1 Introduction
Let be a -finite complete measure space and let be a measurable transformation, that is, for any . If for
each with , then is said to be non-singular.
Any non-singular measurable transformation induces a linear operator from into itself defined by<\/p>\n\n\n\n
(1)<\/p>\n\n\n\n
where denotes the linear space of all equivalence classes of -measurable functions on . Here we identify
any two functions that are equal -almost everywhere on .
Let be the class of all functions in that are finite -almost everywhere on . For , we define the distribution
function of on by<\/p>\n\n\n\n
and the decreasing rearrangement of on by<\/p>\n\n\n\n
The Lorentz space is the set of all classes of -measurable functions on such that the functional , where<\/p>\n\n\n\n
Take , and . Then the Lorentz sequence space is the set of all sequences such that the functional ,
where<\/p>\n\n\n\n
If we take , and is the weight function, then the corresponding Lorentz sequence space with weight is
denoted by and<\/p>\n\n\n\n
Note that the Lorentz spaces are quasi-normed linear spaces and the functional is a norm if and only if
or .
For any -measurable set of finite measure, we have<\/p>\n\n\n\n
For details about Lorentz spaces one can refer to [1, 5] and references therein.
Theorem 1.1 ([6, Theorem 2.3]). Let be a non-singular measurable transformation. Then induces a
bounded composition operator on , , if and only if there exists some constant such that for each .
Moreover,
For Orlicz spaces see [9, 13, 14] and for composition operators on Orlicz spaces one can refer to [8, 14]
and references therein. The work of this paper is motivated by the interesting work of B\u00f6ttcher and
Heidler [3, 4] .
Definition 1.2 Let denote the ring of univariate polynomials with complex coefficients. A polynomial is
said to be monic if .
Definition 1.3 An operator on a Banach space is said to be algebraic if there is a non-zero polynomial
such that and will be called essentially algebraic if there is a non-zero polynomial such that is compact.
Definition 1.4 The monic polynomial of the least degree such that is zero is called the characteristic
polynomial of .
Definition 1.5 The monic polynomial of the least degree such that is compact is called the essentially
characteristic polynomial of .
Let and represent the characteristic and essentially characteristic polynomials associated with the linear
operator . Also for a polynomial , denotes the two sided ideal .
Let be the collection of all bounded operators on a Banach space and be the collection of all compact
operators on . is a Banach algebra under the operator norm and is a two sided ideal in . Let be the
natural map of onto the Calkin algebra . Let and . In case is finite dimensional, is isomorphic to and is
a closed subalgebra of . In case is finite dimensional, is isomorphic to and is a closed subalgebra of .
The essential characteristic polynomial of is actually the characteristic polynomial of the Calkin image of
in the Calkin algebra . See [3, 4] for more details about the above definitions and notations.<\/p>\n\n\n\n
Algebraic and essentially algebraic composition operators on , are studied by B\u00f6ttcher and Heidler [4]
and these results were extended to Orlicz spaces by Kumar and Kumar [5].
2 Weighted Lorentz\u2013Karamata Sequence Spaces
Let , , and let be a weight sequence. Define the weighted counting measure<\/p>\n\n\n\n
(2)<\/p>\n\n\n\n
Let be a self-map. Since , every subset of measure zero is empty, so every self-map is nonsingular. The
composition operator is<\/p>\n\n\n\n
Let be a slowly varying function. For<\/p>\n\n\n\n
the weighted Lorentz\u2013Karamata sequence space is the set of all sequences such that<\/p>\n\n\n\n
(3)<\/p>\n\n\n\n
where is the decreasing rearrangement of with respect to . When , is the usual weighted Lorentz
sequence space .
Assumption 2.1 The function is slowly varying at and at , locally bounded, and positive. We write .
Under Assumption 2.1, the space with the functional is a rearrangement-invariant Banach function
space with the Fatou property (see [11, 12, 10] for details). Throughout, denotes this Banach realization.
Assumption 2.2 The self-map is finite-to-one: for every , the fiber is finite.
Assumption 2.3 There is a constant such that for every periodic cycle of , and for all ,<\/p>\n\n\n\n
Definition 2.4 For , , define the fundamental function analytically by<\/p>\n\n\n\n
Lemma 2.5 For every measurable set of finite measure,
Proof. The distribution function of is for and for . Hence . Substitution in the norm gives the result. \u25fb
Proposition 2.6 Under Assumption 2.1, for , In particular, at both endpoints.
Proof. Write the integrand as . Since is slowly varying and , Karamata\u2019s theorem (see [2, Theorem
1.5.11] ) gives<\/p>\n\n\n\n
Taking -th roots yields the stated asymptotic. \u25fb
Corollary 2.7 Under Assumption 6, satisfies the doubling condition
Proof. By Proposition 2.6, the ratio tends to as and as . Since is continuous and positive on every
compact subinterval , the ratio is bounded above and below by positive constants there. Combining the
two endpoint limits with the compact-interval bound yields global constants with . \u25fb
Lemma 2.8 (Global scaling and inversion estimates). Under Assumption 2.1, the following hold.<\/p>\n\n\n\n
and the reverse inequalities for . These bounds follow from the Uniform Convergence Theorem for
regularly varying functions together with the continuity and positivity of on compact subsets of (see [2,
Theorem1.5.6] and [1, Chapter 2] ).
(i) Fix and let . Applying the upper Potter bound with and (so ), we get<\/p>\n\n\n\n
Since this holds for every , we obtain the stated estimate with .
(ii) Let be given. Choose and let be the constant from the global Potter bound. Suppose for some . If ,
then (since is increasing), so and is trivial for ; if , we may assume without loss of generality, since
otherwise the conclusion fails only for the trivial case which we exclude by choosing .
Assume therefore . Set . Applying the lower Potter bound with , , we get<\/p>\n\n\n\n
Combining with ,<\/p>\n\n\n\n
hence<\/p>\n\n\n\n
Set<\/p>\n\n\n\n
Then as , and . This proves (ii). \u25fb
Theorem 2.9 (Boundedness criterion). Let be a self-map. Then induces a bounded composition
operator if and only if where the supremum is taken over all with . Moreover, the operator norm is
equivalent to this supremum.
Proof. Since is a rearrangement-invariant Banach function space (see [11, 12] ), the composition
operator is bounded if and only if the measure is absolutely continuous with respect to with a bounded
Radon\u2013Nikodym derivative in the associate norm. Concretely:
Suppose first that there is such that for every of finite measure. For and ,<\/p>\n\n\n\n
Hence for all . Therefore<\/p>\n\n\n\n
Substituting ,<\/p>\n\n\n\n
By the doubling property of (equivalently, the slow variation of ), there is a constant depending only on
such that for all . Hence<\/p>\n\n\n\n
so is bounded.
Conversely, suppose is bounded. For every of finite positive measure, and<\/p>\n\n\n\n
while . Hence<\/p>\n\n\n\n
which gives the stated supremum bound. \u25fb
3 Essential Orbit Structure
For a self-map , the set is called the orbit of . If the orbit of is a finite set, then we define the enter length
and the cycle length of the point as<\/p>\n\n\n\n
and<\/p>\n\n\n\n
If the orbit is infinite, then we take .
The maximal enter length of the self-map is denoted by and defined by<\/p>\n\n\n\n
and the set of occurring periods or cycle lengths is<\/p>\n\n\n\n
where<\/p>\n\n\n\n
In case , we say that is an essential period of . The set of all essential periods is denoted by . The set of
all unessential periods is defined as<\/p>\n\n\n\n
and the set of all periodic points with unessential periods is the set<\/p>\n\n\n\n
The essential enter length for the self-map depends on the decreasing rearrangement and on . If the set<\/p>\n\n\n\n
is finite, then we put . If is infinite, then we define as the minimal such that for every there is a finite set
with
(4)
Equivalently, the cofinite limit of the ratios vanishes:<\/p>\n\n\n\n
in the sense that for every , all but finitely many satisfy the inequality. If there is no such that the above
condition is satisfied, then we put .
For details about the above definitions we refer to [3, 4] .
Lemma 3.1 The following are equivalent:
Proof. Suppose . Then is finite. Also<\/p>\n\n\n\n
is a finite union of finite sets, hence finite.
Conversely, suppose and . If were infinite, then since for each and the sets are pairwise disjoint, the
union would be infinite. Contradiction. Hence is finite, so is finite. \u25fb
4 Norm Estimates for Characteristic Functions
Lemma 4.1 Let be disjoint with , , . Put , . Then for , In particular, if , then
Proof. The decreasing rearrangement of is<\/p>\n\n\n\n
Substituting this into the Lorentz\u2013Karamata norm gives the stated formula. \u25fb
Remark 4.2 Because is doubling (Corollary 2.7), Lemma 4.1 implies that normalized differences of
disjoint characteristic functions are bounded away from zero uniformly when and are comparable.
Explicitly, if , then<\/p>\n\n\n\n
independently of .
Lemma 4.3 (Compression estimate). Let be a set of atoms, and let . Let be supported on , and suppose
that Then where is arbitrary and depends only on .
Proof. For , set . Then<\/p>\n\n\n\n
Hence for all . Therefore<\/p>\n\n\n\n
Substituting gives<\/p>\n\n\n\n
By Potter\u2019s bound, for and any . Hence<\/p>\n\n\n\n
Taking -th roots gives the result. \u25fb
5 Disjointness of Preimage Trees
Lemma 5.1 Let be aperiodic, i.e. is not periodic. Then for ,<\/p>\n\n\n\n
Proof. Suppose with . Then . Applying to the first equality gives . Hence , so is periodic with period ,
contradiction. \u25fb
Lemma 5.2 Suppose satisfies Assumption 2.2 and is infinite. Let . Then for every infinite set there
exists a sequence such that the sets are pairwise disjoint.
Proof. We construct the sequence inductively. Since is infinite, choose arbitrarily.
Suppose have been chosen. Define the finite set<\/p>\n\n\n\n
Each is a single point (or empty), and each is finite by Assumption 2.2. Since there are finitely many
and finitely many , the set is finite. Since is infinite, choose .
We claim that the sets for and are pairwise disjoint. Suppose<\/p>\n\n\n\n
for some and . Then and .
If , then . So with , hence , contradiction.
If , then . So with , hence , contradiction.
Therefore the sets are pairwise disjoint. \u25fb
6 Essential Characteristic Polynomial
Lemma 6.1 Let be periodic with cycle length , so that are distinct and . Let . For , put Let , . Then
Moreover, If Assumption 2.3 holds, then Finally, the algebraic identity holds. More precisely, if with ,
then .
Proof. We have<\/p>\n\n\n\n
On the cycle, the coefficient at the point is precisely . Since the points are distinct, the functions are
disjointly supported. Hence<\/p>\n\n\n\n
where is an index at which the maximum is attained. Taking norms and using monotonicity gives the first
bound.
For the second bound, by Assumption 2.3 and Corollary 2.7, iterating the doubling inequality a finite
number of times yields uniformly in , where depends only on and the doubling constant.
For the algebraic identity, divide by with remainder:<\/p>\n\n\n\n
Write . Since for all , the coefficient of in equals the sum of the coefficients with , which is . Hence iff all
, i.e. iff . \u25fb
Lemma 6.2 Suppose . Then there exists a sequence of periodic points with period such that the cycles
are pairwise disjoint. Moreover, if is a polynomial such that , then the sequence satisfies and there is a
constant , independent of , such that for all . Hence cannot be compact.<\/p>\n\n\n\n
Proof. Since , the set is infinite. We extract a sequence of pairwise disjoint -cycles greedily: choose ;
since is infinite, choose outside the finite cycle of ; continue inductively.
For the periodic point , the entire inverse orbit is contained in the cycle:<\/p>\n\n\n\n
Hence the supports of are contained in the disjoint cycles.
For each , Lemma 6.1 and Assumption 2.3 give<\/p>\n\n\n\n
where . If , then not all are zero, so , and the lower bound is uniform in with constant .
Since the cycles are pairwise disjoint, the functions and are supported on disjoint sets for . Applying
Lemma 4.1 and Remark 4.2 with (after normalization), there is a constant , independent of , such that<\/p>\n\n\n\n
Thus the sequence has no convergent subsequence, so is not compact. \u25fb
Lemma 6.3 Suppose . Then there exist infinitely many such that If with , then along a suitable
sequence, Hence cannot be compact. Therefore, if is compact, then for every ; equivalently,
Proof. Since , there is an and an infinite set such that<\/p>\n\n\n\n
Otherwise the cofinite limit defining would vanish for this , contradicting .
Apply Lemma 5.2 with equal to the degree of and with the infinite set , obtaining a sequence such that
the sets<\/p>\n\n\n\n
are pairwise disjoint.
By Lemma 5.1, for each fixed the sets for are pairwise disjoint. Lemma 5.2 ensures disjointness across
different .
For each ,<\/p>\n\n\n\n
The term with has coefficient and is supported on , which is disjoint from all other terms. Hence<\/p>\n\n\n\n
This is a uniform lower bound. The sequence is bounded (norm 1), and the images have disjoint
supports, so by the lattice property,<\/p>\n\n\n\n
Thus no subsequence converges, and is not compact. \u25fb
7 The Key Compactness Lemma
Lemma 7.1 Let be a self-map satisfying Assumptions 2.2 and 2.3.Assume that and that . If then is
compact.<\/p>\n\n\n\n
Proof. Decompose with<\/p>\n\n\n\n
Both sets are forward -invariant: if , then and hence , so ; similarly for . Since is a Banach lattice with
order-continuous norm, the multiplication projections<\/p>\n\n\n\n
are bounded, and<\/p>\n\n\n\n
Periodic part. Decompose by cycle length. For , let<\/p>\n\n\n\n
Then<\/p>\n\n\n\n
Since , this is a finite direct sum of invariant subspaces.
On with , we have , so . Hence<\/p>\n\n\n\n
On with , the set is finite, so is finite-dimensional, and every operator on it is compact. Therefore is
compact.
Aperiodic part. We show that is compact. Since<\/p>\n\n\n\n
and the product is bounded while compact operators form a two-sided ideal, it suffices to prove that is
compact.
For , define<\/p>\n\n\n\n
By the definition of , the set is finite for every .
Let be the projection onto , and set on . The projection has finite rank.
We estimate for . Write with (by definition of , is supported on ). For , we have<\/p>\n\n\n\n
By Lemma 2.8(ii), there exists with as such that<\/p>\n\n\n\n
Set for sufficiently small. By Lemma 4.3,<\/p>\n\n\n\n
Hence<\/p>\n\n\n\n
Therefore<\/p>\n\n\n\n
where the first term has finite rank and the second has norm tending to zero as . Hence is a norm limit of
finite-rank operators and is therefore compact.
Consequently, is compact on both and , hence on all of . \u25fb
Main Classification Theorem
Theorem 25. Let be a self-map satisfying Assumptions 2.2 and 2.3. Assume that and that . Then the
following are equivalent:
Moreover, if is finite-dimensional, then where with .
Proof. The equivalence of the second and third conditions is Lemma 3.1.
Necessity. Suppose . Then there is a nonzero polynomial such that is compact. By Lemma 6.2, must
be divisible by for every . By Lemma 6.3, must be divisible by for every . Hence<\/p>\n\n\n\n
If either or , then no nonzero polynomial can satisfy these divisibility conditions, contradiction. Hence
and . By Lemma 3.1, this is equivalent to the second condition.
Sufficiency. Suppose and . By Lemma 7.1, the polynomial<\/p>\n\n\n\n
satisfies . Hence , so is algebraic in the Calkin algebra. Consider the algebra homomorphism<\/p>\n\n\n\n
Since , the kernel is a nonzero principal ideal of , say . By the necessity part, every annihilating
polynomial is divisible by<\/p>\n\n\n\n
so this polynomial has the least degree among the annihilators and is thus the monic generator of the
kernel. Hence , and by the first isomorphism theorem,<\/p>\n\n\n\n
\u25fb
Corollary 8.2 Under the hypotheses of Theorem 8.1, if and only if Equivalently, is compact if and only
if the minimal annihilating polynomial of in the Calkin algebra is .
Proof. is compact iff iff the minimal annihilating polynomial is . By Theorem 8.1, iff , , and . \u25fb
9 Invariance Under Slowly Varying Perturbation
Proposition 9.1 Let . For positive sequences with and , The same holds if .
Proof. By Proposition 2.6,<\/p>\n\n\n\n
Hence<\/p>\n\n\n\n
By the Potter bounds for slowly varying functions [2, Theorem 1.5.6], for every there is a constant such
that<\/p>\n\n\n\n
Choosing yields<\/p>\n\n\n\n
whenever . The converse follows from the reverse Potter bound<\/p>\n\n\n\n
for some . \u25fb
Proposition 9.2. Let . For positive sequences with for all , there are constants such that for all .
Proof. By the doubling condition (Corollary 2.7), iterating the inequality a finite number of times gives for
any fixed . Similarly, iterating the lower bound gives . Applying these with and to and yields the
result. \u25fb
Theorem 9.3 Let be a self-map satisfying Assumptions 2.2 and 2.3, and assume that for every and
every sequence , Then where is the essential enter length in the classical Lorentz space .
Consequently, the classification of essentially algebraic composition operators on is identical to that on .
Proof. The hypothesis states precisely that the cofinite limit defining vanishes if and only if the cofinite
limit defining vanishes, for every . Hence the minimal such is the same in both cases. The remaining
combinatorial data , , and are independent of the space. Therefore the classification theorem and the
compactness corollary are the same. \u25fb
Corollary 9.4 The hypothesis of Theorem 9.3 is satisfied if, for every , the ratios either tend to , or satisfy
, as along .
Proof. This follows from Propositions 9.1 and 9.2. \u25fb
Remark 9.5 The slowly varying function does not change the zero\/nonzero nature of the asymptotic
ratios that determine compactness and essential algebraicity. The norm itself changes when changes,
but the algebraic classification of composition operators remains the same as in the classical Lorentz
case, under the stated regular-variation hypotheses.
10 Examples
Example 10.1 (Pure periodic case). Let consist of infinitely many cycles of length . Then<\/p>\n\n\n\n
Hence , and<\/p>\n\n\n\n
Thus<\/p>\n\n\n\n
Example 10.2 (Infinite essential enter length). Let<\/p>\n\n\n\n
Then is finite-to-one: , and for . With counting measure,<\/p>\n\n\n\n
so<\/p>\n\n\n\n
Hence the limit defining does not vanish for any , so<\/p>\n\n\n\n
This example shows that finite-to-one maps can have infinite essential enter length.
Example 10.3 (Finite essential enter length). Let for , and for not a power of . Then , , , and so on. The
point is fixed, , , , etc.
Define the weight<\/p>\n\n\n\n
Then is finite-to-one. For ,<\/p>\n\n\n\n
so<\/p>\n\n\n\n
For , the ratio is . Hence the cofinite limit defining vanishes precisely for , so<\/p>\n\n\n\n
To see that is bounded, we check the criterion of Theorem 2.9. For a singleton with not a power of ,
and , so the ratio is . For ,<\/p>\n\n\n\n
For , so<\/p>\n\n\n\n
For a general set , write where contains non-powers of and contains powers of . Then<\/p>\n\n\n\n
Since for all and , we obtain<\/p>\n\n\n\n
Hence the ratio for all , and by monotonicity of , the supremum in Theorem 2.9 is finite. Therefore is
bounded on .
The essential characteristic polynomial is<\/p>\n\n\n\n
since (all fixed points have cycle length , and there are infinitely many of them).
Example 10.4 (Slowly varying perturbation). Let<\/p>\n\n\n\n
Then , and<\/p>\n\n\n\n
For the explicit sequences<\/p>\n\n\n\n
we have and<\/p>\n\n\n\n
So the slowly varying factor does not change the zero\/nonzero nature of the limit.<\/p>\n\n\n\n
References
[1]. C. Bennett and R. Sharpley, Interpolation of Operators, Pure and Applied Mathematics 129,
Academic Press, London, 1988.
[2]. N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Encyclopedia of Mathematics and
its Applications 27, Cambridge University Press, 1987.
[3.] A. B\u00f6ttcher and H. Heidler, Algebraic composition operators, Integr. Equ. Oper. Theory 15 (1992),
389\u2013411.
[4]. A. B\u00f6ttcher and H. Heidler, Classification of finite-dimensional algebras generated by the Calkin
image of a composition operator on with weight, Algebra i Analiz 5 (1993), 69\u201396; English transl., St.
Petersburg Math. J. 5 (1994), 1099\u20131119.
[5]. M. J. Carro, J. A. Rapos and J. Soria, Recent developments in the theory of Lorentz spaces and
weighted inequalities, Mem. Amer. Math. Soc. 187 (2007).
[6]. R. Kumar and R. Kumar, Composition operators on Banach function spaces, Proc. Amer. Math. Soc.
133 (2005), 2109\u20132118.
[7]. R. Kumar and R. Kumar, Compact composition operators on Lorentz spaces, Matemati\u010dki Vesnik 57
(2005), 109\u2013112.
[8]. R. Kumar and R. Kumar, On finite dimensional algebras generated by composition operators on
Orlicz sequence spaces with weight, Houston J. Math. 31 (2005), 1135\u20131152.
[9]. J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I and II, Springer, Heidelberg, 1996.
[10]. C. Merucci, Interpolation, Lorentz spaces and applications, Rend. Mat. Appl. 7 (1987), 261\u2013286.
[11]. J. S. Neves, Lorentz\u2013Karamata spaces, Bessel and Riesz potentials and embeddings,
Dissertationes Math. 405 (2002), 1\u201346.
[12]. B. Opic and L. Pick, On generalized Lorentz\u2013Zygmund spaces, Math. Inequal. Appl. 2 (1999),
391\u2013408.
[13]. M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces, Marcel Dekker, New York, 1991.
[14]. M. M. Rao and Z. D. Ren, Applications of Orlicz Spaces, Marcel Dekker, New York, 2002.
[15]. A. Gupta and N. Bhatia, Composition and weighted composition operators on generalized
Lorentz\u2013Zygmund spaces, J. Math. Anal. 4 (2013), 1\u201313.
[16]. A. Gupta and N. Bhatia, Composition operators on Lorentz\u2013Karamata\u2013Bochner spaces, American
Journal of Mathematical Analysis 3 (2015), 21\u201325.
[17]. I. Eryilmaz, Weighted composition operators on weighted Lorentz\u2013Karamata spaces, Stud. Univ.
Babe\u015f-Bolyai Math. 57 (2012), 111\u2013119.
[18]. D. Pe\u0161a, Lorentz\u2013Karamata spaces, arXiv:2006.14455 (2020).<\/p>\n","protected":false},"excerpt":{"rendered":"
1 IntroductionLet be a -finite complete measure space and let be a measurable transformation, that is, for any . If foreach with , then is said to be non-singular.Any non-singular measurable transformation induces a linear operator from into itself defined by (1) where denotes the linear space of all equivalence classes of -measurable functions on […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"ocean_post_layout":"","ocean_both_sidebars_style":"","ocean_both_sidebars_content_width":0,"ocean_both_sidebars_sidebars_width":0,"ocean_sidebar":"","ocean_second_sidebar":"","ocean_disable_margins":"enable","ocean_add_body_class":"","ocean_shortcode_before_top_bar":"","ocean_shortcode_after_top_bar":"","ocean_shortcode_before_header":"","ocean_shortcode_after_header":"","ocean_has_shortcode":"","ocean_shortcode_after_title":"","ocean_shortcode_before_footer_widgets":"","ocean_shortcode_after_footer_widgets":"","ocean_shortcode_before_footer_bottom":"","ocean_shortcode_after_footer_bottom":"","ocean_display_top_bar":"default","ocean_display_header":"default","ocean_header_style":"","ocean_center_header_left_menu":"","ocean_custom_header_template":"","ocean_custom_logo":0,"ocean_custom_retina_logo":0,"ocean_custom_logo_max_width":0,"ocean_custom_logo_tablet_max_width":0,"ocean_custom_logo_mobile_max_width":0,"ocean_custom_logo_max_height":0,"ocean_custom_logo_tablet_max_height":0,"ocean_custom_logo_mobile_max_height":0,"ocean_header_custom_menu":"","ocean_menu_typo_font_family":"","ocean_menu_typo_font_subset":"","ocean_menu_typo_font_size":0,"ocean_menu_typo_font_size_tablet":0,"ocean_menu_typo_font_size_mobile":0,"ocean_menu_typo_font_size_unit":"px","ocean_menu_typo_font_weight":"","ocean_menu_typo_font_weight_tablet":"","ocean_menu_typo_font_weight_mobile":"","ocean_menu_typo_transform":"","ocean_menu_typo_transform_tablet":"","ocean_menu_typo_transform_mobile":"","ocean_menu_typo_line_height":0,"ocean_menu_typo_line_height_tablet":0,"ocean_menu_typo_line_height_mobile":0,"ocean_menu_typo_line_height_unit":"","ocean_menu_typo_spacing":0,"ocean_menu_typo_spacing_tablet":0,"ocean_menu_typo_spacing_mobile":0,"ocean_menu_typo_spacing_unit":"","ocean_menu_link_color":"","ocean_menu_link_color_hover":"","ocean_menu_link_color_active":"","ocean_menu_link_background":"","ocean_menu_link_hover_background":"","ocean_menu_link_active_background":"","ocean_menu_social_links_bg":"","ocean_menu_social_hover_links_bg":"","ocean_menu_social_links_color":"","ocean_menu_social_hover_links_color":"","ocean_disable_title":"default","ocean_disable_heading":"default","ocean_post_title":"","ocean_post_subheading":"","ocean_post_title_style":"","ocean_post_title_background_color":"","ocean_post_title_background":0,"ocean_post_title_bg_image_position":"","ocean_post_title_bg_image_attachment":"","ocean_post_title_bg_image_repeat":"","ocean_post_title_bg_image_size":"","ocean_post_title_height":0,"ocean_post_title_bg_overlay":0.5,"ocean_post_title_bg_overlay_color":"","ocean_disable_breadcrumbs":"default","ocean_breadcrumbs_color":"","ocean_breadcrumbs_separator_color":"","ocean_breadcrumbs_links_color":"","ocean_breadcrumbs_links_hover_color":"","ocean_display_footer_widgets":"default","ocean_display_footer_bottom":"default","ocean_custom_footer_template":"","omw_enable_modal_window":"enable","ocean_post_oembed":"","ocean_post_self_hosted_media":"","ocean_post_video_embed":"","ocean_link_format":"","ocean_link_format_target":"self","ocean_quote_format":"","ocean_quote_format_link":"post","ocean_gallery_link_images":"on","ocean_gallery_id":[],"footnotes":""},"categories":[25],"tags":[104,101,94,98,93,95,96,99,92,103,97,100,102],"article-archive":[20],"class_list":["post-932","post","type-post","status-publish","format-standard","hentry","category-original-research-article","tag-104","tag-47b33","tag-algebraic","tag-classiication","tag-composition","tag-essentially","tag-lorentz-karamata","tag-mathematics","tag-operators","tag-primary","tag-secondary","tag-spaces-2020","tag-subject","article-archive-volume-6-issue-1-2025","entry"],"acf":[],"_links":{"self":[{"href":"https:\/\/academicsociety.org\/deij\/wp-json\/wp\/v2\/posts\/932","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/academicsociety.org\/deij\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/academicsociety.org\/deij\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/academicsociety.org\/deij\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/academicsociety.org\/deij\/wp-json\/wp\/v2\/comments?post=932"}],"version-history":[{"count":1,"href":"https:\/\/academicsociety.org\/deij\/wp-json\/wp\/v2\/posts\/932\/revisions"}],"predecessor-version":[{"id":936,"href":"https:\/\/academicsociety.org\/deij\/wp-json\/wp\/v2\/posts\/932\/revisions\/936"}],"wp:attachment":[{"href":"https:\/\/academicsociety.org\/deij\/wp-json\/wp\/v2\/media?parent=932"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/academicsociety.org\/deij\/wp-json\/wp\/v2\/categories?post=932"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/academicsociety.org\/deij\/wp-json\/wp\/v2\/tags?post=932"},{"taxonomy":"article-archive","embeddable":true,"href":"https:\/\/academicsociety.org\/deij\/wp-json\/wp\/v2\/article-archive?post=932"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}