivan A. Kapitonov*
Higher School of Tariff Regulation, Plekhanov Russian University of Economics, Moscow, Russian Federation
Department of Energy Policy, Institute of Economics of the Russian Academy of Sciences, Moscow, Russian Federation
*Corresponding Author, E-mail: k4pitonovi@yandex.ru
February 11, 2020 February 25, 2020 February 29, 2020

## ABSTRACT

The concept of low-carbon development involves several interrelated tasks: improving energy efficiency, using renewable energy, protecting and improving the quality of greenhouse gas sinks, limiting or reducing emissions, developing greenhouse gas absorption technologies, eliminating subsidies, and other methods of promoting environmentally destructive activities. The implementation of the principle of low-carbon development is in line with the gradual transformation of the world energy – the transition from fossil fuels as the main source of primary energy resources to other sources of energy. The relevance of the study is determined by the fact that the most important element of development is the possibility of consumption of energy resources and, accordingly, the possibility of their production in the context of economic changes. The methods of statistical analysis, mathematical modelling, the method of criteria estimates were used in the work. Based on the developed mathematical model of behavior of the control object, it is possible to predict the state of stable equilibrium of the energy security territory. The novelty of the article is that the issue of the sustainable development structure of energy security is raised for the first time.

## 1. INTRODUCTION

Today, the security issue reverberates and the reason for its appearance is very transparent – the growing threat to humanity and its multi-vectority. Among threats there are the following: destabilization of the system of international economic relations, high degree of uncertainty of the participants’ actions in the economic space, disintegration tendencies, escalation of international tension, permanent occurrence of global and local crises, limited world resources and their high cost. Recently, the security of the state, the security of the territory and the security of the individual are not considered separately – they are interconnected and interdependent. If any objective or subjective reason (for example, war) represents a threat to national security, it represents a threat both to the enterprise and to the private individual. All this determines the relevance of the study of world experience in solving complex problems of economic security of any subject of management and its institutionalization (the historical process of transition from the phenomenon, which is self-governing and self-organized to managed and organized).

Contributes to the preservation of economy and the use of alternative energy. Low-carbon economy – in the simplest interpretation of the term – is an economy based on energy efficiency, reducing greenhouse gas emissions, and increasing the share of renewable energy sources (Rosa et al., 2018). Therefore, the development of such an economy will contribute to reducing the negative impact on the components of the environment. Combining the desire to improve energy efficiency, to increase the share of renewable energy sources and thus to reduce greenhouse gas emissions, the low-carbon economy is recognized by everybody as a way to implement the concept of sustainable development – a development in which the impact on the environment remains within the economic capacity of the biosphere, so that the natural basis for the reproduction of human life – a source of economic and social well-being – is not destroyed (Buyankin, 2018;Atanelishvili and Silagadze, 2018).

The actions to reduce carbon dioxide emissions into the atmosphere should not only slow down the pace of climate change. This, in its turn, will affect economic and social systems. In the industrial structure, low-carbon agriculture reduces dependence on oil and gas energy and creates an organic, ecological and highly efficient way of development (De Boni, 2017). Low-carbon energy with low carbon emissions creates favorable conditions for the rapid development of low-carbon industry, as a result of which the industry with high carbon emissions will gradually be replaced and its development will slow down. The development of low-carbon logistics will help to improve efficiency and create more efficient logistics lines and routes. Low-carbon services market expects more rapid development.

The most famous foreign scientists, supporters of green economy define a green economy as an economy that is a dependent component of the natural environment within which it exists and is a part of the global ecosystem (Karazijienė and Jurgelevicius, 2017). Since 2009, in the OECD and UN documents, the term “green economy” and its derivatives has become key in determining the directions of sustainable development.

The integrity and sustainability of the state historically depends not only on the effective functioning of the triad of its main systems: political, military and economic, but also from the security system, which unites international, regional, public and state institutions (organizations, structures, authorities) and economic entities, which protect the state from the negative influence of internal and external threats on the basis of the development of norms and laws to ensure the vital interests of citizens, as well as exclusion of the possibility of causing damage to the economy.

## 2. LITERATURE REVIEW

The influence of political, economic, ecological and spiritual crises, the lack of resources and ideas, and the arbitrary nature of aggressive processes necessitate a revision of existing approaches to the issue of ensuring economic security of the country (Hameiri and Jones, 2015;Pirogov and Cherentsov, 2018). Since the regions are the locomotives of the economy of any state, of course they feel this influence (Kosai and Unesaki, 2016). Therefore, the cornerstone of the question is the development of the theoretical and methodological foundations of the synergetic management of economic security in particular of regions or other forms of territories in general (Bialasiewicz et al., 2005).

There are several reasons, which, in fact, concern not so much the general understanding of the economic security of the territory, but it just as a control object, which makes it possible to assume that such specificity exists (Levy, 2009). Therefore, before examining the economic security of the territory as a managed system, it is necessary to study its subject field, that is, territories in terms of a systematic approach (Kononov and Kononov, 2016;Santos et al., 2018). According to the terms of the general theory of systems (systemology), any territory is a system in which the set of inputs into production resources – labor costs (labor items, labor means, labor fources) are transformed into finished products and services (Feng, 2016; Karazijienė and Jurgelevicius, 2017). And the territory also operates within a large system – the foreign policy, economic, social and technical environment with which it con-stantly has complex relationship (Russett, 1979).

In cybernetics the territory can be represented as the intersection of two higher-level systems: the supersystem “society” and the supersystem “resources” (Ivanov, 2016). According to the terms of cybernetics, the founder, the sociotechnical system consists of two subsystems: managed and managing (Treu, 2018). A managed system or managed object is determinant, since it is a set of elements for which control relationships arise (Lappalainen, 2007). That is, the management system of the territory relates to its organizational management structure (Fischhendler and Katz, 2013).

Economic security of the territory as a managed system inherited almost all the properties of the system, as well as its inherent systemic principles: openness, nonlinearity, dynamism, dissipativeness, emergence, homeostaticity, attraction, bifurcation, fractality (Matúš, 2015). In it, processes of self-organization and self-deorganization with such characteristics (according to the scientific content of the synergetics), described by G. Hacken, namely (Froestad and Shearing, 2017), can take place:

• – intensive exchange of energy, information or other resources of the system with the environ-ment, and it is completely chaotic;

• –the behavior of the system is characterized by spontaneity and is described by several values – parameters of order and control parameters (information overload of the system disappears);

• –the presence of a critical value of the control parameter (associated with the arrival of energy, information, knowledge or other re-sources), in which the system is able to lose the state of stable equilibrium (homeostasis); the quantitative accumulation of certain deformations of vital parameters leads to qualitative changes;

• –the new state is due to the harmonious (coherent) behavior of the elements of the system;

• –a new state exists only with a continuous nonstop flow of energy, information, knowledge or other resources (in the presence of internal and external sources of oscillations (fluctuations) of the system). The increase in the intensity of the exchange leads to the critical transitions of the system, resulting in its structure becoming complicated up to the occurrence of turbulent chaos.

## 3. MATERIALS AND METHODS

A significant number of scientific papers became the basis for understanding the economic security of the territory as a managed system (Kazantsev, 2012). But quite often security in the economy is considered chaotically and haphazardly, from which comes the relevance of its research on the part of the system approach, as well as in-depth study of its structure, functions, processes, and goals with the help of iterative analysis.

The article uses the method of comparative statistical analysis, which makes it possible to analyze the energy structure of the states and individual territories. In the framework of the analysis, the statistical method is used to obtain statistical data. The method of criteria assessments is mainly used to form a scale that assesses the adequacy of the efforts made and forms a general framework for understanding how progress in achieving socio-economic parameters is determined. It is also worth noting that the method of mathematical modeling, which is used to build a model of forecasting energy sustainable development of territories, and in a local format, as the model does not consider the context of globalization.

## 4. RESULTS AND DISCUSSION

On the basis of the analysis of the current state and trends of alternative energy development, the criteria were defined for assessing the interaction of the market environment indicators, state regulation of investment and innovation activities, which will reflect the complete set of interdependencies for assessing the development of renewable energy in the countries (Mažylis and Pikšrytė, 2013).

For further research and calculations, a sample of 18 countries has been formed that represent and as a whole characterize most regions of the world, including the USA and Canada, China, India and Brazil, the most developed countries of Europe – France, Norway, Italy, Spain, Germany, Great Britain, countries Eastern Eu-rope – Hungary, Poland, Czech Republic, Slovenia and Slovakia, as well as the Baltic countries – Lithuania and Latvia.

The sample obtained represents 32% of world GDP and 21% of the world population. Therefore, it is sufficiently indicative of the quantitative and qualitative composition in the context of the analysis of the development of renewable energy. It has been decided to collect data for the 10 years (2008-2017) (Asan et al., 2016).

As a result of the statistical data collection, 14 tables with initial values of indicators for the further analysis and calculation of the integral indicator of the effectiveness of the alternative energy development have been formed. Data for the analysis of state support are based on the quantitative assessment of incentives for the development of alternative energy in the state through the following factors:

• - special “green” tariff;

• - investment or tax privileges;

• - reduction of income taxes, energy taxes, VAT, etc.;

• - subsidies for electricity producers;

• - public investment, loans, grants, capital subsidies.

If a country uses one of the given incentives, we assign it a value of 1, if it does not use – 0. In the end, the corresponding parameter will be from 0 to 5, where zero will characterize the lack of state support, and five – maximum state support. The expert estimation method has assigned weight coefficients to the lower-level parameters in such a way that, within each of the criteria, the sum of the coefficients equals one (Table 1).

By analogy, we define the corresponding values for the second level indicators. We determine that the estimation of the market environment will have a coefficient of 0.45, that is, the most will affect the value of the final integral indicator, the state regulation – 0.3 and the investment and innovation environment – 0.25. According to the results of the calculations we have the dynamics of the integral values of the market environment criteria, state regulation of the investment and innovation environment of renewable energy sources (Table 2).

The priority feature in the growth of the “green” economy is a radical increase in energy efficiency, so the term “low-carbon” economy is widely used. It is seen as the basis of the green economy and a model of the economy of the future. Implementation of low-carbon development involves the solution of several interrelated objectives: improving energy efficiency, using renewable energy, protecting and enhancing sinks of greenhouse gases, limitation or reduction of emissions, development of technologies for sequestration of greenhouse gases, phasing out subsidies and other methods of encouraging environmentally destructive activities.

The implementation of the concept of low-carbon development is in line with the gradual transformation of world energy – the transition from fossil fuels as the main source of primary energy resources to other sources of energy. In accordance with the highly probable scenario of human civilization, the structure of energy production and consumption with the absolute predominance of hydrocarbons at the present time (Figure 1) will change in the direction of increasing the share of renewable energy sources in the structure of energy consumption (Figure 2).

In recent decades, the low-carbon economy has become a priority of a number of political programs: the European Union Development Strategy 2020 “Europe 2020”, which includes a number of mandatory targets for the low-carbon economy; the Road Map of the European Commission to move towards a low-carbon economy until 2050 with decarbonization priorities for the energy and transport sectors, etc.

We see that according to the integral estimation of the market environment (Figure 3) Norway has the best characteristics, and the corresponding dynamics is maintained throughout the analyzed period, the values fluctuate within the range of 0.45-0.47. The overall dynamics of development is indicative, as the positive dynamics for the period 2008-2017 is observed in relation to absolutely all the countries being analyzed, which indicates the global tendencies of growth of qualitative market characteristics of the renewable energy sources development (Strielkowski and Lisin, 2016). The leaders are also Spain, Germany, Italy, Latvia, Brazil and the United Kingdom. As could have been predicted, the Czech Republic, Slovakia and Slovenia have the worst performance, ranging from 0.007 to 0.025, which is more than 15 times less than in Norway. It should be noted that in the period from 2008 to 2014, the Czech Republic had the lowest rating, but starting from 2015, the integrated assessment of the market environment outstripped the respective values of Slovakia and Slovenia (Mohapatra, 2017).

Figure 3 shows that Canada has the best integrated assessment of government regulation, showing a value greater than 0.8 within the entire analyzed period. Leaders are also Norway, Great Britain, Germany, the USA and Lithuania. The dynamics of Poland and Slovakia are interesting as they showed the greatest increase of the corresponding estimate for 2008-2017, as well as Slovenia, although it is characterized by the lowest values within the range of 0.28-0.48, also had a significant growth, coming up close to 0.5 in 2017.

According to the calculation of the integral indicator which characterizes the investment and innovation activity in the country, we have two obvious leaders – the USA and China, with the United States characterized by rather stable dynamics at the level of values of 0.68-0.77, while for the analyzed 10 years, the PRC has significantly improved its position by increasing the volume of patents in the field of RES, as well as by reducing the amount of CO2 emissions into the atmosphere. Among the lowest values, we have indicators of such countries as Brazil, Czech Republic, India, Slovakia and Slovenia. Most of the sample is in the middle range, characterizing the investment and innovation environment of the integral estimate from 0.4 to 0.6.

In reality, however, member states do not share the pan-European vision of how the EU energy market should be organized and are making every effort to implement their own national energy policy. Thus, on the development of RES in the EU for the period up to 2030:

• – Denmark supports the targets for 2030;

• – Lithuania believes that RES development targets should be set after a thorough assessment of the impact on industries and specific member states;

• – Austria is strongly in favor of ensuring the safety of the energy system and considering the social consequences;

• – Finland calls for an indicative or moderately binding target;

• – France calls for the consolidation of renewable energy development targets at a later stage, after partial harmonization of the renewable energy support system and approval of the renewable energy integration program in the energy system;

• – Portugal is ready to achieve this goal only in cooperation with other states;

• – Estonia is ready to support renewable energy development targets if the EU's proposed support mechanisms ensure that the country receives a significant economic impact;

• – Romania supports renewable energy development targets to be set by member states.

According to the world economic forum, in 2016, renewable energy became cheaper or equal in price with fossil energy in more than 30 countries. Australia, Brazil, Mexico, Chile, Germany, Israel, New Zealand, Turkey, Japan, etc. are among these states. These reports noted that solar and wind energy have now become quite competitive in many regions. Over the past five to seven years, solar technologies have demonstrated a unique growth rate for the energy sector of installed capacity – at the level of 30-40% per year. According to this indicator, solar energy is confidently ahead of all other energy technologies. Such high rates of development are determined by a significant reduction in the cost of the main technological equipment of solar power plants and, first of all, photovoltaic modules (PVM) for direct conversion of solar radiation into electrical energy.

According to the forecasts of the largest UBS Bank, the rejection of the construction of traditional thermal and nuclear power plants in Europe will occur within the next ten years. In Asian countries, investment in the construction of solar photovoltaic plants (PVP), as noted by General Electric Co., also become more profitable than in projects with traditional thermal power plants. In Russia, according to the calculations produced by LLC “Aliansi”, cost-effective photovoltaic power station may be currently constructed without government subsidies in the South of the European part of Russia, in Altai, in Southern Siberia, the Far East, on Sakhalin. In 2014, the cost of electricity from solar and wind power plants was equal (Figure 4). After 2015, there is already a lower cost of “solar” energy in comparison with the “wind” one. This becomes the basis for the advanced development of solar energy in the future. According to estimates made by analysts in 2011, large-scale development of solar energy was predicted by 2035-2040.

The expected seven- and eight-fold increase in electricity production over the ten-year period (2015-2025) should bring solar energy to the first place among all renewable energy generation technologies. These circumstances are caused by a combination of the following favorable factors: a significant reduction in the cost of basic solar equipment, growing progress in the field of accumulation of large amounts of electricity and the emergence of highly efficient storage devices based on fundamentally new technical solutions, and state support.

Geothermal energy is the fourth largest and fastest among other types of renewable energy. Installed capacity of geothermal power plants increased from 5.83 GW in 1990, 7.97 in 2000 to 10.72 GW in 2018. Average annual growth between 2000 and 2018 was 3.1%. The installed capacity of geothermal plants in the leading countries in 2018: US to 3.09, Philippines – 1.9, Indonesia 1.197, Mexico – 0.958, Italy – 0.843, New Zealand – 0.628 and Japan – 0.536 GW (Sakulyeva and Kseniia, 2019).

In fact, energy security is now defined as the elimination of the threat that the energy aspect will be-come a potential obstacle to the economic growth of states in the long term. While for large net importing countries, lack of energy can be an obstacle to maintaining sufficient growth rates, for a country the development and economic growth of which are significantly tied to energy exports, these are the factors limiting production and exports. High prices, the uncertainty of the forecast for the duration of the period of high prices, the reliability and sufficiency of the energy delivery infrastructure, the reliability of suppliers – all these issues are of great importance for world energy security.

So, according to the results of the integrated assessment of the economic support for the development of alternative energy (Figure 5), Norway is of paramount importance, while the lowest is the Czech Republic, Slovakia and Slovenia. Also, Canada, Germany and the USA have a high level of RES output. China has one of the fastest paces of renewable energy development, which at the beginning of the analyzed period had an integral estimate of 0.32 and increased it by almost 60% – up to the leaders of the sample (0.52). In general, we have global positive dynamics in the period 2008-2017 in the field of alternative energy and stimulating its development through state regulation of the industry and improving the investment and innovation climate of the states as a whole, therefore, we can conclude that in the near future trends will be maintained. Particular attention should be paid to India and China, since, according to many forecasts, these two countries will show the highest growth rates of renewable energy use, as they have significant potential.

The study of economic support for the development of alternative energy as part of the energy security of the territory makes it possible to single out the following points: over the past few years the dynamics of generation and consumption of energy from renewable sources have been steadily growing, the share of alternative energy sources has been increasing, among which water energy dominates and the proportion of solar energy has also been rapidly increasing. At the highest state level, an action plan is being developed to improve the energy efficiency of the economy, in particular, strategies for the development of renewable energy have been developed, and a bilateral dialogue is underway with international agencies on the development of alternative energy, in particular the International Renewable Energy Agency. Also, one of the highest “green” tariffs is currently in force, which remains a powerful incentive for the development of domestic renewable energy, but at the same time there is an imperfection of the legislative base re-garding some aspects of its use, in particular, by house-holds.

In 2004, for the first time, a synergetic model of the stability of an average firm was offered, in which the important role of the control parameters that they play while choosing the path to one or another steady state was revealed. Synergetic modeling of economic processes and phenomena is based on two main points:

• – the choice of order parameters, that is, macroscopic variables, quantitatively characterizing the main connections in the system and control parameters, that is, external conditions in which the system cannot change, and therefore, has to adjust to them.

• – making up the main proportions, which states that the rate of change of the order parameters with the influence of time is proportional to their growth minus their loss. The ratio of proportions is formed on the basis of experience.

Mathematical modeling of the territory sustainability provides the ability to predict the state of stable equilibrium, which allows us to develop appropriate measures to manage the economic security of the territory. The sustainability of the industrial activity of a territory is determined by such parameters as: the number of highly organized personnel, the availability of equity capital and the volume of investments (credit). That is, in this case, the concept of sustainability of a territory is essentially equivalent to the concept of sustainability of a company, since with sustainable development of a territory; there is a consolidation of social and economic policies and a process of structuring the economy of a territory as a corporate structure. The mathematical model of the stability of the territory is presented in the form of a system of three differential equations (Eq. 1):

${ Y ˙ 1 = – γ Y 1 + α Y 2 + α Y 2 Y 3 ≡ f 1 ( Y 1 , Y 2 , Y 3 ) Y ˙ 2 = μ Y 2 – β Y 1 + μ Y 3 – β Y 1 Y 3 ≡ f 2 ( Y 1 , Y 2 , Y 3 ) , Y ˙ 3 = – ω Y 3 + δ Y 2 ≡ ( Y 1 , Y 2 , Y 3 )$
(1)

where $Y 1 = Y 1 ( t )$ – the number of employees at the time, people; $Y 2 = Y 2 ( t )$ – the amount of personal capital at the time t, rub.; $Y 3 = Y 3 ( t )$ – the amount of borrowed capital, i.e. the loan at the time, rub; $Y ˙ 1 , Y ˙ 2 , Y ˙ 3$ – derivatives with respect to the independent variable .

The values of the coefficients are as follows:

• α: the proportionality coefficient, showing how much of its capital a territory can allocate to attract new employees. Namely, promotional costs (including the Internet), informing about the prospects for expanding the scope of activity of the territory, about the well-organized work of employees, about the cohesion of the team, as well as the respectability of external and internal design offices. In short, these are the costs to form a reputation on the labor market. However, from a mathematical point of view, this coefficient does not greatly influence the type of sustainability of the system: it is important that it is not close to zero.

• γ: proportionality coefficient, which summarizes the various reasons as a result of which an employee may be dismissed or he/she quits himself/herself. Employees’ turnover threatens to lose accumulated experience, skills, and competencies, which in turn affects the increase in the cost of finding, selecting and training new employees with lower productivity.

• μ: proportionality coefficient, which shows the effectiveness of capital investments in the territory (taking into account the influence of various taxes, payments, fees – they carry a dissipative effect on the “energy” of the territory and threaten with a decrease in profitability).

• β:proportionality coefficient characterizing the amount of territory expenses per employee. This includes the costs of wages, deductions for social events and the cost of the territory for the maintenance of labor; except for those included in the wage fund (these include expenses of the territory for social security of workers, their cultural and everyday services, for their housing, on vocational training and other labor costs).

• δ:proportionality coefficient, which affects the availability of equity capital. Its sufficient share testifies to a high level of financial health of the territory, solvency in the long term. If the share of own capital is low, then the company faces bankruptcy.

• ω:proportionality ratio indicating the difficulty to obtain a loan. For example, a high interest loan increases the risk of non-repayment, which threatens with bankruptcy.

For an average firm, the coefficients α and β should be relatively large, since both relate to expenses for employees, and the coefficient μ and γ, on the contrary, should not be large because, firstly (in the case of μ), the profit from operations on the market is not too high, otherwise the company would be big, not medium; and secondly (in the case of γ), in a civilized society in the territory, the employee turnover rate is low.

In the model $Y 1 , Y 2 , Y 3$ , variable factors or in the synergetics thesaurus they are called order parameters; α, γ, μ, β, δ, ω are constant factors (proportionality coefficients) or control parameters. We clarified the model of sustainability of the territory as a social structure. The second and third components of the right side of the first equation of the mathematical model (1) indicate that the increase in new employees is proportional to the availability of equity capital and the use of credit for its intended purpose (capital meeting $− α Y 2 Y 3$ ) (people prefer working in a large company, which, in turn, more invests in the formation of its attractive image), and the first member indicates a reduction in the number of employees in the enterprise, mainly due to the lack of funding and due to other, less important reasons .

The members of the right-hand side of the second equation of the mathematical model (2) indicate that the rate of growth of equity is determined in proportion to its value, which is obtained by investing the amount of capital and credit (terms) minus the costs associated with salaries and loan maintenance (summands $μ Y 2 и μ Y 3$ ). The coefficients µ and β of the equation reflect the share of the use of equity and credit for the salary of employees and credit resources.

The third equation of system (2) reflects the fact that the rate of turnover of the credit volume (the possibility of additional lending) is positively affected by the availability of equity capital (the larger the capital of the territory is, the more willingly they are given a loan) and in a negative way – already existing credit obligations $( δ Y 2 )$ . We have also proposed “soft modeling” or a technique for qualitative analysis of a mathematical model (1) first, stationary relation is found, and equating the derivatives to zero, namely, a nonlinear system of algebraic equations (Eq. 2) is considered:

${ − γ Y 1 + α Y 2 + α Y 2 Y 3 = 0 μ Y 2 + μ Y 3 − β Y 1 − β Y 1 Y 3 = 0 , δ Y 2 − ω Y 3 = 0$
(2)

From the third equation it follows: $Y 2 = ω δ Y 3$ , while from the first – $Y 1 = 1 γ ( α ω δ Y 3 + α ω δ Y 3 2 )$ . By substituting these expressions into the second equation of an algebraic system, we get (Eq. 3):

$Y 3 [ Y 3 2 + 2 Y 3 + ( 1 − μ γ β α − μ γ δ β α ω ) ] = 0$
(3)
after obvious transformations. So, the first radical is $Y 3 ( 1 ) = 0$ and we have a trivial singular point (0, 0, 0). Two other radicals are written as follows (Eq. 4):
$Y 3 ( 2 , 3 ) = − 1 ± μ γ β α + μ γ δ β α ω$
(4)

Since the desired stationary outcomes must be in the first quadrant, the relations between the coefficients of the mathematical model (2) are easily established, namely:

$μ γ > β α$
(5)

At the same moment the numerical parameters ω and δ acquire arbitrary positive values, following the economic content:

$μ γ β α ω ω + δ > 1$
(6)

Equities (5, 6) are derived from the requirement that one of the radicals $Y s ( 2 , 3 )$ must be positive, that is not limited only with the trivial stationery solution. For this purpose the condition $μ γ β α ( 1 + δ ω ) > 1$ must be met, the realization of which can be in two ways. The stability of stationary solutions is estimated knowing the radicals of the characteristic equation (Eq. 7).

$[ ( ∂ f 1 ( * ) ∂ Y 1 − λ ) ∂ f 1 ( * ) ∂ Y 2 ∂ f 1 ( * ) ∂ Y 3 ∂ f 2 ( * ) ∂ Y 1 ( ∂ f 2 ( * ) ∂ Y 2 − λ ) ∂ f 2 ( * ) ∂ Y 3 ∂ f 3 ( * ) ∂ Y 1 ∂ f 3 ( * ) ∂ Y 2 ( ∂ f 3 ( * ) ∂ Y 3 − λ ) ] Y 1 = Y 1 s Y 2 = Y 2 s Y 3 = Y 3 s = 0$
(7)

here (Eq. 8):

$λ 3 + α 1 λ 2 + α 2 λ + α 3 = 0$
(8)
Or (Eq. 9, 10, 11):
$α 1 = − ( ∂ f 1 ( * ) ∂ Y 1 + ∂ f 2 ( * ) ∂ Y 2 + ∂ f 3 ( * ) ∂ Y 3 )$
(9)
$α 2 = | ∂ f 1 ( * ) ∂ Y 1 ∂ f 1 ( * ) ∂ Y 2 ∂ f 2 ( * ) ∂ Y 1 ∂ f 2 ( * ) ∂ Y 2 | + | ∂ f 1 ( * ) ∂ Y 1 ∂ f 1 ( * ) ∂ Y 3 ∂ f 3 ( * ) ∂ Y 1 ∂ f 3 ( * ) ∂ Y 3 | + | ∂ f 2 ( * ) ∂ Y 2 ∂ f 2 ( * ) ∂ Y 3 ∂ f 3 ( * ) ∂ Y 2 ∂ f 3 ( * ) ∂ Y 3 |$
(10)
$α 3 = − [ ∂ f 1 ( * ) ∂ Y 1 ∂ f 1 ( * ) ∂ Y 2 ∂ f 1 ( * ) ∂ Y 3 ∂ f 2 ( * ) ∂ Y 1 ∂ f 2 ( * ) ∂ Y 2 ∂ f 2 ( * ) ∂ Y 3 ∂ f 3 ( * ) ∂ Y 1 ∂ f 3 ( * ) ∂ Y 2 ∂ f 3 ( * ) ∂ Y 3 ]$
(11)
where equations $∂ f i ( L ) ∂ Y j ( i , j = 1 , 2 , 3 ; f i ( L )$ – correspondingly the right parts of the equation of the mathematical model (1), representing partial derivatives; designation |•| corresponds to the second order determinant in finding the coefficient $α 2$ and the third order determinant in the case of the coefficient $α 3$ .

If $λ 1 , λ 2 , λ 3$ are the required radicals of the (Eq. 6), then the following equation is satisfied (Eq. 12):

$α 1 = − ( λ 1 + λ 2 + λ 3 ) ; α 2 = λ 1 λ 2 + λ 2 λ 3 ; α 3 = λ 1 λ 2 λ 3$
(12)

Specifically, in our case, the determinant (7) takes the form (Eq. 13):

$[ ( − γ − λ ) α α Y 2 − β − β Y 3 μ − λ μ − β Y 1 0 δ − ω − λ ] = 0$
(13)

This is calculated at the stationary point. It should be noted that the radicals of the characteristic equation can be: real (negative and positive), complex (with negative real parts and positive real parts), and also pure imaginary. To study the nature of the equation radiradicals (6) and their signs, the discriminant of the equation is used (Eq. 14):

(14)

Hurwitz conditions $− α 1 > 0 , α 3 > 0 , α 1 α 2 − α 3 > 0$ ; Descartes’ theorem: the number of positive radicals of an algebraic equation is equal or less by an even number of the number of sign changes in the sequence of coefficients $1 , α 1 , α 2 , α 3$ . The same theorem is applied to the number of negative radicals of the equation and changes in the signs of the sequence, that is, to the radicals of the equation (Eq. 15):

$− λ 3 + α 1 λ 2 − α 2 λ + α 3 = 0$
(15)

Specifically, for the trivial singular point of the mathematical model (2), the determinant is written (Eq. 16):

$[ ( − γ + λ ) λ 0 − β μ − λ μ 0 δ − ( ω + λ ) ] = 0$
(16)

And characteristic (Eq. 17):

$λ 3 + ( γ − μ + ω ) λ 2 + ( − μ γ + ω γ − μ ω − δ μ + β α ) λ − γ μ ω − δ μ γ + β α ω = 0.$
(17)

Based on the developed mathematical model of the control object behavior, it becomes possible to predict the state of stable equilibrium of the territory. There are several types of prediction that requires an understanding of the laws of development:

• – Inertial forecasting or extrapolation is that the process continues the trajectory that was before it began, that is, the future parameters of the system are determined on the basis of previous development trends.

• – Scenario design is to determine the likelihood of possible alternatives for development or the desired state in the future turbulent macroenvironment. Scripts depend on the resources invested in the future.

• – Design management is to predict the impact of decisions. It refers to the structuralist approach, in which there is no dynamic prehistory. In the future, this type of forecasting is applied. As noted above, nonlinear dynamics studies the properties of dynamic systems, which are systems that describe a process in time, namely, the transition from a state of stable equilibrium to unstable and vice versa. The main tasks are: first, the search and classification of a particular equilibrium point, and secondly, the identification of sets that either attract (attractors) or repel (repellers).

## 5. CONCLUSIONS

For the qualitative analysis of the dynamic system, which is the territory, the phase space method is used. It is based on the geometry of time and allows you to visualize the behavior of the territory. A significant simplification of the description of the dynamic system state belongs to the advantages. The phase system space is an abstract multidimensional space in which the change in the coordinates of a point (that is, its movement) forms curves (phase trajectories) over time, which vividly describes the evolution of a non-linear, “chaotic” system in a geometric form (phase portrait). The main characteristic of space is its dimension, i.e. the number of parameters that must be set to determine the state of the system. A phase portrait is a collection of all phase paths describing all possible modes. On its basis, it is possible to determine the particular equilibrium point of the system as a guideline to which it will come with time.

In the economic context, the phase trajectories are the trajectories of the economic development of a business entity, or the trajectories of individual business processes (core and supportive) and interests of stake-holders. If all of them are in the area of attraction of the attractor, then there is no need to make quick management decisions to change the situation. On the contrary, if they are not within the domain of attractor’s attraction, then there is a need to make quick management decisions.

The advantages of introducing green energy are obvious. After all, coal, gas, oil, converted by combustion into electricity are an exhausted resource and can be used only once, each kilogram, liter or barrel. Then they will turn into greenhouse gases and other toxic emissions and will never be available to us again. In the case of renewable energy stations, they only capture and convert the energy around them. Of course, their existence has an impact on the environment: photo panels cast a shadow, and wind turbines create aerodynamic resistance to wind and turbulence. Not all the RES are equally environmentally safe. Some can cause environmental damage. In particular, we are talking about hydroelectric power plants (HPP). Many floodplain ecosystems were destroyed due to hydroelectric power plants, which led to a decrease in species diversity. However, in recent years, hydropower has given way to new types of generation: solar and wind power. According to experts, their share of generation will be equal to the share of hydropower plants by 2030.

The current stage of energy development in the European Union is characterized by the following factors: high dependence on energy imports in a limited number of large energy suppliers; high instability of energy prices; increased risk of energy security coming from the producer states and energy transit countries; increasing threat of climate change, insufficient level of links between the energy systems of the EU countries. The use of renewable energy sources (RES) largely provides a solution to all of the above problems. The achievement of the goals set by the European Union in the field of renewable energy development is based on a system of effective tools and mechanisms to stimulate and support this process.

## Figure

Structure of the World Energy Consumption, 2014.

Structure of the World Energy Consumption, 2050.

Integral figures of the market environment criteria.

Forecast of comparative changes in the cost of electrical energy.

Value of the economic support effectiveness index for the development of RES in 2008-2017.

## Table

Figures of weight coefficients of parameters

Integral values of the economic support criteria for the development of alternative energy of the countries in 2008-2017

## REFERENCES

1. Asan, D. , Batyrova, N. , Aubakirova, A. , Abishov, N. , and Kuralbayev, A. (2016), Implementation of a comparative evaluation method of stable socio-economic development, International Journal of Economics and Financial Issues, 6(2), 9-13.
2. Atanelishvili, T. and Silagadze, A. (2018), Xenophon: Economy finds the name, Bulletin of the Georgian National Academy of Sciences, 12(4), 163-166.
3. Bialasiewicz, L. , Elden, S. , and Painter, J. (2005), The constitution of EU territory, Comparative European Politics, 3, 333-363
4. Buyankin, V. M. (2018), Neuroidentification with neuro self tuning to ensure the operation of the current loop of the electric drive with the desired static and dynamic characteristics, Periodico Tche Quimica, 15(30), 513-519.
5. De Boni, L. A. B. (2017), Empirical theoretical proposal for the production of biodiesel, Periodico Tche Quimica, 14(28), 166-174.
6. Feng, S. (2016), Prospects for European and Eurasian security and China’s future choices, Chinese Political Science Review, 1(4), 670-684.
7. Fischhendler, I. and Katz, D. (2013), The use of “security” jargon in sustainable development discourse: Evidence from UN commission on sustainable development, International Environmental Agreements: Politics, Law and Economics, 13(3), 321-342.
8. Froestad, J. and Shearing, C. (2017), Energy and the anthropocene: Security challenges and solutions, Crime, Law and Social Change, 68(5), 515-28.
9. Hameiri, S. and Jones, L. (2015), Probing the links between political economy and non-traditional security: Themes, approaches and instruments, International Politics, 52(4), 371-388.
10. Ivanov, V. V. (2016), Innovative territory as a basic element in the spatial structure of the national innovation system, Regional Research of Russia, 6(1), 70-79.
11. Karazijienė, Ž. and Jurgelevicius, A. (2017), The impact of intangible resources on economy in the EU, Public Policy and Administration, 16(2), 279-295.
12. Kazantsev, S. V. (2012), Economic security and assessment of economic protectability of regions, Regional Research of Russia, 2(1), 34-40.
13. Kononov, D. Y. and Kononov, Y. D. (2016), Rational aggregation of territory in long-term forecasting of energy prices, Studies on Russian Economic Development, 27(6), 649-655.
14. Kosai, S. and Unesaki, H. (2016), Conceptualizing maritime security for energy transportation security, Journal of Transportation Security, 9(3), 175-90.
15. Lappalainen, V. A. (2007), The new security of energy supply directives. A first response to some big questions, ERA Forum, 8(3), 427-434.
16. Levy, J. K. (2009), A case for sustainable security systems engineering: Integrating national, human, energy and environmental security, Journal of Systems Science and Systems Engineering, 18(4), 385.
17. Matúš, M. (2015), The influence of perception on the preferences of the new member states of the European Union: The case of energy policy, Comparative European Politics, 13(2), 198-221.
18. Mažylis, L. and Pikšrytė, A. (2013), EU regulatory policy models’ application in the renewable energy sector, Public Policy and Administration, 12(1), 120-132.
19. Mohapatra, N. K. (2017), Energy security paradigm, structure of geopolitics and international relations theory: From global south perspectives, Geo Journal, 82(4), 683-700.
20. Pirogov, S. P. and Cherentsov, D. A. (2018), Scientific research using the application software package MATLAB, Periodico Tche Quimica, 15(30), 556-562.
21. Rosa, L. D. S. , Jr., Almeida, H. D. S. , Brasil, S. C. S. D. A. , De Morais, A. B. P. , Saraiva, J. B. , Cordeiro, S. B. , Assunção, F. P. C. , and Pereira, L. M. (2018), Study on methods of determination of an ecological flow for the management of water resources of the river basin on Maracaçumé river, Periodico Tche Quimica, 15(30), 27-34.
22. Russett, B. (1979), World energy demand and world security, Policy Sciences, 11(2), 187-202.
23. Sakulyeva, T. A. and Kseniia, Z. (2019), The single window mechanism in the field of external sector of the economy, International Journal of Civil Engineering and Technology, 10(2), 2205-2212.
24. Santos, A. B. D. , Dias, É., C. , Silva, G. P. C. D. , Ribeiro, R. P. , and Silva, A. M. (2018), Lost water volume in water supply system, in north and central west region, Periodico Tche Quimica, 15(30), 497-503.
25. Strielkowski, W. and Lisin, E. (2016), Economic analysis of renewable energy sources in European union, Journal of Environmental Management and Tourism, 4(16), 553-558.
26. Treu, M. C. (2018), Energy: Territory and new landscapes scenarios, City, Territory and Architecture, 5(1), 14.

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text:
PUMPA - THE SMART LEARNING APP

AI system creates personalised training plan based on your mistakes

### Theory:

Constitution - have you heard this word before? What does it mean? Its importance ?

Lets see, Constitution is like a rule book which specifies the roles, powers and functions for legislative, executive and judiciary. Explanation of these terms will follow in upcoming modules.

Basic explanation is:
The basic principles and laws of a nation, state, or social group that determine the powers and duties of the government and guarantee certain rights to the people in it.

India follows a written constitution, means Indian constitution is a written document.
Dr. B. R. Ambedkar is recognized as the Father of the Constitution of India’.

The major part of the constitution came into force on January $$26$$, $$1950$$. This day is referred to in the Constitution as the ‘date of its commencement’, and celebrated as the Republic Day.

January $$26$$ was specifically chosen as the ‘date of commencement’ of the Constitution because of its historical importance. It was on this day in $$1930$$ that  Purna Swaraj day was celebrated, following the resolution of the Lahore Session (December $$1929$$) of the Indian National Congress.

Having learnt about what Constitution means, lets learn about the type of government our constitution has given us.
Reference:
https://commons.wikimedia.org/wiki/File:Constitution_of_India_(calligraphic)_Cover.png


text:
# Tutorial 3: Customize Models¶

MMEditing supports multiple tasks, each of which has different settings. Fortunately, their customization is similar. Here, we use a super-resolution model, BasicVSR, as an example in this tutorial. You will be able to define your model based on your own needs after this tutorial.

We first need to create BasicVSR in mmedit/models/restorers/basicvsr.py .

from ..registry import MODELS
from .basic_restorer import BasicRestorer

@MODELS.register_module()
class BasicVSR(BasicRestorer):

def __init__(self,
generator,
pixel_loss,
train_cfg=None,
test_cfg=None,
pretrained=None):
super().__init__(generator, pixel_loss, train_cfg, test_cfg,
pretrained)

# fix pre-trained networks
self.fix_iter = train_cfg.get('fix_iter', 0) if train_cfg else 0
self.is_weight_fixed = False

# count training steps
self.register_buffer('step_counter', torch.zeros(1))

## Model Argument¶

The values of these arguments are taken from the configuration file. Let’s have a glance at the model part in the configuration file, you can find the complete config at configs/restorers/basicvsr/basicvsr_reds4.py .

model = dict(
type='BasicVSR',
generator=dict(
type='BasicVSRNet',
mid_channels=64,
num_blocks=30,
'basicvsr/spynet_20210409-c6c1bd09.pth'),
pixel_loss=dict(type='CharbonnierLoss', loss_weight=1.0, reduction='mean'))
train_cfg = dict(fix_iter=5000)
test_cfg = dict(metrics=['PSNR', 'SSIM'], crop_border=0)

We will now go through them one by one.

## generator¶

generator specifies the network architecture, which is called backbone in MMEditing. The definition of the backbone is straightforward, but there is one thing that needs our attention.

### Defining Backbone¶

Create a new file mmedit/models/backbones/basicvsr_net.py . The definition is standard. Please do make sure the line @BACKBONES.register_module() is added for all modules you would like to use.

import torch.nn as nn

from ..builder import BACKBONES

@BACKBONES.register_module()
class BasicVSRNet(nn.Module):

def __init__(self, mid_channels, num_blocks, spynet_pretrained):
pass

def forward(self, x):
pass

### Importing Module¶

This is the part we need to be careful. We need to add the following line to mmedit/models/backbones/__init__.py to use the defined backbone.

from .basicvsr_net import BasicVSRNet

## Specification in Configuration File¶

Given the above model, the specification in the configuration file is straightforward. We see that the argument type is just the name of the backbone, and other arguments correspond to that in the backbone.

generator=dict(
type='BasicVSRNet',
mid_channels=64,
num_blocks=30,
'basicvsr/spynet_20210409-c6c1bd09.pth')

## pixel_loss¶

pixel_loss refers to the loss used in BasicVSR. The specification of the loss is similar to that of the backbone.

### Defining Loss¶

Let’s use Charbonnier loss as an example. We first define the loss in mmedit/models/losses/pixelwise_loss.py . The decorator masked_loss enables the loss to be weighted and masked for each element. Again, do make sure that the line @LOSSES.register_module() is included.

from ..registry import LOSSES

def charbonnier_loss(pred, target, eps=1e-12):

@LOSSES.register_module()
class CharbonnierLoss(nn.Module):
def __init__(self,
loss_weight=1.0,
reduction='mean',
sample_wise=False,
eps=1e-12):
super().__init__()
if reduction not in ['none', 'mean', 'sum']:
raise ValueError(f'Unsupported reduction mode: {reduction}. '
f'Supported ones are: {_reduction_modes}')

self.loss_weight = loss_weight
self.reduction = reduction
self.sample_wise = sample_wise
self.eps = eps

def forward(self, pred, target, weight=None, **kwargs):
return self.loss_weight * charbonnier_loss(
pred,
target,
weight,
eps=self.eps,
reduction=self.reduction,
sample_wise=self.sample_wise)

Similarly, we need to add the following line to mmedit/models/losses/__init__.py .

from .pixelwise_loss import CharbonnierLoss

Then, the specification in the config follows naturally.

pixel_loss=dict(type='CharbonnierLoss', loss_weight=1.0, reduction='mean')

## train_cfg and test_cfg¶

train_cfg and test_cfg are just additional parameters you want to pass to the model. For example, in BasicVSR, a constant is passed to the model to fix a part of the network for a certain number of iterations.

self.fix_iter = train_cfg.get('fix_iter', 0) if train_cfg else 0

## Model Functions¶

The model functions are used to control the training and test. In this tutorial, we will highlight a few important ones. For more details of the functions, you may refer to here.

### train_step¶

This corresponds to the pipeline of each iteration, including forward and backward. In this example, the output and losses are computed. They are then used for backpropagation. More details of the forward process is discussed below.

def train_step(self, data_batch, optimizer):
outputs = self(**data_batch, test_mode=False)
loss, log_vars = self.parse_losses(outputs.pop('losses'))

# optimize
loss.backward()
optimizer['generator'].step()

outputs.update({'log_vars': log_vars})
return outputs

### forward_train¶

This corresponds to the forward process. In this example, we will compute output given lq . Then pixel_loss is computed between output and gt . The computed loss will then be passed to a dictionary for further computations, including backpropagation. If you have any other losses, you should also include them here.

def forward_train(self, lq, gt):
losses = dict()
output = self.generator(lq)
loss_pix = self.pixel_loss(output, gt)
losses['loss_pix'] = loss_pix
outputs = dict(
losses=losses,
num_samples=len(gt.data),
results=dict(lq=lq.cpu(), gt=gt.cpu(), output=output.cpu()))
return outputs

### forward_test¶

This corresponds to the validation and test. For example, you need to specify how you perform evaluation (i.e. calculation of metrics) and how you save the outputs.

def forward_test(self,
lq,
gt=None,
meta=None,
save_image=False,
save_path=None,
iteration=None):

output = self.generator(lq)
if self.test_cfg is not None and self.test_cfg.get('metrics', None):
assert gt is not None, (
'evaluation with metrics must have gt images.')
results = dict(eval_result=self.evaluate(output, gt))
else:
results = dict(lq=lq.cpu(), output=output.cpu())
if gt is not None:
results['gt'] = gt.cpu()

# save image
if save_image:
lq_path = meta[0]['lq_path']
folder_name = osp.splitext(osp.basename(lq_path))[0]
if isinstance(iteration, numbers.Number):
save_path = osp.join(save_path, folder_name,
f'{folder_name}-{iteration + 1:06d}.png')
elif iteration is None:
save_path = osp.join(save_path, f'{folder_name}.png')
else:
raise ValueError('iteration should be number or None, '
f'but got {type(iteration)}')
mmcv.imwrite(tensor2img(output), save_path)

return results


text:
# Solving Higher-Order Kinetic Energy Term (Gross-Pitaevskii equation) [closed]

Consider now propagation of non-linear waves in one-dimensional chain of dimers governed by the non-linear Schrödinger equation for the normalized wave envelope $\Psi(x,t)$, $$i \frac{\partial \Psi}{\partial t}+\frac {1} {2}\frac{\partial^2\Psi}{\partial x^2}+\alpha\vert{\Psi}\vert^2\Psi=0, \tag 1$$ where $t$ is time or propagation variable, $x$ is the spatial coordinate, and $\alpha$ describes the non-linearity or interaction strength. I would like to solve numerically the Gross-Pitaevski Eq. (1).

• Not sure what exactly you are asking for here. – flippiefanus Sep 25 '16 at 11:00
• @flippiefanus, to solve with second order derivatives. – user0322 Sep 25 '16 at 11:20
• @flippiefanus, my questions is to solve a modified equation with a higher-order kinetic energy term. – user0322 Sep 25 '16 at 11:26
• @DavidH you say an awful lot before you ever get to a question (actually you never ask one!). why don't you make it more concise? – anon01 Sep 25 '16 at 12:02
• @ConfusinglyCuriousTheThird, just to make it clear to flippiefanus – user0322 Sep 25 '16 at 12:15


text:
# Permuted Problem Sets

Calvin had prepared the problem sets for Levels 1 to 5 of Geometry and Combinatorics for next week but forgot to label which set was for which level. Since Calvin didn't label them, the computer assigned them labels 1 through 5 randomly, with each label appearing only once. The probability that the problem sets given to each level are within one level of what they were supposed to be can be expressed as $$\frac {a}{b}$$, where $$a$$ and $$b$$ are positive, coprime numbers. What is the value of $$a+b$$?

Details and assumptions

The computer randomly assigns each problem set to a level, and each level has exactly 1 problem set that is assigned. For example, the computer could assign the Level 1 problem set to Level 5 students, the Level 2 problem set to Level 4 students, the Level 3 problem set to Level 3 students, the Level 4 problem set to Level 2 students and the Level 5 problem set to Level 1 students.

×


text:
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`Journal of Atomic and Molecular PhysicsVolume 2013 (2013), Article ID 451715, 4 pageshttp://dx.doi.org/10.1155/2013/451715`
Research Article

## Generation of Multiple Focal Hole Segments Using Double-Ring Shaped Azimuthally Polarized Beam

1Department of Physics, Periyar University, Salem, Tamilnadu 636 011, India
2Department of Physics, Chikkanna Government Arts College, Trippur, Tamilnadu 641 602, India
3Department of Physical Optics, Institute of Applied Optics, 04894 Warsaw, Poland
4National Institute of Telecommunications, 04894 Warsaw, Poland

Received 17 July 2012; Revised 4 January 2013; Accepted 18 February 2013

Academic Editor: Mark D. Havey

Copyright © 2013 K. Lalithambigai et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We investigated the focusing properties of a double-ring-shaped azimuthally polarized beam tightly focused with a high NA lens and a binary phase filter. We observe that by using 3 belts and 5 belts binary phase filters novel focal patterns including splitting of focal rings and multiring focus are obtained. We also found that a suitably designed phase structure can shift the focal rings along optical axis. The author expects such investigation is worthwhile for optical manipulation and material processing technologies.

#### 1. Introduction

Growing interest in the generation of three-dimensional (3D) optical beams that are dark regions in space surrounded by light is driven by wide ranging applications including dark optical traps for atoms [1], manipulation, guiding and binding of microparticles and biological cells [2], and erase beams for super-resolution fluorescence microscopy [3]. Such beams are also known as twice-closed tubular optical structure (TCTOS) [4], optical bubbles [5], and hollow dark spherical spots [6]. Optical beams with oscillating on-axis intensity due to interference resulting in 3D intensity voids and 3D optical chains were proposed recently for trapping of multiple particles along the beam propagation and for controllable particle delivery [7]. Over the past years, a variety of techniques have been proposed for generating such optical bottle beams for applications in optical tweezers and atom traps [815]. Recently, a subwavelength focal hole (~0.5) with a quite long depth of focus (~48) is achieved near the focus by tight focusing of double-ring-shaped azimuthally polarized beam with high NA lens axicon [16]. In this paper we investigate the focal properties of the tightly focused azimuthally polarized double-ring-shaped beam using 3 belts and 5 belts binary phase filters. We observed that by properly designing the binary phase filters one can achieve many novel focal patterns including splitting of focal rings and generation of multiring structures.

#### 2. Theoretical Work

A schematic diagram of the suggested method is shown in Figure 1. The analysis was performed on the basis of Richards and Wolf’s vectorial diffraction method [17] widely used for high-NA focusing systems at arbitrary incident polarization. In the case of the azimuthally incident polarization, adopting the cylindrical coordinates and and the notations of [18], the electric field in the vicinity of the focal region can be written as Here is relative amplitude, = arcsin () is the maximum aperture angle with () is the numerical aperture, and is the index of refraction between the lens and the sample. denotes the Bessel functions of first order, and the function () describes the amplitude modulation.

Figure 1: (Colour online) scheme for a double-ring-shaped azimuthally polarized beam focused by a binary phase plate with high NA lens. The red arrows in (a) shows the direction of polarization. (b) is the DOE and is placed near the lens approximately the distance between lens and the source. The point in (c) represents the radial and axial coordinates of the observation point at the focal plane relative to the focus point, and is the azimuth angle defining the radial direction.

For illumination by a double-ring-shaped R-TEM11* beam with its waist in the pupil, this function is given by [19] where is the parameter that denotes the ratio of pupil diameter to the beam diameter and is the generalized Laguerre polynomial. If , the incident azimuthally polarized beam is a double-ring-shaped azimuthally polarized beam.

#### 3. Analysis of Binary Phase Plate

It is seen from (1) that an additional radial electric field component is produced after introducing the phase modulation to the input double-ring-shaped azimuthally polarized beam [20]. Owing to this radial component, the polarization near the focal plane is rather complicated and space variant. We perform the integration of (1) numerically using parameters , , and of the objective is 0.9. Here, for simplicity, we assume that the refractive index and . The effect of phase modulation on the input double-ring-shaped azimuthally polarized beam is evaluated by replacing the function by where for 3 belts binary phase plate (BPP) is given by

For 5-belt binary phase filter,

When a double-ring-shaped azimuthally polarized beam without any phase modulation is tightly focused, a single-ring focal hole pattern is obtained and is shown in Figure 2(a). We observed that the FWHM of the generated focal hole is 0.782 and its corresponding focal depth is 4. Such a focal ring with subwavelength size can be used for stable trapping of single particle with low refractive index [21]. However, to trap two separates such particles and to manipulate them, we need to split this single-ring focal hole structure to a double ring structure. We observed that this is possible by introducing phase modulation to the input double-ring-shaped azimuthally polarized beam through a 3-belt binary phase plate. We optimized the NA of the first, second, and third ring of the 3-belt binary phase plate as NA1 = 0.1, NA2 = 0.25, and = 0.9 for to achieve the above-mentioned task, and the generated focal segment is shown in Figure 2(b). From Figure 2(b), we observed that the single focal hole is splitted into two focal holes, and the FWHM of each focal hole segment is measured as 0.468 and is separated by the distance of 2.8 between them. We observed that the residual intensity that exists between the two rings is only 35% of the total intensity measured at the focal hole. The ring focus altered by the BPP can be used to move the trapped particle from one place to another. We also observed that it is possible to increase the separation distance between the two focal rings by setting NA1 and NA2 of the BPP as 0.2, 0.85 and by tuning , respectively. The corresponding focal hole segment generated is shown in Figure 2(c). From the figure we measured that the FWHM of each focal holes is 0.468 and is axially separated by the distance of 4 between them. We observed that the residual intensity that exists between the two rings is only 33% of the total intensity measured at the focal hole. It is also possible to achieve 3-ring focal hole segment by setting NA1 and NA2 of the BPP as 0.3, 0.775 and by tuning . The corresponding focal hole segment generated is shown in Figure 2(d). From Figure 2(d), we measured that the FWHM of each focal hole is 0.468λ, and each of them is separated by the distance of 2.8. We observed that the residual intensity that exists between the rings is only 31% of the total intensity measured at the focal hole. We observed that by using 5 belts BPP, it is possible to generate double ring focal segments with FWHM of 0.468and are axially separated by a large distance of 9.4as shown in Figure 3(a).

Figure 2: (Colour online) shows the contour plot of the total intensity distribution in the plane near the focus for the focusing system with 3 belts BPF. (a) Single-ring focus, (b) double-ring focus, (c) double-ring focus with large separation of axial distance, and (d) triple-ring focus.
Figure 3: (Colour online) shows the contour plot of the total intensity distribution in the plane near the focus for the focusing system with 5-belt binary phase filters. (a) Double-ring focus and (b) four-ring focus.

The set of values of NA of BPP optimized to achieve the above-mentioned task is NA1 = 0.34, NA2 = 0.57, NA3 = 0.78, NA4 = 0.85, and = 0.9 for . From Figure 3(b) we also found that it is possible to generate four-ring patterns having almost uniform FWHM of 0.468 and are axially separated by the distance of 2.7 between them by setting NA1 = 0.24, NA2 = 0.354, NA3 = 0.6, NA4 = 0.8, and = 0.9 for . We observed that the residual intensity that exists between the rings is only 30% of the total intensity measured at the focal hole. Thus with the help of liquid crystal-based tunable BPP, it is possible to generate tunable subwavelength focal hole segment with input double-ring-shaped azimuthally polarized beam.

#### 4. Conclusion

In conclusion, we show by vector diffraction theory that the tight focusing of double-ring-shaped azimuthally polarized beam with a high NA lens and tunable BPP, it is possible to generate many novel focal patterns which are potentially useful for optical trapping and manipulation of low refractive index particles.

#### References

1. N. Friedman, A. Kaplan, and N. Davidson, “Dark optical traps for cold atoms,” Advances in Atomic, Molecular, and Optical Physics, vol. 48, pp. 99–151, 2002.
2. T. Cizmar, L. C. D. Romero, K. Dholakia, and D. L. Andrews, “Multiple optical trapping and binding: new routes to self-assembly,” Journal of Physics B, vol. 43, Article ID 102001, 2010.
3. T. Watanabe, Y. Iketaki, T. Omatsu, K. Yamamoto, M. Sakai, and M. Fuji, “Two point separation in super-resolution fluorescence microscope based on up-conversion fluorescence depletion technique,” Optics Express, vol. 11, pp. 3271–3276, 2003.
4. J. Garcia-Sucerquia, F. F. Medina, and G. Matteucci, “Optical tubular structures produced by diffraction of circular apertures,” Optics and Lasers in Engineering, vol. 42, no. 1, pp. 61–70, 2004.
5. W. Chen and Q. Zhan, “Three-dimensional focus shaping with cylindrical vector beams,” Optics Communications, vol. 265, pp. 411–417, 2006.
6. N. Bokor and N. Davidson, “Generation of a hollow dark spherical spot by $4\pi$ focusing of a radially polarized Laguerre-Gaussian beam,” Optics Letters, vol. 31, pp. 149–151, 2006.
7. B. P. S. Ahluwalia, W. C. Cheong, X. C. Yuan et al., “Design and fabrication of a double-axicon for generation of tailorable self-imaged three-dimensional intensity voids,” Optics Letters, vol. 31, no. 7, pp. 987–989, 2006.
8. J. Arlt and M. J. Padgett, “Generation of a beam with a dark focus surrounded by regions of higher intensity: the optical bottle beam,” Optics Letters, vol. 25, no. 4, pp. 191–193, 2000.
9. B. P. S. Ahluwalia, X. C. Yuan, and S. H. Tao, “Transfer of “pure” on-axis spin angular momentum to the absorptive particle using self-imaged bottle beam optical tweezers system,” Optics Express, vol. 12, no. 21, pp. 5172–5177, 2004.
10. D. Yelin, B. E. Bouma, and G. J. Tearney, “Generating an adjustable three-dimensional dark focus,” Optics Letters, vol. 29, no. 7, pp. 661–663, 2004.
11. J. X. Pu, X. Y. Liu, and S. Nemoto, “Partially coherent bottle beams,” Optics Communications, vol. 252, pp. 7–11, 2005.
12. L. Isenhower, W. Williams, A. Dally, and M. Saffman, “Atom trapping in an interferometrically generated bottle beam trap,” Optics Letters, vol. 34, no. 8, pp. 1159–1161, 2009.
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